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577dd3f6e8221b15188983f377f530b823dd582157d754b0476efdb773b52cac | """
This module mainly implements special orthogonal polynomials.
See also functions.combinatorial.numbers which contains some
combinatorial polynomials.
"""
from sympy.core import Rational
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import binomial, factorial, RisingFactorial
from sympy.functions.elementary.complexes import re
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import cos, sec
from sympy.functions.special.gamma_functions import gamma
from sympy.functions.special.hyper import hyper
from sympy.polys.orthopolys import (chebyshevt_poly, chebyshevu_poly,
gegenbauer_poly, hermite_poly, jacobi_poly,
laguerre_poly, legendre_poly)
_x = Dummy('x')
class OrthogonalPolynomial(Function):
"""Base class for orthogonal polynomials.
"""
@classmethod
def _eval_at_order(cls, n, x):
if n.is_integer and n >= 0:
return cls._ortho_poly(int(n), _x).subs(_x, x)
def _eval_conjugate(self):
return self.func(self.args[0], self.args[1].conjugate())
#----------------------------------------------------------------------------
# Jacobi polynomials
#
class jacobi(OrthogonalPolynomial):
r"""
Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.
Explanation
===========
``jacobi(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial
in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.
The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.
Examples
========
>>> from sympy import jacobi, S, conjugate, diff
>>> from sympy.abc import a, b, n, x
>>> jacobi(0, a, b, x)
1
>>> jacobi(1, a, b, x)
a/2 - b/2 + x*(a/2 + b/2 + 1)
>>> jacobi(2, a, b, x)
a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2
>>> jacobi(n, a, b, x)
jacobi(n, a, b, x)
>>> jacobi(n, a, a, x)
RisingFactorial(a + 1, n)*gegenbauer(n,
a + 1/2, x)/RisingFactorial(2*a + 1, n)
>>> jacobi(n, 0, 0, x)
legendre(n, x)
>>> jacobi(n, S(1)/2, S(1)/2, x)
RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)
>>> jacobi(n, -S(1)/2, -S(1)/2, x)
RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)
>>> jacobi(n, a, b, -x)
(-1)**n*jacobi(n, b, a, x)
>>> jacobi(n, a, b, 0)
gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))
>>> jacobi(n, a, b, 1)
RisingFactorial(a + 1, n)/factorial(n)
>>> conjugate(jacobi(n, a, b, x))
jacobi(n, conjugate(a), conjugate(b), conjugate(x))
>>> diff(jacobi(n,a,b,x), x)
(a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)
See Also
========
gegenbauer,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly,
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
.. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/JacobiP/
"""
@classmethod
def eval(cls, n, a, b, x):
# Simplify to other polynomials
# P^{a, a}_n(x)
if a == b:
if a == Rational(-1, 2):
return RisingFactorial(S.Half, n) / factorial(n) * chebyshevt(n, x)
elif a.is_zero:
return legendre(n, x)
elif a == S.Half:
return RisingFactorial(3*S.Half, n) / factorial(n + 1) * chebyshevu(n, x)
else:
return RisingFactorial(a + 1, n) / RisingFactorial(2*a + 1, n) * gegenbauer(n, a + S.Half, x)
elif b == -a:
# P^{a, -a}_n(x)
return gamma(n + a + 1) / gamma(n + 1) * (1 + x)**(a/2) / (1 - x)**(a/2) * assoc_legendre(n, -a, x)
if not n.is_Number:
# Symbolic result P^{a,b}_n(x)
# P^{a,b}_n(-x) ---> (-1)**n * P^{b,a}_n(-x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * jacobi(n, b, a, -x)
# We can evaluate for some special values of x
if x.is_zero:
return (2**(-n) * gamma(a + n + 1) / (gamma(a + 1) * factorial(n)) *
hyper([-b - n, -n], [a + 1], -1))
if x == S.One:
return RisingFactorial(a + 1, n) / factorial(n)
elif x is S.Infinity:
if n.is_positive:
# Make sure a+b+2*n \notin Z
if (a + b + 2*n).is_integer:
raise ValueError("Error. a + b + 2*n should not be an integer.")
return RisingFactorial(a + b + n + 1, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return jacobi_poly(n, a, b, x)
def fdiff(self, argindex=4):
from sympy.concrete.summations import Sum
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt a
n, a, b, x = self.args
k = Dummy("k")
f1 = 1 / (a + b + n + k + 1)
f2 = ((a + b + 2*k + 1) * RisingFactorial(b + k + 1, n - k) /
((n - k) * RisingFactorial(a + b + k + 1, n - k)))
return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
elif argindex == 3:
# Diff wrt b
n, a, b, x = self.args
k = Dummy("k")
f1 = 1 / (a + b + n + k + 1)
f2 = (-1)**(n - k) * ((a + b + 2*k + 1) * RisingFactorial(a + k + 1, n - k) /
((n - k) * RisingFactorial(a + b + k + 1, n - k)))
return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
elif argindex == 4:
# Diff wrt x
n, a, b, x = self.args
return S.Half * (a + b + n + 1) * jacobi(n - 1, a + 1, b + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, a, b, x, **kwargs):
from sympy.concrete.summations import Sum
# Make sure n \in N
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = (RisingFactorial(-n, k) * RisingFactorial(a + b + n + 1, k) * RisingFactorial(a + k + 1, n - k) /
factorial(k) * ((1 - x)/2)**k)
return 1 / factorial(n) * Sum(kern, (k, 0, n))
def _eval_conjugate(self):
n, a, b, x = self.args
return self.func(n, a.conjugate(), b.conjugate(), x.conjugate())
def jacobi_normalized(n, a, b, x):
r"""
Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.
Explanation
===========
``jacobi_normalized(n, alpha, beta, x)`` gives the $n$th
Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.
The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.
This functions returns the polynomials normilzed:
.. math::
\int_{-1}^{1}
P_m^{\left(\alpha, \beta\right)}(x)
P_n^{\left(\alpha, \beta\right)}(x)
(1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x
= \delta_{m,n}
Examples
========
>>> from sympy import jacobi_normalized
>>> from sympy.abc import n,a,b,x
>>> jacobi_normalized(n, a, b, x)
jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))
Parameters
==========
n : integer degree of polynomial
a : alpha value
b : beta value
x : symbol
See Also
========
gegenbauer,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly,
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
.. [2] http://mathworld.wolfram.com/JacobiPolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/JacobiP/
"""
nfactor = (S(2)**(a + b + 1) * (gamma(n + a + 1) * gamma(n + b + 1))
/ (2*n + a + b + 1) / (factorial(n) * gamma(n + a + b + 1)))
return jacobi(n, a, b, x) / sqrt(nfactor)
#----------------------------------------------------------------------------
# Gegenbauer polynomials
#
class gegenbauer(OrthogonalPolynomial):
r"""
Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$.
Explanation
===========
``gegenbauer(n, alpha, x)`` gives the $n$th Gegenbauer polynomial
in $x$, $C_n^{\left(\alpha\right)}(x)$.
The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with
respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$.
Examples
========
>>> from sympy import gegenbauer, conjugate, diff
>>> from sympy.abc import n,a,x
>>> gegenbauer(0, a, x)
1
>>> gegenbauer(1, a, x)
2*a*x
>>> gegenbauer(2, a, x)
-a + x**2*(2*a**2 + 2*a)
>>> gegenbauer(3, a, x)
x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
>>> gegenbauer(n, a, x)
gegenbauer(n, a, x)
>>> gegenbauer(n, a, -x)
(-1)**n*gegenbauer(n, a, x)
>>> gegenbauer(n, a, 0)
2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
>>> gegenbauer(n, a, 1)
gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
>>> conjugate(gegenbauer(n, a, x))
gegenbauer(n, conjugate(a), conjugate(x))
>>> diff(gegenbauer(n, a, x), x)
2*a*gegenbauer(n - 1, a + 1, x)
See Also
========
jacobi,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
.. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/
"""
@classmethod
def eval(cls, n, a, x):
# For negative n the polynomials vanish
# See http://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
if n.is_negative:
return S.Zero
# Some special values for fixed a
if a == S.Half:
return legendre(n, x)
elif a == S.One:
return chebyshevu(n, x)
elif a == S.NegativeOne:
return S.Zero
if not n.is_Number:
# Handle this before the general sign extraction rule
if x == S.NegativeOne:
if (re(a) > S.Half) == True:
return S.ComplexInfinity
else:
return (cos(S.Pi*(a+n)) * sec(S.Pi*a) * gamma(2*a+n) /
(gamma(2*a) * gamma(n+1)))
# Symbolic result C^a_n(x)
# C^a_n(-x) ---> (-1)**n * C^a_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * gegenbauer(n, a, -x)
# We can evaluate for some special values of x
if x.is_zero:
return (2**n * sqrt(S.Pi) * gamma(a + S.Half*n) /
(gamma((1 - n)/2) * gamma(n + 1) * gamma(a)) )
if x == S.One:
return gamma(2*a + n) / (gamma(2*a) * gamma(n + 1))
elif x is S.Infinity:
if n.is_positive:
return RisingFactorial(a, n) * S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
return gegenbauer_poly(n, a, x)
def fdiff(self, argindex=3):
from sympy.concrete.summations import Sum
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt a
n, a, x = self.args
k = Dummy("k")
factor1 = 2 * (1 + (-1)**(n - k)) * (k + a) / ((k +
n + 2*a) * (n - k))
factor2 = 2*(k + 1) / ((k + 2*a) * (2*k + 2*a + 1)) + \
2 / (k + n + 2*a)
kern = factor1*gegenbauer(k, a, x) + factor2*gegenbauer(n, a, x)
return Sum(kern, (k, 0, n - 1))
elif argindex == 3:
# Diff wrt x
n, a, x = self.args
return 2*a*gegenbauer(n - 1, a + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, a, x, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k")
kern = ((-1)**k * RisingFactorial(a, n - k) * (2*x)**(n - 2*k) /
(factorial(k) * factorial(n - 2*k)))
return Sum(kern, (k, 0, floor(n/2)))
def _eval_conjugate(self):
n, a, x = self.args
return self.func(n, a.conjugate(), x.conjugate())
#----------------------------------------------------------------------------
# Chebyshev polynomials of first and second kind
#
class chebyshevt(OrthogonalPolynomial):
r"""
Chebyshev polynomial of the first kind, $T_n(x)$.
Explanation
===========
``chebyshevt(n, x)`` gives the $n$th Chebyshev polynomial (of the first
kind) in $x$, $T_n(x)$.
The Chebyshev polynomials of the first kind are orthogonal on
$[-1, 1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$.
Examples
========
>>> from sympy import chebyshevt, diff
>>> from sympy.abc import n,x
>>> chebyshevt(0, x)
1
>>> chebyshevt(1, x)
x
>>> chebyshevt(2, x)
2*x**2 - 1
>>> chebyshevt(n, x)
chebyshevt(n, x)
>>> chebyshevt(n, -x)
(-1)**n*chebyshevt(n, x)
>>> chebyshevt(-n, x)
chebyshevt(n, x)
>>> chebyshevt(n, 0)
cos(pi*n/2)
>>> chebyshevt(n, -1)
(-1)**n
>>> diff(chebyshevt(n, x), x)
n*chebyshevu(n - 1, x)
See Also
========
jacobi, gegenbauer,
chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/
"""
_ortho_poly = staticmethod(chebyshevt_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result T_n(x)
# T_n(-x) ---> (-1)**n * T_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevt(n, -x)
# T_{-n}(x) ---> T_n(x)
if n.could_extract_minus_sign():
return chebyshevt(-n, x)
# We can evaluate for some special values of x
if x.is_zero:
return cos(S.Half * S.Pi * n)
if x == S.One:
return S.One
elif x is S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# T_{-n}(x) == T_n(x)
return cls._eval_at_order(-n, x)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return n * chebyshevu(n - 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k")
kern = binomial(n, 2*k) * (x**2 - 1)**k * x**(n - 2*k)
return Sum(kern, (k, 0, floor(n/2)))
class chebyshevu(OrthogonalPolynomial):
r"""
Chebyshev polynomial of the second kind, $U_n(x)$.
Explanation
===========
``chebyshevu(n, x)`` gives the $n$th Chebyshev polynomial of the second
kind in x, $U_n(x)$.
The Chebyshev polynomials of the second kind are orthogonal on
$[-1, 1]$ with respect to the weight $\sqrt{1-x^2}$.
Examples
========
>>> from sympy import chebyshevu, diff
>>> from sympy.abc import n,x
>>> chebyshevu(0, x)
1
>>> chebyshevu(1, x)
2*x
>>> chebyshevu(2, x)
4*x**2 - 1
>>> chebyshevu(n, x)
chebyshevu(n, x)
>>> chebyshevu(n, -x)
(-1)**n*chebyshevu(n, x)
>>> chebyshevu(-n, x)
-chebyshevu(n - 2, x)
>>> chebyshevu(n, 0)
cos(pi*n/2)
>>> chebyshevu(n, 1)
n + 1
>>> diff(chebyshevu(n, x), x)
(-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
.. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
.. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
.. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/
.. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/
"""
_ortho_poly = staticmethod(chebyshevu_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result U_n(x)
# U_n(-x) ---> (-1)**n * U_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * chebyshevu(n, -x)
# U_{-n}(x) ---> -U_{n-2}(x)
if n.could_extract_minus_sign():
if n == S.NegativeOne:
# n can not be -1 here
return S.Zero
elif not (-n - 2).could_extract_minus_sign():
return -chebyshevu(-n - 2, x)
# We can evaluate for some special values of x
if x.is_zero:
return cos(S.Half * S.Pi * n)
if x == S.One:
return S.One + n
elif x is S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
# U_{-n}(x) ---> -U_{n-2}(x)
if n == S.NegativeOne:
return S.Zero
else:
return -cls._eval_at_order(-n - 2, x)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return ((n + 1) * chebyshevt(n + 1, x) - x * chebyshevu(n, x)) / (x**2 - 1)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k")
kern = S.NegativeOne**k * factorial(
n - k) * (2*x)**(n - 2*k) / (factorial(k) * factorial(n - 2*k))
return Sum(kern, (k, 0, floor(n/2)))
class chebyshevt_root(Function):
r"""
``chebyshev_root(n, k)`` returns the $k$th root (indexed from zero) of
the $n$th Chebyshev polynomial of the first kind; that is, if
$0 \le k < n$, ``chebyshevt(n, chebyshevt_root(n, k)) == 0``.
Examples
========
>>> from sympy import chebyshevt, chebyshevt_root
>>> chebyshevt_root(3, 2)
-sqrt(3)/2
>>> chebyshevt(3, chebyshevt_root(3, 2))
0
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
"""
@classmethod
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return cos(S.Pi*(2*k + 1)/(2*n))
class chebyshevu_root(Function):
r"""
``chebyshevu_root(n, k)`` returns the $k$th root (indexed from zero) of the
$n$th Chebyshev polynomial of the second kind; that is, if $0 \le k < n$,
``chebyshevu(n, chebyshevu_root(n, k)) == 0``.
Examples
========
>>> from sympy import chebyshevu, chebyshevu_root
>>> chebyshevu_root(3, 2)
-sqrt(2)/2
>>> chebyshevu(3, chebyshevu_root(3, 2))
0
See Also
========
chebyshevt, chebyshevt_root, chebyshevu,
legendre, assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
"""
@classmethod
def eval(cls, n, k):
if not ((0 <= k) and (k < n)):
raise ValueError("must have 0 <= k < n, "
"got k = %s and n = %s" % (k, n))
return cos(S.Pi*(k + 1)/(n + 1))
#----------------------------------------------------------------------------
# Legendre polynomials and Associated Legendre polynomials
#
class legendre(OrthogonalPolynomial):
r"""
``legendre(n, x)`` gives the $n$th Legendre polynomial of $x$, $P_n(x)$
Explanation
===========
The Legendre polynomials are orthogonal on $[-1, 1]$ with respect to
the constant weight 1. They satisfy $P_n(1) = 1$ for all $n$; further,
$P_n$ is odd for odd $n$ and even for even $n$.
Examples
========
>>> from sympy import legendre, diff
>>> from sympy.abc import x, n
>>> legendre(0, x)
1
>>> legendre(1, x)
x
>>> legendre(2, x)
3*x**2/2 - 1/2
>>> legendre(n, x)
legendre(n, x)
>>> diff(legendre(n,x), x)
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
assoc_legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LegendreP/
.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/
"""
_ortho_poly = staticmethod(legendre_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result L_n(x)
# L_n(-x) ---> (-1)**n * L_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * legendre(n, -x)
# L_{-n}(x) ---> L_{n-1}(x)
if n.could_extract_minus_sign() and not(-n - 1).could_extract_minus_sign():
return legendre(-n - S.One, x)
# We can evaluate for some special values of x
if x.is_zero:
return sqrt(S.Pi)/(gamma(S.Half - n/2)*gamma(S.One + n/2))
elif x == S.One:
return S.One
elif x is S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial;
# L_{-n}(x) ---> L_{n-1}(x)
if n.is_negative:
n = -n - S.One
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
# Find better formula, this is unsuitable for x = +/-1
# http://www.autodiff.org/ad16/Oral/Buecker_Legendre.pdf says
# at x = 1:
# n*(n + 1)/2 , m = 0
# oo , m = 1
# -(n-1)*n*(n+1)*(n+2)/4 , m = 2
# 0 , m = 3, 4, ..., n
#
# at x = -1
# (-1)**(n+1)*n*(n + 1)/2 , m = 0
# (-1)**n*oo , m = 1
# (-1)**n*(n-1)*n*(n+1)*(n+2)/4 , m = 2
# 0 , m = 3, 4, ..., n
n, x = self.args
return n/(x**2 - 1)*(x*legendre(n, x) - legendre(n - 1, x))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k")
kern = S.NegativeOne**k*binomial(n, k)**2*((1 + x)/2)**(n - k)*((1 - x)/2)**k
return Sum(kern, (k, 0, n))
class assoc_legendre(Function):
r"""
``assoc_legendre(n, m, x)`` gives $P_n^m(x)$, where $n$ and $m$ are
the degree and order or an expression which is related to the nth
order Legendre polynomial, $P_n(x)$ in the following manner:
.. math::
P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
\frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}
Explanation
===========
Associated Legendre polynomials are orthogonal on $[-1, 1]$ with:
- weight $= 1$ for the same $m$ and different $n$.
- weight $= \frac{1}{1-x^2}$ for the same $n$ and different $m$.
Examples
========
>>> from sympy import assoc_legendre
>>> from sympy.abc import x, m, n
>>> assoc_legendre(0,0, x)
1
>>> assoc_legendre(1,0, x)
x
>>> assoc_legendre(1,1, x)
-sqrt(1 - x**2)
>>> assoc_legendre(n,m,x)
assoc_legendre(n, m, x)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre,
hermite,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
.. [2] http://mathworld.wolfram.com/LegendrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LegendreP/
.. [4] http://functions.wolfram.com/Polynomials/LegendreP2/
"""
@classmethod
def _eval_at_order(cls, n, m):
P = legendre_poly(n, _x, polys=True).diff((_x, m))
return S.NegativeOne**m * (1 - _x**2)**Rational(m, 2) * P.as_expr()
@classmethod
def eval(cls, n, m, x):
if m.could_extract_minus_sign():
# P^{-m}_n ---> F * P^m_n
return S.NegativeOne**(-m) * (factorial(m + n)/factorial(n - m)) * assoc_legendre(n, -m, x)
if m == 0:
# P^0_n ---> L_n
return legendre(n, x)
if x == 0:
return 2**m*sqrt(S.Pi) / (gamma((1 - m - n)/2)*gamma(1 - (m - n)/2))
if n.is_Number and m.is_Number and n.is_integer and m.is_integer:
if n.is_negative:
raise ValueError("%s : 1st index must be nonnegative integer (got %r)" % (cls, n))
if abs(m) > n:
raise ValueError("%s : abs('2nd index') must be <= '1st index' (got %r, %r)" % (cls, n, m))
return cls._eval_at_order(int(n), abs(int(m))).subs(_x, x)
def fdiff(self, argindex=3):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt m
raise ArgumentIndexError(self, argindex)
elif argindex == 3:
# Diff wrt x
# Find better formula, this is unsuitable for x = 1
n, m, x = self.args
return 1/(x**2 - 1)*(x*n*assoc_legendre(n, m, x) - (m + n)*assoc_legendre(n - 1, m, x))
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, m, x, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k")
kern = factorial(2*n - 2*k)/(2**n*factorial(n - k)*factorial(
k)*factorial(n - 2*k - m))*S.NegativeOne**k*x**(n - m - 2*k)
return (1 - x**2)**(m/2) * Sum(kern, (k, 0, floor((n - m)*S.Half)))
def _eval_conjugate(self):
n, m, x = self.args
return self.func(n, m.conjugate(), x.conjugate())
#----------------------------------------------------------------------------
# Hermite polynomials
#
class hermite(OrthogonalPolynomial):
r"""
``hermite(n, x)`` gives the $n$th Hermite polynomial in $x$, $H_n(x)$
Explanation
===========
The Hermite polynomials are orthogonal on $(-\infty, \infty)$
with respect to the weight $\exp\left(-x^2\right)$.
Examples
========
>>> from sympy import hermite, diff
>>> from sympy.abc import x, n
>>> hermite(0, x)
1
>>> hermite(1, x)
2*x
>>> hermite(2, x)
4*x**2 - 2
>>> hermite(n, x)
hermite(n, x)
>>> diff(hermite(n,x), x)
2*n*hermite(n - 1, x)
>>> hermite(n, -x)
(-1)**n*hermite(n, x)
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
laguerre, assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
.. [2] http://mathworld.wolfram.com/HermitePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/HermiteH/
"""
_ortho_poly = staticmethod(hermite_poly)
@classmethod
def eval(cls, n, x):
if not n.is_Number:
# Symbolic result H_n(x)
# H_n(-x) ---> (-1)**n * H_n(x)
if x.could_extract_minus_sign():
return S.NegativeOne**n * hermite(n, -x)
# We can evaluate for some special values of x
if x.is_zero:
return 2**n * sqrt(S.Pi) / gamma((S.One - n)/2)
elif x is S.Infinity:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError(
"The index n must be nonnegative integer (got %r)" % n)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return 2*n*hermite(n - 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy.concrete.summations import Sum
k = Dummy("k")
kern = S.NegativeOne**k / (factorial(k)*factorial(n - 2*k)) * (2*x)**(n - 2*k)
return factorial(n)*Sum(kern, (k, 0, floor(n/2)))
#----------------------------------------------------------------------------
# Laguerre polynomials
#
class laguerre(OrthogonalPolynomial):
r"""
Returns the $n$th Laguerre polynomial in $x$, $L_n(x)$.
Examples
========
>>> from sympy import laguerre, diff
>>> from sympy.abc import x, n
>>> laguerre(0, x)
1
>>> laguerre(1, x)
1 - x
>>> laguerre(2, x)
x**2/2 - 2*x + 1
>>> laguerre(3, x)
-x**3/6 + 3*x**2/2 - 3*x + 1
>>> laguerre(n, x)
laguerre(n, x)
>>> diff(laguerre(n, x), x)
-assoc_laguerre(n - 1, 1, x)
Parameters
==========
n : int
Degree of Laguerre polynomial. Must be `n \ge 0`.
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
assoc_laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
.. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/
"""
_ortho_poly = staticmethod(laguerre_poly)
@classmethod
def eval(cls, n, x):
if n.is_integer is False:
raise ValueError("Error: n should be an integer.")
if not n.is_Number:
# Symbolic result L_n(x)
# L_{n}(-x) ---> exp(-x) * L_{-n-1}(x)
# L_{-n}(x) ---> exp(x) * L_{n-1}(-x)
if n.could_extract_minus_sign() and not(-n - 1).could_extract_minus_sign():
return exp(x)*laguerre(-n - 1, -x)
# We can evaluate for some special values of x
if x.is_zero:
return S.One
elif x is S.NegativeInfinity:
return S.Infinity
elif x is S.Infinity:
return S.NegativeOne**n * S.Infinity
else:
if n.is_negative:
return exp(x)*laguerre(-n - 1, -x)
else:
return cls._eval_at_order(n, x)
def fdiff(self, argindex=2):
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt x
n, x = self.args
return -assoc_laguerre(n - 1, 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
from sympy.concrete.summations import Sum
# Make sure n \in N_0
if n.is_negative:
return exp(x) * self._eval_rewrite_as_polynomial(-n - 1, -x, **kwargs)
if n.is_integer is False:
raise ValueError("Error: n should be an integer.")
k = Dummy("k")
kern = RisingFactorial(-n, k) / factorial(k)**2 * x**k
return Sum(kern, (k, 0, n))
class assoc_laguerre(OrthogonalPolynomial):
r"""
Returns the $n$th generalized Laguerre polynomial in $x$, $L_n(x)$.
Examples
========
>>> from sympy import assoc_laguerre, diff
>>> from sympy.abc import x, n, a
>>> assoc_laguerre(0, a, x)
1
>>> assoc_laguerre(1, a, x)
a - x + 1
>>> assoc_laguerre(2, a, x)
a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
>>> assoc_laguerre(3, a, x)
a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
x*(-a**2/2 - 5*a/2 - 3) + 1
>>> assoc_laguerre(n, a, 0)
binomial(a + n, a)
>>> assoc_laguerre(n, a, x)
assoc_laguerre(n, a, x)
>>> assoc_laguerre(n, 0, x)
laguerre(n, x)
>>> diff(assoc_laguerre(n, a, x), x)
-assoc_laguerre(n - 1, a + 1, x)
>>> diff(assoc_laguerre(n, a, x), a)
Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))
Parameters
==========
n : int
Degree of Laguerre polynomial. Must be `n \ge 0`.
alpha : Expr
Arbitrary expression. For ``alpha=0`` regular Laguerre
polynomials will be generated.
See Also
========
jacobi, gegenbauer,
chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
legendre, assoc_legendre,
hermite,
laguerre,
sympy.polys.orthopolys.jacobi_poly
sympy.polys.orthopolys.gegenbauer_poly
sympy.polys.orthopolys.chebyshevt_poly
sympy.polys.orthopolys.chebyshevu_poly
sympy.polys.orthopolys.hermite_poly
sympy.polys.orthopolys.legendre_poly
sympy.polys.orthopolys.laguerre_poly
References
==========
.. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials
.. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
.. [3] http://functions.wolfram.com/Polynomials/LaguerreL/
.. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/
"""
@classmethod
def eval(cls, n, alpha, x):
# L_{n}^{0}(x) ---> L_{n}(x)
if alpha.is_zero:
return laguerre(n, x)
if not n.is_Number:
# We can evaluate for some special values of x
if x.is_zero:
return binomial(n + alpha, alpha)
elif x is S.Infinity and n > 0:
return S.NegativeOne**n * S.Infinity
elif x is S.NegativeInfinity and n > 0:
return S.Infinity
else:
# n is a given fixed integer, evaluate into polynomial
if n.is_negative:
raise ValueError(
"The index n must be nonnegative integer (got %r)" % n)
else:
return laguerre_poly(n, x, alpha)
def fdiff(self, argindex=3):
from sympy.concrete.summations import Sum
if argindex == 1:
# Diff wrt n
raise ArgumentIndexError(self, argindex)
elif argindex == 2:
# Diff wrt alpha
n, alpha, x = self.args
k = Dummy("k")
return Sum(assoc_laguerre(k, alpha, x) / (n - alpha), (k, 0, n - 1))
elif argindex == 3:
# Diff wrt x
n, alpha, x = self.args
return -assoc_laguerre(n - 1, alpha + 1, x)
else:
raise ArgumentIndexError(self, argindex)
def _eval_rewrite_as_polynomial(self, n, alpha, x, **kwargs):
from sympy.concrete.summations import Sum
# Make sure n \in N_0
if n.is_negative or n.is_integer is False:
raise ValueError("Error: n should be a non-negative integer.")
k = Dummy("k")
kern = RisingFactorial(
-n, k) / (gamma(k + alpha + 1) * factorial(k)) * x**k
return gamma(n + alpha + 1) / factorial(n) * Sum(kern, (k, 0, n))
def _eval_conjugate(self):
n, alpha, x = self.args
return self.func(n, alpha.conjugate(), x.conjugate())
|
5ba76ce32e3c5e3d7a77e3562a46fc58145d82862eb4b78a807d6d47acdf5e6f | from sympy.assumptions.refine import refine
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.function import expand_log
from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (adjoint, conjugate, re, sign, transpose)
from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log)
from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.polys.polytools import gcd
from sympy.series.order import O
from sympy.simplify.simplify import simplify
from sympy.core.parameters import global_parameters
from sympy.functions.elementary.exponential import match_real_imag
from sympy.abc import x, y, z
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
from sympy.testing.pytest import raises, XFAIL, _both_exp_pow
@_both_exp_pow
def test_exp_values():
if global_parameters.exp_is_pow:
assert type(exp(x)) is Pow
else:
assert type(exp(x)) is exp
k = Symbol('k', integer=True)
assert exp(nan) is nan
assert exp(oo) is oo
assert exp(-oo) == 0
assert exp(0) == 1
assert exp(1) == E
assert exp(-1 + x).as_base_exp() == (S.Exp1, x - 1)
assert exp(1 + x).as_base_exp() == (S.Exp1, x + 1)
assert exp(pi*I/2) == I
assert exp(pi*I) == -1
assert exp(pi*I*Rational(3, 2)) == -I
assert exp(2*pi*I) == 1
assert refine(exp(pi*I*2*k)) == 1
assert refine(exp(pi*I*2*(k + S.Half))) == -1
assert refine(exp(pi*I*2*(k + Rational(1, 4)))) == I
assert refine(exp(pi*I*2*(k + Rational(3, 4)))) == -I
assert exp(log(x)) == x
assert exp(2*log(x)) == x**2
assert exp(pi*log(x)) == x**pi
assert exp(17*log(x) + E*log(y)) == x**17 * y**E
assert exp(x*log(x)) != x**x
assert exp(sin(x)*log(x)) != x
assert exp(3*log(x) + oo*x) == exp(oo*x) * x**3
assert exp(4*log(x)*log(y) + 3*log(x)) == x**3 * exp(4*log(x)*log(y))
assert exp(-oo, evaluate=False).is_finite is True
assert exp(oo, evaluate=False).is_finite is False
@_both_exp_pow
def test_exp_period():
assert exp(I*pi*Rational(9, 4)) == exp(I*pi/4)
assert exp(I*pi*Rational(46, 18)) == exp(I*pi*Rational(5, 9))
assert exp(I*pi*Rational(25, 7)) == exp(I*pi*Rational(-3, 7))
assert exp(I*pi*Rational(-19, 3)) == exp(-I*pi/3)
assert exp(I*pi*Rational(37, 8)) - exp(I*pi*Rational(-11, 8)) == 0
assert exp(I*pi*Rational(-5, 3)) / exp(I*pi*Rational(11, 5)) * exp(I*pi*Rational(148, 15)) == 1
assert exp(2 - I*pi*Rational(17, 5)) == exp(2 + I*pi*Rational(3, 5))
assert exp(log(3) + I*pi*Rational(29, 9)) == 3 * exp(I*pi*Rational(-7, 9))
n = Symbol('n', integer=True)
e = Symbol('e', even=True)
assert exp(e*I*pi) == 1
assert exp((e + 1)*I*pi) == -1
assert exp((1 + 4*n)*I*pi/2) == I
assert exp((-1 + 4*n)*I*pi/2) == -I
@_both_exp_pow
def test_exp_log():
x = Symbol("x", real=True)
assert log(exp(x)) == x
assert exp(log(x)) == x
if not global_parameters.exp_is_pow:
assert log(x).inverse() == exp
assert exp(x).inverse() == log
y = Symbol("y", polar=True)
assert log(exp_polar(z)) == z
assert exp(log(y)) == y
@_both_exp_pow
def test_exp_expand():
e = exp(log(Rational(2))*(1 + x) - log(Rational(2))*x)
assert e.expand() == 2
assert exp(x + y) != exp(x)*exp(y)
assert exp(x + y).expand() == exp(x)*exp(y)
@_both_exp_pow
def test_exp__as_base_exp():
assert exp(x).as_base_exp() == (E, x)
assert exp(2*x).as_base_exp() == (E, 2*x)
assert exp(x*y).as_base_exp() == (E, x*y)
assert exp(-x).as_base_exp() == (E, -x)
# Pow( *expr.as_base_exp() ) == expr invariant should hold
assert E**x == exp(x)
assert E**(2*x) == exp(2*x)
assert E**(x*y) == exp(x*y)
assert exp(x).base is S.Exp1
assert exp(x).exp == x
@_both_exp_pow
def test_exp_infinity():
assert exp(I*y) != nan
assert refine(exp(I*oo)) is nan
assert refine(exp(-I*oo)) is nan
assert exp(y*I*oo) != nan
assert exp(zoo) is nan
x = Symbol('x', extended_real=True, finite=False)
assert exp(x).is_complex is None
@_both_exp_pow
def test_exp_subs():
x = Symbol('x')
e = (exp(3*log(x), evaluate=False)) # evaluates to x**3
assert e.subs(x**3, y**3) == e
assert e.subs(x**2, 5) == e
assert (x**3).subs(x**2, y) != y**Rational(3, 2)
assert exp(exp(x) + exp(x**2)).subs(exp(exp(x)), y) == y * exp(exp(x**2))
assert exp(x).subs(E, y) == y**x
x = symbols('x', real=True)
assert exp(5*x).subs(exp(7*x), y) == y**Rational(5, 7)
assert exp(2*x + 7).subs(exp(3*x), y) == y**Rational(2, 3) * exp(7)
x = symbols('x', positive=True)
assert exp(3*log(x)).subs(x**2, y) == y**Rational(3, 2)
# differentiate between E and exp
assert exp(exp(x + E)).subs(exp, 3) == 3**(3**(x + E))
assert exp(exp(x + E)).subs(exp, sin) == sin(sin(x + E))
assert exp(exp(x + E)).subs(E, 3) == 3**(3**(x + 3))
assert exp(3).subs(E, sin) == sin(3)
def test_exp_adjoint():
assert adjoint(exp(x)) == exp(adjoint(x))
def test_exp_conjugate():
assert conjugate(exp(x)) == exp(conjugate(x))
@_both_exp_pow
def test_exp_transpose():
assert transpose(exp(x)) == exp(transpose(x))
@_both_exp_pow
def test_exp_rewrite():
assert exp(x).rewrite(sin) == sinh(x) + cosh(x)
assert exp(x*I).rewrite(cos) == cos(x) + I*sin(x)
assert exp(1).rewrite(cos) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(x).rewrite(tanh) == (1 + tanh(x/2))/(1 - tanh(x/2))
assert exp(pi*I/4).rewrite(sqrt) == sqrt(2)/2 + sqrt(2)*I/2
assert exp(pi*I/3).rewrite(sqrt) == S.Half + sqrt(3)*I/2
if not global_parameters.exp_is_pow:
assert exp(x*log(y)).rewrite(Pow) == y**x
assert exp(log(x)*log(y)).rewrite(Pow) in [x**log(y), y**log(x)]
assert exp(log(log(x))*y).rewrite(Pow) == log(x)**y
n = Symbol('n', integer=True)
assert Sum((exp(pi*I/2)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == Rational(4, 5) + I*2/5
assert Sum((exp(pi*I/4)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == 1/(1 - sqrt(2)*(1 + I)/4)
assert (Sum((exp(pi*I/3)/2)**n, (n, 0, oo)).rewrite(sqrt).doit().cancel()
== 4*I/(sqrt(3) + 3*I))
@_both_exp_pow
def test_exp_leading_term():
assert exp(x).as_leading_term(x) == 1
assert exp(2 + x).as_leading_term(x) == exp(2)
assert exp((2*x + 3) / (x+1)).as_leading_term(x) == exp(3)
# The following tests are commented, since now SymPy returns the
# original function when the leading term in the series expansion does
# not exist.
# raises(NotImplementedError, lambda: exp(1/x).as_leading_term(x))
# raises(NotImplementedError, lambda: exp((x + 1) / x**2).as_leading_term(x))
# raises(NotImplementedError, lambda: exp(x + 1/x).as_leading_term(x))
@_both_exp_pow
def test_exp_taylor_term():
x = symbols('x')
assert exp(x).taylor_term(1, x) == x
assert exp(x).taylor_term(3, x) == x**3/6
assert exp(x).taylor_term(4, x) == x**4/24
assert exp(x).taylor_term(-1, x) is S.Zero
def test_exp_MatrixSymbol():
A = MatrixSymbol("A", 2, 2)
assert exp(A).has(exp)
def test_exp_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: exp(x).fdiff(2))
def test_log_values():
assert log(nan) is nan
assert log(oo) is oo
assert log(-oo) is oo
assert log(zoo) is zoo
assert log(-zoo) is zoo
assert log(0) is zoo
assert log(1) == 0
assert log(-1) == I*pi
assert log(E) == 1
assert log(-E).expand() == 1 + I*pi
assert unchanged(log, pi)
assert log(-pi).expand() == log(pi) + I*pi
assert unchanged(log, 17)
assert log(-17) == log(17) + I*pi
assert log(I) == I*pi/2
assert log(-I) == -I*pi/2
assert log(17*I) == I*pi/2 + log(17)
assert log(-17*I).expand() == -I*pi/2 + log(17)
assert log(oo*I) is oo
assert log(-oo*I) is oo
assert log(0, 2) is zoo
assert log(0, 5) is zoo
assert exp(-log(3))**(-1) == 3
assert log(S.Half) == -log(2)
assert log(2*3).func is log
assert log(2*3**2).func is log
def test_match_real_imag():
x, y = symbols('x,y', real=True)
i = Symbol('i', imaginary=True)
assert match_real_imag(S.One) == (1, 0)
assert match_real_imag(I) == (0, 1)
assert match_real_imag(3 - 5*I) == (3, -5)
assert match_real_imag(-sqrt(3) + S.Half*I) == (-sqrt(3), S.Half)
assert match_real_imag(x + y*I) == (x, y)
assert match_real_imag(x*I + y*I) == (0, x + y)
assert match_real_imag((x + y)*I) == (0, x + y)
assert match_real_imag(Rational(-2, 3)*i*I) == (None, None)
assert match_real_imag(1 - 2*i) == (None, None)
assert match_real_imag(sqrt(2)*(3 - 5*I)) == (None, None)
def test_log_exact():
# check for pi/2, pi/3, pi/4, pi/6, pi/8, pi/12; pi/5, pi/10:
for n in range(-23, 24):
if gcd(n, 24) != 1:
assert log(exp(n*I*pi/24).rewrite(sqrt)) == n*I*pi/24
for n in range(-9, 10):
assert log(exp(n*I*pi/10).rewrite(sqrt)) == n*I*pi/10
assert log(S.Half - I*sqrt(3)/2) == -I*pi/3
assert log(Rational(-1, 2) + I*sqrt(3)/2) == I*pi*Rational(2, 3)
assert log(-sqrt(2)/2 - I*sqrt(2)/2) == -I*pi*Rational(3, 4)
assert log(-sqrt(3)/2 - I*S.Half) == -I*pi*Rational(5, 6)
assert log(Rational(-1, 4) + sqrt(5)/4 - I*sqrt(sqrt(5)/8 + Rational(5, 8))) == -I*pi*Rational(2, 5)
assert log(sqrt(Rational(5, 8) - sqrt(5)/8) + I*(Rational(1, 4) + sqrt(5)/4)) == I*pi*Rational(3, 10)
assert log(-sqrt(sqrt(2)/4 + S.Half) + I*sqrt(S.Half - sqrt(2)/4)) == I*pi*Rational(7, 8)
assert log(-sqrt(6)/4 - sqrt(2)/4 + I*(-sqrt(6)/4 + sqrt(2)/4)) == -I*pi*Rational(11, 12)
assert log(-1 + I*sqrt(3)) == log(2) + I*pi*Rational(2, 3)
assert log(5 + 5*I) == log(5*sqrt(2)) + I*pi/4
assert log(sqrt(-12)) == log(2*sqrt(3)) + I*pi/2
assert log(-sqrt(6) + sqrt(2) - I*sqrt(6) - I*sqrt(2)) == log(4) - I*pi*Rational(7, 12)
assert log(-sqrt(6-3*sqrt(2)) - I*sqrt(6+3*sqrt(2))) == log(2*sqrt(3)) - I*pi*Rational(5, 8)
assert log(1 + I*sqrt(2-sqrt(2))/sqrt(2+sqrt(2))) == log(2/sqrt(sqrt(2) + 2)) + I*pi/8
assert log(cos(pi*Rational(7, 12)) + I*sin(pi*Rational(7, 12))) == I*pi*Rational(7, 12)
assert log(cos(pi*Rational(6, 5)) + I*sin(pi*Rational(6, 5))) == I*pi*Rational(-4, 5)
assert log(5*(1 + I)/sqrt(2)) == log(5) + I*pi/4
assert log(sqrt(2)*(-sqrt(3) + 1 - sqrt(3)*I - I)) == log(4) - I*pi*Rational(7, 12)
assert log(-sqrt(2)*(1 - I*sqrt(3))) == log(2*sqrt(2)) + I*pi*Rational(2, 3)
assert log(sqrt(3)*I*(-sqrt(6 - 3*sqrt(2)) - I*sqrt(3*sqrt(2) + 6))) == log(6) - I*pi/8
zero = (1 + sqrt(2))**2 - 3 - 2*sqrt(2)
assert log(zero - I*sqrt(3)) == log(sqrt(3)) - I*pi/2
assert unchanged(log, zero + I*zero) or log(zero + zero*I) is zoo
# bail quickly if no obvious simplification is possible:
assert unchanged(log, (sqrt(2)-1/sqrt(sqrt(3)+I))**1000)
# beware of non-real coefficients
assert unchanged(log, sqrt(2-sqrt(5))*(1 + I))
def test_log_base():
assert log(1, 2) == 0
assert log(2, 2) == 1
assert log(3, 2) == log(3)/log(2)
assert log(6, 2) == 1 + log(3)/log(2)
assert log(6, 3) == 1 + log(2)/log(3)
assert log(2**3, 2) == 3
assert log(3**3, 3) == 3
assert log(5, 1) is zoo
assert log(1, 1) is nan
assert log(Rational(2, 3), 10) == log(Rational(2, 3))/log(10)
assert log(Rational(2, 3), Rational(1, 3)) == -log(2)/log(3) + 1
assert log(Rational(2, 3), Rational(2, 5)) == \
log(Rational(2, 3))/log(Rational(2, 5))
# issue 17148
assert log(Rational(8, 3), 2) == -log(3)/log(2) + 3
def test_log_symbolic():
assert log(x, exp(1)) == log(x)
assert log(exp(x)) != x
assert log(x, exp(1)) == log(x)
assert log(x*y) != log(x) + log(y)
assert log(x/y).expand() != log(x) - log(y)
assert log(x/y).expand(force=True) == log(x) - log(y)
assert log(x**y).expand() != y*log(x)
assert log(x**y).expand(force=True) == y*log(x)
assert log(x, 2) == log(x)/log(2)
assert log(E, 2) == 1/log(2)
p, q = symbols('p,q', positive=True)
r = Symbol('r', real=True)
assert log(p**2) != 2*log(p)
assert log(p**2).expand() == 2*log(p)
assert log(x**2).expand() != 2*log(x)
assert log(p**q) != q*log(p)
assert log(exp(p)) == p
assert log(p*q) != log(p) + log(q)
assert log(p*q).expand() == log(p) + log(q)
assert log(-sqrt(3)) == log(sqrt(3)) + I*pi
assert log(-exp(p)) != p + I*pi
assert log(-exp(x)).expand() != x + I*pi
assert log(-exp(r)).expand() == r + I*pi
assert log(x**y) != y*log(x)
assert (log(x**-5)**-1).expand() != -1/log(x)/5
assert (log(p**-5)**-1).expand() == -1/log(p)/5
assert log(-x).func is log and log(-x).args[0] == -x
assert log(-p).func is log and log(-p).args[0] == -p
def test_log_exp():
assert log(exp(4*I*pi)) == 0 # exp evaluates
assert log(exp(-5*I*pi)) == I*pi # exp evaluates
assert log(exp(I*pi*Rational(19, 4))) == I*pi*Rational(3, 4)
assert log(exp(I*pi*Rational(25, 7))) == I*pi*Rational(-3, 7)
assert log(exp(-5*I)) == -5*I + 2*I*pi
@_both_exp_pow
def test_exp_assumptions():
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
for e in exp, exp_polar:
assert e(x).is_real is None
assert e(x).is_imaginary is None
assert e(i).is_real is None
assert e(i).is_imaginary is None
assert e(r).is_real is True
assert e(r).is_imaginary is False
assert e(re(x)).is_extended_real is True
assert e(re(x)).is_imaginary is False
assert Pow(E, I*pi, evaluate=False).is_imaginary == False
assert Pow(E, 2*I*pi, evaluate=False).is_imaginary == False
assert Pow(E, I*pi/2, evaluate=False).is_imaginary == True
assert Pow(E, I*pi/3, evaluate=False).is_imaginary is None
assert exp(0, evaluate=False).is_algebraic
a = Symbol('a', algebraic=True)
an = Symbol('an', algebraic=True, nonzero=True)
r = Symbol('r', rational=True)
rn = Symbol('rn', rational=True, nonzero=True)
assert exp(a).is_algebraic is None
assert exp(an).is_algebraic is False
assert exp(pi*r).is_algebraic is None
assert exp(pi*rn).is_algebraic is False
assert exp(0, evaluate=False).is_algebraic is True
assert exp(I*pi/3, evaluate=False).is_algebraic is True
assert exp(I*pi*r, evaluate=False).is_algebraic is True
@_both_exp_pow
def test_exp_AccumBounds():
assert exp(AccumBounds(1, 2)) == AccumBounds(E, E**2)
def test_log_assumptions():
p = symbols('p', positive=True)
n = symbols('n', negative=True)
z = symbols('z', zero=True)
x = symbols('x', infinite=True, extended_positive=True)
assert log(z).is_positive is False
assert log(x).is_extended_positive is True
assert log(2) > 0
assert log(1, evaluate=False).is_zero
assert log(1 + z).is_zero
assert log(p).is_zero is None
assert log(n).is_zero is False
assert log(0.5).is_negative is True
assert log(exp(p) + 1).is_positive
assert log(1, evaluate=False).is_algebraic
assert log(42, evaluate=False).is_algebraic is False
assert log(1 + z).is_rational
def test_log_hashing():
assert x != log(log(x))
assert hash(x) != hash(log(log(x)))
assert log(x) != log(log(log(x)))
e = 1/log(log(x) + log(log(x)))
assert e.base.func is log
e = 1/log(log(x) + log(log(log(x))))
assert e.base.func is log
e = log(log(x))
assert e.func is log
assert x.func is not log
assert hash(log(log(x))) != hash(x)
assert e != x
def test_log_sign():
assert sign(log(2)) == 1
def test_log_expand_complex():
assert log(1 + I).expand(complex=True) == log(2)/2 + I*pi/4
assert log(1 - sqrt(2)).expand(complex=True) == log(sqrt(2) - 1) + I*pi
def test_log_apply_evalf():
value = (log(3)/log(2) - 1).evalf()
assert value.epsilon_eq(Float("0.58496250072115618145373"))
def test_log_nseries():
assert log(x - 1)._eval_nseries(x, 4, None, I) == I*pi - x - x**2/2 - x**3/3 + O(x**4)
assert log(x - 1)._eval_nseries(x, 4, None, -I) == -I*pi - x - x**2/2 - x**3/3 + O(x**4)
assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x + x**2/2 + O(x**3)
assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == -I*pi - I*x + x**2/2 + O(x**3)
assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x**2 + O(x**3)
assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == I*pi - I*x**2 + O(x**3)
def test_log_expand():
w = Symbol("w", positive=True)
e = log(w**(log(5)/log(3)))
assert e.expand() == log(5)/log(3) * log(w)
x, y, z = symbols('x,y,z', positive=True)
assert log(x*(y + z)).expand(mul=False) == log(x) + log(y + z)
assert log(log(x**2)*log(y*z)).expand() in [log(2*log(x)*log(y) +
2*log(x)*log(z)), log(log(x)*log(z) + log(y)*log(x)) + log(2),
log((log(y) + log(z))*log(x)) + log(2)]
assert log(x**log(x**2)).expand(deep=False) == log(x)*log(x**2)
assert log(x**log(x**2)).expand() == 2*log(x)**2
x, y = symbols('x,y')
assert log(x*y).expand(force=True) == log(x) + log(y)
assert log(x**y).expand(force=True) == y*log(x)
assert log(exp(x)).expand(force=True) == x
# there's generally no need to expand out logs since this requires
# factoring and if simplification is sought, it's cheaper to put
# logs together than it is to take them apart.
assert log(2*3**2).expand() != 2*log(3) + log(2)
@XFAIL
def test_log_expand_fail():
x, y, z = symbols('x,y,z', positive=True)
assert (log(x*(y + z))*(x + y)).expand(mul=True, log=True) == y*log(
x) + y*log(y + z) + z*log(x) + z*log(y + z)
def test_log_simplify():
x = Symbol("x", positive=True)
assert log(x**2).expand() == 2*log(x)
assert expand_log(log(x**(2 + log(2)))) == (2 + log(2))*log(x)
z = Symbol('z')
assert log(sqrt(z)).expand() == log(z)/2
assert expand_log(log(z**(log(2) - 1))) == (log(2) - 1)*log(z)
assert log(z**(-1)).expand() != -log(z)
assert log(z**(x/(x+1))).expand() == x*log(z)/(x + 1)
def test_log_AccumBounds():
assert log(AccumBounds(1, E)) == AccumBounds(0, 1)
assert log(AccumBounds(0, E)) == AccumBounds(-oo, 1)
assert log(AccumBounds(-1, E)) == S.NaN
assert log(AccumBounds(0, oo)) == AccumBounds(-oo, oo)
assert log(AccumBounds(-oo, 0)) == S.NaN
assert log(AccumBounds(-oo, oo)) == S.NaN
@_both_exp_pow
def test_lambertw():
k = Symbol('k')
assert LambertW(x, 0) == LambertW(x)
assert LambertW(x, 0, evaluate=False) != LambertW(x)
assert LambertW(0) == 0
assert LambertW(E) == 1
assert LambertW(-1/E) == -1
assert LambertW(-log(2)/2) == -log(2)
assert LambertW(oo) is oo
assert LambertW(0, 1) is -oo
assert LambertW(0, 42) is -oo
assert LambertW(-pi/2, -1) == -I*pi/2
assert LambertW(-1/E, -1) == -1
assert LambertW(-2*exp(-2), -1) == -2
assert LambertW(2*log(2)) == log(2)
assert LambertW(-pi/2) == I*pi/2
assert LambertW(exp(1 + E)) == E
assert LambertW(x**2).diff(x) == 2*LambertW(x**2)/x/(1 + LambertW(x**2))
assert LambertW(x, k).diff(x) == LambertW(x, k)/x/(1 + LambertW(x, k))
assert LambertW(sqrt(2)).evalf(30).epsilon_eq(
Float("0.701338383413663009202120278965", 30), 1e-29)
assert re(LambertW(2, -1)).evalf().epsilon_eq(Float("-0.834310366631110"))
assert LambertW(-1).is_real is False # issue 5215
assert LambertW(2, evaluate=False).is_real
p = Symbol('p', positive=True)
assert LambertW(p, evaluate=False).is_real
assert LambertW(p - 1, evaluate=False).is_real is None
assert LambertW(-p - 2/S.Exp1, evaluate=False).is_real is False
assert LambertW(S.Half, -1, evaluate=False).is_real is False
assert LambertW(Rational(-1, 10), -1, evaluate=False).is_real
assert LambertW(-10, -1, evaluate=False).is_real is False
assert LambertW(-2, 2, evaluate=False).is_real is False
assert LambertW(0, evaluate=False).is_algebraic
na = Symbol('na', nonzero=True, algebraic=True)
assert LambertW(na).is_algebraic is False
assert LambertW(p).is_zero is False
n = Symbol('n', negative=True)
assert LambertW(n).is_zero is False
def test_issue_5673():
e = LambertW(-1)
assert e.is_comparable is False
assert e.is_positive is not True
e2 = 1 - 1/(1 - exp(-1000))
assert e2.is_positive is not True
e3 = -2 + exp(exp(LambertW(log(2)))*LambertW(log(2)))
assert e3.is_nonzero is not True
def test_log_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: log(x).fdiff(2))
def test_log_taylor_term():
x = symbols('x')
assert log(x).taylor_term(0, x) == x
assert log(x).taylor_term(1, x) == -x**2/2
assert log(x).taylor_term(4, x) == x**5/5
assert log(x).taylor_term(-1, x) is S.Zero
def test_exp_expand_NC():
A, B, C = symbols('A,B,C', commutative=False)
assert exp(A + B).expand() == exp(A + B)
assert exp(A + B + C).expand() == exp(A + B + C)
assert exp(x + y).expand() == exp(x)*exp(y)
assert exp(x + y + z).expand() == exp(x)*exp(y)*exp(z)
@_both_exp_pow
def test_as_numer_denom():
n = symbols('n', negative=True)
assert exp(x).as_numer_denom() == (exp(x), 1)
assert exp(-x).as_numer_denom() == (1, exp(x))
assert exp(-2*x).as_numer_denom() == (1, exp(2*x))
assert exp(-2).as_numer_denom() == (1, exp(2))
assert exp(n).as_numer_denom() == (1, exp(-n))
assert exp(-n).as_numer_denom() == (exp(-n), 1)
assert exp(-I*x).as_numer_denom() == (1, exp(I*x))
assert exp(-I*n).as_numer_denom() == (1, exp(I*n))
assert exp(-n).as_numer_denom() == (exp(-n), 1)
@_both_exp_pow
def test_polar():
x, y = symbols('x y', polar=True)
assert abs(exp_polar(I*4)) == 1
assert abs(exp_polar(0)) == 1
assert abs(exp_polar(2 + 3*I)) == exp(2)
assert exp_polar(I*10).n() == exp_polar(I*10)
assert log(exp_polar(z)) == z
assert log(x*y).expand() == log(x) + log(y)
assert log(x**z).expand() == z*log(x)
assert exp_polar(3).exp == 3
# Compare exp(1.0*pi*I).
assert (exp_polar(1.0*pi*I).n(n=5)).as_real_imag()[1] >= 0
assert exp_polar(0).is_rational is True # issue 8008
def test_exp_summation():
w = symbols("w")
m, n, i, j = symbols("m n i j")
expr = exp(Sum(w*i, (i, 0, n), (j, 0, m)))
assert expr.expand() == Product(exp(w*i), (i, 0, n), (j, 0, m))
def test_log_product():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
z = symbols('z', real=True)
w = symbols('w')
expr = log(Product(x**i, (i, 1, n)))
assert simplify(expr) == expr
assert expr.expand() == Sum(i*log(x), (i, 1, n))
expr = log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))
assert simplify(expr) == expr
assert expr.expand() == Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
expr = log(Product(-2, (n, 0, 4)))
assert simplify(expr) == expr
assert expr.expand() == expr
assert expr.expand(force=True) == Sum(log(-2), (n, 0, 4))
expr = log(Product(exp(z*i), (i, 0, n)))
assert expr.expand() == Sum(z*i, (i, 0, n))
expr = log(Product(exp(w*i), (i, 0, n)))
assert expr.expand() == expr
assert expr.expand(force=True) == Sum(w*i, (i, 0, n))
expr = log(Product(i**2*abs(j), (i, 1, n), (j, 1, m)))
assert expr.expand() == Sum(2*log(i) + log(j), (i, 1, n), (j, 1, m))
@XFAIL
def test_log_product_simplify_to_sum():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
assert simplify(log(Product(x**i, (i, 1, n)))) == Sum(i*log(x), (i, 1, n))
assert simplify(log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))) == \
Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
def test_issue_8866():
assert simplify(log(x, 10, evaluate=False)) == simplify(log(x, 10))
assert expand_log(log(x, 10, evaluate=False)) == expand_log(log(x, 10))
y = Symbol('y', positive=True)
l1 = log(exp(y), exp(10))
b1 = log(exp(y), exp(5))
l2 = log(exp(y), exp(10), evaluate=False)
b2 = log(exp(y), exp(5), evaluate=False)
assert simplify(log(l1, b1)) == simplify(log(l2, b2))
assert expand_log(log(l1, b1)) == expand_log(log(l2, b2))
def test_log_expand_factor():
assert (log(18)/log(3) - 2).expand(factor=True) == log(2)/log(3)
assert (log(12)/log(2)).expand(factor=True) == log(3)/log(2) + 2
assert (log(15)/log(3)).expand(factor=True) == 1 + log(5)/log(3)
assert (log(2)/(-log(12) + log(24))).expand(factor=True) == 1
assert expand_log(log(12), factor=True) == log(3) + 2*log(2)
assert expand_log(log(21)/log(7), factor=False) == log(3)/log(7) + 1
assert expand_log(log(45)/log(5) + log(20), factor=False) == \
1 + 2*log(3)/log(5) + log(20)
assert expand_log(log(45)/log(5) + log(26), factor=True) == \
log(2) + log(13) + (log(5) + 2*log(3))/log(5)
def test_issue_9116():
n = Symbol('n', positive=True, integer=True)
assert log(n).is_nonnegative is True
def test_issue_18473():
assert exp(x*log(cos(1/x))).as_leading_term(x) == S.NaN
assert exp(x*log(tan(1/x))).as_leading_term(x) == S.NaN
assert log(cos(1/x)).as_leading_term(x) == S.NaN
assert log(tan(1/x)).as_leading_term(x) == S.NaN
assert log(cos(1/x) + 2).as_leading_term(x) == AccumBounds(0, log(3))
assert exp(x*log(cos(1/x) + 2)).as_leading_term(x) == 1
assert log(cos(1/x) - 2).as_leading_term(x) == S.NaN
assert exp(x*log(cos(1/x) - 2)).as_leading_term(x) == S.NaN
assert log(cos(1/x) + 1).as_leading_term(x) == AccumBounds(-oo, log(2))
assert exp(x*log(cos(1/x) + 1)).as_leading_term(x) == AccumBounds(0, 1)
assert log(sin(1/x)**2).as_leading_term(x) == AccumBounds(-oo, 0)
assert exp(x*log(sin(1/x)**2)).as_leading_term(x) == AccumBounds(0, 1)
assert log(tan(1/x)**2).as_leading_term(x) == AccumBounds(-oo, oo)
assert exp(2*x*(log(tan(1/x)**2))).as_leading_term(x) == AccumBounds(0, oo)
|
016a69904dac35d1574976523635d1ca5fe3f8e2a03b0f417f5211bd4497f646 | from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core.add import Add
from sympy.core.function import (Lambda, diff)
from sympy.core.mul import Mul
from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (arg, conjugate, im, re)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh, coth, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, sinc, tan)
from sympy.functions.special.bessel import (besselj, jn)
from sympy.functions.special.delta_functions import Heaviside
from sympy.matrices.dense import Matrix
from sympy.polys.polytools import (cancel, gcd)
from sympy.series.limits import limit
from sympy.series.order import O
from sympy.series.series import series
from sympy.sets.fancysets import ImageSet
from sympy.sets.sets import (FiniteSet, Interval)
from sympy.simplify.simplify import simplify
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
from sympy.core.relational import Ne, Eq
from sympy.functions.elementary.piecewise import Piecewise
from sympy.sets.setexpr import SetExpr
from sympy.testing.pytest import XFAIL, slow, raises
x, y, z = symbols('x y z')
r = Symbol('r', real=True)
k, m = symbols('k m', integer=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
np = Symbol('p', nonpositive=True)
nn = Symbol('n', nonnegative=True)
nz = Symbol('nz', nonzero=True)
ep = Symbol('ep', extended_positive=True)
en = Symbol('en', extended_negative=True)
enp = Symbol('ep', extended_nonpositive=True)
enn = Symbol('en', extended_nonnegative=True)
enz = Symbol('enz', extended_nonzero=True)
a = Symbol('a', algebraic=True)
na = Symbol('na', nonzero=True, algebraic=True)
def test_sin():
x, y = symbols('x y')
assert sin.nargs == FiniteSet(1)
assert sin(nan) is nan
assert sin(zoo) is nan
assert sin(oo) == AccumBounds(-1, 1)
assert sin(oo) - sin(oo) == AccumBounds(-2, 2)
assert sin(oo*I) == oo*I
assert sin(-oo*I) == -oo*I
assert 0*sin(oo) is S.Zero
assert 0/sin(oo) is S.Zero
assert 0 + sin(oo) == AccumBounds(-1, 1)
assert 5 + sin(oo) == AccumBounds(4, 6)
assert sin(0) == 0
assert sin(asin(x)) == x
assert sin(atan(x)) == x / sqrt(1 + x**2)
assert sin(acos(x)) == sqrt(1 - x**2)
assert sin(acot(x)) == 1 / (sqrt(1 + 1 / x**2) * x)
assert sin(acsc(x)) == 1 / x
assert sin(asec(x)) == sqrt(1 - 1 / x**2)
assert sin(atan2(y, x)) == y / sqrt(x**2 + y**2)
assert sin(pi*I) == sinh(pi)*I
assert sin(-pi*I) == -sinh(pi)*I
assert sin(-2*I) == -sinh(2)*I
assert sin(pi) == 0
assert sin(-pi) == 0
assert sin(2*pi) == 0
assert sin(-2*pi) == 0
assert sin(-3*10**73*pi) == 0
assert sin(7*10**103*pi) == 0
assert sin(pi/2) == 1
assert sin(-pi/2) == -1
assert sin(pi*Rational(5, 2)) == 1
assert sin(pi*Rational(7, 2)) == -1
ne = symbols('ne', integer=True, even=False)
e = symbols('e', even=True)
assert sin(pi*ne/2) == (-1)**(ne/2 - S.Half)
assert sin(pi*k/2).func == sin
assert sin(pi*e/2) == 0
assert sin(pi*k) == 0
assert sin(pi*k).subs(k, 3) == sin(pi*k/2).subs(k, 6) # issue 8298
assert sin(pi/3) == S.Half*sqrt(3)
assert sin(pi*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3)
assert sin(pi/4) == S.Half*sqrt(2)
assert sin(-pi/4) == Rational(-1, 2)*sqrt(2)
assert sin(pi*Rational(17, 4)) == S.Half*sqrt(2)
assert sin(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
assert sin(pi/6) == S.Half
assert sin(-pi/6) == Rational(-1, 2)
assert sin(pi*Rational(7, 6)) == Rational(-1, 2)
assert sin(pi*Rational(-5, 6)) == Rational(-1, 2)
assert sin(pi*Rational(1, 5)) == sqrt((5 - sqrt(5)) / 8)
assert sin(pi*Rational(2, 5)) == sqrt((5 + sqrt(5)) / 8)
assert sin(pi*Rational(3, 5)) == sin(pi*Rational(2, 5))
assert sin(pi*Rational(4, 5)) == sin(pi*Rational(1, 5))
assert sin(pi*Rational(6, 5)) == -sin(pi*Rational(1, 5))
assert sin(pi*Rational(8, 5)) == -sin(pi*Rational(2, 5))
assert sin(pi*Rational(-1273, 5)) == -sin(pi*Rational(2, 5))
assert sin(pi/8) == sqrt((2 - sqrt(2))/4)
assert sin(pi/10) == Rational(-1, 4) + sqrt(5)/4
assert sin(pi/12) == -sqrt(2)/4 + sqrt(6)/4
assert sin(pi*Rational(5, 12)) == sqrt(2)/4 + sqrt(6)/4
assert sin(pi*Rational(-7, 12)) == -sqrt(2)/4 - sqrt(6)/4
assert sin(pi*Rational(-11, 12)) == sqrt(2)/4 - sqrt(6)/4
assert sin(pi*Rational(104, 105)) == sin(pi/105)
assert sin(pi*Rational(106, 105)) == -sin(pi/105)
assert sin(pi*Rational(-104, 105)) == -sin(pi/105)
assert sin(pi*Rational(-106, 105)) == sin(pi/105)
assert sin(x*I) == sinh(x)*I
assert sin(k*pi) == 0
assert sin(17*k*pi) == 0
assert sin(2*k*pi + 4) == sin(4)
assert sin(2*k*pi + m*pi + 1) == (-1)**(m + 2*k)*sin(1)
assert sin(k*pi*I) == sinh(k*pi)*I
assert sin(r).is_real is True
assert sin(0, evaluate=False).is_algebraic
assert sin(a).is_algebraic is None
assert sin(na).is_algebraic is False
q = Symbol('q', rational=True)
assert sin(pi*q).is_algebraic
qn = Symbol('qn', rational=True, nonzero=True)
assert sin(qn).is_rational is False
assert sin(q).is_rational is None # issue 8653
assert isinstance(sin( re(x) - im(y)), sin) is True
assert isinstance(sin(-re(x) + im(y)), sin) is False
assert sin(SetExpr(Interval(0, 1))) == SetExpr(ImageSet(Lambda(x, sin(x)),
Interval(0, 1)))
for d in list(range(1, 22)) + [60, 85]:
for n in range(0, d*2 + 1):
x = n*pi/d
e = abs( float(sin(x)) - sin(float(x)) )
assert e < 1e-12
assert sin(0, evaluate=False).is_zero is True
assert sin(k*pi, evaluate=False).is_zero is True
assert sin(Add(1, -1, evaluate=False), evaluate=False).is_zero is True
def test_sin_cos():
for d in [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 24, 30, 40, 60, 120]: # list is not exhaustive...
for n in range(-2*d, d*2):
x = n*pi/d
assert sin(x + pi/2) == cos(x), "fails for %d*pi/%d" % (n, d)
assert sin(x - pi/2) == -cos(x), "fails for %d*pi/%d" % (n, d)
assert sin(x) == cos(x - pi/2), "fails for %d*pi/%d" % (n, d)
assert -sin(x) == cos(x + pi/2), "fails for %d*pi/%d" % (n, d)
def test_sin_series():
assert sin(x).series(x, 0, 9) == \
x - x**3/6 + x**5/120 - x**7/5040 + O(x**9)
def test_sin_rewrite():
assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2
assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2)
assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2)
assert sin(sinh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
assert sin(cosh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
assert sin(tanh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
assert sin(coth(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
assert sin(sin(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
assert sin(cos(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
assert sin(tan(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
assert sin(cot(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I /2
assert sin(x).rewrite(csc) == 1/csc(x)
assert sin(x).rewrite(cos) == cos(x - pi / 2, evaluate=False)
assert sin(x).rewrite(sec) == 1 / sec(x - pi / 2, evaluate=False)
assert sin(cos(x)).rewrite(Pow) == sin(cos(x))
def _test_extrig(f, i, e):
from sympy.core.function import expand_trig
assert unchanged(f, i)
assert expand_trig(f(i)) == f(i)
# testing directly instead of with .expand(trig=True)
# because the other expansions undo the unevaluated Mul
assert expand_trig(f(Mul(i, 1, evaluate=False))) == e
assert abs(f(i) - e).n() < 1e-10
def test_sin_expansion():
# Note: these formulas are not unique. The ones here come from the
# Chebyshev formulas.
assert sin(x + y).expand(trig=True) == sin(x)*cos(y) + cos(x)*sin(y)
assert sin(x - y).expand(trig=True) == sin(x)*cos(y) - cos(x)*sin(y)
assert sin(y - x).expand(trig=True) == cos(x)*sin(y) - sin(x)*cos(y)
assert sin(2*x).expand(trig=True) == 2*sin(x)*cos(x)
assert sin(3*x).expand(trig=True) == -4*sin(x)**3 + 3*sin(x)
assert sin(4*x).expand(trig=True) == -8*sin(x)**3*cos(x) + 4*sin(x)*cos(x)
_test_extrig(sin, 2, 2*sin(1)*cos(1))
_test_extrig(sin, 3, -4*sin(1)**3 + 3*sin(1))
def test_sin_AccumBounds():
assert sin(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, oo)) == AccumBounds(-1, 1)
assert sin(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, S.Pi*Rational(3, 4))) == AccumBounds(0, 1)
assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(7, 4))) == AccumBounds(-1, sin(S.Pi*Rational(3, 4)))
assert sin(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(sin(S.Pi/4), sin(S.Pi/3))
assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 6))) == AccumBounds(sin(S.Pi*Rational(5, 6)), sin(S.Pi*Rational(3, 4)))
def test_sin_fdiff():
assert sin(x).fdiff() == cos(x)
raises(ArgumentIndexError, lambda: sin(x).fdiff(2))
def test_trig_symmetry():
assert sin(-x) == -sin(x)
assert cos(-x) == cos(x)
assert tan(-x) == -tan(x)
assert cot(-x) == -cot(x)
assert sin(x + pi) == -sin(x)
assert sin(x + 2*pi) == sin(x)
assert sin(x + 3*pi) == -sin(x)
assert sin(x + 4*pi) == sin(x)
assert sin(x - 5*pi) == -sin(x)
assert cos(x + pi) == -cos(x)
assert cos(x + 2*pi) == cos(x)
assert cos(x + 3*pi) == -cos(x)
assert cos(x + 4*pi) == cos(x)
assert cos(x - 5*pi) == -cos(x)
assert tan(x + pi) == tan(x)
assert tan(x - 3*pi) == tan(x)
assert cot(x + pi) == cot(x)
assert cot(x - 3*pi) == cot(x)
assert sin(pi/2 - x) == cos(x)
assert sin(pi*Rational(3, 2) - x) == -cos(x)
assert sin(pi*Rational(5, 2) - x) == cos(x)
assert cos(pi/2 - x) == sin(x)
assert cos(pi*Rational(3, 2) - x) == -sin(x)
assert cos(pi*Rational(5, 2) - x) == sin(x)
assert tan(pi/2 - x) == cot(x)
assert tan(pi*Rational(3, 2) - x) == cot(x)
assert tan(pi*Rational(5, 2) - x) == cot(x)
assert cot(pi/2 - x) == tan(x)
assert cot(pi*Rational(3, 2) - x) == tan(x)
assert cot(pi*Rational(5, 2) - x) == tan(x)
assert sin(pi/2 + x) == cos(x)
assert cos(pi/2 + x) == -sin(x)
assert tan(pi/2 + x) == -cot(x)
assert cot(pi/2 + x) == -tan(x)
def test_cos():
x, y = symbols('x y')
assert cos.nargs == FiniteSet(1)
assert cos(nan) is nan
assert cos(oo) == AccumBounds(-1, 1)
assert cos(oo) - cos(oo) == AccumBounds(-2, 2)
assert cos(oo*I) is oo
assert cos(-oo*I) is oo
assert cos(zoo) is nan
assert cos(0) == 1
assert cos(acos(x)) == x
assert cos(atan(x)) == 1 / sqrt(1 + x**2)
assert cos(asin(x)) == sqrt(1 - x**2)
assert cos(acot(x)) == 1 / sqrt(1 + 1 / x**2)
assert cos(acsc(x)) == sqrt(1 - 1 / x**2)
assert cos(asec(x)) == 1 / x
assert cos(atan2(y, x)) == x / sqrt(x**2 + y**2)
assert cos(pi*I) == cosh(pi)
assert cos(-pi*I) == cosh(pi)
assert cos(-2*I) == cosh(2)
assert cos(pi/2) == 0
assert cos(-pi/2) == 0
assert cos(pi/2) == 0
assert cos(-pi/2) == 0
assert cos((-3*10**73 + 1)*pi/2) == 0
assert cos((7*10**103 + 1)*pi/2) == 0
n = symbols('n', integer=True, even=False)
e = symbols('e', even=True)
assert cos(pi*n/2) == 0
assert cos(pi*e/2) == (-1)**(e/2)
assert cos(pi) == -1
assert cos(-pi) == -1
assert cos(2*pi) == 1
assert cos(5*pi) == -1
assert cos(8*pi) == 1
assert cos(pi/3) == S.Half
assert cos(pi*Rational(-2, 3)) == Rational(-1, 2)
assert cos(pi/4) == S.Half*sqrt(2)
assert cos(-pi/4) == S.Half*sqrt(2)
assert cos(pi*Rational(11, 4)) == Rational(-1, 2)*sqrt(2)
assert cos(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
assert cos(pi/6) == S.Half*sqrt(3)
assert cos(-pi/6) == S.Half*sqrt(3)
assert cos(pi*Rational(7, 6)) == Rational(-1, 2)*sqrt(3)
assert cos(pi*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3)
assert cos(pi*Rational(1, 5)) == (sqrt(5) + 1)/4
assert cos(pi*Rational(2, 5)) == (sqrt(5) - 1)/4
assert cos(pi*Rational(3, 5)) == -cos(pi*Rational(2, 5))
assert cos(pi*Rational(4, 5)) == -cos(pi*Rational(1, 5))
assert cos(pi*Rational(6, 5)) == -cos(pi*Rational(1, 5))
assert cos(pi*Rational(8, 5)) == cos(pi*Rational(2, 5))
assert cos(pi*Rational(-1273, 5)) == -cos(pi*Rational(2, 5))
assert cos(pi/8) == sqrt((2 + sqrt(2))/4)
assert cos(pi/12) == sqrt(2)/4 + sqrt(6)/4
assert cos(pi*Rational(5, 12)) == -sqrt(2)/4 + sqrt(6)/4
assert cos(pi*Rational(7, 12)) == sqrt(2)/4 - sqrt(6)/4
assert cos(pi*Rational(11, 12)) == -sqrt(2)/4 - sqrt(6)/4
assert cos(pi*Rational(104, 105)) == -cos(pi/105)
assert cos(pi*Rational(106, 105)) == -cos(pi/105)
assert cos(pi*Rational(-104, 105)) == -cos(pi/105)
assert cos(pi*Rational(-106, 105)) == -cos(pi/105)
assert cos(x*I) == cosh(x)
assert cos(k*pi*I) == cosh(k*pi)
assert cos(r).is_real is True
assert cos(0, evaluate=False).is_algebraic
assert cos(a).is_algebraic is None
assert cos(na).is_algebraic is False
q = Symbol('q', rational=True)
assert cos(pi*q).is_algebraic
assert cos(pi*Rational(2, 7)).is_algebraic
assert cos(k*pi) == (-1)**k
assert cos(2*k*pi) == 1
for d in list(range(1, 22)) + [60, 85]:
for n in range(0, 2*d + 1):
x = n*pi/d
e = abs( float(cos(x)) - cos(float(x)) )
assert e < 1e-12
def test_issue_6190():
c = Float('123456789012345678901234567890.25', '')
for cls in [sin, cos, tan, cot]:
assert cls(c*pi) == cls(pi/4)
assert cls(4.125*pi) == cls(pi/8)
assert cls(4.7*pi) == cls((4.7 % 2)*pi)
def test_cos_series():
assert cos(x).series(x, 0, 9) == \
1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**9)
def test_cos_rewrite():
assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2
assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2)
assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2)
assert cos(sinh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n()
assert cos(cosh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n()
assert cos(tanh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n()
assert cos(coth(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n()
assert cos(sin(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n()
assert cos(cos(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n()
assert cos(tan(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n()
assert cos(cot(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n()
assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2
assert cos(x).rewrite(sec) == 1/sec(x)
assert cos(x).rewrite(sin) == sin(x + pi/2, evaluate=False)
assert cos(x).rewrite(csc) == 1/csc(-x + pi/2, evaluate=False)
assert cos(sin(x)).rewrite(Pow) == cos(sin(x))
def test_cos_expansion():
assert cos(x + y).expand(trig=True) == cos(x)*cos(y) - sin(x)*sin(y)
assert cos(x - y).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
assert cos(y - x).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
assert cos(2*x).expand(trig=True) == 2*cos(x)**2 - 1
assert cos(3*x).expand(trig=True) == 4*cos(x)**3 - 3*cos(x)
assert cos(4*x).expand(trig=True) == 8*cos(x)**4 - 8*cos(x)**2 + 1
_test_extrig(cos, 2, 2*cos(1)**2 - 1)
_test_extrig(cos, 3, 4*cos(1)**3 - 3*cos(1))
def test_cos_AccumBounds():
assert cos(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
assert cos(AccumBounds(0, oo)) == AccumBounds(-1, 1)
assert cos(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
assert cos(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
assert cos(AccumBounds(-S.Pi/3, S.Pi/4)) == AccumBounds(cos(-S.Pi/3), 1)
assert cos(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 4))) == AccumBounds(-1, cos(S.Pi*Rational(3, 4)))
assert cos(AccumBounds(S.Pi*Rational(5, 4), S.Pi*Rational(4, 3))) == AccumBounds(cos(S.Pi*Rational(5, 4)), cos(S.Pi*Rational(4, 3)))
assert cos(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(cos(S.Pi/3), cos(S.Pi/4))
def test_cos_fdiff():
assert cos(x).fdiff() == -sin(x)
raises(ArgumentIndexError, lambda: cos(x).fdiff(2))
def test_tan():
assert tan(nan) is nan
assert tan(zoo) is nan
assert tan(oo) == AccumBounds(-oo, oo)
assert tan(oo) - tan(oo) == AccumBounds(-oo, oo)
assert tan.nargs == FiniteSet(1)
assert tan(oo*I) == I
assert tan(-oo*I) == -I
assert tan(0) == 0
assert tan(atan(x)) == x
assert tan(asin(x)) == x / sqrt(1 - x**2)
assert tan(acos(x)) == sqrt(1 - x**2) / x
assert tan(acot(x)) == 1 / x
assert tan(acsc(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
assert tan(asec(x)) == sqrt(1 - 1 / x**2) * x
assert tan(atan2(y, x)) == y/x
assert tan(pi*I) == tanh(pi)*I
assert tan(-pi*I) == -tanh(pi)*I
assert tan(-2*I) == -tanh(2)*I
assert tan(pi) == 0
assert tan(-pi) == 0
assert tan(2*pi) == 0
assert tan(-2*pi) == 0
assert tan(-3*10**73*pi) == 0
assert tan(pi/2) is zoo
assert tan(pi*Rational(3, 2)) is zoo
assert tan(pi/3) == sqrt(3)
assert tan(pi*Rational(-2, 3)) == sqrt(3)
assert tan(pi/4) is S.One
assert tan(-pi/4) is S.NegativeOne
assert tan(pi*Rational(17, 4)) is S.One
assert tan(pi*Rational(-3, 4)) is S.One
assert tan(pi/5) == sqrt(5 - 2*sqrt(5))
assert tan(pi*Rational(2, 5)) == sqrt(5 + 2*sqrt(5))
assert tan(pi*Rational(18, 5)) == -sqrt(5 + 2*sqrt(5))
assert tan(pi*Rational(-16, 5)) == -sqrt(5 - 2*sqrt(5))
assert tan(pi/6) == 1/sqrt(3)
assert tan(-pi/6) == -1/sqrt(3)
assert tan(pi*Rational(7, 6)) == 1/sqrt(3)
assert tan(pi*Rational(-5, 6)) == 1/sqrt(3)
assert tan(pi/8) == -1 + sqrt(2)
assert tan(pi*Rational(3, 8)) == 1 + sqrt(2) # issue 15959
assert tan(pi*Rational(5, 8)) == -1 - sqrt(2)
assert tan(pi*Rational(7, 8)) == 1 - sqrt(2)
assert tan(pi/10) == sqrt(1 - 2*sqrt(5)/5)
assert tan(pi*Rational(3, 10)) == sqrt(1 + 2*sqrt(5)/5)
assert tan(pi*Rational(17, 10)) == -sqrt(1 + 2*sqrt(5)/5)
assert tan(pi*Rational(-31, 10)) == -sqrt(1 - 2*sqrt(5)/5)
assert tan(pi/12) == -sqrt(3) + 2
assert tan(pi*Rational(5, 12)) == sqrt(3) + 2
assert tan(pi*Rational(7, 12)) == -sqrt(3) - 2
assert tan(pi*Rational(11, 12)) == sqrt(3) - 2
assert tan(pi/24).radsimp() == -2 - sqrt(3) + sqrt(2) + sqrt(6)
assert tan(pi*Rational(5, 24)).radsimp() == -2 + sqrt(3) - sqrt(2) + sqrt(6)
assert tan(pi*Rational(7, 24)).radsimp() == 2 - sqrt(3) - sqrt(2) + sqrt(6)
assert tan(pi*Rational(11, 24)).radsimp() == 2 + sqrt(3) + sqrt(2) + sqrt(6)
assert tan(pi*Rational(13, 24)).radsimp() == -2 - sqrt(3) - sqrt(2) - sqrt(6)
assert tan(pi*Rational(17, 24)).radsimp() == -2 + sqrt(3) + sqrt(2) - sqrt(6)
assert tan(pi*Rational(19, 24)).radsimp() == 2 - sqrt(3) + sqrt(2) - sqrt(6)
assert tan(pi*Rational(23, 24)).radsimp() == 2 + sqrt(3) - sqrt(2) - sqrt(6)
assert tan(x*I) == tanh(x)*I
assert tan(k*pi) == 0
assert tan(17*k*pi) == 0
assert tan(k*pi*I) == tanh(k*pi)*I
assert tan(r).is_real is None
assert tan(r).is_extended_real is True
assert tan(0, evaluate=False).is_algebraic
assert tan(a).is_algebraic is None
assert tan(na).is_algebraic is False
assert tan(pi*Rational(10, 7)) == tan(pi*Rational(3, 7))
assert tan(pi*Rational(11, 7)) == -tan(pi*Rational(3, 7))
assert tan(pi*Rational(-11, 7)) == tan(pi*Rational(3, 7))
assert tan(pi*Rational(15, 14)) == tan(pi/14)
assert tan(pi*Rational(-15, 14)) == -tan(pi/14)
assert tan(r).is_finite is None
assert tan(I*r).is_finite is True
# https://github.com/sympy/sympy/issues/21177
f = tan(pi*(x + S(3)/2))/(3*x)
assert f.as_leading_term(x) == -1/(3*pi*x**2)
def test_tan_series():
assert tan(x).series(x, 0, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315 + O(x**9)
def test_tan_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
assert tan(x).rewrite(cos) == cos(x - S.Pi/2, evaluate=False)/cos(x)
assert tan(x).rewrite(cot) == 1/cot(x)
assert tan(sinh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
assert tan(cosh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
assert tan(tanh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
assert tan(coth(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
assert tan(sin(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
assert tan(cos(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
assert tan(tan(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
assert tan(cot(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
assert 0 == (cos(pi/34)*tan(pi/34) - sin(pi/34)).rewrite(pow)
assert 0 == (cos(pi/17)*tan(pi/17) - sin(pi/17)).rewrite(pow)
assert tan(pi/19).rewrite(pow) == tan(pi/19)
assert tan(pi*Rational(8, 19)).rewrite(sqrt) == tan(pi*Rational(8, 19))
assert tan(x).rewrite(sec) == sec(x)/sec(x - pi/2, evaluate=False)
assert tan(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)/csc(x)
assert tan(sin(x)).rewrite(Pow) == tan(sin(x))
assert tan(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == sqrt(sqrt(5)/8 +
Rational(5, 8))/(Rational(-1, 4) + sqrt(5)/4)
def test_tan_subs():
assert tan(x).subs(tan(x), y) == y
assert tan(x).subs(x, y) == tan(y)
assert tan(x).subs(x, S.Pi/2) is zoo
assert tan(x).subs(x, S.Pi*Rational(3, 2)) is zoo
def test_tan_expansion():
assert tan(x + y).expand(trig=True) == ((tan(x) + tan(y))/(1 - tan(x)*tan(y))).expand()
assert tan(x - y).expand(trig=True) == ((tan(x) - tan(y))/(1 + tan(x)*tan(y))).expand()
assert tan(x + y + z).expand(trig=True) == (
(tan(x) + tan(y) + tan(z) - tan(x)*tan(y)*tan(z))/
(1 - tan(x)*tan(y) - tan(x)*tan(z) - tan(y)*tan(z))).expand()
assert 0 == tan(2*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 7))])*24 - 7
assert 0 == tan(3*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*55 - 37
assert 0 == tan(4*x - pi/4).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*239 - 1
_test_extrig(tan, 2, 2*tan(1)/(1 - tan(1)**2))
_test_extrig(tan, 3, (-tan(1)**3 + 3*tan(1))/(1 - 3*tan(1)**2))
def test_tan_AccumBounds():
assert tan(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
assert tan(AccumBounds(S.Pi/3, S.Pi*Rational(2, 3))) == AccumBounds(-oo, oo)
assert tan(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(tan(S.Pi/6), tan(S.Pi/3))
def test_tan_fdiff():
assert tan(x).fdiff() == tan(x)**2 + 1
raises(ArgumentIndexError, lambda: tan(x).fdiff(2))
def test_cot():
assert cot(nan) is nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(zoo) is nan
assert cot(0) is zoo
assert cot(2*pi) is zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(acsc(x)) == sqrt(1 - 1 / x**2) * x
assert cot(asec(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(pi*Rational(5, 2)) == 0
assert cot(pi*Rational(7, 2)) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(pi*Rational(-2, 3)) == 1/sqrt(3)
assert cot(pi/4) is S.One
assert cot(-pi/4) is S.NegativeOne
assert cot(pi*Rational(17, 4)) is S.One
assert cot(pi*Rational(-3, 4)) is S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(pi*Rational(7, 6)) == sqrt(3)
assert cot(pi*Rational(-5, 6)) == sqrt(3)
assert cot(pi/8) == 1 + sqrt(2)
assert cot(pi*Rational(3, 8)) == -1 + sqrt(2)
assert cot(pi*Rational(5, 8)) == 1 - sqrt(2)
assert cot(pi*Rational(7, 8)) == -1 - sqrt(2)
assert cot(pi/12) == sqrt(3) + 2
assert cot(pi*Rational(5, 12)) == -sqrt(3) + 2
assert cot(pi*Rational(7, 12)) == sqrt(3) - 2
assert cot(pi*Rational(11, 12)) == -sqrt(3) - 2
assert cot(pi/24).radsimp() == sqrt(2) + sqrt(3) + 2 + sqrt(6)
assert cot(pi*Rational(5, 24)).radsimp() == -sqrt(2) - sqrt(3) + 2 + sqrt(6)
assert cot(pi*Rational(7, 24)).radsimp() == -sqrt(2) + sqrt(3) - 2 + sqrt(6)
assert cot(pi*Rational(11, 24)).radsimp() == sqrt(2) - sqrt(3) - 2 + sqrt(6)
assert cot(pi*Rational(13, 24)).radsimp() == -sqrt(2) + sqrt(3) + 2 - sqrt(6)
assert cot(pi*Rational(17, 24)).radsimp() == sqrt(2) - sqrt(3) + 2 - sqrt(6)
assert cot(pi*Rational(19, 24)).radsimp() == sqrt(2) + sqrt(3) - 2 - sqrt(6)
assert cot(pi*Rational(23, 24)).radsimp() == -sqrt(2) - sqrt(3) - 2 - sqrt(6)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is None
assert cot(r).is_extended_real is True
assert cot(a).is_algebraic is None
assert cot(na).is_algebraic is False
assert cot(pi*Rational(10, 7)) == cot(pi*Rational(3, 7))
assert cot(pi*Rational(11, 7)) == -cot(pi*Rational(3, 7))
assert cot(pi*Rational(-11, 7)) == cot(pi*Rational(3, 7))
assert cot(pi*Rational(39, 34)) == cot(pi*Rational(5, 34))
assert cot(pi*Rational(-41, 34)) == -cot(pi*Rational(7, 34))
assert cot(x).is_finite is None
assert cot(r).is_finite is None
i = Symbol('i', imaginary=True)
assert cot(i).is_finite is True
assert cot(x).subs(x, 3*pi) is zoo
# https://github.com/sympy/sympy/issues/21177
f = cot(pi*(x + 4))/(3*x)
assert f.as_leading_term(x) == 1/(3*pi*x**2)
def test_tan_cot_sin_cos_evalf():
assert abs((tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15)) - 1).evalf()) < 1e-14
assert abs((cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15)) - 1).evalf()) < 1e-14
@XFAIL
def test_tan_cot_sin_cos_ratsimp():
assert 1 == (tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15))).ratsimp()
assert 1 == (cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15))).ratsimp()
def test_cot_series():
assert cot(x).series(x, 0, 9) == \
1/x - x/3 - x**3/45 - 2*x**5/945 - x**7/4725 + O(x**9)
# issue 6210
assert cot(x**4 + x**5).series(x, 0, 1) == \
x**(-4) - 1/x**3 + x**(-2) - 1/x + 1 + O(x)
assert cot(pi*(1-x)).series(x, 0, 3) == -1/(pi*x) + pi*x/3 + O(x**3)
assert cot(x).taylor_term(0, x) == 1/x
assert cot(x).taylor_term(2, x) is S.Zero
assert cot(x).taylor_term(3, x) == -x**3/45
def test_cot_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert cot(x).rewrite(exp) == I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
assert cot(x).rewrite(sin) == sin(2*x)/(2*(sin(x)**2))
assert cot(x).rewrite(cos) == cos(x)/cos(x - pi/2, evaluate=False)
assert cot(x).rewrite(tan) == 1/tan(x)
def check(func):
z = cot(func(x)).rewrite(exp
) - cot(x).rewrite(exp).subs(x, func(x))
assert z.rewrite(exp).expand() == 0
check(sinh)
check(cosh)
check(tanh)
check(coth)
check(sin)
check(cos)
check(tan)
assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)
assert cot(pi*Rational(4, 34)).rewrite(pow).ratsimp() == (cos(pi*Rational(4, 34))/sin(pi*Rational(4, 34))).rewrite(pow).ratsimp()
assert cot(pi*Rational(4, 17)).rewrite(pow) == (cos(pi*Rational(4, 17))/sin(pi*Rational(4, 17))).rewrite(pow)
assert cot(pi/19).rewrite(pow) == cot(pi/19)
assert cot(pi/19).rewrite(sqrt) == cot(pi/19)
assert cot(x).rewrite(sec) == sec(x - pi / 2, evaluate=False) / sec(x)
assert cot(x).rewrite(csc) == csc(x) / csc(- x + pi / 2, evaluate=False)
assert cot(sin(x)).rewrite(Pow) == cot(sin(x))
assert cot(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == (Rational(-1, 4) + sqrt(5)/4)/\
sqrt(sqrt(5)/8 + Rational(5, 8))
def test_cot_subs():
assert cot(x).subs(cot(x), y) == y
assert cot(x).subs(x, y) == cot(y)
assert cot(x).subs(x, 0) is zoo
assert cot(x).subs(x, S.Pi) is zoo
def test_cot_expansion():
assert cot(x + y).expand(trig=True).together() == (
(cot(x)*cot(y) - 1)/(cot(x) + cot(y)))
assert cot(x - y).expand(trig=True).together() == (
cot(x)*cot(-y) - 1)/(cot(x) + cot(-y))
assert cot(x + y + z).expand(trig=True).together() == (
(cot(x)*cot(y)*cot(z) - cot(x) - cot(y) - cot(z))/
(-1 + cot(x)*cot(y) + cot(x)*cot(z) + cot(y)*cot(z)))
assert cot(3*x).expand(trig=True).together() == (
(cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1))
assert cot(2*x).expand(trig=True) == cot(x)/2 - 1/(2*cot(x))
assert cot(3*x).expand(trig=True).together() == (
cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1)
assert cot(4*x - pi/4).expand(trig=True).cancel() == (
-tan(x)**4 + 4*tan(x)**3 + 6*tan(x)**2 - 4*tan(x) - 1
)/(tan(x)**4 + 4*tan(x)**3 - 6*tan(x)**2 - 4*tan(x) + 1)
_test_extrig(cot, 2, (-1 + cot(1)**2)/(2*cot(1)))
_test_extrig(cot, 3, (-3*cot(1) + cot(1)**3)/(-1 + 3*cot(1)**2))
def test_cot_AccumBounds():
assert cot(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
assert cot(AccumBounds(-S.Pi/3, S.Pi/3)) == AccumBounds(-oo, oo)
assert cot(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(cot(S.Pi/3), cot(S.Pi/6))
def test_cot_fdiff():
assert cot(x).fdiff() == -cot(x)**2 - 1
raises(ArgumentIndexError, lambda: cot(x).fdiff(2))
def test_sinc():
assert isinstance(sinc(x), sinc)
s = Symbol('s', zero=True)
assert sinc(s) is S.One
assert sinc(S.Infinity) is S.Zero
assert sinc(S.NegativeInfinity) is S.Zero
assert sinc(S.NaN) is S.NaN
assert sinc(S.ComplexInfinity) is S.NaN
n = Symbol('n', integer=True, nonzero=True)
assert sinc(n*pi) is S.Zero
assert sinc(-n*pi) is S.Zero
assert sinc(pi/2) == 2 / pi
assert sinc(-pi/2) == 2 / pi
assert sinc(pi*Rational(5, 2)) == 2 / (5*pi)
assert sinc(pi*Rational(7, 2)) == -2 / (7*pi)
assert sinc(-x) == sinc(x)
assert sinc(x).diff(x) == cos(x)/x - sin(x)/x**2
assert sinc(x).diff(x) == (sin(x)/x).diff(x)
assert sinc(x).diff(x, x) == (-sin(x) - 2*cos(x)/x + 2*sin(x)/x**2)/x
assert sinc(x).diff(x, x) == (sin(x)/x).diff(x, x)
assert limit(sinc(x).diff(x), x, 0) == 0
assert limit(sinc(x).diff(x, x), x, 0) == -S(1)/3
# https://github.com/sympy/sympy/issues/11402
#
# assert sinc(x).diff(x) == Piecewise(((x*cos(x) - sin(x)) / x**2, Ne(x, 0)), (0, True))
#
# assert sinc(x).diff(x).equals(sinc(x).rewrite(sin).diff(x))
#
# assert sinc(x).diff(x).subs(x, 0) is S.Zero
assert sinc(x).series() == 1 - x**2/6 + x**4/120 + O(x**6)
assert sinc(x).rewrite(jn) == jn(0, x)
assert sinc(x).rewrite(sin) == Piecewise((sin(x)/x, Ne(x, 0)), (1, True))
assert sinc(pi, evaluate=False).is_zero is True
assert sinc(0, evaluate=False).is_zero is False
assert sinc(n*pi, evaluate=False).is_zero is True
assert sinc(x).is_zero is None
xr = Symbol('xr', real=True, nonzero=True)
assert sinc(x).is_real is None
assert sinc(xr).is_real is True
assert sinc(I*xr).is_real is True
assert sinc(I*100).is_real is True
assert sinc(x).is_finite is None
assert sinc(xr).is_finite is True
def test_asin():
assert asin(nan) is nan
assert asin.nargs == FiniteSet(1)
assert asin(oo) == -I*oo
assert asin(-oo) == I*oo
assert asin(zoo) is zoo
# Note: asin(-x) = - asin(x)
assert asin(0) == 0
assert asin(1) == pi/2
assert asin(-1) == -pi/2
assert asin(sqrt(3)/2) == pi/3
assert asin(-sqrt(3)/2) == -pi/3
assert asin(sqrt(2)/2) == pi/4
assert asin(-sqrt(2)/2) == -pi/4
assert asin(sqrt((5 - sqrt(5))/8)) == pi/5
assert asin(-sqrt((5 - sqrt(5))/8)) == -pi/5
assert asin(S.Half) == pi/6
assert asin(Rational(-1, 2)) == -pi/6
assert asin((sqrt(2 - sqrt(2)))/2) == pi/8
assert asin(-(sqrt(2 - sqrt(2)))/2) == -pi/8
assert asin((sqrt(5) - 1)/4) == pi/10
assert asin(-(sqrt(5) - 1)/4) == -pi/10
assert asin((sqrt(3) - 1)/sqrt(2**3)) == pi/12
assert asin(-(sqrt(3) - 1)/sqrt(2**3)) == -pi/12
# check round-trip for exact values:
for d in [5, 6, 8, 10, 12]:
for n in range(-(d//2), d//2 + 1):
if gcd(n, d) == 1:
assert asin(sin(n*pi/d)) == n*pi/d
assert asin(x).diff(x) == 1/sqrt(1 - x**2)
assert asin(1/x).as_leading_term(x) == I*log(1/x)
assert asin(0.2, evaluate=False).is_real is True
assert asin(-2).is_real is False
assert asin(r).is_real is None
assert asin(-2*I) == -I*asinh(2)
assert asin(Rational(1, 7), evaluate=False).is_positive is True
assert asin(Rational(-1, 7), evaluate=False).is_positive is False
assert asin(p).is_positive is None
assert asin(sin(Rational(7, 2))) == Rational(-7, 2) + pi
assert asin(sin(Rational(-7, 4))) == Rational(7, 4) - pi
assert unchanged(asin, cos(x))
def test_asin_series():
assert asin(x).series(x, 0, 9) == \
x + x**3/6 + 3*x**5/40 + 5*x**7/112 + O(x**9)
t5 = asin(x).taylor_term(5, x)
assert t5 == 3*x**5/40
assert asin(x).taylor_term(7, x, t5, 0) == 5*x**7/112
def test_asin_rewrite():
assert asin(x).rewrite(log) == -I*log(I*x + sqrt(1 - x**2))
assert asin(x).rewrite(atan) == 2*atan(x/(1 + sqrt(1 - x**2)))
assert asin(x).rewrite(acos) == S.Pi/2 - acos(x)
assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x)
assert asin(x).rewrite(asec) == -asec(1/x) + pi/2
assert asin(x).rewrite(acsc) == acsc(1/x)
def test_asin_fdiff():
assert asin(x).fdiff() == 1/sqrt(1 - x**2)
raises(ArgumentIndexError, lambda: asin(x).fdiff(2))
def test_acos():
assert acos(nan) is nan
assert acos(zoo) is zoo
assert acos.nargs == FiniteSet(1)
assert acos(oo) == I*oo
assert acos(-oo) == -I*oo
# Note: acos(-x) = pi - acos(x)
assert acos(0) == pi/2
assert acos(S.Half) == pi/3
assert acos(Rational(-1, 2)) == pi*Rational(2, 3)
assert acos(1) == 0
assert acos(-1) == pi
assert acos(sqrt(2)/2) == pi/4
assert acos(-sqrt(2)/2) == pi*Rational(3, 4)
# check round-trip for exact values:
for d in [5, 6, 8, 10, 12]:
for num in range(d):
if gcd(num, d) == 1:
assert acos(cos(num*pi/d)) == num*pi/d
assert acos(2*I) == pi/2 - asin(2*I)
assert acos(x).diff(x) == -1/sqrt(1 - x**2)
assert acos(1/x).as_leading_term(x) == I*log(1/x)
assert acos(0.2).is_real is True
assert acos(-2).is_real is False
assert acos(r).is_real is None
assert acos(Rational(1, 7), evaluate=False).is_positive is True
assert acos(Rational(-1, 7), evaluate=False).is_positive is True
assert acos(Rational(3, 2), evaluate=False).is_positive is False
assert acos(p).is_positive is None
assert acos(2 + p).conjugate() != acos(10 + p)
assert acos(-3 + n).conjugate() != acos(-3 + n)
assert acos(Rational(1, 3)).conjugate() == acos(Rational(1, 3))
assert acos(Rational(-1, 3)).conjugate() == acos(Rational(-1, 3))
assert acos(p + n*I).conjugate() == acos(p - n*I)
assert acos(z).conjugate() != acos(conjugate(z))
def test_acos_series():
assert acos(x).series(x, 0, 8) == \
pi/2 - x - x**3/6 - 3*x**5/40 - 5*x**7/112 + O(x**8)
assert acos(x).series(x, 0, 8) == pi/2 - asin(x).series(x, 0, 8)
t5 = acos(x).taylor_term(5, x)
assert t5 == -3*x**5/40
assert acos(x).taylor_term(7, x, t5, 0) == -5*x**7/112
assert acos(x).taylor_term(0, x) == pi/2
assert acos(x).taylor_term(2, x) is S.Zero
def test_acos_rewrite():
assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2))
assert acos(x).rewrite(atan) == \
atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))
assert acos(0).rewrite(atan) == S.Pi/2
assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
assert acos(x).rewrite(asin) == S.Pi/2 - asin(x)
assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2
assert acos(x).rewrite(asec) == asec(1/x)
assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2
def test_acos_fdiff():
assert acos(x).fdiff() == -1/sqrt(1 - x**2)
raises(ArgumentIndexError, lambda: acos(x).fdiff(2))
def test_atan():
assert atan(nan) is nan
assert atan.nargs == FiniteSet(1)
assert atan(oo) == pi/2
assert atan(-oo) == -pi/2
assert atan(zoo) == AccumBounds(-pi/2, pi/2)
assert atan(0) == 0
assert atan(1) == pi/4
assert atan(sqrt(3)) == pi/3
assert atan(-(1 + sqrt(2))) == pi*Rational(-3, 8)
assert atan(sqrt(5 - 2 * sqrt(5))) == pi/5
assert atan(-sqrt(1 - 2 * sqrt(5)/ 5)) == -pi/10
assert atan(sqrt(1 + 2 * sqrt(5) / 5)) == pi*Rational(3, 10)
assert atan(-2 + sqrt(3)) == -pi/12
assert atan(2 + sqrt(3)) == pi*Rational(5, 12)
assert atan(-2 - sqrt(3)) == pi*Rational(-5, 12)
# check round-trip for exact values:
for d in [5, 6, 8, 10, 12]:
for num in range(-(d//2), d//2 + 1):
if gcd(num, d) == 1:
assert atan(tan(num*pi/d)) == num*pi/d
assert atan(oo) == pi/2
assert atan(x).diff(x) == 1/(1 + x**2)
assert atan(1/x).as_leading_term(x) == pi/2
assert atan(r).is_real is True
assert atan(-2*I) == -I*atanh(2)
assert unchanged(atan, cot(x))
assert atan(cot(Rational(1, 4))) == Rational(-1, 4) + pi/2
assert acot(Rational(1, 4)).is_rational is False
for s in (x, p, n, np, nn, nz, ep, en, enp, enn, enz):
if s.is_real or s.is_extended_real is None:
assert s.is_nonzero is atan(s).is_nonzero
assert s.is_positive is atan(s).is_positive
assert s.is_negative is atan(s).is_negative
assert s.is_nonpositive is atan(s).is_nonpositive
assert s.is_nonnegative is atan(s).is_nonnegative
else:
assert s.is_extended_nonzero is atan(s).is_nonzero
assert s.is_extended_positive is atan(s).is_positive
assert s.is_extended_negative is atan(s).is_negative
assert s.is_extended_nonpositive is atan(s).is_nonpositive
assert s.is_extended_nonnegative is atan(s).is_nonnegative
assert s.is_extended_nonzero is atan(s).is_extended_nonzero
assert s.is_extended_positive is atan(s).is_extended_positive
assert s.is_extended_negative is atan(s).is_extended_negative
assert s.is_extended_nonpositive is atan(s).is_extended_nonpositive
assert s.is_extended_nonnegative is atan(s).is_extended_nonnegative
def test_atan_rewrite():
assert atan(x).rewrite(log) == I*(log(1 - I*x)-log(1 + I*x))/2
assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x
assert atan(x).rewrite(acot) == acot(1/x)
assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x
assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(-5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:-5*I})
assert atan(5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:5*I})
def test_atan_fdiff():
assert atan(x).fdiff() == 1/(x**2 + 1)
raises(ArgumentIndexError, lambda: atan(x).fdiff(2))
def test_atan2():
assert atan2.nargs == FiniteSet(2)
assert atan2(0, 0) is S.NaN
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == pi*Rational(3, 4)
assert atan2(0, -1) == pi
assert atan2(-1, -1) == pi*Rational(-3, 4)
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
eq = atan2(r, i)
ans = -I*log((i + I*r)/sqrt(i**2 + r**2))
reps = ((r, 2), (i, I))
assert eq.subs(reps) == ans.subs(reps)
x = Symbol('x', negative=True)
y = Symbol('y', negative=True)
assert atan2(y, x) == atan(y/x) - pi
y = Symbol('y', nonnegative=True)
assert atan2(y, x) == atan(y/x) + pi
y = Symbol('y')
assert atan2(y, x) == atan2(y, x, evaluate=False)
u = Symbol("u", positive=True)
assert atan2(0, u) == 0
u = Symbol("u", negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo)== 2*pi*Heaviside(re(y), S.Half) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(0, 0) is S.NaN
ex = atan2(y, x) - arg(x + I*y)
assert ex.subs({x:2, y:3}).rewrite(arg) == 0
assert ex.subs({x:2, y:3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5)
assert ex.subs({x:2*I, y:3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I)
assert ex.subs({x:2*I, y:3*I}).rewrite(arg) == -pi + atan(Rational(2, 3)) + atan(Rational(3, 2))
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
e = atan2(i, r)
rewrite = e.rewrite(arg)
reps = {i: I, r: -2}
assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2))
assert (e - rewrite).subs(reps).equals(0)
assert atan2(0, x).rewrite(atan) == Piecewise((pi, re(x) < 0),
(0, Ne(x, 0)),
(nan, True))
assert atan2(0, r).rewrite(atan) == Piecewise((pi, r < 0), (0, Ne(r, 0)), (S.NaN, True))
assert atan2(0, i),rewrite(atan) == 0
assert atan2(0, r + i).rewrite(atan) == Piecewise((pi, r < 0), (0, True))
assert atan2(y, x).rewrite(atan) == Piecewise(
(2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)),
(pi, re(x) < 0),
(0, (re(x) > 0) | Ne(im(x), 0)),
(nan, True))
assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
assert str(atan2(1, 2).evalf(5)) == '0.46365'
raises(ArgumentIndexError, lambda: atan2(x, y).fdiff(3))
def test_issue_17461():
class A(Symbol):
is_extended_real = True
def _eval_evalf(self, prec):
return Float(5.0)
x = A('X')
y = A('Y')
assert abs(atan2(x, y).evalf() - 0.785398163397448) <= 1e-10
def test_acot():
assert acot(nan) is nan
assert acot.nargs == FiniteSet(1)
assert acot(-oo) == 0
assert acot(oo) == 0
assert acot(zoo) == 0
assert acot(1) == pi/4
assert acot(0) == pi/2
assert acot(sqrt(3)/3) == pi/3
assert acot(1/sqrt(3)) == pi/3
assert acot(-1/sqrt(3)) == -pi/3
assert acot(x).diff(x) == -1/(1 + x**2)
assert acot(1/x).as_leading_term(x) == x
assert acot(r).is_extended_real is True
assert acot(I*pi) == -I*acoth(pi)
assert acot(-2*I) == I*acoth(2)
assert acot(x).is_positive is None
assert acot(n).is_positive is False
assert acot(p).is_positive is True
assert acot(I).is_positive is False
assert acot(Rational(1, 4)).is_rational is False
assert unchanged(acot, cot(x))
assert unchanged(acot, tan(x))
assert acot(cot(Rational(1, 4))) == Rational(1, 4)
assert acot(tan(Rational(-1, 4))) == Rational(1, 4) - pi/2
def test_acot_rewrite():
assert acot(x).rewrite(log) == I*(log(1 - I/x)-log(1 + I/x))/2
assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2))
assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1))
assert acot(x).rewrite(atan) == atan(1/x)
assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2))
assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2))
assert acot(-I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:-I/5})
assert acot(I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:I/5})
def test_acot_fdiff():
assert acot(x).fdiff() == -1/(x**2 + 1)
raises(ArgumentIndexError, lambda: acot(x).fdiff(2))
def test_attributes():
assert sin(x).args == (x,)
def test_sincos_rewrite():
assert sin(pi/2 - x) == cos(x)
assert sin(pi - x) == sin(x)
assert cos(pi/2 - x) == sin(x)
assert cos(pi - x) == -cos(x)
def _check_even_rewrite(func, arg):
"""Checks that the expr has been rewritten using f(-x) -> f(x)
arg : -x
"""
return func(arg).args[0] == -arg
def _check_odd_rewrite(func, arg):
"""Checks that the expr has been rewritten using f(-x) -> -f(x)
arg : -x
"""
return func(arg).func.is_Mul
def _check_no_rewrite(func, arg):
"""Checks that the expr is not rewritten"""
return func(arg).args[0] == arg
def test_evenodd_rewrite():
a = cos(2) # negative
b = sin(1) # positive
even = [cos]
odd = [sin, tan, cot, asin, atan, acot]
with_minus = [-1, -2**1024 * E, -pi/105, -x*y, -x - y]
for func in even:
for expr in with_minus:
assert _check_even_rewrite(func, expr)
assert _check_no_rewrite(func, a*b)
assert func(
x - y) == func(y - x) # it doesn't matter which form is canonical
for func in odd:
for expr in with_minus:
assert _check_odd_rewrite(func, expr)
assert _check_no_rewrite(func, a*b)
assert func(
x - y) == -func(y - x) # it doesn't matter which form is canonical
def test_issue_4547():
assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2)
assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2)
assert tan(x).rewrite(cot) == 1/cot(x)
assert cot(x).fdiff() == -1 - cot(x)**2
def test_as_leading_term_issue_5272():
assert sin(x).as_leading_term(x) == x
assert cos(x).as_leading_term(x) == 1
assert tan(x).as_leading_term(x) == x
assert cot(x).as_leading_term(x) == 1/x
assert asin(x).as_leading_term(x) == x
assert acos(x).as_leading_term(x) == pi/2
assert atan(x).as_leading_term(x) == x
assert acot(x).as_leading_term(x) == pi/2
def test_leading_terms():
assert sin(1/x).as_leading_term(x) == AccumBounds(-1, 1)
assert sin(S.Half).as_leading_term(x) == sin(S.Half)
assert cos(1/x).as_leading_term(x) == AccumBounds(-1, 1)
assert cos(S.Half).as_leading_term(x) == cos(S.Half)
assert sec(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert csc(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert tan(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert cot(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
# https://github.com/sympy/sympy/issues/21038
f = sin(pi*(x + 4))/(3*x)
assert f.as_leading_term(x) == pi/3
def test_atan2_expansion():
assert cancel(atan2(x**2, x + 1).diff(x) - atan(x**2/(x + 1)).diff(x)) == 0
assert cancel(atan(y/x).series(y, 0, 5) - atan2(y, x).series(y, 0, 5)
+ atan2(0, x) - atan(0)) == O(y**5)
assert cancel(atan(y/x).series(x, 1, 4) - atan2(y, x).series(x, 1, 4)
+ atan2(y, 1) - atan(y)) == O((x - 1)**4, (x, 1))
assert cancel(atan((y + x)/x).series(x, 1, 3) - atan2(y + x, x).series(x, 1, 3)
+ atan2(1 + y, 1) - atan(1 + y)) == O((x - 1)**3, (x, 1))
assert Matrix([atan2(y, x)]).jacobian([y, x]) == \
Matrix([[x/(y**2 + x**2), -y/(y**2 + x**2)]])
def test_aseries():
def t(n, v, d, e):
assert abs(
n(1/v).evalf() - n(1/x).series(x, dir=d).removeO().subs(x, v)) < e
t(atan, 0.1, '+', 1e-5)
t(atan, -0.1, '-', 1e-5)
t(acot, 0.1, '+', 1e-5)
t(acot, -0.1, '-', 1e-5)
def test_issue_4420():
i = Symbol('i', integer=True)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
# unknown parity for variable
assert cos(4*i*pi) == 1
assert sin(4*i*pi) == 0
assert tan(4*i*pi) == 0
assert cot(4*i*pi) is zoo
assert cos(3*i*pi) == cos(pi*i) # +/-1
assert sin(3*i*pi) == 0
assert tan(3*i*pi) == 0
assert cot(3*i*pi) is zoo
assert cos(4.0*i*pi) == 1
assert sin(4.0*i*pi) == 0
assert tan(4.0*i*pi) == 0
assert cot(4.0*i*pi) is zoo
assert cos(3.0*i*pi) == cos(pi*i) # +/-1
assert sin(3.0*i*pi) == 0
assert tan(3.0*i*pi) == 0
assert cot(3.0*i*pi) is zoo
assert cos(4.5*i*pi) == cos(0.5*pi*i)
assert sin(4.5*i*pi) == sin(0.5*pi*i)
assert tan(4.5*i*pi) == tan(0.5*pi*i)
assert cot(4.5*i*pi) == cot(0.5*pi*i)
# parity of variable is known
assert cos(4*e*pi) == 1
assert sin(4*e*pi) == 0
assert tan(4*e*pi) == 0
assert cot(4*e*pi) is zoo
assert cos(3*e*pi) == 1
assert sin(3*e*pi) == 0
assert tan(3*e*pi) == 0
assert cot(3*e*pi) is zoo
assert cos(4.0*e*pi) == 1
assert sin(4.0*e*pi) == 0
assert tan(4.0*e*pi) == 0
assert cot(4.0*e*pi) is zoo
assert cos(3.0*e*pi) == 1
assert sin(3.0*e*pi) == 0
assert tan(3.0*e*pi) == 0
assert cot(3.0*e*pi) is zoo
assert cos(4.5*e*pi) == cos(0.5*pi*e)
assert sin(4.5*e*pi) == sin(0.5*pi*e)
assert tan(4.5*e*pi) == tan(0.5*pi*e)
assert cot(4.5*e*pi) == cot(0.5*pi*e)
assert cos(4*o*pi) == 1
assert sin(4*o*pi) == 0
assert tan(4*o*pi) == 0
assert cot(4*o*pi) is zoo
assert cos(3*o*pi) == -1
assert sin(3*o*pi) == 0
assert tan(3*o*pi) == 0
assert cot(3*o*pi) is zoo
assert cos(4.0*o*pi) == 1
assert sin(4.0*o*pi) == 0
assert tan(4.0*o*pi) == 0
assert cot(4.0*o*pi) is zoo
assert cos(3.0*o*pi) == -1
assert sin(3.0*o*pi) == 0
assert tan(3.0*o*pi) == 0
assert cot(3.0*o*pi) is zoo
assert cos(4.5*o*pi) == cos(0.5*pi*o)
assert sin(4.5*o*pi) == sin(0.5*pi*o)
assert tan(4.5*o*pi) == tan(0.5*pi*o)
assert cot(4.5*o*pi) == cot(0.5*pi*o)
# x could be imaginary
assert cos(4*x*pi) == cos(4*pi*x)
assert sin(4*x*pi) == sin(4*pi*x)
assert tan(4*x*pi) == tan(4*pi*x)
assert cot(4*x*pi) == cot(4*pi*x)
assert cos(3*x*pi) == cos(3*pi*x)
assert sin(3*x*pi) == sin(3*pi*x)
assert tan(3*x*pi) == tan(3*pi*x)
assert cot(3*x*pi) == cot(3*pi*x)
assert cos(4.0*x*pi) == cos(4.0*pi*x)
assert sin(4.0*x*pi) == sin(4.0*pi*x)
assert tan(4.0*x*pi) == tan(4.0*pi*x)
assert cot(4.0*x*pi) == cot(4.0*pi*x)
assert cos(3.0*x*pi) == cos(3.0*pi*x)
assert sin(3.0*x*pi) == sin(3.0*pi*x)
assert tan(3.0*x*pi) == tan(3.0*pi*x)
assert cot(3.0*x*pi) == cot(3.0*pi*x)
assert cos(4.5*x*pi) == cos(4.5*pi*x)
assert sin(4.5*x*pi) == sin(4.5*pi*x)
assert tan(4.5*x*pi) == tan(4.5*pi*x)
assert cot(4.5*x*pi) == cot(4.5*pi*x)
def test_inverses():
raises(AttributeError, lambda: sin(x).inverse())
raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
raises(AttributeError, lambda: csc(x).inverse())
raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def test_real_imag():
a, b = symbols('a b', real=True)
z = a + b*I
for deep in [True, False]:
assert sin(
z).as_real_imag(deep=deep) == (sin(a)*cosh(b), cos(a)*sinh(b))
assert cos(
z).as_real_imag(deep=deep) == (cos(a)*cosh(b), -sin(a)*sinh(b))
assert tan(z).as_real_imag(deep=deep) == (sin(2*a)/(cos(2*a) +
cosh(2*b)), sinh(2*b)/(cos(2*a) + cosh(2*b)))
assert cot(z).as_real_imag(deep=deep) == (-sin(2*a)/(cos(2*a) -
cosh(2*b)), sinh(2*b)/(cos(2*a) - cosh(2*b)))
assert sin(a).as_real_imag(deep=deep) == (sin(a), 0)
assert cos(a).as_real_imag(deep=deep) == (cos(a), 0)
assert tan(a).as_real_imag(deep=deep) == (tan(a), 0)
assert cot(a).as_real_imag(deep=deep) == (cot(a), 0)
@XFAIL
def test_sin_cos_with_infinity():
# Test for issue 5196
# https://github.com/sympy/sympy/issues/5196
assert sin(oo) is S.NaN
assert cos(oo) is S.NaN
@slow
def test_sincos_rewrite_sqrt():
# equivalent to testing rewrite(pow)
for p in [1, 3, 5, 17]:
for t in [1, 8]:
n = t*p
# The vertices `exp(i*pi/n)` of a regular `n`-gon can
# be expressed by means of nested square roots if and
# only if `n` is a product of Fermat primes, `p`, and
# powers of 2, `t'. The code aims to check all vertices
# not belonging to an `m`-gon for `m < n`(`gcd(i, n) == 1`).
# For large `n` this makes the test too slow, therefore
# the vertices are limited to those of index `i < 10`.
for i in range(1, min((n + 1)//2 + 1, 10)):
if 1 == gcd(i, n):
x = i*pi/n
s1 = sin(x).rewrite(sqrt)
c1 = cos(x).rewrite(sqrt)
assert not s1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
assert not c1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs(sin(x.evalf(5)) - s1.evalf(2)), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs(cos(x.evalf(5)) - c1.evalf(2)), "fails for %d*pi/%d" % (i, n)
assert cos(pi/14).rewrite(sqrt) == sqrt(cos(pi/7)/2 + S.Half)
assert cos(pi/257).rewrite(sqrt).evalf(64) == cos(pi/257).evalf(64)
assert cos(pi*Rational(-15, 2)/11, evaluate=False).rewrite(
sqrt) == -sqrt(-cos(pi*Rational(4, 11))/2 + S.Half)
assert cos(Mul(2, pi, S.Half, evaluate=False), evaluate=False).rewrite(
sqrt) == -1
e = cos(pi/3/17) # don't use pi/15 since that is caught at instantiation
a = (
-3*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17) + 17)/64 -
3*sqrt(34)*sqrt(sqrt(17) + 17)/128 - sqrt(sqrt(17) +
17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+ sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 - sqrt(-sqrt(17)
+ 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 - Rational(1, 32) +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
3*sqrt(2)*sqrt(sqrt(17) + 17)/128 + sqrt(34)*sqrt(-sqrt(17) + 17)/128
+ 13*sqrt(2)*sqrt(-sqrt(17) + 17)/128 + sqrt(17)*sqrt(-sqrt(17) +
17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+ sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 + 5*sqrt(17)/32
+ sqrt(3)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/8 -
5*sqrt(2)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 -
3*sqrt(2)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/32
+ sqrt(34)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/2 +
S.Half + sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) +
17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
6*sqrt(17) + 34)/32 + Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
6*sqrt(17) + 34)/32 + sqrt(34)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
Rational(15, 32))/32)/2)
assert e.rewrite(sqrt) == a
assert e.n() == a.n()
# coverage of fermatCoords: multiplicity > 1; the following could be
# different but that portion of the code should be tested in some way
assert cos(pi/9/17).rewrite(sqrt) == \
sin(pi/9)*sin(pi*Rational(2, 17)) + cos(pi/9)*cos(pi*Rational(2, 17))
@slow
def test_tancot_rewrite_sqrt():
# equivalent to testing rewrite(pow)
for p in [1, 3, 5, 17]:
for t in [1, 8]:
n = t*p
for i in range(1, min((n + 1)//2 + 1, 10)):
if 1 == gcd(i, n):
x = i*pi/n
if 2*i != n and 3*i != 2*n:
t1 = tan(x).rewrite(sqrt)
assert not t1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs( tan(x.evalf(7)) - t1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
if i != 0 and i != n:
c1 = cot(x).rewrite(sqrt)
assert not c1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs( cot(x.evalf(7)) - c1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
def test_sec():
x = symbols('x', real=True)
z = symbols('z')
assert sec.nargs == FiniteSet(1)
assert sec(zoo) is nan
assert sec(0) == 1
assert sec(pi) == -1
assert sec(pi/2) is zoo
assert sec(-pi/2) is zoo
assert sec(pi/6) == 2*sqrt(3)/3
assert sec(pi/3) == 2
assert sec(pi*Rational(5, 2)) is zoo
assert sec(pi*Rational(9, 7)) == -sec(pi*Rational(2, 7))
assert sec(pi*Rational(3, 4)) == -sqrt(2) # issue 8421
assert sec(I) == 1/cosh(1)
assert sec(x*I) == 1/cosh(x)
assert sec(-x) == sec(x)
assert sec(asec(x)) == x
assert sec(z).conjugate() == sec(conjugate(z))
assert (sec(z).as_real_imag() ==
(cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2),
sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2)))
assert sec(x).expand(trig=True) == 1/cos(x)
assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1)
assert sec(x).is_extended_real == True
assert sec(z).is_real == None
assert sec(a).is_algebraic is None
assert sec(na).is_algebraic is False
assert sec(x).as_leading_term() == sec(x)
assert sec(0, evaluate=False).is_finite == True
assert sec(x).is_finite == None
assert sec(pi/2, evaluate=False).is_finite == False
assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6)
# https://github.com/sympy/sympy/issues/7166
assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6)
# https://github.com/sympy/sympy/issues/7167
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - pi*Rational(3, 2)) + (x - pi*Rational(3, 2))**Rational(3, 2)/12 +
(x - pi*Rational(3, 2))**Rational(7, 2)/160 + O((x - pi*Rational(3, 2))**4, (x, pi*Rational(3, 2))))
assert sec(x).diff(x) == tan(x)*sec(x)
# Taylor Term checks
assert sec(z).taylor_term(4, z) == 5*z**4/24
assert sec(z).taylor_term(6, z) == 61*z**6/720
assert sec(z).taylor_term(5, z) == 0
def test_sec_rewrite():
assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2)
assert sec(x).rewrite(cos) == 1/cos(x)
assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1)
assert sec(x).rewrite(pow) == sec(x)
assert sec(x).rewrite(sqrt) == sec(x)
assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1)
assert sec(x).rewrite(sin) == 1 / sin(x + pi / 2, evaluate=False)
assert sec(x).rewrite(tan) == (tan(x / 2)**2 + 1) / (-tan(x / 2)**2 + 1)
assert sec(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)
def test_sec_fdiff():
assert sec(x).fdiff() == tan(x)*sec(x)
raises(ArgumentIndexError, lambda: sec(x).fdiff(2))
def test_csc():
x = symbols('x', real=True)
z = symbols('z')
# https://github.com/sympy/sympy/issues/6707
cosecant = csc('x')
alternate = 1/sin('x')
assert cosecant.equals(alternate) == True
assert alternate.equals(cosecant) == True
assert csc.nargs == FiniteSet(1)
assert csc(0) is zoo
assert csc(pi) is zoo
assert csc(zoo) is nan
assert csc(pi/2) == 1
assert csc(-pi/2) == -1
assert csc(pi/6) == 2
assert csc(pi/3) == 2*sqrt(3)/3
assert csc(pi*Rational(5, 2)) == 1
assert csc(pi*Rational(9, 7)) == -csc(pi*Rational(2, 7))
assert csc(pi*Rational(3, 4)) == sqrt(2) # issue 8421
assert csc(I) == -I/sinh(1)
assert csc(x*I) == -I/sinh(x)
assert csc(-x) == -csc(x)
assert csc(acsc(x)) == x
assert csc(z).conjugate() == csc(conjugate(z))
assert (csc(z).as_real_imag() ==
(sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2),
-cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2)))
assert csc(x).expand(trig=True) == 1/sin(x)
assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x))
assert csc(x).is_extended_real == True
assert csc(z).is_real == None
assert csc(a).is_algebraic is None
assert csc(na).is_algebraic is False
assert csc(x).as_leading_term() == csc(x)
assert csc(0, evaluate=False).is_finite == False
assert csc(x).is_finite == None
assert csc(pi/2, evaluate=False).is_finite == True
assert series(csc(x), x, x0=pi/2, n=6) == \
1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2))
assert series(csc(x), x, x0=0, n=6) == \
1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6)
assert csc(x).diff(x) == -cot(x)*csc(x)
assert csc(x).taylor_term(2, x) == 0
assert csc(x).taylor_term(3, x) == 7*x**3/360
assert csc(x).taylor_term(5, x) == 31*x**5/15120
raises(ArgumentIndexError, lambda: csc(x).fdiff(2))
def test_asec():
z = Symbol('z', zero=True)
assert asec(z) is zoo
assert asec(nan) is nan
assert asec(1) == 0
assert asec(-1) == pi
assert asec(oo) == pi/2
assert asec(-oo) == pi/2
assert asec(zoo) == pi/2
assert asec(sec(pi*Rational(13, 4))) == pi*Rational(3, 4)
assert asec(1 + sqrt(5)) == pi*Rational(2, 5)
assert asec(2/sqrt(3)) == pi/6
assert asec(sqrt(4 - 2*sqrt(2))) == pi/8
assert asec(-sqrt(4 + 2*sqrt(2))) == pi*Rational(5, 8)
assert asec(sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(3, 10)
assert asec(-sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(7, 10)
assert asec(sqrt(2) - sqrt(6)) == pi*Rational(11, 12)
assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
assert asec(x).as_leading_term(x) == I*log(x)
assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2
assert asec(x).rewrite(asin) == -asin(1/x) + pi/2
assert asec(x).rewrite(acos) == acos(1/x)
assert asec(x).rewrite(atan) == (2*atan(x + sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x
assert asec(x).rewrite(acot) == (2*acot(x - sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x
assert asec(x).rewrite(acsc) == -acsc(x) + pi/2
raises(ArgumentIndexError, lambda: asec(x).fdiff(2))
def test_asec_is_real():
assert asec(S.Half).is_real is False
n = Symbol('n', positive=True, integer=True)
assert asec(n).is_extended_real is True
assert asec(x).is_real is None
assert asec(r).is_real is None
t = Symbol('t', real=False, finite=True)
assert asec(t).is_real is False
def test_acsc():
assert acsc(nan) is nan
assert acsc(1) == pi/2
assert acsc(-1) == -pi/2
assert acsc(oo) == 0
assert acsc(-oo) == 0
assert acsc(zoo) == 0
assert acsc(0) is zoo
assert acsc(csc(3)) == -3 + pi
assert acsc(csc(4)) == -4 + pi
assert acsc(csc(6)) == 6 - 2*pi
assert unchanged(acsc, csc(x))
assert unchanged(acsc, sec(x))
assert acsc(2/sqrt(3)) == pi/3
assert acsc(csc(pi*Rational(13, 4))) == -pi/4
assert acsc(sqrt(2 + 2*sqrt(5)/5)) == pi/5
assert acsc(-sqrt(2 + 2*sqrt(5)/5)) == -pi/5
assert acsc(-2) == -pi/6
assert acsc(-sqrt(4 + 2*sqrt(2))) == -pi/8
assert acsc(sqrt(4 - 2*sqrt(2))) == pi*Rational(3, 8)
assert acsc(1 + sqrt(5)) == pi/10
assert acsc(sqrt(2) - sqrt(6)) == pi*Rational(-5, 12)
assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2))
assert acsc(x).as_leading_term(x) == I*log(x)
assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x)
assert acsc(x).rewrite(asin) == asin(1/x)
assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2
assert acsc(x).rewrite(atan) == (-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(asec) == -asec(x) + pi/2
raises(ArgumentIndexError, lambda: acsc(x).fdiff(2))
def test_csc_rewrite():
assert csc(x).rewrite(pow) == csc(x)
assert csc(x).rewrite(sqrt) == csc(x)
assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x))
assert csc(x).rewrite(sin) == 1/sin(x)
assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2))
assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2))
assert csc(x).rewrite(cos) == 1/cos(x - pi/2, evaluate=False)
assert csc(x).rewrite(sec) == sec(-x + pi/2, evaluate=False)
# issue 17349
assert csc(1 - exp(-besselj(I, I))).rewrite(cos) == \
-1/cos(-pi/2 - 1 + cos(I*besselj(I, I)) +
I*cos(-pi/2 + I*besselj(I, I), evaluate=False), evaluate=False)
def test_inverses_nseries():
assert asin(x + 2)._eval_nseries(x, 4, None, I) == -asin(2) + pi + sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 + \
sqrt(3)*I*x**3/18 + O(x**4)
assert asin(x + 2)._eval_nseries(x, 4, None, -I) == asin(2) - sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert asin(x - 2)._eval_nseries(x, 4, None, I) == -asin(2) - sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert asin(x - 2)._eval_nseries(x, 4, None, -I) == asin(2) - pi + sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 + \
sqrt(3)*I*x**3/18 + O(x**4)
assert asin(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -asin(2) + pi - sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3)
assert asin(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == asin(2) + sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3)
assert asin(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == -asin(2) + sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3)
assert asin(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == asin(2) - pi - sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3)
assert asin(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -asin(2) + pi - sqrt(3)*x**2/3 + O(x**3)
assert asin(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == -asin(2) + pi - sqrt(3)*x**2/3 + O(x**3)
assert asin(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == -asin(2) + sqrt(3)*x**2/3 + O(x**3)
assert asin(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == -asin(2) + sqrt(3)*x**2/3 + O(x**3)
assert asin(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) - \
sqrt(2)*(-x)**(S(3)/2)/12 - 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
assert asin(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) + \
sqrt(2)*x**(S(3)/2)/12 + 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
assert asin(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) + \
sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
assert asin(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - \
sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
assert acos(x + 2)._eval_nseries(x, 4, None, I) == -acos(2) - sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert acos(x + 2)._eval_nseries(x, 4, None, -I) == acos(2) + sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert acos(x - 2)._eval_nseries(x, 4, None, I) == acos(-2) + sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert acos(x - 2)._eval_nseries(x, 4, None, -I) == -acos(-2) + 2*pi - sqrt(3)*I*x/3 - \
sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
# assert acos(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -acos(2) + sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3)
# assert acos(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == acos(2) - sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3)
# assert acos(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == acos(-2) - sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3)
# assert acos(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == -acos(-2) + 2*pi + sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3)
# assert acos(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -acos(2) + sqrt(3)*x**2/3 + O(x**3)
# assert acos(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == -acos(2) + sqrt(3)*x**2/3 + O(x**3)
# assert acos(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == acos(-2) - sqrt(3)*x**2/3 + O(x**3)
# assert acos(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == acos(-2) - sqrt(3)*x**2/3 + O(x**3)
# assert acos(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) + sqrt(2)*(-x)**(S(3)/2)/12 + 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
# assert acos(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) - sqrt(2)*x**(S(3)/2)/12 - 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
# assert acos(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) - sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
# assert acos(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
assert atan(x + 2*I)._eval_nseries(x, 4, None, 1) == I*atanh(2) - x/3 - 2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(x + 2*I)._eval_nseries(x, 4, None, -1) == I*atanh(2) - pi - x/3 - 2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(x - 2*I)._eval_nseries(x, 4, None, 1) == -I*atanh(2) + pi - x/3 + 2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(x - 2*I)._eval_nseries(x, 4, None, -1) == -I*atanh(2) - x/3 + 2*I*x**2/9 + 13*x**3/81 + O(x**4)
# assert atan(x**2 + 2*I)._eval_nseries(x, 3, None, 1) == I*atanh(2) - x**2/3 + O(x**3)
# assert atan(x**2 + 2*I)._eval_nseries(x, 3, None, -1) == I*atanh(2) - x**2/3 + O(x**3)
# assert atan(x**2 - 2*I)._eval_nseries(x, 3, None, 1) == -I*atanh(2) + pi - x**2/3 + O(x**3)
# assert atan(x**2 - 2*I)._eval_nseries(x, 3, None, -1) == -I*atanh(2) + pi - x**2/3 + O(x**3)
assert atan(1/x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2)
assert atan(1/x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2)
assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, 1) == -I*acoth(S(1)/2) + pi - 4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, -1) == -I*acoth(S(1)/2) - 4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, 1) == I*acoth(S(1)/2) - 4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, -1) == I*acoth(S(1)/2) - pi - 4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4)
# assert acot(x**2 + S(1)/2*I)._eval_nseries(x, 3, None, 1) == -I*acoth(S(1)/2) + pi - 4*x**2/3 + O(x**3)
# assert acot(x**2 + S(1)/2*I)._eval_nseries(x, 3, None, -1) == -I*acoth(S(1)/2) + pi - 4*x**2/3 + O(x**3)
# assert acot(x**2 - S(1)/2*I)._eval_nseries(x, 3, None, 1) == I*acoth(S(1)/2) - 4*x**2/3 + O(x**3)
# assert acot(x**2 - S(1)/2*I)._eval_nseries(x, 3, None, -1) == I*acoth(S(1)/2) - 4*x**2/3 + O(x**3)
# assert acot(x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2)
# assert acot(x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2)
assert asec(x + S(1)/2)._eval_nseries(x, 4, None, I) == asec(S(1)/2) - 4*sqrt(3)*I*x/3 + \
8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert asec(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -asec(S(1)/2) + 4*sqrt(3)*I*x/3 - \
8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert asec(x - S(1)/2)._eval_nseries(x, 4, None, I) == -asec(-S(1)/2) + 2*pi + 4*sqrt(3)*I*x/3 + \
8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert asec(x - S(1)/2)._eval_nseries(x, 4, None, -I) == asec(-S(1)/2) - 4*sqrt(3)*I*x/3 - \
8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
# assert asec(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == asec(S(1)/2) + 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert asec(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == -asec(S(1)/2) - 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert asec(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == -asec(-S(1)/2) + 2*pi - 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert asec(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == asec(-S(1)/2) + 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert asec(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == asec(S(1)/2) + 4*sqrt(3)*x**2/3 + O(x**3)
# assert asec(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == asec(S(1)/2) + 4*sqrt(3)*x**2/3 + O(x**3)
# assert asec(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == -asec(-S(1)/2) + 2*pi - 4*sqrt(3)*x**2/3 + O(x**3)
# assert asec(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == -asec(-S(1)/2) + 2*pi - 4*sqrt(3)*x**2/3 + O(x**3)
# assert asec(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - 5*sqrt(2)*x**(S(3)/2)/12 + 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
# assert asec(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + 5*sqrt(2)*(-x)**(S(3)/2)/12 - 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
# assert asec(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3)
# assert asec(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) + sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3)
assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) + 4*sqrt(3)*I*x/3 - \
8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + pi - 4*sqrt(3)*I*x/3 + \
8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) - pi - 4*sqrt(3)*I*x/3 - \
8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + 4*sqrt(3)*I*x/3 + \
8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
# assert acsc(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert acsc(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == -acsc(S(1)/2) + pi + 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert acsc(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - pi + 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert acsc(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == -acsc(S(1)/2) - 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3)
# assert acsc(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - 4*sqrt(3)*x**2/3 + O(x**3)
# assert acsc(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == acsc(S(1)/2) - 4*sqrt(3)*x**2/3 + O(x**3)
# assert acsc(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - pi + 4*sqrt(3)*x**2/3 + O(x**3)
# assert acsc(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == acsc(S(1)/2) - pi + 4*sqrt(3)*x**2/3 + O(x**3)
# assert acsc(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + 5*sqrt(2)*x**(S(3)/2)/12 - 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
# assert acsc(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - 5*sqrt(2)*(-x)**(S(3)/2)/12 + 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
# assert acsc(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3)
# assert acsc(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) - sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3)
def test_issue_8653():
n = Symbol('n', integer=True)
assert sin(n).is_irrational is None
assert cos(n).is_irrational is None
assert tan(n).is_irrational is None
def test_issue_9157():
n = Symbol('n', integer=True, positive=True)
assert atan(n - 1).is_nonnegative is True
def test_trig_period():
x, y = symbols('x, y')
assert sin(x).period() == 2*pi
assert cos(x).period() == 2*pi
assert tan(x).period() == pi
assert cot(x).period() == pi
assert sec(x).period() == 2*pi
assert csc(x).period() == 2*pi
assert sin(2*x).period() == pi
assert cot(4*x - 6).period() == pi/4
assert cos((-3)*x).period() == pi*Rational(2, 3)
assert cos(x*y).period(x) == 2*pi/abs(y)
assert sin(3*x*y + 2*pi).period(y) == 2*pi/abs(3*x)
assert tan(3*x).period(y) is S.Zero
raises(NotImplementedError, lambda: sin(x**2).period(x))
def test_issue_7171():
assert sin(x).rewrite(sqrt) == sin(x)
assert sin(x).rewrite(pow) == sin(x)
def test_issue_11864():
w, k = symbols('w, k', real=True)
F = Piecewise((1, Eq(2*pi*k, 0)), (sin(pi*k)/(pi*k), True))
soln = Piecewise((1, Eq(2*pi*k, 0)), (sinc(pi*k), True))
assert F.rewrite(sinc) == soln
def test_real_assumptions():
z = Symbol('z', real=False, finite=True)
assert sin(z).is_real is None
assert cos(z).is_real is None
assert tan(z).is_real is False
assert sec(z).is_real is None
assert csc(z).is_real is None
assert cot(z).is_real is False
assert asin(p).is_real is None
assert asin(n).is_real is None
assert asec(p).is_real is None
assert asec(n).is_real is None
assert acos(p).is_real is None
assert acos(n).is_real is None
assert acsc(p).is_real is None
assert acsc(n).is_real is None
assert atan(p).is_positive is True
assert atan(n).is_negative is True
assert acot(p).is_positive is True
assert acot(n).is_negative is True
def test_issue_14320():
assert asin(sin(2)) == -2 + pi and (-pi/2 <= -2 + pi <= pi/2) and sin(2) == sin(-2 + pi)
assert asin(cos(2)) == -2 + pi/2 and (-pi/2 <= -2 + pi/2 <= pi/2) and cos(2) == sin(-2 + pi/2)
assert acos(sin(2)) == -pi/2 + 2 and (0 <= -pi/2 + 2 <= pi) and sin(2) == cos(-pi/2 + 2)
assert acos(cos(20)) == -6*pi + 20 and (0 <= -6*pi + 20 <= pi) and cos(20) == cos(-6*pi + 20)
assert acos(cos(30)) == -30 + 10*pi and (0 <= -30 + 10*pi <= pi) and cos(30) == cos(-30 + 10*pi)
assert atan(tan(17)) == -5*pi + 17 and (-pi/2 < -5*pi + 17 < pi/2) and tan(17) == tan(-5*pi + 17)
assert atan(tan(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 < pi/2) and tan(15) == tan(-5*pi + 15)
assert atan(cot(12)) == -12 + pi*Rational(7, 2) and (-pi/2 < -12 + pi*Rational(7, 2) < pi/2) and cot(12) == tan(-12 + pi*Rational(7, 2))
assert acot(cot(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 <= pi/2) and cot(15) == cot(-5*pi + 15)
assert acot(tan(19)) == -19 + pi*Rational(13, 2) and (-pi/2 < -19 + pi*Rational(13, 2) <= pi/2) and tan(19) == cot(-19 + pi*Rational(13, 2))
assert asec(sec(11)) == -11 + 4*pi and (0 <= -11 + 4*pi <= pi) and cos(11) == cos(-11 + 4*pi)
assert asec(csc(13)) == -13 + pi*Rational(9, 2) and (0 <= -13 + pi*Rational(9, 2) <= pi) and sin(13) == cos(-13 + pi*Rational(9, 2))
assert acsc(csc(14)) == -4*pi + 14 and (-pi/2 <= -4*pi + 14 <= pi/2) and sin(14) == sin(-4*pi + 14)
assert acsc(sec(10)) == pi*Rational(-7, 2) + 10 and (-pi/2 <= pi*Rational(-7, 2) + 10 <= pi/2) and cos(10) == sin(pi*Rational(-7, 2) + 10)
def test_issue_14543():
assert sec(2*pi + 11) == sec(11)
assert sec(2*pi - 11) == sec(11)
assert sec(pi + 11) == -sec(11)
assert sec(pi - 11) == -sec(11)
assert csc(2*pi + 17) == csc(17)
assert csc(2*pi - 17) == -csc(17)
assert csc(pi + 17) == -csc(17)
assert csc(pi - 17) == csc(17)
x = Symbol('x')
assert csc(pi/2 + x) == sec(x)
assert csc(pi/2 - x) == sec(x)
assert csc(pi*Rational(3, 2) + x) == -sec(x)
assert csc(pi*Rational(3, 2) - x) == -sec(x)
assert sec(pi/2 - x) == csc(x)
assert sec(pi/2 + x) == -csc(x)
assert sec(pi*Rational(3, 2) + x) == csc(x)
assert sec(pi*Rational(3, 2) - x) == -csc(x)
def test_as_real_imag():
# This is for https://github.com/sympy/sympy/issues/17142
# If it start failing again in irrelevant builds or in the master
# please open up the issue again.
expr = atan(I/(I + I*tan(1)))
assert expr.as_real_imag() == (expr, 0)
def test_issue_18746():
e3 = cos(S.Pi*(x/4 + 1/4))
assert e3.period() == 8
|
1e4221c675a5ac3af8364a1f1e7e54799f69055bdd00abbcd1fef936eb416540 | from sympy.core.numbers import (I, nan, oo, pi)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (adjoint, conjugate, sign, transpose)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.simplify.simplify import signsimp
from sympy.testing.pytest import raises
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
x, y = symbols('x y')
i = symbols('t', nonzero=True)
j = symbols('j', positive=True)
k = symbols('k', negative=True)
def test_DiracDelta():
assert DiracDelta(1) == 0
assert DiracDelta(5.1) == 0
assert DiracDelta(-pi) == 0
assert DiracDelta(5, 7) == 0
assert DiracDelta(i) == 0
assert DiracDelta(j) == 0
assert DiracDelta(k) == 0
assert DiracDelta(nan) is nan
assert DiracDelta(0).func is DiracDelta
assert DiracDelta(x).func is DiracDelta
# FIXME: this is generally undefined @ x=0
# But then limit(Delta(c)*Heaviside(x),x,-oo)
# need's to be implemented.
# assert 0*DiracDelta(x) == 0
assert adjoint(DiracDelta(x)) == DiracDelta(x)
assert adjoint(DiracDelta(x - y)) == DiracDelta(x - y)
assert conjugate(DiracDelta(x)) == DiracDelta(x)
assert conjugate(DiracDelta(x - y)) == DiracDelta(x - y)
assert transpose(DiracDelta(x)) == DiracDelta(x)
assert transpose(DiracDelta(x - y)) == DiracDelta(x - y)
assert DiracDelta(x).diff(x) == DiracDelta(x, 1)
assert DiracDelta(x, 1).diff(x) == DiracDelta(x, 2)
assert DiracDelta(x).is_simple(x) is True
assert DiracDelta(3*x).is_simple(x) is True
assert DiracDelta(x**2).is_simple(x) is False
assert DiracDelta(sqrt(x)).is_simple(x) is False
assert DiracDelta(x).is_simple(y) is False
assert DiracDelta(x*y).expand(diracdelta=True, wrt=x) == DiracDelta(x)/abs(y)
assert DiracDelta(x*y).expand(diracdelta=True, wrt=y) == DiracDelta(y)/abs(x)
assert DiracDelta(x**2*y).expand(diracdelta=True, wrt=x) == DiracDelta(x**2*y)
assert DiracDelta(y).expand(diracdelta=True, wrt=x) == DiracDelta(y)
assert DiracDelta((x - 1)*(x - 2)*(x - 3)).expand(diracdelta=True, wrt=x) == (
DiracDelta(x - 3)/2 + DiracDelta(x - 2) + DiracDelta(x - 1)/2)
assert DiracDelta(2*x) != DiracDelta(x) # scaling property
assert DiracDelta(x) == DiracDelta(-x) # even function
assert DiracDelta(-x, 2) == DiracDelta(x, 2)
assert DiracDelta(-x, 1) == -DiracDelta(x, 1) # odd deriv is odd
assert DiracDelta(-oo*x) == DiracDelta(oo*x)
assert DiracDelta(x - y) != DiracDelta(y - x)
assert signsimp(DiracDelta(x - y) - DiracDelta(y - x)) == 0
assert DiracDelta(x*y).expand(diracdelta=True, wrt=x) == DiracDelta(x)/abs(y)
assert DiracDelta(x*y).expand(diracdelta=True, wrt=y) == DiracDelta(y)/abs(x)
assert DiracDelta(x**2*y).expand(diracdelta=True, wrt=x) == DiracDelta(x**2*y)
assert DiracDelta(y).expand(diracdelta=True, wrt=x) == DiracDelta(y)
assert DiracDelta((x - 1)*(x - 2)*(x - 3)).expand(diracdelta=True) == (
DiracDelta(x - 3)/2 + DiracDelta(x - 2) + DiracDelta(x - 1)/2)
raises(ArgumentIndexError, lambda: DiracDelta(x).fdiff(2))
raises(ValueError, lambda: DiracDelta(x, -1))
raises(ValueError, lambda: DiracDelta(I))
raises(ValueError, lambda: DiracDelta(2 + 3*I))
def test_heaviside():
assert Heaviside(-5) == 0
assert Heaviside(1) == 1
assert Heaviside(0) == S.Half
assert Heaviside(0, x) == x
assert unchanged(Heaviside,x, nan)
assert Heaviside(0, nan) == nan
h0 = Heaviside(x, 0)
h12 = Heaviside(x, S.Half)
h1 = Heaviside(x, 1)
assert h0.args == h0.pargs == (x, 0)
assert h1.args == h1.pargs == (x, 1)
assert h12.args == (x, S.Half)
assert h12.pargs == (x,) # default 1/2 suppressed
assert adjoint(Heaviside(x)) == Heaviside(x)
assert adjoint(Heaviside(x - y)) == Heaviside(x - y)
assert conjugate(Heaviside(x)) == Heaviside(x)
assert conjugate(Heaviside(x - y)) == Heaviside(x - y)
assert transpose(Heaviside(x)) == Heaviside(x)
assert transpose(Heaviside(x - y)) == Heaviside(x - y)
assert Heaviside(x).diff(x) == DiracDelta(x)
assert Heaviside(x + I).is_Function is True
assert Heaviside(I*x).is_Function is True
raises(ArgumentIndexError, lambda: Heaviside(x).fdiff(2))
raises(ValueError, lambda: Heaviside(I))
raises(ValueError, lambda: Heaviside(2 + 3*I))
def test_rewrite():
x, y = Symbol('x', real=True), Symbol('y')
assert Heaviside(x).rewrite(Piecewise) == (
Piecewise((0, x < 0), (Heaviside(0), Eq(x, 0)), (1, x > 0)))
assert Heaviside(y).rewrite(Piecewise) == (
Piecewise((0, y < 0), (Heaviside(0), Eq(y, 0)), (1, y > 0)))
assert Heaviside(x, y).rewrite(Piecewise) == (
Piecewise((0, x < 0), (y, Eq(x, 0)), (1, x > 0)))
assert Heaviside(x, 0).rewrite(Piecewise) == (
Piecewise((0, x <= 0), (1, x > 0)))
assert Heaviside(x, 1).rewrite(Piecewise) == (
Piecewise((0, x < 0), (1, x >= 0)))
assert Heaviside(x, nan).rewrite(Piecewise) == (
Piecewise((0, x < 0), (nan, Eq(x, 0)), (1, x > 0)))
assert Heaviside(x).rewrite(sign) == \
Heaviside(x, H0=Heaviside(0)).rewrite(sign) == \
Piecewise(
(sign(x)/2 + S(1)/2, Eq(Heaviside(0), S(1)/2)),
(Piecewise(
(sign(x)/2 + S(1)/2, Ne(x, 0)), (Heaviside(0), True)), True)
)
assert Heaviside(y).rewrite(sign) == Heaviside(y)
assert Heaviside(x, S.Half).rewrite(sign) == (sign(x)+1)/2
assert Heaviside(x, y).rewrite(sign) == \
Piecewise(
(sign(x)/2 + S(1)/2, Eq(y, S(1)/2)),
(Piecewise(
(sign(x)/2 + S(1)/2, Ne(x, 0)), (y, True)), True)
)
assert DiracDelta(y).rewrite(Piecewise) == Piecewise((DiracDelta(0), Eq(y, 0)), (0, True))
assert DiracDelta(y, 1).rewrite(Piecewise) == DiracDelta(y, 1)
assert DiracDelta(x - 5).rewrite(Piecewise) == (
Piecewise((DiracDelta(0), Eq(x - 5, 0)), (0, True)))
assert (x*DiracDelta(x - 10)).rewrite(SingularityFunction) == x*SingularityFunction(x, 10, -1)
assert 5*x*y*DiracDelta(y, 1).rewrite(SingularityFunction) == 5*x*y*SingularityFunction(y, 0, -2)
assert DiracDelta(0).rewrite(SingularityFunction) == SingularityFunction(0, 0, -1)
assert DiracDelta(0, 1).rewrite(SingularityFunction) == SingularityFunction(0, 0, -2)
assert Heaviside(x).rewrite(SingularityFunction) == SingularityFunction(x, 0, 0)
assert 5*x*y*Heaviside(y + 1).rewrite(SingularityFunction) == 5*x*y*SingularityFunction(y, -1, 0)
assert ((x - 3)**3*Heaviside(x - 3)).rewrite(SingularityFunction) == (x - 3)**3*SingularityFunction(x, 3, 0)
assert Heaviside(0).rewrite(SingularityFunction) == S.Half
|
87b8d5043fd305381272adc27e054e0c7b9145fe144c0aed0b094250d07b4d53 | from itertools import product
from sympy.concrete.summations import Sum
from sympy.core.function import (diff, expand_func)
from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (conjugate, polar_lift)
from sympy.functions.elementary.exponential import (exp, exp_polar, log)
from sympy.functions.elementary.hyperbolic import (cosh, sinh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely, hankel1, hankel2, hn1, hn2, jn, jn_zeros, yn)
from sympy.functions.special.gamma_functions import (gamma, uppergamma)
from sympy.functions.special.hyper import hyper
from sympy.integrals.integrals import Integral
from sympy.series.order import O
from sympy.series.series import series
from sympy.functions.special.bessel import (airyai, airybi,
airyaiprime, airybiprime, marcumq)
from sympy.core.random import (random_complex_number as randcplx,
verify_numerically as tn,
test_derivative_numerically as td,
_randint)
from sympy.simplify import besselsimp
from sympy.testing.pytest import raises, slow
from sympy.abc import z, n, k, x
randint = _randint()
def test_bessel_rand():
for f in [besselj, bessely, besseli, besselk, hankel1, hankel2]:
assert td(f(randcplx(), z), z)
for f in [jn, yn, hn1, hn2]:
assert td(f(randint(-10, 10), z), z)
def test_bessel_twoinputs():
for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, jn, yn]:
raises(TypeError, lambda: f(1))
raises(TypeError, lambda: f(1, 2, 3))
def test_besselj_leading_term():
assert besselj(0, x).as_leading_term(x) == 1
assert besselj(1, sin(x)).as_leading_term(x) == x/2
assert besselj(1, 2*sqrt(x)).as_leading_term(x) == sqrt(x)
# https://github.com/sympy/sympy/issues/21701
assert (besselj(z, x)/x**z).as_leading_term(x) == 1/(2**z*gamma(z + 1))
def test_bessely_leading_term():
assert bessely(0, x).as_leading_term(x) == (2*log(x) - 2*log(2))/pi
assert bessely(1, sin(x)).as_leading_term(x) == (x*log(x) - x*log(2))/pi
assert bessely(1, 2*sqrt(x)).as_leading_term(x) == sqrt(x)*log(x)/pi
def test_besselj_series():
assert besselj(0, x).series(x) == 1 - x**2/4 + x**4/64 + O(x**6)
assert besselj(0, x**(1.1)).series(x) == 1 + x**4.4/64 - x**2.2/4 + O(x**6)
assert besselj(0, x**2 + x).series(x) == 1 - x**2/4 - x**3/2\
- 15*x**4/64 + x**5/16 + O(x**6)
assert besselj(0, sqrt(x) + x).series(x, n=4) == 1 - x/4 - 15*x**2/64\
+ 215*x**3/2304 - x**Rational(3, 2)/2 + x**Rational(5, 2)/16\
+ 23*x**Rational(7, 2)/384 + O(x**4)
assert besselj(0, x/(1 - x)).series(x) == 1 - x**2/4 - x**3/2 - 47*x**4/64\
- 15*x**5/16 + O(x**6)
assert besselj(0, log(1 + x)).series(x) == 1 - x**2/4 + x**3/4\
- 41*x**4/192 + 17*x**5/96 + O(x**6)
assert besselj(1, sin(x)).series(x) == x/2 - 7*x**3/48 + 73*x**5/1920 + O(x**6)
assert besselj(1, 2*sqrt(x)).series(x) == sqrt(x) - x**Rational(3, 2)/2\
+ x**Rational(5, 2)/12 - x**Rational(7, 2)/144 + x**Rational(9, 2)/2880\
- x**Rational(11, 2)/86400 + O(x**6)
assert besselj(-2, sin(x)).series(x, n=4) == besselj(2, sin(x)).series(x, n=4)
def test_bessely_series():
const = 2*S.EulerGamma/pi - 2*log(2)/pi + 2*log(x)/pi
assert bessely(0, x).series(x, n=4) == const + x**2*(-log(x)/(2*pi)\
+ (2 - 2*S.EulerGamma)/(4*pi) + log(2)/(2*pi)) + O(x**4*log(x))
assert bessely(0, x**(1.1)).series(x, n=4) == 2*S.EulerGamma/pi\
- 2*log(2)/pi + 2.2*log(x)/pi + x**2.2*(-0.55*log(x)/pi\
+ (2 - 2*S.EulerGamma)/(4*pi) + log(2)/(2*pi)) + O(x**4*log(x))
assert bessely(0, x**2 + x).series(x, n=4) == \
const - (2 - 2*S.EulerGamma)*(-x**3/(2*pi) - x**2/(4*pi)) + 2*x/pi\
+ x**2*(-log(x)/(2*pi) - 1/pi + log(2)/(2*pi))\
+ x**3*(-log(x)/pi + 1/(6*pi) + log(2)/pi) + O(x**4*log(x))
assert bessely(0, x/(1 - x)).series(x, n=3) == const\
+ 2*x/pi + x**2*(-log(x)/(2*pi) + (2 - 2*S.EulerGamma)/(4*pi)\
+ log(2)/(2*pi) + 1/pi) + O(x**3*log(x))
assert bessely(0, log(1 + x)).series(x, n=3) == const\
- x/pi + x**2*(-log(x)/(2*pi) + (2 - 2*S.EulerGamma)/(4*pi)\
+ log(2)/(2*pi) + 5/(12*pi)) + O(x**3*log(x))
assert bessely(1, sin(x)).series(x, n=4) == -(1/pi)*(1 - 2*S.EulerGamma)\
* (-x**3/12 + x/2) + x*(log(x)/pi - log(2)/pi) + x**3*(-7*log(x)\
/ (24*pi) - 1/(6*pi) + (Rational(5, 2) - 2*S.EulerGamma)/(16*pi)\
+ 7*log(2)/(24*pi)) + O(x**4*log(x))
assert bessely(1, 2*sqrt(x)).series(x, n=3) == sqrt(x)*(log(x)/pi \
- (1 - 2*S.EulerGamma)/pi) + x**Rational(3, 2)*(-log(x)/(2*pi)\
+ (Rational(5, 2) - 2*S.EulerGamma)/(2*pi))\
+ x**Rational(5, 2)*(log(x)/(12*pi)\
- (Rational(10, 3) - 2*S.EulerGamma)/(12*pi)) + O(x**3*log(x))
assert bessely(-2, sin(x)).series(x, n=4) == bessely(2, sin(x)).series(x, n=4)
def test_diff():
assert besselj(n, z).diff(z) == besselj(n - 1, z)/2 - besselj(n + 1, z)/2
assert bessely(n, z).diff(z) == bessely(n - 1, z)/2 - bessely(n + 1, z)/2
assert besseli(n, z).diff(z) == besseli(n - 1, z)/2 + besseli(n + 1, z)/2
assert besselk(n, z).diff(z) == -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
assert hankel1(n, z).diff(z) == hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
assert hankel2(n, z).diff(z) == hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2
def test_rewrite():
assert besselj(n, z).rewrite(jn) == sqrt(2*z/pi)*jn(n - S.Half, z)
assert bessely(n, z).rewrite(yn) == sqrt(2*z/pi)*yn(n - S.Half, z)
assert besseli(n, z).rewrite(besselj) == \
exp(-I*n*pi/2)*besselj(n, polar_lift(I)*z)
assert besselj(n, z).rewrite(besseli) == \
exp(I*n*pi/2)*besseli(n, polar_lift(-I)*z)
nu = randcplx()
assert tn(besselj(nu, z), besselj(nu, z).rewrite(besseli), z)
assert tn(besselj(nu, z), besselj(nu, z).rewrite(bessely), z)
assert tn(besseli(nu, z), besseli(nu, z).rewrite(besselj), z)
assert tn(besseli(nu, z), besseli(nu, z).rewrite(bessely), z)
assert tn(bessely(nu, z), bessely(nu, z).rewrite(besselj), z)
assert tn(bessely(nu, z), bessely(nu, z).rewrite(besseli), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(besselj), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(besseli), z)
assert tn(besselk(nu, z), besselk(nu, z).rewrite(bessely), z)
# check that a rewrite was triggered, when the order is set to a generic
# symbol 'nu'
assert yn(nu, z) != yn(nu, z).rewrite(jn)
assert hn1(nu, z) != hn1(nu, z).rewrite(jn)
assert hn2(nu, z) != hn2(nu, z).rewrite(jn)
assert jn(nu, z) != jn(nu, z).rewrite(yn)
assert hn1(nu, z) != hn1(nu, z).rewrite(yn)
assert hn2(nu, z) != hn2(nu, z).rewrite(yn)
# rewriting spherical bessel functions (SBFs) w.r.t. besselj, bessely is
# not allowed if a generic symbol 'nu' is used as the order of the SBFs
# to avoid inconsistencies (the order of bessel[jy] is allowed to be
# complex-valued, whereas SBFs are defined only for integer orders)
order = nu
for f in (besselj, bessely):
assert hn1(order, z) == hn1(order, z).rewrite(f)
assert hn2(order, z) == hn2(order, z).rewrite(f)
assert jn(order, z).rewrite(besselj) == sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(order + S.Half, z)/2
assert jn(order, z).rewrite(bessely) == (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-order - S.Half, z)/2
# for integral orders rewriting SBFs w.r.t bessel[jy] is allowed
N = Symbol('n', integer=True)
ri = randint(-11, 10)
for order in (ri, N):
for f in (besselj, bessely):
assert yn(order, z) != yn(order, z).rewrite(f)
assert jn(order, z) != jn(order, z).rewrite(f)
assert hn1(order, z) != hn1(order, z).rewrite(f)
assert hn2(order, z) != hn2(order, z).rewrite(f)
for func, refunc in product((yn, jn, hn1, hn2),
(jn, yn, besselj, bessely)):
assert tn(func(ri, z), func(ri, z).rewrite(refunc), z)
def test_expand():
assert expand_func(besselj(S.Half, z).rewrite(jn)) == \
sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert expand_func(bessely(S.Half, z).rewrite(yn)) == \
-sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
# XXX: teach sin/cos to work around arguments like
# x*exp_polar(I*pi*n/2). Then change besselsimp -> expand_func
assert besselsimp(besselj(S.Half, z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(Rational(-1, 2), z)) == sqrt(2)*cos(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besselj(Rational(5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselj(Rational(-5, 2), z)) == \
-sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(S.Half, z)) == \
-(sqrt(2)*cos(z))/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(-1, 2), z)) == sqrt(2)*sin(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(bessely(Rational(5, 2), z)) == \
sqrt(2)*(z**2*cos(z) - 3*z*sin(z) - 3*cos(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(bessely(Rational(-5, 2), z)) == \
-sqrt(2)*(z**2*sin(z) + 3*z*cos(z) - 3*sin(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(S.Half, z)) == sqrt(2)*sinh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(-1, 2), z)) == \
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
assert besselsimp(besseli(Rational(5, 2), z)) == \
sqrt(2)*(z**2*sinh(z) - 3*z*cosh(z) + 3*sinh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besseli(Rational(-5, 2), z)) == \
sqrt(2)*(z**2*cosh(z) - 3*z*sinh(z) + 3*cosh(z))/(sqrt(pi)*z**Rational(5, 2))
assert besselsimp(besselk(S.Half, z)) == \
besselsimp(besselk(Rational(-1, 2), z)) == sqrt(pi)*exp(-z)/(sqrt(2)*sqrt(z))
assert besselsimp(besselk(Rational(5, 2), z)) == \
besselsimp(besselk(Rational(-5, 2), z)) == \
sqrt(2)*sqrt(pi)*(z**2 + 3*z + 3)*exp(-z)/(2*z**Rational(5, 2))
n = Symbol('n', integer=True, positive=True)
assert expand_func(besseli(n + 2, z)) == \
besseli(n, z) + (-2*n - 2)*(-2*n*besseli(n, z)/z + besseli(n - 1, z))/z
assert expand_func(besselj(n + 2, z)) == \
-besselj(n, z) + (2*n + 2)*(2*n*besselj(n, z)/z - besselj(n - 1, z))/z
assert expand_func(besselk(n + 2, z)) == \
besselk(n, z) + (2*n + 2)*(2*n*besselk(n, z)/z + besselk(n - 1, z))/z
assert expand_func(bessely(n + 2, z)) == \
-bessely(n, z) + (2*n + 2)*(2*n*bessely(n, z)/z - bessely(n - 1, z))/z
assert expand_func(besseli(n + S.Half, z).rewrite(jn)) == \
(sqrt(2)*sqrt(z)*exp(-I*pi*(n + S.Half)/2) *
exp_polar(I*pi/4)*jn(n, z*exp_polar(I*pi/2))/sqrt(pi))
assert expand_func(besselj(n + S.Half, z).rewrite(jn)) == \
sqrt(2)*sqrt(z)*jn(n, z)/sqrt(pi)
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
for besselx in [besselj, bessely, besseli, besselk]:
assert besselx(i, p).is_extended_real is True
assert besselx(i, x).is_extended_real is None
assert besselx(x, z).is_extended_real is None
for besselx in [besselj, besseli]:
assert besselx(i, r).is_extended_real is True
for besselx in [bessely, besselk]:
assert besselx(i, r).is_extended_real is None
for besselx in [besselj, bessely, besseli, besselk]:
assert expand_func(besselx(oo, x)) == besselx(oo, x, evaluate=False)
assert expand_func(besselx(-oo, x)) == besselx(-oo, x, evaluate=False)
# Quite varying time, but often really slow
@slow
def test_slow_expand():
def check(eq, ans):
return tn(eq, ans) and eq == ans
rn = randcplx(a=1, b=0, d=0, c=2)
for besselx in [besselj, bessely, besseli, besselk]:
ri = S(2*randint(-11, 10) + 1) / 2 # half integer in [-21/2, 21/2]
assert tn(besselsimp(besselx(ri, z)), besselx(ri, z))
assert check(expand_func(besseli(rn, x)),
besseli(rn - 2, x) - 2*(rn - 1)*besseli(rn - 1, x)/x)
assert check(expand_func(besseli(-rn, x)),
besseli(-rn + 2, x) + 2*(-rn + 1)*besseli(-rn + 1, x)/x)
assert check(expand_func(besselj(rn, x)),
-besselj(rn - 2, x) + 2*(rn - 1)*besselj(rn - 1, x)/x)
assert check(expand_func(besselj(-rn, x)),
-besselj(-rn + 2, x) + 2*(-rn + 1)*besselj(-rn + 1, x)/x)
assert check(expand_func(besselk(rn, x)),
besselk(rn - 2, x) + 2*(rn - 1)*besselk(rn - 1, x)/x)
assert check(expand_func(besselk(-rn, x)),
besselk(-rn + 2, x) - 2*(-rn + 1)*besselk(-rn + 1, x)/x)
assert check(expand_func(bessely(rn, x)),
-bessely(rn - 2, x) + 2*(rn - 1)*bessely(rn - 1, x)/x)
assert check(expand_func(bessely(-rn, x)),
-bessely(-rn + 2, x) + 2*(-rn + 1)*bessely(-rn + 1, x)/x)
def mjn(n, z):
return expand_func(jn(n, z))
def myn(n, z):
return expand_func(yn(n, z))
def test_jn():
z = symbols("z")
assert jn(0, 0) == 1
assert jn(1, 0) == 0
assert jn(-1, 0) == S.ComplexInfinity
assert jn(z, 0) == jn(z, 0, evaluate=False)
assert jn(0, oo) == 0
assert jn(0, -oo) == 0
assert mjn(0, z) == sin(z)/z
assert mjn(1, z) == sin(z)/z**2 - cos(z)/z
assert mjn(2, z) == (3/z**3 - 1/z)*sin(z) - (3/z**2) * cos(z)
assert mjn(3, z) == (15/z**4 - 6/z**2)*sin(z) + (1/z - 15/z**3)*cos(z)
assert mjn(4, z) == (1/z + 105/z**5 - 45/z**3)*sin(z) + \
(-105/z**4 + 10/z**2)*cos(z)
assert mjn(5, z) == (945/z**6 - 420/z**4 + 15/z**2)*sin(z) + \
(-1/z - 945/z**5 + 105/z**3)*cos(z)
assert mjn(6, z) == (-1/z + 10395/z**7 - 4725/z**5 + 210/z**3)*sin(z) + \
(-10395/z**6 + 1260/z**4 - 21/z**2)*cos(z)
assert expand_func(jn(n, z)) == jn(n, z)
# SBFs not defined for complex-valued orders
assert jn(2+3j, 5.2+0.3j).evalf() == jn(2+3j, 5.2+0.3j)
assert eq([jn(2, 5.2+0.3j).evalf(10)],
[0.09941975672 - 0.05452508024*I])
def test_yn():
z = symbols("z")
assert myn(0, z) == -cos(z)/z
assert myn(1, z) == -cos(z)/z**2 - sin(z)/z
assert myn(2, z) == -((3/z**3 - 1/z)*cos(z) + (3/z**2)*sin(z))
assert expand_func(yn(n, z)) == yn(n, z)
# SBFs not defined for complex-valued orders
assert yn(2+3j, 5.2+0.3j).evalf() == yn(2+3j, 5.2+0.3j)
assert eq([yn(2, 5.2+0.3j).evalf(10)],
[0.185250342 + 0.01489557397*I])
def test_sympify_yn():
assert S(15) in myn(3, pi).atoms()
assert myn(3, pi) == 15/pi**4 - 6/pi**2
def eq(a, b, tol=1e-6):
for u, v in zip(a, b):
if not (abs(u - v) < tol):
return False
return True
def test_jn_zeros():
assert eq(jn_zeros(0, 4), [3.141592, 6.283185, 9.424777, 12.566370])
assert eq(jn_zeros(1, 4), [4.493409, 7.725251, 10.904121, 14.066193])
assert eq(jn_zeros(2, 4), [5.763459, 9.095011, 12.322940, 15.514603])
assert eq(jn_zeros(3, 4), [6.987932, 10.417118, 13.698023, 16.923621])
assert eq(jn_zeros(4, 4), [8.182561, 11.704907, 15.039664, 18.301255])
def test_bessel_eval():
n, m, k = Symbol('n', integer=True), Symbol('m'), Symbol('k', integer=True, zero=False)
for f in [besselj, besseli]:
assert f(0, 0) is S.One
assert f(2.1, 0) is S.Zero
assert f(-3, 0) is S.Zero
assert f(-10.2, 0) is S.ComplexInfinity
assert f(1 + 3*I, 0) is S.Zero
assert f(-3 + I, 0) is S.ComplexInfinity
assert f(-2*I, 0) is S.NaN
assert f(n, 0) != S.One and f(n, 0) != S.Zero
assert f(m, 0) != S.One and f(m, 0) != S.Zero
assert f(k, 0) is S.Zero
assert bessely(0, 0) is S.NegativeInfinity
assert besselk(0, 0) is S.Infinity
for f in [bessely, besselk]:
assert f(1 + I, 0) is S.ComplexInfinity
assert f(I, 0) is S.NaN
for f in [besselj, bessely]:
assert f(m, S.Infinity) is S.Zero
assert f(m, S.NegativeInfinity) is S.Zero
for f in [besseli, besselk]:
assert f(m, I*S.Infinity) is S.Zero
assert f(m, I*S.NegativeInfinity) is S.Zero
for f in [besseli, besselk]:
assert f(-4, z) == f(4, z)
assert f(-3, z) == f(3, z)
assert f(-n, z) == f(n, z)
assert f(-m, z) != f(m, z)
for f in [besselj, bessely]:
assert f(-4, z) == f(4, z)
assert f(-3, z) == -f(3, z)
assert f(-n, z) == (-1)**n*f(n, z)
assert f(-m, z) != (-1)**m*f(m, z)
for f in [besselj, besseli]:
assert f(m, -z) == (-z)**m*z**(-m)*f(m, z)
assert besseli(2, -z) == besseli(2, z)
assert besseli(3, -z) == -besseli(3, z)
assert besselj(0, -z) == besselj(0, z)
assert besselj(1, -z) == -besselj(1, z)
assert besseli(0, I*z) == besselj(0, z)
assert besseli(1, I*z) == I*besselj(1, z)
assert besselj(3, I*z) == -I*besseli(3, z)
def test_bessel_nan():
# FIXME: could have these return NaN; for now just fix infinite recursion
for f in [besselj, bessely, besseli, besselk, hankel1, hankel2, yn, jn]:
assert f(1, S.NaN) == f(1, S.NaN, evaluate=False)
def test_meromorphic():
assert besselj(2, x).is_meromorphic(x, 1) == True
assert besselj(2, x).is_meromorphic(x, 0) == True
assert besselj(2, x).is_meromorphic(x, oo) == False
assert besselj(S(2)/3, x).is_meromorphic(x, 1) == True
assert besselj(S(2)/3, x).is_meromorphic(x, 0) == False
assert besselj(S(2)/3, x).is_meromorphic(x, oo) == False
assert besselj(x, 2*x).is_meromorphic(x, 2) == False
assert besselk(0, x).is_meromorphic(x, 1) == True
assert besselk(2, x).is_meromorphic(x, 0) == True
assert besseli(0, x).is_meromorphic(x, 1) == True
assert besseli(2, x).is_meromorphic(x, 0) == True
assert bessely(0, x).is_meromorphic(x, 1) == True
assert bessely(0, x).is_meromorphic(x, 0) == False
assert bessely(2, x).is_meromorphic(x, 0) == True
assert hankel1(3, x**2 + 2*x).is_meromorphic(x, 1) == True
assert hankel1(0, x).is_meromorphic(x, 0) == False
assert hankel2(11, 4).is_meromorphic(x, 5) == True
assert hn1(6, 7*x**3 + 4).is_meromorphic(x, 7) == True
assert hn2(3, 2*x).is_meromorphic(x, 9) == True
assert jn(5, 2*x + 7).is_meromorphic(x, 4) == True
assert yn(8, x**2 + 11).is_meromorphic(x, 6) == True
def test_conjugate():
n = Symbol('n')
z = Symbol('z', extended_real=False)
x = Symbol('x', extended_real=True)
y = Symbol('y', positive=True)
t = Symbol('t', negative=True)
for f in [besseli, besselj, besselk, bessely, hankel1, hankel2]:
assert f(n, -1).conjugate() != f(conjugate(n), -1)
assert f(n, x).conjugate() != f(conjugate(n), x)
assert f(n, t).conjugate() != f(conjugate(n), t)
rz = randcplx(b=0.5)
for f in [besseli, besselj, besselk, bessely]:
assert f(n, 1 + I).conjugate() == f(conjugate(n), 1 - I)
assert f(n, 0).conjugate() == f(conjugate(n), 0)
assert f(n, 1).conjugate() == f(conjugate(n), 1)
assert f(n, z).conjugate() == f(conjugate(n), conjugate(z))
assert f(n, y).conjugate() == f(conjugate(n), y)
assert tn(f(n, rz).conjugate(), f(conjugate(n), conjugate(rz)))
assert hankel1(n, 1 + I).conjugate() == hankel2(conjugate(n), 1 - I)
assert hankel1(n, 0).conjugate() == hankel2(conjugate(n), 0)
assert hankel1(n, 1).conjugate() == hankel2(conjugate(n), 1)
assert hankel1(n, y).conjugate() == hankel2(conjugate(n), y)
assert hankel1(n, z).conjugate() == hankel2(conjugate(n), conjugate(z))
assert tn(hankel1(n, rz).conjugate(), hankel2(conjugate(n), conjugate(rz)))
assert hankel2(n, 1 + I).conjugate() == hankel1(conjugate(n), 1 - I)
assert hankel2(n, 0).conjugate() == hankel1(conjugate(n), 0)
assert hankel2(n, 1).conjugate() == hankel1(conjugate(n), 1)
assert hankel2(n, y).conjugate() == hankel1(conjugate(n), y)
assert hankel2(n, z).conjugate() == hankel1(conjugate(n), conjugate(z))
assert tn(hankel2(n, rz).conjugate(), hankel1(conjugate(n), conjugate(rz)))
def test_branching():
assert besselj(polar_lift(k), x) == besselj(k, x)
assert besseli(polar_lift(k), x) == besseli(k, x)
n = Symbol('n', integer=True)
assert besselj(n, exp_polar(2*pi*I)*x) == besselj(n, x)
assert besselj(n, polar_lift(x)) == besselj(n, x)
assert besseli(n, exp_polar(2*pi*I)*x) == besseli(n, x)
assert besseli(n, polar_lift(x)) == besseli(n, x)
def tn(func, s):
from sympy.core.random import uniform
c = uniform(1, 5)
expr = func(s, c*exp_polar(I*pi)) - func(s, c*exp_polar(-I*pi))
eps = 1e-15
expr2 = func(s + eps, -c + eps*I) - func(s + eps, -c - eps*I)
return abs(expr.n() - expr2.n()).n() < 1e-10
nu = Symbol('nu')
assert besselj(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besselj(nu, x)
assert besseli(nu, exp_polar(2*pi*I)*x) == exp(2*pi*I*nu)*besseli(nu, x)
assert tn(besselj, 2)
assert tn(besselj, pi)
assert tn(besselj, I)
assert tn(besseli, 2)
assert tn(besseli, pi)
assert tn(besseli, I)
def test_airy_base():
z = Symbol('z')
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert conjugate(airyai(z)) == airyai(conjugate(z))
assert airyai(x).is_extended_real
assert airyai(x+I*y).as_real_imag() == (
airyai(x - I*y)/2 + airyai(x + I*y)/2,
I*(airyai(x - I*y) - airyai(x + I*y))/2)
def test_airyai():
z = Symbol('z', real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyai(z), airyai)
assert airyai(0) == 3**Rational(1, 3)/(3*gamma(Rational(2, 3)))
assert airyai(oo) == 0
assert airyai(-oo) == 0
assert diff(airyai(z), z) == airyaiprime(z)
assert series(airyai(z), z, 0, 3) == (
3**Rational(5, 6)*gamma(Rational(1, 3))/(6*pi) - 3**Rational(1, 6)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
assert airyai(z).rewrite(hyper) == (
-3**Rational(2, 3)*z*hyper((), (Rational(4, 3),), z**3/9)/(3*gamma(Rational(1, 3))) +
3**Rational(1, 3)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
assert isinstance(airyai(z).rewrite(besselj), airyai)
assert airyai(t).rewrite(besselj) == (
sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airyai(z).rewrite(besseli) == (
-z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(3*(z**Rational(3, 2))**Rational(1, 3)) +
(z**Rational(3, 2))**Rational(1, 3)*besseli(Rational(-1, 3), 2*z**Rational(3, 2)/3)/3)
assert airyai(p).rewrite(besseli) == (
sqrt(p)*(besseli(Rational(-1, 3), 2*p**Rational(3, 2)/3) -
besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
assert expand_func(airyai(2*(3*z**5)**Rational(1, 3))) == (
-sqrt(3)*(-1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airybi(2*3**Rational(1, 3)*z**Rational(5, 3))/6 +
(1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airyai(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
def test_airybi():
z = Symbol('z', real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybi(z), airybi)
assert airybi(0) == 3**Rational(5, 6)/(3*gamma(Rational(2, 3)))
assert airybi(oo) is oo
assert airybi(-oo) == 0
assert diff(airybi(z), z) == airybiprime(z)
assert series(airybi(z), z, 0, 3) == (
3**Rational(1, 3)*gamma(Rational(1, 3))/(2*pi) + 3**Rational(2, 3)*z*gamma(Rational(2, 3))/(2*pi) + O(z**3))
assert airybi(z).rewrite(hyper) == (
3**Rational(1, 6)*z*hyper((), (Rational(4, 3),), z**3/9)/gamma(Rational(1, 3)) +
3**Rational(5, 6)*hyper((), (Rational(2, 3),), z**3/9)/(3*gamma(Rational(2, 3))))
assert isinstance(airybi(z).rewrite(besselj), airybi)
assert airyai(t).rewrite(besselj) == (
sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airybi(z).rewrite(besseli) == (
sqrt(3)*(z*besseli(Rational(1, 3), 2*z**Rational(3, 2)/3)/(z**Rational(3, 2))**Rational(1, 3) +
(z**Rational(3, 2))**Rational(1, 3)*besseli(Rational(-1, 3), 2*z**Rational(3, 2)/3))/3)
assert airybi(p).rewrite(besseli) == (
sqrt(3)*sqrt(p)*(besseli(Rational(-1, 3), 2*p**Rational(3, 2)/3) +
besseli(Rational(1, 3), 2*p**Rational(3, 2)/3))/3)
assert expand_func(airybi(2*(3*z**5)**Rational(1, 3))) == (
sqrt(3)*(1 - (z**5)**Rational(1, 3)/z**Rational(5, 3))*airyai(2*3**Rational(1, 3)*z**Rational(5, 3))/2 +
(1 + (z**5)**Rational(1, 3)/z**Rational(5, 3))*airybi(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
def test_airyaiprime():
z = Symbol('z', real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airyaiprime(z), airyaiprime)
assert airyaiprime(0) == -3**Rational(2, 3)/(3*gamma(Rational(1, 3)))
assert airyaiprime(oo) == 0
assert diff(airyaiprime(z), z) == z*airyai(z)
assert series(airyaiprime(z), z, 0, 3) == (
-3**Rational(2, 3)/(3*gamma(Rational(1, 3))) + 3**Rational(1, 3)*z**2/(6*gamma(Rational(2, 3))) + O(z**3))
assert airyaiprime(z).rewrite(hyper) == (
3**Rational(1, 3)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) -
3**Rational(2, 3)*hyper((), (Rational(1, 3),), z**3/9)/(3*gamma(Rational(1, 3))))
assert isinstance(airyaiprime(z).rewrite(besselj), airyaiprime)
assert airyai(t).rewrite(besselj) == (
sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airyaiprime(z).rewrite(besseli) == (
z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(3*(z**Rational(3, 2))**Rational(2, 3)) -
(z**Rational(3, 2))**Rational(2, 3)*besseli(Rational(-1, 3), 2*z**Rational(3, 2)/3)/3)
assert airyaiprime(p).rewrite(besseli) == (
p*(-besseli(Rational(-2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3)
assert expand_func(airyaiprime(2*(3*z**5)**Rational(1, 3))) == (
sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/6 +
(z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
def test_airybiprime():
z = Symbol('z', real=False)
t = Symbol('t', negative=True)
p = Symbol('p', positive=True)
assert isinstance(airybiprime(z), airybiprime)
assert airybiprime(0) == 3**Rational(1, 6)/gamma(Rational(1, 3))
assert airybiprime(oo) is oo
assert airybiprime(-oo) == 0
assert diff(airybiprime(z), z) == z*airybi(z)
assert series(airybiprime(z), z, 0, 3) == (
3**Rational(1, 6)/gamma(Rational(1, 3)) + 3**Rational(5, 6)*z**2/(6*gamma(Rational(2, 3))) + O(z**3))
assert airybiprime(z).rewrite(hyper) == (
3**Rational(5, 6)*z**2*hyper((), (Rational(5, 3),), z**3/9)/(6*gamma(Rational(2, 3))) +
3**Rational(1, 6)*hyper((), (Rational(1, 3),), z**3/9)/gamma(Rational(1, 3)))
assert isinstance(airybiprime(z).rewrite(besselj), airybiprime)
assert airyai(t).rewrite(besselj) == (
sqrt(-t)*(besselj(Rational(-1, 3), 2*(-t)**Rational(3, 2)/3) +
besselj(Rational(1, 3), 2*(-t)**Rational(3, 2)/3))/3)
assert airybiprime(z).rewrite(besseli) == (
sqrt(3)*(z**2*besseli(Rational(2, 3), 2*z**Rational(3, 2)/3)/(z**Rational(3, 2))**Rational(2, 3) +
(z**Rational(3, 2))**Rational(2, 3)*besseli(Rational(-2, 3), 2*z**Rational(3, 2)/3))/3)
assert airybiprime(p).rewrite(besseli) == (
sqrt(3)*p*(besseli(Rational(-2, 3), 2*p**Rational(3, 2)/3) + besseli(Rational(2, 3), 2*p**Rational(3, 2)/3))/3)
assert expand_func(airybiprime(2*(3*z**5)**Rational(1, 3))) == (
sqrt(3)*(z**Rational(5, 3)/(z**5)**Rational(1, 3) - 1)*airyaiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2 +
(z**Rational(5, 3)/(z**5)**Rational(1, 3) + 1)*airybiprime(2*3**Rational(1, 3)*z**Rational(5, 3))/2)
def test_marcumq():
m = Symbol('m')
a = Symbol('a')
b = Symbol('b')
assert marcumq(0, 0, 0) == 0
assert marcumq(m, 0, b) == uppergamma(m, b**2/2)/gamma(m)
assert marcumq(2, 0, 5) == 27*exp(Rational(-25, 2))/2
assert marcumq(0, a, 0) == 1 - exp(-a**2/2)
assert marcumq(0, pi, 0) == 1 - exp(-pi**2/2)
assert marcumq(1, a, a) == S.Half + exp(-a**2)*besseli(0, a**2)/2
assert marcumq(2, a, a) == S.Half + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)
assert diff(marcumq(1, a, 3), a) == a*(-marcumq(1, a, 3) + marcumq(2, a, 3))
assert diff(marcumq(2, 3, b), b) == -b**2*exp(-b**2/2 - Rational(9, 2))*besseli(1, 3*b)/3
x = Symbol('x')
assert marcumq(2, 3, 4).rewrite(Integral, x=x) == \
Integral(x**2*exp(-x**2/2 - Rational(9, 2))*besseli(1, 3*x), (x, 4, oo))/3
assert eq([marcumq(5, -2, 3).rewrite(Integral).evalf(10)],
[0.7905769565])
k = Symbol('k')
assert marcumq(-3, -5, -7).rewrite(Sum, k=k) == \
exp(-37)*Sum((Rational(5, 7))**k*besseli(k, 35), (k, 4, oo))
assert eq([marcumq(1, 3, 1).rewrite(Sum).evalf(10)],
[0.9891705502])
assert marcumq(1, a, a, evaluate=False).rewrite(besseli) == S.Half + exp(-a**2)*besseli(0, a**2)/2
assert marcumq(2, a, a, evaluate=False).rewrite(besseli) == S.Half + exp(-a**2)*besseli(0, a**2)/2 + \
exp(-a**2)*besseli(1, a**2)
assert marcumq(3, a, a).rewrite(besseli) == (besseli(1, a**2) + besseli(2, a**2))*exp(-a**2) + \
S.Half + exp(-a**2)*besseli(0, a**2)/2
assert marcumq(5, 8, 8).rewrite(besseli) == exp(-64)*besseli(0, 64)/2 + \
(besseli(4, 64) + besseli(3, 64) + besseli(2, 64) + besseli(1, 64))*exp(-64) + S.Half
assert marcumq(m, a, a).rewrite(besseli) == marcumq(m, a, a)
x = Symbol('x', integer=True)
assert marcumq(x, a, a).rewrite(besseli) == marcumq(x, a, a)
|
4b8940b8ca2094320c186f43104fbdfbc8299c47c72048e574c5d9b5d2f19dbb | from sympy.core.add import Add
from sympy.core.assumptions import check_assumptions
from sympy.core.containers import Tuple
from sympy.core.exprtools import factor_terms
from sympy.core.function import _mexpand
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.numbers import igcdex, ilcm, igcd
from sympy.core.power import integer_nthroot, isqrt
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import Symbol, symbols
from sympy.core.sympify import _sympify
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.ntheory.factor_ import (
divisors, factorint, multiplicity, perfect_power)
from sympy.ntheory.generate import nextprime
from sympy.ntheory.primetest import is_square, isprime
from sympy.ntheory.residue_ntheory import sqrt_mod
from sympy.polys.polyerrors import GeneratorsNeeded
from sympy.polys.polytools import Poly, factor_list
from sympy.simplify.simplify import signsimp
from sympy.solvers.solveset import solveset_real
from sympy.utilities import numbered_symbols
from sympy.utilities.misc import as_int, filldedent
from sympy.utilities.iterables import (is_sequence, subsets, permute_signs,
signed_permutations, ordered_partitions)
# these are imported with 'from sympy.solvers.diophantine import *
__all__ = ['diophantine', 'classify_diop']
class DiophantineSolutionSet(set):
"""
Container for a set of solutions to a particular diophantine equation.
The base representation is a set of tuples representing each of the solutions.
Parameters
==========
symbols : list
List of free symbols in the original equation.
parameters: list
List of parameters to be used in the solution.
Examples
========
Adding solutions:
>>> from sympy.solvers.diophantine.diophantine import DiophantineSolutionSet
>>> from sympy.abc import x, y, t, u
>>> s1 = DiophantineSolutionSet([x, y], [t, u])
>>> s1
set()
>>> s1.add((2, 3))
>>> s1.add((-1, u))
>>> s1
{(-1, u), (2, 3)}
>>> s2 = DiophantineSolutionSet([x, y], [t, u])
>>> s2.add((3, 4))
>>> s1.update(*s2)
>>> s1
{(-1, u), (2, 3), (3, 4)}
Conversion of solutions into dicts:
>>> list(s1.dict_iterator())
[{x: -1, y: u}, {x: 2, y: 3}, {x: 3, y: 4}]
Substituting values:
>>> s3 = DiophantineSolutionSet([x, y], [t, u])
>>> s3.add((t**2, t + u))
>>> s3
{(t**2, t + u)}
>>> s3.subs({t: 2, u: 3})
{(4, 5)}
>>> s3.subs(t, -1)
{(1, u - 1)}
>>> s3.subs(t, 3)
{(9, u + 3)}
Evaluation at specific values. Positional arguments are given in the same order as the parameters:
>>> s3(-2, 3)
{(4, 1)}
>>> s3(5)
{(25, u + 5)}
>>> s3(None, 2)
{(t**2, t + 2)}
"""
def __init__(self, symbols_seq, parameters):
super().__init__()
if not is_sequence(symbols_seq):
raise ValueError("Symbols must be given as a sequence.")
if not is_sequence(parameters):
raise ValueError("Parameters must be given as a sequence.")
self.symbols = tuple(symbols_seq)
self.parameters = tuple(parameters)
def add(self, solution):
if len(solution) != len(self.symbols):
raise ValueError("Solution should have a length of %s, not %s" % (len(self.symbols), len(solution)))
super().add(Tuple(*solution))
def update(self, *solutions):
for solution in solutions:
self.add(solution)
def dict_iterator(self):
for solution in ordered(self):
yield dict(zip(self.symbols, solution))
def subs(self, *args, **kwargs):
result = DiophantineSolutionSet(self.symbols, self.parameters)
for solution in self:
result.add(solution.subs(*args, **kwargs))
return result
def __call__(self, *args):
if len(args) > len(self.parameters):
raise ValueError("Evaluation should have at most %s values, not %s" % (len(self.parameters), len(args)))
return self.subs(list(zip(self.parameters, args)))
class DiophantineEquationType:
"""
Internal representation of a particular diophantine equation type.
Parameters
==========
equation :
The diophantine equation that is being solved.
free_symbols : list (optional)
The symbols being solved for.
Attributes
==========
total_degree :
The maximum of the degrees of all terms in the equation
homogeneous :
Does the equation contain a term of degree 0
homogeneous_order :
Does the equation contain any coefficient that is in the symbols being solved for
dimension :
The number of symbols being solved for
"""
name = None # type: str
def __init__(self, equation, free_symbols=None):
self.equation = _sympify(equation).expand(force=True)
if free_symbols is not None:
self.free_symbols = free_symbols
else:
self.free_symbols = list(self.equation.free_symbols)
self.free_symbols.sort(key=default_sort_key)
if not self.free_symbols:
raise ValueError('equation should have 1 or more free symbols')
self.coeff = self.equation.as_coefficients_dict()
if not all(_is_int(c) for c in self.coeff.values()):
raise TypeError("Coefficients should be Integers")
self.total_degree = Poly(self.equation).total_degree()
self.homogeneous = 1 not in self.coeff
self.homogeneous_order = not (set(self.coeff) & set(self.free_symbols))
self.dimension = len(self.free_symbols)
self._parameters = None
def matches(self):
"""
Determine whether the given equation can be matched to the particular equation type.
"""
return False
@property
def n_parameters(self):
return self.dimension
@property
def parameters(self):
if self._parameters is None:
self._parameters = symbols('t_:%i' % (self.n_parameters,), integer=True)
return self._parameters
def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet:
raise NotImplementedError('No solver has been written for %s.' % self.name)
def pre_solve(self, parameters=None):
if not self.matches():
raise ValueError("This equation does not match the %s equation type." % self.name)
if parameters is not None:
if len(parameters) != self.n_parameters:
raise ValueError("Expected %s parameter(s) but got %s" % (self.n_parameters, len(parameters)))
self._parameters = parameters
class Univariate(DiophantineEquationType):
"""
Representation of a univariate diophantine equation.
A univariate diophantine equation is an equation of the form
`a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x` is an integer variable.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import Univariate
>>> from sympy.abc import x
>>> Univariate((x - 2)*(x - 3)**2).solve() # solves equation (x - 2)*(x - 3)**2 == 0
{(2,), (3,)}
"""
name = 'univariate'
def matches(self):
return self.dimension == 1
def solve(self, parameters=None, limit=None):
self.pre_solve(parameters)
result = DiophantineSolutionSet(self.free_symbols, parameters=self.parameters)
for i in solveset_real(self.equation, self.free_symbols[0]).intersect(S.Integers):
result.add((i,))
return result
class Linear(DiophantineEquationType):
"""
Representation of a linear diophantine equation.
A linear diophantine equation is an equation of the form `a_{1}x_{1} +
a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import Linear
>>> from sympy.abc import x, y, z
>>> l1 = Linear(2*x - 3*y - 5)
>>> l1.matches() # is this equation linear
True
>>> l1.solve() # solves equation 2*x - 3*y - 5 == 0
{(3*t_0 - 5, 2*t_0 - 5)}
Here x = -3*t_0 - 5 and y = -2*t_0 - 5
>>> Linear(2*x - 3*y - 4*z -3).solve()
{(t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3)}
"""
name = 'linear'
def matches(self):
return self.total_degree == 1
def solve(self, parameters=None, limit=None):
self.pre_solve(parameters)
coeff = self.coeff
var = self.free_symbols
if 1 in coeff:
# negate coeff[] because input is of the form: ax + by + c == 0
# but is used as: ax + by == -c
c = -coeff[1]
else:
c = 0
result = DiophantineSolutionSet(var, parameters=self.parameters)
params = result.parameters
if len(var) == 1:
q, r = divmod(c, coeff[var[0]])
if not r:
result.add((q,))
return result
else:
return result
'''
base_solution_linear() can solve diophantine equations of the form:
a*x + b*y == c
We break down multivariate linear diophantine equations into a
series of bivariate linear diophantine equations which can then
be solved individually by base_solution_linear().
Consider the following:
a_0*x_0 + a_1*x_1 + a_2*x_2 == c
which can be re-written as:
a_0*x_0 + g_0*y_0 == c
where
g_0 == gcd(a_1, a_2)
and
y == (a_1*x_1)/g_0 + (a_2*x_2)/g_0
This leaves us with two binary linear diophantine equations.
For the first equation:
a == a_0
b == g_0
c == c
For the second:
a == a_1/g_0
b == a_2/g_0
c == the solution we find for y_0 in the first equation.
The arrays A and B are the arrays of integers used for
'a' and 'b' in each of the n-1 bivariate equations we solve.
'''
A = [coeff[v] for v in var]
B = []
if len(var) > 2:
B.append(igcd(A[-2], A[-1]))
A[-2] = A[-2] // B[0]
A[-1] = A[-1] // B[0]
for i in range(len(A) - 3, 0, -1):
gcd = igcd(B[0], A[i])
B[0] = B[0] // gcd
A[i] = A[i] // gcd
B.insert(0, gcd)
B.append(A[-1])
'''
Consider the trivariate linear equation:
4*x_0 + 6*x_1 + 3*x_2 == 2
This can be re-written as:
4*x_0 + 3*y_0 == 2
where
y_0 == 2*x_1 + x_2
(Note that gcd(3, 6) == 3)
The complete integral solution to this equation is:
x_0 == 2 + 3*t_0
y_0 == -2 - 4*t_0
where 't_0' is any integer.
Now that we have a solution for 'x_0', find 'x_1' and 'x_2':
2*x_1 + x_2 == -2 - 4*t_0
We can then solve for '-2' and '-4' independently,
and combine the results:
2*x_1a + x_2a == -2
x_1a == 0 + t_0
x_2a == -2 - 2*t_0
2*x_1b + x_2b == -4*t_0
x_1b == 0*t_0 + t_1
x_2b == -4*t_0 - 2*t_1
==>
x_1 == t_0 + t_1
x_2 == -2 - 6*t_0 - 2*t_1
where 't_0' and 't_1' are any integers.
Note that:
4*(2 + 3*t_0) + 6*(t_0 + t_1) + 3*(-2 - 6*t_0 - 2*t_1) == 2
for any integral values of 't_0', 't_1'; as required.
This method is generalised for many variables, below.
'''
solutions = []
for i in range(len(B)):
tot_x, tot_y = [], []
for j, arg in enumerate(Add.make_args(c)):
if arg.is_Integer:
# example: 5 -> k = 5
k, p = arg, S.One
pnew = params[0]
else: # arg is a Mul or Symbol
# example: 3*t_1 -> k = 3
# example: t_0 -> k = 1
k, p = arg.as_coeff_Mul()
pnew = params[params.index(p) + 1]
sol = sol_x, sol_y = base_solution_linear(k, A[i], B[i], pnew)
if p is S.One:
if None in sol:
return result
else:
# convert a + b*pnew -> a*p + b*pnew
if isinstance(sol_x, Add):
sol_x = sol_x.args[0]*p + sol_x.args[1]
if isinstance(sol_y, Add):
sol_y = sol_y.args[0]*p + sol_y.args[1]
tot_x.append(sol_x)
tot_y.append(sol_y)
solutions.append(Add(*tot_x))
c = Add(*tot_y)
solutions.append(c)
result.add(solutions)
return result
class BinaryQuadratic(DiophantineEquationType):
"""
Representation of a binary quadratic diophantine equation.
A binary quadratic diophantine equation is an equation of the
form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`, where `A, B, C, D, E,
F` are integer constants and `x` and `y` are integer variables.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import BinaryQuadratic
>>> b1 = BinaryQuadratic(x**3 + y**2 + 1)
>>> b1.matches()
False
>>> b2 = BinaryQuadratic(x**2 + y**2 + 2*x + 2*y + 2)
>>> b2.matches()
True
>>> b2.solve()
{(-1, -1)}
References
==========
.. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online],
Available: http://www.alpertron.com.ar/METHODS.HTM
.. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online],
Available: https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
"""
name = 'binary_quadratic'
def matches(self):
return self.total_degree == 2 and self.dimension == 2
def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet:
self.pre_solve(parameters)
var = self.free_symbols
coeff = self.coeff
x, y = var
A = coeff[x**2]
B = coeff[x*y]
C = coeff[y**2]
D = coeff[x]
E = coeff[y]
F = coeff[S.One]
A, B, C, D, E, F = [as_int(i) for i in _remove_gcd(A, B, C, D, E, F)]
# (1) Simple-Hyperbolic case: A = C = 0, B != 0
# In this case equation can be converted to (Bx + E)(By + D) = DE - BF
# We consider two cases; DE - BF = 0 and DE - BF != 0
# More details, http://www.alpertron.com.ar/METHODS.HTM#SHyperb
result = DiophantineSolutionSet(var, self.parameters)
t, u = result.parameters
discr = B**2 - 4*A*C
if A == 0 and C == 0 and B != 0:
if D*E - B*F == 0:
q, r = divmod(E, B)
if not r:
result.add((-q, t))
q, r = divmod(D, B)
if not r:
result.add((t, -q))
else:
div = divisors(D*E - B*F)
div = div + [-term for term in div]
for d in div:
x0, r = divmod(d - E, B)
if not r:
q, r = divmod(D*E - B*F, d)
if not r:
y0, r = divmod(q - D, B)
if not r:
result.add((x0, y0))
# (2) Parabolic case: B**2 - 4*A*C = 0
# There are two subcases to be considered in this case.
# sqrt(c)D - sqrt(a)E = 0 and sqrt(c)D - sqrt(a)E != 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Parabol
elif discr == 0:
if A == 0:
s = BinaryQuadratic(self.equation, free_symbols=[y, x]).solve(parameters=[t, u])
for soln in s:
result.add((soln[1], soln[0]))
else:
g = sign(A)*igcd(A, C)
a = A // g
c = C // g
e = sign(B / A)
sqa = isqrt(a)
sqc = isqrt(c)
_c = e*sqc*D - sqa*E
if not _c:
z = symbols("z", real=True)
eq = sqa*g*z**2 + D*z + sqa*F
roots = solveset_real(eq, z).intersect(S.Integers)
for root in roots:
ans = diop_solve(sqa*x + e*sqc*y - root)
result.add((ans[0], ans[1]))
elif _is_int(c):
solve_x = lambda u: -e*sqc*g*_c*t**2 - (E + 2*e*sqc*g*u)*t \
- (e*sqc*g*u**2 + E*u + e*sqc*F) // _c
solve_y = lambda u: sqa*g*_c*t**2 + (D + 2*sqa*g*u)*t \
+ (sqa*g*u**2 + D*u + sqa*F) // _c
for z0 in range(0, abs(_c)):
# Check if the coefficients of y and x obtained are integers or not
if (divisible(sqa*g*z0**2 + D*z0 + sqa*F, _c) and
divisible(e*sqc*g*z0**2 + E*z0 + e*sqc*F, _c)):
result.add((solve_x(z0), solve_y(z0)))
# (3) Method used when B**2 - 4*A*C is a square, is described in p. 6 of the below paper
# by John P. Robertson.
# https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
elif is_square(discr):
if A != 0:
r = sqrt(discr)
u, v = symbols("u, v", integer=True)
eq = _mexpand(
4*A*r*u*v + 4*A*D*(B*v + r*u + r*v - B*u) +
2*A*4*A*E*(u - v) + 4*A*r*4*A*F)
solution = diop_solve(eq, t)
for s0, t0 in solution:
num = B*t0 + r*s0 + r*t0 - B*s0
x_0 = S(num) / (4*A*r)
y_0 = S(s0 - t0) / (2*r)
if isinstance(s0, Symbol) or isinstance(t0, Symbol):
if len(check_param(x_0, y_0, 4*A*r, parameters)) > 0:
ans = check_param(x_0, y_0, 4*A*r, parameters)
result.update(*ans)
elif x_0.is_Integer and y_0.is_Integer:
if is_solution_quad(var, coeff, x_0, y_0):
result.add((x_0, y_0))
else:
s = BinaryQuadratic(self.equation, free_symbols=var[::-1]).solve(parameters=[t, u]) # Interchange x and y
while s:
result.add(s.pop()[::-1]) # and solution <--------+
# (4) B**2 - 4*A*C > 0 and B**2 - 4*A*C not a square or B**2 - 4*A*C < 0
else:
P, Q = _transformation_to_DN(var, coeff)
D, N = _find_DN(var, coeff)
solns_pell = diop_DN(D, N)
if D < 0:
for x0, y0 in solns_pell:
for x in [-x0, x0]:
for y in [-y0, y0]:
s = P*Matrix([x, y]) + Q
try:
result.add([as_int(_) for _ in s])
except ValueError:
pass
else:
# In this case equation can be transformed into a Pell equation
solns_pell = set(solns_pell)
for X, Y in list(solns_pell):
solns_pell.add((-X, -Y))
a = diop_DN(D, 1)
T = a[0][0]
U = a[0][1]
if all(_is_int(_) for _ in P[:4] + Q[:2]):
for r, s in solns_pell:
_a = (r + s*sqrt(D))*(T + U*sqrt(D))**t
_b = (r - s*sqrt(D))*(T - U*sqrt(D))**t
x_n = _mexpand(S(_a + _b) / 2)
y_n = _mexpand(S(_a - _b) / (2*sqrt(D)))
s = P*Matrix([x_n, y_n]) + Q
result.add(s)
else:
L = ilcm(*[_.q for _ in P[:4] + Q[:2]])
k = 1
T_k = T
U_k = U
while (T_k - 1) % L != 0 or U_k % L != 0:
T_k, U_k = T_k*T + D*U_k*U, T_k*U + U_k*T
k += 1
for X, Y in solns_pell:
for i in range(k):
if all(_is_int(_) for _ in P*Matrix([X, Y]) + Q):
_a = (X + sqrt(D)*Y)*(T_k + sqrt(D)*U_k)**t
_b = (X - sqrt(D)*Y)*(T_k - sqrt(D)*U_k)**t
Xt = S(_a + _b) / 2
Yt = S(_a - _b) / (2*sqrt(D))
s = P*Matrix([Xt, Yt]) + Q
result.add(s)
X, Y = X*T + D*U*Y, X*U + Y*T
return result
class InhomogeneousTernaryQuadratic(DiophantineEquationType):
"""
Representation of an inhomogeneous ternary quadratic.
No solver is currently implemented for this equation type.
"""
name = 'inhomogeneous_ternary_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
return not self.homogeneous_order
class HomogeneousTernaryQuadraticNormal(DiophantineEquationType):
"""
Representation of a homogeneous ternary quadratic normal diophantine equation.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import HomogeneousTernaryQuadraticNormal
>>> HomogeneousTernaryQuadraticNormal(4*x**2 - 5*y**2 + z**2).solve()
{(1, 2, 4)}
"""
name = 'homogeneous_ternary_quadratic_normal'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
if not self.homogeneous_order:
return False
nonzero = [k for k in self.coeff if self.coeff[k]]
return len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols)
def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet:
self.pre_solve(parameters)
var = self.free_symbols
coeff = self.coeff
x, y, z = var
a = coeff[x**2]
b = coeff[y**2]
c = coeff[z**2]
(sqf_of_a, sqf_of_b, sqf_of_c), (a_1, b_1, c_1), (a_2, b_2, c_2) = \
sqf_normal(a, b, c, steps=True)
A = -a_2*c_2
B = -b_2*c_2
result = DiophantineSolutionSet(var, parameters=self.parameters)
# If following two conditions are satisfied then there are no solutions
if A < 0 and B < 0:
return result
if (
sqrt_mod(-b_2*c_2, a_2) is None or
sqrt_mod(-c_2*a_2, b_2) is None or
sqrt_mod(-a_2*b_2, c_2) is None):
return result
z_0, x_0, y_0 = descent(A, B)
z_0, q = _rational_pq(z_0, abs(c_2))
x_0 *= q
y_0 *= q
x_0, y_0, z_0 = _remove_gcd(x_0, y_0, z_0)
# Holzer reduction
if sign(a) == sign(b):
x_0, y_0, z_0 = holzer(x_0, y_0, z_0, abs(a_2), abs(b_2), abs(c_2))
elif sign(a) == sign(c):
x_0, z_0, y_0 = holzer(x_0, z_0, y_0, abs(a_2), abs(c_2), abs(b_2))
else:
y_0, z_0, x_0 = holzer(y_0, z_0, x_0, abs(b_2), abs(c_2), abs(a_2))
x_0 = reconstruct(b_1, c_1, x_0)
y_0 = reconstruct(a_1, c_1, y_0)
z_0 = reconstruct(a_1, b_1, z_0)
sq_lcm = ilcm(sqf_of_a, sqf_of_b, sqf_of_c)
x_0 = abs(x_0*sq_lcm // sqf_of_a)
y_0 = abs(y_0*sq_lcm // sqf_of_b)
z_0 = abs(z_0*sq_lcm // sqf_of_c)
result.add(_remove_gcd(x_0, y_0, z_0))
return result
class HomogeneousTernaryQuadratic(DiophantineEquationType):
"""
Representation of a homogeneous ternary quadratic diophantine equation.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import HomogeneousTernaryQuadratic
>>> HomogeneousTernaryQuadratic(x**2 + y**2 - 3*z**2 + x*y).solve()
{(-1, 2, 1)}
>>> HomogeneousTernaryQuadratic(3*x**2 + y**2 - 3*z**2 + 5*x*y + y*z).solve()
{(3, 12, 13)}
"""
name = 'homogeneous_ternary_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
if not self.homogeneous_order:
return False
nonzero = [k for k in self.coeff if self.coeff[k]]
return not (len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols))
def solve(self, parameters=None, limit=None):
self.pre_solve(parameters)
_var = self.free_symbols
coeff = self.coeff
x, y, z = _var
var = [x, y, z]
# Equations of the form B*x*y + C*z*x + E*y*z = 0 and At least two of the
# coefficients A, B, C are non-zero.
# There are infinitely many solutions for the equation.
# Ex: (0, 0, t), (0, t, 0), (t, 0, 0)
# Equation can be re-written as y*(B*x + E*z) = -C*x*z and we can find rather
# unobvious solutions. Set y = -C and B*x + E*z = x*z. The latter can be solved by
# using methods for binary quadratic diophantine equations. Let's select the
# solution which minimizes |x| + |z|
result = DiophantineSolutionSet(var, parameters=self.parameters)
def unpack_sol(sol):
if len(sol) > 0:
return list(sol)[0]
return None, None, None
if not any(coeff[i**2] for i in var):
if coeff[x*z]:
sols = diophantine(coeff[x*y]*x + coeff[y*z]*z - x*z)
s = sols.pop()
min_sum = abs(s[0]) + abs(s[1])
for r in sols:
m = abs(r[0]) + abs(r[1])
if m < min_sum:
s = r
min_sum = m
result.add(_remove_gcd(s[0], -coeff[x*z], s[1]))
return result
else:
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
if x_0 is not None:
result.add((x_0, y_0, z_0))
return result
if coeff[x**2] == 0:
# If the coefficient of x is zero change the variables
if coeff[y**2] == 0:
var[0], var[2] = _var[2], _var[0]
z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
if coeff[x*y] or coeff[x*z]:
# Apply the transformation x --> X - (B*y + C*z)/(2*A)
A = coeff[x**2]
B = coeff[x*y]
C = coeff[x*z]
D = coeff[y**2]
E = coeff[y*z]
F = coeff[z**2]
_coeff = dict()
_coeff[x**2] = 4*A**2
_coeff[y**2] = 4*A*D - B**2
_coeff[z**2] = 4*A*F - C**2
_coeff[y*z] = 4*A*E - 2*B*C
_coeff[x*y] = 0
_coeff[x*z] = 0
x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, _coeff))
if x_0 is None:
return result
p, q = _rational_pq(B*y_0 + C*z_0, 2*A)
x_0, y_0, z_0 = x_0*q - p, y_0*q, z_0*q
elif coeff[z*y] != 0:
if coeff[y**2] == 0:
if coeff[z**2] == 0:
# Equations of the form A*x**2 + E*yz = 0.
A = coeff[x**2]
E = coeff[y*z]
b, a = _rational_pq(-E, A)
x_0, y_0, z_0 = b, a, b
else:
# Ax**2 + E*y*z + F*z**2 = 0
var[0], var[2] = _var[2], _var[0]
z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
# A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, C may be zero
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
# Ax**2 + D*y**2 + F*z**2 = 0, C may be zero
x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic_normal(var, coeff))
if x_0 is None:
return result
result.add(_remove_gcd(x_0, y_0, z_0))
return result
class InhomogeneousGeneralQuadratic(DiophantineEquationType):
"""
Representation of an inhomogeneous general quadratic.
No solver is currently implemented for this equation type.
"""
name = 'inhomogeneous_general_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return True
else:
# there may be Pow keys like x**2 or Mul keys like x*y
if any(k.is_Mul for k in self.coeff): # cross terms
return not self.homogeneous
return False
class HomogeneousGeneralQuadratic(DiophantineEquationType):
"""
Representation of a homogeneous general quadratic.
No solver is currently implemented for this equation type.
"""
name = 'homogeneous_general_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
else:
# there may be Pow keys like x**2 or Mul keys like x*y
if any(k.is_Mul for k in self.coeff): # cross terms
return self.homogeneous
return False
class GeneralSumOfSquares(DiophantineEquationType):
r"""
Representation of the diophantine equation
`x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Details
=======
When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be
no solutions. Refer [1]_ for more details.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import GeneralSumOfSquares
>>> from sympy.abc import a, b, c, d, e
>>> GeneralSumOfSquares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345).solve()
{(15, 22, 22, 24, 24)}
By default only 1 solution is returned. Use the `limit` keyword for more:
>>> sorted(GeneralSumOfSquares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345).solve(limit=3))
[(15, 22, 22, 24, 24), (16, 19, 24, 24, 24), (16, 20, 22, 23, 26)]
References
==========
.. [1] Representing an integer as a sum of three squares, [online],
Available:
http://www.proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares
"""
name = 'general_sum_of_squares'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
if any(k.is_Mul for k in self.coeff):
return False
return all(self.coeff[k] == 1 for k in self.coeff if k != 1)
def solve(self, parameters=None, limit=1):
self.pre_solve(parameters)
var = self.free_symbols
k = -int(self.coeff[1])
n = self.dimension
result = DiophantineSolutionSet(var, parameters=self.parameters)
if k < 0 or limit < 1:
return result
signs = [-1 if x.is_nonpositive else 1 for x in var]
negs = signs.count(-1) != 0
took = 0
for t in sum_of_squares(k, n, zeros=True):
if negs:
result.add([signs[i]*j for i, j in enumerate(t)])
else:
result.add(t)
took += 1
if took == limit:
break
return result
class GeneralPythagorean(DiophantineEquationType):
"""
Representation of the general pythagorean equation,
`a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import GeneralPythagorean
>>> from sympy.abc import a, b, c, d, e, x, y, z, t
>>> GeneralPythagorean(a**2 + b**2 + c**2 - d**2).solve()
{(t_0**2 + t_1**2 - t_2**2, 2*t_0*t_2, 2*t_1*t_2, t_0**2 + t_1**2 + t_2**2)}
>>> GeneralPythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2).solve(parameters=[x, y, z, t])
{(-10*t**2 + 10*x**2 + 10*y**2 + 10*z**2, 15*t**2 + 15*x**2 + 15*y**2 + 15*z**2, 15*t*x, 12*t*y, 60*t*z)}
"""
name = 'general_pythagorean'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
if any(k.is_Mul for k in self.coeff):
return False
if all(self.coeff[k] == 1 for k in self.coeff if k != 1):
return False
if not all(is_square(abs(self.coeff[k])) for k in self.coeff):
return False
# all but one has the same sign
# e.g. 4*x**2 + y**2 - 4*z**2
return abs(sum(sign(self.coeff[k]) for k in self.coeff)) == self.dimension - 2
@property
def n_parameters(self):
return self.dimension - 1
def solve(self, parameters=None, limit=1):
self.pre_solve(parameters)
coeff = self.coeff
var = self.free_symbols
n = self.dimension
if sign(coeff[var[0] ** 2]) + sign(coeff[var[1] ** 2]) + sign(coeff[var[2] ** 2]) < 0:
for key in coeff.keys():
coeff[key] = -coeff[key]
result = DiophantineSolutionSet(var, parameters=self.parameters)
index = 0
for i, v in enumerate(var):
if sign(coeff[v ** 2]) == -1:
index = i
m = result.parameters
ith = sum(m_i ** 2 for m_i in m)
L = [ith - 2 * m[n - 2] ** 2]
L.extend([2 * m[i] * m[n - 2] for i in range(n - 2)])
sol = L[:index] + [ith] + L[index:]
lcm = 1
for i, v in enumerate(var):
if i == index or (index > 0 and i == 0) or (index == 0 and i == 1):
lcm = ilcm(lcm, sqrt(abs(coeff[v ** 2])))
else:
s = sqrt(coeff[v ** 2])
lcm = ilcm(lcm, s if _odd(s) else s // 2)
for i, v in enumerate(var):
sol[i] = (lcm * sol[i]) / sqrt(abs(coeff[v ** 2]))
result.add(sol)
return result
class CubicThue(DiophantineEquationType):
"""
Representation of a cubic Thue diophantine equation.
A cubic Thue diophantine equation is a polynomial of the form
`f(x, y) = r` of degree 3, where `x` and `y` are integers
and `r` is a rational number.
No solver is currently implemented for this equation type.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import CubicThue
>>> c1 = CubicThue(x**3 + y**2 + 1)
>>> c1.matches()
True
"""
name = 'cubic_thue'
def matches(self):
return self.total_degree == 3 and self.dimension == 2
class GeneralSumOfEvenPowers(DiophantineEquationType):
"""
Representation of the diophantine equation
`x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`
where `e` is an even, integer power.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import GeneralSumOfEvenPowers
>>> from sympy.abc import a, b
>>> GeneralSumOfEvenPowers(a**4 + b**4 - (2**4 + 3**4)).solve()
{(2, 3)}
"""
name = 'general_sum_of_even_powers'
def matches(self):
if not self.total_degree > 3:
return False
if self.total_degree % 2 != 0:
return False
if not all(k.is_Pow and k.exp == self.total_degree for k in self.coeff if k != 1):
return False
return all(self.coeff[k] == 1 for k in self.coeff if k != 1)
def solve(self, parameters=None, limit=1):
self.pre_solve(parameters)
var = self.free_symbols
coeff = self.coeff
p = None
for q in coeff.keys():
if q.is_Pow and coeff[q]:
p = q.exp
k = len(var)
n = -coeff[1]
result = DiophantineSolutionSet(var, parameters=self.parameters)
if n < 0 or limit < 1:
return result
sign = [-1 if x.is_nonpositive else 1 for x in var]
negs = sign.count(-1) != 0
took = 0
for t in power_representation(n, p, k):
if negs:
result.add([sign[i]*j for i, j in enumerate(t)])
else:
result.add(t)
took += 1
if took == limit:
break
return result
# these types are known (but not necessarily handled)
# note that order is important here (in the current solver state)
all_diop_classes = [
Linear,
Univariate,
BinaryQuadratic,
InhomogeneousTernaryQuadratic,
HomogeneousTernaryQuadraticNormal,
HomogeneousTernaryQuadratic,
InhomogeneousGeneralQuadratic,
HomogeneousGeneralQuadratic,
GeneralSumOfSquares,
GeneralPythagorean,
CubicThue,
GeneralSumOfEvenPowers,
]
diop_known = {diop_class.name for diop_class in all_diop_classes}
def _is_int(i):
try:
as_int(i)
return True
except ValueError:
pass
def _sorted_tuple(*i):
return tuple(sorted(i))
def _remove_gcd(*x):
try:
g = igcd(*x)
except ValueError:
fx = list(filter(None, x))
if len(fx) < 2:
return x
g = igcd(*[i.as_content_primitive()[0] for i in fx])
except TypeError:
raise TypeError('_remove_gcd(a,b,c) or _remove_gcd(*container)')
if g == 1:
return x
return tuple([i//g for i in x])
def _rational_pq(a, b):
# return `(numer, denom)` for a/b; sign in numer and gcd removed
return _remove_gcd(sign(b)*a, abs(b))
def _nint_or_floor(p, q):
# return nearest int to p/q; in case of tie return floor(p/q)
w, r = divmod(p, q)
if abs(r) <= abs(q)//2:
return w
return w + 1
def _odd(i):
return i % 2 != 0
def _even(i):
return i % 2 == 0
def diophantine(eq, param=symbols("t", integer=True), syms=None,
permute=False):
"""
Simplify the solution procedure of diophantine equation ``eq`` by
converting it into a product of terms which should equal zero.
Explanation
===========
For example, when solving, `x^2 - y^2 = 0` this is treated as
`(x + y)(x - y) = 0` and `x + y = 0` and `x - y = 0` are solved
independently and combined. Each term is solved by calling
``diop_solve()``. (Although it is possible to call ``diop_solve()``
directly, one must be careful to pass an equation in the correct
form and to interpret the output correctly; ``diophantine()`` is
the public-facing function to use in general.)
Output of ``diophantine()`` is a set of tuples. The elements of the
tuple are the solutions for each variable in the equation and
are arranged according to the alphabetic ordering of the variables.
e.g. For an equation with two variables, `a` and `b`, the first
element of the tuple is the solution for `a` and the second for `b`.
Usage
=====
``diophantine(eq, t, syms)``: Solve the diophantine
equation ``eq``.
``t`` is the optional parameter to be used by ``diop_solve()``.
``syms`` is an optional list of symbols which determines the
order of the elements in the returned tuple.
By default, only the base solution is returned. If ``permute`` is set to
True then permutations of the base solution and/or permutations of the
signs of the values will be returned when applicable.
Examples
========
>>> from sympy import diophantine
>>> from sympy.abc import a, b
>>> eq = a**4 + b**4 - (2**4 + 3**4)
>>> diophantine(eq)
{(2, 3)}
>>> diophantine(eq, permute=True)
{(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
Details
=======
``eq`` should be an expression which is assumed to be zero.
``t`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, z
>>> diophantine(x**2 - y**2)
{(t_0, -t_0), (t_0, t_0)}
>>> diophantine(x*(2*x + 3*y - z))
{(0, n1, n2), (t_0, t_1, 2*t_0 + 3*t_1)}
>>> diophantine(x**2 + 3*x*y + 4*x)
{(0, n1), (3*t_0 - 4, -t_0)}
See Also
========
diop_solve()
sympy.utilities.iterables.permute_signs
sympy.utilities.iterables.signed_permutations
"""
eq = _sympify(eq)
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs
try:
var = list(eq.expand(force=True).free_symbols)
var.sort(key=default_sort_key)
if syms:
if not is_sequence(syms):
raise TypeError(
'syms should be given as a sequence, e.g. a list')
syms = [i for i in syms if i in var]
if syms != var:
dict_sym_index = dict(zip(syms, range(len(syms))))
return {tuple([t[dict_sym_index[i]] for i in var])
for t in diophantine(eq, param, permute=permute)}
n, d = eq.as_numer_denom()
if n.is_number:
return set()
if not d.is_number:
dsol = diophantine(d)
good = diophantine(n) - dsol
return {s for s in good if _mexpand(d.subs(zip(var, s)))}
else:
eq = n
eq = factor_terms(eq)
assert not eq.is_number
eq = eq.as_independent(*var, as_Add=False)[1]
p = Poly(eq)
assert not any(g.is_number for g in p.gens)
eq = p.as_expr()
assert eq.is_polynomial()
except (GeneratorsNeeded, AssertionError):
raise TypeError(filldedent('''
Equation should be a polynomial with Rational coefficients.'''))
# permute only sign
do_permute_signs = False
# permute sign and values
do_permute_signs_var = False
# permute few signs
permute_few_signs = False
try:
# if we know that factoring should not be attempted, skip
# the factoring step
v, c, t = classify_diop(eq)
# check for permute sign
if permute:
len_var = len(v)
permute_signs_for = [
GeneralSumOfSquares.name,
GeneralSumOfEvenPowers.name]
permute_signs_check = [
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name,
BinaryQuadratic.name]
if t in permute_signs_for:
do_permute_signs_var = True
elif t in permute_signs_check:
# if all the variables in eq have even powers
# then do_permute_sign = True
if len_var == 3:
var_mul = list(subsets(v, 2))
# here var_mul is like [(x, y), (x, z), (y, z)]
xy_coeff = True
x_coeff = True
var1_mul_var2 = map(lambda a: a[0]*a[1], var_mul)
# if coeff(y*z), coeff(y*x), coeff(x*z) is not 0 then
# `xy_coeff` => True and do_permute_sign => False.
# Means no permuted solution.
for v1_mul_v2 in var1_mul_var2:
try:
coeff = c[v1_mul_v2]
except KeyError:
coeff = 0
xy_coeff = bool(xy_coeff) and bool(coeff)
var_mul = list(subsets(v, 1))
# here var_mul is like [(x,), (y, )]
for v1 in var_mul:
try:
coeff = c[v1[0]]
except KeyError:
coeff = 0
x_coeff = bool(x_coeff) and bool(coeff)
if not any((xy_coeff, x_coeff)):
# means only x**2, y**2, z**2, const is present
do_permute_signs = True
elif not x_coeff:
permute_few_signs = True
elif len_var == 2:
var_mul = list(subsets(v, 2))
# here var_mul is like [(x, y)]
xy_coeff = True
x_coeff = True
var1_mul_var2 = map(lambda x: x[0]*x[1], var_mul)
for v1_mul_v2 in var1_mul_var2:
try:
coeff = c[v1_mul_v2]
except KeyError:
coeff = 0
xy_coeff = bool(xy_coeff) and bool(coeff)
var_mul = list(subsets(v, 1))
# here var_mul is like [(x,), (y, )]
for v1 in var_mul:
try:
coeff = c[v1[0]]
except KeyError:
coeff = 0
x_coeff = bool(x_coeff) and bool(coeff)
if not any((xy_coeff, x_coeff)):
# means only x**2, y**2 and const is present
# so we can get more soln by permuting this soln.
do_permute_signs = True
elif not x_coeff:
# when coeff(x), coeff(y) is not present then signs of
# x, y can be permuted such that their sign are same
# as sign of x*y.
# e.g 1. (x_val,y_val)=> (x_val,y_val), (-x_val,-y_val)
# 2. (-x_vall, y_val)=> (-x_val,y_val), (x_val,-y_val)
permute_few_signs = True
if t == 'general_sum_of_squares':
# trying to factor such expressions will sometimes hang
terms = [(eq, 1)]
else:
raise TypeError
except (TypeError, NotImplementedError):
fl = factor_list(eq)
if fl[0].is_Rational and fl[0] != 1:
return diophantine(eq/fl[0], param=param, syms=syms, permute=permute)
terms = fl[1]
sols = set()
for term in terms:
base, _ = term
var_t, _, eq_type = classify_diop(base, _dict=False)
_, base = signsimp(base, evaluate=False).as_coeff_Mul()
solution = diop_solve(base, param)
if eq_type in [
Linear.name,
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name,
GeneralPythagorean.name]:
sols.add(merge_solution(var, var_t, solution))
elif eq_type in [
BinaryQuadratic.name,
GeneralSumOfSquares.name,
GeneralSumOfEvenPowers.name,
Univariate.name]:
for sol in solution:
sols.add(merge_solution(var, var_t, sol))
else:
raise NotImplementedError('unhandled type: %s' % eq_type)
# remove null merge results
if () in sols:
sols.remove(())
null = tuple([0]*len(var))
# if there is no solution, return trivial solution
if not sols and eq.subs(zip(var, null)).is_zero:
sols.add(null)
final_soln = set()
for sol in sols:
if all(_is_int(s) for s in sol):
if do_permute_signs:
permuted_sign = set(permute_signs(sol))
final_soln.update(permuted_sign)
elif permute_few_signs:
lst = list(permute_signs(sol))
lst = list(filter(lambda x: x[0]*x[1] == sol[1]*sol[0], lst))
permuted_sign = set(lst)
final_soln.update(permuted_sign)
elif do_permute_signs_var:
permuted_sign_var = set(signed_permutations(sol))
final_soln.update(permuted_sign_var)
else:
final_soln.add(sol)
else:
final_soln.add(sol)
return final_soln
def merge_solution(var, var_t, solution):
"""
This is used to construct the full solution from the solutions of sub
equations.
Explanation
===========
For example when solving the equation `(x - y)(x^2 + y^2 - z^2) = 0`,
solutions for each of the equations `x - y = 0` and `x^2 + y^2 - z^2` are
found independently. Solutions for `x - y = 0` are `(x, y) = (t, t)`. But
we should introduce a value for z when we output the solution for the
original equation. This function converts `(t, t)` into `(t, t, n_{1})`
where `n_{1}` is an integer parameter.
"""
sol = []
if None in solution:
return ()
solution = iter(solution)
params = numbered_symbols("n", integer=True, start=1)
for v in var:
if v in var_t:
sol.append(next(solution))
else:
sol.append(next(params))
for val, symb in zip(sol, var):
if check_assumptions(val, **symb.assumptions0) is False:
return tuple()
return tuple(sol)
def _diop_solve(eq, params=None):
for diop_type in all_diop_classes:
if diop_type(eq).matches():
return diop_type(eq).solve(parameters=params)
def diop_solve(eq, param=symbols("t", integer=True)):
"""
Solves the diophantine equation ``eq``.
Explanation
===========
Unlike ``diophantine()``, factoring of ``eq`` is not attempted. Uses
``classify_diop()`` to determine the type of the equation and calls
the appropriate solver function.
Use of ``diophantine()`` is recommended over other helper functions.
``diop_solve()`` can return either a set or a tuple depending on the
nature of the equation.
Usage
=====
``diop_solve(eq, t)``: Solve diophantine equation, ``eq`` using ``t``
as a parameter if needed.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``t`` is a parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine import diop_solve
>>> from sympy.abc import x, y, z, w
>>> diop_solve(2*x + 3*y - 5)
(3*t_0 - 5, 5 - 2*t_0)
>>> diop_solve(4*x + 3*y - 4*z + 5)
(t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5)
>>> diop_solve(x + 3*y - 4*z + w - 6)
(t_0, t_0 + t_1, 6*t_0 + 5*t_1 + 4*t_2 - 6, 5*t_0 + 4*t_1 + 3*t_2 - 6)
>>> diop_solve(x**2 + y**2 - 5)
{(-2, -1), (-2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2), (2, -1), (2, 1)}
See Also
========
diophantine()
"""
var, coeff, eq_type = classify_diop(eq, _dict=False)
if eq_type == Linear.name:
return diop_linear(eq, param)
elif eq_type == BinaryQuadratic.name:
return diop_quadratic(eq, param)
elif eq_type == HomogeneousTernaryQuadratic.name:
return diop_ternary_quadratic(eq, parameterize=True)
elif eq_type == HomogeneousTernaryQuadraticNormal.name:
return diop_ternary_quadratic_normal(eq, parameterize=True)
elif eq_type == GeneralPythagorean.name:
return diop_general_pythagorean(eq, param)
elif eq_type == Univariate.name:
return diop_univariate(eq)
elif eq_type == GeneralSumOfSquares.name:
return diop_general_sum_of_squares(eq, limit=S.Infinity)
elif eq_type == GeneralSumOfEvenPowers.name:
return diop_general_sum_of_even_powers(eq, limit=S.Infinity)
if eq_type is not None and eq_type not in diop_known:
raise ValueError(filldedent('''
Alhough this type of equation was identified, it is not yet
handled. It should, however, be listed in `diop_known` at the
top of this file. Developers should see comments at the end of
`classify_diop`.
''')) # pragma: no cover
else:
raise NotImplementedError(
'No solver has been written for %s.' % eq_type)
def classify_diop(eq, _dict=True):
# docstring supplied externally
matched = False
diop_type = None
for diop_class in all_diop_classes:
diop_type = diop_class(eq)
if diop_type.matches():
matched = True
break
if matched:
return diop_type.free_symbols, dict(diop_type.coeff) if _dict else diop_type.coeff, diop_type.name
# new diop type instructions
# --------------------------
# if this error raises and the equation *can* be classified,
# * it should be identified in the if-block above
# * the type should be added to the diop_known
# if a solver can be written for it,
# * a dedicated handler should be written (e.g. diop_linear)
# * it should be passed to that handler in diop_solve
raise NotImplementedError(filldedent('''
This equation is not yet recognized or else has not been
simplified sufficiently to put it in a form recognized by
diop_classify().'''))
classify_diop.func_doc = ( # type: ignore
'''
Helper routine used by diop_solve() to find information about ``eq``.
Explanation
===========
Returns a tuple containing the type of the diophantine equation
along with the variables (free symbols) and their coefficients.
Variables are returned as a list and coefficients are returned
as a dict with the key being the respective term and the constant
term is keyed to 1. The type is one of the following:
* %s
Usage
=====
``classify_diop(eq)``: Return variables, coefficients and type of the
``eq``.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``_dict`` is for internal use: when True (default) a dict is returned,
otherwise a defaultdict which supplies 0 for missing keys is returned.
Examples
========
>>> from sympy.solvers.diophantine import classify_diop
>>> from sympy.abc import x, y, z, w, t
>>> classify_diop(4*x + 6*y - 4)
([x, y], {1: -4, x: 4, y: 6}, 'linear')
>>> classify_diop(x + 3*y -4*z + 5)
([x, y, z], {1: 5, x: 1, y: 3, z: -4}, 'linear')
>>> classify_diop(x**2 + y**2 - x*y + x + 5)
([x, y], {1: 5, x: 1, x**2: 1, y**2: 1, x*y: -1}, 'binary_quadratic')
''' % ('\n * '.join(sorted(diop_known))))
def diop_linear(eq, param=symbols("t", integer=True)):
"""
Solves linear diophantine equations.
A linear diophantine equation is an equation of the form `a_{1}x_{1} +
a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables.
Usage
=====
``diop_linear(eq)``: Returns a tuple containing solutions to the
diophantine equation ``eq``. Values in the tuple is arranged in the same
order as the sorted variables.
Details
=======
``eq`` is a linear diophantine equation which is assumed to be zero.
``param`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_linear
>>> from sympy.abc import x, y, z
>>> diop_linear(2*x - 3*y - 5) # solves equation 2*x - 3*y - 5 == 0
(3*t_0 - 5, 2*t_0 - 5)
Here x = -3*t_0 - 5 and y = -2*t_0 - 5
>>> diop_linear(2*x - 3*y - 4*z -3)
(t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3)
See Also
========
diop_quadratic(), diop_ternary_quadratic(), diop_general_pythagorean(),
diop_general_sum_of_squares()
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == Linear.name:
parameters = None
if param is not None:
parameters = symbols('%s_0:%i' % (param, len(var)), integer=True)
result = Linear(eq).solve(parameters=parameters)
if param is None:
result = result(*[0]*len(result.parameters))
if len(result) > 0:
return list(result)[0]
else:
return tuple([None]*len(result.parameters))
def base_solution_linear(c, a, b, t=None):
"""
Return the base solution for the linear equation, `ax + by = c`.
Explanation
===========
Used by ``diop_linear()`` to find the base solution of a linear
Diophantine equation. If ``t`` is given then the parametrized solution is
returned.
Usage
=====
``base_solution_linear(c, a, b, t)``: ``a``, ``b``, ``c`` are coefficients
in `ax + by = c` and ``t`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import base_solution_linear
>>> from sympy.abc import t
>>> base_solution_linear(5, 2, 3) # equation 2*x + 3*y = 5
(-5, 5)
>>> base_solution_linear(0, 5, 7) # equation 5*x + 7*y = 0
(0, 0)
>>> base_solution_linear(5, 2, 3, t) # equation 2*x + 3*y = 5
(3*t - 5, 5 - 2*t)
>>> base_solution_linear(0, 5, 7, t) # equation 5*x + 7*y = 0
(7*t, -5*t)
"""
a, b, c = _remove_gcd(a, b, c)
if c == 0:
if t is not None:
if b < 0:
t = -t
return (b*t, -a*t)
else:
return (0, 0)
else:
x0, y0, d = igcdex(abs(a), abs(b))
x0 *= sign(a)
y0 *= sign(b)
if divisible(c, d):
if t is not None:
if b < 0:
t = -t
return (c*x0 + b*t, c*y0 - a*t)
else:
return (c*x0, c*y0)
else:
return (None, None)
def diop_univariate(eq):
"""
Solves a univariate diophantine equations.
Explanation
===========
A univariate diophantine equation is an equation of the form
`a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x` is an integer variable.
Usage
=====
``diop_univariate(eq)``: Returns a set containing solutions to the
diophantine equation ``eq``.
Details
=======
``eq`` is a univariate diophantine equation which is assumed to be zero.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_univariate
>>> from sympy.abc import x
>>> diop_univariate((x - 2)*(x - 3)**2) # solves equation (x - 2)*(x - 3)**2 == 0
{(2,), (3,)}
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == Univariate.name:
return {(int(i),) for i in solveset_real(
eq, var[0]).intersect(S.Integers)}
def divisible(a, b):
"""
Returns `True` if ``a`` is divisible by ``b`` and `False` otherwise.
"""
return not a % b
def diop_quadratic(eq, param=symbols("t", integer=True)):
"""
Solves quadratic diophantine equations.
i.e. equations of the form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`. Returns a
set containing the tuples `(x, y)` which contains the solutions. If there
are no solutions then `(None, None)` is returned.
Usage
=====
``diop_quadratic(eq, param)``: ``eq`` is a quadratic binary diophantine
equation. ``param`` is used to indicate the parameter to be used in the
solution.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``param`` is a parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, t
>>> from sympy.solvers.diophantine.diophantine import diop_quadratic
>>> diop_quadratic(x**2 + y**2 + 2*x + 2*y + 2, t)
{(-1, -1)}
References
==========
.. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online],
Available: http://www.alpertron.com.ar/METHODS.HTM
.. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online],
Available: https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
See Also
========
diop_linear(), diop_ternary_quadratic(), diop_general_sum_of_squares(),
diop_general_pythagorean()
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
if param is not None:
parameters = [param, Symbol("u", integer=True)]
else:
parameters = None
return set(BinaryQuadratic(eq).solve(parameters=parameters))
def is_solution_quad(var, coeff, u, v):
"""
Check whether `(u, v)` is solution to the quadratic binary diophantine
equation with the variable list ``var`` and coefficient dictionary
``coeff``.
Not intended for use by normal users.
"""
reps = dict(zip(var, (u, v)))
eq = Add(*[j*i.xreplace(reps) for i, j in coeff.items()])
return _mexpand(eq) == 0
def diop_DN(D, N, t=symbols("t", integer=True)):
"""
Solves the equation `x^2 - Dy^2 = N`.
Explanation
===========
Mainly concerned with the case `D > 0, D` is not a perfect square,
which is the same as the generalized Pell equation. The LMM
algorithm [1]_ is used to solve this equation.
Returns one solution tuple, (`x, y)` for each class of the solutions.
Other solutions of the class can be constructed according to the
values of ``D`` and ``N``.
Usage
=====
``diop_DN(D, N, t)``: D and N are integers as in `x^2 - Dy^2 = N` and
``t`` is the parameter to be used in the solutions.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
``t`` is the parameter to be used in the solutions.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_DN
>>> diop_DN(13, -4) # Solves equation x**2 - 13*y**2 = -4
[(3, 1), (393, 109), (36, 10)]
The output can be interpreted as follows: There are three fundamental
solutions to the equation `x^2 - 13y^2 = -4` given by (3, 1), (393, 109)
and (36, 10). Each tuple is in the form (x, y), i.e. solution (3, 1) means
that `x = 3` and `y = 1`.
>>> diop_DN(986, 1) # Solves equation x**2 - 986*y**2 = 1
[(49299, 1570)]
See Also
========
find_DN(), diop_bf_DN()
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Pages 16 - 17. [online], Available:
https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
if D < 0:
if N == 0:
return [(0, 0)]
elif N < 0:
return []
elif N > 0:
sol = []
for d in divisors(square_factor(N)):
sols = cornacchia(1, -D, N // d**2)
if sols:
for x, y in sols:
sol.append((d*x, d*y))
if D == -1:
sol.append((d*y, d*x))
return sol
elif D == 0:
if N < 0:
return []
if N == 0:
return [(0, t)]
sN, _exact = integer_nthroot(N, 2)
if _exact:
return [(sN, t)]
else:
return []
else: # D > 0
sD, _exact = integer_nthroot(D, 2)
if _exact:
if N == 0:
return [(sD*t, t)]
else:
sol = []
for y in range(floor(sign(N)*(N - 1)/(2*sD)) + 1):
try:
sq, _exact = integer_nthroot(D*y**2 + N, 2)
except ValueError:
_exact = False
if _exact:
sol.append((sq, y))
return sol
elif 1 < N**2 < D:
# It is much faster to call `_special_diop_DN`.
return _special_diop_DN(D, N)
else:
if N == 0:
return [(0, 0)]
elif abs(N) == 1:
pqa = PQa(0, 1, D)
j = 0
G = []
B = []
for i in pqa:
a = i[2]
G.append(i[5])
B.append(i[4])
if j != 0 and a == 2*sD:
break
j = j + 1
if _odd(j):
if N == -1:
x = G[j - 1]
y = B[j - 1]
else:
count = j
while count < 2*j - 1:
i = next(pqa)
G.append(i[5])
B.append(i[4])
count += 1
x = G[count]
y = B[count]
else:
if N == 1:
x = G[j - 1]
y = B[j - 1]
else:
return []
return [(x, y)]
else:
fs = []
sol = []
div = divisors(N)
for d in div:
if divisible(N, d**2):
fs.append(d)
for f in fs:
m = N // f**2
zs = sqrt_mod(D, abs(m), all_roots=True)
zs = [i for i in zs if i <= abs(m) // 2 ]
if abs(m) != 2:
zs = zs + [-i for i in zs if i] # omit dupl 0
for z in zs:
pqa = PQa(z, abs(m), D)
j = 0
G = []
B = []
for i in pqa:
G.append(i[5])
B.append(i[4])
if j != 0 and abs(i[1]) == 1:
r = G[j-1]
s = B[j-1]
if r**2 - D*s**2 == m:
sol.append((f*r, f*s))
elif diop_DN(D, -1) != []:
a = diop_DN(D, -1)
sol.append((f*(r*a[0][0] + a[0][1]*s*D), f*(r*a[0][1] + s*a[0][0])))
break
j = j + 1
if j == length(z, abs(m), D):
break
return sol
def _special_diop_DN(D, N):
"""
Solves the equation `x^2 - Dy^2 = N` for the special case where
`1 < N**2 < D` and `D` is not a perfect square.
It is better to call `diop_DN` rather than this function, as
the former checks the condition `1 < N**2 < D`, and calls the latter only
if appropriate.
Usage
=====
WARNING: Internal method. Do not call directly!
``_special_diop_DN(D, N)``: D and N are integers as in `x^2 - Dy^2 = N`.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import _special_diop_DN
>>> _special_diop_DN(13, -3) # Solves equation x**2 - 13*y**2 = -3
[(7, 2), (137, 38)]
The output can be interpreted as follows: There are two fundamental
solutions to the equation `x^2 - 13y^2 = -3` given by (7, 2) and
(137, 38). Each tuple is in the form (x, y), i.e. solution (7, 2) means
that `x = 7` and `y = 2`.
>>> _special_diop_DN(2445, -20) # Solves equation x**2 - 2445*y**2 = -20
[(445, 9), (17625560, 356454), (698095554475, 14118073569)]
See Also
========
diop_DN()
References
==========
.. [1] Section 4.4.4 of the following book:
Quadratic Diophantine Equations, T. Andreescu and D. Andrica,
Springer, 2015.
"""
# The following assertion was removed for efficiency, with the understanding
# that this method is not called directly. The parent method, `diop_DN`
# is responsible for performing the appropriate checks.
#
# assert (1 < N**2 < D) and (not integer_nthroot(D, 2)[1])
sqrt_D = sqrt(D)
F = [(N, 1)]
f = 2
while True:
f2 = f**2
if f2 > abs(N):
break
n, r = divmod(N, f2)
if r == 0:
F.append((n, f))
f += 1
P = 0
Q = 1
G0, G1 = 0, 1
B0, B1 = 1, 0
solutions = []
i = 0
while True:
a = floor((P + sqrt_D) / Q)
P = a*Q - P
Q = (D - P**2) // Q
G2 = a*G1 + G0
B2 = a*B1 + B0
for n, f in F:
if G2**2 - D*B2**2 == n:
solutions.append((f*G2, f*B2))
i += 1
if Q == 1 and i % 2 == 0:
break
G0, G1 = G1, G2
B0, B1 = B1, B2
return solutions
def cornacchia(a, b, m):
r"""
Solves `ax^2 + by^2 = m` where `\gcd(a, b) = 1 = gcd(a, m)` and `a, b > 0`.
Explanation
===========
Uses the algorithm due to Cornacchia. The method only finds primitive
solutions, i.e. ones with `\gcd(x, y) = 1`. So this method cannot be used to
find the solutions of `x^2 + y^2 = 20` since the only solution to former is
`(x, y) = (4, 2)` and it is not primitive. When `a = b`, only the
solutions with `x \leq y` are found. For more details, see the References.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import cornacchia
>>> cornacchia(2, 3, 35) # equation 2x**2 + 3y**2 = 35
{(2, 3), (4, 1)}
>>> cornacchia(1, 1, 25) # equation x**2 + y**2 = 25
{(4, 3)}
References
===========
.. [1] A. Nitaj, "L'algorithme de Cornacchia"
.. [2] Solving the diophantine equation ax**2 + by**2 = m by Cornacchia's
method, [online], Available:
http://www.numbertheory.org/php/cornacchia.html
See Also
========
sympy.utilities.iterables.signed_permutations
"""
sols = set()
a1 = igcdex(a, m)[0]
v = sqrt_mod(-b*a1, m, all_roots=True)
if not v:
return None
for t in v:
if t < m // 2:
continue
u, r = t, m
while True:
u, r = r, u % r
if a*r**2 < m:
break
m1 = m - a*r**2
if m1 % b == 0:
m1 = m1 // b
s, _exact = integer_nthroot(m1, 2)
if _exact:
if a == b and r < s:
r, s = s, r
sols.add((int(r), int(s)))
return sols
def PQa(P_0, Q_0, D):
r"""
Returns useful information needed to solve the Pell equation.
Explanation
===========
There are six sequences of integers defined related to the continued
fraction representation of `\\frac{P + \sqrt{D}}{Q}`, namely {`P_{i}`},
{`Q_{i}`}, {`a_{i}`},{`A_{i}`}, {`B_{i}`}, {`G_{i}`}. ``PQa()`` Returns
these values as a 6-tuple in the same order as mentioned above. Refer [1]_
for more detailed information.
Usage
=====
``PQa(P_0, Q_0, D)``: ``P_0``, ``Q_0`` and ``D`` are integers corresponding
to `P_{0}`, `Q_{0}` and `D` in the continued fraction
`\\frac{P_{0} + \sqrt{D}}{Q_{0}}`.
Also it's assumed that `P_{0}^2 == D mod(|Q_{0}|)` and `D` is square free.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import PQa
>>> pqa = PQa(13, 4, 5) # (13 + sqrt(5))/4
>>> next(pqa) # (P_0, Q_0, a_0, A_0, B_0, G_0)
(13, 4, 3, 3, 1, -1)
>>> next(pqa) # (P_1, Q_1, a_1, A_1, B_1, G_1)
(-1, 1, 1, 4, 1, 3)
References
==========
.. [1] Solving the generalized Pell equation x^2 - Dy^2 = N, John P.
Robertson, July 31, 2004, Pages 4 - 8. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
A_i_2 = B_i_1 = 0
A_i_1 = B_i_2 = 1
G_i_2 = -P_0
G_i_1 = Q_0
P_i = P_0
Q_i = Q_0
while True:
a_i = floor((P_i + sqrt(D))/Q_i)
A_i = a_i*A_i_1 + A_i_2
B_i = a_i*B_i_1 + B_i_2
G_i = a_i*G_i_1 + G_i_2
yield P_i, Q_i, a_i, A_i, B_i, G_i
A_i_1, A_i_2 = A_i, A_i_1
B_i_1, B_i_2 = B_i, B_i_1
G_i_1, G_i_2 = G_i, G_i_1
P_i = a_i*Q_i - P_i
Q_i = (D - P_i**2)/Q_i
def diop_bf_DN(D, N, t=symbols("t", integer=True)):
r"""
Uses brute force to solve the equation, `x^2 - Dy^2 = N`.
Explanation
===========
Mainly concerned with the generalized Pell equation which is the case when
`D > 0, D` is not a perfect square. For more information on the case refer
[1]_. Let `(t, u)` be the minimal positive solution of the equation
`x^2 - Dy^2 = 1`. Then this method requires
`\sqrt{\\frac{\mid N \mid (t \pm 1)}{2D}}` to be small.
Usage
=====
``diop_bf_DN(D, N, t)``: ``D`` and ``N`` are coefficients in
`x^2 - Dy^2 = N` and ``t`` is the parameter to be used in the solutions.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
``t`` is the parameter to be used in the solutions.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_bf_DN
>>> diop_bf_DN(13, -4)
[(3, 1), (-3, 1), (36, 10)]
>>> diop_bf_DN(986, 1)
[(49299, 1570)]
See Also
========
diop_DN()
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Page 15. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
D = as_int(D)
N = as_int(N)
sol = []
a = diop_DN(D, 1)
u = a[0][0]
if abs(N) == 1:
return diop_DN(D, N)
elif N > 1:
L1 = 0
L2 = integer_nthroot(int(N*(u - 1)/(2*D)), 2)[0] + 1
elif N < -1:
L1, _exact = integer_nthroot(-int(N/D), 2)
if not _exact:
L1 += 1
L2 = integer_nthroot(-int(N*(u + 1)/(2*D)), 2)[0] + 1
else: # N = 0
if D < 0:
return [(0, 0)]
elif D == 0:
return [(0, t)]
else:
sD, _exact = integer_nthroot(D, 2)
if _exact:
return [(sD*t, t), (-sD*t, t)]
else:
return [(0, 0)]
for y in range(L1, L2):
try:
x, _exact = integer_nthroot(N + D*y**2, 2)
except ValueError:
_exact = False
if _exact:
sol.append((x, y))
if not equivalent(x, y, -x, y, D, N):
sol.append((-x, y))
return sol
def equivalent(u, v, r, s, D, N):
"""
Returns True if two solutions `(u, v)` and `(r, s)` of `x^2 - Dy^2 = N`
belongs to the same equivalence class and False otherwise.
Explanation
===========
Two solutions `(u, v)` and `(r, s)` to the above equation fall to the same
equivalence class iff both `(ur - Dvs)` and `(us - vr)` are divisible by
`N`. See reference [1]_. No test is performed to test whether `(u, v)` and
`(r, s)` are actually solutions to the equation. User should take care of
this.
Usage
=====
``equivalent(u, v, r, s, D, N)``: `(u, v)` and `(r, s)` are two solutions
of the equation `x^2 - Dy^2 = N` and all parameters involved are integers.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import equivalent
>>> equivalent(18, 5, -18, -5, 13, -1)
True
>>> equivalent(3, 1, -18, 393, 109, -4)
False
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Page 12. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
return divisible(u*r - D*v*s, N) and divisible(u*s - v*r, N)
def length(P, Q, D):
r"""
Returns the (length of aperiodic part + length of periodic part) of
continued fraction representation of `\\frac{P + \sqrt{D}}{Q}`.
It is important to remember that this does NOT return the length of the
periodic part but the sum of the lengths of the two parts as mentioned
above.
Usage
=====
``length(P, Q, D)``: ``P``, ``Q`` and ``D`` are integers corresponding to
the continued fraction `\\frac{P + \sqrt{D}}{Q}`.
Details
=======
``P``, ``D`` and ``Q`` corresponds to P, D and Q in the continued fraction,
`\\frac{P + \sqrt{D}}{Q}`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import length
>>> length(-2, 4, 5) # (-2 + sqrt(5))/4
3
>>> length(-5, 4, 17) # (-5 + sqrt(17))/4
4
See Also
========
sympy.ntheory.continued_fraction.continued_fraction_periodic
"""
from sympy.ntheory.continued_fraction import continued_fraction_periodic
v = continued_fraction_periodic(P, Q, D)
if isinstance(v[-1], list):
rpt = len(v[-1])
nonrpt = len(v) - 1
else:
rpt = 0
nonrpt = len(v)
return rpt + nonrpt
def transformation_to_DN(eq):
"""
This function transforms general quadratic,
`ax^2 + bxy + cy^2 + dx + ey + f = 0`
to more easy to deal with `X^2 - DY^2 = N` form.
Explanation
===========
This is used to solve the general quadratic equation by transforming it to
the latter form. Refer to [1]_ for more detailed information on the
transformation. This function returns a tuple (A, B) where A is a 2 X 2
matrix and B is a 2 X 1 matrix such that,
Transpose([x y]) = A * Transpose([X Y]) + B
Usage
=====
``transformation_to_DN(eq)``: where ``eq`` is the quadratic to be
transformed.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import transformation_to_DN
>>> A, B = transformation_to_DN(x**2 - 3*x*y - y**2 - 2*y + 1)
>>> A
Matrix([
[1/26, 3/26],
[ 0, 1/13]])
>>> B
Matrix([
[-6/13],
[-4/13]])
A, B returned are such that Transpose((x y)) = A * Transpose((X Y)) + B.
Substituting these values for `x` and `y` and a bit of simplifying work
will give an equation of the form `x^2 - Dy^2 = N`.
>>> from sympy.abc import X, Y
>>> from sympy import Matrix, simplify
>>> u = (A*Matrix([X, Y]) + B)[0] # Transformation for x
>>> u
X/26 + 3*Y/26 - 6/13
>>> v = (A*Matrix([X, Y]) + B)[1] # Transformation for y
>>> v
Y/13 - 4/13
Next we will substitute these formulas for `x` and `y` and do
``simplify()``.
>>> eq = simplify((x**2 - 3*x*y - y**2 - 2*y + 1).subs(zip((x, y), (u, v))))
>>> eq
X**2/676 - Y**2/52 + 17/13
By multiplying the denominator appropriately, we can get a Pell equation
in the standard form.
>>> eq * 676
X**2 - 13*Y**2 + 884
If only the final equation is needed, ``find_DN()`` can be used.
See Also
========
find_DN()
References
==========
.. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0,
John P.Robertson, May 8, 2003, Page 7 - 11.
https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return _transformation_to_DN(var, coeff)
def _transformation_to_DN(var, coeff):
x, y = var
a = coeff[x**2]
b = coeff[x*y]
c = coeff[y**2]
d = coeff[x]
e = coeff[y]
f = coeff[1]
a, b, c, d, e, f = [as_int(i) for i in _remove_gcd(a, b, c, d, e, f)]
X, Y = symbols("X, Y", integer=True)
if b:
B, C = _rational_pq(2*a, b)
A, T = _rational_pq(a, B**2)
# eq_1 = A*B*X**2 + B*(c*T - A*C**2)*Y**2 + d*T*X + (B*e*T - d*T*C)*Y + f*T*B
coeff = {X**2: A*B, X*Y: 0, Y**2: B*(c*T - A*C**2), X: d*T, Y: B*e*T - d*T*C, 1: f*T*B}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*A_0, Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*B_0
else:
if d:
B, C = _rational_pq(2*a, d)
A, T = _rational_pq(a, B**2)
# eq_2 = A*X**2 + c*T*Y**2 + e*T*Y + f*T - A*C**2
coeff = {X**2: A, X*Y: 0, Y**2: c*T, X: 0, Y: e*T, 1: f*T - A*C**2}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [S.One/B, 0, 0, 1])*A_0, Matrix(2, 2, [S.One/B, 0, 0, 1])*B_0 + Matrix([-S(C)/B, 0])
else:
if e:
B, C = _rational_pq(2*c, e)
A, T = _rational_pq(c, B**2)
# eq_3 = a*T*X**2 + A*Y**2 + f*T - A*C**2
coeff = {X**2: a*T, X*Y: 0, Y**2: A, X: 0, Y: 0, 1: f*T - A*C**2}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [1, 0, 0, S.One/B])*A_0, Matrix(2, 2, [1, 0, 0, S.One/B])*B_0 + Matrix([0, -S(C)/B])
else:
# TODO: pre-simplification: Not necessary but may simplify
# the equation.
return Matrix(2, 2, [S.One/a, 0, 0, 1]), Matrix([0, 0])
def find_DN(eq):
"""
This function returns a tuple, `(D, N)` of the simplified form,
`x^2 - Dy^2 = N`, corresponding to the general quadratic,
`ax^2 + bxy + cy^2 + dx + ey + f = 0`.
Solving the general quadratic is then equivalent to solving the equation
`X^2 - DY^2 = N` and transforming the solutions by using the transformation
matrices returned by ``transformation_to_DN()``.
Usage
=====
``find_DN(eq)``: where ``eq`` is the quadratic to be transformed.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import find_DN
>>> find_DN(x**2 - 3*x*y - y**2 - 2*y + 1)
(13, -884)
Interpretation of the output is that we get `X^2 -13Y^2 = -884` after
transforming `x^2 - 3xy - y^2 - 2y + 1` using the transformation returned
by ``transformation_to_DN()``.
See Also
========
transformation_to_DN()
References
==========
.. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0,
John P.Robertson, May 8, 2003, Page 7 - 11.
https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return _find_DN(var, coeff)
def _find_DN(var, coeff):
x, y = var
X, Y = symbols("X, Y", integer=True)
A, B = _transformation_to_DN(var, coeff)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
eq = x**2*coeff[x**2] + x*y*coeff[x*y] + y**2*coeff[y**2] + x*coeff[x] + y*coeff[y] + coeff[1]
simplified = _mexpand(eq.subs(zip((x, y), (u, v))))
coeff = simplified.as_coefficients_dict()
return -coeff[Y**2]/coeff[X**2], -coeff[1]/coeff[X**2]
def check_param(x, y, a, params):
"""
If there is a number modulo ``a`` such that ``x`` and ``y`` are both
integers, then return a parametric representation for ``x`` and ``y``
else return (None, None).
Here ``x`` and ``y`` are functions of ``t``.
"""
from sympy.simplify.simplify import clear_coefficients
if x.is_number and not x.is_Integer:
return DiophantineSolutionSet([x, y], parameters=params)
if y.is_number and not y.is_Integer:
return DiophantineSolutionSet([x, y], parameters=params)
m, n = symbols("m, n", integer=True)
c, p = (m*x + n*y).as_content_primitive()
if a % c.q:
return DiophantineSolutionSet([x, y], parameters=params)
# clear_coefficients(mx + b, R)[1] -> (R - b)/m
eq = clear_coefficients(x, m)[1] - clear_coefficients(y, n)[1]
junk, eq = eq.as_content_primitive()
return _diop_solve(eq, params=params)
def diop_ternary_quadratic(eq, parameterize=False):
"""
Solves the general quadratic ternary form,
`ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`.
Returns a tuple `(x, y, z)` which is a base solution for the above
equation. If there are no solutions, `(None, None, None)` is returned.
Usage
=====
``diop_ternary_quadratic(eq)``: Return a tuple containing a basic solution
to ``eq``.
Details
=======
``eq`` should be an homogeneous expression of degree two in three variables
and it is assumed to be zero.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic
>>> diop_ternary_quadratic(x**2 + 3*y**2 - z**2)
(1, 0, 1)
>>> diop_ternary_quadratic(4*x**2 + 5*y**2 - z**2)
(1, 0, 2)
>>> diop_ternary_quadratic(45*x**2 - 7*y**2 - 8*x*y - z**2)
(28, 45, 105)
>>> diop_ternary_quadratic(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y)
(9, 1, 5)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name):
sol = _diop_ternary_quadratic(var, coeff)
if len(sol) > 0:
x_0, y_0, z_0 = list(sol)[0]
else:
x_0, y_0, z_0 = None, None, None
if parameterize:
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
return x_0, y_0, z_0
def _diop_ternary_quadratic(_var, coeff):
eq = sum([i*coeff[i] for i in coeff])
if HomogeneousTernaryQuadratic(eq).matches():
return HomogeneousTernaryQuadratic(eq, free_symbols=_var).solve()
elif HomogeneousTernaryQuadraticNormal(eq).matches():
return HomogeneousTernaryQuadraticNormal(eq, free_symbols=_var).solve()
def transformation_to_normal(eq):
"""
Returns the transformation Matrix that converts a general ternary
quadratic equation ``eq`` (`ax^2 + by^2 + cz^2 + dxy + eyz + fxz`)
to a form without cross terms: `ax^2 + by^2 + cz^2 = 0`. This is
not used in solving ternary quadratics; it is only implemented for
the sake of completeness.
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
return _transformation_to_normal(var, coeff)
def _transformation_to_normal(var, coeff):
_var = list(var) # copy
x, y, z = var
if not any(coeff[i**2] for i in var):
# https://math.stackexchange.com/questions/448051/transform-quadratic-ternary-form-to-normal-form/448065#448065
a = coeff[x*y]
b = coeff[y*z]
c = coeff[x*z]
swap = False
if not a: # b can't be 0 or else there aren't 3 vars
swap = True
a, b = b, a
T = Matrix(((1, 1, -b/a), (1, -1, -c/a), (0, 0, 1)))
if swap:
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
if coeff[x**2] == 0:
# If the coefficient of x is zero change the variables
if coeff[y**2] == 0:
_var[0], _var[2] = var[2], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 2)
T.col_swap(0, 2)
return T
else:
_var[0], _var[1] = var[1], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
# Apply the transformation x --> X - (B*Y + C*Z)/(2*A)
if coeff[x*y] != 0 or coeff[x*z] != 0:
A = coeff[x**2]
B = coeff[x*y]
C = coeff[x*z]
D = coeff[y**2]
E = coeff[y*z]
F = coeff[z**2]
_coeff = dict()
_coeff[x**2] = 4*A**2
_coeff[y**2] = 4*A*D - B**2
_coeff[z**2] = 4*A*F - C**2
_coeff[y*z] = 4*A*E - 2*B*C
_coeff[x*y] = 0
_coeff[x*z] = 0
T_0 = _transformation_to_normal(_var, _coeff)
return Matrix(3, 3, [1, S(-B)/(2*A), S(-C)/(2*A), 0, 1, 0, 0, 0, 1])*T_0
elif coeff[y*z] != 0:
if coeff[y**2] == 0:
if coeff[z**2] == 0:
# Equations of the form A*x**2 + E*yz = 0.
# Apply transformation y -> Y + Z ans z -> Y - Z
return Matrix(3, 3, [1, 0, 0, 0, 1, 1, 0, 1, -1])
else:
# Ax**2 + E*y*z + F*z**2 = 0
_var[0], _var[2] = var[2], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 2)
T.col_swap(0, 2)
return T
else:
# A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, F may be zero
_var[0], _var[1] = var[1], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
else:
return Matrix.eye(3)
def parametrize_ternary_quadratic(eq):
"""
Returns the parametrized general solution for the ternary quadratic
equation ``eq`` which has the form
`ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`.
Examples
========
>>> from sympy import Tuple, ordered
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import parametrize_ternary_quadratic
The parametrized solution may be returned with three parameters:
>>> parametrize_ternary_quadratic(2*x**2 + y**2 - 2*z**2)
(p**2 - 2*q**2, -2*p**2 + 4*p*q - 4*p*r - 4*q**2, p**2 - 4*p*q + 2*q**2 - 4*q*r)
There might also be only two parameters:
>>> parametrize_ternary_quadratic(4*x**2 + 2*y**2 - 3*z**2)
(2*p**2 - 3*q**2, -4*p**2 + 12*p*q - 6*q**2, 4*p**2 - 8*p*q + 6*q**2)
Notes
=====
Consider ``p`` and ``q`` in the previous 2-parameter
solution and observe that more than one solution can be represented
by a given pair of parameters. If `p` and ``q`` are not coprime, this is
trivially true since the common factor will also be a common factor of the
solution values. But it may also be true even when ``p`` and
``q`` are coprime:
>>> sol = Tuple(*_)
>>> p, q = ordered(sol.free_symbols)
>>> sol.subs([(p, 3), (q, 2)])
(6, 12, 12)
>>> sol.subs([(q, 1), (p, 1)])
(-1, 2, 2)
>>> sol.subs([(q, 0), (p, 1)])
(2, -4, 4)
>>> sol.subs([(q, 1), (p, 0)])
(-3, -6, 6)
Except for sign and a common factor, these are equivalent to
the solution of (1, 2, 2).
References
==========
.. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart,
London Mathematical Society Student Texts 41, Cambridge University
Press, Cambridge, 1998.
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
x_0, y_0, z_0 = list(_diop_ternary_quadratic(var, coeff))[0]
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
def _parametrize_ternary_quadratic(solution, _var, coeff):
# called for a*x**2 + b*y**2 + c*z**2 + d*x*y + e*y*z + f*x*z = 0
assert 1 not in coeff
x_0, y_0, z_0 = solution
v = list(_var) # copy
if x_0 is None:
return (None, None, None)
if solution.count(0) >= 2:
# if there are 2 zeros the equation reduces
# to k*X**2 == 0 where X is x, y, or z so X must
# be zero, too. So there is only the trivial
# solution.
return (None, None, None)
if x_0 == 0:
v[0], v[1] = v[1], v[0]
y_p, x_p, z_p = _parametrize_ternary_quadratic(
(y_0, x_0, z_0), v, coeff)
return x_p, y_p, z_p
x, y, z = v
r, p, q = symbols("r, p, q", integer=True)
eq = sum(k*v for k, v in coeff.items())
eq_1 = _mexpand(eq.subs(zip(
(x, y, z), (r*x_0, r*y_0 + p, r*z_0 + q))))
A, B = eq_1.as_independent(r, as_Add=True)
x = A*x_0
y = (A*y_0 - _mexpand(B/r*p))
z = (A*z_0 - _mexpand(B/r*q))
return _remove_gcd(x, y, z)
def diop_ternary_quadratic_normal(eq, parameterize=False):
"""
Solves the quadratic ternary diophantine equation,
`ax^2 + by^2 + cz^2 = 0`.
Explanation
===========
Here the coefficients `a`, `b`, and `c` should be non zero. Otherwise the
equation will be a quadratic binary or univariate equation. If solvable,
returns a tuple `(x, y, z)` that satisfies the given equation. If the
equation does not have integer solutions, `(None, None, None)` is returned.
Usage
=====
``diop_ternary_quadratic_normal(eq)``: where ``eq`` is an equation of the form
`ax^2 + by^2 + cz^2 = 0`.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic_normal
>>> diop_ternary_quadratic_normal(x**2 + 3*y**2 - z**2)
(1, 0, 1)
>>> diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2)
(1, 0, 2)
>>> diop_ternary_quadratic_normal(34*x**2 - 3*y**2 - 301*z**2)
(4, 9, 1)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == HomogeneousTernaryQuadraticNormal.name:
sol = _diop_ternary_quadratic_normal(var, coeff)
if len(sol) > 0:
x_0, y_0, z_0 = list(sol)[0]
else:
x_0, y_0, z_0 = None, None, None
if parameterize:
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
return x_0, y_0, z_0
def _diop_ternary_quadratic_normal(var, coeff):
eq = sum([i * coeff[i] for i in coeff])
return HomogeneousTernaryQuadraticNormal(eq, free_symbols=var).solve()
def sqf_normal(a, b, c, steps=False):
"""
Return `a', b', c'`, the coefficients of the square-free normal
form of `ax^2 + by^2 + cz^2 = 0`, where `a', b', c'` are pairwise
prime. If `steps` is True then also return three tuples:
`sq`, `sqf`, and `(a', b', c')` where `sq` contains the square
factors of `a`, `b` and `c` after removing the `gcd(a, b, c)`;
`sqf` contains the values of `a`, `b` and `c` after removing
both the `gcd(a, b, c)` and the square factors.
The solutions for `ax^2 + by^2 + cz^2 = 0` can be
recovered from the solutions of `a'x^2 + b'y^2 + c'z^2 = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sqf_normal
>>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11)
(11, 1, 5)
>>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11, True)
((3, 1, 7), (5, 55, 11), (11, 1, 5))
References
==========
.. [1] Legendre's Theorem, Legrange's Descent,
http://public.csusm.edu/aitken_html/notes/legendre.pdf
See Also
========
reconstruct()
"""
ABC = _remove_gcd(a, b, c)
sq = tuple(square_factor(i) for i in ABC)
sqf = A, B, C = tuple([i//j**2 for i,j in zip(ABC, sq)])
pc = igcd(A, B)
A /= pc
B /= pc
pa = igcd(B, C)
B /= pa
C /= pa
pb = igcd(A, C)
A /= pb
B /= pb
A *= pa
B *= pb
C *= pc
if steps:
return (sq, sqf, (A, B, C))
else:
return A, B, C
def square_factor(a):
r"""
Returns an integer `c` s.t. `a = c^2k, \ c,k \in Z`. Here `k` is square
free. `a` can be given as an integer or a dictionary of factors.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import square_factor
>>> square_factor(24)
2
>>> square_factor(-36*3)
6
>>> square_factor(1)
1
>>> square_factor({3: 2, 2: 1, -1: 1}) # -18
3
See Also
========
sympy.ntheory.factor_.core
"""
f = a if isinstance(a, dict) else factorint(a)
return Mul(*[p**(e//2) for p, e in f.items()])
def reconstruct(A, B, z):
"""
Reconstruct the `z` value of an equivalent solution of `ax^2 + by^2 + cz^2`
from the `z` value of a solution of the square-free normal form of the
equation, `a'*x^2 + b'*y^2 + c'*z^2`, where `a'`, `b'` and `c'` are square
free and `gcd(a', b', c') == 1`.
"""
f = factorint(igcd(A, B))
for p, e in f.items():
if e != 1:
raise ValueError('a and b should be square-free')
z *= p
return z
def ldescent(A, B):
"""
Return a non-trivial solution to `w^2 = Ax^2 + By^2` using
Lagrange's method; return None if there is no such solution.
.
Here, `A \\neq 0` and `B \\neq 0` and `A` and `B` are square free. Output a
tuple `(w_0, x_0, y_0)` which is a solution to the above equation.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import ldescent
>>> ldescent(1, 1) # w^2 = x^2 + y^2
(1, 1, 0)
>>> ldescent(4, -7) # w^2 = 4x^2 - 7y^2
(2, -1, 0)
This means that `x = -1, y = 0` and `w = 2` is a solution to the equation
`w^2 = 4x^2 - 7y^2`
>>> ldescent(5, -1) # w^2 = 5x^2 - y^2
(2, 1, -1)
References
==========
.. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart,
London Mathematical Society Student Texts 41, Cambridge University
Press, Cambridge, 1998.
.. [2] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
[online], Available:
http://eprints.nottingham.ac.uk/60/1/kvxefz87.pdf
"""
if abs(A) > abs(B):
w, y, x = ldescent(B, A)
return w, x, y
if A == 1:
return (1, 1, 0)
if B == 1:
return (1, 0, 1)
if B == -1: # and A == -1
return
r = sqrt_mod(A, B)
Q = (r**2 - A) // B
if Q == 0:
B_0 = 1
d = 0
else:
div = divisors(Q)
B_0 = None
for i in div:
sQ, _exact = integer_nthroot(abs(Q) // i, 2)
if _exact:
B_0, d = sign(Q)*i, sQ
break
if B_0 is not None:
W, X, Y = ldescent(A, B_0)
return _remove_gcd((-A*X + r*W), (r*X - W), Y*(B_0*d))
def descent(A, B):
"""
Returns a non-trivial solution, (x, y, z), to `x^2 = Ay^2 + Bz^2`
using Lagrange's descent method with lattice-reduction. `A` and `B`
are assumed to be valid for such a solution to exist.
This is faster than the normal Lagrange's descent algorithm because
the Gaussian reduction is used.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import descent
>>> descent(3, 1) # x**2 = 3*y**2 + z**2
(1, 0, 1)
`(x, y, z) = (1, 0, 1)` is a solution to the above equation.
>>> descent(41, -113)
(-16, -3, 1)
References
==========
.. [1] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
"""
if abs(A) > abs(B):
x, y, z = descent(B, A)
return x, z, y
if B == 1:
return (1, 0, 1)
if A == 1:
return (1, 1, 0)
if B == -A:
return (0, 1, 1)
if B == A:
x, z, y = descent(-1, A)
return (A*y, z, x)
w = sqrt_mod(A, B)
x_0, z_0 = gaussian_reduce(w, A, B)
t = (x_0**2 - A*z_0**2) // B
t_2 = square_factor(t)
t_1 = t // t_2**2
x_1, z_1, y_1 = descent(A, t_1)
return _remove_gcd(x_0*x_1 + A*z_0*z_1, z_0*x_1 + x_0*z_1, t_1*t_2*y_1)
def gaussian_reduce(w, a, b):
r"""
Returns a reduced solution `(x, z)` to the congruence
`X^2 - aZ^2 \equiv 0 \ (mod \ b)` so that `x^2 + |a|z^2` is minimal.
Details
=======
Here ``w`` is a solution of the congruence `x^2 \equiv a \ (mod \ b)`
References
==========
.. [1] Gaussian lattice Reduction [online]. Available:
http://home.ie.cuhk.edu.hk/~wkshum/wordpress/?p=404
.. [2] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
"""
u = (0, 1)
v = (1, 0)
if dot(u, v, w, a, b) < 0:
v = (-v[0], -v[1])
if norm(u, w, a, b) < norm(v, w, a, b):
u, v = v, u
while norm(u, w, a, b) > norm(v, w, a, b):
k = dot(u, v, w, a, b) // dot(v, v, w, a, b)
u, v = v, (u[0]- k*v[0], u[1]- k*v[1])
u, v = v, u
if dot(u, v, w, a, b) < dot(v, v, w, a, b)/2 or norm((u[0]-v[0], u[1]-v[1]), w, a, b) > norm(v, w, a, b):
c = v
else:
c = (u[0] - v[0], u[1] - v[1])
return c[0]*w + b*c[1], c[0]
def dot(u, v, w, a, b):
r"""
Returns a special dot product of the vectors `u = (u_{1}, u_{2})` and
`v = (v_{1}, v_{2})` which is defined in order to reduce solution of
the congruence equation `X^2 - aZ^2 \equiv 0 \ (mod \ b)`.
"""
u_1, u_2 = u
v_1, v_2 = v
return (w*u_1 + b*u_2)*(w*v_1 + b*v_2) + abs(a)*u_1*v_1
def norm(u, w, a, b):
r"""
Returns the norm of the vector `u = (u_{1}, u_{2})` under the dot product
defined by `u \cdot v = (wu_{1} + bu_{2})(w*v_{1} + bv_{2}) + |a|*u_{1}*v_{1}`
where `u = (u_{1}, u_{2})` and `v = (v_{1}, v_{2})`.
"""
u_1, u_2 = u
return sqrt(dot((u_1, u_2), (u_1, u_2), w, a, b))
def holzer(x, y, z, a, b, c):
r"""
Simplify the solution `(x, y, z)` of the equation
`ax^2 + by^2 = cz^2` with `a, b, c > 0` and `z^2 \geq \mid ab \mid` to
a new reduced solution `(x', y', z')` such that `z'^2 \leq \mid ab \mid`.
The algorithm is an interpretation of Mordell's reduction as described
on page 8 of Cremona and Rusin's paper [1]_ and the work of Mordell in
reference [2]_.
References
==========
.. [1] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
.. [2] Diophantine Equations, L. J. Mordell, page 48.
"""
if _odd(c):
k = 2*c
else:
k = c//2
small = a*b*c
step = 0
while True:
t1, t2, t3 = a*x**2, b*y**2, c*z**2
# check that it's a solution
if t1 + t2 != t3:
if step == 0:
raise ValueError('bad starting solution')
break
x_0, y_0, z_0 = x, y, z
if max(t1, t2, t3) <= small:
# Holzer condition
break
uv = u, v = base_solution_linear(k, y_0, -x_0)
if None in uv:
break
p, q = -(a*u*x_0 + b*v*y_0), c*z_0
r = Rational(p, q)
if _even(c):
w = _nint_or_floor(p, q)
assert abs(w - r) <= S.Half
else:
w = p//q # floor
if _odd(a*u + b*v + c*w):
w += 1
assert abs(w - r) <= S.One
A = (a*u**2 + b*v**2 + c*w**2)
B = (a*u*x_0 + b*v*y_0 + c*w*z_0)
x = Rational(x_0*A - 2*u*B, k)
y = Rational(y_0*A - 2*v*B, k)
z = Rational(z_0*A - 2*w*B, k)
assert all(i.is_Integer for i in (x, y, z))
step += 1
return tuple([int(i) for i in (x_0, y_0, z_0)])
def diop_general_pythagorean(eq, param=symbols("m", integer=True)):
"""
Solves the general pythagorean equation,
`a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`.
Returns a tuple which contains a parametrized solution to the equation,
sorted in the same order as the input variables.
Usage
=====
``diop_general_pythagorean(eq, param)``: where ``eq`` is a general
pythagorean equation which is assumed to be zero and ``param`` is the base
parameter used to construct other parameters by subscripting.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_pythagorean
>>> from sympy.abc import a, b, c, d, e
>>> diop_general_pythagorean(a**2 + b**2 + c**2 - d**2)
(m1**2 + m2**2 - m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2)
>>> diop_general_pythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2)
(10*m1**2 + 10*m2**2 + 10*m3**2 - 10*m4**2, 15*m1**2 + 15*m2**2 + 15*m3**2 + 15*m4**2, 15*m1*m4, 12*m2*m4, 60*m3*m4)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralPythagorean.name:
if param is None:
params = None
else:
params = symbols('%s1:%i' % (param, len(var)), integer=True)
return list(GeneralPythagorean(eq).solve(parameters=params))[0]
def diop_general_sum_of_squares(eq, limit=1):
r"""
Solves the equation `x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Returns at most ``limit`` number of solutions.
Usage
=====
``general_sum_of_squares(eq, limit)`` : Here ``eq`` is an expression which
is assumed to be zero. Also, ``eq`` should be in the form,
`x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Details
=======
When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be
no solutions. Refer to [1]_ for more details.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_squares
>>> from sympy.abc import a, b, c, d, e
>>> diop_general_sum_of_squares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345)
{(15, 22, 22, 24, 24)}
Reference
=========
.. [1] Representing an integer as a sum of three squares, [online],
Available:
http://www.proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralSumOfSquares.name:
return set(GeneralSumOfSquares(eq).solve(limit=limit))
def diop_general_sum_of_even_powers(eq, limit=1):
"""
Solves the equation `x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`
where `e` is an even, integer power.
Returns at most ``limit`` number of solutions.
Usage
=====
``general_sum_of_even_powers(eq, limit)`` : Here ``eq`` is an expression which
is assumed to be zero. Also, ``eq`` should be in the form,
`x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_even_powers
>>> from sympy.abc import a, b
>>> diop_general_sum_of_even_powers(a**4 + b**4 - (2**4 + 3**4))
{(2, 3)}
See Also
========
power_representation
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralSumOfEvenPowers.name:
return set(GeneralSumOfEvenPowers(eq).solve(limit=limit))
## Functions below this comment can be more suitably grouped under
## an Additive number theory module rather than the Diophantine
## equation module.
def partition(n, k=None, zeros=False):
"""
Returns a generator that can be used to generate partitions of an integer
`n`.
Explanation
===========
A partition of `n` is a set of positive integers which add up to `n`. For
example, partitions of 3 are 3, 1 + 2, 1 + 1 + 1. A partition is returned
as a tuple. If ``k`` equals None, then all possible partitions are returned
irrespective of their size, otherwise only the partitions of size ``k`` are
returned. If the ``zero`` parameter is set to True then a suitable
number of zeros are added at the end of every partition of size less than
``k``.
``zero`` parameter is considered only if ``k`` is not None. When the
partitions are over, the last `next()` call throws the ``StopIteration``
exception, so this function should always be used inside a try - except
block.
Details
=======
``partition(n, k)``: Here ``n`` is a positive integer and ``k`` is the size
of the partition which is also positive integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import partition
>>> f = partition(5)
>>> next(f)
(1, 1, 1, 1, 1)
>>> next(f)
(1, 1, 1, 2)
>>> g = partition(5, 3)
>>> next(g)
(1, 1, 3)
>>> next(g)
(1, 2, 2)
>>> g = partition(5, 3, zeros=True)
>>> next(g)
(0, 0, 5)
"""
if not zeros or k is None:
for i in ordered_partitions(n, k):
yield tuple(i)
else:
for m in range(1, k + 1):
for i in ordered_partitions(n, m):
i = tuple(i)
yield (0,)*(k - len(i)) + i
def prime_as_sum_of_two_squares(p):
"""
Represent a prime `p` as a unique sum of two squares; this can
only be done if the prime is congruent to 1 mod 4.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import prime_as_sum_of_two_squares
>>> prime_as_sum_of_two_squares(7) # can't be done
>>> prime_as_sum_of_two_squares(5)
(1, 2)
Reference
=========
.. [1] Representing a number as a sum of four squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
if not p % 4 == 1:
return
if p % 8 == 5:
b = 2
else:
b = 3
while pow(b, (p - 1) // 2, p) == 1:
b = nextprime(b)
b = pow(b, (p - 1) // 4, p)
a = p
while b**2 > p:
a, b = b, a % b
return (int(a % b), int(b)) # convert from long
def sum_of_three_squares(n):
r"""
Returns a 3-tuple $(a, b, c)$ such that $a^2 + b^2 + c^2 = n$ and
$a, b, c \geq 0$.
Returns None if $n = 4^a(8m + 7)$ for some `a, m \in \mathbb{Z}`. See
[1]_ for more details.
Usage
=====
``sum_of_three_squares(n)``: Here ``n`` is a non-negative integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_three_squares
>>> sum_of_three_squares(44542)
(18, 37, 207)
References
==========
.. [1] Representing a number as a sum of three squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
special = {1:(1, 0, 0), 2:(1, 1, 0), 3:(1, 1, 1), 10: (1, 3, 0), 34: (3, 3, 4), 58:(3, 7, 0),
85:(6, 7, 0), 130:(3, 11, 0), 214:(3, 6, 13), 226:(8, 9, 9), 370:(8, 9, 15),
526:(6, 7, 21), 706:(15, 15, 16), 730:(1, 27, 0), 1414:(6, 17, 33), 1906:(13, 21, 36),
2986: (21, 32, 39), 9634: (56, 57, 57)}
v = 0
if n == 0:
return (0, 0, 0)
v = multiplicity(4, n)
n //= 4**v
if n % 8 == 7:
return
if n in special.keys():
x, y, z = special[n]
return _sorted_tuple(2**v*x, 2**v*y, 2**v*z)
s, _exact = integer_nthroot(n, 2)
if _exact:
return (2**v*s, 0, 0)
x = None
if n % 8 == 3:
s = s if _odd(s) else s - 1
for x in range(s, -1, -2):
N = (n - x**2) // 2
if isprime(N):
y, z = prime_as_sum_of_two_squares(N)
return _sorted_tuple(2**v*x, 2**v*(y + z), 2**v*abs(y - z))
return
if n % 8 in (2, 6):
s = s if _odd(s) else s - 1
else:
s = s - 1 if _odd(s) else s
for x in range(s, -1, -2):
N = n - x**2
if isprime(N):
y, z = prime_as_sum_of_two_squares(N)
return _sorted_tuple(2**v*x, 2**v*y, 2**v*z)
def sum_of_four_squares(n):
r"""
Returns a 4-tuple `(a, b, c, d)` such that `a^2 + b^2 + c^2 + d^2 = n`.
Here `a, b, c, d \geq 0`.
Usage
=====
``sum_of_four_squares(n)``: Here ``n`` is a non-negative integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_four_squares
>>> sum_of_four_squares(3456)
(8, 8, 32, 48)
>>> sum_of_four_squares(1294585930293)
(0, 1234, 2161, 1137796)
References
==========
.. [1] Representing a number as a sum of four squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
if n == 0:
return (0, 0, 0, 0)
v = multiplicity(4, n)
n //= 4**v
if n % 8 == 7:
d = 2
n = n - 4
elif n % 8 in (2, 6):
d = 1
n = n - 1
else:
d = 0
x, y, z = sum_of_three_squares(n)
return _sorted_tuple(2**v*d, 2**v*x, 2**v*y, 2**v*z)
def power_representation(n, p, k, zeros=False):
r"""
Returns a generator for finding k-tuples of integers,
`(n_{1}, n_{2}, . . . n_{k})`, such that
`n = n_{1}^p + n_{2}^p + . . . n_{k}^p`.
Usage
=====
``power_representation(n, p, k, zeros)``: Represent non-negative number
``n`` as a sum of ``k`` ``p``\ th powers. If ``zeros`` is true, then the
solutions is allowed to contain zeros.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import power_representation
Represent 1729 as a sum of two cubes:
>>> f = power_representation(1729, 3, 2)
>>> next(f)
(9, 10)
>>> next(f)
(1, 12)
If the flag `zeros` is True, the solution may contain tuples with
zeros; any such solutions will be generated after the solutions
without zeros:
>>> list(power_representation(125, 2, 3, zeros=True))
[(5, 6, 8), (3, 4, 10), (0, 5, 10), (0, 2, 11)]
For even `p` the `permute_sign` function can be used to get all
signed values:
>>> from sympy.utilities.iterables import permute_signs
>>> list(permute_signs((1, 12)))
[(1, 12), (-1, 12), (1, -12), (-1, -12)]
All possible signed permutations can also be obtained:
>>> from sympy.utilities.iterables import signed_permutations
>>> list(signed_permutations((1, 12)))
[(1, 12), (-1, 12), (1, -12), (-1, -12), (12, 1), (-12, 1), (12, -1), (-12, -1)]
"""
n, p, k = [as_int(i) for i in (n, p, k)]
if n < 0:
if p % 2:
for t in power_representation(-n, p, k, zeros):
yield tuple(-i for i in t)
return
if p < 1 or k < 1:
raise ValueError(filldedent('''
Expecting positive integers for `(p, k)`, but got `(%s, %s)`'''
% (p, k)))
if n == 0:
if zeros:
yield (0,)*k
return
if k == 1:
if p == 1:
yield (n,)
else:
be = perfect_power(n)
if be:
b, e = be
d, r = divmod(e, p)
if not r:
yield (b**d,)
return
if p == 1:
for t in partition(n, k, zeros=zeros):
yield t
return
if p == 2:
feasible = _can_do_sum_of_squares(n, k)
if not feasible:
return
if not zeros and n > 33 and k >= 5 and k <= n and n - k in (
13, 10, 7, 5, 4, 2, 1):
'''Todd G. Will, "When Is n^2 a Sum of k Squares?", [online].
Available: https://www.maa.org/sites/default/files/Will-MMz-201037918.pdf'''
return
if feasible is not True: # it's prime and k == 2
yield prime_as_sum_of_two_squares(n)
return
if k == 2 and p > 2:
be = perfect_power(n)
if be and be[1] % p == 0:
return # Fermat: a**n + b**n = c**n has no solution for n > 2
if n >= k:
a = integer_nthroot(n - (k - 1), p)[0]
for t in pow_rep_recursive(a, k, n, [], p):
yield tuple(reversed(t))
if zeros:
a = integer_nthroot(n, p)[0]
for i in range(1, k):
for t in pow_rep_recursive(a, i, n, [], p):
yield tuple(reversed(t + (0,)*(k - i)))
sum_of_powers = power_representation
def pow_rep_recursive(n_i, k, n_remaining, terms, p):
if k == 0 and n_remaining == 0:
yield tuple(terms)
else:
if n_i >= 1 and k > 0:
yield from pow_rep_recursive(n_i - 1, k, n_remaining, terms, p)
residual = n_remaining - pow(n_i, p)
if residual >= 0:
yield from pow_rep_recursive(n_i, k - 1, residual, terms + [n_i], p)
def sum_of_squares(n, k, zeros=False):
"""Return a generator that yields the k-tuples of nonnegative
values, the squares of which sum to n. If zeros is False (default)
then the solution will not contain zeros. The nonnegative
elements of a tuple are sorted.
* If k == 1 and n is square, (n,) is returned.
* If k == 2 then n can only be written as a sum of squares if
every prime in the factorization of n that has the form
4*k + 3 has an even multiplicity. If n is prime then
it can only be written as a sum of two squares if it is
in the form 4*k + 1.
* if k == 3 then n can be written as a sum of squares if it does
not have the form 4**m*(8*k + 7).
* all integers can be written as the sum of 4 squares.
* if k > 4 then n can be partitioned and each partition can
be written as a sum of 4 squares; if n is not evenly divisible
by 4 then n can be written as a sum of squares only if the
an additional partition can be written as sum of squares.
For example, if k = 6 then n is partitioned into two parts,
the first being written as a sum of 4 squares and the second
being written as a sum of 2 squares -- which can only be
done if the condition above for k = 2 can be met, so this will
automatically reject certain partitions of n.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_squares
>>> list(sum_of_squares(25, 2))
[(3, 4)]
>>> list(sum_of_squares(25, 2, True))
[(3, 4), (0, 5)]
>>> list(sum_of_squares(25, 4))
[(1, 2, 2, 4)]
See Also
========
sympy.utilities.iterables.signed_permutations
"""
yield from power_representation(n, 2, k, zeros)
def _can_do_sum_of_squares(n, k):
"""Return True if n can be written as the sum of k squares,
False if it cannot, or 1 if ``k == 2`` and ``n`` is prime (in which
case it *can* be written as a sum of two squares). A False
is returned only if it cannot be written as ``k``-squares, even
if 0s are allowed.
"""
if k < 1:
return False
if n < 0:
return False
if n == 0:
return True
if k == 1:
return is_square(n)
if k == 2:
if n in (1, 2):
return True
if isprime(n):
if n % 4 == 1:
return 1 # signal that it was prime
return False
else:
f = factorint(n)
for p, m in f.items():
# we can proceed iff no prime factor in the form 4*k + 3
# has an odd multiplicity
if (p % 4 == 3) and m % 2:
return False
return True
if k == 3:
if (n//4**multiplicity(4, n)) % 8 == 7:
return False
# every number can be written as a sum of 4 squares; for k > 4 partitions
# can be 0
return True
|
17d1a09741606d589f24d3171c2d6e1a47cdcf101e5acb2035b83d9f6d64d053 | r"""
This module contains :py:meth:`~sympy.solvers.ode.dsolve` and different helper
functions that it uses.
:py:meth:`~sympy.solvers.ode.dsolve` solves ordinary differential equations.
See the docstring on the various functions for their uses. Note that partial
differential equations support is in ``pde.py``. Note that hint functions
have docstrings describing their various methods, but they are intended for
internal use. Use ``dsolve(ode, func, hint=hint)`` to solve an ODE using a
specific hint. See also the docstring on
:py:meth:`~sympy.solvers.ode.dsolve`.
**Functions in this module**
These are the user functions in this module:
- :py:meth:`~sympy.solvers.ode.dsolve` - Solves ODEs.
- :py:meth:`~sympy.solvers.ode.classify_ode` - Classifies ODEs into
possible hints for :py:meth:`~sympy.solvers.ode.dsolve`.
- :py:meth:`~sympy.solvers.ode.checkodesol` - Checks if an equation is the
solution to an ODE.
- :py:meth:`~sympy.solvers.ode.homogeneous_order` - Returns the
homogeneous order of an expression.
- :py:meth:`~sympy.solvers.ode.infinitesimals` - Returns the infinitesimals
of the Lie group of point transformations of an ODE, such that it is
invariant.
- :py:meth:`~sympy.solvers.ode.checkinfsol` - Checks if the given infinitesimals
are the actual infinitesimals of a first order ODE.
These are the non-solver helper functions that are for internal use. The
user should use the various options to
:py:meth:`~sympy.solvers.ode.dsolve` to obtain the functionality provided
by these functions:
- :py:meth:`~sympy.solvers.ode.ode.odesimp` - Does all forms of ODE
simplification.
- :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity` - A key function for
comparing solutions by simplicity.
- :py:meth:`~sympy.solvers.ode.constantsimp` - Simplifies arbitrary
constants.
- :py:meth:`~sympy.solvers.ode.ode.constant_renumber` - Renumber arbitrary
constants.
- :py:meth:`~sympy.solvers.ode.ode._handle_Integral` - Evaluate unevaluated
Integrals.
See also the docstrings of these functions.
**Currently implemented solver methods**
The following methods are implemented for solving ordinary differential
equations. See the docstrings of the various hint functions for more
information on each (run ``help(ode)``):
- 1st order separable differential equations.
- 1st order differential equations whose coefficients or `dx` and `dy` are
functions homogeneous of the same order.
- 1st order exact differential equations.
- 1st order linear differential equations.
- 1st order Bernoulli differential equations.
- Power series solutions for first order differential equations.
- Lie Group method of solving first order differential equations.
- 2nd order Liouville differential equations.
- Power series solutions for second order differential equations
at ordinary and regular singular points.
- `n`\th order differential equation that can be solved with algebraic
rearrangement and integration.
- `n`\th order linear homogeneous differential equation with constant
coefficients.
- `n`\th order linear inhomogeneous differential equation with constant
coefficients using the method of undetermined coefficients.
- `n`\th order linear inhomogeneous differential equation with constant
coefficients using the method of variation of parameters.
**Philosophy behind this module**
This module is designed to make it easy to add new ODE solving methods without
having to mess with the solving code for other methods. The idea is that
there is a :py:meth:`~sympy.solvers.ode.classify_ode` function, which takes in
an ODE and tells you what hints, if any, will solve the ODE. It does this
without attempting to solve the ODE, so it is fast. Each solving method is a
hint, and it has its own function, named ``ode_<hint>``. That function takes
in the ODE and any match expression gathered by
:py:meth:`~sympy.solvers.ode.classify_ode` and returns a solved result. If
this result has any integrals in it, the hint function will return an
unevaluated :py:class:`~sympy.integrals.integrals.Integral` class.
:py:meth:`~sympy.solvers.ode.dsolve`, which is the user wrapper function
around all of this, will then call :py:meth:`~sympy.solvers.ode.ode.odesimp` on
the result, which, among other things, will attempt to solve the equation for
the dependent variable (the function we are solving for), simplify the
arbitrary constants in the expression, and evaluate any integrals, if the hint
allows it.
**How to add new solution methods**
If you have an ODE that you want :py:meth:`~sympy.solvers.ode.dsolve` to be
able to solve, try to avoid adding special case code here. Instead, try
finding a general method that will solve your ODE, as well as others. This
way, the :py:mod:`~sympy.solvers.ode` module will become more robust, and
unhindered by special case hacks. WolphramAlpha and Maple's
DETools[odeadvisor] function are two resources you can use to classify a
specific ODE. It is also better for a method to work with an `n`\th order ODE
instead of only with specific orders, if possible.
To add a new method, there are a few things that you need to do. First, you
need a hint name for your method. Try to name your hint so that it is
unambiguous with all other methods, including ones that may not be implemented
yet. If your method uses integrals, also include a ``hint_Integral`` hint.
If there is more than one way to solve ODEs with your method, include a hint
for each one, as well as a ``<hint>_best`` hint. Your ``ode_<hint>_best()``
function should choose the best using min with ``ode_sol_simplicity`` as the
key argument. See
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest`, for example.
The function that uses your method will be called ``ode_<hint>()``, so the
hint must only use characters that are allowed in a Python function name
(alphanumeric characters and the underscore '``_``' character). Include a
function for every hint, except for ``_Integral`` hints
(:py:meth:`~sympy.solvers.ode.dsolve` takes care of those automatically).
Hint names should be all lowercase, unless a word is commonly capitalized
(such as Integral or Bernoulli). If you have a hint that you do not want to
run with ``all_Integral`` that does not have an ``_Integral`` counterpart (such
as a best hint that would defeat the purpose of ``all_Integral``), you will
need to remove it manually in the :py:meth:`~sympy.solvers.ode.dsolve` code.
See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
guidelines on writing a hint name.
Determine *in general* how the solutions returned by your method compare with
other methods that can potentially solve the same ODEs. Then, put your hints
in the :py:data:`~sympy.solvers.ode.allhints` tuple in the order that they
should be called. The ordering of this tuple determines which hints are
default. Note that exceptions are ok, because it is easy for the user to
choose individual hints with :py:meth:`~sympy.solvers.ode.dsolve`. In
general, ``_Integral`` variants should go at the end of the list, and
``_best`` variants should go before the various hints they apply to. For
example, the ``undetermined_coefficients`` hint comes before the
``variation_of_parameters`` hint because, even though variation of parameters
is more general than undetermined coefficients, undetermined coefficients
generally returns cleaner results for the ODEs that it can solve than
variation of parameters does, and it does not require integration, so it is
much faster.
Next, you need to have a match expression or a function that matches the type
of the ODE, which you should put in :py:meth:`~sympy.solvers.ode.classify_ode`
(if the match function is more than just a few lines. It should match the
ODE without solving for it as much as possible, so that
:py:meth:`~sympy.solvers.ode.classify_ode` remains fast and is not hindered by
bugs in solving code. Be sure to consider corner cases. For example, if your
solution method involves dividing by something, make sure you exclude the case
where that division will be 0.
In most cases, the matching of the ODE will also give you the various parts
that you need to solve it. You should put that in a dictionary (``.match()``
will do this for you), and add that as ``matching_hints['hint'] = matchdict``
in the relevant part of :py:meth:`~sympy.solvers.ode.classify_ode`.
:py:meth:`~sympy.solvers.ode.classify_ode` will then send this to
:py:meth:`~sympy.solvers.ode.dsolve`, which will send it to your function as
the ``match`` argument. Your function should be named ``ode_<hint>(eq, func,
order, match)`. If you need to send more information, put it in the ``match``
dictionary. For example, if you had to substitute in a dummy variable in
:py:meth:`~sympy.solvers.ode.classify_ode` to match the ODE, you will need to
pass it to your function using the `match` dict to access it. You can access
the independent variable using ``func.args[0]``, and the dependent variable
(the function you are trying to solve for) as ``func.func``. If, while trying
to solve the ODE, you find that you cannot, raise ``NotImplementedError``.
:py:meth:`~sympy.solvers.ode.dsolve` will catch this error with the ``all``
meta-hint, rather than causing the whole routine to fail.
Add a docstring to your function that describes the method employed. Like
with anything else in SymPy, you will need to add a doctest to the docstring,
in addition to real tests in ``test_ode.py``. Try to maintain consistency
with the other hint functions' docstrings. Add your method to the list at the
top of this docstring. Also, add your method to ``ode.rst`` in the
``docs/src`` directory, so that the Sphinx docs will pull its docstring into
the main SymPy documentation. Be sure to make the Sphinx documentation by
running ``make html`` from within the doc directory to verify that the
docstring formats correctly.
If your solution method involves integrating, use :py:obj:`~.Integral` instead of
:py:meth:`~sympy.core.expr.Expr.integrate`. This allows the user to bypass
hard/slow integration by using the ``_Integral`` variant of your hint. In
most cases, calling :py:meth:`sympy.core.basic.Basic.doit` will integrate your
solution. If this is not the case, you will need to write special code in
:py:meth:`~sympy.solvers.ode.ode._handle_Integral`. Arbitrary constants should be
symbols named ``C1``, ``C2``, and so on. All solution methods should return
an equality instance. If you need an arbitrary number of arbitrary constants,
you can use ``constants = numbered_symbols(prefix='C', cls=Symbol, start=1)``.
If it is possible to solve for the dependent function in a general way, do so.
Otherwise, do as best as you can, but do not call solve in your
``ode_<hint>()`` function. :py:meth:`~sympy.solvers.ode.ode.odesimp` will attempt
to solve the solution for you, so you do not need to do that. Lastly, if your
ODE has a common simplification that can be applied to your solutions, you can
add a special case in :py:meth:`~sympy.solvers.ode.ode.odesimp` for it. For
example, solutions returned from the ``1st_homogeneous_coeff`` hints often
have many :obj:`~sympy.functions.elementary.exponential.log` terms, so
:py:meth:`~sympy.solvers.ode.ode.odesimp` calls
:py:meth:`~sympy.simplify.simplify.logcombine` on them (it also helps to write
the arbitrary constant as ``log(C1)`` instead of ``C1`` in this case). Also
consider common ways that you can rearrange your solution to have
:py:meth:`~sympy.solvers.ode.constantsimp` take better advantage of it. It is
better to put simplification in :py:meth:`~sympy.solvers.ode.ode.odesimp` than in
your method, because it can then be turned off with the simplify flag in
:py:meth:`~sympy.solvers.ode.dsolve`. If you have any extraneous
simplification in your function, be sure to only run it using ``if
match.get('simplify', True):``, especially if it can be slow or if it can
reduce the domain of the solution.
Finally, as with every contribution to SymPy, your method will need to be
tested. Add a test for each method in ``test_ode.py``. Follow the
conventions there, i.e., test the solver using ``dsolve(eq, f(x),
hint=your_hint)``, and also test the solution using
:py:meth:`~sympy.solvers.ode.checkodesol` (you can put these in a separate
tests and skip/XFAIL if it runs too slow/does not work). Be sure to call your
hint specifically in :py:meth:`~sympy.solvers.ode.dsolve`, that way the test
will not be broken simply by the introduction of another matching hint. If your
method works for higher order (>1) ODEs, you will need to run ``sol =
constant_renumber(sol, 'C', 1, order)`` for each solution, where ``order`` is
the order of the ODE. This is because ``constant_renumber`` renumbers the
arbitrary constants by printing order, which is platform dependent. Try to
test every corner case of your solver, including a range of orders if it is a
`n`\th order solver, but if your solver is slow, such as if it involves hard
integration, try to keep the test run time down.
Feel free to refactor existing hints to avoid duplicating code or creating
inconsistencies. If you can show that your method exactly duplicates an
existing method, including in the simplicity and speed of obtaining the
solutions, then you can remove the old, less general method. The existing
code is tested extensively in ``test_ode.py``, so if anything is broken, one
of those tests will surely fail.
"""
from sympy.core import Add, S, Mul, Pow, oo
from sympy.core.containers import Tuple
from sympy.core.expr import AtomicExpr, Expr
from sympy.core.function import (Function, Derivative, AppliedUndef, diff,
expand, expand_mul, Subs)
from sympy.core.multidimensional import vectorize
from sympy.core.numbers import nan, zoo, Number
from sympy.core.relational import Equality, Eq
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import Symbol, Wild, Dummy, symbols
from sympy.core.sympify import sympify
from sympy.core.traversal import preorder_traversal
from sympy.logic.boolalg import (BooleanAtom, BooleanTrue,
BooleanFalse)
from sympy.functions import exp, log, sqrt
from sympy.functions.combinatorial.factorials import factorial
from sympy.integrals.integrals import Integral
from sympy.polys import (Poly, terms_gcd, PolynomialError, lcm)
from sympy.polys.polytools import cancel
from sympy.series import Order
from sympy.series.series import series
from sympy.simplify import (collect, logcombine, powsimp, # type: ignore
separatevars, simplify, cse)
from sympy.simplify.radsimp import collect_const
from sympy.solvers import checksol, solve
from sympy.utilities import numbered_symbols
from sympy.utilities.iterables import uniq, sift, iterable
from sympy.solvers.deutils import _preprocess, ode_order, _desolve
#: This is a list of hints in the order that they should be preferred by
#: :py:meth:`~sympy.solvers.ode.classify_ode`. In general, hints earlier in the
#: list should produce simpler solutions than those later in the list (for
#: ODEs that fit both). For now, the order of this list is based on empirical
#: observations by the developers of SymPy.
#:
#: The hint used by :py:meth:`~sympy.solvers.ode.dsolve` for a specific ODE
#: can be overridden (see the docstring).
#:
#: In general, ``_Integral`` hints are grouped at the end of the list, unless
#: there is a method that returns an unevaluable integral most of the time
#: (which go near the end of the list anyway). ``default``, ``all``,
#: ``best``, and ``all_Integral`` meta-hints should not be included in this
#: list, but ``_best`` and ``_Integral`` hints should be included.
allhints = (
"factorable",
"nth_algebraic",
"separable",
"1st_exact",
"1st_linear",
"Bernoulli",
"1st_rational_riccati",
"Riccati_special_minus2",
"1st_homogeneous_coeff_best",
"1st_homogeneous_coeff_subs_indep_div_dep",
"1st_homogeneous_coeff_subs_dep_div_indep",
"almost_linear",
"linear_coefficients",
"separable_reduced",
"1st_power_series",
"lie_group",
"nth_linear_constant_coeff_homogeneous",
"nth_linear_euler_eq_homogeneous",
"nth_linear_constant_coeff_undetermined_coefficients",
"nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients",
"nth_linear_constant_coeff_variation_of_parameters",
"nth_linear_euler_eq_nonhomogeneous_variation_of_parameters",
"Liouville",
"2nd_linear_airy",
"2nd_linear_bessel",
"2nd_hypergeometric",
"2nd_hypergeometric_Integral",
"nth_order_reducible",
"2nd_power_series_ordinary",
"2nd_power_series_regular",
"nth_algebraic_Integral",
"separable_Integral",
"1st_exact_Integral",
"1st_linear_Integral",
"Bernoulli_Integral",
"1st_homogeneous_coeff_subs_indep_div_dep_Integral",
"1st_homogeneous_coeff_subs_dep_div_indep_Integral",
"almost_linear_Integral",
"linear_coefficients_Integral",
"separable_reduced_Integral",
"nth_linear_constant_coeff_variation_of_parameters_Integral",
"nth_linear_euler_eq_nonhomogeneous_variation_of_parameters_Integral",
"Liouville_Integral",
"2nd_nonlinear_autonomous_conserved",
"2nd_nonlinear_autonomous_conserved_Integral",
)
def get_numbered_constants(eq, num=1, start=1, prefix='C'):
"""
Returns a list of constants that do not occur
in eq already.
"""
ncs = iter_numbered_constants(eq, start, prefix)
Cs = [next(ncs) for i in range(num)]
return (Cs[0] if num == 1 else tuple(Cs))
def iter_numbered_constants(eq, start=1, prefix='C'):
"""
Returns an iterator of constants that do not occur
in eq already.
"""
if isinstance(eq, (Expr, Eq)):
eq = [eq]
elif not iterable(eq):
raise ValueError("Expected Expr or iterable but got %s" % eq)
atom_set = set().union(*[i.free_symbols for i in eq])
func_set = set().union(*[i.atoms(Function) for i in eq])
if func_set:
atom_set |= {Symbol(str(f.func)) for f in func_set}
return numbered_symbols(start=start, prefix=prefix, exclude=atom_set)
def dsolve(eq, func=None, hint="default", simplify=True,
ics= None, xi=None, eta=None, x0=0, n=6, **kwargs):
r"""
Solves any (supported) kind of ordinary differential equation and
system of ordinary differential equations.
For single ordinary differential equation
=========================================
It is classified under this when number of equation in ``eq`` is one.
**Usage**
``dsolve(eq, f(x), hint)`` -> Solve ordinary differential equation
``eq`` for function ``f(x)``, using method ``hint``.
**Details**
``eq`` can be any supported ordinary differential equation (see the
:py:mod:`~sympy.solvers.ode` docstring for supported methods).
This can either be an :py:class:`~sympy.core.relational.Equality`,
or an expression, which is assumed to be equal to ``0``.
``f(x)`` is a function of one variable whose derivatives in that
variable make up the ordinary differential equation ``eq``. In
many cases it is not necessary to provide this; it will be
autodetected (and an error raised if it could not be detected).
``hint`` is the solving method that you want dsolve to use. Use
``classify_ode(eq, f(x))`` to get all of the possible hints for an
ODE. The default hint, ``default``, will use whatever hint is
returned first by :py:meth:`~sympy.solvers.ode.classify_ode`. See
Hints below for more options that you can use for hint.
``simplify`` enables simplification by
:py:meth:`~sympy.solvers.ode.ode.odesimp`. See its docstring for more
information. Turn this off, for example, to disable solving of
solutions for ``func`` or simplification of arbitrary constants.
It will still integrate with this hint. Note that the solution may
contain more arbitrary constants than the order of the ODE with
this option enabled.
``xi`` and ``eta`` are the infinitesimal functions of an ordinary
differential equation. They are the infinitesimals of the Lie group
of point transformations for which the differential equation is
invariant. The user can specify values for the infinitesimals. If
nothing is specified, ``xi`` and ``eta`` are calculated using
:py:meth:`~sympy.solvers.ode.infinitesimals` with the help of various
heuristics.
``ics`` is the set of initial/boundary conditions for the differential equation.
It should be given in the form of ``{f(x0): x1, f(x).diff(x).subs(x, x2):
x3}`` and so on. For power series solutions, if no initial
conditions are specified ``f(0)`` is assumed to be ``C0`` and the power
series solution is calculated about 0.
``x0`` is the point about which the power series solution of a differential
equation is to be evaluated.
``n`` gives the exponent of the dependent variable up to which the power series
solution of a differential equation is to be evaluated.
**Hints**
Aside from the various solving methods, there are also some meta-hints
that you can pass to :py:meth:`~sympy.solvers.ode.dsolve`:
``default``:
This uses whatever hint is returned first by
:py:meth:`~sympy.solvers.ode.classify_ode`. This is the
default argument to :py:meth:`~sympy.solvers.ode.dsolve`.
``all``:
To make :py:meth:`~sympy.solvers.ode.dsolve` apply all
relevant classification hints, use ``dsolve(ODE, func,
hint="all")``. This will return a dictionary of
``hint:solution`` terms. If a hint causes dsolve to raise the
``NotImplementedError``, value of that hint's key will be the
exception object raised. The dictionary will also include
some special keys:
- ``order``: The order of the ODE. See also
:py:meth:`~sympy.solvers.deutils.ode_order` in
``deutils.py``.
- ``best``: The simplest hint; what would be returned by
``best`` below.
- ``best_hint``: The hint that would produce the solution
given by ``best``. If more than one hint produces the best
solution, the first one in the tuple returned by
:py:meth:`~sympy.solvers.ode.classify_ode` is chosen.
- ``default``: The solution that would be returned by default.
This is the one produced by the hint that appears first in
the tuple returned by
:py:meth:`~sympy.solvers.ode.classify_ode`.
``all_Integral``:
This is the same as ``all``, except if a hint also has a
corresponding ``_Integral`` hint, it only returns the
``_Integral`` hint. This is useful if ``all`` causes
:py:meth:`~sympy.solvers.ode.dsolve` to hang because of a
difficult or impossible integral. This meta-hint will also be
much faster than ``all``, because
:py:meth:`~sympy.core.expr.Expr.integrate` is an expensive
routine.
``best``:
To have :py:meth:`~sympy.solvers.ode.dsolve` try all methods
and return the simplest one. This takes into account whether
the solution is solvable in the function, whether it contains
any Integral classes (i.e. unevaluatable integrals), and
which one is the shortest in size.
See also the :py:meth:`~sympy.solvers.ode.classify_ode` docstring for
more info on hints, and the :py:mod:`~sympy.solvers.ode` docstring for
a list of all supported hints.
**Tips**
- You can declare the derivative of an unknown function this way:
>>> from sympy import Function, Derivative
>>> from sympy.abc import x # x is the independent variable
>>> f = Function("f")(x) # f is a function of x
>>> # f_ will be the derivative of f with respect to x
>>> f_ = Derivative(f, x)
- See ``test_ode.py`` for many tests, which serves also as a set of
examples for how to use :py:meth:`~sympy.solvers.ode.dsolve`.
- :py:meth:`~sympy.solvers.ode.dsolve` always returns an
:py:class:`~sympy.core.relational.Equality` class (except for the
case when the hint is ``all`` or ``all_Integral``). If possible, it
solves the solution explicitly for the function being solved for.
Otherwise, it returns an implicit solution.
- Arbitrary constants are symbols named ``C1``, ``C2``, and so on.
- Because all solutions should be mathematically equivalent, some
hints may return the exact same result for an ODE. Often, though,
two different hints will return the same solution formatted
differently. The two should be equivalent. Also note that sometimes
the values of the arbitrary constants in two different solutions may
not be the same, because one constant may have "absorbed" other
constants into it.
- Do ``help(ode.ode_<hintname>)`` to get help more information on a
specific hint, where ``<hintname>`` is the name of a hint without
``_Integral``.
For system of ordinary differential equations
=============================================
**Usage**
``dsolve(eq, func)`` -> Solve a system of ordinary differential
equations ``eq`` for ``func`` being list of functions including
`x(t)`, `y(t)`, `z(t)` where number of functions in the list depends
upon the number of equations provided in ``eq``.
**Details**
``eq`` can be any supported system of ordinary differential equations
This can either be an :py:class:`~sympy.core.relational.Equality`,
or an expression, which is assumed to be equal to ``0``.
``func`` holds ``x(t)`` and ``y(t)`` being functions of one variable which
together with some of their derivatives make up the system of ordinary
differential equation ``eq``. It is not necessary to provide this; it
will be autodetected (and an error raised if it could not be detected).
**Hints**
The hints are formed by parameters returned by classify_sysode, combining
them give hints name used later for forming method name.
Examples
========
>>> from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(Derivative(f(x), x, x) + 9*f(x), f(x))
Eq(f(x), C1*sin(3*x) + C2*cos(3*x))
>>> eq = sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x)
>>> dsolve(eq, hint='1st_exact')
[Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
>>> dsolve(eq, hint='almost_linear')
[Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))]
>>> t = symbols('t')
>>> x, y = symbols('x, y', cls=Function)
>>> eq = (Eq(Derivative(x(t),t), 12*t*x(t) + 8*y(t)), Eq(Derivative(y(t),t), 21*x(t) + 7*t*y(t)))
>>> dsolve(eq)
[Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t)),
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(8*exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)**2, t) +
exp(Integral(7*t, t))*exp(Integral(12*t, t))/x0(t)))]
>>> eq = (Eq(Derivative(x(t),t),x(t)*y(t)*sin(t)), Eq(Derivative(y(t),t),y(t)**2*sin(t)))
>>> dsolve(eq)
{Eq(x(t), -exp(C1)/(C2*exp(C1) - cos(t))), Eq(y(t), -1/(C1 - cos(t)))}
"""
if iterable(eq):
from sympy.solvers.ode.systems import dsolve_system
# This may have to be changed in future
# when we have weakly and strongly
# connected components. This have to
# changed to show the systems that haven't
# been solved.
try:
sol = dsolve_system(eq, funcs=func, ics=ics, doit=True)
return sol[0] if len(sol) == 1 else sol
except NotImplementedError:
pass
match = classify_sysode(eq, func)
eq = match['eq']
order = match['order']
func = match['func']
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
# keep highest order term coefficient positive
for i in range(len(eq)):
for func_ in func:
if isinstance(func_, list):
pass
else:
if eq[i].coeff(diff(func[i],t,ode_order(eq[i], func[i]))).is_negative:
eq[i] = -eq[i]
match['eq'] = eq
if len(set(order.values()))!=1:
raise ValueError("It solves only those systems of equations whose orders are equal")
match['order'] = list(order.values())[0]
def recur_len(l):
return sum(recur_len(item) if isinstance(item,list) else 1 for item in l)
if recur_len(func) != len(eq):
raise ValueError("dsolve() and classify_sysode() work with "
"number of functions being equal to number of equations")
if match['type_of_equation'] is None:
raise NotImplementedError
else:
if match['is_linear'] == True:
solvefunc = globals()['sysode_linear_%(no_of_equation)seq_order%(order)s' % match]
else:
solvefunc = globals()['sysode_nonlinear_%(no_of_equation)seq_order%(order)s' % match]
sols = solvefunc(match)
if ics:
constants = Tuple(*sols).free_symbols - Tuple(*eq).free_symbols
solved_constants = solve_ics(sols, func, constants, ics)
return [sol.subs(solved_constants) for sol in sols]
return sols
else:
given_hint = hint # hint given by the user
# See the docstring of _desolve for more details.
hints = _desolve(eq, func=func,
hint=hint, simplify=True, xi=xi, eta=eta, type='ode', ics=ics,
x0=x0, n=n, **kwargs)
eq = hints.pop('eq', eq)
all_ = hints.pop('all', False)
if all_:
retdict = {}
failed_hints = {}
gethints = classify_ode(eq, dict=True, hint='all')
orderedhints = gethints['ordered_hints']
for hint in hints:
try:
rv = _helper_simplify(eq, hint, hints[hint], simplify)
except NotImplementedError as detail:
failed_hints[hint] = detail
else:
retdict[hint] = rv
func = hints[hint]['func']
retdict['best'] = min(list(retdict.values()), key=lambda x:
ode_sol_simplicity(x, func, trysolving=not simplify))
if given_hint == 'best':
return retdict['best']
for i in orderedhints:
if retdict['best'] == retdict.get(i, None):
retdict['best_hint'] = i
break
retdict['default'] = gethints['default']
retdict['order'] = gethints['order']
retdict.update(failed_hints)
return retdict
else:
# The key 'hint' stores the hint needed to be solved for.
hint = hints['hint']
return _helper_simplify(eq, hint, hints, simplify, ics=ics)
def _helper_simplify(eq, hint, match, simplify=True, ics=None, **kwargs):
r"""
Helper function of dsolve that calls the respective
:py:mod:`~sympy.solvers.ode` functions to solve for the ordinary
differential equations. This minimizes the computation in calling
:py:meth:`~sympy.solvers.deutils._desolve` multiple times.
"""
r = match
func = r['func']
order = r['order']
match = r[hint]
if isinstance(match, SingleODESolver):
solvefunc = match
elif hint.endswith('_Integral'):
solvefunc = globals()['ode_' + hint[:-len('_Integral')]]
else:
solvefunc = globals()['ode_' + hint]
free = eq.free_symbols
cons = lambda s: s.free_symbols.difference(free)
if simplify:
# odesimp() will attempt to integrate, if necessary, apply constantsimp(),
# attempt to solve for func, and apply any other hint specific
# simplifications
if isinstance(solvefunc, SingleODESolver):
sols = solvefunc.get_general_solution()
else:
sols = solvefunc(eq, func, order, match)
if iterable(sols):
rv = [odesimp(eq, s, func, hint) for s in sols]
else:
rv = odesimp(eq, sols, func, hint)
else:
# We still want to integrate (you can disable it separately with the hint)
if isinstance(solvefunc, SingleODESolver):
exprs = solvefunc.get_general_solution(simplify=False)
else:
match['simplify'] = False # Some hints can take advantage of this option
exprs = solvefunc(eq, func, order, match)
if isinstance(exprs, list):
rv = [_handle_Integral(expr, func, hint) for expr in exprs]
else:
rv = _handle_Integral(exprs, func, hint)
if isinstance(rv, list):
if simplify:
rv = _remove_redundant_solutions(eq, rv, order, func.args[0])
if len(rv) == 1:
rv = rv[0]
if ics and 'power_series' not in hint:
if isinstance(rv, (Expr, Eq)):
solved_constants = solve_ics([rv], [r['func']], cons(rv), ics)
rv = rv.subs(solved_constants)
else:
rv1 = []
for s in rv:
try:
solved_constants = solve_ics([s], [r['func']], cons(s), ics)
except ValueError:
continue
rv1.append(s.subs(solved_constants))
if len(rv1) == 1:
return rv1[0]
rv = rv1
return rv
def solve_ics(sols, funcs, constants, ics):
"""
Solve for the constants given initial conditions
``sols`` is a list of solutions.
``funcs`` is a list of functions.
``constants`` is a list of constants.
``ics`` is the set of initial/boundary conditions for the differential
equation. It should be given in the form of ``{f(x0): x1,
f(x).diff(x).subs(x, x2): x3}`` and so on.
Returns a dictionary mapping constants to values.
``solution.subs(constants)`` will replace the constants in ``solution``.
Example
=======
>>> # From dsolve(f(x).diff(x) - f(x), f(x))
>>> from sympy import symbols, Eq, exp, Function
>>> from sympy.solvers.ode.ode import solve_ics
>>> f = Function('f')
>>> x, C1 = symbols('x C1')
>>> sols = [Eq(f(x), C1*exp(x))]
>>> funcs = [f(x)]
>>> constants = [C1]
>>> ics = {f(0): 2}
>>> solved_constants = solve_ics(sols, funcs, constants, ics)
>>> solved_constants
{C1: 2}
>>> sols[0].subs(solved_constants)
Eq(f(x), 2*exp(x))
"""
# Assume ics are of the form f(x0): value or Subs(diff(f(x), x, n), (x,
# x0)): value (currently checked by classify_ode). To solve, replace x
# with x0, f(x0) with value, then solve for constants. For f^(n)(x0),
# differentiate the solution n times, so that f^(n)(x) appears.
x = funcs[0].args[0]
diff_sols = []
subs_sols = []
diff_variables = set()
for funcarg, value in ics.items():
if isinstance(funcarg, AppliedUndef):
x0 = funcarg.args[0]
matching_func = [f for f in funcs if f.func == funcarg.func][0]
S = sols
elif isinstance(funcarg, (Subs, Derivative)):
if isinstance(funcarg, Subs):
# Make sure it stays a subs. Otherwise subs below will produce
# a different looking term.
funcarg = funcarg.doit()
if isinstance(funcarg, Subs):
deriv = funcarg.expr
x0 = funcarg.point[0]
variables = funcarg.expr.variables
matching_func = deriv
elif isinstance(funcarg, Derivative):
deriv = funcarg
x0 = funcarg.variables[0]
variables = (x,)*len(funcarg.variables)
matching_func = deriv.subs(x0, x)
for sol in sols:
if sol.has(deriv.expr.func):
diff_sols.append(Eq(sol.lhs.diff(*variables), sol.rhs.diff(*variables)))
diff_variables.add(variables)
S = diff_sols
else:
raise NotImplementedError("Unrecognized initial condition")
for sol in S:
if sol.has(matching_func):
sol2 = sol
sol2 = sol2.subs(x, x0)
sol2 = sol2.subs(funcarg, value)
# This check is necessary because of issue #15724
if not isinstance(sol2, BooleanAtom) or not subs_sols:
subs_sols = [s for s in subs_sols if not isinstance(s, BooleanAtom)]
subs_sols.append(sol2)
# TODO: Use solveset here
try:
solved_constants = solve(subs_sols, constants, dict=True)
except NotImplementedError:
solved_constants = []
# XXX: We can't differentiate between the solution not existing because of
# invalid initial conditions, and not existing because solve is not smart
# enough. If we could use solveset, this might be improvable, but for now,
# we use NotImplementedError in this case.
if not solved_constants:
raise ValueError("Couldn't solve for initial conditions")
if solved_constants == True:
raise ValueError("Initial conditions did not produce any solutions for constants. Perhaps they are degenerate.")
if len(solved_constants) > 1:
raise NotImplementedError("Initial conditions produced too many solutions for constants")
return solved_constants[0]
def classify_ode(eq, func=None, dict=False, ics=None, *, prep=True, xi=None, eta=None, n=None, **kwargs):
r"""
Returns a tuple of possible :py:meth:`~sympy.solvers.ode.dsolve`
classifications for an ODE.
The tuple is ordered so that first item is the classification that
:py:meth:`~sympy.solvers.ode.dsolve` uses to solve the ODE by default. In
general, classifications at the near the beginning of the list will
produce better solutions faster than those near the end, thought there are
always exceptions. To make :py:meth:`~sympy.solvers.ode.dsolve` use a
different classification, use ``dsolve(ODE, func,
hint=<classification>)``. See also the
:py:meth:`~sympy.solvers.ode.dsolve` docstring for different meta-hints
you can use.
If ``dict`` is true, :py:meth:`~sympy.solvers.ode.classify_ode` will
return a dictionary of ``hint:match`` expression terms. This is intended
for internal use by :py:meth:`~sympy.solvers.ode.dsolve`. Note that
because dictionaries are ordered arbitrarily, this will most likely not be
in the same order as the tuple.
You can get help on different hints by executing
``help(ode.ode_hintname)``, where ``hintname`` is the name of the hint
without ``_Integral``.
See :py:data:`~sympy.solvers.ode.allhints` or the
:py:mod:`~sympy.solvers.ode` docstring for a list of all supported hints
that can be returned from :py:meth:`~sympy.solvers.ode.classify_ode`.
Notes
=====
These are remarks on hint names.
``_Integral``
If a classification has ``_Integral`` at the end, it will return the
expression with an unevaluated :py:class:`~.Integral`
class in it. Note that a hint may do this anyway if
:py:meth:`~sympy.core.expr.Expr.integrate` cannot do the integral,
though just using an ``_Integral`` will do so much faster. Indeed, an
``_Integral`` hint will always be faster than its corresponding hint
without ``_Integral`` because
:py:meth:`~sympy.core.expr.Expr.integrate` is an expensive routine.
If :py:meth:`~sympy.solvers.ode.dsolve` hangs, it is probably because
:py:meth:`~sympy.core.expr.Expr.integrate` is hanging on a tough or
impossible integral. Try using an ``_Integral`` hint or
``all_Integral`` to get it return something.
Note that some hints do not have ``_Integral`` counterparts. This is
because :py:func:`~sympy.integrals.integrals.integrate` is not used in
solving the ODE for those method. For example, `n`\th order linear
homogeneous ODEs with constant coefficients do not require integration
to solve, so there is no
``nth_linear_homogeneous_constant_coeff_Integrate`` hint. You can
easily evaluate any unevaluated
:py:class:`~sympy.integrals.integrals.Integral`\s in an expression by
doing ``expr.doit()``.
Ordinals
Some hints contain an ordinal such as ``1st_linear``. This is to help
differentiate them from other hints, as well as from other methods
that may not be implemented yet. If a hint has ``nth`` in it, such as
the ``nth_linear`` hints, this means that the method used to applies
to ODEs of any order.
``indep`` and ``dep``
Some hints contain the words ``indep`` or ``dep``. These reference
the independent variable and the dependent function, respectively. For
example, if an ODE is in terms of `f(x)`, then ``indep`` will refer to
`x` and ``dep`` will refer to `f`.
``subs``
If a hints has the word ``subs`` in it, it means that the ODE is solved
by substituting the expression given after the word ``subs`` for a
single dummy variable. This is usually in terms of ``indep`` and
``dep`` as above. The substituted expression will be written only in
characters allowed for names of Python objects, meaning operators will
be spelled out. For example, ``indep``/``dep`` will be written as
``indep_div_dep``.
``coeff``
The word ``coeff`` in a hint refers to the coefficients of something
in the ODE, usually of the derivative terms. See the docstring for
the individual methods for more info (``help(ode)``). This is
contrast to ``coefficients``, as in ``undetermined_coefficients``,
which refers to the common name of a method.
``_best``
Methods that have more than one fundamental way to solve will have a
hint for each sub-method and a ``_best`` meta-classification. This
will evaluate all hints and return the best, using the same
considerations as the normal ``best`` meta-hint.
Examples
========
>>> from sympy import Function, classify_ode, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> classify_ode(Eq(f(x).diff(x), 0), f(x))
('nth_algebraic',
'separable',
'1st_exact',
'1st_linear',
'Bernoulli',
'1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep',
'1st_homogeneous_coeff_subs_dep_div_indep',
'1st_power_series', 'lie_group', 'nth_linear_constant_coeff_homogeneous',
'nth_linear_euler_eq_homogeneous',
'nth_algebraic_Integral', 'separable_Integral', '1st_exact_Integral',
'1st_linear_Integral', 'Bernoulli_Integral',
'1st_homogeneous_coeff_subs_indep_div_dep_Integral',
'1st_homogeneous_coeff_subs_dep_div_indep_Integral')
>>> classify_ode(f(x).diff(x, 2) + 3*f(x).diff(x) + 2*f(x) - 4)
('factorable', 'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_constant_coeff_variation_of_parameters_Integral')
"""
ics = sympify(ics)
if func and len(func.args) != 1:
raise ValueError("dsolve() and classify_ode() only "
"work with functions of one variable, not %s" % func)
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
# Some methods want the unprocessed equation
eq_orig = eq
if prep or func is None:
eq, func_ = _preprocess(eq, func)
if func is None:
func = func_
x = func.args[0]
f = func.func
y = Dummy('y')
terms = 5 if n is None else n
order = ode_order(eq, f(x))
# hint:matchdict or hint:(tuple of matchdicts)
# Also will contain "default":<default hint> and "order":order items.
matching_hints = {"order": order}
df = f(x).diff(x)
a = Wild('a', exclude=[f(x)])
d = Wild('d', exclude=[df, f(x).diff(x, 2)])
e = Wild('e', exclude=[df])
n = Wild('n', exclude=[x, f(x), df])
c1 = Wild('c1', exclude=[x])
a3 = Wild('a3', exclude=[f(x), df, f(x).diff(x, 2)])
b3 = Wild('b3', exclude=[f(x), df, f(x).diff(x, 2)])
c3 = Wild('c3', exclude=[f(x), df, f(x).diff(x, 2)])
boundary = {} # Used to extract initial conditions
C1 = Symbol("C1")
# Preprocessing to get the initial conditions out
if ics is not None:
for funcarg in ics:
# Separating derivatives
if isinstance(funcarg, (Subs, Derivative)):
# f(x).diff(x).subs(x, 0) is a Subs, but f(x).diff(x).subs(x,
# y) is a Derivative
if isinstance(funcarg, Subs):
deriv = funcarg.expr
old = funcarg.variables[0]
new = funcarg.point[0]
elif isinstance(funcarg, Derivative):
deriv = funcarg
# No information on this. Just assume it was x
old = x
new = funcarg.variables[0]
if (isinstance(deriv, Derivative) and isinstance(deriv.args[0],
AppliedUndef) and deriv.args[0].func == f and
len(deriv.args[0].args) == 1 and old == x and not
new.has(x) and all(i == deriv.variables[0] for i in
deriv.variables) and not ics[funcarg].has(f)):
dorder = ode_order(deriv, x)
temp = 'f' + str(dorder)
boundary.update({temp: new, temp + 'val': ics[funcarg]})
else:
raise ValueError("Enter valid boundary conditions for Derivatives")
# Separating functions
elif isinstance(funcarg, AppliedUndef):
if (funcarg.func == f and len(funcarg.args) == 1 and
not funcarg.args[0].has(x) and not ics[funcarg].has(f)):
boundary.update({'f0': funcarg.args[0], 'f0val': ics[funcarg]})
else:
raise ValueError("Enter valid boundary conditions for Function")
else:
raise ValueError("Enter boundary conditions of the form ics={f(point): value, f(x).diff(x, order).subs(x, point): value}")
ode = SingleODEProblem(eq_orig, func, x, prep=prep, xi=xi, eta=eta)
user_hint = kwargs.get('hint', 'default')
# Used when dsolve is called without an explicit hint.
# We exit early to return the first valid match
early_exit = (user_hint=='default')
if user_hint.endswith('_Integral'):
user_hint = user_hint[:-len('_Integral')]
user_map = solver_map
# An explicit hint has been given to dsolve
# Skip matching code for other hints
if user_hint not in ['default', 'all', 'all_Integral', 'best'] and user_hint in solver_map:
user_map = {user_hint: solver_map[user_hint]}
for hint in user_map:
solver = user_map[hint](ode)
if solver.matches():
matching_hints[hint] = solver
if user_map[hint].has_integral:
matching_hints[hint + "_Integral"] = solver
if dict and early_exit:
matching_hints["default"] = hint
return matching_hints
eq = expand(eq)
# Precondition to try remove f(x) from highest order derivative
reduced_eq = None
if eq.is_Add:
deriv_coef = eq.coeff(f(x).diff(x, order))
if deriv_coef not in (1, 0):
r = deriv_coef.match(a*f(x)**c1)
if r and r[c1]:
den = f(x)**r[c1]
reduced_eq = Add(*[arg/den for arg in eq.args])
if not reduced_eq:
reduced_eq = eq
if order == 1:
# NON-REDUCED FORM OF EQUATION matches
r = collect(eq, df, exact=True).match(d + e * df)
if r:
r['d'] = d
r['e'] = e
r['y'] = y
r[d] = r[d].subs(f(x), y)
r[e] = r[e].subs(f(x), y)
# FIRST ORDER POWER SERIES WHICH NEEDS INITIAL CONDITIONS
# TODO: Hint first order series should match only if d/e is analytic.
# For now, only d/e and (d/e).diff(arg) is checked for existence at
# at a given point.
# This is currently done internally in ode_1st_power_series.
point = boundary.get('f0', 0)
value = boundary.get('f0val', C1)
check = cancel(r[d]/r[e])
check1 = check.subs({x: point, y: value})
if not check1.has(oo) and not check1.has(zoo) and \
not check1.has(nan) and not check1.has(-oo):
check2 = (check1.diff(x)).subs({x: point, y: value})
if not check2.has(oo) and not check2.has(zoo) and \
not check2.has(nan) and not check2.has(-oo):
rseries = r.copy()
rseries.update({'terms': terms, 'f0': point, 'f0val': value})
matching_hints["1st_power_series"] = rseries
elif order == 2:
# Homogeneous second order differential equation of the form
# a3*f(x).diff(x, 2) + b3*f(x).diff(x) + c3
# It has a definite power series solution at point x0 if, b3/a3 and c3/a3
# are analytic at x0.
deq = a3*(f(x).diff(x, 2)) + b3*df + c3*f(x)
r = collect(reduced_eq,
[f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
ordinary = False
if r:
if not all(r[key].is_polynomial() for key in r):
n, d = reduced_eq.as_numer_denom()
reduced_eq = expand(n)
r = collect(reduced_eq,
[f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
if r and r[a3] != 0:
p = cancel(r[b3]/r[a3]) # Used below
q = cancel(r[c3]/r[a3]) # Used below
point = kwargs.get('x0', 0)
check = p.subs(x, point)
if not check.has(oo, nan, zoo, -oo):
check = q.subs(x, point)
if not check.has(oo, nan, zoo, -oo):
ordinary = True
r.update({'a3': a3, 'b3': b3, 'c3': c3, 'x0': point, 'terms': terms})
matching_hints["2nd_power_series_ordinary"] = r
# Checking if the differential equation has a regular singular point
# at x0. It has a regular singular point at x0, if (b3/a3)*(x - x0)
# and (c3/a3)*((x - x0)**2) are analytic at x0.
if not ordinary:
p = cancel((x - point)*p)
check = p.subs(x, point)
if not check.has(oo, nan, zoo, -oo):
q = cancel(((x - point)**2)*q)
check = q.subs(x, point)
if not check.has(oo, nan, zoo, -oo):
coeff_dict = {'p': p, 'q': q, 'x0': point, 'terms': terms}
matching_hints["2nd_power_series_regular"] = coeff_dict
# Order keys based on allhints.
retlist = [i for i in allhints if i in matching_hints]
if dict:
# Dictionaries are ordered arbitrarily, so make note of which
# hint would come first for dsolve(). Use an ordered dict in Py 3.
matching_hints["default"] = retlist[0] if retlist else None
matching_hints["ordered_hints"] = tuple(retlist)
return matching_hints
else:
return tuple(retlist)
def classify_sysode(eq, funcs=None, **kwargs):
r"""
Returns a dictionary of parameter names and values that define the system
of ordinary differential equations in ``eq``.
The parameters are further used in
:py:meth:`~sympy.solvers.ode.dsolve` for solving that system.
Some parameter names and values are:
'is_linear' (boolean), which tells whether the given system is linear.
Note that "linear" here refers to the operator: terms such as ``x*diff(x,t)`` are
nonlinear, whereas terms like ``sin(t)*diff(x,t)`` are still linear operators.
'func' (list) contains the :py:class:`~sympy.core.function.Function`s that
appear with a derivative in the ODE, i.e. those that we are trying to solve
the ODE for.
'order' (dict) with the maximum derivative for each element of the 'func'
parameter.
'func_coeff' (dict or Matrix) with the coefficient for each triple ``(equation number,
function, order)```. The coefficients are those subexpressions that do not
appear in 'func', and hence can be considered constant for purposes of ODE
solving. The value of this parameter can also be a Matrix if the system of ODEs are
linear first order of the form X' = AX where X is the vector of dependent variables.
Here, this function returns the coefficient matrix A.
'eq' (list) with the equations from ``eq``, sympified and transformed into
expressions (we are solving for these expressions to be zero).
'no_of_equations' (int) is the number of equations (same as ``len(eq)``).
'type_of_equation' (string) is an internal classification of the type of
ODE.
'is_constant' (boolean), which tells if the system of ODEs is constant coefficient
or not. This key is temporary addition for now and is in the match dict only when
the system of ODEs is linear first order constant coefficient homogeneous. So, this
key's value is True for now if it is available else it does not exist.
'is_homogeneous' (boolean), which tells if the system of ODEs is homogeneous. Like the
key 'is_constant', this key is a temporary addition and it is True since this key value
is available only when the system is linear first order constant coefficient homogeneous.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode-toc1.htm
-A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists
Examples
========
>>> from sympy import Function, Eq, symbols, diff
>>> from sympy.solvers.ode.ode import classify_sysode
>>> from sympy.abc import t
>>> f, x, y = symbols('f, x, y', cls=Function)
>>> k, l, m, n = symbols('k, l, m, n', Integer=True)
>>> x1 = diff(x(t), t) ; y1 = diff(y(t), t)
>>> x2 = diff(x(t), t, t) ; y2 = diff(y(t), t, t)
>>> eq = (Eq(x1, 12*x(t) - 6*y(t)), Eq(y1, 11*x(t) + 3*y(t)))
>>> classify_sysode(eq)
{'eq': [-12*x(t) + 6*y(t) + Derivative(x(t), t), -11*x(t) - 3*y(t) + Derivative(y(t), t)], 'func': [x(t), y(t)],
'func_coeff': {(0, x(t), 0): -12, (0, x(t), 1): 1, (0, y(t), 0): 6, (0, y(t), 1): 0, (1, x(t), 0): -11, (1, x(t), 1): 0, (1, y(t), 0): -3, (1, y(t), 1): 1}, 'is_linear': True, 'no_of_equation': 2, 'order': {x(t): 1, y(t): 1}, 'type_of_equation': None}
>>> eq = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t) + 2), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
>>> classify_sysode(eq)
{'eq': [-t**2*y(t) - 5*t*x(t) + Derivative(x(t), t) - 2, t**2*x(t) - 5*t*y(t) + Derivative(y(t), t)],
'func': [x(t), y(t)], 'func_coeff': {(0, x(t), 0): -5*t, (0, x(t), 1): 1, (0, y(t), 0): -t**2, (0, y(t), 1): 0,
(1, x(t), 0): t**2, (1, x(t), 1): 0, (1, y(t), 0): -5*t, (1, y(t), 1): 1}, 'is_linear': True, 'no_of_equation': 2,
'order': {x(t): 1, y(t): 1}, 'type_of_equation': None}
"""
# Sympify equations and convert iterables of equations into
# a list of equations
def _sympify(eq):
return list(map(sympify, eq if iterable(eq) else [eq]))
eq, funcs = (_sympify(w) for w in [eq, funcs])
for i, fi in enumerate(eq):
if isinstance(fi, Equality):
eq[i] = fi.lhs - fi.rhs
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
matching_hints = {"no_of_equation":i+1}
matching_hints['eq'] = eq
if i==0:
raise ValueError("classify_sysode() works for systems of ODEs. "
"For scalar ODEs, classify_ode should be used")
# find all the functions if not given
order = dict()
if funcs==[None]:
funcs = _extract_funcs(eq)
funcs = list(set(funcs))
if len(funcs) != len(eq):
raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs)
# This logic of list of lists in funcs to
# be replaced later.
func_dict = dict()
for func in funcs:
if not order.get(func, False):
max_order = 0
for i, eqs_ in enumerate(eq):
order_ = ode_order(eqs_,func)
if max_order < order_:
max_order = order_
eq_no = i
if eq_no in func_dict:
func_dict[eq_no] = [func_dict[eq_no], func]
else:
func_dict[eq_no] = func
order[func] = max_order
funcs = [func_dict[i] for i in range(len(func_dict))]
matching_hints['func'] = funcs
for func in funcs:
if isinstance(func, list):
for func_elem in func:
if len(func_elem.args) != 1:
raise ValueError("dsolve() and classify_sysode() work with "
"functions of one variable only, not %s" % func)
else:
if func and len(func.args) != 1:
raise ValueError("dsolve() and classify_sysode() work with "
"functions of one variable only, not %s" % func)
# find the order of all equation in system of odes
matching_hints["order"] = order
# find coefficients of terms f(t), diff(f(t),t) and higher derivatives
# and similarly for other functions g(t), diff(g(t),t) in all equations.
# Here j denotes the equation number, funcs[l] denotes the function about
# which we are talking about and k denotes the order of function funcs[l]
# whose coefficient we are calculating.
def linearity_check(eqs, j, func, is_linear_):
for k in range(order[func] + 1):
func_coef[j, func, k] = collect(eqs.expand(), [diff(func, t, k)]).coeff(diff(func, t, k))
if is_linear_ == True:
if func_coef[j, func, k] == 0:
if k == 0:
coef = eqs.as_independent(func, as_Add=True)[1]
for xr in range(1, ode_order(eqs,func) + 1):
coef -= eqs.as_independent(diff(func, t, xr), as_Add=True)[1]
if coef != 0:
is_linear_ = False
else:
if eqs.as_independent(diff(func, t, k), as_Add=True)[1]:
is_linear_ = False
else:
for func_ in funcs:
if isinstance(func_, list):
for elem_func_ in func_:
dep = func_coef[j, func, k].as_independent(elem_func_, as_Add=True)[1]
if dep != 0:
is_linear_ = False
else:
dep = func_coef[j, func, k].as_independent(func_, as_Add=True)[1]
if dep != 0:
is_linear_ = False
return is_linear_
func_coef = {}
is_linear = True
for j, eqs in enumerate(eq):
for func in funcs:
if isinstance(func, list):
for func_elem in func:
is_linear = linearity_check(eqs, j, func_elem, is_linear)
else:
is_linear = linearity_check(eqs, j, func, is_linear)
matching_hints['func_coeff'] = func_coef
matching_hints['is_linear'] = is_linear
if len(set(order.values())) == 1:
order_eq = list(matching_hints['order'].values())[0]
if matching_hints['is_linear'] == True:
if matching_hints['no_of_equation'] == 2:
if order_eq == 1:
type_of_equation = check_linear_2eq_order1(eq, funcs, func_coef)
else:
type_of_equation = None
# If the equation does not match up with any of the
# general case solvers in systems.py and the number
# of equations is greater than 2, then NotImplementedError
# should be raised.
else:
type_of_equation = None
else:
if matching_hints['no_of_equation'] == 2:
if order_eq == 1:
type_of_equation = check_nonlinear_2eq_order1(eq, funcs, func_coef)
else:
type_of_equation = None
elif matching_hints['no_of_equation'] == 3:
if order_eq == 1:
type_of_equation = check_nonlinear_3eq_order1(eq, funcs, func_coef)
else:
type_of_equation = None
else:
type_of_equation = None
else:
type_of_equation = None
matching_hints['type_of_equation'] = type_of_equation
return matching_hints
def check_linear_2eq_order1(eq, func, func_coef):
x = func[0].func
y = func[1].func
fc = func_coef
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
r = dict()
# for equations Eq(a1*diff(x(t),t), b1*x(t) + c1*y(t) + d1)
# and Eq(a2*diff(y(t),t), b2*x(t) + c2*y(t) + d2)
r['a1'] = fc[0,x(t),1] ; r['a2'] = fc[1,y(t),1]
r['b1'] = -fc[0,x(t),0]/fc[0,x(t),1] ; r['b2'] = -fc[1,x(t),0]/fc[1,y(t),1]
r['c1'] = -fc[0,y(t),0]/fc[0,x(t),1] ; r['c2'] = -fc[1,y(t),0]/fc[1,y(t),1]
forcing = [S.Zero,S.Zero]
for i in range(2):
for j in Add.make_args(eq[i]):
if not j.has(x(t), y(t)):
forcing[i] += j
if not (forcing[0].has(t) or forcing[1].has(t)):
# We can handle homogeneous case and simple constant forcings
r['d1'] = forcing[0]
r['d2'] = forcing[1]
else:
# Issue #9244: nonhomogeneous linear systems are not supported
return None
# Conditions to check for type 6 whose equations are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and
# Eq(diff(y(t),t), a*[f(t) + a*h(t)]x(t) + a*[g(t) - h(t)]*y(t))
p = 0
q = 0
p1 = cancel(r['b2']/(cancel(r['b2']/r['c2']).as_numer_denom()[0]))
p2 = cancel(r['b1']/(cancel(r['b1']/r['c1']).as_numer_denom()[0]))
for n, i in enumerate([p1, p2]):
for j in Mul.make_args(collect_const(i)):
if not j.has(t):
q = j
if q and n==0:
if ((r['b2']/j - r['b1'])/(r['c1'] - r['c2']/j)) == j:
p = 1
elif q and n==1:
if ((r['b1']/j - r['b2'])/(r['c2'] - r['c1']/j)) == j:
p = 2
# End of condition for type 6
if r['d1']!=0 or r['d2']!=0:
return None
else:
if not any(r[k].has(t) for k in 'a1 a2 b1 b2 c1 c2'.split()):
return None
else:
r['b1'] = r['b1']/r['a1'] ; r['b2'] = r['b2']/r['a2']
r['c1'] = r['c1']/r['a1'] ; r['c2'] = r['c2']/r['a2']
if p:
return "type6"
else:
# Equations for type 7 are Eq(diff(x(t),t), f(t)*x(t) + g(t)*y(t)) and Eq(diff(y(t),t), h(t)*x(t) + p(t)*y(t))
return "type7"
def check_nonlinear_2eq_order1(eq, func, func_coef):
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
f = Wild('f')
g = Wild('g')
u, v = symbols('u, v', cls=Dummy)
def check_type(x, y):
r1 = eq[0].match(t*diff(x(t),t) - x(t) + f)
r2 = eq[1].match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t)
r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t)
if not (r1 and r2):
r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f)
r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t)
r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t)
if r1 and r2 and not (r1[f].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t) \
or r2[g].subs(diff(x(t),t),u).subs(diff(y(t),t),v).has(t)):
return 'type5'
else:
return None
for func_ in func:
if isinstance(func_, list):
x = func[0][0].func
y = func[0][1].func
eq_type = check_type(x, y)
if not eq_type:
eq_type = check_type(y, x)
return eq_type
x = func[0].func
y = func[1].func
fc = func_coef
n = Wild('n', exclude=[x(t),y(t)])
f1 = Wild('f1', exclude=[v,t])
f2 = Wild('f2', exclude=[v,t])
g1 = Wild('g1', exclude=[u,t])
g2 = Wild('g2', exclude=[u,t])
for i in range(2):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
r = eq[0].match(diff(x(t),t) - x(t)**n*f)
if r:
g = (diff(y(t),t) - eq[1])/r[f]
if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)):
return 'type1'
r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f)
if r:
g = (diff(y(t),t) - eq[1])/r[f]
if r and not (g.has(x(t)) or g.subs(y(t),v).has(t) or r[f].subs(x(t),u).subs(y(t),v).has(t)):
return 'type2'
g = Wild('g')
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
if r1 and r2 and not (r1[f].subs(x(t),u).subs(y(t),v).has(t) or \
r2[g].subs(x(t),u).subs(y(t),v).has(t)):
return 'type3'
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
num, den = (
(r1[f].subs(x(t),u).subs(y(t),v))/
(r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom()
R1 = num.match(f1*g1)
R2 = den.match(f2*g2)
# phi = (r1[f].subs(x(t),u).subs(y(t),v))/num
if R1 and R2:
return 'type4'
return None
def check_nonlinear_2eq_order2(eq, func, func_coef):
return None
def check_nonlinear_3eq_order1(eq, func, func_coef):
x = func[0].func
y = func[1].func
z = func[2].func
fc = func_coef
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
u, v, w = symbols('u, v, w', cls=Dummy)
a = Wild('a', exclude=[x(t), y(t), z(t), t])
b = Wild('b', exclude=[x(t), y(t), z(t), t])
c = Wild('c', exclude=[x(t), y(t), z(t), t])
f = Wild('f')
F1 = Wild('F1')
F2 = Wild('F2')
F3 = Wild('F3')
for i in range(3):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
r1 = eq[0].match(diff(x(t),t) - a*y(t)*z(t))
r2 = eq[1].match(diff(y(t),t) - b*z(t)*x(t))
r3 = eq[2].match(diff(z(t),t) - c*x(t)*y(t))
if r1 and r2 and r3:
num1, den1 = r1[a].as_numer_denom()
num2, den2 = r2[b].as_numer_denom()
num3, den3 = r3[c].as_numer_denom()
if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]):
return 'type1'
r = eq[0].match(diff(x(t),t) - y(t)*z(t)*f)
if r:
r1 = collect_const(r[f]).match(a*f)
r2 = ((diff(y(t),t) - eq[1])/r1[f]).match(b*z(t)*x(t))
r3 = ((diff(z(t),t) - eq[2])/r1[f]).match(c*x(t)*y(t))
if r1 and r2 and r3:
num1, den1 = r1[a].as_numer_denom()
num2, den2 = r2[b].as_numer_denom()
num3, den3 = r3[c].as_numer_denom()
if solve([num1*u-den1*(v-w), num2*v-den2*(w-u), num3*w-den3*(u-v)],[u, v]):
return 'type2'
r = eq[0].match(diff(x(t),t) - (F2-F3))
if r:
r1 = collect_const(r[F2]).match(c*F2)
r1.update(collect_const(r[F3]).match(b*F3))
if r1:
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
r1[F2], r1[F3] = r1[F3], r1[F2]
r1[c], r1[b] = -r1[b], -r1[c]
r2 = eq[1].match(diff(y(t),t) - a*r1[F3] + r1[c]*F1)
if r2:
r3 = (eq[2] == diff(z(t),t) - r1[b]*r2[F1] + r2[a]*r1[F2])
if r1 and r2 and r3:
return 'type3'
r = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3)
if r:
r1 = collect_const(r[F2]).match(c*F2)
r1.update(collect_const(r[F3]).match(b*F3))
if r1:
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
r1[F2], r1[F3] = r1[F3], r1[F2]
r1[c], r1[b] = -r1[b], -r1[c]
r2 = (diff(y(t),t) - eq[1]).match(a*x(t)*r1[F3] - r1[c]*z(t)*F1)
if r2:
r3 = (diff(z(t),t) - eq[2] == r1[b]*y(t)*r2[F1] - r2[a]*x(t)*r1[F2])
if r1 and r2 and r3:
return 'type4'
r = (diff(x(t),t) - eq[0]).match(x(t)*(F2 - F3))
if r:
r1 = collect_const(r[F2]).match(c*F2)
r1.update(collect_const(r[F3]).match(b*F3))
if r1:
if eq[1].has(r1[F2]) and not eq[1].has(r1[F3]):
r1[F2], r1[F3] = r1[F3], r1[F2]
r1[c], r1[b] = -r1[b], -r1[c]
r2 = (diff(y(t),t) - eq[1]).match(y(t)*(a*r1[F3] - r1[c]*F1))
if r2:
r3 = (diff(z(t),t) - eq[2] == z(t)*(r1[b]*r2[F1] - r2[a]*r1[F2]))
if r1 and r2 and r3:
return 'type5'
return None
def check_nonlinear_3eq_order2(eq, func, func_coef):
return None
@vectorize(0)
def odesimp(ode, eq, func, hint):
r"""
Simplifies solutions of ODEs, including trying to solve for ``func`` and
running :py:meth:`~sympy.solvers.ode.constantsimp`.
It may use knowledge of the type of solution that the hint returns to
apply additional simplifications.
It also attempts to integrate any :py:class:`~sympy.integrals.integrals.Integral`\s
in the expression, if the hint is not an ``_Integral`` hint.
This function should have no effect on expressions returned by
:py:meth:`~sympy.solvers.ode.dsolve`, as
:py:meth:`~sympy.solvers.ode.dsolve` already calls
:py:meth:`~sympy.solvers.ode.ode.odesimp`, but the individual hint functions
do not call :py:meth:`~sympy.solvers.ode.ode.odesimp` (because the
:py:meth:`~sympy.solvers.ode.dsolve` wrapper does). Therefore, this
function is designed for mainly internal use.
Examples
========
>>> from sympy import sin, symbols, dsolve, pprint, Function
>>> from sympy.solvers.ode.ode import odesimp
>>> x, u2, C1= symbols('x,u2,C1')
>>> f = Function('f')
>>> eq = dsolve(x*f(x).diff(x) - f(x) - x*sin(f(x)/x), f(x),
... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral',
... simplify=False)
>>> pprint(eq, wrap_line=False)
x
----
f(x)
/
|
| / 1 \
| -|u1 + -------|
| | /1 \|
| | sin|--||
| \ \u1//
log(f(x)) = log(C1) + | ---------------- d(u1)
| 2
| u1
|
/
>>> pprint(odesimp(eq, f(x), 1, {C1},
... hint='1st_homogeneous_coeff_subs_indep_div_dep'
... )) #doctest: +SKIP
x
--------- = C1
/f(x)\
tan|----|
\2*x /
"""
x = func.args[0]
f = func.func
C1 = get_numbered_constants(eq, num=1)
constants = eq.free_symbols - ode.free_symbols
# First, integrate if the hint allows it.
eq = _handle_Integral(eq, func, hint)
if hint.startswith("nth_linear_euler_eq_nonhomogeneous"):
eq = simplify(eq)
if not isinstance(eq, Equality):
raise TypeError("eq should be an instance of Equality")
# Second, clean up the arbitrary constants.
# Right now, nth linear hints can put as many as 2*order constants in an
# expression. If that number grows with another hint, the third argument
# here should be raised accordingly, or constantsimp() rewritten to handle
# an arbitrary number of constants.
eq = constantsimp(eq, constants)
# Lastly, now that we have cleaned up the expression, try solving for func.
# When CRootOf is implemented in solve(), we will want to return a CRootOf
# every time instead of an Equality.
# Get the f(x) on the left if possible.
if eq.rhs == func and not eq.lhs.has(func):
eq = [Eq(eq.rhs, eq.lhs)]
# make sure we are working with lists of solutions in simplified form.
if eq.lhs == func and not eq.rhs.has(func):
# The solution is already solved
eq = [eq]
else:
# The solution is not solved, so try to solve it
try:
floats = any(i.is_Float for i in eq.atoms(Number))
eqsol = solve(eq, func, force=True, rational=False if floats else None)
if not eqsol:
raise NotImplementedError
except (NotImplementedError, PolynomialError):
eq = [eq]
else:
def _expand(expr):
numer, denom = expr.as_numer_denom()
if denom.is_Add:
return expr
else:
return powsimp(expr.expand(), combine='exp', deep=True)
# XXX: the rest of odesimp() expects each ``t`` to be in a
# specific normal form: rational expression with numerator
# expanded, but with combined exponential functions (at
# least in this setup all tests pass).
eq = [Eq(f(x), _expand(t)) for t in eqsol]
# special simplification of the lhs.
if hint.startswith("1st_homogeneous_coeff"):
for j, eqi in enumerate(eq):
newi = logcombine(eqi, force=True)
if isinstance(newi.lhs, log) and newi.rhs == 0:
newi = Eq(newi.lhs.args[0]/C1, C1)
eq[j] = newi
# We cleaned up the constants before solving to help the solve engine with
# a simpler expression, but the solved expression could have introduced
# things like -C1, so rerun constantsimp() one last time before returning.
for i, eqi in enumerate(eq):
eq[i] = constantsimp(eqi, constants)
eq[i] = constant_renumber(eq[i], ode.free_symbols)
# If there is only 1 solution, return it;
# otherwise return the list of solutions.
if len(eq) == 1:
eq = eq[0]
return eq
def ode_sol_simplicity(sol, func, trysolving=True):
r"""
Returns an extended integer representing how simple a solution to an ODE
is.
The following things are considered, in order from most simple to least:
- ``sol`` is solved for ``func``.
- ``sol`` is not solved for ``func``, but can be if passed to solve (e.g.,
a solution returned by ``dsolve(ode, func, simplify=False``).
- If ``sol`` is not solved for ``func``, then base the result on the
length of ``sol``, as computed by ``len(str(sol))``.
- If ``sol`` has any unevaluated :py:class:`~sympy.integrals.integrals.Integral`\s,
this will automatically be considered less simple than any of the above.
This function returns an integer such that if solution A is simpler than
solution B by above metric, then ``ode_sol_simplicity(sola, func) <
ode_sol_simplicity(solb, func)``.
Currently, the following are the numbers returned, but if the heuristic is
ever improved, this may change. Only the ordering is guaranteed.
+----------------------------------------------+-------------------+
| Simplicity | Return |
+==============================================+===================+
| ``sol`` solved for ``func`` | ``-2`` |
+----------------------------------------------+-------------------+
| ``sol`` not solved for ``func`` but can be | ``-1`` |
+----------------------------------------------+-------------------+
| ``sol`` is not solved nor solvable for | ``len(str(sol))`` |
| ``func`` | |
+----------------------------------------------+-------------------+
| ``sol`` contains an | ``oo`` |
| :obj:`~sympy.integrals.integrals.Integral` | |
+----------------------------------------------+-------------------+
``oo`` here means the SymPy infinity, which should compare greater than
any integer.
If you already know :py:meth:`~sympy.solvers.solvers.solve` cannot solve
``sol``, you can use ``trysolving=False`` to skip that step, which is the
only potentially slow step. For example,
:py:meth:`~sympy.solvers.ode.dsolve` with the ``simplify=False`` flag
should do this.
If ``sol`` is a list of solutions, if the worst solution in the list
returns ``oo`` it returns that, otherwise it returns ``len(str(sol))``,
that is, the length of the string representation of the whole list.
Examples
========
This function is designed to be passed to ``min`` as the key argument,
such as ``min(listofsolutions, key=lambda i: ode_sol_simplicity(i,
f(x)))``.
>>> from sympy import symbols, Function, Eq, tan, Integral
>>> from sympy.solvers.ode.ode import ode_sol_simplicity
>>> x, C1, C2 = symbols('x, C1, C2')
>>> f = Function('f')
>>> ode_sol_simplicity(Eq(f(x), C1*x**2), f(x))
-2
>>> ode_sol_simplicity(Eq(x**2 + f(x), C1), f(x))
-1
>>> ode_sol_simplicity(Eq(f(x), C1*Integral(2*x, x)), f(x))
oo
>>> eq1 = Eq(f(x)/tan(f(x)/(2*x)), C1)
>>> eq2 = Eq(f(x)/tan(f(x)/(2*x) + f(x)), C2)
>>> [ode_sol_simplicity(eq, f(x)) for eq in [eq1, eq2]]
[28, 35]
>>> min([eq1, eq2], key=lambda i: ode_sol_simplicity(i, f(x)))
Eq(f(x)/tan(f(x)/(2*x)), C1)
"""
# TODO: if two solutions are solved for f(x), we still want to be
# able to get the simpler of the two
# See the docstring for the coercion rules. We check easier (faster)
# things here first, to save time.
if iterable(sol):
# See if there are Integrals
for i in sol:
if ode_sol_simplicity(i, func, trysolving=trysolving) == oo:
return oo
return len(str(sol))
if sol.has(Integral):
return oo
# Next, try to solve for func. This code will change slightly when CRootOf
# is implemented in solve(). Probably a CRootOf solution should fall
# somewhere between a normal solution and an unsolvable expression.
# First, see if they are already solved
if sol.lhs == func and not sol.rhs.has(func) or \
sol.rhs == func and not sol.lhs.has(func):
return -2
# We are not so lucky, try solving manually
if trysolving:
try:
sols = solve(sol, func)
if not sols:
raise NotImplementedError
except NotImplementedError:
pass
else:
return -1
# Finally, a naive computation based on the length of the string version
# of the expression. This may favor combined fractions because they
# will not have duplicate denominators, and may slightly favor expressions
# with fewer additions and subtractions, as those are separated by spaces
# by the printer.
# Additional ideas for simplicity heuristics are welcome, like maybe
# checking if a equation has a larger domain, or if constantsimp has
# introduced arbitrary constants numbered higher than the order of a
# given ODE that sol is a solution of.
return len(str(sol))
def _extract_funcs(eqs):
funcs = []
for eq in eqs:
derivs = [node for node in preorder_traversal(eq) if isinstance(node, Derivative)]
func = []
for d in derivs:
func += list(d.atoms(AppliedUndef))
for func_ in func:
funcs.append(func_)
funcs = list(uniq(funcs))
return funcs
def _get_constant_subexpressions(expr, Cs):
Cs = set(Cs)
Ces = []
def _recursive_walk(expr):
expr_syms = expr.free_symbols
if expr_syms and expr_syms.issubset(Cs):
Ces.append(expr)
else:
if expr.func == exp:
expr = expr.expand(mul=True)
if expr.func in (Add, Mul):
d = sift(expr.args, lambda i : i.free_symbols.issubset(Cs))
if len(d[True]) > 1:
x = expr.func(*d[True])
if not x.is_number:
Ces.append(x)
elif isinstance(expr, Integral):
if expr.free_symbols.issubset(Cs) and \
all(len(x) == 3 for x in expr.limits):
Ces.append(expr)
for i in expr.args:
_recursive_walk(i)
return
_recursive_walk(expr)
return Ces
def __remove_linear_redundancies(expr, Cs):
cnts = {i: expr.count(i) for i in Cs}
Cs = [i for i in Cs if cnts[i] > 0]
def _linear(expr):
if isinstance(expr, Add):
xs = [i for i in Cs if expr.count(i)==cnts[i] \
and 0 == expr.diff(i, 2)]
d = {}
for x in xs:
y = expr.diff(x)
if y not in d:
d[y]=[]
d[y].append(x)
for y in d:
if len(d[y]) > 1:
d[y].sort(key=str)
for x in d[y][1:]:
expr = expr.subs(x, 0)
return expr
def _recursive_walk(expr):
if len(expr.args) != 0:
expr = expr.func(*[_recursive_walk(i) for i in expr.args])
expr = _linear(expr)
return expr
if isinstance(expr, Equality):
lhs, rhs = [_recursive_walk(i) for i in expr.args]
f = lambda i: isinstance(i, Number) or i in Cs
if isinstance(lhs, Symbol) and lhs in Cs:
rhs, lhs = lhs, rhs
if lhs.func in (Add, Symbol) and rhs.func in (Add, Symbol):
dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f)
drhs = sift([rhs] if isinstance(rhs, AtomicExpr) else rhs.args, f)
for i in [True, False]:
for hs in [dlhs, drhs]:
if i not in hs:
hs[i] = [0]
# this calculation can be simplified
lhs = Add(*dlhs[False]) - Add(*drhs[False])
rhs = Add(*drhs[True]) - Add(*dlhs[True])
elif lhs.func in (Mul, Symbol) and rhs.func in (Mul, Symbol):
dlhs = sift([lhs] if isinstance(lhs, AtomicExpr) else lhs.args, f)
if True in dlhs:
if False not in dlhs:
dlhs[False] = [1]
lhs = Mul(*dlhs[False])
rhs = rhs/Mul(*dlhs[True])
return Eq(lhs, rhs)
else:
return _recursive_walk(expr)
@vectorize(0)
def constantsimp(expr, constants):
r"""
Simplifies an expression with arbitrary constants in it.
This function is written specifically to work with
:py:meth:`~sympy.solvers.ode.dsolve`, and is not intended for general use.
Simplification is done by "absorbing" the arbitrary constants into other
arbitrary constants, numbers, and symbols that they are not independent
of.
The symbols must all have the same name with numbers after it, for
example, ``C1``, ``C2``, ``C3``. The ``symbolname`` here would be
'``C``', the ``startnumber`` would be 1, and the ``endnumber`` would be 3.
If the arbitrary constants are independent of the variable ``x``, then the
independent symbol would be ``x``. There is no need to specify the
dependent function, such as ``f(x)``, because it already has the
independent symbol, ``x``, in it.
Because terms are "absorbed" into arbitrary constants and because
constants are renumbered after simplifying, the arbitrary constants in
expr are not necessarily equal to the ones of the same name in the
returned result.
If two or more arbitrary constants are added, multiplied, or raised to the
power of each other, they are first absorbed together into a single
arbitrary constant. Then the new constant is combined into other terms if
necessary.
Absorption of constants is done with limited assistance:
1. terms of :py:class:`~sympy.core.add.Add`\s are collected to try join
constants so `e^x (C_1 \cos(x) + C_2 \cos(x))` will simplify to `e^x
C_1 \cos(x)`;
2. powers with exponents that are :py:class:`~sympy.core.add.Add`\s are
expanded so `e^{C_1 + x}` will be simplified to `C_1 e^x`.
Use :py:meth:`~sympy.solvers.ode.ode.constant_renumber` to renumber constants
after simplification or else arbitrary numbers on constants may appear,
e.g. `C_1 + C_3 x`.
In rare cases, a single constant can be "simplified" into two constants.
Every differential equation solution should have as many arbitrary
constants as the order of the differential equation. The result here will
be technically correct, but it may, for example, have `C_1` and `C_2` in
an expression, when `C_1` is actually equal to `C_2`. Use your discretion
in such situations, and also take advantage of the ability to use hints in
:py:meth:`~sympy.solvers.ode.dsolve`.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers.ode.ode import constantsimp
>>> C1, C2, C3, x, y = symbols('C1, C2, C3, x, y')
>>> constantsimp(2*C1*x, {C1, C2, C3})
C1*x
>>> constantsimp(C1 + 2 + x, {C1, C2, C3})
C1 + x
>>> constantsimp(C1*C2 + 2 + C2 + C3*x, {C1, C2, C3})
C1 + C3*x
"""
# This function works recursively. The idea is that, for Mul,
# Add, Pow, and Function, if the class has a constant in it, then
# we can simplify it, which we do by recursing down and
# simplifying up. Otherwise, we can skip that part of the
# expression.
Cs = constants
orig_expr = expr
constant_subexprs = _get_constant_subexpressions(expr, Cs)
for xe in constant_subexprs:
xes = list(xe.free_symbols)
if not xes:
continue
if all(expr.count(c) == xe.count(c) for c in xes):
xes.sort(key=str)
expr = expr.subs(xe, xes[0])
# try to perform common sub-expression elimination of constant terms
try:
commons, rexpr = cse(expr)
commons.reverse()
rexpr = rexpr[0]
for s in commons:
cs = list(s[1].atoms(Symbol))
if len(cs) == 1 and cs[0] in Cs and \
cs[0] not in rexpr.atoms(Symbol) and \
not any(cs[0] in ex for ex in commons if ex != s):
rexpr = rexpr.subs(s[0], cs[0])
else:
rexpr = rexpr.subs(*s)
expr = rexpr
except IndexError:
pass
expr = __remove_linear_redundancies(expr, Cs)
def _conditional_term_factoring(expr):
new_expr = terms_gcd(expr, clear=False, deep=True, expand=False)
# we do not want to factor exponentials, so handle this separately
if new_expr.is_Mul:
infac = False
asfac = False
for m in new_expr.args:
if isinstance(m, exp):
asfac = True
elif m.is_Add:
infac = any(isinstance(fi, exp) for t in m.args
for fi in Mul.make_args(t))
if asfac and infac:
new_expr = expr
break
return new_expr
expr = _conditional_term_factoring(expr)
# call recursively if more simplification is possible
if orig_expr != expr:
return constantsimp(expr, Cs)
return expr
def constant_renumber(expr, variables=None, newconstants=None):
r"""
Renumber arbitrary constants in ``expr`` to use the symbol names as given
in ``newconstants``. In the process, this reorders expression terms in a
standard way.
If ``newconstants`` is not provided then the new constant names will be
``C1``, ``C2`` etc. Otherwise ``newconstants`` should be an iterable
giving the new symbols to use for the constants in order.
The ``variables`` argument is a list of non-constant symbols. All other
free symbols found in ``expr`` are assumed to be constants and will be
renumbered. If ``variables`` is not given then any numbered symbol
beginning with ``C`` (e.g. ``C1``) is assumed to be a constant.
Symbols are renumbered based on ``.sort_key()``, so they should be
numbered roughly in the order that they appear in the final, printed
expression. Note that this ordering is based in part on hashes, so it can
produce different results on different machines.
The structure of this function is very similar to that of
:py:meth:`~sympy.solvers.ode.constantsimp`.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers.ode.ode import constant_renumber
>>> x, C1, C2, C3 = symbols('x,C1:4')
>>> expr = C3 + C2*x + C1*x**2
>>> expr
C1*x**2 + C2*x + C3
>>> constant_renumber(expr)
C1 + C2*x + C3*x**2
The ``variables`` argument specifies which are constants so that the
other symbols will not be renumbered:
>>> constant_renumber(expr, [C1, x])
C1*x**2 + C2 + C3*x
The ``newconstants`` argument is used to specify what symbols to use when
replacing the constants:
>>> constant_renumber(expr, [x], newconstants=symbols('E1:4'))
E1 + E2*x + E3*x**2
"""
# System of expressions
if isinstance(expr, (set, list, tuple)):
return type(expr)(constant_renumber(Tuple(*expr),
variables=variables, newconstants=newconstants))
# Symbols in solution but not ODE are constants
if variables is not None:
variables = set(variables)
free_symbols = expr.free_symbols
constantsymbols = list(free_symbols - variables)
# Any Cn is a constant...
else:
variables = set()
isconstant = lambda s: s.startswith('C') and s[1:].isdigit()
constantsymbols = [sym for sym in expr.free_symbols if isconstant(sym.name)]
# Find new constants checking that they aren't already in the ODE
if newconstants is None:
iter_constants = numbered_symbols(start=1, prefix='C', exclude=variables)
else:
iter_constants = (sym for sym in newconstants if sym not in variables)
constants_found = []
# make a mapping to send all constantsymbols to S.One and use
# that to make sure that term ordering is not dependent on
# the indexed value of C
C_1 = [(ci, S.One) for ci in constantsymbols]
sort_key=lambda arg: default_sort_key(arg.subs(C_1))
def _constant_renumber(expr):
r"""
We need to have an internal recursive function
"""
# For system of expressions
if isinstance(expr, Tuple):
renumbered = [_constant_renumber(e) for e in expr]
return Tuple(*renumbered)
if isinstance(expr, Equality):
return Eq(
_constant_renumber(expr.lhs),
_constant_renumber(expr.rhs))
if type(expr) not in (Mul, Add, Pow) and not expr.is_Function and \
not expr.has(*constantsymbols):
# Base case, as above. Hope there aren't constants inside
# of some other class, because they won't be renumbered.
return expr
elif expr.is_Piecewise:
return expr
elif expr in constantsymbols:
if expr not in constants_found:
constants_found.append(expr)
return expr
elif expr.is_Function or expr.is_Pow:
return expr.func(
*[_constant_renumber(x) for x in expr.args])
else:
sortedargs = list(expr.args)
sortedargs.sort(key=sort_key)
return expr.func(*[_constant_renumber(x) for x in sortedargs])
expr = _constant_renumber(expr)
# Don't renumber symbols present in the ODE.
constants_found = [c for c in constants_found if c not in variables]
# Renumbering happens here
subs_dict = {var: cons for var, cons in zip(constants_found, iter_constants)}
expr = expr.subs(subs_dict, simultaneous=True)
return expr
def _handle_Integral(expr, func, hint):
r"""
Converts a solution with Integrals in it into an actual solution.
For most hints, this simply runs ``expr.doit()``.
"""
if hint == "nth_linear_constant_coeff_homogeneous":
sol = expr
elif not hint.endswith("_Integral"):
sol = expr.doit()
else:
sol = expr
return sol
# XXX: Should this function maybe go somewhere else?
def homogeneous_order(eq, *symbols):
r"""
Returns the order `n` if `g` is homogeneous and ``None`` if it is not
homogeneous.
Determines if a function is homogeneous and if so of what order. A
function `f(x, y, \cdots)` is homogeneous of order `n` if `f(t x, t y,
\cdots) = t^n f(x, y, \cdots)`.
If the function is of two variables, `F(x, y)`, then `f` being homogeneous
of any order is equivalent to being able to rewrite `F(x, y)` as `G(x/y)`
or `H(y/x)`. This fact is used to solve 1st order ordinary differential
equations whose coefficients are homogeneous of the same order (see the
docstrings of
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep` and
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`).
Symbols can be functions, but every argument of the function must be a
symbol, and the arguments of the function that appear in the expression
must match those given in the list of symbols. If a declared function
appears with different arguments than given in the list of symbols,
``None`` is returned.
Examples
========
>>> from sympy import Function, homogeneous_order, sqrt
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> homogeneous_order(f(x), f(x)) is None
True
>>> homogeneous_order(f(x,y), f(y, x), x, y) is None
True
>>> homogeneous_order(f(x), f(x), x)
1
>>> homogeneous_order(x**2*f(x)/sqrt(x**2+f(x)**2), x, f(x))
2
>>> homogeneous_order(x**2+f(x), x, f(x)) is None
True
"""
if not symbols:
raise ValueError("homogeneous_order: no symbols were given.")
symset = set(symbols)
eq = sympify(eq)
# The following are not supported
if eq.has(Order, Derivative):
return None
# These are all constants
if (eq.is_Number or
eq.is_NumberSymbol or
eq.is_number
):
return S.Zero
# Replace all functions with dummy variables
dum = numbered_symbols(prefix='d', cls=Dummy)
newsyms = set()
for i in [j for j in symset if getattr(j, 'is_Function')]:
iargs = set(i.args)
if iargs.difference(symset):
return None
else:
dummyvar = next(dum)
eq = eq.subs(i, dummyvar)
symset.remove(i)
newsyms.add(dummyvar)
symset.update(newsyms)
if not eq.free_symbols & symset:
return None
# assuming order of a nested function can only be equal to zero
if isinstance(eq, Function):
return None if homogeneous_order(
eq.args[0], *tuple(symset)) != 0 else S.Zero
# make the replacement of x with x*t and see if t can be factored out
t = Dummy('t', positive=True) # It is sufficient that t > 0
eqs = separatevars(eq.subs([(i, t*i) for i in symset]), [t], dict=True)[t]
if eqs is S.One:
return S.Zero # there was no term with only t
i, d = eqs.as_independent(t, as_Add=False)
b, e = d.as_base_exp()
if b == t:
return e
def ode_2nd_power_series_ordinary(eq, func, order, match):
r"""
Gives a power series solution to a second order homogeneous differential
equation with polynomial coefficients at an ordinary point. A homogeneous
differential equation is of the form
.. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) y(x) = 0
For simplicity it is assumed that `P(x)`, `Q(x)` and `R(x)` are polynomials,
it is sufficient that `\frac{Q(x)}{P(x)}` and `\frac{R(x)}{P(x)}` exists at
`x_{0}`. A recurrence relation is obtained by substituting `y` as `\sum_{n=0}^\infty a_{n}x^{n}`,
in the differential equation, and equating the nth term. Using this relation
various terms can be generated.
Examples
========
>>> from sympy import dsolve, Function, pprint
>>> from sympy.abc import x
>>> f = Function("f")
>>> eq = f(x).diff(x, 2) + f(x)
>>> pprint(dsolve(eq, hint='2nd_power_series_ordinary'))
/ 4 2 \ / 2\
|x x | | x | / 6\
f(x) = C2*|-- - -- + 1| + C1*x*|1 - --| + O\x /
\24 2 / \ 6 /
References
==========
- http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx
- George E. Simmons, "Differential Equations with Applications and
Historical Notes", p.p 176 - 184
"""
x = func.args[0]
f = func.func
C0, C1 = get_numbered_constants(eq, num=2)
n = Dummy("n", integer=True)
s = Wild("s")
k = Wild("k", exclude=[x])
x0 = match['x0']
terms = match['terms']
p = match[match['a3']]
q = match[match['b3']]
r = match[match['c3']]
seriesdict = {}
recurr = Function("r")
# Generating the recurrence relation which works this way:
# for the second order term the summation begins at n = 2. The coefficients
# p is multiplied with an*(n - 1)*(n - 2)*x**n-2 and a substitution is made such that
# the exponent of x becomes n.
# For example, if p is x, then the second degree recurrence term is
# an*(n - 1)*(n - 2)*x**n-1, substituting (n - 1) as n, it transforms to
# an+1*n*(n - 1)*x**n.
# A similar process is done with the first order and zeroth order term.
coefflist = [(recurr(n), r), (n*recurr(n), q), (n*(n - 1)*recurr(n), p)]
for index, coeff in enumerate(coefflist):
if coeff[1]:
f2 = powsimp(expand((coeff[1]*(x - x0)**(n - index)).subs(x, x + x0)))
if f2.is_Add:
addargs = f2.args
else:
addargs = [f2]
for arg in addargs:
powm = arg.match(s*x**k)
term = coeff[0]*powm[s]
if not powm[k].is_Symbol:
term = term.subs(n, n - powm[k].as_independent(n)[0])
startind = powm[k].subs(n, index)
# Seeing if the startterm can be reduced further.
# If it vanishes for n lesser than startind, it is
# equal to summation from n.
if startind:
for i in reversed(range(startind)):
if not term.subs(n, i):
seriesdict[term] = i
else:
seriesdict[term] = i + 1
break
else:
seriesdict[term] = S.Zero
# Stripping of terms so that the sum starts with the same number.
teq = S.Zero
suminit = seriesdict.values()
rkeys = seriesdict.keys()
req = Add(*rkeys)
if any(suminit):
maxval = max(suminit)
for term in seriesdict:
val = seriesdict[term]
if val != maxval:
for i in range(val, maxval):
teq += term.subs(n, val)
finaldict = {}
if teq:
fargs = teq.atoms(AppliedUndef)
if len(fargs) == 1:
finaldict[fargs.pop()] = 0
else:
maxf = max(fargs, key = lambda x: x.args[0])
sol = solve(teq, maxf)
if isinstance(sol, list):
sol = sol[0]
finaldict[maxf] = sol
# Finding the recurrence relation in terms of the largest term.
fargs = req.atoms(AppliedUndef)
maxf = max(fargs, key = lambda x: x.args[0])
minf = min(fargs, key = lambda x: x.args[0])
if minf.args[0].is_Symbol:
startiter = 0
else:
startiter = -minf.args[0].as_independent(n)[0]
lhs = maxf
rhs = solve(req, maxf)
if isinstance(rhs, list):
rhs = rhs[0]
# Checking how many values are already present
tcounter = len([t for t in finaldict.values() if t])
for _ in range(tcounter, terms - 3): # Assuming c0 and c1 to be arbitrary
check = rhs.subs(n, startiter)
nlhs = lhs.subs(n, startiter)
nrhs = check.subs(finaldict)
finaldict[nlhs] = nrhs
startiter += 1
# Post processing
series = C0 + C1*(x - x0)
for term in finaldict:
if finaldict[term]:
fact = term.args[0]
series += (finaldict[term].subs([(recurr(0), C0), (recurr(1), C1)])*(
x - x0)**fact)
series = collect(expand_mul(series), [C0, C1]) + Order(x**terms)
return Eq(f(x), series)
def ode_2nd_power_series_regular(eq, func, order, match):
r"""
Gives a power series solution to a second order homogeneous differential
equation with polynomial coefficients at a regular point. A second order
homogeneous differential equation is of the form
.. math :: P(x)\frac{d^2y}{dx^2} + Q(x)\frac{dy}{dx} + R(x) y(x) = 0
A point is said to regular singular at `x0` if `x - x0\frac{Q(x)}{P(x)}`
and `(x - x0)^{2}\frac{R(x)}{P(x)}` are analytic at `x0`. For simplicity
`P(x)`, `Q(x)` and `R(x)` are assumed to be polynomials. The algorithm for
finding the power series solutions is:
1. Try expressing `(x - x0)P(x)` and `((x - x0)^{2})Q(x)` as power series
solutions about x0. Find `p0` and `q0` which are the constants of the
power series expansions.
2. Solve the indicial equation `f(m) = m(m - 1) + m*p0 + q0`, to obtain the
roots `m1` and `m2` of the indicial equation.
3. If `m1 - m2` is a non integer there exists two series solutions. If
`m1 = m2`, there exists only one solution. If `m1 - m2` is an integer,
then the existence of one solution is confirmed. The other solution may
or may not exist.
The power series solution is of the form `x^{m}\sum_{n=0}^\infty a_{n}x^{n}`. The
coefficients are determined by the following recurrence relation.
`a_{n} = -\frac{\sum_{k=0}^{n-1} q_{n-k} + (m + k)p_{n-k}}{f(m + n)}`. For the case
in which `m1 - m2` is an integer, it can be seen from the recurrence relation
that for the lower root `m`, when `n` equals the difference of both the
roots, the denominator becomes zero. So if the numerator is not equal to zero,
a second series solution exists.
Examples
========
>>> from sympy import dsolve, Function, pprint
>>> from sympy.abc import x
>>> f = Function("f")
>>> eq = x*(f(x).diff(x, 2)) + 2*(f(x).diff(x)) + x*f(x)
>>> pprint(dsolve(eq, hint='2nd_power_series_regular'))
/ 6 4 2 \
| x x x |
/ 4 2 \ C1*|- --- + -- - -- + 1|
| x x | \ 720 24 2 / / 6\
f(x) = C2*|--- - -- + 1| + ------------------------ + O\x /
\120 6 / x
References
==========
- George E. Simmons, "Differential Equations with Applications and
Historical Notes", p.p 176 - 184
"""
x = func.args[0]
f = func.func
C0, C1 = get_numbered_constants(eq, num=2)
m = Dummy("m") # for solving the indicial equation
x0 = match['x0']
terms = match['terms']
p = match['p']
q = match['q']
# Generating the indicial equation
indicial = []
for term in [p, q]:
if not term.has(x):
indicial.append(term)
else:
term = series(term, x=x, n=1, x0=x0)
if isinstance(term, Order):
indicial.append(S.Zero)
else:
for arg in term.args:
if not arg.has(x):
indicial.append(arg)
break
p0, q0 = indicial
sollist = solve(m*(m - 1) + m*p0 + q0, m)
if sollist and isinstance(sollist, list) and all(
sol.is_real for sol in sollist):
serdict1 = {}
serdict2 = {}
if len(sollist) == 1:
# Only one series solution exists in this case.
m1 = m2 = sollist.pop()
if terms-m1-1 <= 0:
return Eq(f(x), Order(terms))
serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0)
else:
m1 = sollist[0]
m2 = sollist[1]
if m1 < m2:
m1, m2 = m2, m1
# Irrespective of whether m1 - m2 is an integer or not, one
# Frobenius series solution exists.
serdict1 = _frobenius(terms-m1-1, m1, p0, q0, p, q, x0, x, C0)
if not (m1 - m2).is_integer:
# Second frobenius series solution exists.
serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1)
else:
# Check if second frobenius series solution exists.
serdict2 = _frobenius(terms-m2-1, m2, p0, q0, p, q, x0, x, C1, check=m1)
if serdict1:
finalseries1 = C0
for key in serdict1:
power = int(key.name[1:])
finalseries1 += serdict1[key]*(x - x0)**power
finalseries1 = (x - x0)**m1*finalseries1
finalseries2 = S.Zero
if serdict2:
for key in serdict2:
power = int(key.name[1:])
finalseries2 += serdict2[key]*(x - x0)**power
finalseries2 += C1
finalseries2 = (x - x0)**m2*finalseries2
return Eq(f(x), collect(finalseries1 + finalseries2,
[C0, C1]) + Order(x**terms))
def _frobenius(n, m, p0, q0, p, q, x0, x, c, check=None):
r"""
Returns a dict with keys as coefficients and values as their values in terms of C0
"""
n = int(n)
# In cases where m1 - m2 is not an integer
m2 = check
d = Dummy("d")
numsyms = numbered_symbols("C", start=0)
numsyms = [next(numsyms) for i in range(n + 1)]
serlist = []
for ser in [p, q]:
# Order term not present
if ser.is_polynomial(x) and Poly(ser, x).degree() <= n:
if x0:
ser = ser.subs(x, x + x0)
dict_ = Poly(ser, x).as_dict()
# Order term present
else:
tseries = series(ser, x=x0, n=n+1)
# Removing order
dict_ = Poly(list(ordered(tseries.args))[: -1], x).as_dict()
# Fill in with zeros, if coefficients are zero.
for i in range(n + 1):
if (i,) not in dict_:
dict_[(i,)] = S.Zero
serlist.append(dict_)
pseries = serlist[0]
qseries = serlist[1]
indicial = d*(d - 1) + d*p0 + q0
frobdict = {}
for i in range(1, n + 1):
num = c*(m*pseries[(i,)] + qseries[(i,)])
for j in range(1, i):
sym = Symbol("C" + str(j))
num += frobdict[sym]*((m + j)*pseries[(i - j,)] + qseries[(i - j,)])
# Checking for cases when m1 - m2 is an integer. If num equals zero
# then a second Frobenius series solution cannot be found. If num is not zero
# then set constant as zero and proceed.
if m2 is not None and i == m2 - m:
if num:
return False
else:
frobdict[numsyms[i]] = S.Zero
else:
frobdict[numsyms[i]] = -num/(indicial.subs(d, m+i))
return frobdict
def _remove_redundant_solutions(eq, solns, order, var):
r"""
Remove redundant solutions from the set of solutions.
This function is needed because otherwise dsolve can return
redundant solutions. As an example consider:
eq = Eq((f(x).diff(x, 2))*f(x).diff(x), 0)
There are two ways to find solutions to eq. The first is to solve f(x).diff(x, 2) = 0
leading to solution f(x)=C1 + C2*x. The second is to solve the equation f(x).diff(x) = 0
leading to the solution f(x) = C1. In this particular case we then see
that the second solution is a special case of the first and we do not
want to return it.
This does not always happen. If we have
eq = Eq((f(x)**2-4)*(f(x).diff(x)-4), 0)
then we get the algebraic solution f(x) = [-2, 2] and the integral solution
f(x) = x + C1 and in this case the two solutions are not equivalent wrt
initial conditions so both should be returned.
"""
def is_special_case_of(soln1, soln2):
return _is_special_case_of(soln1, soln2, eq, order, var)
unique_solns = []
for soln1 in solns:
for soln2 in unique_solns[:]:
if is_special_case_of(soln1, soln2):
break
elif is_special_case_of(soln2, soln1):
unique_solns.remove(soln2)
else:
unique_solns.append(soln1)
return unique_solns
def _is_special_case_of(soln1, soln2, eq, order, var):
r"""
True if soln1 is found to be a special case of soln2 wrt some value of the
constants that appear in soln2. False otherwise.
"""
# The solutions returned by dsolve may be given explicitly or implicitly.
# We will equate the sol1=(soln1.rhs - soln1.lhs), sol2=(soln2.rhs - soln2.lhs)
# of the two solutions.
#
# Since this is supposed to hold for all x it also holds for derivatives.
# For an order n ode we should be able to differentiate
# each solution n times to get n+1 equations.
#
# We then try to solve those n+1 equations for the integrations constants
# in sol2. If we can find a solution that does not depend on x then it
# means that some value of the constants in sol1 is a special case of
# sol2 corresponding to a particular choice of the integration constants.
# In case the solution is in implicit form we subtract the sides
soln1 = soln1.rhs - soln1.lhs
soln2 = soln2.rhs - soln2.lhs
# Work for the series solution
if soln1.has(Order) and soln2.has(Order):
if soln1.getO() == soln2.getO():
soln1 = soln1.removeO()
soln2 = soln2.removeO()
else:
return False
elif soln1.has(Order) or soln2.has(Order):
return False
constants1 = soln1.free_symbols.difference(eq.free_symbols)
constants2 = soln2.free_symbols.difference(eq.free_symbols)
constants1_new = get_numbered_constants(Tuple(soln1, soln2), len(constants1))
if len(constants1) == 1:
constants1_new = {constants1_new}
for c_old, c_new in zip(constants1, constants1_new):
soln1 = soln1.subs(c_old, c_new)
# n equations for sol1 = sol2, sol1'=sol2', ...
lhs = soln1
rhs = soln2
eqns = [Eq(lhs, rhs)]
for n in range(1, order):
lhs = lhs.diff(var)
rhs = rhs.diff(var)
eq = Eq(lhs, rhs)
eqns.append(eq)
# BooleanTrue/False awkwardly show up for trivial equations
if any(isinstance(eq, BooleanFalse) for eq in eqns):
return False
eqns = [eq for eq in eqns if not isinstance(eq, BooleanTrue)]
try:
constant_solns = solve(eqns, constants2)
except NotImplementedError:
return False
# Sometimes returns a dict and sometimes a list of dicts
if isinstance(constant_solns, dict):
constant_solns = [constant_solns]
# after solving the issue 17418, maybe we don't need the following checksol code.
for constant_soln in constant_solns:
for eq in eqns:
eq=eq.rhs-eq.lhs
if checksol(eq, constant_soln) is not True:
return False
# If any solution gives all constants as expressions that don't depend on
# x then there exists constants for soln2 that give soln1
for constant_soln in constant_solns:
if not any(c.has(var) for c in constant_soln.values()):
return True
return False
def ode_1st_power_series(eq, func, order, match):
r"""
The power series solution is a method which gives the Taylor series expansion
to the solution of a differential equation.
For a first order differential equation `\frac{dy}{dx} = h(x, y)`, a power
series solution exists at a point `x = x_{0}` if `h(x, y)` is analytic at `x_{0}`.
The solution is given by
.. math:: y(x) = y(x_{0}) + \sum_{n = 1}^{\infty} \frac{F_{n}(x_{0},b)(x - x_{0})^n}{n!},
where `y(x_{0}) = b` is the value of y at the initial value of `x_{0}`.
To compute the values of the `F_{n}(x_{0},b)` the following algorithm is
followed, until the required number of terms are generated.
1. `F_1 = h(x_{0}, b)`
2. `F_{n+1} = \frac{\partial F_{n}}{\partial x} + \frac{\partial F_{n}}{\partial y}F_{1}`
Examples
========
>>> from sympy import Function, pprint, exp, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = exp(x)*(f(x).diff(x)) - f(x)
>>> pprint(dsolve(eq, hint='1st_power_series'))
3 4 5
C1*x C1*x C1*x / 6\
f(x) = C1 + C1*x - ----- + ----- + ----- + O\x /
6 24 60
References
==========
- Travis W. Walker, Analytic power series technique for solving first-order
differential equations, p.p 17, 18
"""
x = func.args[0]
y = match['y']
f = func.func
h = -match[match['d']]/match[match['e']]
point = match['f0']
value = match['f0val']
terms = match['terms']
# First term
F = h
if not h:
return Eq(f(x), value)
# Initialization
series = value
if terms > 1:
hc = h.subs({x: point, y: value})
if hc.has(oo) or hc.has(nan) or hc.has(zoo):
# Derivative does not exist, not analytic
return Eq(f(x), oo)
elif hc:
series += hc*(x - point)
for factcount in range(2, terms):
Fnew = F.diff(x) + F.diff(y)*h
Fnewc = Fnew.subs({x: point, y: value})
# Same logic as above
if Fnewc.has(oo) or Fnewc.has(nan) or Fnewc.has(-oo) or Fnewc.has(zoo):
return Eq(f(x), oo)
series += Fnewc*((x - point)**factcount)/factorial(factcount)
F = Fnew
series += Order(x**terms)
return Eq(f(x), series)
def checkinfsol(eq, infinitesimals, func=None, order=None):
r"""
This function is used to check if the given infinitesimals are the
actual infinitesimals of the given first order differential equation.
This method is specific to the Lie Group Solver of ODEs.
As of now, it simply checks, by substituting the infinitesimals in the
partial differential equation.
.. math:: \frac{\partial \eta}{\partial x} + \left(\frac{\partial \eta}{\partial y}
- \frac{\partial \xi}{\partial x}\right)*h
- \frac{\partial \xi}{\partial y}*h^{2}
- \xi\frac{\partial h}{\partial x} - \eta\frac{\partial h}{\partial y} = 0
where `\eta`, and `\xi` are the infinitesimals and `h(x,y) = \frac{dy}{dx}`
The infinitesimals should be given in the form of a list of dicts
``[{xi(x, y): inf, eta(x, y): inf}]``, corresponding to the
output of the function infinitesimals. It returns a list
of values of the form ``[(True/False, sol)]`` where ``sol`` is the value
obtained after substituting the infinitesimals in the PDE. If it
is ``True``, then ``sol`` would be 0.
"""
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
if not func:
eq, func = _preprocess(eq)
variables = func.args
if len(variables) != 1:
raise ValueError("ODE's have only one independent variable")
else:
x = variables[0]
if not order:
order = ode_order(eq, func)
if order != 1:
raise NotImplementedError("Lie groups solver has been implemented "
"only for first order differential equations")
else:
df = func.diff(x)
a = Wild('a', exclude = [df])
b = Wild('b', exclude = [df])
match = collect(expand(eq), df).match(a*df + b)
if match:
h = -simplify(match[b]/match[a])
else:
try:
sol = solve(eq, df)
except NotImplementedError:
raise NotImplementedError("Infinitesimals for the "
"first order ODE could not be found")
else:
h = sol[0] # Find infinitesimals for one solution
y = Dummy('y')
h = h.subs(func, y)
xi = Function('xi')(x, y)
eta = Function('eta')(x, y)
dxi = Function('xi')(x, func)
deta = Function('eta')(x, func)
pde = (eta.diff(x) + (eta.diff(y) - xi.diff(x))*h -
(xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y)))
soltup = []
for sol in infinitesimals:
tsol = {xi: S(sol[dxi]).subs(func, y),
eta: S(sol[deta]).subs(func, y)}
sol = simplify(pde.subs(tsol).doit())
if sol:
soltup.append((False, sol.subs(y, func)))
else:
soltup.append((True, 0))
return soltup
def sysode_linear_2eq_order1(match_):
x = match_['func'][0].func
y = match_['func'][1].func
func = match_['func']
fc = match_['func_coeff']
eq = match_['eq']
r = dict()
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
for i in range(2):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
# for equations Eq(a1*diff(x(t),t), a*x(t) + b*y(t) + k1)
# and Eq(a2*diff(x(t),t), c*x(t) + d*y(t) + k2)
r['a'] = -fc[0,x(t),0]/fc[0,x(t),1]
r['c'] = -fc[1,x(t),0]/fc[1,y(t),1]
r['b'] = -fc[0,y(t),0]/fc[0,x(t),1]
r['d'] = -fc[1,y(t),0]/fc[1,y(t),1]
forcing = [S.Zero,S.Zero]
for i in range(2):
for j in Add.make_args(eq[i]):
if not j.has(x(t), y(t)):
forcing[i] += j
if not (forcing[0].has(t) or forcing[1].has(t)):
r['k1'] = forcing[0]
r['k2'] = forcing[1]
else:
raise NotImplementedError("Only homogeneous problems are supported" +
" (and constant inhomogeneity)")
if match_['type_of_equation'] == 'type6':
sol = _linear_2eq_order1_type6(x, y, t, r, eq)
if match_['type_of_equation'] == 'type7':
sol = _linear_2eq_order1_type7(x, y, t, r, eq)
return sol
def _linear_2eq_order1_type6(x, y, t, r, eq):
r"""
The equations of this type of ode are .
.. math:: x' = f(t) x + g(t) y
.. math:: y' = a [f(t) + a h(t)] x + a [g(t) - h(t)] y
This is solved by first multiplying the first equation by `-a` and adding
it to the second equation to obtain
.. math:: y' - a x' = -a h(t) (y - a x)
Setting `U = y - ax` and integrating the equation we arrive at
.. math:: y - ax = C_1 e^{-a \int h(t) \,dt}
and on substituting the value of y in first equation give rise to first order ODEs. After solving for
`x`, we can obtain `y` by substituting the value of `x` in second equation.
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
p = 0
q = 0
p1 = cancel(r['c']/cancel(r['c']/r['d']).as_numer_denom()[0])
p2 = cancel(r['a']/cancel(r['a']/r['b']).as_numer_denom()[0])
for n, i in enumerate([p1, p2]):
for j in Mul.make_args(collect_const(i)):
if not j.has(t):
q = j
if q!=0 and n==0:
if ((r['c']/j - r['a'])/(r['b'] - r['d']/j)) == j:
p = 1
s = j
break
if q!=0 and n==1:
if ((r['a']/j - r['c'])/(r['d'] - r['b']/j)) == j:
p = 2
s = j
break
if p == 1:
equ = diff(x(t),t) - r['a']*x(t) - r['b']*(s*x(t) + C1*exp(-s*Integral(r['b'] - r['d']/s, t)))
hint1 = classify_ode(equ)[1]
sol1 = dsolve(equ, hint=hint1+'_Integral').rhs
sol2 = s*sol1 + C1*exp(-s*Integral(r['b'] - r['d']/s, t))
elif p ==2:
equ = diff(y(t),t) - r['c']*y(t) - r['d']*s*y(t) + C1*exp(-s*Integral(r['d'] - r['b']/s, t))
hint1 = classify_ode(equ)[1]
sol2 = dsolve(equ, hint=hint1+'_Integral').rhs
sol1 = s*sol2 + C1*exp(-s*Integral(r['d'] - r['b']/s, t))
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def _linear_2eq_order1_type7(x, y, t, r, eq):
r"""
The equations of this type of ode are .
.. math:: x' = f(t) x + g(t) y
.. math:: y' = h(t) x + p(t) y
Differentiating the first equation and substituting the value of `y`
from second equation will give a second-order linear equation
.. math:: g x'' - (fg + gp + g') x' + (fgp - g^{2} h + f g' - f' g) x = 0
This above equation can be easily integrated if following conditions are satisfied.
1. `fgp - g^{2} h + f g' - f' g = 0`
2. `fgp - g^{2} h + f g' - f' g = ag, fg + gp + g' = bg`
If first condition is satisfied then it is solved by current dsolve solver and in second case it becomes
a constant coefficient differential equation which is also solved by current solver.
Otherwise if the above condition fails then,
a particular solution is assumed as `x = x_0(t)` and `y = y_0(t)`
Then the general solution is expressed as
.. math:: x = C_1 x_0(t) + C_2 x_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt
.. math:: y = C_1 y_0(t) + C_2 [\frac{F(t) P(t)}{x_0(t)} + y_0(t) \int \frac{g(t) F(t) P(t)}{x_0^{2}(t)} \,dt]
where C1 and C2 are arbitrary constants and
.. math:: F(t) = e^{\int f(t) \,dt}, P(t) = e^{\int p(t) \,dt}
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
e1 = r['a']*r['b']*r['c'] - r['b']**2*r['c'] + r['a']*diff(r['b'],t) - diff(r['a'],t)*r['b']
e2 = r['a']*r['c']*r['d'] - r['b']*r['c']**2 + diff(r['c'],t)*r['d'] - r['c']*diff(r['d'],t)
m1 = r['a']*r['b'] + r['b']*r['d'] + diff(r['b'],t)
m2 = r['a']*r['c'] + r['c']*r['d'] + diff(r['c'],t)
if e1 == 0:
sol1 = dsolve(r['b']*diff(x(t),t,t) - m1*diff(x(t),t)).rhs
sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs
elif e2 == 0:
sol2 = dsolve(r['c']*diff(y(t),t,t) - m2*diff(y(t),t)).rhs
sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs
elif not (e1/r['b']).has(t) and not (m1/r['b']).has(t):
sol1 = dsolve(diff(x(t),t,t) - (m1/r['b'])*diff(x(t),t) - (e1/r['b'])*x(t)).rhs
sol2 = dsolve(diff(y(t),t) - r['c']*sol1 - r['d']*y(t)).rhs
elif not (e2/r['c']).has(t) and not (m2/r['c']).has(t):
sol2 = dsolve(diff(y(t),t,t) - (m2/r['c'])*diff(y(t),t) - (e2/r['c'])*y(t)).rhs
sol1 = dsolve(diff(x(t),t) - r['a']*x(t) - r['b']*sol2).rhs
else:
x0 = Function('x0')(t) # x0 and y0 being particular solutions
y0 = Function('y0')(t)
F = exp(Integral(r['a'],t))
P = exp(Integral(r['d'],t))
sol1 = C1*x0 + C2*x0*Integral(r['b']*F*P/x0**2, t)
sol2 = C1*y0 + C2*(F*P/x0 + y0*Integral(r['b']*F*P/x0**2, t))
return [Eq(x(t), sol1), Eq(y(t), sol2)]
def sysode_nonlinear_2eq_order1(match_):
func = match_['func']
eq = match_['eq']
fc = match_['func_coeff']
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
if match_['type_of_equation'] == 'type5':
sol = _nonlinear_2eq_order1_type5(func, t, eq)
return sol
x = func[0].func
y = func[1].func
for i in range(2):
eqs = 0
for terms in Add.make_args(eq[i]):
eqs += terms/fc[i,func[i],1]
eq[i] = eqs
if match_['type_of_equation'] == 'type1':
sol = _nonlinear_2eq_order1_type1(x, y, t, eq)
elif match_['type_of_equation'] == 'type2':
sol = _nonlinear_2eq_order1_type2(x, y, t, eq)
elif match_['type_of_equation'] == 'type3':
sol = _nonlinear_2eq_order1_type3(x, y, t, eq)
elif match_['type_of_equation'] == 'type4':
sol = _nonlinear_2eq_order1_type4(x, y, t, eq)
return sol
def _nonlinear_2eq_order1_type1(x, y, t, eq):
r"""
Equations:
.. math:: x' = x^n F(x,y)
.. math:: y' = g(y) F(x,y)
Solution:
.. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2
where
if `n \neq 1`
.. math:: \varphi = [C_1 + (1-n) \int \frac{1}{g(y)} \,dy]^{\frac{1}{1-n}}
if `n = 1`
.. math:: \varphi = C_1 e^{\int \frac{1}{g(y)} \,dy}
where `C_1` and `C_2` are arbitrary constants.
"""
C1, C2 = get_numbered_constants(eq, num=2)
n = Wild('n', exclude=[x(t),y(t)])
f = Wild('f')
u, v = symbols('u, v')
r = eq[0].match(diff(x(t),t) - x(t)**n*f)
g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v)
F = r[f].subs(x(t),u).subs(y(t),v)
n = r[n]
if n!=1:
phi = (C1 + (1-n)*Integral(1/g, v))**(1/(1-n))
else:
phi = C1*exp(Integral(1/g, v))
phi = phi.doit()
sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v)
sol = []
for sols in sol2:
sol.append(Eq(x(t),phi.subs(v, sols)))
sol.append(Eq(y(t), sols))
return sol
def _nonlinear_2eq_order1_type2(x, y, t, eq):
r"""
Equations:
.. math:: x' = e^{\lambda x} F(x,y)
.. math:: y' = g(y) F(x,y)
Solution:
.. math:: x = \varphi(y), \int \frac{1}{g(y) F(\varphi(y),y)} \,dy = t + C_2
where
if `\lambda \neq 0`
.. math:: \varphi = -\frac{1}{\lambda} log(C_1 - \lambda \int \frac{1}{g(y)} \,dy)
if `\lambda = 0`
.. math:: \varphi = C_1 + \int \frac{1}{g(y)} \,dy
where `C_1` and `C_2` are arbitrary constants.
"""
C1, C2 = get_numbered_constants(eq, num=2)
n = Wild('n', exclude=[x(t),y(t)])
f = Wild('f')
u, v = symbols('u, v')
r = eq[0].match(diff(x(t),t) - exp(n*x(t))*f)
g = ((diff(y(t),t) - eq[1])/r[f]).subs(y(t),v)
F = r[f].subs(x(t),u).subs(y(t),v)
n = r[n]
if n:
phi = -1/n*log(C1 - n*Integral(1/g, v))
else:
phi = C1 + Integral(1/g, v)
phi = phi.doit()
sol2 = solve(Integral(1/(g*F.subs(u,phi)), v).doit() - t - C2, v)
sol = []
for sols in sol2:
sol.append(Eq(x(t),phi.subs(v, sols)))
sol.append(Eq(y(t), sols))
return sol
def _nonlinear_2eq_order1_type3(x, y, t, eq):
r"""
Autonomous system of general form
.. math:: x' = F(x,y)
.. math:: y' = G(x,y)
Assuming `y = y(x, C_1)` where `C_1` is an arbitrary constant is the general
solution of the first-order equation
.. math:: F(x,y) y'_x = G(x,y)
Then the general solution of the original system of equations has the form
.. math:: \int \frac{1}{F(x,y(x,C_1))} \,dx = t + C_1
"""
C1, C2, C3, C4 = get_numbered_constants(eq, num=4)
v = Function('v')
u = Symbol('u')
f = Wild('f')
g = Wild('g')
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
F = r1[f].subs(x(t), u).subs(y(t), v(u))
G = r2[g].subs(x(t), u).subs(y(t), v(u))
sol2r = dsolve(Eq(diff(v(u), u), G/F))
if isinstance(sol2r, Equality):
sol2r = [sol2r]
for sol2s in sol2r:
sol1 = solve(Integral(1/F.subs(v(u), sol2s.rhs), u).doit() - t - C2, u)
sol = []
for sols in sol1:
sol.append(Eq(x(t), sols))
sol.append(Eq(y(t), (sol2s.rhs).subs(u, sols)))
return sol
def _nonlinear_2eq_order1_type4(x, y, t, eq):
r"""
Equation:
.. math:: x' = f_1(x) g_1(y) \phi(x,y,t)
.. math:: y' = f_2(x) g_2(y) \phi(x,y,t)
First integral:
.. math:: \int \frac{f_2(x)}{f_1(x)} \,dx - \int \frac{g_1(y)}{g_2(y)} \,dy = C
where `C` is an arbitrary constant.
On solving the first integral for `x` (resp., `y` ) and on substituting the
resulting expression into either equation of the original solution, one
arrives at a first-order equation for determining `y` (resp., `x` ).
"""
C1, C2 = get_numbered_constants(eq, num=2)
u, v = symbols('u, v')
U, V = symbols('U, V', cls=Function)
f = Wild('f')
g = Wild('g')
f1 = Wild('f1', exclude=[v,t])
f2 = Wild('f2', exclude=[v,t])
g1 = Wild('g1', exclude=[u,t])
g2 = Wild('g2', exclude=[u,t])
r1 = eq[0].match(diff(x(t),t) - f)
r2 = eq[1].match(diff(y(t),t) - g)
num, den = (
(r1[f].subs(x(t),u).subs(y(t),v))/
(r2[g].subs(x(t),u).subs(y(t),v))).as_numer_denom()
R1 = num.match(f1*g1)
R2 = den.match(f2*g2)
phi = (r1[f].subs(x(t),u).subs(y(t),v))/num
F1 = R1[f1]; F2 = R2[f2]
G1 = R1[g1]; G2 = R2[g2]
sol1r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, u)
sol2r = solve(Integral(F2/F1, u).doit() - Integral(G1/G2,v).doit() - C1, v)
sol = []
for sols in sol1r:
sol.append(Eq(y(t), dsolve(diff(V(t),t) - F2.subs(u,sols).subs(v,V(t))*G2.subs(v,V(t))*phi.subs(u,sols).subs(v,V(t))).rhs))
for sols in sol2r:
sol.append(Eq(x(t), dsolve(diff(U(t),t) - F1.subs(u,U(t))*G1.subs(v,sols).subs(u,U(t))*phi.subs(v,sols).subs(u,U(t))).rhs))
return set(sol)
def _nonlinear_2eq_order1_type5(func, t, eq):
r"""
Clairaut system of ODEs
.. math:: x = t x' + F(x',y')
.. math:: y = t y' + G(x',y')
The following are solutions of the system
`(i)` straight lines:
.. math:: x = C_1 t + F(C_1, C_2), y = C_2 t + G(C_1, C_2)
where `C_1` and `C_2` are arbitrary constants;
`(ii)` envelopes of the above lines;
`(iii)` continuously differentiable lines made up from segments of the lines
`(i)` and `(ii)`.
"""
C1, C2 = get_numbered_constants(eq, num=2)
f = Wild('f')
g = Wild('g')
def check_type(x, y):
r1 = eq[0].match(t*diff(x(t),t) - x(t) + f)
r2 = eq[1].match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = eq[0].match(diff(x(t),t) - x(t)/t + f/t)
r2 = eq[1].match(diff(y(t),t) - y(t)/t + g/t)
if not (r1 and r2):
r1 = (-eq[0]).match(t*diff(x(t),t) - x(t) + f)
r2 = (-eq[1]).match(t*diff(y(t),t) - y(t) + g)
if not (r1 and r2):
r1 = (-eq[0]).match(diff(x(t),t) - x(t)/t + f/t)
r2 = (-eq[1]).match(diff(y(t),t) - y(t)/t + g/t)
return [r1, r2]
for func_ in func:
if isinstance(func_, list):
x = func[0][0].func
y = func[0][1].func
[r1, r2] = check_type(x, y)
if not (r1 and r2):
[r1, r2] = check_type(y, x)
x, y = y, x
x1 = diff(x(t),t); y1 = diff(y(t),t)
return {Eq(x(t), C1*t + r1[f].subs(x1,C1).subs(y1,C2)), Eq(y(t), C2*t + r2[g].subs(x1,C1).subs(y1,C2))}
def sysode_nonlinear_3eq_order1(match_):
x = match_['func'][0].func
y = match_['func'][1].func
z = match_['func'][2].func
eq = match_['eq']
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
if match_['type_of_equation'] == 'type1':
sol = _nonlinear_3eq_order1_type1(x, y, z, t, eq)
if match_['type_of_equation'] == 'type2':
sol = _nonlinear_3eq_order1_type2(x, y, z, t, eq)
if match_['type_of_equation'] == 'type3':
sol = _nonlinear_3eq_order1_type3(x, y, z, t, eq)
if match_['type_of_equation'] == 'type4':
sol = _nonlinear_3eq_order1_type4(x, y, z, t, eq)
if match_['type_of_equation'] == 'type5':
sol = _nonlinear_3eq_order1_type5(x, y, z, t, eq)
return sol
def _nonlinear_3eq_order1_type1(x, y, z, t, eq):
r"""
Equations:
.. math:: a x' = (b - c) y z, \enspace b y' = (c - a) z x, \enspace c z' = (a - b) x y
First Integrals:
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
.. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2
where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and
`z` and on substituting the resulting expressions into the first equation of the
system, we arrives at a separable first-order equation on `x`. Similarly doing that
for other two equations, we will arrive at first order equation on `y` and `z` too.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0401.pdf
"""
C1, C2 = get_numbered_constants(eq, num=2)
u, v, w = symbols('u, v, w')
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
r = (diff(x(t),t) - eq[0]).match(p*y(t)*z(t))
r.update((diff(y(t),t) - eq[1]).match(q*z(t)*x(t)))
r.update((diff(z(t),t) - eq[2]).match(s*x(t)*y(t)))
n1, d1 = r[p].as_numer_denom()
n2, d2 = r[q].as_numer_denom()
n3, d3 = r[s].as_numer_denom()
val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, d3*u-d3*v-n3*w],[u,v])
vals = [val[v], val[u]]
c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1])
b = vals[0].subs(w, c)
a = vals[1].subs(w, c)
y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b)))
z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c)))
z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c)))
x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a)))
x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a)))
y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b)))
sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x)
sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y)
sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z)
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type2(x, y, z, t, eq):
r"""
Equations:
.. math:: a x' = (b - c) y z f(x, y, z, t)
.. math:: b y' = (c - a) z x f(x, y, z, t)
.. math:: c z' = (a - b) x y f(x, y, z, t)
First Integrals:
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
.. math:: a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2} = C_2
where `C_1` and `C_2` are arbitrary constants. On solving the integrals for `y` and
`z` and on substituting the resulting expressions into the first equation of the
system, we arrives at a first-order differential equations on `x`. Similarly doing
that for other two equations we will arrive at first order equation on `y` and `z`.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0402.pdf
"""
C1, C2 = get_numbered_constants(eq, num=2)
u, v, w = symbols('u, v, w')
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
f = Wild('f')
r1 = (diff(x(t),t) - eq[0]).match(y(t)*z(t)*f)
r = collect_const(r1[f]).match(p*f)
r.update(((diff(y(t),t) - eq[1])/r[f]).match(q*z(t)*x(t)))
r.update(((diff(z(t),t) - eq[2])/r[f]).match(s*x(t)*y(t)))
n1, d1 = r[p].as_numer_denom()
n2, d2 = r[q].as_numer_denom()
n3, d3 = r[s].as_numer_denom()
val = solve([n1*u-d1*v+d1*w, d2*u+n2*v-d2*w, -d3*u+d3*v+n3*w],[u,v])
vals = [val[v], val[u]]
c = lcm(vals[0].as_numer_denom()[1], vals[1].as_numer_denom()[1])
a = vals[0].subs(w, c)
b = vals[1].subs(w, c)
y_x = sqrt(((c*C1-C2) - a*(c-a)*x(t)**2)/(b*(c-b)))
z_x = sqrt(((b*C1-C2) - a*(b-a)*x(t)**2)/(c*(b-c)))
z_y = sqrt(((a*C1-C2) - b*(a-b)*y(t)**2)/(c*(a-c)))
x_y = sqrt(((c*C1-C2) - b*(c-b)*y(t)**2)/(a*(c-a)))
x_z = sqrt(((b*C1-C2) - c*(b-c)*z(t)**2)/(a*(b-a)))
y_z = sqrt(((a*C1-C2) - c*(a-c)*z(t)**2)/(b*(a-b)))
sol1 = dsolve(a*diff(x(t),t) - (b-c)*y_x*z_x*r[f])
sol2 = dsolve(b*diff(y(t),t) - (c-a)*z_y*x_y*r[f])
sol3 = dsolve(c*diff(z(t),t) - (a-b)*x_z*y_z*r[f])
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type3(x, y, z, t, eq):
r"""
Equations:
.. math:: x' = c F_2 - b F_3, \enspace y' = a F_3 - c F_1, \enspace z' = b F_1 - a F_2
where `F_n = F_n(x, y, z, t)`.
1. First Integral:
.. math:: a x + b y + c z = C_1,
where C is an arbitrary constant.
2. If we assume function `F_n` to be independent of `t`,i.e, `F_n` = `F_n (x, y, z)`
Then, on eliminating `t` and `z` from the first two equation of the system, one
arrives at the first-order equation
.. math:: \frac{dy}{dx} = \frac{a F_3 (x, y, z) - c F_1 (x, y, z)}{c F_2 (x, y, z) -
b F_3 (x, y, z)}
where `z = \frac{1}{c} (C_1 - a x - b y)`
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0404.pdf
"""
C1 = get_numbered_constants(eq, num=1)
u, v, w = symbols('u, v, w')
fu, fv, fw = symbols('u, v, w', cls=Function)
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
r1 = (diff(x(t), t) - eq[0]).match(F2-F3)
r = collect_const(r1[F2]).match(s*F2)
r.update(collect_const(r1[F3]).match(q*F3))
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
r[F2], r[F3] = r[F3], r[F2]
r[s], r[q] = -r[q], -r[s]
r.update((diff(y(t), t) - eq[1]).match(p*r[F3] - r[s]*F1))
a = r[p]; b = r[q]; c = r[s]
F1 = r[F1].subs(x(t), u).subs(y(t),v).subs(z(t), w)
F2 = r[F2].subs(x(t), u).subs(y(t),v).subs(z(t), w)
F3 = r[F3].subs(x(t), u).subs(y(t),v).subs(z(t), w)
z_xy = (C1-a*u-b*v)/c
y_zx = (C1-a*u-c*w)/b
x_yz = (C1-b*v-c*w)/a
y_x = dsolve(diff(fv(u),u) - ((a*F3-c*F1)/(c*F2-b*F3)).subs(w,z_xy).subs(v,fv(u))).rhs
z_x = dsolve(diff(fw(u),u) - ((b*F1-a*F2)/(c*F2-b*F3)).subs(v,y_zx).subs(w,fw(u))).rhs
z_y = dsolve(diff(fw(v),v) - ((b*F1-a*F2)/(a*F3-c*F1)).subs(u,x_yz).subs(w,fw(v))).rhs
x_y = dsolve(diff(fu(v),v) - ((c*F2-b*F3)/(a*F3-c*F1)).subs(w,z_xy).subs(u,fu(v))).rhs
y_z = dsolve(diff(fv(w),w) - ((a*F3-c*F1)/(b*F1-a*F2)).subs(u,x_yz).subs(v,fv(w))).rhs
x_z = dsolve(diff(fu(w),w) - ((c*F2-b*F3)/(b*F1-a*F2)).subs(v,y_zx).subs(u,fu(w))).rhs
sol1 = dsolve(diff(fu(t),t) - (c*F2 - b*F3).subs(v,y_x).subs(w,z_x).subs(u,fu(t))).rhs
sol2 = dsolve(diff(fv(t),t) - (a*F3 - c*F1).subs(u,x_y).subs(w,z_y).subs(v,fv(t))).rhs
sol3 = dsolve(diff(fw(t),t) - (b*F1 - a*F2).subs(u,x_z).subs(v,y_z).subs(w,fw(t))).rhs
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type4(x, y, z, t, eq):
r"""
Equations:
.. math:: x' = c z F_2 - b y F_3, \enspace y' = a x F_3 - c z F_1, \enspace z' = b y F_1 - a x F_2
where `F_n = F_n (x, y, z, t)`
1. First integral:
.. math:: a x^{2} + b y^{2} + c z^{2} = C_1
where `C` is an arbitrary constant.
2. Assuming the function `F_n` is independent of `t`: `F_n = F_n (x, y, z)`. Then on
eliminating `t` and `z` from the first two equations of the system, one arrives at
the first-order equation
.. math:: \frac{dy}{dx} = \frac{a x F_3 (x, y, z) - c z F_1 (x, y, z)}
{c z F_2 (x, y, z) - b y F_3 (x, y, z)}
where `z = \pm \sqrt{\frac{1}{c} (C_1 - a x^{2} - b y^{2})}`
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0405.pdf
"""
C1 = get_numbered_constants(eq, num=1)
u, v, w = symbols('u, v, w')
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
r1 = eq[0].match(diff(x(t),t) - z(t)*F2 + y(t)*F3)
r = collect_const(r1[F2]).match(s*F2)
r.update(collect_const(r1[F3]).match(q*F3))
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
r[F2], r[F3] = r[F3], r[F2]
r[s], r[q] = -r[q], -r[s]
r.update((diff(y(t),t) - eq[1]).match(p*x(t)*r[F3] - r[s]*z(t)*F1))
a = r[p]; b = r[q]; c = r[s]
F1 = r[F1].subs(x(t),u).subs(y(t),v).subs(z(t),w)
F2 = r[F2].subs(x(t),u).subs(y(t),v).subs(z(t),w)
F3 = r[F3].subs(x(t),u).subs(y(t),v).subs(z(t),w)
x_yz = sqrt((C1 - b*v**2 - c*w**2)/a)
y_zx = sqrt((C1 - c*w**2 - a*u**2)/b)
z_xy = sqrt((C1 - a*u**2 - b*v**2)/c)
y_x = dsolve(diff(v(u),u) - ((a*u*F3-c*w*F1)/(c*w*F2-b*v*F3)).subs(w,z_xy).subs(v,v(u))).rhs
z_x = dsolve(diff(w(u),u) - ((b*v*F1-a*u*F2)/(c*w*F2-b*v*F3)).subs(v,y_zx).subs(w,w(u))).rhs
z_y = dsolve(diff(w(v),v) - ((b*v*F1-a*u*F2)/(a*u*F3-c*w*F1)).subs(u,x_yz).subs(w,w(v))).rhs
x_y = dsolve(diff(u(v),v) - ((c*w*F2-b*v*F3)/(a*u*F3-c*w*F1)).subs(w,z_xy).subs(u,u(v))).rhs
y_z = dsolve(diff(v(w),w) - ((a*u*F3-c*w*F1)/(b*v*F1-a*u*F2)).subs(u,x_yz).subs(v,v(w))).rhs
x_z = dsolve(diff(u(w),w) - ((c*w*F2-b*v*F3)/(b*v*F1-a*u*F2)).subs(v,y_zx).subs(u,u(w))).rhs
sol1 = dsolve(diff(u(t),t) - (c*w*F2 - b*v*F3).subs(v,y_x).subs(w,z_x).subs(u,u(t))).rhs
sol2 = dsolve(diff(v(t),t) - (a*u*F3 - c*w*F1).subs(u,x_y).subs(w,z_y).subs(v,v(t))).rhs
sol3 = dsolve(diff(w(t),t) - (b*v*F1 - a*u*F2).subs(u,x_z).subs(v,y_z).subs(w,w(t))).rhs
return [sol1, sol2, sol3]
def _nonlinear_3eq_order1_type5(x, y, z, t, eq):
r"""
.. math:: x' = x (c F_2 - b F_3), \enspace y' = y (a F_3 - c F_1), \enspace z' = z (b F_1 - a F_2)
where `F_n = F_n (x, y, z, t)` and are arbitrary functions.
First Integral:
.. math:: \left|x\right|^{a} \left|y\right|^{b} \left|z\right|^{c} = C_1
where `C` is an arbitrary constant. If the function `F_n` is independent of `t`,
then, by eliminating `t` and `z` from the first two equations of the system, one
arrives at a first-order equation.
References
==========
-http://eqworld.ipmnet.ru/en/solutions/sysode/sode0406.pdf
"""
C1 = get_numbered_constants(eq, num=1)
u, v, w = symbols('u, v, w')
fu, fv, fw = symbols('u, v, w', cls=Function)
p = Wild('p', exclude=[x(t), y(t), z(t), t])
q = Wild('q', exclude=[x(t), y(t), z(t), t])
s = Wild('s', exclude=[x(t), y(t), z(t), t])
F1, F2, F3 = symbols('F1, F2, F3', cls=Wild)
r1 = eq[0].match(diff(x(t), t) - x(t)*F2 + x(t)*F3)
r = collect_const(r1[F2]).match(s*F2)
r.update(collect_const(r1[F3]).match(q*F3))
if eq[1].has(r[F2]) and not eq[1].has(r[F3]):
r[F2], r[F3] = r[F3], r[F2]
r[s], r[q] = -r[q], -r[s]
r.update((diff(y(t), t) - eq[1]).match(y(t)*(p*r[F3] - r[s]*F1)))
a = r[p]; b = r[q]; c = r[s]
F1 = r[F1].subs(x(t), u).subs(y(t), v).subs(z(t), w)
F2 = r[F2].subs(x(t), u).subs(y(t), v).subs(z(t), w)
F3 = r[F3].subs(x(t), u).subs(y(t), v).subs(z(t), w)
x_yz = (C1*v**-b*w**-c)**-a
y_zx = (C1*w**-c*u**-a)**-b
z_xy = (C1*u**-a*v**-b)**-c
y_x = dsolve(diff(fv(u), u) - ((v*(a*F3 - c*F1))/(u*(c*F2 - b*F3))).subs(w, z_xy).subs(v, fv(u))).rhs
z_x = dsolve(diff(fw(u), u) - ((w*(b*F1 - a*F2))/(u*(c*F2 - b*F3))).subs(v, y_zx).subs(w, fw(u))).rhs
z_y = dsolve(diff(fw(v), v) - ((w*(b*F1 - a*F2))/(v*(a*F3 - c*F1))).subs(u, x_yz).subs(w, fw(v))).rhs
x_y = dsolve(diff(fu(v), v) - ((u*(c*F2 - b*F3))/(v*(a*F3 - c*F1))).subs(w, z_xy).subs(u, fu(v))).rhs
y_z = dsolve(diff(fv(w), w) - ((v*(a*F3 - c*F1))/(w*(b*F1 - a*F2))).subs(u, x_yz).subs(v, fv(w))).rhs
x_z = dsolve(diff(fu(w), w) - ((u*(c*F2 - b*F3))/(w*(b*F1 - a*F2))).subs(v, y_zx).subs(u, fu(w))).rhs
sol1 = dsolve(diff(fu(t), t) - (u*(c*F2 - b*F3)).subs(v, y_x).subs(w, z_x).subs(u, fu(t))).rhs
sol2 = dsolve(diff(fv(t), t) - (v*(a*F3 - c*F1)).subs(u, x_y).subs(w, z_y).subs(v, fv(t))).rhs
sol3 = dsolve(diff(fw(t), t) - (w*(b*F1 - a*F2)).subs(u, x_z).subs(v, y_z).subs(w, fw(t))).rhs
return [sol1, sol2, sol3]
#This import is written at the bottom to avoid circular imports.
from .single import SingleODEProblem, SingleODESolver, solver_map
|
dfcb576a50f707f9ca82bcefc838fd7e918577c2b97656eb479f362f7df6bc80 | #
# This is the module for ODE solver classes for single ODEs.
#
import typing
if typing.TYPE_CHECKING:
from typing import ClassVar
from typing import Dict as tDict, Type, Iterator, List, Optional
from .riccati import match_riccati, solve_riccati
from sympy.core import Add, S, Pow, Rational
from sympy.core.exprtools import factor_terms
from sympy.core.expr import Expr
from sympy.core.function import AppliedUndef, Derivative, diff, Function, expand, Subs, _mexpand
from sympy.core.numbers import zoo
from sympy.core.relational import Equality, Eq
from sympy.core.symbol import Symbol, Dummy, Wild
from sympy.core.mul import Mul
from sympy.functions import exp, tan, log, sqrt, besselj, bessely, cbrt, airyai, airybi
from sympy.integrals import Integral
from sympy.polys import Poly
from sympy.polys.polytools import cancel, factor, degree
from sympy.simplify import collect, simplify, separatevars, logcombine, posify # type: ignore
from sympy.simplify.radsimp import fraction
from sympy.utilities import numbered_symbols
from sympy.solvers.solvers import solve
from sympy.solvers.deutils import ode_order, _preprocess
from sympy.polys.matrices.linsolve import _lin_eq2dict
from sympy.polys.solvers import PolyNonlinearError
from .hypergeometric import equivalence_hypergeometric, match_2nd_2F1_hypergeometric, \
get_sol_2F1_hypergeometric, match_2nd_hypergeometric
from .nonhomogeneous import _get_euler_characteristic_eq_sols, _get_const_characteristic_eq_sols, \
_solve_undetermined_coefficients, _solve_variation_of_parameters, _test_term, _undetermined_coefficients_match, \
_get_simplified_sol
from .lie_group import _ode_lie_group
class ODEMatchError(NotImplementedError):
"""Raised if a SingleODESolver is asked to solve an ODE it does not match"""
pass
def cached_property(func):
'''Decorator to cache property method'''
attrname = '_' + func.__name__
def propfunc(self):
val = getattr(self, attrname, None)
if val is None:
val = func(self)
setattr(self, attrname, val)
return val
return property(propfunc)
class SingleODEProblem:
"""Represents an ordinary differential equation (ODE)
This class is used internally in the by dsolve and related
functions/classes so that properties of an ODE can be computed
efficiently.
Examples
========
This class is used internally by dsolve. To instantiate an instance
directly first define an ODE problem:
>>> from sympy import Function, Symbol
>>> x = Symbol('x')
>>> f = Function('f')
>>> eq = f(x).diff(x, 2)
Now you can create a SingleODEProblem instance and query its properties:
>>> from sympy.solvers.ode.single import SingleODEProblem
>>> problem = SingleODEProblem(f(x).diff(x), f(x), x)
>>> problem.eq
Derivative(f(x), x)
>>> problem.func
f(x)
>>> problem.sym
x
"""
# Instance attributes:
eq = None # type: Expr
func = None # type: AppliedUndef
sym = None # type: Symbol
_order = None # type: int
_eq_expanded = None # type: Expr
_eq_preprocessed = None # type: Expr
_eq_high_order_free = None
def __init__(self, eq, func, sym, prep=True, **kwargs):
assert isinstance(eq, Expr)
assert isinstance(func, AppliedUndef)
assert isinstance(sym, Symbol)
assert isinstance(prep, bool)
self.eq = eq
self.func = func
self.sym = sym
self.prep = prep
self.params = kwargs
@cached_property
def order(self) -> int:
return ode_order(self.eq, self.func)
@cached_property
def eq_preprocessed(self) -> Expr:
return self._get_eq_preprocessed()
@cached_property
def eq_high_order_free(self) -> Expr:
a = Wild('a', exclude=[self.func])
c1 = Wild('c1', exclude=[self.sym])
# Precondition to try remove f(x) from highest order derivative
reduced_eq = None
if self.eq.is_Add:
deriv_coef = self.eq.coeff(self.func.diff(self.sym, self.order))
if deriv_coef not in (1, 0):
r = deriv_coef.match(a*self.func**c1)
if r and r[c1]:
den = self.func**r[c1]
reduced_eq = Add(*[arg/den for arg in self.eq.args])
if not reduced_eq:
reduced_eq = expand(self.eq)
return reduced_eq
@cached_property
def eq_expanded(self) -> Expr:
return expand(self.eq_preprocessed)
def _get_eq_preprocessed(self) -> Expr:
if self.prep:
process_eq, process_func = _preprocess(self.eq, self.func)
if process_func != self.func:
raise ValueError
else:
process_eq = self.eq
return process_eq
def get_numbered_constants(self, num=1, start=1, prefix='C') -> List[Symbol]:
"""
Returns a list of constants that do not occur
in eq already.
"""
ncs = self.iter_numbered_constants(start, prefix)
Cs = [next(ncs) for i in range(num)]
return Cs
def iter_numbered_constants(self, start=1, prefix='C') -> Iterator[Symbol]:
"""
Returns an iterator of constants that do not occur
in eq already.
"""
atom_set = self.eq.free_symbols
func_set = self.eq.atoms(Function)
if func_set:
atom_set |= {Symbol(str(f.func)) for f in func_set}
return numbered_symbols(start=start, prefix=prefix, exclude=atom_set)
@cached_property
def is_autonomous(self):
u = Dummy('u')
x = self.sym
syms = self.eq.subs(self.func, u).free_symbols
return x not in syms
def get_linear_coefficients(self, eq, func, order):
r"""
Matches a differential equation to the linear form:
.. math:: a_n(x) y^{(n)} + \cdots + a_1(x)y' + a_0(x) y + B(x) = 0
Returns a dict of order:coeff terms, where order is the order of the
derivative on each term, and coeff is the coefficient of that derivative.
The key ``-1`` holds the function `B(x)`. Returns ``None`` if the ODE is
not linear. This function assumes that ``func`` has already been checked
to be good.
Examples
========
>>> from sympy import Function, cos, sin
>>> from sympy.abc import x
>>> from sympy.solvers.ode.single import SingleODEProblem
>>> f = Function('f')
>>> eq = f(x).diff(x, 3) + 2*f(x).diff(x) + \
... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) - \
... sin(x)
>>> obj = SingleODEProblem(eq, f(x), x)
>>> obj.get_linear_coefficients(eq, f(x), 3)
{-1: x - sin(x), 0: -1, 1: cos(x) + 2, 2: x, 3: 1}
>>> eq = f(x).diff(x, 3) + 2*f(x).diff(x) + \
... x*f(x).diff(x, 2) + cos(x)*f(x).diff(x) + x - f(x) - \
... sin(f(x))
>>> obj = SingleODEProblem(eq, f(x), x)
>>> obj.get_linear_coefficients(eq, f(x), 3) == None
True
"""
f = func.func
x = func.args[0]
symset = {Derivative(f(x), x, i) for i in range(order+1)}
try:
rhs, lhs_terms = _lin_eq2dict(eq, symset)
except PolyNonlinearError:
return None
if rhs.has(func) or any(c.has(func) for c in lhs_terms.values()):
return None
terms = {i: lhs_terms.get(f(x).diff(x, i), S.Zero) for i in range(order+1)}
terms[-1] = rhs
return terms
# TODO: Add methods that can be used by many ODE solvers:
# order
# is_linear()
# get_linear_coefficients()
# eq_prepared (the ODE in prepared form)
class SingleODESolver:
"""
Base class for Single ODE solvers.
Subclasses should implement the _matches and _get_general_solution
methods. This class is not intended to be instantiated directly but its
subclasses are as part of dsolve.
Examples
========
You can use a subclass of SingleODEProblem to solve a particular type of
ODE. We first define a particular ODE problem:
>>> from sympy import Function, Symbol
>>> x = Symbol('x')
>>> f = Function('f')
>>> eq = f(x).diff(x, 2)
Now we solve this problem using the NthAlgebraic solver which is a
subclass of SingleODESolver:
>>> from sympy.solvers.ode.single import NthAlgebraic, SingleODEProblem
>>> problem = SingleODEProblem(eq, f(x), x)
>>> solver = NthAlgebraic(problem)
>>> solver.get_general_solution()
[Eq(f(x), _C*x + _C)]
The normal way to solve an ODE is to use dsolve (which would use
NthAlgebraic and other solvers internally). When using dsolve a number of
other things are done such as evaluating integrals, simplifying the
solution and renumbering the constants:
>>> from sympy import dsolve
>>> dsolve(eq, hint='nth_algebraic')
Eq(f(x), C1 + C2*x)
"""
# Subclasses should store the hint name (the argument to dsolve) in this
# attribute
hint = None # type: ClassVar[str]
# Subclasses should define this to indicate if they support an _Integral
# hint.
has_integral = None # type: ClassVar[bool]
# The ODE to be solved
ode_problem = None # type: SingleODEProblem
# Cache whether or not the equation has matched the method
_matched = None # type: Optional[bool]
# Subclasses should store in this attribute the list of order(s) of ODE
# that subclass can solve or leave it to None if not specific to any order
order = None # type: Optional[list]
def __init__(self, ode_problem):
self.ode_problem = ode_problem
def matches(self) -> bool:
if self.order is not None and self.ode_problem.order not in self.order:
self._matched = False
return self._matched
if self._matched is None:
self._matched = self._matches()
return self._matched
def get_general_solution(self, *, simplify: bool = True) -> List[Equality]:
if not self.matches():
msg = "%s solver cannot solve:\n%s"
raise ODEMatchError(msg % (self.hint, self.ode_problem.eq))
return self._get_general_solution(simplify_flag=simplify)
def _matches(self) -> bool:
msg = "Subclasses of SingleODESolver should implement matches."
raise NotImplementedError(msg)
def _get_general_solution(self, *, simplify_flag: bool = True) -> List[Equality]:
msg = "Subclasses of SingleODESolver should implement get_general_solution."
raise NotImplementedError(msg)
class SinglePatternODESolver(SingleODESolver):
'''Superclass for ODE solvers based on pattern matching'''
def wilds(self):
prob = self.ode_problem
f = prob.func.func
x = prob.sym
order = prob.order
return self._wilds(f, x, order)
def wilds_match(self):
match = self._wilds_match
return [match.get(w, S.Zero) for w in self.wilds()]
def _matches(self):
eq = self.ode_problem.eq_expanded
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
df = f(x).diff(x, order)
if order not in [1, 2]:
return False
pattern = self._equation(f(x), x, order)
if not pattern.coeff(df).has(Wild):
eq = expand(eq / eq.coeff(df))
eq = eq.collect([f(x).diff(x), f(x)], func = cancel)
self._wilds_match = match = eq.match(pattern)
if match is not None:
return self._verify(f(x))
return False
def _verify(self, fx) -> bool:
return True
def _wilds(self, f, x, order):
msg = "Subclasses of SingleODESolver should implement _wilds"
raise NotImplementedError(msg)
def _equation(self, fx, x, order):
msg = "Subclasses of SingleODESolver should implement _equation"
raise NotImplementedError(msg)
class NthAlgebraic(SingleODESolver):
r"""
Solves an `n`\th order ordinary differential equation using algebra and
integrals.
There is no general form for the kind of equation that this can solve. The
the equation is solved algebraically treating differentiation as an
invertible algebraic function.
Examples
========
>>> from sympy import Function, dsolve, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = Eq(f(x) * (f(x).diff(x)**2 - 1), 0)
>>> dsolve(eq, f(x), hint='nth_algebraic')
[Eq(f(x), 0), Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
Note that this solver can return algebraic solutions that do not have any
integration constants (f(x) = 0 in the above example).
"""
hint = 'nth_algebraic'
has_integral = True # nth_algebraic_Integral hint
def _matches(self):
r"""
Matches any differential equation that nth_algebraic can solve. Uses
`sympy.solve` but teaches it how to integrate derivatives.
This involves calling `sympy.solve` and does most of the work of finding a
solution (apart from evaluating the integrals).
"""
eq = self.ode_problem.eq
func = self.ode_problem.func
var = self.ode_problem.sym
# Derivative that solve can handle:
diffx = self._get_diffx(var)
# Replace derivatives wrt the independent variable with diffx
def replace(eq, var):
def expand_diffx(*args):
differand, diffs = args[0], args[1:]
toreplace = differand
for v, n in diffs:
for _ in range(n):
if v == var:
toreplace = diffx(toreplace)
else:
toreplace = Derivative(toreplace, v)
return toreplace
return eq.replace(Derivative, expand_diffx)
# Restore derivatives in solution afterwards
def unreplace(eq, var):
return eq.replace(diffx, lambda e: Derivative(e, var))
subs_eqn = replace(eq, var)
try:
# turn off simplification to protect Integrals that have
# _t instead of fx in them and would otherwise factor
# as t_*Integral(1, x)
solns = solve(subs_eqn, func, simplify=False)
except NotImplementedError:
solns = []
solns = [simplify(unreplace(soln, var)) for soln in solns]
solns = [Equality(func, soln) for soln in solns]
self.solutions = solns
return len(solns) != 0
def _get_general_solution(self, *, simplify_flag: bool = True):
return self.solutions
# This needs to produce an invertible function but the inverse depends
# which variable we are integrating with respect to. Since the class can
# be stored in cached results we need to ensure that we always get the
# same class back for each particular integration variable so we store these
# classes in a global dict:
_diffx_stored = {} # type: tDict[Symbol, Type[Function]]
@staticmethod
def _get_diffx(var):
diffcls = NthAlgebraic._diffx_stored.get(var, None)
if diffcls is None:
# A class that behaves like Derivative wrt var but is "invertible".
class diffx(Function):
def inverse(self):
# don't use integrate here because fx has been replaced by _t
# in the equation; integrals will not be correct while solve
# is at work.
return lambda expr: Integral(expr, var) + Dummy('C')
diffcls = NthAlgebraic._diffx_stored.setdefault(var, diffx)
return diffcls
class FirstExact(SinglePatternODESolver):
r"""
Solves 1st order exact ordinary differential equations.
A 1st order differential equation is called exact if it is the total
differential of a function. That is, the differential equation
.. math:: P(x, y) \,\partial{}x + Q(x, y) \,\partial{}y = 0
is exact if there is some function `F(x, y)` such that `P(x, y) =
\partial{}F/\partial{}x` and `Q(x, y) = \partial{}F/\partial{}y`. It can
be shown that a necessary and sufficient condition for a first order ODE
to be exact is that `\partial{}P/\partial{}y = \partial{}Q/\partial{}x`.
Then, the solution will be as given below::
>>> from sympy import Function, Eq, Integral, symbols, pprint
>>> x, y, t, x0, y0, C1= symbols('x,y,t,x0,y0,C1')
>>> P, Q, F= map(Function, ['P', 'Q', 'F'])
>>> pprint(Eq(Eq(F(x, y), Integral(P(t, y), (t, x0, x)) +
... Integral(Q(x0, t), (t, y0, y))), C1))
x y
/ /
| |
F(x, y) = | P(t, y) dt + | Q(x0, t) dt = C1
| |
/ /
x0 y0
Where the first partials of `P` and `Q` exist and are continuous in a
simply connected region.
A note: SymPy currently has no way to represent inert substitution on an
expression, so the hint ``1st_exact_Integral`` will return an integral
with `dy`. This is supposed to represent the function that you are
solving for.
Examples
========
>>> from sympy import Function, dsolve, cos, sin
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x),
... f(x), hint='1st_exact')
Eq(x*cos(f(x)) + f(x)**3/3, C1)
References
==========
- https://en.wikipedia.org/wiki/Exact_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 73
# indirect doctest
"""
hint = "1st_exact"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x).diff(x)])
Q = Wild('Q', exclude=[f(x).diff(x)])
return P, Q
def _equation(self, fx, x, order):
P, Q = self.wilds()
return P + Q*fx.diff(x)
def _verify(self, fx) -> bool:
P, Q = self.wilds()
x = self.ode_problem.sym
y = Dummy('y')
m, n = self.wilds_match()
m = m.subs(fx, y)
n = n.subs(fx, y)
numerator = cancel(m.diff(y) - n.diff(x))
if numerator.is_zero:
# Is exact
return True
else:
# The following few conditions try to convert a non-exact
# differential equation into an exact one.
# References:
# 1. Differential equations with applications
# and historical notes - George E. Simmons
# 2. https://math.okstate.edu/people/binegar/2233-S99/2233-l12.pdf
factor_n = cancel(numerator/n)
factor_m = cancel(-numerator/m)
if y not in factor_n.free_symbols:
# If (dP/dy - dQ/dx) / Q = f(x)
# then exp(integral(f(x))*equation becomes exact
factor = factor_n
integration_variable = x
elif x not in factor_m.free_symbols:
# If (dP/dy - dQ/dx) / -P = f(y)
# then exp(integral(f(y))*equation becomes exact
factor = factor_m
integration_variable = y
else:
# Couldn't convert to exact
return False
factor = exp(Integral(factor, integration_variable))
m *= factor
n *= factor
self._wilds_match[P] = m.subs(y, fx)
self._wilds_match[Q] = n.subs(y, fx)
return True
def _get_general_solution(self, *, simplify_flag: bool = True):
m, n = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
y = Dummy('y')
m = m.subs(fx, y)
n = n.subs(fx, y)
gen_sol = Eq(Subs(Integral(m, x)
+ Integral(n - Integral(m, x).diff(y), y), y, fx), C1)
return [gen_sol]
class FirstLinear(SinglePatternODESolver):
r"""
Solves 1st order linear differential equations.
These are differential equations of the form
.. math:: dy/dx + P(x) y = Q(x)\text{.}
These kinds of differential equations can be solved in a general way. The
integrating factor `e^{\int P(x) \,dx}` will turn the equation into a
separable equation. The general solution is::
>>> from sympy import Function, dsolve, Eq, pprint, diff, sin
>>> from sympy.abc import x
>>> f, P, Q = map(Function, ['f', 'P', 'Q'])
>>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x))
>>> pprint(genform)
d
P(x)*f(x) + --(f(x)) = Q(x)
dx
>>> pprint(dsolve(genform, f(x), hint='1st_linear_Integral'))
/ / \
| | |
| | / | /
| | | | |
| | | P(x) dx | - | P(x) dx
| | | | |
| | / | /
f(x) = |C1 + | Q(x)*e dx|*e
| | |
\ / /
Examples
========
>>> f = Function('f')
>>> pprint(dsolve(Eq(x*diff(f(x), x) - f(x), x**2*sin(x)),
... f(x), '1st_linear'))
f(x) = x*(C1 - cos(x))
References
==========
- https://en.wikipedia.org/wiki/Linear_differential_equation#First_order_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 92
# indirect doctest
"""
hint = '1st_linear'
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x)])
Q = Wild('Q', exclude=[f(x), f(x).diff(x)])
return P, Q
def _equation(self, fx, x, order):
P, Q = self.wilds()
return fx.diff(x) + P*fx - Q
def _get_general_solution(self, *, simplify_flag: bool = True):
P, Q = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
gensol = Eq(fx, ((C1 + Integral(Q*exp(Integral(P, x)), x))
* exp(-Integral(P, x))))
return [gensol]
class AlmostLinear(SinglePatternODESolver):
r"""
Solves an almost-linear differential equation.
The general form of an almost linear differential equation is
.. math:: a(x) g'(f(x)) f'(x) + b(x) g(f(x)) + c(x)
Here `f(x)` is the function to be solved for (the dependent variable).
The substitution `g(f(x)) = u(x)` leads to a linear differential equation
for `u(x)` of the form `a(x) u' + b(x) u + c(x) = 0`. This can be solved
for `u(x)` by the `first_linear` hint and then `f(x)` is found by solving
`g(f(x)) = u(x)`.
See Also
========
:obj:`sympy.solvers.ode.single.FirstLinear`
Examples
========
>>> from sympy import dsolve, Function, pprint, sin, cos
>>> from sympy.abc import x
>>> f = Function('f')
>>> d = f(x).diff(x)
>>> eq = x*d + x*f(x) + 1
>>> dsolve(eq, f(x), hint='almost_linear')
Eq(f(x), (C1 - Ei(x))*exp(-x))
>>> pprint(dsolve(eq, f(x), hint='almost_linear'))
-x
f(x) = (C1 - Ei(x))*e
>>> example = cos(f(x))*f(x).diff(x) + sin(f(x)) + 1
>>> pprint(example)
d
sin(f(x)) + cos(f(x))*--(f(x)) + 1
dx
>>> pprint(dsolve(example, f(x), hint='almost_linear'))
/ -x \ / -x \
[f(x) = pi - asin\C1*e - 1/, f(x) = asin\C1*e - 1/]
References
==========
- Joel Moses, "Symbolic Integration - The Stormy Decade", Communications
of the ACM, Volume 14, Number 8, August 1971, pp. 558
"""
hint = "almost_linear"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x).diff(x)])
Q = Wild('Q', exclude=[f(x).diff(x)])
return P, Q
def _equation(self, fx, x, order):
P, Q = self.wilds()
return P*fx.diff(x) + Q
def _verify(self, fx):
a, b = self.wilds_match()
c, b = b.as_independent(fx) if b.is_Add else (S.Zero, b)
# a, b and c are the function a(x), b(x) and c(x) respectively.
# c(x) is obtained by separating out b as terms with and without fx i.e, l(y)
# The following conditions checks if the given equation is an almost-linear differential equation using the fact that
# a(x)*(l(y))' / l(y)' is independent of l(y)
if b.diff(fx) != 0 and not simplify(b.diff(fx)/a).has(fx):
self.ly = factor_terms(b).as_independent(fx, as_Add=False)[1] # Gives the term containing fx i.e., l(y)
self.ax = a / self.ly.diff(fx)
self.cx = -c # cx is taken as -c(x) to simplify expression in the solution integral
self.bx = factor_terms(b) / self.ly
return True
return False
def _get_general_solution(self, *, simplify_flag: bool = True):
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
gensol = Eq(self.ly, ((C1 + Integral((self.cx/self.ax)*exp(Integral(self.bx/self.ax, x)), x))
* exp(-Integral(self.bx/self.ax, x))))
return [gensol]
class Bernoulli(SinglePatternODESolver):
r"""
Solves Bernoulli differential equations.
These are equations of the form
.. math:: dy/dx + P(x) y = Q(x) y^n\text{, }n \ne 1`\text{.}
The substitution `w = 1/y^{1-n}` will transform an equation of this form
into one that is linear (see the docstring of
:obj:`~sympy.solvers.ode.single.FirstLinear`). The general solution is::
>>> from sympy import Function, dsolve, Eq, pprint
>>> from sympy.abc import x, n
>>> f, P, Q = map(Function, ['f', 'P', 'Q'])
>>> genform = Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)**n)
>>> pprint(genform)
d n
P(x)*f(x) + --(f(x)) = Q(x)*f (x)
dx
>>> pprint(dsolve(genform, f(x), hint='Bernoulli_Integral'), num_columns=110)
-1
-----
n - 1
// / / \ \
|| | | | |
|| | / | / | / |
|| | | | | | | |
|| | -(n - 1)* | P(x) dx | -(n - 1)* | P(x) dx | (n - 1)* | P(x) dx|
|| | | | | | | |
|| | / | / | / |
f(x) = ||C1 - n* | Q(x)*e dx + | Q(x)*e dx|*e |
|| | | | |
\\ / / / /
Note that the equation is separable when `n = 1` (see the docstring of
:obj:`~sympy.solvers.ode.single.Separable`).
>>> pprint(dsolve(Eq(f(x).diff(x) + P(x)*f(x), Q(x)*f(x)), f(x),
... hint='separable_Integral'))
f(x)
/
| /
| 1 |
| - dy = C1 + | (-P(x) + Q(x)) dx
| y |
| /
/
Examples
========
>>> from sympy import Function, dsolve, Eq, pprint, log
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(Eq(x*f(x).diff(x) + f(x), log(x)*f(x)**2),
... f(x), hint='Bernoulli'))
1
f(x) = -----------------
C1*x + log(x) + 1
References
==========
- https://en.wikipedia.org/wiki/Bernoulli_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 95
# indirect doctest
"""
hint = "Bernoulli"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
P = Wild('P', exclude=[f(x)])
Q = Wild('Q', exclude=[f(x)])
n = Wild('n', exclude=[x, f(x), f(x).diff(x)])
return P, Q, n
def _equation(self, fx, x, order):
P, Q, n = self.wilds()
return fx.diff(x) + P*fx - Q*fx**n
def _get_general_solution(self, *, simplify_flag: bool = True):
P, Q, n = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
if n==1:
gensol = Eq(log(fx), (
C1 + Integral((-P + Q), x)
))
else:
gensol = Eq(fx**(1-n), (
(C1 - (n - 1) * Integral(Q*exp(-n*Integral(P, x))
* exp(Integral(P, x)), x)
) * exp(-(1 - n)*Integral(P, x)))
)
return [gensol]
class Factorable(SingleODESolver):
r"""
Solves equations having a solvable factor.
This function is used to solve the equation having factors. Factors may be of type algebraic or ode. It
will try to solve each factor independently. Factors will be solved by calling dsolve. We will return the
list of solutions.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = (f(x)**2-4)*(f(x).diff(x)+f(x))
>>> pprint(dsolve(eq, f(x)))
-x
[f(x) = 2, f(x) = -2, f(x) = C1*e ]
"""
hint = "factorable"
has_integral = False
def _matches(self):
eq_orig = self.ode_problem.eq
f = self.ode_problem.func.func
x = self.ode_problem.sym
df = f(x).diff(x)
self.eqs = []
eq = eq_orig.collect(f(x), func = cancel)
eq = fraction(factor(eq))[0]
factors = Mul.make_args(factor(eq))
roots = [fac.as_base_exp() for fac in factors if len(fac.args)!=0]
if len(roots)>1 or roots[0][1]>1:
for base, expo in roots:
if base.has(f(x)):
self.eqs.append(base)
if len(self.eqs)>0:
return True
roots = solve(eq, df)
if len(roots)>0:
self.eqs = [(df - root) for root in roots]
# Avoid infinite recursion
matches = self.eqs != [eq_orig]
return matches
for i in factors:
if i.has(f(x)):
self.eqs.append(i)
return len(self.eqs)>0 and len(factors)>1
def _get_general_solution(self, *, simplify_flag: bool = True):
func = self.ode_problem.func.func
x = self.ode_problem.sym
eqns = self.eqs
sols = []
for eq in eqns:
try:
sol = dsolve(eq, func(x))
except NotImplementedError:
continue
else:
if isinstance(sol, list):
sols.extend(sol)
else:
sols.append(sol)
if sols == []:
raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by"
+ " the factorable group method")
return sols
class RiccatiSpecial(SinglePatternODESolver):
r"""
The general Riccati equation has the form
.. math:: dy/dx = f(x) y^2 + g(x) y + h(x)\text{.}
While it does not have a general solution [1], the "special" form, `dy/dx
= a y^2 - b x^c`, does have solutions in many cases [2]. This routine
returns a solution for `a(dy/dx) = b y^2 + c y/x + d/x^2` that is obtained
by using a suitable change of variables to reduce it to the special form
and is valid when neither `a` nor `b` are zero and either `c` or `d` is
zero.
>>> from sympy.abc import x, a, b, c, d
>>> from sympy import dsolve, checkodesol, pprint, Function
>>> f = Function('f')
>>> y = f(x)
>>> genform = a*y.diff(x) - (b*y**2 + c*y/x + d/x**2)
>>> sol = dsolve(genform, y, hint="Riccati_special_minus2")
>>> pprint(sol, wrap_line=False)
/ / __________________ \\
| __________________ | / 2 ||
| / 2 | \/ 4*b*d - (a + c) *log(x)||
-|a + c - \/ 4*b*d - (a + c) *tan|C1 + ----------------------------||
\ \ 2*a //
f(x) = ------------------------------------------------------------------------
2*b*x
>>> checkodesol(genform, sol, order=1)[0]
True
References
==========
- http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Riccati
- http://eqworld.ipmnet.ru/en/solutions/ode/ode0106.pdf -
http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf
"""
hint = "Riccati_special_minus2"
has_integral = False
order = [1]
def _wilds(self, f, x, order):
a = Wild('a', exclude=[x, f(x), f(x).diff(x), 0])
b = Wild('b', exclude=[x, f(x), f(x).diff(x), 0])
c = Wild('c', exclude=[x, f(x), f(x).diff(x)])
d = Wild('d', exclude=[x, f(x), f(x).diff(x)])
return a, b, c, d
def _equation(self, fx, x, order):
a, b, c, d = self.wilds()
return a*fx.diff(x) + b*fx**2 + c*fx/x + d/x**2
def _get_general_solution(self, *, simplify_flag: bool = True):
a, b, c, d = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
(C1,) = self.ode_problem.get_numbered_constants(num=1)
mu = sqrt(4*d*b - (a - c)**2)
gensol = Eq(fx, (a - c - mu*tan(mu/(2*a)*log(x) + C1))/(2*b*x))
return [gensol]
class RationalRiccati(SinglePatternODESolver):
r"""
Gives general solutions to the first order Riccati differential
equations that have atleast one rational particular solution.
.. math :: y' = b_0(x) + b_1(x) y + b_2(x) y^2
where `b_0`, `b_1` and `b_2` are rational functions of `x`
with `b_2 \ne 0` (`b_2 = 0` would make it a Bernoulli equation).
Examples
========
>>> from sympy import Symbol, Function, dsolve, checkodesol
>>> f = Function('f')
>>> x = Symbol('x')
>>> eq = -x**4*f(x)**2 + x**3*f(x).diff(x) + x**2*f(x) + 20
>>> sol = dsolve(eq, hint="1st_rational_riccati")
>>> sol
Eq(f(x), (4*C1 - 5*x**9 - 4)/(x**2*(C1 + x**9 - 1)))
>>> checkodesol(eq, sol)
(True, 0)
References
==========
- Riccati ODE: https://en.wikipedia.org/wiki/Riccati_equation
- N. Thieu Vo - Rational and Algebraic Solutions of First-Order Algebraic ODEs:
Algorithm 11, pp. 78 - https://www3.risc.jku.at/publications/download/risc_5387/PhDThesisThieu.pdf
"""
has_integral = False
hint = "1st_rational_riccati"
order = [1]
def _wilds(self, f, x, order):
b0 = Wild('b0', exclude=[f(x), f(x).diff(x)])
b1 = Wild('b1', exclude=[f(x), f(x).diff(x)])
b2 = Wild('b2', exclude=[f(x), f(x).diff(x)])
return (b0, b1, b2)
def _equation(self, fx, x, order):
b0, b1, b2 = self.wilds()
return fx.diff(x) - b0 - b1*fx - b2*fx**2
def _matches(self):
eq = self.ode_problem.eq_expanded
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
if order != 1:
return False
match, funcs = match_riccati(eq, f, x)
if not match:
return False
_b0, _b1, _b2 = funcs
b0, b1, b2 = self.wilds()
self._wilds_match = match = {b0: _b0, b1: _b1, b2: _b2}
return True
def _get_general_solution(self, *, simplify_flag: bool = True):
# Match the equation
b0, b1, b2 = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
return solve_riccati(fx, x, b0, b1, b2, gensol=True)
class SecondNonlinearAutonomousConserved(SinglePatternODESolver):
r"""
Gives solution for the autonomous second order nonlinear
differential equation of the form
.. math :: f''(x) = g(f(x))
The solution for this differential equation can be computed
by multiplying by `f'(x)` and integrating on both sides,
converting it into a first order differential equation.
Examples
========
>>> from sympy import Function, symbols, dsolve
>>> f, g = symbols('f g', cls=Function)
>>> x = symbols('x')
>>> eq = f(x).diff(x, 2) - g(f(x))
>>> dsolve(eq, simplify=False)
[Eq(Integral(1/sqrt(C1 + 2*Integral(g(_u), _u)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(C1 + 2*Integral(g(_u), _u)), (_u, f(x))), C2 - x)]
>>> from sympy import exp, log
>>> eq = f(x).diff(x, 2) - exp(f(x)) + log(f(x))
>>> dsolve(eq, simplify=False)
[Eq(Integral(1/sqrt(-2*_u*log(_u) + 2*_u + C1 + 2*exp(_u)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(-2*_u*log(_u) + 2*_u + C1 + 2*exp(_u)), (_u, f(x))), C2 - x)]
References
==========
- http://eqworld.ipmnet.ru/en/solutions/ode/ode0301.pdf
"""
hint = "2nd_nonlinear_autonomous_conserved"
has_integral = True
order = [2]
def _wilds(self, f, x, order):
fy = Wild('fy', exclude=[0, f(x).diff(x), f(x).diff(x, 2)])
return (fy, )
def _equation(self, fx, x, order):
fy = self.wilds()[0]
return fx.diff(x, 2) + fy
def _verify(self, fx):
return self.ode_problem.is_autonomous
def _get_general_solution(self, *, simplify_flag: bool = True):
g = self.wilds_match()[0]
fx = self.ode_problem.func
x = self.ode_problem.sym
u = Dummy('u')
g = g.subs(fx, u)
C1, C2 = self.ode_problem.get_numbered_constants(num=2)
inside = -2*Integral(g, u) + C1
lhs = Integral(1/sqrt(inside), (u, fx))
return [Eq(lhs, C2 + x), Eq(lhs, C2 - x)]
class Liouville(SinglePatternODESolver):
r"""
Solves 2nd order Liouville differential equations.
The general form of a Liouville ODE is
.. math:: \frac{d^2 y}{dx^2} + g(y) \left(\!
\frac{dy}{dx}\!\right)^2 + h(x)
\frac{dy}{dx}\text{.}
The general solution is:
>>> from sympy import Function, dsolve, Eq, pprint, diff
>>> from sympy.abc import x
>>> f, g, h = map(Function, ['f', 'g', 'h'])
>>> genform = Eq(diff(f(x),x,x) + g(f(x))*diff(f(x),x)**2 +
... h(x)*diff(f(x),x), 0)
>>> pprint(genform)
2 2
/d \ d d
g(f(x))*|--(f(x))| + h(x)*--(f(x)) + ---(f(x)) = 0
\dx / dx 2
dx
>>> pprint(dsolve(genform, f(x), hint='Liouville_Integral'))
f(x)
/ /
| |
| / | /
| | | |
| - | h(x) dx | | g(y) dy
| | | |
| / | /
C1 + C2* | e dx + | e dy = 0
| |
/ /
Examples
========
>>> from sympy import Function, dsolve, Eq, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(diff(f(x), x, x) + diff(f(x), x)**2/f(x) +
... diff(f(x), x)/x, f(x), hint='Liouville'))
________________ ________________
[f(x) = -\/ C1 + C2*log(x) , f(x) = \/ C1 + C2*log(x) ]
References
==========
- Goldstein and Braun, "Advanced Methods for the Solution of Differential
Equations", pp. 98
- http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Liouville
# indirect doctest
"""
hint = "Liouville"
has_integral = True
order = [2]
def _wilds(self, f, x, order):
d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)])
e = Wild('e', exclude=[f(x).diff(x)])
k = Wild('k', exclude=[f(x).diff(x)])
return d, e, k
def _equation(self, fx, x, order):
# Liouville ODE in the form
# f(x).diff(x, 2) + g(f(x))*(f(x).diff(x))**2 + h(x)*f(x).diff(x)
# See Goldstein and Braun, "Advanced Methods for the Solution of
# Differential Equations", pg. 98
d, e, k = self.wilds()
return d*fx.diff(x, 2) + e*fx.diff(x)**2 + k*fx.diff(x)
def _verify(self, fx):
d, e, k = self.wilds_match()
self.y = Dummy('y')
x = self.ode_problem.sym
self.g = simplify(e/d).subs(fx, self.y)
self.h = simplify(k/d).subs(fx, self.y)
if self.y in self.h.free_symbols or x in self.g.free_symbols:
return False
return True
def _get_general_solution(self, *, simplify_flag: bool = True):
d, e, k = self.wilds_match()
fx = self.ode_problem.func
x = self.ode_problem.sym
C1, C2 = self.ode_problem.get_numbered_constants(num=2)
int = Integral(exp(Integral(self.g, self.y)), (self.y, None, fx))
gen_sol = Eq(int + C1*Integral(exp(-Integral(self.h, x)), x) + C2, 0)
return [gen_sol]
class Separable(SinglePatternODESolver):
r"""
Solves separable 1st order differential equations.
This is any differential equation that can be written as `P(y)
\tfrac{dy}{dx} = Q(x)`. The solution can then just be found by
rearranging terms and integrating: `\int P(y) \,dy = \int Q(x) \,dx`.
This hint uses :py:meth:`sympy.simplify.simplify.separatevars` as its back
end, so if a separable equation is not caught by this solver, it is most
likely the fault of that function.
:py:meth:`~sympy.simplify.simplify.separatevars` is
smart enough to do most expansion and factoring necessary to convert a
separable equation `F(x, y)` into the proper form `P(x)\cdot{}Q(y)`. The
general solution is::
>>> from sympy import Function, dsolve, Eq, pprint
>>> from sympy.abc import x
>>> a, b, c, d, f = map(Function, ['a', 'b', 'c', 'd', 'f'])
>>> genform = Eq(a(x)*b(f(x))*f(x).diff(x), c(x)*d(f(x)))
>>> pprint(genform)
d
a(x)*b(f(x))*--(f(x)) = c(x)*d(f(x))
dx
>>> pprint(dsolve(genform, f(x), hint='separable_Integral'))
f(x)
/ /
| |
| b(y) | c(x)
| ---- dy = C1 + | ---- dx
| d(y) | a(x)
| |
/ /
Examples
========
>>> from sympy import Function, dsolve, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(Eq(f(x)*f(x).diff(x) + x, 3*x*f(x)**2), f(x),
... hint='separable', simplify=False))
/ 2 \ 2
log\3*f (x) - 1/ x
---------------- = C1 + --
6 2
References
==========
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 52
# indirect doctest
"""
hint = "separable"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)])
e = Wild('e', exclude=[f(x).diff(x)])
return d, e
def _equation(self, fx, x, order):
d, e = self.wilds()
return d + e*fx.diff(x)
def _verify(self, fx):
d, e = self.wilds_match()
self.y = Dummy('y')
x = self.ode_problem.sym
d = separatevars(d.subs(fx, self.y))
e = separatevars(e.subs(fx, self.y))
# m1[coeff]*m1[x]*m1[y] + m2[coeff]*m2[x]*m2[y]*y'
self.m1 = separatevars(d, dict=True, symbols=(x, self.y))
self.m2 = separatevars(e, dict=True, symbols=(x, self.y))
if self.m1 and self.m2:
return True
return False
def _get_match_object(self):
fx = self.ode_problem.func
x = self.ode_problem.sym
return self.m1, self.m2, x, fx
def _get_general_solution(self, *, simplify_flag: bool = True):
m1, m2, x, fx = self._get_match_object()
(C1,) = self.ode_problem.get_numbered_constants(num=1)
int = Integral(m2['coeff']*m2[self.y]/m1[self.y],
(self.y, None, fx))
gen_sol = Eq(int, Integral(-m1['coeff']*m1[x]/
m2[x], x) + C1)
return [gen_sol]
class SeparableReduced(Separable):
r"""
Solves a differential equation that can be reduced to the separable form.
The general form of this equation is
.. math:: y' + (y/x) H(x^n y) = 0\text{}.
This can be solved by substituting `u(y) = x^n y`. The equation then
reduces to the separable form `\frac{u'}{u (\mathrm{power} - H(u))} -
\frac{1}{x} = 0`.
The general solution is:
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x, n
>>> f, g = map(Function, ['f', 'g'])
>>> genform = f(x).diff(x) + (f(x)/x)*g(x**n*f(x))
>>> pprint(genform)
/ n \
d f(x)*g\x *f(x)/
--(f(x)) + ---------------
dx x
>>> pprint(dsolve(genform, hint='separable_reduced'))
n
x *f(x)
/
|
| 1
| ------------ dy = C1 + log(x)
| y*(n - g(y))
|
/
See Also
========
:obj:`sympy.solvers.ode.single.Separable`
Examples
========
>>> from sympy import dsolve, Function, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> d = f(x).diff(x)
>>> eq = (x - x**2*f(x))*d - f(x)
>>> dsolve(eq, hint='separable_reduced')
[Eq(f(x), (1 - sqrt(C1*x**2 + 1))/x), Eq(f(x), (sqrt(C1*x**2 + 1) + 1)/x)]
>>> pprint(dsolve(eq, hint='separable_reduced'))
___________ ___________
/ 2 / 2
1 - \/ C1*x + 1 \/ C1*x + 1 + 1
[f(x) = ------------------, f(x) = ------------------]
x x
References
==========
- Joel Moses, "Symbolic Integration - The Stormy Decade", Communications
of the ACM, Volume 14, Number 8, August 1971, pp. 558
"""
hint = "separable_reduced"
has_integral = True
order = [1]
def _degree(self, expr, x):
# Made this function to calculate the degree of
# x in an expression. If expr will be of form
# x**p*y, (wheare p can be variables/rationals) then it
# will return p.
for val in expr:
if val.has(x):
if isinstance(val, Pow) and val.as_base_exp()[0] == x:
return (val.as_base_exp()[1])
elif val == x:
return (val.as_base_exp()[1])
else:
return self._degree(val.args, x)
return 0
def _powers(self, expr):
# this function will return all the different relative power of x w.r.t f(x).
# expr = x**p * f(x)**q then it will return {p/q}.
pows = set()
fx = self.ode_problem.func
x = self.ode_problem.sym
self.y = Dummy('y')
if isinstance(expr, Add):
exprs = expr.atoms(Add)
elif isinstance(expr, Mul):
exprs = expr.atoms(Mul)
elif isinstance(expr, Pow):
exprs = expr.atoms(Pow)
else:
exprs = {expr}
for arg in exprs:
if arg.has(x):
_, u = arg.as_independent(x, fx)
pow = self._degree((u.subs(fx, self.y), ), x)/self._degree((u.subs(fx, self.y), ), self.y)
pows.add(pow)
return pows
def _verify(self, fx):
num, den = self.wilds_match()
x = self.ode_problem.sym
factor = simplify(x/fx*num/den)
# Try representing factor in terms of x^n*y
# where n is lowest power of x in factor;
# first remove terms like sqrt(2)*3 from factor.atoms(Mul)
num, dem = factor.as_numer_denom()
num = expand(num)
dem = expand(dem)
pows = self._powers(num)
pows.update(self._powers(dem))
pows = list(pows)
if(len(pows)==1) and pows[0]!=zoo:
self.t = Dummy('t')
self.r2 = {'t': self.t}
num = num.subs(x**pows[0]*fx, self.t)
dem = dem.subs(x**pows[0]*fx, self.t)
test = num/dem
free = test.free_symbols
if len(free) == 1 and free.pop() == self.t:
self.r2.update({'power' : pows[0], 'u' : test})
return True
return False
return False
def _get_match_object(self):
fx = self.ode_problem.func
x = self.ode_problem.sym
u = self.r2['u'].subs(self.r2['t'], self.y)
ycoeff = 1/(self.y*(self.r2['power'] - u))
m1 = {self.y: 1, x: -1/x, 'coeff': 1}
m2 = {self.y: ycoeff, x: 1, 'coeff': 1}
return m1, m2, x, x**self.r2['power']*fx
class HomogeneousCoeffSubsDepDivIndep(SinglePatternODESolver):
r"""
Solves a 1st order differential equation with homogeneous coefficients
using the substitution `u_1 = \frac{\text{<dependent
variable>}}{\text{<independent variable>}}`.
This is a differential equation
.. math:: P(x, y) + Q(x, y) dy/dx = 0
such that `P` and `Q` are homogeneous and of the same order. A function
`F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`.
Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See
also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`.
If the coefficients `P` and `Q` in the differential equation above are
homogeneous functions of the same order, then it can be shown that the
substitution `y = u_1 x` (i.e. `u_1 = y/x`) will turn the differential
equation into an equation separable in the variables `x` and `u`. If
`h(u_1)` is the function that results from making the substitution `u_1 =
f(x)/x` on `P(x, f(x))` and `g(u_2)` is the function that results from the
substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) +
Q(x, f(x)) f'(x) = 0`, then the general solution is::
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f, g, h = map(Function, ['f', 'g', 'h'])
>>> genform = g(f(x)/x) + h(f(x)/x)*f(x).diff(x)
>>> pprint(genform)
/f(x)\ /f(x)\ d
g|----| + h|----|*--(f(x))
\ x / \ x / dx
>>> pprint(dsolve(genform, f(x),
... hint='1st_homogeneous_coeff_subs_dep_div_indep_Integral'))
f(x)
----
x
/
|
| -h(u1)
log(x) = C1 + | ---------------- d(u1)
| u1*h(u1) + g(u1)
|
/
Where `u_1 h(u_1) + g(u_1) \ne 0` and `x \ne 0`.
See also the docstrings of
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest` and
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`.
Examples
========
>>> from sympy import Function, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),
... hint='1st_homogeneous_coeff_subs_dep_div_indep', simplify=False))
/ 3 \
|3*f(x) f (x)|
log|------ + -----|
| x 3 |
\ x /
log(x) = log(C1) - -------------------
3
References
==========
- https://en.wikipedia.org/wiki/Homogeneous_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 59
# indirect doctest
"""
hint = "1st_homogeneous_coeff_subs_dep_div_indep"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)])
e = Wild('e', exclude=[f(x).diff(x)])
return d, e
def _equation(self, fx, x, order):
d, e = self.wilds()
return d + e*fx.diff(x)
def _verify(self, fx):
self.d, self.e = self.wilds_match()
self.y = Dummy('y')
x = self.ode_problem.sym
self.d = separatevars(self.d.subs(fx, self.y))
self.e = separatevars(self.e.subs(fx, self.y))
ordera = homogeneous_order(self.d, x, self.y)
orderb = homogeneous_order(self.e, x, self.y)
if ordera == orderb and ordera is not None:
self.u = Dummy('u')
if simplify((self.d + self.u*self.e).subs({x: 1, self.y: self.u})) != 0:
return True
return False
return False
def _get_match_object(self):
fx = self.ode_problem.func
x = self.ode_problem.sym
self.u1 = Dummy('u1')
xarg = 0
yarg = 0
return [self.d, self.e, fx, x, self.u, self.u1, self.y, xarg, yarg]
def _get_general_solution(self, *, simplify_flag: bool = True):
d, e, fx, x, u, u1, y, xarg, yarg = self._get_match_object()
(C1,) = self.ode_problem.get_numbered_constants(num=1)
int = Integral(
(-e/(d + u1*e)).subs({x: 1, y: u1}),
(u1, None, fx/x))
sol = logcombine(Eq(log(x), int + log(C1)), force=True)
gen_sol = sol.subs(fx, u).subs(((u, u - yarg), (x, x - xarg), (u, fx)))
return [gen_sol]
class HomogeneousCoeffSubsIndepDivDep(SinglePatternODESolver):
r"""
Solves a 1st order differential equation with homogeneous coefficients
using the substitution `u_2 = \frac{\text{<independent
variable>}}{\text{<dependent variable>}}`.
This is a differential equation
.. math:: P(x, y) + Q(x, y) dy/dx = 0
such that `P` and `Q` are homogeneous and of the same order. A function
`F(x, y)` is homogeneous of order `n` if `F(x t, y t) = t^n F(x, y)`.
Equivalently, `F(x, y)` can be rewritten as `G(y/x)` or `H(x/y)`. See
also the docstring of :py:meth:`~sympy.solvers.ode.homogeneous_order`.
If the coefficients `P` and `Q` in the differential equation above are
homogeneous functions of the same order, then it can be shown that the
substitution `x = u_2 y` (i.e. `u_2 = x/y`) will turn the differential
equation into an equation separable in the variables `y` and `u_2`. If
`h(u_2)` is the function that results from making the substitution `u_2 =
x/f(x)` on `P(x, f(x))` and `g(u_2)` is the function that results from the
substitution on `Q(x, f(x))` in the differential equation `P(x, f(x)) +
Q(x, f(x)) f'(x) = 0`, then the general solution is:
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f, g, h = map(Function, ['f', 'g', 'h'])
>>> genform = g(x/f(x)) + h(x/f(x))*f(x).diff(x)
>>> pprint(genform)
/ x \ / x \ d
g|----| + h|----|*--(f(x))
\f(x)/ \f(x)/ dx
>>> pprint(dsolve(genform, f(x),
... hint='1st_homogeneous_coeff_subs_indep_div_dep_Integral'))
x
----
f(x)
/
|
| -g(u1)
| ---------------- d(u1)
| u1*g(u1) + h(u1)
|
/
<BLANKLINE>
f(x) = C1*e
Where `u_1 g(u_1) + h(u_1) \ne 0` and `f(x) \ne 0`.
See also the docstrings of
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffBest` and
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep`.
Examples
========
>>> from sympy import Function, pprint, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),
... hint='1st_homogeneous_coeff_subs_indep_div_dep',
... simplify=False))
/ 2 \
| 3*x |
log|----- + 1|
| 2 |
\f (x) /
log(f(x)) = log(C1) - --------------
3
References
==========
- https://en.wikipedia.org/wiki/Homogeneous_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 59
# indirect doctest
"""
hint = "1st_homogeneous_coeff_subs_indep_div_dep"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)])
e = Wild('e', exclude=[f(x).diff(x)])
return d, e
def _equation(self, fx, x, order):
d, e = self.wilds()
return d + e*fx.diff(x)
def _verify(self, fx):
self.d, self.e = self.wilds_match()
self.y = Dummy('y')
x = self.ode_problem.sym
self.d = separatevars(self.d.subs(fx, self.y))
self.e = separatevars(self.e.subs(fx, self.y))
ordera = homogeneous_order(self.d, x, self.y)
orderb = homogeneous_order(self.e, x, self.y)
if ordera == orderb and ordera is not None:
self.u = Dummy('u')
if simplify((self.e + self.u*self.d).subs({x: self.u, self.y: 1})) != 0:
return True
return False
return False
def _get_match_object(self):
fx = self.ode_problem.func
x = self.ode_problem.sym
self.u1 = Dummy('u1')
xarg = 0
yarg = 0
return [self.d, self.e, fx, x, self.u, self.u1, self.y, xarg, yarg]
def _get_general_solution(self, *, simplify_flag: bool = True):
d, e, fx, x, u, u1, y, xarg, yarg = self._get_match_object()
(C1,) = self.ode_problem.get_numbered_constants(num=1)
int = Integral(simplify((-d/(e + u1*d)).subs({x: u1, y: 1})), (u1, None, x/fx)) # type: ignore
sol = logcombine(Eq(log(fx), int + log(C1)), force=True)
gen_sol = sol.subs(fx, u).subs(((u, u - yarg), (x, x - xarg), (u, fx)))
return [gen_sol]
class HomogeneousCoeffBest(HomogeneousCoeffSubsIndepDivDep, HomogeneousCoeffSubsDepDivIndep):
r"""
Returns the best solution to an ODE from the two hints
``1st_homogeneous_coeff_subs_dep_div_indep`` and
``1st_homogeneous_coeff_subs_indep_div_dep``.
This is as determined by :py:meth:`~sympy.solvers.ode.ode.ode_sol_simplicity`.
See the
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`
and
:obj:`~sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep`
docstrings for more information on these hints. Note that there is no
``ode_1st_homogeneous_coeff_best_Integral`` hint.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x), f(x),
... hint='1st_homogeneous_coeff_best', simplify=False))
/ 2 \
| 3*x |
log|----- + 1|
| 2 |
\f (x) /
log(f(x)) = log(C1) - --------------
3
References
==========
- https://en.wikipedia.org/wiki/Homogeneous_differential_equation
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 59
# indirect doctest
"""
hint = "1st_homogeneous_coeff_best"
has_integral = False
order = [1]
def _verify(self, fx):
if HomogeneousCoeffSubsIndepDivDep._verify(self, fx) and HomogeneousCoeffSubsDepDivIndep._verify(self, fx):
return True
return False
def _get_general_solution(self, *, simplify_flag: bool = True):
# There are two substitutions that solve the equation, u1=y/x and u2=x/y
# # They produce different integrals, so try them both and see which
# # one is easier
sol1 = HomogeneousCoeffSubsIndepDivDep._get_general_solution(self)
sol2 = HomogeneousCoeffSubsDepDivIndep._get_general_solution(self)
fx = self.ode_problem.func
if simplify_flag:
sol1 = odesimp(self.ode_problem.eq, *sol1, fx, "1st_homogeneous_coeff_subs_indep_div_dep")
sol2 = odesimp(self.ode_problem.eq, *sol2, fx, "1st_homogeneous_coeff_subs_dep_div_indep")
return min([sol1, sol2], key=lambda x: ode_sol_simplicity(x, fx, trysolving=not simplify))
class LinearCoefficients(HomogeneousCoeffBest):
r"""
Solves a differential equation with linear coefficients.
The general form of a differential equation with linear coefficients is
.. math:: y' + F\left(\!\frac{a_1 x + b_1 y + c_1}{a_2 x + b_2 y +
c_2}\!\right) = 0\text{,}
where `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are constants and `a_1 b_2
- a_2 b_1 \ne 0`.
This can be solved by substituting:
.. math:: x = x' + \frac{b_2 c_1 - b_1 c_2}{a_2 b_1 - a_1 b_2}
y = y' + \frac{a_1 c_2 - a_2 c_1}{a_2 b_1 - a_1
b_2}\text{.}
This substitution reduces the equation to a homogeneous differential
equation.
See Also
========
:obj:`sympy.solvers.ode.single.HomogeneousCoeffBest`
:obj:`sympy.solvers.ode.single.HomogeneousCoeffSubsIndepDivDep`
:obj:`sympy.solvers.ode.single.HomogeneousCoeffSubsDepDivIndep`
Examples
========
>>> from sympy import dsolve, Function, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> df = f(x).diff(x)
>>> eq = (x + f(x) + 1)*df + (f(x) - 6*x + 1)
>>> dsolve(eq, hint='linear_coefficients')
[Eq(f(x), -x - sqrt(C1 + 7*x**2) - 1), Eq(f(x), -x + sqrt(C1 + 7*x**2) - 1)]
>>> pprint(dsolve(eq, hint='linear_coefficients'))
___________ ___________
/ 2 / 2
[f(x) = -x - \/ C1 + 7*x - 1, f(x) = -x + \/ C1 + 7*x - 1]
References
==========
- Joel Moses, "Symbolic Integration - The Stormy Decade", Communications
of the ACM, Volume 14, Number 8, August 1971, pp. 558
"""
hint = "linear_coefficients"
has_integral = True
order = [1]
def _wilds(self, f, x, order):
d = Wild('d', exclude=[f(x).diff(x), f(x).diff(x, 2)])
e = Wild('e', exclude=[f(x).diff(x)])
return d, e
def _equation(self, fx, x, order):
d, e = self.wilds()
return d + e*fx.diff(x)
def _verify(self, fx):
self.d, self.e = self.wilds_match()
a, b = self.wilds()
F = self.d/self.e
x = self.ode_problem.sym
params = self._linear_coeff_match(F, fx)
if params:
self.xarg, self.yarg = params
u = Dummy('u')
t = Dummy('t')
self.y = Dummy('y')
# Dummy substitution for df and f(x).
dummy_eq = self.ode_problem.eq.subs(((fx.diff(x), t), (fx, u)))
reps = ((x, x + self.xarg), (u, u + self.yarg), (t, fx.diff(x)), (u, fx))
dummy_eq = simplify(dummy_eq.subs(reps))
# get the re-cast values for e and d
r2 = collect(expand(dummy_eq), [fx.diff(x), fx]).match(a*fx.diff(x) + b)
if r2:
self.d, self.e = r2[b], r2[a]
orderd = homogeneous_order(self.d, x, fx)
ordere = homogeneous_order(self.e, x, fx)
if orderd == ordere and orderd is not None:
self.d = self.d.subs(fx, self.y)
self.e = self.e.subs(fx, self.y)
return True
return False
return False
def _linear_coeff_match(self, expr, func):
r"""
Helper function to match hint ``linear_coefficients``.
Matches the expression to the form `(a_1 x + b_1 f(x) + c_1)/(a_2 x + b_2
f(x) + c_2)` where the following conditions hold:
1. `a_1`, `b_1`, `c_1`, `a_2`, `b_2`, `c_2` are Rationals;
2. `c_1` or `c_2` are not equal to zero;
3. `a_2 b_1 - a_1 b_2` is not equal to zero.
Return ``xarg``, ``yarg`` where
1. ``xarg`` = `(b_2 c_1 - b_1 c_2)/(a_2 b_1 - a_1 b_2)`
2. ``yarg`` = `(a_1 c_2 - a_2 c_1)/(a_2 b_1 - a_1 b_2)`
Examples
========
>>> from sympy import Function, sin
>>> from sympy.abc import x
>>> from sympy.solvers.ode.single import LinearCoefficients
>>> f = Function('f')
>>> eq = (-25*f(x) - 8*x + 62)/(4*f(x) + 11*x - 11)
>>> obj = LinearCoefficients(eq)
>>> obj._linear_coeff_match(eq, f(x))
(1/9, 22/9)
>>> eq = sin((-5*f(x) - 8*x + 6)/(4*f(x) + x - 1))
>>> obj = LinearCoefficients(eq)
>>> obj._linear_coeff_match(eq, f(x))
(19/27, 2/27)
>>> eq = sin(f(x)/x)
>>> obj = LinearCoefficients(eq)
>>> obj._linear_coeff_match(eq, f(x))
"""
f = func.func
x = func.args[0]
def abc(eq):
r'''
Internal function of _linear_coeff_match
that returns Rationals a, b, c
if eq is a*x + b*f(x) + c, else None.
'''
eq = _mexpand(eq)
c = eq.as_independent(x, f(x), as_Add=True)[0]
if not c.is_Rational:
return
a = eq.coeff(x)
if not a.is_Rational:
return
b = eq.coeff(f(x))
if not b.is_Rational:
return
if eq == a*x + b*f(x) + c:
return a, b, c
def match(arg):
r'''
Internal function of _linear_coeff_match that returns Rationals a1,
b1, c1, a2, b2, c2 and a2*b1 - a1*b2 of the expression (a1*x + b1*f(x)
+ c1)/(a2*x + b2*f(x) + c2) if one of c1 or c2 and a2*b1 - a1*b2 is
non-zero, else None.
'''
n, d = arg.together().as_numer_denom()
m = abc(n)
if m is not None:
a1, b1, c1 = m
m = abc(d)
if m is not None:
a2, b2, c2 = m
d = a2*b1 - a1*b2
if (c1 or c2) and d:
return a1, b1, c1, a2, b2, c2, d
m = [fi.args[0] for fi in expr.atoms(Function) if fi.func != f and
len(fi.args) == 1 and not fi.args[0].is_Function] or {expr}
m1 = match(m.pop())
if m1 and all(match(mi) == m1 for mi in m):
a1, b1, c1, a2, b2, c2, denom = m1
return (b2*c1 - b1*c2)/denom, (a1*c2 - a2*c1)/denom
def _get_match_object(self):
fx = self.ode_problem.func
x = self.ode_problem.sym
self.u1 = Dummy('u1')
u = Dummy('u')
return [self.d, self.e, fx, x, u, self.u1, self.y, self.xarg, self.yarg]
class NthOrderReducible(SingleODESolver):
r"""
Solves ODEs that only involve derivatives of the dependent variable using
a substitution of the form `f^n(x) = g(x)`.
For example any second order ODE of the form `f''(x) = h(f'(x), x)` can be
transformed into a pair of 1st order ODEs `g'(x) = h(g(x), x)` and
`f'(x) = g(x)`. Usually the 1st order ODE for `g` is easier to solve. If
that gives an explicit solution for `g` then `f` is found simply by
integration.
Examples
========
>>> from sympy import Function, dsolve, Eq
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = Eq(x*f(x).diff(x)**2 + f(x).diff(x, 2), 0)
>>> dsolve(eq, f(x), hint='nth_order_reducible')
... # doctest: +NORMALIZE_WHITESPACE
Eq(f(x), C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) + sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))
"""
hint = "nth_order_reducible"
has_integral = False
def _matches(self):
# Any ODE that can be solved with a substitution and
# repeated integration e.g.:
# `d^2/dx^2(y) + x*d/dx(y) = constant
#f'(x) must be finite for this to work
eq = self.ode_problem.eq_preprocessed
func = self.ode_problem.func
x = self.ode_problem.sym
r"""
Matches any differential equation that can be rewritten with a smaller
order. Only derivatives of ``func`` alone, wrt a single variable,
are considered, and only in them should ``func`` appear.
"""
# ODE only handles functions of 1 variable so this affirms that state
assert len(func.args) == 1
vc = [d.variable_count[0] for d in eq.atoms(Derivative)
if d.expr == func and len(d.variable_count) == 1]
ords = [c for v, c in vc if v == x]
if len(ords) < 2:
return False
self.smallest = min(ords)
# make sure func does not appear outside of derivatives
D = Dummy()
if eq.subs(func.diff(x, self.smallest), D).has(func):
return False
return True
def _get_general_solution(self, *, simplify_flag: bool = True):
eq = self.ode_problem.eq
f = self.ode_problem.func.func
x = self.ode_problem.sym
n = self.smallest
# get a unique function name for g
names = [a.name for a in eq.atoms(AppliedUndef)]
while True:
name = Dummy().name
if name not in names:
g = Function(name)
break
w = f(x).diff(x, n)
geq = eq.subs(w, g(x))
gsol = dsolve(geq, g(x))
if not isinstance(gsol, list):
gsol = [gsol]
# Might be multiple solutions to the reduced ODE:
fsol = []
for gsoli in gsol:
fsoli = dsolve(gsoli.subs(g(x), w), f(x)) # or do integration n times
fsol.append(fsoli)
return fsol
class SecondHypergeometric(SingleODESolver):
r"""
Solves 2nd order linear differential equations.
It computes special function solutions which can be expressed using the
2F1, 1F1 or 0F1 hypergeometric functions.
.. math:: y'' + A(x) y' + B(x) y = 0\text{,}
where `A` and `B` are rational functions.
These kinds of differential equations have solution of non-Liouvillian form.
Given linear ODE can be obtained from 2F1 given by
.. math:: (x^2 - x) y'' + ((a + b + 1) x - c) y' + b a y = 0\text{,}
where {a, b, c} are arbitrary constants.
Notes
=====
The algorithm should find any solution of the form
.. math:: y = P(x) _pF_q(..; ..;\frac{\alpha x^k + \beta}{\gamma x^k + \delta})\text{,}
where pFq is any of 2F1, 1F1 or 0F1 and `P` is an "arbitrary function".
Currently only the 2F1 case is implemented in SymPy but the other cases are
described in the paper and could be implemented in future (contributions
welcome!).
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = (x*x - x)*f(x).diff(x,2) + (5*x - 1)*f(x).diff(x) + 4*f(x)
>>> pprint(dsolve(eq, f(x), '2nd_hypergeometric'))
_
/ / 4 \\ |_ /-1, -1 | \
|C1 + C2*|log(x) + -----||* | | | x|
\ \ x + 1// 2 1 \ 1 | /
f(x) = --------------------------------------------
3
(x - 1)
References
==========
- "Non-Liouvillian solutions for second order linear ODEs" by L. Chan, E.S. Cheb-Terrab
"""
hint = "2nd_hypergeometric"
has_integral = True
def _matches(self):
eq = self.ode_problem.eq_preprocessed
func = self.ode_problem.func
r = match_2nd_hypergeometric(eq, func)
self.match_object = None
if r:
A, B = r
d = equivalence_hypergeometric(A, B, func)
if d:
if d['type'] == "2F1":
self.match_object = match_2nd_2F1_hypergeometric(d['I0'], d['k'], d['sing_point'], func)
if self.match_object is not None:
self.match_object.update({'A':A, 'B':B})
# We can extend it for 1F1 and 0F1 type also.
return self.match_object is not None
def _get_general_solution(self, *, simplify_flag: bool = True):
eq = self.ode_problem.eq
func = self.ode_problem.func
if self.match_object['type'] == "2F1":
sol = get_sol_2F1_hypergeometric(eq, func, self.match_object)
if sol is None:
raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by"
+ " the hypergeometric method")
return [sol]
class NthLinearConstantCoeffHomogeneous(SingleODESolver):
r"""
Solves an `n`\th order linear homogeneous differential equation with
constant coefficients.
This is an equation of the form
.. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x)
+ a_0 f(x) = 0\text{.}
These equations can be solved in a general manner, by taking the roots of
the characteristic equation `a_n m^n + a_{n-1} m^{n-1} + \cdots + a_1 m +
a_0 = 0`. The solution will then be the sum of `C_n x^i e^{r x}` terms,
for each where `C_n` is an arbitrary constant, `r` is a root of the
characteristic equation and `i` is one of each from 0 to the multiplicity
of the root - 1 (for example, a root 3 of multiplicity 2 would create the
terms `C_1 e^{3 x} + C_2 x e^{3 x}`). The exponential is usually expanded
for complex roots using Euler's equation `e^{I x} = \cos(x) + I \sin(x)`.
Complex roots always come in conjugate pairs in polynomials with real
coefficients, so the two roots will be represented (after simplifying the
constants) as `e^{a x} \left(C_1 \cos(b x) + C_2 \sin(b x)\right)`.
If SymPy cannot find exact roots to the characteristic equation, a
:py:class:`~sympy.polys.rootoftools.ComplexRootOf` instance will be return
instead.
>>> from sympy import Function, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(f(x).diff(x, 5) + 10*f(x).diff(x) - 2*f(x), f(x),
... hint='nth_linear_constant_coeff_homogeneous')
... # doctest: +NORMALIZE_WHITESPACE
Eq(f(x), C5*exp(x*CRootOf(_x**5 + 10*_x - 2, 0))
+ (C1*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 1)))
+ C2*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 1))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 1)))
+ (C3*sin(x*im(CRootOf(_x**5 + 10*_x - 2, 3)))
+ C4*cos(x*im(CRootOf(_x**5 + 10*_x - 2, 3))))*exp(x*re(CRootOf(_x**5 + 10*_x - 2, 3))))
Note that because this method does not involve integration, there is no
``nth_linear_constant_coeff_homogeneous_Integral`` hint.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x, 4) + 2*f(x).diff(x, 3) -
... 2*f(x).diff(x, 2) - 6*f(x).diff(x) + 5*f(x), f(x),
... hint='nth_linear_constant_coeff_homogeneous'))
x -2*x
f(x) = (C1 + C2*x)*e + (C3*sin(x) + C4*cos(x))*e
References
==========
- https://en.wikipedia.org/wiki/Linear_differential_equation section:
Nonhomogeneous_equation_with_constant_coefficients
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 211
# indirect doctest
"""
hint = "nth_linear_constant_coeff_homogeneous"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq_high_order_free
func = self.ode_problem.func
order = self.ode_problem.order
x = self.ode_problem.sym
self.r = self.ode_problem.get_linear_coefficients(eq, func, order)
if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0):
if not self.r[-1]:
return True
else:
return False
return False
def _get_general_solution(self, *, simplify_flag: bool = True):
fx = self.ode_problem.func
order = self.ode_problem.order
roots, collectterms = _get_const_characteristic_eq_sols(self.r, fx, order)
# A generator of constants
constants = self.ode_problem.get_numbered_constants(num=len(roots))
gsol = Add(*[i*j for (i, j) in zip(constants, roots)])
gsol = Eq(fx, gsol)
if simplify_flag:
gsol = _get_simplified_sol([gsol], fx, collectterms)
return [gsol]
class NthLinearConstantCoeffVariationOfParameters(SingleODESolver):
r"""
Solves an `n`\th order linear differential equation with constant
coefficients using the method of variation of parameters.
This method works on any differential equations of the form
.. math:: f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x) + a_0
f(x) = P(x)\text{.}
This method works by assuming that the particular solution takes the form
.. math:: \sum_{x=1}^{n} c_i(x) y_i(x)\text{,}
where `y_i` is the `i`\th solution to the homogeneous equation. The
solution is then solved using Wronskian's and Cramer's Rule. The
particular solution is given by
.. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \,dx
\right) y_i(x) \text{,}
where `W(x)` is the Wronskian of the fundamental system (the system of `n`
linearly independent solutions to the homogeneous equation), and `W_i(x)`
is the Wronskian of the fundamental system with the `i`\th column replaced
with `[0, 0, \cdots, 0, P(x)]`.
This method is general enough to solve any `n`\th order inhomogeneous
linear differential equation with constant coefficients, but sometimes
SymPy cannot simplify the Wronskian well enough to integrate it. If this
method hangs, try using the
``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and
simplifying the integrals manually. Also, prefer using
``nth_linear_constant_coeff_undetermined_coefficients`` when it
applies, because it does not use integration, making it faster and more
reliable.
Warning, using simplify=False with
'nth_linear_constant_coeff_variation_of_parameters' in
:py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will
not attempt to simplify the Wronskian before integrating. It is
recommended that you only use simplify=False with
'nth_linear_constant_coeff_variation_of_parameters_Integral' for this
method, especially if the solution to the homogeneous equation has
trigonometric functions in it.
Examples
========
>>> from sympy import Function, dsolve, pprint, exp, log
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x, 3) - 3*f(x).diff(x, 2) +
... 3*f(x).diff(x) - f(x) - exp(x)*log(x), f(x),
... hint='nth_linear_constant_coeff_variation_of_parameters'))
/ / / x*log(x) 11*x\\\ x
f(x) = |C1 + x*|C2 + x*|C3 + -------- - ----|||*e
\ \ \ 6 36 ///
References
==========
- https://en.wikipedia.org/wiki/Variation_of_parameters
- http://planetmath.org/VariationOfParameters
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 233
# indirect doctest
"""
hint = "nth_linear_constant_coeff_variation_of_parameters"
has_integral = True
def _matches(self):
eq = self.ode_problem.eq_high_order_free
func = self.ode_problem.func
order = self.ode_problem.order
x = self.ode_problem.sym
self.r = self.ode_problem.get_linear_coefficients(eq, func, order)
if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0):
if self.r[-1]:
return True
else:
return False
return False
def _get_general_solution(self, *, simplify_flag: bool = True):
eq = self.ode_problem.eq_high_order_free
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
roots, collectterms = _get_const_characteristic_eq_sols(self.r, f(x), order)
# A generator of constants
constants = self.ode_problem.get_numbered_constants(num=len(roots))
homogen_sol = Add(*[i*j for (i, j) in zip(constants, roots)])
homogen_sol = Eq(f(x), homogen_sol)
homogen_sol = _solve_variation_of_parameters(eq, f(x), roots, homogen_sol, order, self.r, simplify_flag)
if simplify_flag:
homogen_sol = _get_simplified_sol([homogen_sol], f(x), collectterms)
return [homogen_sol]
class NthLinearConstantCoeffUndeterminedCoefficients(SingleODESolver):
r"""
Solves an `n`\th order linear differential equation with constant
coefficients using the method of undetermined coefficients.
This method works on differential equations of the form
.. math:: a_n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \cdots + a_1 f'(x)
+ a_0 f(x) = P(x)\text{,}
where `P(x)` is a function that has a finite number of linearly
independent derivatives.
Functions that fit this requirement are finite sums functions of the form
`a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i`
is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For
example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`,
and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have
a finite number of derivatives, because they can be expanded into `\sin(a
x)` and `\cos(b x)` terms. However, SymPy currently cannot do that
expansion, so you will need to manually rewrite the expression in terms of
the above to use this method. So, for example, you will need to manually
convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method
of undetermined coefficients on it.
This method works by creating a trial function from the expression and all
of its linear independent derivatives and substituting them into the
original ODE. The coefficients for each term will be a system of linear
equations, which are be solved for and substituted, giving the solution.
If any of the trial functions are linearly dependent on the solution to
the homogeneous equation, they are multiplied by sufficient `x` to make
them linearly independent.
Examples
========
>>> from sympy import Function, dsolve, pprint, exp, cos
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x, 2) + 2*f(x).diff(x) + f(x) -
... 4*exp(-x)*x**2 + cos(2*x), f(x),
... hint='nth_linear_constant_coeff_undetermined_coefficients'))
/ / 3\\
| | x || -x 4*sin(2*x) 3*cos(2*x)
f(x) = |C1 + x*|C2 + --||*e - ---------- + ----------
\ \ 3 // 25 25
References
==========
- https://en.wikipedia.org/wiki/Method_of_undetermined_coefficients
- M. Tenenbaum & H. Pollard, "Ordinary Differential Equations",
Dover 1963, pp. 221
# indirect doctest
"""
hint = "nth_linear_constant_coeff_undetermined_coefficients"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq_high_order_free
func = self.ode_problem.func
order = self.ode_problem.order
x = self.ode_problem.sym
self.r = self.ode_problem.get_linear_coefficients(eq, func, order)
does_match = False
if order and self.r and not any(self.r[i].has(x) for i in self.r if i >= 0):
if self.r[-1]:
eq_homogeneous = Add(eq, -self.r[-1])
undetcoeff = _undetermined_coefficients_match(self.r[-1], x, func, eq_homogeneous)
if undetcoeff['test']:
self.trialset = undetcoeff['trialset']
does_match = True
return does_match
def _get_general_solution(self, *, simplify_flag: bool = True):
eq = self.ode_problem.eq
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
roots, collectterms = _get_const_characteristic_eq_sols(self.r, f(x), order)
# A generator of constants
constants = self.ode_problem.get_numbered_constants(num=len(roots))
homogen_sol = Add(*[i*j for (i, j) in zip(constants, roots)])
homogen_sol = Eq(f(x), homogen_sol)
self.r.update({'list': roots, 'sol': homogen_sol, 'simpliy_flag': simplify_flag})
gsol = _solve_undetermined_coefficients(eq, f(x), order, self.r, self.trialset)
if simplify_flag:
gsol = _get_simplified_sol([gsol], f(x), collectterms)
return [gsol]
class NthLinearEulerEqHomogeneous(SingleODESolver):
r"""
Solves an `n`\th order linear homogeneous variable-coefficient
Cauchy-Euler equidimensional ordinary differential equation.
This is an equation with form `0 = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x)
\cdots`.
These equations can be solved in a general manner, by substituting
solutions of the form `f(x) = x^r`, and deriving a characteristic equation
for `r`. When there are repeated roots, we include extra terms of the
form `C_{r k} \ln^k(x) x^r`, where `C_{r k}` is an arbitrary integration
constant, `r` is a root of the characteristic equation, and `k` ranges
over the multiplicity of `r`. In the cases where the roots are complex,
solutions of the form `C_1 x^a \sin(b \log(x)) + C_2 x^a \cos(b \log(x))`
are returned, based on expansions with Euler's formula. The general
solution is the sum of the terms found. If SymPy cannot find exact roots
to the characteristic equation, a
:py:obj:`~.ComplexRootOf` instance will be returned
instead.
>>> from sympy import Function, dsolve
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(4*x**2*f(x).diff(x, 2) + f(x), f(x),
... hint='nth_linear_euler_eq_homogeneous')
... # doctest: +NORMALIZE_WHITESPACE
Eq(f(x), sqrt(x)*(C1 + C2*log(x)))
Note that because this method does not involve integration, there is no
``nth_linear_euler_eq_homogeneous_Integral`` hint.
The following is for internal use:
- ``returns = 'sol'`` returns the solution to the ODE.
- ``returns = 'list'`` returns a list of linearly independent solutions,
corresponding to the fundamental solution set, for use with non
homogeneous solution methods like variation of parameters and
undetermined coefficients. Note that, though the solutions should be
linearly independent, this function does not explicitly check that. You
can do ``assert simplify(wronskian(sollist)) != 0`` to check for linear
independence. Also, ``assert len(sollist) == order`` will need to pass.
- ``returns = 'both'``, return a dictionary ``{'sol': <solution to ODE>,
'list': <list of linearly independent solutions>}``.
Examples
========
>>> from sympy import Function, dsolve, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = f(x).diff(x, 2)*x**2 - 4*f(x).diff(x)*x + 6*f(x)
>>> pprint(dsolve(eq, f(x),
... hint='nth_linear_euler_eq_homogeneous'))
2
f(x) = x *(C1 + C2*x)
References
==========
- https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation
- C. Bender & S. Orszag, "Advanced Mathematical Methods for Scientists and
Engineers", Springer 1999, pp. 12
# indirect doctest
"""
hint = "nth_linear_euler_eq_homogeneous"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq_preprocessed
f = self.ode_problem.func.func
order = self.ode_problem.order
x = self.ode_problem.sym
match = self.ode_problem.get_linear_coefficients(eq, f(x), order)
self.r = None
does_match = False
if order and match:
coeff = match[order]
factor = x**order / coeff
self.r = {i: factor*match[i] for i in match}
if self.r and all(_test_term(self.r[i], f(x), i) for i in
self.r if i >= 0):
if not self.r[-1]:
does_match = True
return does_match
def _get_general_solution(self, *, simplify_flag: bool = True):
fx = self.ode_problem.func
eq = self.ode_problem.eq
homogen_sol = _get_euler_characteristic_eq_sols(eq, fx, self.r)[0]
return [homogen_sol]
class NthLinearEulerEqNonhomogeneousVariationOfParameters(SingleODESolver):
r"""
Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional
ordinary differential equation using variation of parameters.
This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x)
\cdots`.
This method works by assuming that the particular solution takes the form
.. math:: \sum_{x=1}^{n} c_i(x) y_i(x) {a_n} {x^n} \text{, }
where `y_i` is the `i`\th solution to the homogeneous equation. The
solution is then solved using Wronskian's and Cramer's Rule. The
particular solution is given by multiplying eq given below with `a_n x^{n}`
.. math:: \sum_{x=1}^n \left( \int \frac{W_i(x)}{W(x)} \, dx
\right) y_i(x) \text{, }
where `W(x)` is the Wronskian of the fundamental system (the system of `n`
linearly independent solutions to the homogeneous equation), and `W_i(x)`
is the Wronskian of the fundamental system with the `i`\th column replaced
with `[0, 0, \cdots, 0, \frac{x^{- n}}{a_n} g{\left(x \right)}]`.
This method is general enough to solve any `n`\th order inhomogeneous
linear differential equation, but sometimes SymPy cannot simplify the
Wronskian well enough to integrate it. If this method hangs, try using the
``nth_linear_constant_coeff_variation_of_parameters_Integral`` hint and
simplifying the integrals manually. Also, prefer using
``nth_linear_constant_coeff_undetermined_coefficients`` when it
applies, because it does not use integration, making it faster and more
reliable.
Warning, using simplify=False with
'nth_linear_constant_coeff_variation_of_parameters' in
:py:meth:`~sympy.solvers.ode.dsolve` may cause it to hang, because it will
not attempt to simplify the Wronskian before integrating. It is
recommended that you only use simplify=False with
'nth_linear_constant_coeff_variation_of_parameters_Integral' for this
method, especially if the solution to the homogeneous equation has
trigonometric functions in it.
Examples
========
>>> from sympy import Function, dsolve, Derivative
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - x**4
>>> dsolve(eq, f(x),
... hint='nth_linear_euler_eq_nonhomogeneous_variation_of_parameters').expand()
Eq(f(x), C1*x + C2*x**2 + x**4/6)
"""
hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters"
has_integral = True
def _matches(self):
eq = self.ode_problem.eq_preprocessed
f = self.ode_problem.func.func
order = self.ode_problem.order
x = self.ode_problem.sym
match = self.ode_problem.get_linear_coefficients(eq, f(x), order)
self.r = None
does_match = False
if order and match:
coeff = match[order]
factor = x**order / coeff
self.r = {i: factor*match[i] for i in match}
if self.r and all(_test_term(self.r[i], f(x), i) for i in
self.r if i >= 0):
if self.r[-1]:
does_match = True
return does_match
def _get_general_solution(self, *, simplify_flag: bool = True):
eq = self.ode_problem.eq
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
homogen_sol, roots = _get_euler_characteristic_eq_sols(eq, f(x), self.r)
self.r[-1] = self.r[-1]/self.r[order]
sol = _solve_variation_of_parameters(eq, f(x), roots, homogen_sol, order, self.r, simplify_flag)
return [Eq(f(x), homogen_sol.rhs + (sol.rhs - homogen_sol.rhs)*self.r[order])]
class NthLinearEulerEqNonhomogeneousUndeterminedCoefficients(SingleODESolver):
r"""
Solves an `n`\th order linear non homogeneous Cauchy-Euler equidimensional
ordinary differential equation using undetermined coefficients.
This is an equation with form `g(x) = a_0 f(x) + a_1 x f'(x) + a_2 x^2 f''(x)
\cdots`.
These equations can be solved in a general manner, by substituting
solutions of the form `x = exp(t)`, and deriving a characteristic equation
of form `g(exp(t)) = b_0 f(t) + b_1 f'(t) + b_2 f''(t) \cdots` which can
be then solved by nth_linear_constant_coeff_undetermined_coefficients if
g(exp(t)) has finite number of linearly independent derivatives.
Functions that fit this requirement are finite sums functions of the form
`a x^i e^{b x} \sin(c x + d)` or `a x^i e^{b x} \cos(c x + d)`, where `i`
is a non-negative integer and `a`, `b`, `c`, and `d` are constants. For
example any polynomial in `x`, functions like `x^2 e^{2 x}`, `x \sin(x)`,
and `e^x \cos(x)` can all be used. Products of `\sin`'s and `\cos`'s have
a finite number of derivatives, because they can be expanded into `\sin(a
x)` and `\cos(b x)` terms. However, SymPy currently cannot do that
expansion, so you will need to manually rewrite the expression in terms of
the above to use this method. So, for example, you will need to manually
convert `\sin^2(x)` into `(1 + \cos(2 x))/2` to properly apply the method
of undetermined coefficients on it.
After replacement of x by exp(t), this method works by creating a trial function
from the expression and all of its linear independent derivatives and
substituting them into the original ODE. The coefficients for each term
will be a system of linear equations, which are be solved for and
substituted, giving the solution. If any of the trial functions are linearly
dependent on the solution to the homogeneous equation, they are multiplied
by sufficient `x` to make them linearly independent.
Examples
========
>>> from sympy import dsolve, Function, Derivative, log
>>> from sympy.abc import x
>>> f = Function('f')
>>> eq = x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x)
>>> dsolve(eq, f(x),
... hint='nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients').expand()
Eq(f(x), C1*x + C2*x**2 + log(x)/2 + 3/4)
"""
hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq_high_order_free
f = self.ode_problem.func.func
order = self.ode_problem.order
x = self.ode_problem.sym
match = self.ode_problem.get_linear_coefficients(eq, f(x), order)
self.r = None
does_match = False
if order and match:
coeff = match[order]
factor = x**order / coeff
self.r = {i: factor*match[i] for i in match}
if self.r and all(_test_term(self.r[i], f(x), i) for i in
self.r if i >= 0):
if self.r[-1]:
e, re = posify(self.r[-1].subs(x, exp(x)))
undetcoeff = _undetermined_coefficients_match(e.subs(re), x)
if undetcoeff['test']:
does_match = True
return does_match
def _get_general_solution(self, *, simplify_flag: bool = True):
f = self.ode_problem.func.func
x = self.ode_problem.sym
chareq, eq, symbol = S.Zero, S.Zero, Dummy('x')
for i in self.r.keys():
if i >= 0:
chareq += (self.r[i]*diff(x**symbol, x, i)*x**-symbol).expand()
for i in range(1, degree(Poly(chareq, symbol))+1):
eq += chareq.coeff(symbol**i)*diff(f(x), x, i)
if chareq.as_coeff_add(symbol)[0]:
eq += chareq.as_coeff_add(symbol)[0]*f(x)
e, re = posify(self.r[-1].subs(x, exp(x)))
eq += e.subs(re)
self.const_undet_instance = NthLinearConstantCoeffUndeterminedCoefficients(SingleODEProblem(eq, f(x), x))
sol = self.const_undet_instance.get_general_solution(simplify = simplify_flag)[0]
sol = sol.subs(x, log(x))
sol = sol.subs(f(log(x)), f(x)).expand()
return [sol]
class SecondLinearBessel(SingleODESolver):
r"""
Gives solution of the Bessel differential equation
.. math :: x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} y(x) + (x^2-n^2) y(x)
if `n` is integer then the solution is of the form ``Eq(f(x), C0 besselj(n,x)
+ C1 bessely(n,x))`` as both the solutions are linearly independent else if
`n` is a fraction then the solution is of the form ``Eq(f(x), C0 besselj(n,x)
+ C1 besselj(-n,x))`` which can also transform into ``Eq(f(x), C0 besselj(n,x)
+ C1 bessely(n,x))``.
Examples
========
>>> from sympy.abc import x
>>> from sympy import Symbol
>>> v = Symbol('v', positive=True)
>>> from sympy import dsolve, Function
>>> f = Function('f')
>>> y = f(x)
>>> genform = x**2*y.diff(x, 2) + x*y.diff(x) + (x**2 - v**2)*y
>>> dsolve(genform)
Eq(f(x), C1*besselj(v, x) + C2*bessely(v, x))
References
==========
https://www.math24.net/bessel-differential-equation/
"""
hint = "2nd_linear_bessel"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq_high_order_free
f = self.ode_problem.func
order = self.ode_problem.order
x = self.ode_problem.sym
df = f.diff(x)
a = Wild('a', exclude=[f,df])
b = Wild('b', exclude=[x, f,df])
a4 = Wild('a4', exclude=[x,f,df])
b4 = Wild('b4', exclude=[x,f,df])
c4 = Wild('c4', exclude=[x,f,df])
d4 = Wild('d4', exclude=[x,f,df])
a3 = Wild('a3', exclude=[f, df, f.diff(x, 2)])
b3 = Wild('b3', exclude=[f, df, f.diff(x, 2)])
c3 = Wild('c3', exclude=[f, df, f.diff(x, 2)])
deq = a3*(f.diff(x, 2)) + b3*df + c3*f
r = collect(eq,
[f.diff(x, 2), df, f]).match(deq)
if order == 2 and r:
if not all(r[key].is_polynomial() for key in r):
n, d = eq.as_numer_denom()
eq = expand(n)
r = collect(eq,
[f.diff(x, 2), df, f]).match(deq)
if r and r[a3] != 0:
# leading coeff of f(x).diff(x, 2)
coeff = factor(r[a3]).match(a4*(x-b)**b4)
if coeff:
# if coeff[b4] = 0 means constant coefficient
if coeff[b4] == 0:
return False
point = coeff[b]
else:
return False
if point:
r[a3] = simplify(r[a3].subs(x, x+point))
r[b3] = simplify(r[b3].subs(x, x+point))
r[c3] = simplify(r[c3].subs(x, x+point))
# making a3 in the form of x**2
r[a3] = cancel(r[a3]/(coeff[a4]*(x)**(-2+coeff[b4])))
r[b3] = cancel(r[b3]/(coeff[a4]*(x)**(-2+coeff[b4])))
r[c3] = cancel(r[c3]/(coeff[a4]*(x)**(-2+coeff[b4])))
# checking if b3 is of form c*(x-b)
coeff1 = factor(r[b3]).match(a4*(x))
if coeff1 is None:
return False
# c3 maybe of very complex form so I am simply checking (a - b) form
# if yes later I will match with the standerd form of bessel in a and b
# a, b are wild variable defined above.
_coeff2 = r[c3].match(a - b)
if _coeff2 is None:
return False
# matching with standerd form for c3
coeff2 = factor(_coeff2[a]).match(c4**2*(x)**(2*a4))
if coeff2 is None:
return False
if _coeff2[b] == 0:
coeff2[d4] = 0
else:
coeff2[d4] = factor(_coeff2[b]).match(d4**2)[d4]
self.rn = {'n':coeff2[d4], 'a4':coeff2[c4], 'd4':coeff2[a4]}
self.rn['c4'] = coeff1[a4]
self.rn['b4'] = point
return True
return False
def _get_general_solution(self, *, simplify_flag: bool = True):
f = self.ode_problem.func.func
x = self.ode_problem.sym
n = self.rn['n']
a4 = self.rn['a4']
c4 = self.rn['c4']
d4 = self.rn['d4']
b4 = self.rn['b4']
n = sqrt(n**2 + Rational(1, 4)*(c4 - 1)**2)
(C1, C2) = self.ode_problem.get_numbered_constants(num=2)
return [Eq(f(x), ((x**(Rational(1-c4,2)))*(C1*besselj(n/d4,a4*x**d4/d4)
+ C2*bessely(n/d4,a4*x**d4/d4))).subs(x, x-b4))]
class SecondLinearAiry(SingleODESolver):
r"""
Gives solution of the Airy differential equation
.. math :: \frac{d^2y}{dx^2} + (a + b x) y(x) = 0
in terms of Airy special functions airyai and airybi.
Examples
========
>>> from sympy import dsolve, Function
>>> from sympy.abc import x
>>> f = Function("f")
>>> eq = f(x).diff(x, 2) - x*f(x)
>>> dsolve(eq)
Eq(f(x), C1*airyai(x) + C2*airybi(x))
"""
hint = "2nd_linear_airy"
has_integral = False
def _matches(self):
eq = self.ode_problem.eq_high_order_free
f = self.ode_problem.func
order = self.ode_problem.order
x = self.ode_problem.sym
df = f.diff(x)
a4 = Wild('a4', exclude=[x,f,df])
b4 = Wild('b4', exclude=[x,f,df])
match = self.ode_problem.get_linear_coefficients(eq, f, order)
does_match = False
if order == 2 and match and match[2] != 0:
if match[1].is_zero:
self.rn = cancel(match[0]/match[2]).match(a4+b4*x)
if self.rn and self.rn[b4] != 0:
self.rn = {'b':self.rn[a4],'m':self.rn[b4]}
does_match = True
return does_match
def _get_general_solution(self, *, simplify_flag: bool = True):
f = self.ode_problem.func.func
x = self.ode_problem.sym
(C1, C2) = self.ode_problem.get_numbered_constants(num=2)
b = self.rn['b']
m = self.rn['m']
if m.is_positive:
arg = - b/cbrt(m)**2 - cbrt(m)*x
elif m.is_negative:
arg = - b/cbrt(-m)**2 + cbrt(-m)*x
else:
arg = - b/cbrt(-m)**2 + cbrt(-m)*x
return [Eq(f(x), C1*airyai(arg) + C2*airybi(arg))]
class LieGroup(SingleODESolver):
r"""
This hint implements the Lie group method of solving first order differential
equations. The aim is to convert the given differential equation from the
given coordinate system into another coordinate system where it becomes
invariant under the one-parameter Lie group of translations. The converted
ODE can be easily solved by quadrature. It makes use of the
:py:meth:`sympy.solvers.ode.infinitesimals` function which returns the
infinitesimals of the transformation.
The coordinates `r` and `s` can be found by solving the following Partial
Differential Equations.
.. math :: \xi\frac{\partial r}{\partial x} + \eta\frac{\partial r}{\partial y}
= 0
.. math :: \xi\frac{\partial s}{\partial x} + \eta\frac{\partial s}{\partial y}
= 1
The differential equation becomes separable in the new coordinate system
.. math :: \frac{ds}{dr} = \frac{\frac{\partial s}{\partial x} +
h(x, y)\frac{\partial s}{\partial y}}{
\frac{\partial r}{\partial x} + h(x, y)\frac{\partial r}{\partial y}}
After finding the solution by integration, it is then converted back to the original
coordinate system by substituting `r` and `s` in terms of `x` and `y` again.
Examples
========
>>> from sympy import Function, dsolve, exp, pprint
>>> from sympy.abc import x
>>> f = Function('f')
>>> pprint(dsolve(f(x).diff(x) + 2*x*f(x) - x*exp(-x**2), f(x),
... hint='lie_group'))
/ 2\ 2
| x | -x
f(x) = |C1 + --|*e
\ 2 /
References
==========
- Solving differential equations by Symmetry Groups,
John Starrett, pp. 1 - pp. 14
"""
hint = "lie_group"
has_integral = False
def _has_additional_params(self):
return 'xi' in self.ode_problem.params and 'eta' in self.ode_problem.params
def _matches(self):
eq = self.ode_problem.eq
f = self.ode_problem.func.func
order = self.ode_problem.order
x = self.ode_problem.sym
df = f(x).diff(x)
y = Dummy('y')
d = Wild('d', exclude=[df, f(x).diff(x, 2)])
e = Wild('e', exclude=[df])
does_match = False
if self._has_additional_params() and order == 1:
xi = self.ode_problem.params['xi']
eta = self.ode_problem.params['eta']
self.r3 = {'xi': xi, 'eta': eta}
r = collect(eq, df, exact=True).match(d + e * df)
if r:
r['d'] = d
r['e'] = e
r['y'] = y
r[d] = r[d].subs(f(x), y)
r[e] = r[e].subs(f(x), y)
self.r3.update(r)
does_match = True
return does_match
def _get_general_solution(self, *, simplify_flag: bool = True):
eq = self.ode_problem.eq
x = self.ode_problem.sym
func = self.ode_problem.func
order = self.ode_problem.order
df = func.diff(x)
try:
eqsol = solve(eq, df)
except NotImplementedError:
eqsol = []
desols = []
for s in eqsol:
sol = _ode_lie_group(s, func, order, match=self.r3)
if sol:
desols.extend(sol)
if desols == []:
raise NotImplementedError("The given ODE " + str(eq) + " cannot be solved by"
+ " the lie group method")
return desols
solver_map = {
'factorable': Factorable,
'nth_linear_constant_coeff_homogeneous': NthLinearConstantCoeffHomogeneous,
'nth_linear_euler_eq_homogeneous': NthLinearEulerEqHomogeneous,
'nth_linear_constant_coeff_undetermined_coefficients': NthLinearConstantCoeffUndeterminedCoefficients,
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients': NthLinearEulerEqNonhomogeneousUndeterminedCoefficients,
'separable': Separable,
'1st_exact': FirstExact,
'1st_linear': FirstLinear,
'Bernoulli': Bernoulli,
'Riccati_special_minus2': RiccatiSpecial,
'1st_rational_riccati': RationalRiccati,
'1st_homogeneous_coeff_best': HomogeneousCoeffBest,
'1st_homogeneous_coeff_subs_indep_div_dep': HomogeneousCoeffSubsIndepDivDep,
'1st_homogeneous_coeff_subs_dep_div_indep': HomogeneousCoeffSubsDepDivIndep,
'almost_linear': AlmostLinear,
'linear_coefficients': LinearCoefficients,
'separable_reduced': SeparableReduced,
'nth_linear_constant_coeff_variation_of_parameters': NthLinearConstantCoeffVariationOfParameters,
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters': NthLinearEulerEqNonhomogeneousVariationOfParameters,
'Liouville': Liouville,
'2nd_linear_airy': SecondLinearAiry,
'2nd_linear_bessel': SecondLinearBessel,
'2nd_hypergeometric': SecondHypergeometric,
'nth_order_reducible': NthOrderReducible,
'2nd_nonlinear_autonomous_conserved': SecondNonlinearAutonomousConserved,
'nth_algebraic': NthAlgebraic,
'lie_group': LieGroup,
}
# Avoid circular import:
from .ode import dsolve, ode_sol_simplicity, odesimp, homogeneous_order
|
368c1f81d4d5f17f4880ab030de75efc62d9cd369371ba9e37e78f1c33267960 | from sympy.core import Add, Mul, S
from sympy.core.containers import Tuple
from sympy.core.exprtools import factor_terms
from sympy.core.numbers import I
from sympy.core.relational import Eq, Equality
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import Dummy, Symbol
from sympy.core.function import (expand_mul, expand, Derivative,
AppliedUndef, Function, Subs)
from sympy.functions import (exp, im, cos, sin, re, Piecewise,
piecewise_fold, sqrt, log)
from sympy.functions.combinatorial.factorials import factorial
from sympy.matrices import zeros, Matrix, NonSquareMatrixError, MatrixBase, eye
from sympy.polys import Poly, together
from sympy.simplify import collect, radsimp, signsimp # type: ignore
from sympy.simplify.powsimp import powdenest, powsimp
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import simplify
from sympy.sets.sets import FiniteSet
from sympy.solvers.deutils import ode_order
from sympy.solvers.solveset import NonlinearError, solveset
from sympy.utilities.iterables import (connected_components, iterable,
strongly_connected_components)
from sympy.utilities.misc import filldedent
from sympy.integrals.integrals import Integral, integrate
def _get_func_order(eqs, funcs):
return {func: max(ode_order(eq, func) for eq in eqs) for func in funcs}
class ODEOrderError(ValueError):
"""Raised by linear_ode_to_matrix if the system has the wrong order"""
pass
class ODENonlinearError(NonlinearError):
"""Raised by linear_ode_to_matrix if the system is nonlinear"""
pass
def _simpsol(soleq):
lhs = soleq.lhs
sol = soleq.rhs
sol = powsimp(sol)
gens = list(sol.atoms(exp))
p = Poly(sol, *gens, expand=False)
gens = [factor_terms(g) for g in gens]
if not gens:
gens = p.gens
syms = [Symbol('C1'), Symbol('C2')]
terms = []
for coeff, monom in zip(p.coeffs(), p.monoms()):
coeff = piecewise_fold(coeff)
if isinstance(coeff, Piecewise):
coeff = Piecewise(*((ratsimp(coef).collect(syms), cond) for coef, cond in coeff.args))
else:
coeff = ratsimp(coeff).collect(syms)
monom = Mul(*(g ** i for g, i in zip(gens, monom)))
terms.append(coeff * monom)
return Eq(lhs, Add(*terms))
def _solsimp(e, t):
no_t, has_t = powsimp(expand_mul(e)).as_independent(t)
no_t = ratsimp(no_t)
has_t = has_t.replace(exp, lambda a: exp(factor_terms(a)))
return no_t + has_t
def simpsol(sol, wrt1, wrt2, doit=True):
"""Simplify solutions from dsolve_system."""
# The parameter sol is the solution as returned by dsolve (list of Eq).
#
# The parameters wrt1 and wrt2 are lists of symbols to be collected for
# with those in wrt1 being collected for first. This allows for collecting
# on any factors involving the independent variable before collecting on
# the integration constants or vice versa using e.g.:
#
# sol = simpsol(sol, [t], [C1, C2]) # t first, constants after
# sol = simpsol(sol, [C1, C2], [t]) # constants first, t after
#
# If doit=True (default) then simpsol will begin by evaluating any
# unevaluated integrals. Since many integrals will appear multiple times
# in the solutions this is done intelligently by computing each integral
# only once.
#
# The strategy is to first perform simple cancellation with factor_terms
# and then multiply out all brackets with expand_mul. This gives an Add
# with many terms.
#
# We split each term into two multiplicative factors dep and coeff where
# all factors that involve wrt1 are in dep and any constant factors are in
# coeff e.g.
# sqrt(2)*C1*exp(t) -> ( exp(t), sqrt(2)*C1 )
#
# The dep factors are simplified using powsimp to combine expanded
# exponential factors e.g.
# exp(a*t)*exp(b*t) -> exp(t*(a+b))
#
# We then collect coefficients for all terms having the same (simplified)
# dep. The coefficients are then simplified using together and ratsimp and
# lastly by recursively applying the same transformation to the
# coefficients to collect on wrt2.
#
# Finally the result is recombined into an Add and signsimp is used to
# normalise any minus signs.
def simprhs(rhs, rep, wrt1, wrt2):
"""Simplify the rhs of an ODE solution"""
if rep:
rhs = rhs.subs(rep)
rhs = factor_terms(rhs)
rhs = simp_coeff_dep(rhs, wrt1, wrt2)
rhs = signsimp(rhs)
return rhs
def simp_coeff_dep(expr, wrt1, wrt2=None):
"""Split rhs into terms, split terms into dep and coeff and collect on dep"""
add_dep_terms = lambda e: e.is_Add and e.has(*wrt1)
expandable = lambda e: e.is_Mul and any(map(add_dep_terms, e.args))
expand_func = lambda e: expand_mul(e, deep=False)
expand_mul_mod = lambda e: e.replace(expandable, expand_func)
terms = Add.make_args(expand_mul_mod(expr))
dc = {}
for term in terms:
coeff, dep = term.as_independent(*wrt1, as_Add=False)
# Collect together the coefficients for terms that have the same
# dependence on wrt1 (after dep is normalised using simpdep).
dep = simpdep(dep, wrt1)
# See if the dependence on t cancels out...
if dep is not S.One:
dep2 = factor_terms(dep)
if not dep2.has(*wrt1):
coeff *= dep2
dep = S.One
if dep not in dc:
dc[dep] = coeff
else:
dc[dep] += coeff
# Apply the method recursively to the coefficients but this time
# collecting on wrt2 rather than wrt2.
termpairs = ((simpcoeff(c, wrt2), d) for d, c in dc.items())
if wrt2 is not None:
termpairs = ((simp_coeff_dep(c, wrt2), d) for c, d in termpairs)
return Add(*(c * d for c, d in termpairs))
def simpdep(term, wrt1):
"""Normalise factors involving t with powsimp and recombine exp"""
def canonicalise(a):
# Using factor_terms here isn't quite right because it leads to things
# like exp(t*(1+t)) that we don't want. We do want to cancel factors
# and pull out a common denominator but ideally the numerator would be
# expressed as a standard form polynomial in t so we expand_mul
# and collect afterwards.
a = factor_terms(a)
num, den = a.as_numer_denom()
num = expand_mul(num)
num = collect(num, wrt1)
return num / den
term = powsimp(term)
rep = {e: exp(canonicalise(e.args[0])) for e in term.atoms(exp)}
term = term.subs(rep)
return term
def simpcoeff(coeff, wrt2):
"""Bring to a common fraction and cancel with ratsimp"""
coeff = together(coeff)
if coeff.is_polynomial():
# Calling ratsimp can be expensive. The main reason is to simplify
# sums of terms with irrational denominators so we limit ourselves
# to the case where the expression is polynomial in any symbols.
# Maybe there's a better approach...
coeff = ratsimp(radsimp(coeff))
# collect on secondary variables first and any remaining symbols after
if wrt2 is not None:
syms = list(wrt2) + list(ordered(coeff.free_symbols - set(wrt2)))
else:
syms = list(ordered(coeff.free_symbols))
coeff = collect(coeff, syms)
coeff = together(coeff)
return coeff
# There are often repeated integrals. Collect unique integrals and
# evaluate each once and then substitute into the final result to replace
# all occurrences in each of the solution equations.
if doit:
integrals = set().union(*(s.atoms(Integral) for s in sol))
rep = {i: factor_terms(i).doit() for i in integrals}
else:
rep = {}
sol = [Eq(s.lhs, simprhs(s.rhs, rep, wrt1, wrt2)) for s in sol]
return sol
def linodesolve_type(A, t, b=None):
r"""
Helper function that determines the type of the system of ODEs for solving with :obj:`sympy.solvers.ode.systems.linodesolve()`
Explanation
===========
This function takes in the coefficient matrix and/or the non-homogeneous term
and returns the type of the equation that can be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`.
If the system is constant coefficient homogeneous, then "type1" is returned
If the system is constant coefficient non-homogeneous, then "type2" is returned
If the system is non-constant coefficient homogeneous, then "type3" is returned
If the system is non-constant coefficient non-homogeneous, then "type4" is returned
If the system has a non-constant coefficient matrix which can be factorized into constant
coefficient matrix, then "type5" or "type6" is returned for when the system is homogeneous or
non-homogeneous respectively.
Note that, if the system of ODEs is of "type3" or "type4", then along with the type,
the commutative antiderivative of the coefficient matrix is also returned.
If the system cannot be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`, then
NotImplementedError is raised.
Parameters
==========
A : Matrix
Coefficient matrix of the system of ODEs
b : Matrix or None
Non-homogeneous term of the system. The default value is None.
If this argument is None, then the system is assumed to be homogeneous.
Examples
========
>>> from sympy import symbols, Matrix
>>> from sympy.solvers.ode.systems import linodesolve_type
>>> t = symbols("t")
>>> A = Matrix([[1, 1], [2, 3]])
>>> b = Matrix([t, 1])
>>> linodesolve_type(A, t)
{'antiderivative': None, 'type_of_equation': 'type1'}
>>> linodesolve_type(A, t, b=b)
{'antiderivative': None, 'type_of_equation': 'type2'}
>>> A_t = Matrix([[1, t], [-t, 1]])
>>> linodesolve_type(A_t, t)
{'antiderivative': Matrix([
[ t, t**2/2],
[-t**2/2, t]]), 'type_of_equation': 'type3'}
>>> linodesolve_type(A_t, t, b=b)
{'antiderivative': Matrix([
[ t, t**2/2],
[-t**2/2, t]]), 'type_of_equation': 'type4'}
>>> A_non_commutative = Matrix([[1, t], [t, -1]])
>>> linodesolve_type(A_non_commutative, t)
Traceback (most recent call last):
...
NotImplementedError:
The system does not have a commutative antiderivative, it cannot be
solved by linodesolve.
Returns
=======
Dict
Raises
======
NotImplementedError
When the coefficient matrix does not have a commutative antiderivative
See Also
========
linodesolve: Function for which linodesolve_type gets the information
"""
match = {}
is_non_constant = not _matrix_is_constant(A, t)
is_non_homogeneous = not (b is None or b.is_zero_matrix)
type = "type{}".format(int("{}{}".format(int(is_non_constant), int(is_non_homogeneous)), 2) + 1)
B = None
match.update({"type_of_equation": type, "antiderivative": B})
if is_non_constant:
B, is_commuting = _is_commutative_anti_derivative(A, t)
if not is_commuting:
raise NotImplementedError(filldedent('''
The system does not have a commutative antiderivative, it cannot be solved
by linodesolve.
'''))
match['antiderivative'] = B
match.update(_first_order_type5_6_subs(A, t, b=b))
return match
def _first_order_type5_6_subs(A, t, b=None):
match = {}
factor_terms = _factor_matrix(A, t)
is_homogeneous = b is None or b.is_zero_matrix
if factor_terms is not None:
t_ = Symbol("{}_".format(t))
F_t = integrate(factor_terms[0], t)
inverse = solveset(Eq(t_, F_t), t)
# Note: A simple way to check if a function is invertible
# or not.
if isinstance(inverse, FiniteSet) and not inverse.has(Piecewise)\
and len(inverse) == 1:
A = factor_terms[1]
if not is_homogeneous:
b = b / factor_terms[0]
b = b.subs(t, list(inverse)[0])
type = "type{}".format(5 + (not is_homogeneous))
match.update({'func_coeff': A, 'tau': F_t,
't_': t_, 'type_of_equation': type, 'rhs': b})
return match
def linear_ode_to_matrix(eqs, funcs, t, order):
r"""
Convert a linear system of ODEs to matrix form
Explanation
===========
Express a system of linear ordinary differential equations as a single
matrix differential equation [1]. For example the system $x' = x + y + 1$
and $y' = x - y$ can be represented as
.. math:: A_1 X' = A_0 X + b
where $A_1$ and $A_0$ are $2 \times 2$ matrices and $b$, $X$ and $X'$ are
$2 \times 1$ matrices with $X = [x, y]^T$.
Higher-order systems are represented with additional matrices e.g. a
second-order system would look like
.. math:: A_2 X'' = A_1 X' + A_0 X + b
Examples
========
>>> from sympy import Function, Symbol, Matrix, Eq
>>> from sympy.solvers.ode.systems import linear_ode_to_matrix
>>> t = Symbol('t')
>>> x = Function('x')
>>> y = Function('y')
We can create a system of linear ODEs like
>>> eqs = [
... Eq(x(t).diff(t), x(t) + y(t) + 1),
... Eq(y(t).diff(t), x(t) - y(t)),
... ]
>>> funcs = [x(t), y(t)]
>>> order = 1 # 1st order system
Now ``linear_ode_to_matrix`` can represent this as a matrix
differential equation.
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, order)
>>> A1
Matrix([
[1, 0],
[0, 1]])
>>> A0
Matrix([
[1, 1],
[1, -1]])
>>> b
Matrix([
[1],
[0]])
The original equations can be recovered from these matrices:
>>> eqs_mat = Matrix([eq.lhs - eq.rhs for eq in eqs])
>>> X = Matrix(funcs)
>>> A1 * X.diff(t) - A0 * X - b == eqs_mat
True
If the system of equations has a maximum order greater than the
order of the system specified, a ODEOrderError exception is raised.
>>> eqs = [Eq(x(t).diff(t, 2), x(t).diff(t) + x(t)), Eq(y(t).diff(t), y(t) + x(t))]
>>> linear_ode_to_matrix(eqs, funcs, t, 1)
Traceback (most recent call last):
...
ODEOrderError: Cannot represent system in 1-order form
If the system of equations is nonlinear, then ODENonlinearError is
raised.
>>> eqs = [Eq(x(t).diff(t), x(t) + y(t)), Eq(y(t).diff(t), y(t)**2 + x(t))]
>>> linear_ode_to_matrix(eqs, funcs, t, 1)
Traceback (most recent call last):
...
ODENonlinearError: The system of ODEs is nonlinear.
Parameters
==========
eqs : list of SymPy expressions or equalities
The equations as expressions (assumed equal to zero).
funcs : list of applied functions
The dependent variables of the system of ODEs.
t : symbol
The independent variable.
order : int
The order of the system of ODEs.
Returns
=======
The tuple ``(As, b)`` where ``As`` is a tuple of matrices and ``b`` is the
the matrix representing the rhs of the matrix equation.
Raises
======
ODEOrderError
When the system of ODEs have an order greater than what was specified
ODENonlinearError
When the system of ODEs is nonlinear
See Also
========
linear_eq_to_matrix: for systems of linear algebraic equations.
References
==========
.. [1] https://en.wikipedia.org/wiki/Matrix_differential_equation
"""
from sympy.solvers.solveset import linear_eq_to_matrix
if any(ode_order(eq, func) > order for eq in eqs for func in funcs):
msg = "Cannot represent system in {}-order form"
raise ODEOrderError(msg.format(order))
As = []
for o in range(order, -1, -1):
# Work from the highest derivative down
funcs_deriv = [func.diff(t, o) for func in funcs]
# linear_eq_to_matrix expects a proper symbol so substitute e.g.
# Derivative(x(t), t) for a Dummy.
rep = {func_deriv: Dummy() for func_deriv in funcs_deriv}
eqs = [eq.subs(rep) for eq in eqs]
syms = [rep[func_deriv] for func_deriv in funcs_deriv]
# Ai is the matrix for X(t).diff(t, o)
# eqs is minus the remainder of the equations.
try:
Ai, b = linear_eq_to_matrix(eqs, syms)
except NonlinearError:
raise ODENonlinearError("The system of ODEs is nonlinear.")
Ai = Ai.applyfunc(expand_mul)
As.append(Ai if o == order else -Ai)
if o:
eqs = [-eq for eq in b]
else:
rhs = b
return As, rhs
def matrix_exp(A, t):
r"""
Matrix exponential $\exp(A*t)$ for the matrix ``A`` and scalar ``t``.
Explanation
===========
This functions returns the $\exp(A*t)$ by doing a simple
matrix multiplication:
.. math:: \exp(A*t) = P * expJ * P^{-1}
where $expJ$ is $\exp(J*t)$. $J$ is the Jordan normal
form of $A$ and $P$ is matrix such that:
.. math:: A = P * J * P^{-1}
The matrix exponential $\exp(A*t)$ appears in the solution of linear
differential equations. For example if $x$ is a vector and $A$ is a matrix
then the initial value problem
.. math:: \frac{dx(t)}{dt} = A \times x(t), x(0) = x0
has the unique solution
.. math:: x(t) = \exp(A t) x0
Examples
========
>>> from sympy import Symbol, Matrix, pprint
>>> from sympy.solvers.ode.systems import matrix_exp
>>> t = Symbol('t')
We will consider a 2x2 matrix for comupting the exponential
>>> A = Matrix([[2, -5], [2, -4]])
>>> pprint(A)
[2 -5]
[ ]
[2 -4]
Now, exp(A*t) is given as follows:
>>> pprint(matrix_exp(A, t))
[ -t -t -t ]
[3*e *sin(t) + e *cos(t) -5*e *sin(t) ]
[ ]
[ -t -t -t ]
[ 2*e *sin(t) - 3*e *sin(t) + e *cos(t)]
Parameters
==========
A : Matrix
The matrix $A$ in the expression $\exp(A*t)$
t : Symbol
The independent variable
See Also
========
matrix_exp_jordan_form: For exponential of Jordan normal form
References
==========
.. [1] https://en.wikipedia.org/wiki/Jordan_normal_form
.. [2] https://en.wikipedia.org/wiki/Matrix_exponential
"""
P, expJ = matrix_exp_jordan_form(A, t)
return P * expJ * P.inv()
def matrix_exp_jordan_form(A, t):
r"""
Matrix exponential $\exp(A*t)$ for the matrix *A* and scalar *t*.
Explanation
===========
Returns the Jordan form of the $\exp(A*t)$ along with the matrix $P$ such that:
.. math::
\exp(A*t) = P * expJ * P^{-1}
Examples
========
>>> from sympy import Matrix, Symbol
>>> from sympy.solvers.ode.systems import matrix_exp, matrix_exp_jordan_form
>>> t = Symbol('t')
We will consider a 2x2 defective matrix. This shows that our method
works even for defective matrices.
>>> A = Matrix([[1, 1], [0, 1]])
It can be observed that this function gives us the Jordan normal form
and the required invertible matrix P.
>>> P, expJ = matrix_exp_jordan_form(A, t)
Here, it is shown that P and expJ returned by this function is correct
as they satisfy the formula: P * expJ * P_inverse = exp(A*t).
>>> P * expJ * P.inv() == matrix_exp(A, t)
True
Parameters
==========
A : Matrix
The matrix $A$ in the expression $\exp(A*t)$
t : Symbol
The independent variable
References
==========
.. [1] https://en.wikipedia.org/wiki/Defective_matrix
.. [2] https://en.wikipedia.org/wiki/Jordan_matrix
.. [3] https://en.wikipedia.org/wiki/Jordan_normal_form
"""
N, M = A.shape
if N != M:
raise ValueError('Needed square matrix but got shape (%s, %s)' % (N, M))
elif A.has(t):
raise ValueError('Matrix A should not depend on t')
def jordan_chains(A):
'''Chains from Jordan normal form analogous to M.eigenvects().
Returns a dict with eignevalues as keys like:
{e1: [[v111,v112,...], [v121, v122,...]], e2:...}
where vijk is the kth vector in the jth chain for eigenvalue i.
'''
P, blocks = A.jordan_cells()
basis = [P[:,i] for i in range(P.shape[1])]
n = 0
chains = {}
for b in blocks:
eigval = b[0, 0]
size = b.shape[0]
if eigval not in chains:
chains[eigval] = []
chains[eigval].append(basis[n:n+size])
n += size
return chains
eigenchains = jordan_chains(A)
# Needed for consistency across Python versions
eigenchains_iter = sorted(eigenchains.items(), key=default_sort_key)
isreal = not A.has(I)
blocks = []
vectors = []
seen_conjugate = set()
for e, chains in eigenchains_iter:
for chain in chains:
n = len(chain)
if isreal and e != e.conjugate() and e.conjugate() in eigenchains:
if e in seen_conjugate:
continue
seen_conjugate.add(e.conjugate())
exprt = exp(re(e) * t)
imrt = im(e) * t
imblock = Matrix([[cos(imrt), sin(imrt)],
[-sin(imrt), cos(imrt)]])
expJblock2 = Matrix(n, n, lambda i,j:
imblock * t**(j-i) / factorial(j-i) if j >= i
else zeros(2, 2))
expJblock = Matrix(2*n, 2*n, lambda i,j: expJblock2[i//2,j//2][i%2,j%2])
blocks.append(exprt * expJblock)
for i in range(n):
vectors.append(re(chain[i]))
vectors.append(im(chain[i]))
else:
vectors.extend(chain)
fun = lambda i,j: t**(j-i)/factorial(j-i) if j >= i else 0
expJblock = Matrix(n, n, fun)
blocks.append(exp(e * t) * expJblock)
expJ = Matrix.diag(*blocks)
P = Matrix(N, N, lambda i,j: vectors[j][i])
return P, expJ
# Note: To add a docstring example with tau
def linodesolve(A, t, b=None, B=None, type="auto", doit=False,
tau=None):
r"""
System of n equations linear first-order differential equations
Explanation
===========
This solver solves the system of ODEs of the follwing form:
.. math::
X'(t) = A(t) X(t) + b(t)
Here, $A(t)$ is the coefficient matrix, $X(t)$ is the vector of n independent variables,
$b(t)$ is the non-homogeneous term and $X'(t)$ is the derivative of $X(t)$
Depending on the properties of $A(t)$ and $b(t)$, this solver evaluates the solution
differently.
When $A(t)$ is constant coefficient matrix and $b(t)$ is zero vector i.e. system is homogeneous,
the system is "type1". The solution is:
.. math::
X(t) = \exp(A t) C
Here, $C$ is a vector of constants and $A$ is the constant coefficient matrix.
When $A(t)$ is constant coefficient matrix and $b(t)$ is non-zero i.e. system is non-homogeneous,
the system is "type2". The solution is:
.. math::
X(t) = e^{A t} ( \int e^{- A t} b \,dt + C)
When $A(t)$ is coefficient matrix such that its commutative with its antiderivative $B(t)$ and
$b(t)$ is a zero vector i.e. system is homogeneous, the system is "type3". The solution is:
.. math::
X(t) = \exp(B(t)) C
When $A(t)$ is commutative with its antiderivative $B(t)$ and $b(t)$ is non-zero i.e. system is
non-homogeneous, the system is "type4". The solution is:
.. math::
X(t) = e^{B(t)} ( \int e^{-B(t)} b(t) \,dt + C)
When $A(t)$ is a coefficient matrix such that it can be factorized into a scalar and a constant
coefficient matrix:
.. math::
A(t) = f(t) * A
Where $f(t)$ is a scalar expression in the independent variable $t$ and $A$ is a constant matrix,
then we can do the following substitutions:
.. math::
tau = \int f(t) dt, X(t) = Y(tau), b(t) = b(f^{-1}(tau))
Here, the substitution for the non-homogeneous term is done only when its non-zero.
Using these substitutions, our original system becomes:
.. math::
Y'(tau) = A * Y(tau) + b(tau)/f(tau)
The above system can be easily solved using the solution for "type1" or "type2" depending
on the homogeneity of the system. After we get the solution for $Y(tau)$, we substitute the
solution for $tau$ as $t$ to get back $X(t)$
.. math::
X(t) = Y(tau)
Systems of "type5" and "type6" have a commutative antiderivative but we use this solution
because its faster to compute.
The final solution is the general solution for all the four equations since a constant coefficient
matrix is always commutative with its antidervative.
An additional feature of this function is, if someone wants to substitute for value of the independent
variable, they can pass the substitution `tau` and the solution will have the independent variable
substituted with the passed expression(`tau`).
Parameters
==========
A : Matrix
Coefficient matrix of the system of linear first order ODEs.
t : Symbol
Independent variable in the system of ODEs.
b : Matrix or None
Non-homogeneous term in the system of ODEs. If None is passed,
a homogeneous system of ODEs is assumed.
B : Matrix or None
Antiderivative of the coefficient matrix. If the antiderivative
is not passed and the solution requires the term, then the solver
would compute it internally.
type : String
Type of the system of ODEs passed. Depending on the type, the
solution is evaluated. The type values allowed and the corresponding
system it solves are: "type1" for constant coefficient homogeneous
"type2" for constant coefficient non-homogeneous, "type3" for non-constant
coefficient homogeneous, "type4" for non-constant coefficient non-homogeneous,
"type5" and "type6" for non-constant coefficient homogeneous and non-homogeneous
systems respectively where the coefficient matrix can be factorized to a constant
coefficient matrix.
The default value is "auto" which will let the solver decide the correct type of
the system passed.
doit : Boolean
Evaluate the solution if True, default value is False
tau: Expression
Used to substitute for the value of `t` after we get the solution of the system.
Examples
========
To solve the system of ODEs using this function directly, several things must be
done in the right order. Wrong inputs to the function will lead to incorrect results.
>>> from sympy import symbols, Function, Eq
>>> from sympy.solvers.ode.systems import canonical_odes, linear_ode_to_matrix, linodesolve, linodesolve_type
>>> from sympy.solvers.ode.subscheck import checkodesol
>>> f, g = symbols("f, g", cls=Function)
>>> x, a = symbols("x, a")
>>> funcs = [f(x), g(x)]
>>> eqs = [Eq(f(x).diff(x) - f(x), a*g(x) + 1), Eq(g(x).diff(x) + g(x), a*f(x))]
Here, it is important to note that before we derive the coefficient matrix, it is
important to get the system of ODEs into the desired form. For that we will use
:obj:`sympy.solvers.ode.systems.canonical_odes()`.
>>> eqs = canonical_odes(eqs, funcs, x)
>>> eqs
[[Eq(Derivative(f(x), x), a*g(x) + f(x) + 1), Eq(Derivative(g(x), x), a*f(x) - g(x))]]
Now, we will use :obj:`sympy.solvers.ode.systems.linear_ode_to_matrix()` to get the coefficient matrix and the
non-homogeneous term if it is there.
>>> eqs = eqs[0]
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1)
>>> A = A0
We have the coefficient matrices and the non-homogeneous term ready. Now, we can use
:obj:`sympy.solvers.ode.systems.linodesolve_type()` to get the information for the system of ODEs
to finally pass it to the solver.
>>> system_info = linodesolve_type(A, x, b=b)
>>> sol_vector = linodesolve(A, x, b=b, B=system_info['antiderivative'], type=system_info['type_of_equation'])
Now, we can prove if the solution is correct or not by using :obj:`sympy.solvers.ode.checkodesol()`
>>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
>>> checkodesol(eqs, sol)
(True, [0, 0])
We can also use the doit method to evaluate the solutions passed by the function.
>>> sol_vector_evaluated = linodesolve(A, x, b=b, type="type2", doit=True)
Now, we will look at a system of ODEs which is non-constant.
>>> eqs = [Eq(f(x).diff(x), f(x) + x*g(x)), Eq(g(x).diff(x), -x*f(x) + g(x))]
The system defined above is already in the desired form, so we do not have to convert it.
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1)
>>> A = A0
A user can also pass the commutative antiderivative required for type3 and type4 system of ODEs.
Passing an incorrect one will lead to incorrect results. If the coefficient matrix is not commutative
with its antiderivative, then :obj:`sympy.solvers.ode.systems.linodesolve_type()` raises a NotImplementedError.
If it does have a commutative antiderivative, then the function just returns the information about the system.
>>> system_info = linodesolve_type(A, x, b=b)
Now, we can pass the antiderivative as an argument to get the solution. If the system information is not
passed, then the solver will compute the required arguments internally.
>>> sol_vector = linodesolve(A, x, b=b)
Once again, we can verify the solution obtained.
>>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
>>> checkodesol(eqs, sol)
(True, [0, 0])
Returns
=======
List
Raises
======
ValueError
This error is raised when the coefficient matrix, non-homogeneous term
or the antiderivative, if passed, are not a matrix or
do not have correct dimensions
NonSquareMatrixError
When the coefficient matrix or its antiderivative, if passed is not a
square matrix
NotImplementedError
If the coefficient matrix does not have a commutative antiderivative
See Also
========
linear_ode_to_matrix: Coefficient matrix computation function
canonical_odes: System of ODEs representation change
linodesolve_type: Getting information about systems of ODEs to pass in this solver
"""
if not isinstance(A, MatrixBase):
raise ValueError(filldedent('''\
The coefficients of the system of ODEs should be of type Matrix
'''))
if not A.is_square:
raise NonSquareMatrixError(filldedent('''\
The coefficient matrix must be a square
'''))
if b is not None:
if not isinstance(b, MatrixBase):
raise ValueError(filldedent('''\
The non-homogeneous terms of the system of ODEs should be of type Matrix
'''))
if A.rows != b.rows:
raise ValueError(filldedent('''\
The system of ODEs should have the same number of non-homogeneous terms and the number of
equations
'''))
if B is not None:
if not isinstance(B, MatrixBase):
raise ValueError(filldedent('''\
The antiderivative of coefficients of the system of ODEs should be of type Matrix
'''))
if not B.is_square:
raise NonSquareMatrixError(filldedent('''\
The antiderivative of the coefficient matrix must be a square
'''))
if A.rows != B.rows:
raise ValueError(filldedent('''\
The coefficient matrix and its antiderivative should have same dimensions
'''))
if not any(type == "type{}".format(i) for i in range(1, 7)) and not type == "auto":
raise ValueError(filldedent('''\
The input type should be a valid one
'''))
n = A.rows
# constants = numbered_symbols(prefix='C', cls=Dummy, start=const_idx+1)
Cvect = Matrix(list(Dummy() for _ in range(n)))
if any(type == typ for typ in ["type2", "type4", "type6"]) and b is None:
b = zeros(n, 1)
is_transformed = tau is not None
passed_type = type
if type == "auto":
system_info = linodesolve_type(A, t, b=b)
type = system_info["type_of_equation"]
B = system_info["antiderivative"]
if type in ("type5", "type6"):
is_transformed = True
if passed_type != "auto":
if tau is None:
system_info = _first_order_type5_6_subs(A, t, b=b)
if not system_info:
raise ValueError(filldedent('''
The system passed isn't {}.
'''.format(type)))
tau = system_info['tau']
t = system_info['t_']
A = system_info['A']
b = system_info['b']
if type in ("type1", "type2", "type5", "type6"):
P, J = matrix_exp_jordan_form(A, t)
P = simplify(P)
if type in ("type1", "type5"):
sol_vector = P * (J * Cvect)
else:
Jinv = J.subs(t, -t)
sol_vector = P * J * ((Jinv * P.inv() * b).applyfunc(lambda x: Integral(x, t)) + Cvect)
else:
if B is None:
B, _ = _is_commutative_anti_derivative(A, t)
if type == "type3":
sol_vector = B.exp() * Cvect
else:
sol_vector = B.exp() * (((-B).exp() * b).applyfunc(lambda x: Integral(x, t)) + Cvect)
if is_transformed:
sol_vector = sol_vector.subs(t, tau)
gens = sol_vector.atoms(exp)
if type != "type1":
sol_vector = [expand_mul(s) for s in sol_vector]
sol_vector = [collect(s, ordered(gens), exact=True) for s in sol_vector]
if doit:
sol_vector = [s.doit() for s in sol_vector]
return sol_vector
def _matrix_is_constant(M, t):
"""Checks if the matrix M is independent of t or not."""
return all(coef.as_independent(t, as_Add=True)[1] == 0 for coef in M)
def canonical_odes(eqs, funcs, t):
r"""
Function that solves for highest order derivatives in a system
Explanation
===========
This function inputs a system of ODEs and based on the system,
the dependent variables and their highest order, returns the system
in the following form:
.. math::
X'(t) = A(t) X(t) + b(t)
Here, $X(t)$ is the vector of dependent variables of lower order, $A(t)$ is
the coefficient matrix, $b(t)$ is the non-homogeneous term and $X'(t)$ is the
vector of dependent variables in their respective highest order. We use the term
canonical form to imply the system of ODEs which is of the above form.
If the system passed has a non-linear term with multiple solutions, then a list of
systems is returned in its canonical form.
Parameters
==========
eqs : List
List of the ODEs
funcs : List
List of dependent variables
t : Symbol
Independent variable
Examples
========
>>> from sympy import symbols, Function, Eq, Derivative
>>> from sympy.solvers.ode.systems import canonical_odes
>>> f, g = symbols("f g", cls=Function)
>>> x, y = symbols("x y")
>>> funcs = [f(x), g(x)]
>>> eqs = [Eq(f(x).diff(x) - 7*f(x), 12*g(x)), Eq(g(x).diff(x) + g(x), 20*f(x))]
>>> canonical_eqs = canonical_odes(eqs, funcs, x)
>>> canonical_eqs
[[Eq(Derivative(f(x), x), 7*f(x) + 12*g(x)), Eq(Derivative(g(x), x), 20*f(x) - g(x))]]
>>> system = [Eq(Derivative(f(x), x)**2 - 2*Derivative(f(x), x) + 1, 4), Eq(-y*f(x) + Derivative(g(x), x), 0)]
>>> canonical_system = canonical_odes(system, funcs, x)
>>> canonical_system
[[Eq(Derivative(f(x), x), -1), Eq(Derivative(g(x), x), y*f(x))], [Eq(Derivative(f(x), x), 3), Eq(Derivative(g(x), x), y*f(x))]]
Returns
=======
List
"""
from sympy.solvers.solvers import solve
order = _get_func_order(eqs, funcs)
canon_eqs = solve(eqs, *[func.diff(t, order[func]) for func in funcs], dict=True)
systems = []
for eq in canon_eqs:
system = [Eq(func.diff(t, order[func]), eq[func.diff(t, order[func])]) for func in funcs]
systems.append(system)
return systems
def _is_commutative_anti_derivative(A, t):
r"""
Helper function for determining if the Matrix passed is commutative with its antiderivative
Explanation
===========
This function checks if the Matrix $A$ passed is commutative with its antiderivative with respect
to the independent variable $t$.
.. math::
B(t) = \int A(t) dt
The function outputs two values, first one being the antiderivative $B(t)$, second one being a
boolean value, if True, then the matrix $A(t)$ passed is commutative with $B(t)$, else the matrix
passed isn't commutative with $B(t)$.
Parameters
==========
A : Matrix
The matrix which has to be checked
t : Symbol
Independent variable
Examples
========
>>> from sympy import symbols, Matrix
>>> from sympy.solvers.ode.systems import _is_commutative_anti_derivative
>>> t = symbols("t")
>>> A = Matrix([[1, t], [-t, 1]])
>>> B, is_commuting = _is_commutative_anti_derivative(A, t)
>>> is_commuting
True
Returns
=======
Matrix, Boolean
"""
B = integrate(A, t)
is_commuting = (B*A - A*B).applyfunc(expand).applyfunc(factor_terms).is_zero_matrix
is_commuting = False if is_commuting is None else is_commuting
return B, is_commuting
def _factor_matrix(A, t):
term = None
for element in A:
temp_term = element.as_independent(t)[1]
if temp_term.has(t):
term = temp_term
break
if term is not None:
A_factored = (A/term).applyfunc(ratsimp)
can_factor = _matrix_is_constant(A_factored, t)
term = (term, A_factored) if can_factor else None
return term
def _is_second_order_type2(A, t):
term = _factor_matrix(A, t)
is_type2 = False
if term is not None:
term = 1/term[0]
is_type2 = term.is_polynomial()
if is_type2:
poly = Poly(term.expand(), t)
monoms = poly.monoms()
if monoms[0][0] in (2, 4):
cs = _get_poly_coeffs(poly, 4)
a, b, c, d, e = cs
a1 = powdenest(sqrt(a), force=True)
c1 = powdenest(sqrt(e), force=True)
b1 = powdenest(sqrt(c - 2*a1*c1), force=True)
is_type2 = (b == 2*a1*b1) and (d == 2*b1*c1)
term = a1*t**2 + b1*t + c1
else:
is_type2 = False
return is_type2, term
def _get_poly_coeffs(poly, order):
cs = [0 for _ in range(order+1)]
for c, m in zip(poly.coeffs(), poly.monoms()):
cs[-1-m[0]] = c
return cs
def _match_second_order_type(A1, A0, t, b=None):
r"""
Works only for second order system in its canonical form.
Type 0: Constant coefficient matrix, can be simply solved by
introducing dummy variables.
Type 1: When the substitution: $U = t*X' - X$ works for reducing
the second order system to first order system.
Type 2: When the system is of the form: $poly * X'' = A*X$ where
$poly$ is square of a quadratic polynomial with respect to
*t* and $A$ is a constant coefficient matrix.
"""
match = {"type_of_equation": "type0"}
n = A1.shape[0]
if _matrix_is_constant(A1, t) and _matrix_is_constant(A0, t):
return match
if (A1 + A0*t).applyfunc(expand_mul).is_zero_matrix:
match.update({"type_of_equation": "type1", "A1": A1})
elif A1.is_zero_matrix and (b is None or b.is_zero_matrix):
is_type2, term = _is_second_order_type2(A0, t)
if is_type2:
a, b, c = _get_poly_coeffs(Poly(term, t), 2)
A = (A0*(term**2).expand()).applyfunc(ratsimp) + (b**2/4 - a*c)*eye(n, n)
tau = integrate(1/term, t)
t_ = Symbol("{}_".format(t))
match.update({"type_of_equation": "type2", "A0": A,
"g(t)": sqrt(term), "tau": tau, "is_transformed": True,
"t_": t_})
return match
def _second_order_subs_type1(A, b, funcs, t):
r"""
For a linear, second order system of ODEs, a particular substitution.
A system of the below form can be reduced to a linear first order system of
ODEs:
.. math::
X'' = A(t) * (t*X' - X) + b(t)
By substituting:
.. math:: U = t*X' - X
To get the system:
.. math:: U' = t*(A(t)*U + b(t))
Where $U$ is the vector of dependent variables, $X$ is the vector of dependent
variables in `funcs` and $X'$ is the first order derivative of $X$ with respect to
$t$. It may or may not reduce the system into linear first order system of ODEs.
Then a check is made to determine if the system passed can be reduced or not, if
this substitution works, then the system is reduced and its solved for the new
substitution. After we get the solution for $U$:
.. math:: U = a(t)
We substitute and return the reduced system:
.. math::
a(t) = t*X' - X
Parameters
==========
A: Matrix
Coefficient matrix($A(t)*t$) of the second order system of this form.
b: Matrix
Non-homogeneous term($b(t)$) of the system of ODEs.
funcs: List
List of dependent variables
t: Symbol
Independent variable of the system of ODEs.
Returns
=======
List
"""
U = Matrix([t*func.diff(t) - func for func in funcs])
sol = linodesolve(A, t, t*b)
reduced_eqs = [Eq(u, s) for s, u in zip(sol, U)]
reduced_eqs = canonical_odes(reduced_eqs, funcs, t)[0]
return reduced_eqs
def _second_order_subs_type2(A, funcs, t_):
r"""
Returns a second order system based on the coefficient matrix passed.
Explanation
===========
This function returns a system of second order ODE of the following form:
.. math::
X'' = A * X
Here, $X$ is the vector of dependent variables, but a bit modified, $A$ is the
coefficient matrix passed.
Along with returning the second order system, this function also returns the new
dependent variables with the new independent variable `t_` passed.
Parameters
==========
A: Matrix
Coefficient matrix of the system
funcs: List
List of old dependent variables
t_: Symbol
New independent variable
Returns
=======
List, List
"""
func_names = [func.func.__name__ for func in funcs]
new_funcs = [Function(Dummy("{}_".format(name)))(t_) for name in func_names]
rhss = A * Matrix(new_funcs)
new_eqs = [Eq(func.diff(t_, 2), rhs) for func, rhs in zip(new_funcs, rhss)]
return new_eqs, new_funcs
def _is_euler_system(As, t):
return all(_matrix_is_constant((A*t**i).applyfunc(ratsimp), t) for i, A in enumerate(As))
def _classify_linear_system(eqs, funcs, t, is_canon=False):
r"""
Returns a dictionary with details of the eqs if the system passed is linear
and can be classified by this function else returns None
Explanation
===========
This function takes the eqs, converts it into a form Ax = b where x is a vector of terms
containing dependent variables and their derivatives till their maximum order. If it is
possible to convert eqs into Ax = b, then all the equations in eqs are linear otherwise
they are non-linear.
To check if the equations are constant coefficient, we need to check if all the terms in
A obtained above are constant or not.
To check if the equations are homogeneous or not, we need to check if b is a zero matrix
or not.
Parameters
==========
eqs: List
List of ODEs
funcs: List
List of dependent variables
t: Symbol
Independent variable of the equations in eqs
is_canon: Boolean
If True, then this function will not try to get the
system in canonical form. Default value is False
Returns
=======
match = {
'no_of_equation': len(eqs),
'eq': eqs,
'func': funcs,
'order': order,
'is_linear': is_linear,
'is_constant': is_constant,
'is_homogeneous': is_homogeneous,
}
Dict or list of Dicts or None
Dict with values for keys:
1. no_of_equation: Number of equations
2. eq: The set of equations
3. func: List of dependent variables
4. order: A dictionary that gives the order of the
dependent variable in eqs
5. is_linear: Boolean value indicating if the set of
equations are linear or not.
6. is_constant: Boolean value indicating if the set of
equations have constant coefficients or not.
7. is_homogeneous: Boolean value indicating if the set of
equations are homogeneous or not.
8. commutative_antiderivative: Antiderivative of the coefficient
matrix if the coefficient matrix is non-constant
and commutative with its antiderivative. This key
may or may not exist.
9. is_general: Boolean value indicating if the system of ODEs is
solvable using one of the general case solvers or not.
10. rhs: rhs of the non-homogeneous system of ODEs in Matrix form. This
key may or may not exist.
11. is_higher_order: True if the system passed has an order greater than 1.
This key may or may not exist.
12. is_second_order: True if the system passed is a second order ODE. This
key may or may not exist.
This Dict is the answer returned if the eqs are linear and constant
coefficient. Otherwise, None is returned.
"""
# Error for i == 0 can be added but isn't for now
# Check for len(funcs) == len(eqs)
if len(funcs) != len(eqs):
raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs)
# ValueError when functions have more than one arguments
for func in funcs:
if len(func.args) != 1:
raise ValueError("dsolve() and classify_sysode() work with "
"functions of one variable only, not %s" % func)
# Getting the func_dict and order using the helper
# function
order = _get_func_order(eqs, funcs)
system_order = max(order[func] for func in funcs)
is_higher_order = system_order > 1
is_second_order = system_order == 2 and all(order[func] == 2 for func in funcs)
# Not adding the check if the len(func.args) for
# every func in funcs is 1
# Linearity check
try:
canon_eqs = canonical_odes(eqs, funcs, t) if not is_canon else [eqs]
if len(canon_eqs) == 1:
As, b = linear_ode_to_matrix(canon_eqs[0], funcs, t, system_order)
else:
match = {
'is_implicit': True,
'canon_eqs': canon_eqs
}
return match
# When the system of ODEs is non-linear, an ODENonlinearError is raised.
# This function catches the error and None is returned.
except ODENonlinearError:
return None
is_linear = True
# Homogeneous check
is_homogeneous = True if b.is_zero_matrix else False
# Is general key is used to identify if the system of ODEs can be solved by
# one of the general case solvers or not.
match = {
'no_of_equation': len(eqs),
'eq': eqs,
'func': funcs,
'order': order,
'is_linear': is_linear,
'is_homogeneous': is_homogeneous,
'is_general': True
}
if not is_homogeneous:
match['rhs'] = b
is_constant = all(_matrix_is_constant(A_, t) for A_ in As)
# The match['is_linear'] check will be added in the future when this
# function becomes ready to deal with non-linear systems of ODEs
if not is_higher_order:
A = As[1]
match['func_coeff'] = A
# Constant coefficient check
is_constant = _matrix_is_constant(A, t)
match['is_constant'] = is_constant
try:
system_info = linodesolve_type(A, t, b=b)
except NotImplementedError:
return None
match.update(system_info)
antiderivative = match.pop("antiderivative")
if not is_constant:
match['commutative_antiderivative'] = antiderivative
return match
else:
match['type_of_equation'] = "type0"
if is_second_order:
A1, A0 = As[1:]
match_second_order = _match_second_order_type(A1, A0, t)
match.update(match_second_order)
match['is_second_order'] = True
# If system is constant, then no need to check if its in euler
# form or not. It will be easier and faster to directly proceed
# to solve it.
if match['type_of_equation'] == "type0" and not is_constant:
is_euler = _is_euler_system(As, t)
if is_euler:
t_ = Symbol('{}_'.format(t))
match.update({'is_transformed': True, 'type_of_equation': 'type1',
't_': t_})
else:
is_jordan = lambda M: M == Matrix.jordan_block(M.shape[0], M[0, 0])
terms = _factor_matrix(As[-1], t)
if all(A.is_zero_matrix for A in As[1:-1]) and terms is not None and not is_jordan(terms[1]):
P, J = terms[1].jordan_form()
match.update({'type_of_equation': 'type2', 'J': J,
'f(t)': terms[0], 'P': P, 'is_transformed': True})
if match['type_of_equation'] != 'type0' and is_second_order:
match.pop('is_second_order', None)
match['is_higher_order'] = is_higher_order
return match
def _preprocess_eqs(eqs):
processed_eqs = []
for eq in eqs:
processed_eqs.append(eq if isinstance(eq, Equality) else Eq(eq, 0))
return processed_eqs
def _eqs2dict(eqs, funcs):
eqsorig = {}
eqsmap = {}
funcset = set(funcs)
for eq in eqs:
f1, = eq.lhs.atoms(AppliedUndef)
f2s = (eq.rhs.atoms(AppliedUndef) - {f1}) & funcset
eqsmap[f1] = f2s
eqsorig[f1] = eq
return eqsmap, eqsorig
def _dict2graph(d):
nodes = list(d)
edges = [(f1, f2) for f1, f2s in d.items() for f2 in f2s]
G = (nodes, edges)
return G
def _is_type1(scc, t):
eqs, funcs = scc
try:
(A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, 1)
except (ODENonlinearError, ODEOrderError):
return False
if _matrix_is_constant(A0, t) and b.is_zero_matrix:
return True
return False
def _combine_type1_subsystems(subsystem, funcs, t):
indices = [i for i, sys in enumerate(zip(subsystem, funcs)) if _is_type1(sys, t)]
remove = set()
for ip, i in enumerate(indices):
for j in indices[ip+1:]:
if any(eq2.has(funcs[i]) for eq2 in subsystem[j]):
subsystem[j] = subsystem[i] + subsystem[j]
remove.add(i)
subsystem = [sys for i, sys in enumerate(subsystem) if i not in remove]
return subsystem
def _component_division(eqs, funcs, t):
# Assuming that each eq in eqs is in canonical form,
# that is, [f(x).diff(x) = .., g(x).diff(x) = .., etc]
# and that the system passed is in its first order
eqsmap, eqsorig = _eqs2dict(eqs, funcs)
subsystems = []
for cc in connected_components(_dict2graph(eqsmap)):
eqsmap_c = {f: eqsmap[f] for f in cc}
sccs = strongly_connected_components(_dict2graph(eqsmap_c))
subsystem = [[eqsorig[f] for f in scc] for scc in sccs]
subsystem = _combine_type1_subsystems(subsystem, sccs, t)
subsystems.append(subsystem)
return subsystems
# Returns: List of equations
def _linear_ode_solver(match):
t = match['t']
funcs = match['func']
rhs = match.get('rhs', None)
tau = match.get('tau', None)
t = match['t_'] if 't_' in match else t
A = match['func_coeff']
# Note: To make B None when the matrix has constant
# coefficient
B = match.get('commutative_antiderivative', None)
type = match['type_of_equation']
sol_vector = linodesolve(A, t, b=rhs, B=B,
type=type, tau=tau)
sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
return sol
def _select_equations(eqs, funcs, key=lambda x: x):
eq_dict = {e.lhs: e.rhs for e in eqs}
return [Eq(f, eq_dict[key(f)]) for f in funcs]
def _higher_order_ode_solver(match):
eqs = match["eq"]
funcs = match["func"]
t = match["t"]
sysorder = match['order']
type = match.get('type_of_equation', "type0")
is_second_order = match.get('is_second_order', False)
is_transformed = match.get('is_transformed', False)
is_euler = is_transformed and type == "type1"
is_higher_order_type2 = is_transformed and type == "type2" and 'P' in match
if is_second_order:
new_eqs, new_funcs = _second_order_to_first_order(eqs, funcs, t,
A1=match.get("A1", None), A0=match.get("A0", None),
b=match.get("rhs", None), type=type,
t_=match.get("t_", None))
else:
new_eqs, new_funcs = _higher_order_to_first_order(eqs, sysorder, t, funcs=funcs,
type=type, J=match.get('J', None),
f_t=match.get('f(t)', None),
P=match.get('P', None), b=match.get('rhs', None))
if is_transformed:
t = match.get('t_', t)
if not is_higher_order_type2:
new_eqs = _select_equations(new_eqs, [f.diff(t) for f in new_funcs])
sol = None
# NotImplementedError may be raised when the system may be actually
# solvable if it can be just divided into sub-systems
try:
if not is_higher_order_type2:
sol = _strong_component_solver(new_eqs, new_funcs, t)
except NotImplementedError:
sol = None
# Dividing the system only when it becomes essential
if sol is None:
try:
sol = _component_solver(new_eqs, new_funcs, t)
except NotImplementedError:
sol = None
if sol is None:
return sol
is_second_order_type2 = is_second_order and type == "type2"
underscores = '__' if is_transformed else '_'
sol = _select_equations(sol, funcs,
key=lambda x: Function(Dummy('{}{}0'.format(x.func.__name__, underscores)))(t))
if match.get("is_transformed", False):
if is_second_order_type2:
g_t = match["g(t)"]
tau = match["tau"]
sol = [Eq(s.lhs, s.rhs.subs(t, tau) * g_t) for s in sol]
elif is_euler:
t = match['t']
tau = match['t_']
sol = [s.subs(tau, log(t)) for s in sol]
elif is_higher_order_type2:
P = match['P']
sol_vector = P * Matrix([s.rhs for s in sol])
sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
return sol
# Returns: List of equations or None
# If None is returned by this solver, then the system
# of ODEs cannot be solved directly by dsolve_system.
def _strong_component_solver(eqs, funcs, t):
from sympy.solvers.ode.ode import dsolve, constant_renumber
match = _classify_linear_system(eqs, funcs, t, is_canon=True)
sol = None
# Assuming that we can't get an implicit system
# since we are already canonical equations from
# dsolve_system
if match:
match['t'] = t
if match.get('is_higher_order', False):
sol = _higher_order_ode_solver(match)
elif match.get('is_linear', False):
sol = _linear_ode_solver(match)
# Note: For now, only linear systems are handled by this function
# hence, the match condition is added. This can be removed later.
if sol is None and len(eqs) == 1:
sol = dsolve(eqs[0], func=funcs[0])
variables = Tuple(eqs[0]).free_symbols
new_constants = [Dummy() for _ in range(ode_order(eqs[0], funcs[0]))]
sol = constant_renumber(sol, variables=variables, newconstants=new_constants)
sol = [sol]
# To add non-linear case here in future
return sol
def _get_funcs_from_canon(eqs):
return [eq.lhs.args[0] for eq in eqs]
# Returns: List of Equations(a solution)
def _weak_component_solver(wcc, t):
# We will divide the systems into sccs
# only when the wcc cannot be solved as
# a whole
eqs = []
for scc in wcc:
eqs += scc
funcs = _get_funcs_from_canon(eqs)
sol = _strong_component_solver(eqs, funcs, t)
if sol:
return sol
sol = []
for j, scc in enumerate(wcc):
eqs = scc
funcs = _get_funcs_from_canon(eqs)
# Substituting solutions for the dependent
# variables solved in previous SCC, if any solved.
comp_eqs = [eq.subs({s.lhs: s.rhs for s in sol}) for eq in eqs]
scc_sol = _strong_component_solver(comp_eqs, funcs, t)
if scc_sol is None:
raise NotImplementedError(filldedent('''
The system of ODEs passed cannot be solved by dsolve_system.
'''))
# scc_sol: List of equations
# scc_sol is a solution
sol += scc_sol
return sol
# Returns: List of Equations(a solution)
def _component_solver(eqs, funcs, t):
components = _component_division(eqs, funcs, t)
sol = []
for wcc in components:
# wcc_sol: List of Equations
sol += _weak_component_solver(wcc, t)
# sol: List of Equations
return sol
def _second_order_to_first_order(eqs, funcs, t, type="auto", A1=None,
A0=None, b=None, t_=None):
r"""
Expects the system to be in second order and in canonical form
Explanation
===========
Reduces a second order system into a first order one depending on the type of second
order system.
1. "type0": If this is passed, then the system will be reduced to first order by
introducing dummy variables.
2. "type1": If this is passed, then a particular substitution will be used to reduce the
the system into first order.
3. "type2": If this is passed, then the system will be transformed with new dependent
variables and independent variables. This transformation is a part of solving
the corresponding system of ODEs.
`A1` and `A0` are the coefficient matrices from the system and it is assumed that the
second order system has the form given below:
.. math::
A2 * X'' = A1 * X' + A0 * X + b
Here, $A2$ is the coefficient matrix for the vector $X''$ and $b$ is the non-homogeneous
term.
Default value for `b` is None but if `A1` and `A0` are passed and `b` is not passed, then the
system will be assumed homogeneous.
"""
is_a1 = A1 is None
is_a0 = A0 is None
if (type == "type1" and is_a1) or (type == "type2" and is_a0)\
or (type == "auto" and (is_a1 or is_a0)):
(A2, A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, 2)
if not A2.is_Identity:
raise ValueError(filldedent('''
The system must be in its canonical form.
'''))
if type == "auto":
match = _match_second_order_type(A1, A0, t)
type = match["type_of_equation"]
A1 = match.get("A1", None)
A0 = match.get("A0", None)
sys_order = {func: 2 for func in funcs}
if type == "type1":
if b is None:
b = zeros(len(eqs))
eqs = _second_order_subs_type1(A1, b, funcs, t)
sys_order = {func: 1 for func in funcs}
if type == "type2":
if t_ is None:
t_ = Symbol("{}_".format(t))
t = t_
eqs, funcs = _second_order_subs_type2(A0, funcs, t_)
sys_order = {func: 2 for func in funcs}
return _higher_order_to_first_order(eqs, sys_order, t, funcs=funcs)
def _higher_order_type2_to_sub_systems(J, f_t, funcs, t, max_order, b=None, P=None):
# Note: To add a test for this ValueError
if J is None or f_t is None or not _matrix_is_constant(J, t):
raise ValueError(filldedent('''
Correctly input for args 'A' and 'f_t' for Linear, Higher Order,
Type 2
'''))
if P is None and b is not None and not b.is_zero_matrix:
raise ValueError(filldedent('''
Provide the keyword 'P' for matrix P in A = P * J * P-1.
'''))
new_funcs = Matrix([Function(Dummy('{}__0'.format(f.func.__name__)))(t) for f in funcs])
new_eqs = new_funcs.diff(t, max_order) - f_t * J * new_funcs
if b is not None and not b.is_zero_matrix:
new_eqs -= P.inv() * b
new_eqs = canonical_odes(new_eqs, new_funcs, t)[0]
return new_eqs, new_funcs
def _higher_order_to_first_order(eqs, sys_order, t, funcs=None, type="type0", **kwargs):
if funcs is None:
funcs = sys_order.keys()
# Standard Cauchy Euler system
if type == "type1":
t_ = Symbol('{}_'.format(t))
new_funcs = [Function(Dummy('{}_'.format(f.func.__name__)))(t_) for f in funcs]
max_order = max(sys_order[func] for func in funcs)
subs_dict = {func: new_func for func, new_func in zip(funcs, new_funcs)}
subs_dict[t] = exp(t_)
free_function = Function(Dummy())
def _get_coeffs_from_subs_expression(expr):
if isinstance(expr, Subs):
free_symbol = expr.args[1][0]
term = expr.args[0]
return {ode_order(term, free_symbol): 1}
if isinstance(expr, Mul):
coeff = expr.args[0]
order = list(_get_coeffs_from_subs_expression(expr.args[1]).keys())[0]
return {order: coeff}
if isinstance(expr, Add):
coeffs = {}
for arg in expr.args:
if isinstance(arg, Mul):
coeffs.update(_get_coeffs_from_subs_expression(arg))
else:
order = list(_get_coeffs_from_subs_expression(arg).keys())[0]
coeffs[order] = 1
return coeffs
for o in range(1, max_order + 1):
expr = free_function(log(t_)).diff(t_, o)*t_**o
coeff_dict = _get_coeffs_from_subs_expression(expr)
coeffs = [coeff_dict[order] if order in coeff_dict else 0 for order in range(o + 1)]
expr_to_subs = sum(free_function(t_).diff(t_, i) * c for i, c in
enumerate(coeffs)) / t**o
subs_dict.update({f.diff(t, o): expr_to_subs.subs(free_function(t_), nf)
for f, nf in zip(funcs, new_funcs)})
new_eqs = [eq.subs(subs_dict) for eq in eqs]
new_sys_order = {nf: sys_order[f] for f, nf in zip(funcs, new_funcs)}
new_eqs = canonical_odes(new_eqs, new_funcs, t_)[0]
return _higher_order_to_first_order(new_eqs, new_sys_order, t_, funcs=new_funcs)
# Systems of the form: X(n)(t) = f(t)*A*X + b
# where X(n)(t) is the nth derivative of the vector of dependent variables
# with respect to the independent variable and A is a constant matrix.
if type == "type2":
J = kwargs.get('J', None)
f_t = kwargs.get('f_t', None)
b = kwargs.get('b', None)
P = kwargs.get('P', None)
max_order = max(sys_order[func] for func in funcs)
return _higher_order_type2_to_sub_systems(J, f_t, funcs, t, max_order, P=P, b=b)
# Note: To be changed to this after doit option is disabled for default cases
# new_sysorder = _get_func_order(new_eqs, new_funcs)
#
# return _higher_order_to_first_order(new_eqs, new_sysorder, t, funcs=new_funcs)
new_funcs = []
for prev_func in funcs:
func_name = prev_func.func.__name__
func = Function(Dummy('{}_0'.format(func_name)))(t)
new_funcs.append(func)
subs_dict = {prev_func: func}
new_eqs = []
for i in range(1, sys_order[prev_func]):
new_func = Function(Dummy('{}_{}'.format(func_name, i)))(t)
subs_dict[prev_func.diff(t, i)] = new_func
new_funcs.append(new_func)
prev_f = subs_dict[prev_func.diff(t, i-1)]
new_eq = Eq(prev_f.diff(t), new_func)
new_eqs.append(new_eq)
eqs = [eq.subs(subs_dict) for eq in eqs] + new_eqs
return eqs, new_funcs
def dsolve_system(eqs, funcs=None, t=None, ics=None, doit=False, simplify=True):
r"""
Solves any(supported) system of Ordinary Differential Equations
Explanation
===========
This function takes a system of ODEs as an input, determines if the
it is solvable by this function, and returns the solution if found any.
This function can handle:
1. Linear, First Order, Constant coefficient homogeneous system of ODEs
2. Linear, First Order, Constant coefficient non-homogeneous system of ODEs
3. Linear, First Order, non-constant coefficient homogeneous system of ODEs
4. Linear, First Order, non-constant coefficient non-homogeneous system of ODEs
5. Any implicit system which can be divided into system of ODEs which is of the above 4 forms
6. Any higher order linear system of ODEs that can be reduced to one of the 5 forms of systems described above.
The types of systems described above are not limited by the number of equations, i.e. this
function can solve the above types irrespective of the number of equations in the system passed.
But, the bigger the system, the more time it will take to solve the system.
This function returns a list of solutions. Each solution is a list of equations where LHS is
the dependent variable and RHS is an expression in terms of the independent variable.
Among the non constant coefficient types, not all the systems are solvable by this function. Only
those which have either a coefficient matrix with a commutative antiderivative or those systems which
may be divided further so that the divided systems may have coefficient matrix with commutative antiderivative.
Parameters
==========
eqs : List
system of ODEs to be solved
funcs : List or None
List of dependent variables that make up the system of ODEs
t : Symbol or None
Independent variable in the system of ODEs
ics : Dict or None
Set of initial boundary/conditions for the system of ODEs
doit : Boolean
Evaluate the solutions if True. Default value is True. Can be
set to false if the integral evaluation takes too much time and/or
is not required.
simplify: Boolean
Simplify the solutions for the systems. Default value is True.
Can be set to false if simplification takes too much time and/or
is not required.
Examples
========
>>> from sympy import symbols, Eq, Function
>>> from sympy.solvers.ode.systems import dsolve_system
>>> f, g = symbols("f g", cls=Function)
>>> x = symbols("x")
>>> eqs = [Eq(f(x).diff(x), g(x)), Eq(g(x).diff(x), f(x))]
>>> dsolve_system(eqs)
[[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]]
You can also pass the initial conditions for the system of ODEs:
>>> dsolve_system(eqs, ics={f(0): 1, g(0): 0})
[[Eq(f(x), exp(x)/2 + exp(-x)/2), Eq(g(x), exp(x)/2 - exp(-x)/2)]]
Optionally, you can pass the dependent variables and the independent
variable for which the system is to be solved:
>>> funcs = [f(x), g(x)]
>>> dsolve_system(eqs, funcs=funcs, t=x)
[[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]]
Lets look at an implicit system of ODEs:
>>> eqs = [Eq(f(x).diff(x)**2, g(x)**2), Eq(g(x).diff(x), g(x))]
>>> dsolve_system(eqs)
[[Eq(f(x), C1 - C2*exp(x)), Eq(g(x), C2*exp(x))], [Eq(f(x), C1 + C2*exp(x)), Eq(g(x), C2*exp(x))]]
Returns
=======
List of List of Equations
Raises
======
NotImplementedError
When the system of ODEs is not solvable by this function.
ValueError
When the parameters passed are not in the required form.
"""
from sympy.solvers.ode.ode import solve_ics, _extract_funcs, constant_renumber
if not iterable(eqs):
raise ValueError(filldedent('''
List of equations should be passed. The input is not valid.
'''))
eqs = _preprocess_eqs(eqs)
if funcs is not None and not isinstance(funcs, list):
raise ValueError(filldedent('''
Input to the funcs should be a list of functions.
'''))
if funcs is None:
funcs = _extract_funcs(eqs)
if any(len(func.args) != 1 for func in funcs):
raise ValueError(filldedent('''
dsolve_system can solve a system of ODEs with only one independent
variable.
'''))
if len(eqs) != len(funcs):
raise ValueError(filldedent('''
Number of equations and number of functions do not match
'''))
if t is not None and not isinstance(t, Symbol):
raise ValueError(filldedent('''
The indepedent variable must be of type Symbol
'''))
if t is None:
t = list(list(eqs[0].atoms(Derivative))[0].atoms(Symbol))[0]
sols = []
canon_eqs = canonical_odes(eqs, funcs, t)
for canon_eq in canon_eqs:
try:
sol = _strong_component_solver(canon_eq, funcs, t)
except NotImplementedError:
sol = None
if sol is None:
sol = _component_solver(canon_eq, funcs, t)
sols.append(sol)
if sols:
final_sols = []
variables = Tuple(*eqs).free_symbols
for sol in sols:
sol = _select_equations(sol, funcs)
sol = constant_renumber(sol, variables=variables)
if ics:
constants = Tuple(*sol).free_symbols - variables
solved_constants = solve_ics(sol, funcs, constants, ics)
sol = [s.subs(solved_constants) for s in sol]
if simplify:
constants = Tuple(*sol).free_symbols - variables
sol = simpsol(sol, [t], constants, doit=doit)
final_sols.append(sol)
sols = final_sols
return sols
|
c58eae4734eb19b1860488b5f997807b3c1e9151bc29cab8298a3cca4219bee4 | from sympy.assumptions.ask import (Q, ask)
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import (Derivative, Function, diff)
from sympy.core.mul import Mul
from sympy.core import (GoldenRatio, TribonacciConstant)
from sympy.core.numbers import (E, Float, I, Rational, oo, pi)
from sympy.core.relational import (Eq, Gt, Lt, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, Wild, symbols)
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import binomial
from sympy.functions.elementary.complexes import (Abs, arg, conjugate, im, re)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (atanh, cosh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import (cbrt, root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, asin, atan, atan2, cos, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc, erfcinv, erfinv)
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (And, Or)
from sympy.matrices.dense import Matrix
from sympy.matrices import SparseMatrix
from sympy.polys.polytools import Poly
from sympy.printing.str import sstr
from sympy.simplify.radsimp import denom
from sympy.solvers.solvers import (nsolve, solve, solve_linear)
from sympy.core.function import nfloat
from sympy.solvers import solve_linear_system, solve_linear_system_LU, \
solve_undetermined_coeffs
from sympy.solvers.bivariate import _filtered_gens, _solve_lambert, _lambert
from sympy.solvers.solvers import _invert, unrad, checksol, posify, _ispow, \
det_quick, det_perm, det_minor, _simple_dens, denoms
from sympy.physics.units import cm
from sympy.polys.rootoftools import CRootOf
from sympy.testing.pytest import slow, XFAIL, SKIP, raises
from sympy.core.random import verify_numerically as tn
from sympy.abc import a, b, c, d, e, k, h, p, x, y, z, t, q, m, R
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_swap_back():
f, g = map(Function, 'fg')
fx, gx = f(x), g(x)
assert solve([fx + y - 2, fx - gx - 5], fx, y, gx) == \
{fx: gx + 5, y: -gx - 3}
assert solve(fx + gx*x - 2, [fx, gx], dict=True)[0] == {fx: 2, gx: 0}
assert solve(fx + gx**2*x - y, [fx, gx], dict=True) == [{fx: y - gx**2*x}]
assert solve([f(1) - 2, x + 2], dict=True) == [{x: -2, f(1): 2}]
def guess_solve_strategy(eq, symbol):
try:
solve(eq, symbol)
return True
except (TypeError, NotImplementedError):
return False
def test_guess_poly():
# polynomial equations
assert guess_solve_strategy( S(4), x ) # == GS_POLY
assert guess_solve_strategy( x, x ) # == GS_POLY
assert guess_solve_strategy( x + a, x ) # == GS_POLY
assert guess_solve_strategy( 2*x, x ) # == GS_POLY
assert guess_solve_strategy( x + sqrt(2), x) # == GS_POLY
assert guess_solve_strategy( x + 2**Rational(1, 4), x) # == GS_POLY
assert guess_solve_strategy( x**2 + 1, x ) # == GS_POLY
assert guess_solve_strategy( x**2 - 1, x ) # == GS_POLY
assert guess_solve_strategy( x*y + y, x ) # == GS_POLY
assert guess_solve_strategy( x*exp(y) + y, x) # == GS_POLY
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), x) # == GS_POLY
def test_guess_poly_cv():
# polynomial equations via a change of variable
assert guess_solve_strategy( sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy(
x**Rational(1, 3) + sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy( 4*x*(1 - sqrt(x)), x ) # == GS_POLY_CV_1
# polynomial equation multiplying both sides by x**n
assert guess_solve_strategy( x + 1/x + y, x ) # == GS_POLY_CV_2
def test_guess_rational_cv():
# rational functions
assert guess_solve_strategy( (x + 1)/(x**2 + 2), x) # == GS_RATIONAL
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), y) # == GS_RATIONAL_CV_1
# rational functions via the change of variable y -> x**n
assert guess_solve_strategy( (sqrt(x) + 1)/(x**Rational(1, 3) + sqrt(x) + 1), x ) \
#== GS_RATIONAL_CV_1
def test_guess_transcendental():
#transcendental functions
assert guess_solve_strategy( exp(x) + 1, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy( 2*cos(x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(
exp(x) + exp(-x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(3**x - 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(-3**x + 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(a*x**b - y, x) # == GS_TRANSCENDENTAL
def test_solve_args():
# equation container, issue 5113
ans = {x: -3, y: 1}
eqs = (x + 5*y - 2, -3*x + 6*y - 15)
assert all(solve(container(eqs), x, y) == ans for container in
(tuple, list, set, frozenset))
assert solve(Tuple(*eqs), x, y) == ans
# implicit symbol to solve for
assert set(solve(x**2 - 4)) == {S(2), -S(2)}
assert solve([x + y - 3, x - y - 5]) == {x: 4, y: -1}
assert solve(x - exp(x), x, implicit=True) == [exp(x)]
# no symbol to solve for
assert solve(42) == solve(42, x) == []
assert solve([1, 2]) == []
assert solve([sqrt(2)],[x]) == []
# duplicate symbols removed
assert solve((x - 3, y + 2), x, y, x) == {x: 3, y: -2}
# unordered symbols
# only 1
assert solve(y - 3, {y}) == [3]
# more than 1
assert solve(y - 3, {x, y}) == [{y: 3}]
# multiple symbols: take the first linear solution+
# - return as tuple with values for all requested symbols
assert solve(x + y - 3, [x, y]) == [(3 - y, y)]
# - unless dict is True
assert solve(x + y - 3, [x, y], dict=True) == [{x: 3 - y}]
# - or no symbols are given
assert solve(x + y - 3) == [{x: 3 - y}]
# multiple symbols might represent an undetermined coefficients system
assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0}
args = (a + b)*x - b**2 + 2, a, b
assert solve(*args) == \
[(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))]
assert solve(*args, set=True) == \
([a, b], {(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))})
assert solve(*args, dict=True) == \
[{b: sqrt(2), a: -sqrt(2)}, {b: -sqrt(2), a: sqrt(2)}]
eq = a*x**2 + b*x + c - ((x - h)**2 + 4*p*k)/4/p
flags = dict(dict=True)
assert solve(eq, [h, p, k], exclude=[a, b, c], **flags) == \
[{k: c - b**2/(4*a), h: -b/(2*a), p: 1/(4*a)}]
flags.update(dict(simplify=False))
assert solve(eq, [h, p, k], exclude=[a, b, c], **flags) == \
[{k: (4*a*c - b**2)/(4*a), h: -b/(2*a), p: 1/(4*a)}]
# failing undetermined system
assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b, dict=True) == \
[{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}]
# failed single equation
assert solve(1/(1/x - y + exp(y))) == []
raises(
NotImplementedError, lambda: solve(exp(x) + sin(x) + exp(y) + sin(y)))
# failed system
# -- when no symbols given, 1 fails
assert solve([y, exp(x) + x]) == {x: -LambertW(1), y: 0}
# both fail
assert solve(
(exp(x) - x, exp(y) - y)) == {x: -LambertW(-1), y: -LambertW(-1)}
# -- when symbols given
assert solve([y, exp(x) + x], x, y) == {y: 0, x: -LambertW(1)}
# symbol is a number
assert solve(x**2 - pi, pi) == [x**2]
# no equations
assert solve([], [x]) == []
# overdetermined system
# - nonlinear
assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}]
# - linear
assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2}
# When one or more args are Boolean
assert solve(Eq(x**2, 0.0)) == [0] # issue 19048
assert solve([True, Eq(x, 0)], [x], dict=True) == [{x: 0}]
assert solve([Eq(x, x), Eq(x, 0), Eq(x, x+1)], [x], dict=True) == []
assert not solve([Eq(x, x+1), x < 2], x)
assert solve([Eq(x, 0), x+1<2]) == Eq(x, 0)
assert solve([Eq(x, x), Eq(x, x+1)], x) == []
assert solve(True, x) == []
assert solve([x - 1, False], [x], set=True) == ([], set())
assert solve([-y*(x + y - 1)/2, (y - 1)/x/y + 1/y],
set=True, check=False) == ([x, y], {(1 - y, y), (x, 0)})
def test_solve_polynomial1():
assert solve(3*x - 2, x) == [Rational(2, 3)]
assert solve(Eq(3*x, 2), x) == [Rational(2, 3)]
assert set(solve(x**2 - 1, x)) == {-S.One, S.One}
assert set(solve(Eq(x**2, 1), x)) == {-S.One, S.One}
assert solve(x - y**3, x) == [y**3]
rx = root(x, 3)
assert solve(x - y**3, y) == [
rx, -rx/2 - sqrt(3)*I*rx/2, -rx/2 + sqrt(3)*I*rx/2]
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
assert solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) == \
{
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
solution = {y: S.Zero, x: S.Zero}
assert solve((x - y, x + y), x, y ) == solution
assert solve((x - y, x + y), (x, y)) == solution
assert solve((x - y, x + y), [x, y]) == solution
assert set(solve(x**3 - 15*x - 4, x)) == {
-2 + 3**S.Half,
S(4),
-2 - 3**S.Half
}
assert set(solve((x**2 - 1)**2 - a, x)) == \
{sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))}
def test_solve_polynomial2():
assert solve(4, x) == []
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to a polynomial equation
using the change of variable y -> x**Rational(p, q)
"""
assert solve( sqrt(x) - 1, x) == [1]
assert solve( sqrt(x) - 2, x) == [4]
assert solve( x**Rational(1, 4) - 2, x) == [16]
assert solve( x**Rational(1, 3) - 3, x) == [27]
assert solve(sqrt(x) + x**Rational(1, 3) + x**Rational(1, 4), x) == [0]
def test_solve_polynomial_cv_1b():
assert set(solve(4*x*(1 - a*sqrt(x)), x)) == {S.Zero, 1/a**2}
assert set(solve(x*(root(x, 3) - 3), x)) == {S.Zero, S(27)}
def test_solve_polynomial_cv_2():
"""
Test for solving on equations that can be converted to a polynomial equation
multiplying both sides of the equation by x**m
"""
assert solve(x + 1/x - 1, x) in \
[[ S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2],
[ S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]]
def test_quintics_1():
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solve(f, check=False)
for r in s:
res = f.subs(x, r.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for r in s:
assert r.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get RootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(solve(x**5 + 3*x**3 + 7)[0], exponent=False) == \
CRootOf(x**5 + 3*x**3 + 7, 0).n()
def test_quintics_2():
f = x**5 + 15*x + 12
s = solve(f, check=False)
for r in s:
res = f.subs(x, r.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for r in s:
assert r.func == CRootOf
assert solve(x**5 - 6*x**3 - 6*x**2 + x - 6) == [
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 0),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 1),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 2),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 3),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 4)]
def test_quintics_3():
y = x**5 + x**3 - 2**Rational(1, 3)
assert solve(y) == solve(-y) == []
def test_highorder_poly():
# just testing that the uniq generator is unpacked
sol = solve(x**6 - 2*x + 2)
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
def test_solve_rational():
"""Test solve for rational functions"""
assert solve( ( x - y**3 )/( (y**2)*sqrt(1 - y**2) ), x) == [y**3]
def test_solve_conjugate():
"""Test solve for simple conjugate functions"""
assert solve(conjugate(x) -3 + I) == [3 + I]
def test_solve_nonlinear():
assert solve(x**2 - y**2, x, y, dict=True) == [{x: -y}, {x: y}]
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: -x*sqrt(exp(x))},
{y: x*sqrt(exp(x))}]
def test_issue_8666():
x = symbols('x')
assert solve(Eq(x**2 - 1/(x**2 - 4), 4 - 1/(x**2 - 4)), x) == []
assert solve(Eq(x + 1/x, 1/x), x) == []
def test_issue_7228():
assert solve(4**(2*(x**2) + 2*x) - 8, x) == [Rational(-3, 2), S.Half]
def test_issue_7190():
assert solve(log(x-3) + log(x+3), x) == [sqrt(10)]
def test_issue_21004():
x = symbols('x')
f = x/sqrt(x**2+1)
f_diff = f.diff(x)
assert solve(f_diff, x) == []
def test_linear_system():
x, y, z, t, n = symbols('x, y, z, t, n')
assert solve([x - 1, x - y, x - 2*y, y - 1], [x, y]) == []
assert solve([x - 1, x - y, x - 2*y, x - 1], [x, y]) == []
assert solve([x - 1, x - 1, x - y, x - 2*y], [x, y]) == []
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == {x: -3, y: 1}
M = Matrix([[0, 0, n*(n + 1), (n + 1)**2, 0],
[n + 1, n + 1, -2*n - 1, -(n + 1), 0],
[-1, 0, 1, 0, 0]])
assert solve_linear_system(M, x, y, z, t) == \
{x: t*(-n-1)/n, z: t*(-n-1)/n, y: 0}
assert solve([x + y + z + t, -z - t], x, y, z, t) == {x: -y, z: -t}
@XFAIL
def test_linear_system_xfail():
# https://github.com/sympy/sympy/issues/6420
M = Matrix([[0, 15.0, 10.0, 700.0],
[1, 1, 1, 100.0],
[0, 10.0, 5.0, 200.0],
[-5.0, 0, 0, 0 ]])
assert solve_linear_system(M, x, y, z) == {x: 0, y: -60.0, z: 160.0}
def test_linear_system_function():
a = Function('a')
assert solve([a(0, 0) + a(0, 1) + a(1, 0) + a(1, 1), -a(1, 0) - a(1, 1)],
a(0, 0), a(0, 1), a(1, 0), a(1, 1)) == {a(1, 0): -a(1, 1), a(0, 0): -a(0, 1)}
def test_linear_system_symbols_doesnt_hang_1():
def _mk_eqs(wy):
# Equations for fitting a wy*2 - 1 degree polynomial between two points,
# at end points derivatives are known up to order: wy - 1
order = 2*wy - 1
x, x0, x1 = symbols('x, x0, x1', real=True)
y0s = symbols('y0_:{}'.format(wy), real=True)
y1s = symbols('y1_:{}'.format(wy), real=True)
c = symbols('c_:{}'.format(order+1), real=True)
expr = sum([coeff*x**o for o, coeff in enumerate(c)])
eqs = []
for i in range(wy):
eqs.append(expr.diff(x, i).subs({x: x0}) - y0s[i])
eqs.append(expr.diff(x, i).subs({x: x1}) - y1s[i])
return eqs, c
#
# The purpose of this test is just to see that these calls don't hang. The
# expressions returned are complicated so are not included here. Testing
# their correctness takes longer than solving the system.
#
for n in range(1, 7+1):
eqs, c = _mk_eqs(n)
solve(eqs, c)
def test_linear_system_symbols_doesnt_hang_2():
M = Matrix([
[66, 24, 39, 50, 88, 40, 37, 96, 16, 65, 31, 11, 37, 72, 16, 19, 55, 37, 28, 76],
[10, 93, 34, 98, 59, 44, 67, 74, 74, 94, 71, 61, 60, 23, 6, 2, 57, 8, 29, 78],
[19, 91, 57, 13, 64, 65, 24, 53, 77, 34, 85, 58, 87, 39, 39, 7, 36, 67, 91, 3],
[74, 70, 15, 53, 68, 43, 86, 83, 81, 72, 25, 46, 67, 17, 59, 25, 78, 39, 63, 6],
[69, 40, 67, 21, 67, 40, 17, 13, 93, 44, 46, 89, 62, 31, 30, 38, 18, 20, 12, 81],
[50, 22, 74, 76, 34, 45, 19, 76, 28, 28, 11, 99, 97, 82, 8, 46, 99, 57, 68, 35],
[58, 18, 45, 88, 10, 64, 9, 34, 90, 82, 17, 41, 43, 81, 45, 83, 22, 88, 24, 39],
[42, 21, 70, 68, 6, 33, 64, 81, 83, 15, 86, 75, 86, 17, 77, 34, 62, 72, 20, 24],
[ 7, 8, 2, 72, 71, 52, 96, 5, 32, 51, 31, 36, 79, 88, 25, 77, 29, 26, 33, 13],
[19, 31, 30, 85, 81, 39, 63, 28, 19, 12, 16, 49, 37, 66, 38, 13, 3, 71, 61, 51],
[29, 82, 80, 49, 26, 85, 1, 37, 2, 74, 54, 82, 26, 47, 54, 9, 35, 0, 99, 40],
[15, 49, 82, 91, 93, 57, 45, 25, 45, 97, 15, 98, 48, 52, 66, 24, 62, 54, 97, 37],
[62, 23, 73, 53, 52, 86, 28, 38, 0, 74, 92, 38, 97, 70, 71, 29, 26, 90, 67, 45],
[ 2, 32, 23, 24, 71, 37, 25, 71, 5, 41, 97, 65, 93, 13, 65, 45, 25, 88, 69, 50],
[40, 56, 1, 29, 79, 98, 79, 62, 37, 28, 45, 47, 3, 1, 32, 74, 98, 35, 84, 32],
[33, 15, 87, 79, 65, 9, 14, 63, 24, 19, 46, 28, 74, 20, 29, 96, 84, 91, 93, 1],
[97, 18, 12, 52, 1, 2, 50, 14, 52, 76, 19, 82, 41, 73, 51, 79, 13, 3, 82, 96],
[40, 28, 52, 10, 10, 71, 56, 78, 82, 5, 29, 48, 1, 26, 16, 18, 50, 76, 86, 52],
[38, 89, 83, 43, 29, 52, 90, 77, 57, 0, 67, 20, 81, 88, 48, 96, 88, 58, 14, 3]])
syms = x0,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18 = symbols('x:19')
sol = {
x0: -S(1967374186044955317099186851240896179)/3166636564687820453598895768302256588,
x1: -S(84268280268757263347292368432053826)/791659141171955113399723942075564147,
x2: -S(229962957341664730974463872411844965)/1583318282343910226799447884151128294,
x3: S(990156781744251750886760432229180537)/6333273129375640907197791536604513176,
x4: -S(2169830351210066092046760299593096265)/18999819388126922721593374609813539528,
x5: S(4680868883477577389628494526618745355)/9499909694063461360796687304906769764,
x6: -S(1590820774344371990683178396480879213)/3166636564687820453598895768302256588,
x7: -S(54104723404825537735226491634383072)/339282489073695048599881689460956063,
x8: S(3182076494196560075964847771774733847)/6333273129375640907197791536604513176,
x9: -S(10870817431029210431989147852497539675)/18999819388126922721593374609813539528,
x10: -S(13118019242576506476316318268573312603)/18999819388126922721593374609813539528,
x11: -S(5173852969886775824855781403820641259)/4749954847031730680398343652453384882,
x12: S(4261112042731942783763341580651820563)/4749954847031730680398343652453384882,
x13: -S(821833082694661608993818117038209051)/6333273129375640907197791536604513176,
x14: S(906881575107250690508618713632090559)/904753304196520129599684505229216168,
x15: -S(732162528717458388995329317371283987)/6333273129375640907197791536604513176,
x16: S(4524215476705983545537087360959896817)/9499909694063461360796687304906769764,
x17: -S(3898571347562055611881270844646055217)/6333273129375640907197791536604513176,
x18: S(7513502486176995632751685137907442269)/18999819388126922721593374609813539528
}
eqs = list(M * Matrix(syms + (1,)))
assert solve(eqs, syms) == sol
y = Symbol('y')
eqs = list(y * M * Matrix(syms + (1,)))
assert solve(eqs, syms) == sol
def test_linear_systemLU():
n = Symbol('n')
M = Matrix([[1, 2, 0, 1], [1, 3, 2*n, 1], [4, -1, n**2, 1]])
assert solve_linear_system_LU(M, [x, y, z]) == {z: -3/(n**2 + 18*n),
x: 1 - 12*n/(n**2 + 18*n),
y: 6*n/(n**2 + 18*n)}
# Note: multiple solutions exist for some of these equations, so the tests
# should be expected to break if the implementation of the solver changes
# in such a way that a different branch is chosen
@slow
def test_solve_transcendental():
from sympy.abc import a, b
assert solve(exp(x) - 3, x) == [log(3)]
assert set(solve((a*x + b)*(exp(x) - 3), x)) == {-b/a, log(3)}
assert solve(cos(x) - y, x) == [-acos(y) + 2*pi, acos(y)]
assert solve(2*cos(x) - y, x) == [-acos(y/2) + 2*pi, acos(y/2)]
assert solve(Eq(cos(x), sin(x)), x) == [pi/4]
assert set(solve(exp(x) + exp(-x) - y, x)) in [{
log(y/2 - sqrt(y**2 - 4)/2),
log(y/2 + sqrt(y**2 - 4)/2),
}, {
log(y - sqrt(y**2 - 4)) - log(2),
log(y + sqrt(y**2 - 4)) - log(2)},
{
log(y/2 - sqrt((y - 2)*(y + 2))/2),
log(y/2 + sqrt((y - 2)*(y + 2))/2)}]
assert solve(exp(x) - 3, x) == [log(3)]
assert solve(Eq(exp(x), 3), x) == [log(3)]
assert solve(log(x) - 3, x) == [exp(3)]
assert solve(sqrt(3*x) - 4, x) == [Rational(16, 3)]
assert solve(3**(x + 2), x) == []
assert solve(3**(2 - x), x) == []
assert solve(x + 2**x, x) == [-LambertW(log(2))/log(2)]
assert solve(2*x + 5 + log(3*x - 2), x) == \
[Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2]
assert solve(3*x + log(4*x), x) == [LambertW(Rational(3, 4))/3]
assert set(solve((2*x + 8)*(8 + exp(x)), x)) == {S(-4), log(8) + pi*I}
eq = 2*exp(3*x + 4) - 3
ans = solve(eq, x) # this generated a failure in flatten
assert len(ans) == 3 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(2*log(3*x + 4) - 3, x) == [(exp(Rational(3, 2)) - 4)/3]
assert solve(exp(x) + 1, x) == [pi*I]
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solve(eq, x)
x0 = -log(2401)
x1 = 3**Rational(1, 5)
x2 = log(7**(7*x1/20))
x3 = sqrt(2)
x4 = sqrt(5)
x5 = x3*sqrt(x4 - 5)
x6 = x4 + 1
x7 = 1/(3*log(7))
x8 = -x4
x9 = x3*sqrt(x8 - 5)
x10 = x8 + 1
ans = [x7*(x0 - 5*LambertW(x2*(-x5 + x6))),
x7*(x0 - 5*LambertW(x2*(x5 + x6))),
x7*(x0 - 5*LambertW(x2*(x10 - x9))),
x7*(x0 - 5*LambertW(x2*(x10 + x9))),
x7*(x0 - 5*LambertW(-log(7**(7*x1/5))))]
assert result == ans, result
# it works if expanded, too
assert solve(eq.expand(), x) == result
assert solve(z*cos(x) - y, x) == [-acos(y/z) + 2*pi, acos(y/z)]
assert solve(z*cos(2*x) - y, x) == [-acos(y/z)/2 + pi, acos(y/z)/2]
assert solve(z*cos(sin(x)) - y, x) == [
pi - asin(acos(y/z)), asin(acos(y/z) - 2*pi) + pi,
-asin(acos(y/z) - 2*pi), asin(acos(y/z))]
assert solve(z*cos(x), x) == [pi/2, pi*Rational(3, 2)]
# issue 4508
assert solve(y - b*x/(a + x), x) in [[-a*y/(y - b)], [a*y/(b - y)]]
assert solve(y - b*exp(a/x), x) == [a/log(y/b)]
# issue 4507
assert solve(y - b/(1 + a*x), x) in [[(b - y)/(a*y)], [-((y - b)/(a*y))]]
# issue 4506
assert solve(y - a*x**b, x) == [(y/a)**(1/b)]
# issue 4505
assert solve(z**x - y, x) == [log(y)/log(z)]
# issue 4504
assert solve(2**x - 10, x) == [1 + log(5)/log(2)]
# issue 6744
assert solve(x*y) == [{x: 0}, {y: 0}]
assert solve([x*y]) == [{x: 0}, {y: 0}]
assert solve(x**y - 1) == [{x: 1}, {y: 0}]
assert solve([x**y - 1]) == [{x: 1}, {y: 0}]
assert solve(x*y*(x**2 - y**2)) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
assert solve([x*y*(x**2 - y**2)]) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
# issue 4739
assert solve(exp(log(5)*x) - 2**x, x) == [0]
# issue 14791
assert solve(exp(log(5)*x) - exp(log(2)*x), x) == [0]
f = Function('f')
assert solve(y*f(log(5)*x) - y*f(log(2)*x), x) == [0]
assert solve(f(x) - f(0), x) == [0]
assert solve(f(x) - f(2 - x), x) == [1]
raises(NotImplementedError, lambda: solve(f(x, y) - f(1, 2), x))
raises(NotImplementedError, lambda: solve(f(x, y) - f(2 - x, 2), x))
raises(ValueError, lambda: solve(f(x, y) - f(1 - x), x))
raises(ValueError, lambda: solve(f(x, y) - f(1), x))
# misc
# make sure that the right variables is picked up in tsolve
# shouldn't generate a GeneratorsNeeded error in _tsolve when the NaN is generated
# for eq_down. Actual answers, as determined numerically are approx. +/- 0.83
raises(NotImplementedError, lambda:
solve(sinh(x)*sinh(sinh(x)) + cosh(x)*cosh(sinh(x)) - 3))
# watch out for recursive loop in tsolve
raises(NotImplementedError, lambda: solve((x + 2)**y*x - 3, x))
# issue 7245
assert solve(sin(sqrt(x))) == [0, pi**2]
# issue 7602
a, b = symbols('a, b', real=True, negative=False)
assert str(solve(Eq(a, 0.5 - cos(pi*b)/2), b)) == \
'[2.0 - 0.318309886183791*acos(1.0 - 2.0*a), 0.318309886183791*acos(1.0 - 2.0*a)]'
# issue 15325
assert solve(y**(1/x) - z, x) == [log(y)/log(z)]
def test_solve_for_functions_derivatives():
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y)
assert soln == {
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
assert solve(x - 1, x) == [1]
assert solve(3*x - 2, x) == [Rational(2, 3)]
soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) +
a22*y.diff(t) - b2], x.diff(t), y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x.diff(t): (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
assert solve(x.diff(t) - 1, x.diff(t)) == [1]
assert solve(3*x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)]
eqns = {3*x - 1, 2*y - 4}
assert solve(eqns, {x, y}) == { x: Rational(1, 3), y: 2 }
x = Symbol('x')
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2)/f(x)]
# Mixed cased with a Symbol and a Function
x = Symbol('x')
y = Function('y')(t)
soln = solve([a11*x + a12*y.diff(t) - b1, a21*x +
a22*y.diff(t) - b2], x, y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
# issue 13263
x = Symbol('x')
f = Function('f')
soln = solve([f(x).diff(x) + f(x).diff(x, 2) - 1, f(x).diff(x) - f(x).diff(x, 2)],
f(x).diff(x), f(x).diff(x, 2))
assert soln == { f(x).diff(x, 2): 1/2, f(x).diff(x): 1/2 }
soln = solve([f(x).diff(x, 2) + f(x).diff(x, 3) - 1, 1 - f(x).diff(x, 2) -
f(x).diff(x, 3), 1 - f(x).diff(x,3)], f(x).diff(x, 2), f(x).diff(x, 3))
assert soln == { f(x).diff(x, 2): 0, f(x).diff(x, 3): 1 }
def test_issue_3725():
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
e = F.diff(x)
assert solve(e, f(x).diff(x)) in [[(2 - x)/f(x)], [-((x - 2)/f(x))]]
def test_issue_3870():
a, b, c, d = symbols('a b c d')
A = Matrix(2, 2, [a, b, c, d])
B = Matrix(2, 2, [0, 2, -3, 0])
C = Matrix(2, 2, [1, 2, 3, 4])
assert solve(A*B - C, [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve([A*B - C], [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve(Eq(A*B, C), [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve([A*B - B*A], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c}
assert solve([A*C - C*A], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c}
assert solve([A*B - B*A, A*C - C*A], [a, b, c, d]) == {a: d, b: 0, c: 0}
assert solve([Eq(A*B, B*A)], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c}
assert solve([Eq(A*C, C*A)], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c}
assert solve([Eq(A*B, B*A), Eq(A*C, C*A)], [a, b, c, d]) == {a: d, b: 0, c: 0}
def test_solve_linear():
w = Wild('w')
assert solve_linear(x, x) == (0, 1)
assert solve_linear(x, exclude=[x]) == (0, 1)
assert solve_linear(x, symbols=[w]) == (0, 1)
assert solve_linear(x, y - 2*x) in [(x, y/3), (y, 3*x)]
assert solve_linear(x, y - 2*x, exclude=[x]) == (y, 3*x)
assert solve_linear(3*x - y, 0) in [(x, y/3), (y, 3*x)]
assert solve_linear(3*x - y, 0, [x]) == (x, y/3)
assert solve_linear(3*x - y, 0, [y]) == (y, 3*x)
assert solve_linear(x**2/y, 1) == (y, x**2)
assert solve_linear(w, x) in [(w, x), (x, w)]
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y) == \
(y, -2 - cos(x)**2 - sin(x)**2)
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y, symbols=[x]) == (0, 1)
assert solve_linear(Eq(x, 3)) == (x, 3)
assert solve_linear(1/(1/x - 2)) == (0, 0)
assert solve_linear((x + 1)*exp(-x), symbols=[x]) == (x, -1)
assert solve_linear((x + 1)*exp(x), symbols=[x]) == ((x + 1)*exp(x), 1)
assert solve_linear(x*exp(-x**2), symbols=[x]) == (x, 0)
assert solve_linear(0**x - 1) == (0**x - 1, 1)
assert solve_linear(1 + 1/(x - 1)) == (x, 0)
eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0
assert solve_linear(eq) == (0, 1)
eq = cos(x)**2 + sin(x)**2 # = 1
assert solve_linear(eq) == (0, 1)
raises(ValueError, lambda: solve_linear(Eq(x, 3), 3))
def test_solve_undetermined_coeffs():
assert solve_undetermined_coeffs(a*x**2 + b*x**2 + b*x + 2*c*x + c + 1, [a, b, c], x) == \
{a: -2, b: 2, c: -1}
# Test that rational functions work
assert solve_undetermined_coeffs(a/x + b/(x + 1) - (2*x + 1)/(x**2 + x), [a, b], x) == \
{a: 1, b: 1}
# Test cancellation in rational functions
assert solve_undetermined_coeffs(((c + 1)*a*x**2 + (c + 1)*b*x**2 +
(c + 1)*b*x + (c + 1)*2*c*x + (c + 1)**2)/(c + 1), [a, b, c], x) == \
{a: -2, b: 2, c: -1}
def test_solve_inequalities():
x = Symbol('x')
sol = And(S.Zero < x, x < oo)
assert solve(x + 1 > 1) == sol
assert solve([x + 1 > 1]) == sol
assert solve([x + 1 > 1], x) == sol
assert solve([x + 1 > 1], [x]) == sol
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
And(Or(And(Lt(-sqrt(2), x), Lt(x, -1)),
And(Lt(1, x), Lt(x, sqrt(2)))), Eq(0, 0))
x = Symbol('x', real=True)
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
Or(And(Lt(-sqrt(2), x), Lt(x, -1)), And(Lt(1, x), Lt(x, sqrt(2))))
# issues 6627, 3448
assert solve((x - 3)/(x - 2) < 0, x) == And(Lt(2, x), Lt(x, 3))
assert solve(x/(x + 1) > 1, x) == And(Lt(-oo, x), Lt(x, -1))
assert solve(sin(x) > S.Half) == And(pi/6 < x, x < pi*Rational(5, 6))
assert solve(Eq(False, x < 1)) == (S.One <= x) & (x < oo)
assert solve(Eq(True, x < 1)) == (-oo < x) & (x < 1)
assert solve(Eq(x < 1, False)) == (S.One <= x) & (x < oo)
assert solve(Eq(x < 1, True)) == (-oo < x) & (x < 1)
assert solve(Eq(False, x)) == False
assert solve(Eq(0, x)) == [0]
assert solve(Eq(True, x)) == True
assert solve(Eq(1, x)) == [1]
assert solve(Eq(False, ~x)) == True
assert solve(Eq(True, ~x)) == False
assert solve(Ne(True, x)) == False
assert solve(Ne(1, x)) == (x > -oo) & (x < oo) & Ne(x, 1)
def test_issue_4793():
assert solve(1/x) == []
assert solve(x*(1 - 5/x)) == [5]
assert solve(x + sqrt(x) - 2) == [1]
assert solve(-(1 + x)/(2 + x)**2 + 1/(2 + x)) == []
assert solve(-x**2 - 2*x + (x + 1)**2 - 1) == []
assert solve((x/(x + 1) + 3)**(-2)) == []
assert solve(x/sqrt(x**2 + 1), x) == [0]
assert solve(exp(x) - y, x) == [log(y)]
assert solve(exp(x)) == []
assert solve(x**2 + x + sin(y)**2 + cos(y)**2 - 1, x) in [[0, -1], [-1, 0]]
eq = 4*3**(5*x + 2) - 7
ans = solve(eq, x)
assert len(ans) == 5 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(log(x**2) - y**2/exp(x), x, y, set=True) == (
[x, y],
{(x, sqrt(exp(x) * log(x ** 2))), (x, -sqrt(exp(x) * log(x ** 2)))})
assert solve(x**2*z**2 - z**2*y**2) == [{x: -y}, {x: y}, {z: 0}]
assert solve((x - 1)/(1 + 1/(x - 1))) == []
assert solve(x**(y*z) - x, x) == [1]
raises(NotImplementedError, lambda: solve(log(x) - exp(x), x))
raises(NotImplementedError, lambda: solve(2**x - exp(x) - 3))
def test_PR1964():
# issue 5171
assert solve(sqrt(x)) == solve(sqrt(x**3)) == [0]
assert solve(sqrt(x - 1)) == [1]
# issue 4462
a = Symbol('a')
assert solve(-3*a/sqrt(x), x) == []
# issue 4486
assert solve(2*x/(x + 2) - 1, x) == [2]
# issue 4496
assert set(solve((x**2/(7 - x)).diff(x))) == {S.Zero, S(14)}
# issue 4695
f = Function('f')
assert solve((3 - 5*x/f(x))*f(x), f(x)) == [x*Rational(5, 3)]
# issue 4497
assert solve(1/root(5 + x, 5) - 9, x) == [Rational(-295244, 59049)]
assert solve(sqrt(x) + sqrt(sqrt(x)) - 4) == [(Rational(-1, 2) + sqrt(17)/2)**4]
assert set(solve(Poly(sqrt(exp(x)) + sqrt(exp(-x)) - 4))) in \
[
{log((-sqrt(3) + 2)**2), log((sqrt(3) + 2)**2)},
{2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2)},
{log(-4*sqrt(3) + 7), log(4*sqrt(3) + 7)},
]
assert set(solve(Poly(exp(x) + exp(-x) - 4))) == \
{log(-sqrt(3) + 2), log(sqrt(3) + 2)}
assert set(solve(x**y + x**(2*y) - 1, x)) == \
{(Rational(-1, 2) + sqrt(5)/2)**(1/y), (Rational(-1, 2) - sqrt(5)/2)**(1/y)}
assert solve(exp(x/y)*exp(-z/y) - 2, y) == [(x - z)/log(2)]
assert solve(
x**z*y**z - 2, z) in [[log(2)/(log(x) + log(y))], [log(2)/(log(x*y))]]
# if you do inversion too soon then multiple roots (as for the following)
# will be missed, e.g. if exp(3*x) = exp(3) -> 3*x = 3
E = S.Exp1
assert solve(exp(3*x) - exp(3), x) in [
[1, log(E*(Rational(-1, 2) - sqrt(3)*I/2)), log(E*(Rational(-1, 2) + sqrt(3)*I/2))],
[1, log(-E/2 - sqrt(3)*E*I/2), log(-E/2 + sqrt(3)*E*I/2)],
]
# coverage test
p = Symbol('p', positive=True)
assert solve((1/p + 1)**(p + 1)) == []
def test_issue_5197():
x = Symbol('x', real=True)
assert solve(x**2 + 1, x) == []
n = Symbol('n', integer=True, positive=True)
assert solve((n - 1)*(n + 2)*(2*n - 1), n) == [1]
x = Symbol('x', positive=True)
y = Symbol('y')
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == []
# not {x: -3, y: 1} b/c x is positive
# The solution following should not contain (-sqrt(2), sqrt(2))
assert solve((x + y)*n - y**2 + 2, x, y) == [(sqrt(2), -sqrt(2))]
y = Symbol('y', positive=True)
# The solution following should not contain {y: -x*exp(x/2)}
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: x*exp(x/2)}]
x, y, z = symbols('x y z', positive=True)
assert solve(z**2*x**2 - z**2*y**2/exp(x), y, x, z, dict=True) == [{y: x*exp(x/2)}]
def test_checking():
assert set(
solve(x*(x - y/x), x, check=False)) == {sqrt(y), S.Zero, -sqrt(y)}
assert set(solve(x*(x - y/x), x, check=True)) == {sqrt(y), -sqrt(y)}
# {x: 0, y: 4} sets denominator to 0 in the following so system should return None
assert solve((1/(1/x + 2), 1/(y - 3) - 1)) == []
# 0 sets denominator of 1/x to zero so None is returned
assert solve(1/(1/x + 2)) == []
def test_issue_4671_4463_4467():
assert solve(sqrt(x**2 - 1) - 2) in ([sqrt(5), -sqrt(5)],
[-sqrt(5), sqrt(5)])
assert solve((2**exp(y**2/x) + 2)/(x**2 + 15), y) == [
-sqrt(x*log(1 + I*pi/log(2))), sqrt(x*log(1 + I*pi/log(2)))]
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
a = Symbol('a')
E = S.Exp1
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2]
)
assert solve(log(a**(-3) - x**2)/a, x) in (
[-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
[sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2],)
assert solve((a**2 + 1)*(sin(a*x) + cos(a*x)), x) == [-pi/(4*a)]
assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [log(3)/a]
assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \
{log(-2 + sqrt(5))/a, log(-sqrt(2) + 1)/a,
log(-sqrt(5) - 2)/a, log(1 + sqrt(2))/a}
assert solve(atan(x) - 1) == [tan(1)]
def test_issue_5132():
r, t = symbols('r,t')
assert set(solve([r - x**2 - y**2, tan(t) - y/x], [x, y])) == \
{(
-sqrt(r*cos(t)**2), -1*sqrt(r*cos(t)**2)*tan(t)),
(sqrt(r*cos(t)**2), sqrt(r*cos(t)**2)*tan(t))}
assert solve([exp(x) - sin(y), 1/y - 3], [x, y]) == \
[(log(sin(Rational(1, 3))), Rational(1, 3))]
assert solve([exp(x) - sin(y), 1/exp(y) - 3], [x, y]) == \
[(log(-sin(log(3))), -log(3))]
assert set(solve([exp(x) - sin(y), y**2 - 4], [x, y])) == \
{(log(-sin(2)), -S(2)), (log(sin(2)), S(2))}
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
assert solve(eqs, set=True) == \
([y, z], {
(-log(3), sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), -sqrt(-exp(2*x) - sin(log(3))))})
assert solve(eqs, x, z, set=True) == (
[x, z],
{(x, sqrt(-exp(2*x) + sin(y))), (x, -sqrt(-exp(2*x) + sin(y)))})
assert set(solve(eqs, x, y)) == \
{
(log(-sqrt(-z**2 - sin(log(3)))), -log(3)),
(log(-z**2 - sin(log(3)))/2, -log(3))}
assert set(solve(eqs, y, z)) == \
{
(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), sqrt(-exp(2*x) - sin(log(3))))}
eqs = [exp(x)**2 - sin(y) + z, 1/exp(y) - 3]
assert solve(eqs, set=True) == ([y, z], {
(-log(3), -exp(2*x) - sin(log(3)))})
assert solve(eqs, x, z, set=True) == (
[x, z], {(x, -exp(2*x) + sin(y))})
assert set(solve(eqs, x, y)) == {
(log(-sqrt(-z - sin(log(3)))), -log(3)),
(log(-z - sin(log(3)))/2, -log(3))}
assert solve(eqs, z, y) == \
[(-exp(2*x) - sin(log(3)), -log(3))]
assert solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), set=True) == (
[x, y], {(S.One, S(3)), (S(3), S.One)})
assert set(solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), x, y)) == \
{(S.One, S(3)), (S(3), S.One)}
def test_issue_5335():
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions obtained manually but only two are valid
assert len(solve(eqs, sym, manual=True, minimal=True)) == 2
assert len(solve(eqs, sym)) == 2 # cf below with rational=False
@SKIP("Hangs")
def _test_issue_5335_float():
# gives ZeroDivisionError: polynomial division
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
assert len(solve(eqs, sym, rational=False)) == 2
def test_issue_5767():
assert set(solve([x**2 + y + 4], [x])) == \
{(-sqrt(-y - 4),), (sqrt(-y - 4),)}
def test_polysys():
assert set(solve([x**2 + 2/y - 2, x + y - 3], [x, y])) == \
{(S.One, S(2)), (1 + sqrt(5), 2 - sqrt(5)),
(1 - sqrt(5), 2 + sqrt(5))}
assert solve([x**2 + y - 2, x**2 + y]) == []
# the ordering should be whatever the user requested
assert solve([x**2 + y - 3, x - y - 4], (x, y)) != solve([x**2 +
y - 3, x - y - 4], (y, x))
@slow
def test_unrad1():
raises(NotImplementedError, lambda:
unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(NotImplementedError, lambda:
unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y)))
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 5203)
def check(rv, ans):
assert bool(rv[1]) == bool(ans[1])
if ans[1]:
return s_check(rv, ans)
e = rv[0].expand()
a = ans[0].expand()
return e in [a, -a] and rv[1] == ans[1]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = list(zip(d, [s]*len(d)))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)])
ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)])
return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \
str(rv[1]) == str(ans[1])
assert unrad(1) is None
assert check(unrad(sqrt(x)),
(x, []))
assert check(unrad(sqrt(x) + 1),
(x - 1, []))
assert check(unrad(sqrt(x) + root(x, 3) + 2),
(s**3 + s**2 + 2, [s, s**6 - x]))
assert check(unrad(sqrt(x)*root(x, 3) + 2),
(x**5 - 64, []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(x**3 - (x + 1)**2, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)),
(-2*sqrt(2)*x - 2*x + 1, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + 2),
(16*x - 9, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)),
(5*x**2 - 4*x, []))
assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)),
((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, []))
assert check(unrad(sqrt(x) + sqrt(1 - x)),
(2*x - 1, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) - 3),
(x**2 - x + 16, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(5*x**2 - 2*x + 1, []))
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [
(25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []),
(25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])]
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \
(41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S.One, []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**2 - 9*x, []))
assert set(solve(eq, check=False)) == {S.Zero, Rational(9, 16)}
assert solve(eq) == []
# but this one really does have those solutions
assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \
{S.Zero, Rational(9, 16)}
assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y),
(S('2*sqrt(x)*(x + 1)**(1/3) + x - 4*y + (x + 1)**(2/3)'), []))
assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1, 3)),
(x**5 - x**4 - x**3 + 2*x**2 + x - 1, []))
assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(4*x*y + x - 4*y, []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x**2 - x + 4, []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert set(solve(Eq(sqrt(5*x + 6) - 2, x))) == {-S.One, S(2)}
assert set(solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))) == {S(5), S(13)}
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x - 5)**Rational(1, 3) - 3) == [16]
assert set(solve(x + 1 - root(x**4 + 4*x**3 - x, 4))) == \
{Rational(-1, 2), Rational(-1, 3)}
assert set(solve(sqrt(2*x**2 - 7) - (3 - x))) == {-S(8), S(2)}
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and do use _mexpand
z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x)
p = posify(z)[0]
assert solve(p) == []
assert solve(z) == []
assert solve(z + 6*I) == [Rational(-1, 11)]
assert solve(p + 6*I) == []
# issue 8622
assert unrad(root(x + 1, 5) - root(x, 3)) == (
-(x**5 - x**3 - 3*x**2 - 3*x - 1), [])
# issue #8679
assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x),
(s**3 + s**2 + s + sqrt(y), [s, s**3 - x]))
# for coverage
assert check(unrad(sqrt(x) + root(x, 3) + y),
(s**3 + s**2 + y, [s, s**6 - x]))
assert solve(sqrt(x) + root(x, 3) - 2) == [1]
raises(NotImplementedError, lambda:
solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2))
# fails through a different code path
raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x))
# unrad some
assert solve(sqrt(x + root(x, 3))+root(x - y, 5), y) == [
x + (x**Rational(1, 3) + x)**Rational(5, 2)]
assert check(unrad(sqrt(x) - root(x + 1, 3)*sqrt(x + 2) + 2),
(s**10 + 8*s**8 + 24*s**6 - 12*s**5 - 22*s**4 - 160*s**3 - 212*s**2 -
192*s - 56, [s, s**2 - x]))
e = root(x + 1, 3) + root(x, 3)
assert unrad(e) == (2*x + 1, [])
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
(15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, []))
assert check(unrad(root(x, 4) + root(x, 4)**3 - 1),
(s**3 + s - 1, [s, s**4 - x]))
assert check(unrad(root(x, 2) + root(x, 2)**3 - 1),
(x**3 + 2*x**2 + x - 1, []))
assert unrad(x**0.5) is None
assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3),
(s**3 + s + t, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y),
(s**3 + s + x, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x),
(s**5 + s**3 + s - y, [s, s**5 - x - y]))
assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)),
(s**5 + 5*2**Rational(1, 5)*s**4 + s**3 + 10*2**Rational(2, 5)*s**3 +
10*2**Rational(3, 5)*s**2 + 5*2**Rational(4, 5)*s + 4, [s, s**3 - x + 1]))
raises(NotImplementedError, lambda:
unrad((root(x, 2) + root(x, 3) + root(x, 4)).subs(x, x**5 - x + 1)))
# the simplify flag should be reset to False for unrad results;
# if it's not then this next test will take a long time
assert solve(root(x, 3) + root(x, 5) - 2) == [1]
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), []))
ans = S('''
[4/5, -1484/375 + 172564/(140625*(114*sqrt(12657)/78125 +
12459439/52734375)**(1/3)) +
4*(114*sqrt(12657)/78125 + 12459439/52734375)**(1/3)]''')
assert solve(eq) == ans
# duplicate radical handling
assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2),
(s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1]))
# cov post-processing
e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2
assert check(unrad(e),
(s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30,
[s, s**3 - x**2 - 1]))
e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2
assert check(unrad(e),
(s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25,
[s, s**3 - x - 1]))
assert check(unrad(e, _reverse=True),
(s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89,
[s, s**2 - x - sqrt(x + 1)]))
# this one needs r0, r1 reversal to work
assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2),
(s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 +
32*s + 17, [s, s**6 - x]))
# why does this pass
assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == (
-(x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5
- cosh(x)**5), [])
# and this fail?
#assert unrad(sqrt(cosh(x)/x) + root(x + 1, 3)*sqrt(x) - 1) == (
# -s**6 + 6*s**5 - 15*s**4 + 20*s**3 - 15*s**2 + 6*s + x**5 +
# 2*x**4 + x**3 - 1, [s, s**2 - cosh(x)/x])
# watch for symbols in exponents
assert unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1')) is None
assert check(unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1'), x),
(s**(2*y) + s + 1, [s, s**3 - x - y]))
# should _Q be so lenient?
assert unrad(x**(S.Half/y) + y, x) == (x**(1/y) - y**2, [])
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests that the use of
# composite
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 -
1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3
# watch out for when the cov doesn't involve the symbol of interest
eq = S('-x + (7*y/8 - (27*x/2 + 27*sqrt(x**2)/2)**(1/3)/3)**3 - 1')
assert solve(eq, y) == [
2**(S(2)/3)*(27*x + 27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 +
S(512)/343)**(S(1)/3)*(-S(1)/2 - sqrt(3)*I/2), 2**(S(2)/3)*(27*x +
27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 +
S(512)/343)**(S(1)/3)*(-S(1)/2 + sqrt(3)*I/2), 2**(S(2)/3)*(27*x +
27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 + S(512)/343)**(S(1)/3)]
eq = root(x + 1, 3) - (root(x, 3) + root(x, 5))
assert check(unrad(eq),
(3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x]))
assert check(unrad(eq - 2),
(3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 +
12*s**3 + 7, [s, s**15 - x]))
assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)),
(s*(4096*s**9 + 960*s**8 + 48*s**7 - s**6 - 1728),
[s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389
assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2),
(343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 -
3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x -
1])) # orig expr has one real root: -0.048
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)),
(729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 -
3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x -
1])) # orig expr has 2 real roots: -0.91, -0.15
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2),
(729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 +
453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3
- 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1]))
# orig expr has 1 real root: 19.53
ans = solve(sqrt(x) + sqrt(x + 1) -
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# https://github.com/sympy/sympy/issues/4793#issuecomment-36994519
F = Symbol('F')
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
ans = F*Rational(2, 7) - sqrt(2)*F/14
X = solve(eq, x, check=False)
for xi in reversed(X): # reverse since currently, ans is the 2nd one
Y = solve((x*y).subs(x, xi).diff(y), y, simplify=False, check=False)
if any((a - ans).expand().is_zero for a in Y):
break
else:
assert None # no answer was found
assert solve(sqrt(x + 1) + root(x, 3) - 2) == S('''
[(-11/(9*(47/54 + sqrt(93)/6)**(1/3)) + 1/3 + (47/54 +
sqrt(93)/6)**(1/3))**3]''')
assert solve(sqrt(sqrt(x + 1)) + x**Rational(1, 3) - 2) == S('''
[(-sqrt(-2*(-1/16 + sqrt(6913)/16)**(1/3) + 6/(-1/16 +
sqrt(6913)/16)**(1/3) + 17/2 + 121/(4*sqrt(-6/(-1/16 +
sqrt(6913)/16)**(1/3) + 2*(-1/16 + sqrt(6913)/16)**(1/3) + 17/4)))/2 +
sqrt(-6/(-1/16 + sqrt(6913)/16)**(1/3) + 2*(-1/16 +
sqrt(6913)/16)**(1/3) + 17/4)/2 + 9/4)**3]''')
assert solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == S('''
[(-(81/2 + 3*sqrt(741)/2)**(1/3)/3 + (81/2 + 3*sqrt(741)/2)**(-1/3) +
2)**2]''')
eq = S('''
-x + (1/2 - sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3
+ x*(3*x**2 - 34) + 90)**2/4 - 39304/27) - 45)**(1/3) + 34/(3*(1/2 -
sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2
- 34) + 90)**2/4 - 39304/27) - 45)**(1/3))''')
assert check(unrad(eq),
(s*-(-s**6 + sqrt(3)*s**6*I - 153*2**Rational(2, 3)*3**Rational(1, 3)*s**4 +
51*12**Rational(1, 3)*s**4 - 102*2**Rational(2, 3)*3**Rational(5, 6)*s**4*I - 1620*s**3 +
1620*sqrt(3)*s**3*I + 13872*18**Rational(1, 3)*s**2 - 471648 +
471648*sqrt(3)*I), [s, s**3 - 306*x - sqrt(3)*sqrt(31212*x**2 -
165240*x + 61484) + 810]))
assert solve(eq) == [] # not other code errors
eq = root(x, 3) - root(y, 3) + root(x, 5)
assert check(unrad(eq),
(s**15 + 3*s**13 + 3*s**11 + s**9 - y, [s, s**15 - x]))
eq = root(x, 3) + root(y, 3) + root(x*y, 4)
assert check(unrad(eq),
(s*y*(-s**12 - 3*s**11*y - 3*s**10*y**2 - s**9*y**3 -
3*s**8*y**2 + 21*s**7*y**3 - 3*s**6*y**4 - 3*s**4*y**4 -
3*s**3*y**5 - y**6), [s, s**4 - x*y]))
raises(NotImplementedError,
lambda: unrad(root(x, 3) + root(y, 3) + root(x*y, 5)))
# Test unrad with an Equality
eq = Eq(-x**(S(1)/5) + x**(S(1)/3), -3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5))
assert check(unrad(eq),
(-s**5 + s**3 - 3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5), [s, s**15 - x]))
# make sure buried radicals are exposed
s = sqrt(x) - 1
assert unrad(s**2 - s**3) == (x**3 - 6*x**2 + 9*x - 4, [])
# make sure numerators which are already polynomial are rejected
assert unrad((x/(x + 1) + 3)**(-2), x) is None
@slow
def test_unrad_slow():
# this has roots with multiplicity > 1; there should be no
# repeats in roots obtained, however
eq = (sqrt(1 + sqrt(1 - 4*x**2)) - x*(1 + sqrt(1 + 2*sqrt(1 - 4*x**2))))
assert solve(eq) == [S.Half]
@XFAIL
def test_unrad_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
assert solve(root(x**3 - 3*x**2, 3) + 1 - x) == [Rational(1, 3)]
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == [
-1, -1 + CRootOf(x**5 + x**4 + 5*x**3 + 8*x**2 + 10*x + 5, 0)**3]
def test_checksol():
x, y, r, t = symbols('x, y, r, t')
eq = r - x**2 - y**2
dict_var_soln = {y: - sqrt(r) / sqrt(tan(t)**2 + 1),
x: -sqrt(r)*tan(t)/sqrt(tan(t)**2 + 1)}
assert checksol(eq, dict_var_soln) == True
assert checksol(Eq(x, False), {x: False}) is True
assert checksol(Ne(x, False), {x: False}) is False
assert checksol(Eq(x < 1, True), {x: 0}) is True
assert checksol(Eq(x < 1, True), {x: 1}) is False
assert checksol(Eq(x < 1, False), {x: 1}) is True
assert checksol(Eq(x < 1, False), {x: 0}) is False
assert checksol(Eq(x + 1, x**2 + 1), {x: 1}) is True
assert checksol([x - 1, x**2 - 1], x, 1) is True
assert checksol([x - 1, x**2 - 2], x, 1) is False
assert checksol(Poly(x**2 - 1), x, 1) is True
assert checksol(0, {}) is True
assert checksol([1e-10, x - 2], x, 2) is False
assert checksol([0.5, 0, x], x, 0) is False
assert checksol(y, x, 2) is False
assert checksol(x+1e-10, x, 0, numerical=True) is True
assert checksol(x+1e-10, x, 0, numerical=False) is False
raises(ValueError, lambda: checksol(x, 1))
raises(ValueError, lambda: checksol([], x, 1))
def test__invert():
assert _invert(x - 2) == (2, x)
assert _invert(2) == (2, 0)
assert _invert(exp(1/x) - 3, x) == (1/log(3), x)
assert _invert(exp(1/x + a/x) - 3, x) == ((a + 1)/log(3), x)
assert _invert(a, x) == (a, 0)
def test_issue_4463():
assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
assert solve(x**x) == []
assert solve(x**x - 2) == [exp(LambertW(log(2)))]
assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
@slow
def test_issue_5114_solvers():
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r')
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = a, b, c, f, h, k, n
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(solve(eqs, syms, manual=True, check=False, simplify=False)) == 1
def test_issue_5849():
#
# XXX: This system does not have a solution for most values of the
# parameters. Generally solve returns the empty set for systems that are
# generically inconsistent.
#
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
ans = [{
I1: I2 + I3,
dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24,
I4: I3 - I5,
dQ4: I3 - I5,
Q4: -I3/2 + 3*I5/2 - dI4/2,
dQ2: I2,
Q2: 2*I3 + 2*I5 + 3*I6}]
v = I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4
assert solve(e, *v, manual=True, check=False, dict=True) == ans
assert solve(e, *v, manual=True, check=False) == ans[0]
assert solve(e, *v, manual=True) == []
assert solve(e, *v) == []
# the matrix solver (tested below) doesn't like this because it produces
# a zero row in the matrix. Is this related to issue 4551?
assert [ei.subs(
ans[0]) for ei in e] == [0, 0, I3 - I6, -I3 + I6, 0, 0, 0, 0, 0]
def test_issue_5849_matrix():
'''Same as test_issue_5849 but solved with the matrix solver.
A solution only exists if I3 == I6 which is not generically true,
but `solve` does not return conditions under which the solution is
valid, only a solution that is canonical and consistent with the input.
'''
# a simple example with the same issue
# assert solve([x+y+z, x+y], [x, y]) == {x: y}
# the longer example
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
assert solve(e, I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4) == []
def test_issue_21882():
a, b, c, d, f, g, k = unknowns = symbols('a, b, c, d, f, g, k')
equations = [
-k*a + b + 5*f/6 + 2*c/9 + 5*d/6 + 4*a/3,
-k*f + 4*f/3 + d/2,
-k*d + f/6 + d,
13*b/18 + 13*c/18 + 13*a/18,
-k*c + b/2 + 20*c/9 + a,
-k*b + b + c/18 + a/6,
5*b/3 + c/3 + a,
2*b/3 + 2*c + 4*a/3,
-g,
]
answer = [
{a: 0, f: 0, b: 0, d: 0, c: 0, g: 0},
{a: 0, f: -d, b: 0, k: S(5)/6, c: 0, g: 0},
{a: -2*c, f: 0, b: c, d: 0, k: S(13)/18, g: 0},
]
assert solve(equations, unknowns, dict=True) == answer
def test_issue_5901():
f, g, h = map(Function, 'fgh')
a = Symbol('a')
D = Derivative(f(x), x)
G = Derivative(g(a), a)
assert solve(f(x) + f(x).diff(x), f(x)) == \
[-D]
assert solve(f(x) - 3, f(x)) == \
[3]
assert solve(f(x) - 3*f(x).diff(x), f(x)) == \
[3*D]
assert solve([f(x) - 3*f(x).diff(x)], f(x)) == \
{f(x): 3*D}
assert solve([f(x) - 3*f(x).diff(x), f(x)**2 - y + 4], f(x), y) == \
[{f(x): 3*D, y: 9*D**2 + 4}]
assert solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a),
h(a), g(a), set=True) == \
([g(a)], {
(-sqrt(h(a)**2*f(a)**2 + G)/f(a),),
(sqrt(h(a)**2*f(a)**2+ G)/f(a),)})
args = [f(x).diff(x, 2)*(f(x) + g(x)) - g(x)**2 + 2, f(x), g(x)]
assert set(solve(*args)) == \
{(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))}
eqs = [f(x)**2 + g(x) - 2*f(x).diff(x), g(x)**2 - 4]
assert solve(eqs, f(x), g(x), set=True) == \
([f(x), g(x)], {
(-sqrt(2*D - 2), S(2)),
(sqrt(2*D - 2), S(2)),
(-sqrt(2*D + 2), -S(2)),
(sqrt(2*D + 2), -S(2))})
# the underlying problem was in solve_linear that was not masking off
# anything but a Mul or Add; it now raises an error if it gets anything
# but a symbol and solve handles the substitutions necessary so solve_linear
# won't make this error
raises(
ValueError, lambda: solve_linear(f(x) + f(x).diff(x), symbols=[f(x)]))
assert solve_linear(f(x) + f(x).diff(x), symbols=[x]) == \
(f(x) + Derivative(f(x), x), 1)
assert solve_linear(f(x) + Integral(x, (x, y)), symbols=[x]) == \
(f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(x) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x + f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(y) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x, -f(y) - Integral(x, (x, y)))
assert solve_linear(x - f(x)/a + (f(x) - 1)/a, symbols=[x]) == \
(x, 1/a)
assert solve_linear(x + Derivative(2*x, x)) == \
(x, -2)
assert solve_linear(x + Integral(x, y), symbols=[x]) == \
(x, 0)
assert solve_linear(x + Integral(x, y) - 2, symbols=[x]) == \
(x, 2/(y + 1))
assert set(solve(x + exp(x)**2, exp(x))) == \
{-sqrt(-x), sqrt(-x)}
assert solve(x + exp(x), x, implicit=True) == \
[-exp(x)]
assert solve(cos(x) - sin(x), x, implicit=True) == []
assert solve(x - sin(x), x, implicit=True) == \
[sin(x)]
assert solve(x**2 + x - 3, x, implicit=True) == \
[-x**2 + 3]
assert solve(x**2 + x - 3, x**2, implicit=True) == \
[-x + 3]
def test_issue_5912():
assert set(solve(x**2 - x - 0.1, rational=True)) == \
{S.Half + sqrt(35)/10, -sqrt(35)/10 + S.Half}
ans = solve(x**2 - x - 0.1, rational=False)
assert len(ans) == 2 and all(a.is_Number for a in ans)
ans = solve(x**2 - x - 0.1)
assert len(ans) == 2 and all(a.is_Number for a in ans)
def test_float_handling():
def test(e1, e2):
return len(e1.atoms(Float)) == len(e2.atoms(Float))
assert solve(x - 0.5, rational=True)[0].is_Rational
assert solve(x - 0.5, rational=False)[0].is_Float
assert solve(x - S.Half, rational=False)[0].is_Rational
assert solve(x - 0.5, rational=None)[0].is_Float
assert solve(x - S.Half, rational=None)[0].is_Rational
assert test(nfloat(1 + 2*x), 1.0 + 2.0*x)
for contain in [list, tuple, set]:
ans = nfloat(contain([1 + 2*x]))
assert type(ans) is contain and test(list(ans)[0], 1.0 + 2.0*x)
k, v = list(nfloat({2*x: [1 + 2*x]}).items())[0]
assert test(k, 2*x) and test(v[0], 1.0 + 2.0*x)
assert test(nfloat(cos(2*x)), cos(2.0*x))
assert test(nfloat(3*x**2), 3.0*x**2)
assert test(nfloat(3*x**2, exponent=True), 3.0*x**2.0)
assert test(nfloat(exp(2*x)), exp(2.0*x))
assert test(nfloat(x/3), x/3.0)
assert test(nfloat(x**4 + 2*x + cos(Rational(1, 3)) + 1),
x**4 + 2.0*x + 1.94495694631474)
# don't call nfloat if there is no solution
tot = 100 + c + z + t
assert solve(((.7 + c)/tot - .6, (.2 + z)/tot - .3, t/tot - .1)) == []
def test_check_assumptions():
x = symbols('x', positive=True)
assert solve(x**2 - 1) == [1]
def test_issue_6056():
assert solve(tanh(x + 3)*tanh(x - 3) - 1) == []
assert solve(tanh(x - 1)*tanh(x + 1) + 1) == \
[I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)]
assert solve((tanh(x + 3)*tanh(x - 3) + 1)**2) == \
[I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)]
def test_issue_5673():
eq = -x + exp(exp(LambertW(log(x)))*LambertW(log(x)))
assert checksol(eq, x, 2) is True
assert checksol(eq, x, 2, numerical=False) is None
def test_exclude():
R, C, Ri, Vout, V1, Vminus, Vplus, s = \
symbols('R, C, Ri, Vout, V1, Vminus, Vplus, s')
Rf = symbols('Rf', positive=True) # to eliminate Rf = 0 soln
eqs = [C*V1*s + Vplus*(-2*C*s - 1/R),
Vminus*(-1/Ri - 1/Rf) + Vout/Rf,
C*Vplus*s + V1*(-C*s - 1/R) + Vout/R,
-Vminus + Vplus]
assert solve(eqs, exclude=s*C*R) == [
{
Rf: Ri*(C*R*s + 1)**2/(C*R*s),
Vminus: Vplus,
V1: 2*Vplus + Vplus/(C*R*s),
Vout: C*R*Vplus*s + 3*Vplus + Vplus/(C*R*s)},
{
Vplus: 0,
Vminus: 0,
V1: 0,
Vout: 0},
]
# TODO: Investigate why currently solution [0] is preferred over [1].
assert solve(eqs, exclude=[Vplus, s, C]) in [[{
Vminus: Vplus,
V1: Vout/2 + Vplus/2 + sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus - sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}, {
Vminus: Vplus,
V1: Vout/2 + Vplus/2 - sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus + sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}], [{
Vminus: Vplus,
Vout: (V1**2 - V1*Vplus - Vplus**2)/(V1 - 2*Vplus),
Rf: Ri*(V1 - Vplus)**2/(Vplus*(V1 - 2*Vplus)),
R: Vplus/(C*s*(V1 - 2*Vplus)),
}]]
def test_high_order_roots():
s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4)
assert set(solve(s)) == set(Poly(s*4, domain='ZZ').all_roots())
def test_minsolve_linear_system():
pqt = dict(quick=True, particular=True)
pqf = dict(quick=False, particular=True)
assert solve([x + y - 5, 2*x - y - 1], **pqt) == {x: 2, y: 3}
assert solve([x + y - 5, 2*x - y - 1], **pqf) == {x: 2, y: 3}
def count(dic):
return len([x for x in dic.values() if x == 0])
assert count(solve([x + y + z, y + z + a + t], **pqt)) == 3
assert count(solve([x + y + z, y + z + a + t], **pqf)) == 3
assert count(solve([x + y + z, y + z + a], **pqt)) == 1
assert count(solve([x + y + z, y + z + a], **pqf)) == 2
# issue 22718
A = Matrix([
[ 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0],
[ 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, -1, -1, 0, 0],
[-1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 1],
[ 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, -1, 0, -1, 0],
[-1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, 1],
[-1, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1],
[ 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, -1, -1, 0],
[ 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1],
[ 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1],
[ 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1],
[ 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0],
[ 0, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0]])
v = Matrix(symbols("v:14", integer=True))
B = Matrix([[2], [-2], [0], [0], [0], [0], [0], [0], [0],
[0], [0], [0]])
eqs = A@v-B
assert solve(eqs) == []
assert solve(eqs, particular=True) == [] # assumption violated
assert all(v for v in solve([x + y + z, y + z + a]).values())
for _q in (True, False):
assert not all(v for v in solve(
[x + y + z, y + z + a], quick=_q,
particular=True).values())
# raise error if quick used w/o particular=True
raises(ValueError, lambda: solve([x + 1], quick=_q))
raises(ValueError, lambda: solve([x + 1], quick=_q, particular=False))
# and give a good error message if someone tries to use
# particular with a single equation
raises(ValueError, lambda: solve(x + 1, particular=True))
def test_real_roots():
# cf. issue 6650
x = Symbol('x', real=True)
assert len(solve(x**5 + x**3 + 1)) == 1
def test_issue_6528():
eqs = [
327600995*x**2 - 37869137*x + 1809975124*y**2 - 9998905626,
895613949*x**2 - 273830224*x*y + 530506983*y**2 - 10000000000]
# two expressions encountered are > 1400 ops long so if this hangs
# it is likely because simplification is being done
assert len(solve(eqs, y, x, check=False)) == 4
def test_overdetermined():
x = symbols('x', real=True)
eqs = [Abs(4*x - 7) - 5, Abs(3 - 8*x) - 1]
assert solve(eqs, x) == [(S.Half,)]
assert solve(eqs, x, manual=True) == [(S.Half,)]
assert solve(eqs, x, manual=True, check=False) == [(S.Half,), (S(3),)]
def test_issue_6605():
x = symbols('x')
assert solve(4**(x/2) - 2**(x/3)) == [0, 3*I*pi/log(2)]
# while the first one passed, this one failed
x = symbols('x', real=True)
assert solve(5**(x/2) - 2**(x/3)) == [0]
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solve(5**(x/2) - 2**(3/x)) == [-b, b]
def test__ispow():
assert _ispow(x**2)
assert not _ispow(x)
assert not _ispow(True)
def test_issue_6644():
eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2)
sol = solve(eq, q, simplify=False, check=False)
assert len(sol) == 5
def test_issue_6752():
assert solve([a**2 + a, a - b], [a, b]) == [(-1, -1), (0, 0)]
assert solve([a**2 + a*c, a - b], [a, b]) == [(0, 0), (-c, -c)]
def test_issue_6792():
assert solve(x*(x - 1)**2*(x + 1)*(x**6 - x + 1)) == [
-1, 0, 1, CRootOf(x**6 - x + 1, 0), CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2), CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4), CRootOf(x**6 - x + 1, 5)]
def test_issues_6819_6820_6821_6248_8692():
# issue 6821
x, y = symbols('x y', real=True)
assert solve(abs(x + 3) - 2*abs(x - 3)) == [1, 9]
assert solve([abs(x) - 2, arg(x) - pi], x) == [(-2,)]
assert set(solve(abs(x - 7) - 8)) == {-S.One, S(15)}
# issue 8692
assert solve(Eq(Abs(x + 1) + Abs(x**2 - 7), 9), x) == [
Rational(-1, 2) + sqrt(61)/2, -sqrt(69)/2 + S.Half]
# issue 7145
assert solve(2*abs(x) - abs(x - 1)) == [-1, Rational(1, 3)]
x = symbols('x')
assert solve([re(x) - 1, im(x) - 2], x) == [
{re(x): 1, x: 1 + 2*I, im(x): 2}]
# check for 'dict' handling of solution
eq = sqrt(re(x)**2 + im(x)**2) - 3
assert solve(eq) == solve(eq, x)
i = symbols('i', imaginary=True)
assert solve(abs(i) - 3) == [-3*I, 3*I]
raises(NotImplementedError, lambda: solve(abs(x) - 3))
w = symbols('w', integer=True)
assert solve(2*x**w - 4*y**w, w) == solve((x/y)**w - 2, w)
x, y = symbols('x y', real=True)
assert solve(x + y*I + 3) == {y: 0, x: -3}
# issue 2642
assert solve(x*(1 + I)) == [0]
x, y = symbols('x y', imaginary=True)
assert solve(x + y*I + 3 + 2*I) == {x: -2*I, y: 3*I}
x = symbols('x', real=True)
assert solve(x + y + 3 + 2*I) == {x: -3, y: -2*I}
# issue 6248
f = Function('f')
assert solve(f(x + 1) - f(2*x - 1)) == [2]
assert solve(log(x + 1) - log(2*x - 1)) == [2]
x = symbols('x')
assert solve(2**x + 4**x) == [I*pi/log(2)]
def test_issue_14607():
# issue 14607
s, tau_c, tau_1, tau_2, phi, K = symbols(
's, tau_c, tau_1, tau_2, phi, K')
target = (s**2*tau_1*tau_2 + s*tau_1 + s*tau_2 + 1)/(K*s*(-phi + tau_c))
K_C, tau_I, tau_D = symbols('K_C, tau_I, tau_D',
positive=True, nonzero=True)
PID = K_C*(1 + 1/(tau_I*s) + tau_D*s)
eq = (target - PID).together()
eq *= denom(eq).simplify()
eq = Poly(eq, s)
c = eq.coeffs()
vars = [K_C, tau_I, tau_D]
s = solve(c, vars, dict=True)
assert len(s) == 1
knownsolution = {K_C: -(tau_1 + tau_2)/(K*(phi - tau_c)),
tau_I: tau_1 + tau_2,
tau_D: tau_1*tau_2/(tau_1 + tau_2)}
for var in vars:
assert s[0][var].simplify() == knownsolution[var].simplify()
def test_lambert_multivariate():
from sympy.abc import x, y
assert _filtered_gens(Poly(x + 1/x + exp(x) + y), x) == {x, exp(x)}
assert _lambert(x, x) == []
assert solve((x**2 - 2*x + 1).subs(x, log(x) + 3*x)) == [LambertW(3*S.Exp1)/3]
assert solve((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1)) == \
[LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3]
assert solve((x**2 - 2*x - 2).subs(x, log(x) + 3*x)) == \
[LambertW(3*exp(1 - sqrt(3)))/3, LambertW(3*exp(1 + sqrt(3)))/3]
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solve(eq) == [LambertW(3*exp(-LambertW(3)))]
# coverage test
raises(NotImplementedError, lambda: solve(x - sin(x)*log(y - x), x))
ans = [3, -3*LambertW(-log(3)/3)/log(3)] # 3 and 2.478...
assert solve(x**3 - 3**x, x) == ans
assert set(solve(3*log(x) - x*log(3))) == set(ans)
assert solve(LambertW(2*x) - y, x) == [y*exp(y)/2]
@XFAIL
def test_other_lambert():
assert solve(3*sin(x) - x*sin(3), x) == [3]
assert set(solve(x**a - a**x), x) == {
a, -a*LambertW(-log(a)/a)/log(a)}
@slow
def test_lambert_bivariate():
# tests passing current implementation
assert solve((x**2 + x)*exp(x**2 + x) - 1) == [
Rational(-1, 2) + sqrt(1 + 4*LambertW(1))/2,
Rational(-1, 2) - sqrt(1 + 4*LambertW(1))/2]
assert solve((x**2 + x)*exp((x**2 + x)*2) - 1) == [
Rational(-1, 2) + sqrt(1 + 2*LambertW(2))/2,
Rational(-1, 2) - sqrt(1 + 2*LambertW(2))/2]
assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
assert solve((a/x + exp(x/2)).diff(x), x) == \
[4*LambertW(-sqrt(2)*sqrt(a)/4), 4*LambertW(sqrt(2)*sqrt(a)/4)]
assert solve((1/x + exp(x/2)).diff(x), x) == \
[4*LambertW(-sqrt(2)/4),
4*LambertW(sqrt(2)/4), # nsimplifies as 2*2**(141/299)*3**(206/299)*5**(205/299)*7**(37/299)/21
4*LambertW(-sqrt(2)/4, -1)]
assert solve(x*log(x) + 3*x + 1, x) == \
[exp(-3 + LambertW(-exp(3)))]
assert solve(-x**2 + 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)]
assert solve(x**2 - 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)]
ans = solve(3*x + 5 + 2**(-5*x + 3), x)
assert len(ans) == 1 and ans[0].expand() == \
Rational(-5, 3) + LambertW(-10240*root(2, 3)*log(2)/3)/(5*log(2))
assert solve(5*x - 1 + 3*exp(2 - 7*x), x) == \
[Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7]
assert solve((log(x) + x).subs(x, x**2 + 1)) == [
-I*sqrt(-LambertW(1) + 1), sqrt(-1 + LambertW(1))]
# check collection
ax = a**(3*x + 5)
ans = solve(3*log(ax) + b*log(ax) + ax, x)
x0 = 1/log(a)
x1 = sqrt(3)*I
x2 = b + 3
x3 = x2*LambertW(1/x2)/a**5
x4 = x3**Rational(1, 3)/2
assert ans == [
x0*log(x4*(-x1 - 1)),
x0*log(x4*(x1 - 1)),
x0*log(x3)/3]
x1 = LambertW(Rational(1, 3))
x2 = a**(-5)
x3 = -3**Rational(1, 3)
x4 = 3**Rational(5, 6)*I
x5 = x1**Rational(1, 3)*x2**Rational(1, 3)/2
ans = solve(3*log(ax) + ax, x)
assert ans == [
x0*log(3*x1*x2)/3,
x0*log(x5*(x3 - x4)),
x0*log(x5*(x3 + x4))]
# coverage
p = symbols('p', positive=True)
eq = 4*2**(2*p + 3) - 2*p - 3
assert _solve_lambert(eq, p, _filtered_gens(Poly(eq), p)) == [
Rational(-3, 2) - LambertW(-4*log(2))/(2*log(2))]
assert set(solve(3**cos(x) - cos(x)**3)) == {
acos(3), acos(-3*LambertW(-log(3)/3)/log(3))}
# should give only one solution after using `uniq`
assert solve(2*log(x) - 2*log(z) + log(z + log(x) + log(z)), x) == [
exp(-z + LambertW(2*z**4*exp(2*z))/2)/z]
# cases when p != S.One
# issue 4271
ans = solve((a/x + exp(x/2)).diff(x, 2), x)
x0 = (-a)**Rational(1, 3)
x1 = sqrt(3)*I
x2 = x0/6
assert ans == [
6*LambertW(x0/3),
6*LambertW(x2*(-x1 - 1)),
6*LambertW(x2*(x1 - 1))]
assert solve((1/x + exp(x/2)).diff(x, 2), x) == \
[6*LambertW(Rational(-1, 3)), 6*LambertW(Rational(1, 6) - sqrt(3)*I/6), \
6*LambertW(Rational(1, 6) + sqrt(3)*I/6), 6*LambertW(Rational(-1, 3), -1)]
assert solve(x**2 - y**2/exp(x), x, y, dict=True) == \
[{x: 2*LambertW(-y/2)}, {x: 2*LambertW(y/2)}]
# this is slow but not exceedingly slow
assert solve((x**3)**(x/2) + pi/2, x) == [
exp(LambertW(-2*log(2)/3 + 2*log(pi)/3 + I*pi*Rational(2, 3)))]
def test_rewrite_trig():
assert solve(sin(x) + tan(x)) == [0, -pi, pi, 2*pi]
assert solve(sin(x) + sec(x)) == [
-2*atan(Rational(-1, 2) + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2),
2*atan(S.Half - sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half
+ sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half -
sqrt(3)*I/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2)]
assert solve(sinh(x) + tanh(x)) == [0, I*pi]
# issue 6157
assert solve(2*sin(x) - cos(x), x) == [atan(S.Half)]
@XFAIL
def test_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy.functions.elementary.hyperbolic import sech
assert solve(sinh(x) + sech(x)) == [
2*atanh(Rational(-1, 2) + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(Rational(-1, 2) + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half)]
def test_uselogcombine():
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solve(eq, x, force=True) == [-sqrt(y*(y - exp(z))), sqrt(y*(y - exp(z)))]
assert solve(log(x + 3) + log(1 + 3/x) - 3) in [
[-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2],
[-3 + sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2,
-3 - sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2],
]
assert solve(log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)) == []
def test_atan2():
assert solve(atan2(x, 2) - pi/3, x) == [2*sqrt(3)]
def test_errorinverses():
assert solve(erf(x) - y, x) == [erfinv(y)]
assert solve(erfinv(x) - y, x) == [erf(y)]
assert solve(erfc(x) - y, x) == [erfcinv(y)]
assert solve(erfcinv(x) - y, x) == [erfc(y)]
def test_issue_2725():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solve(eq, R, set=True)[1]
assert sol == {(Rational(5, 3) + (Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3) + 40/(9*((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3))),), (Rational(5, 3) + 40/(9*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3)) + (Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3),)}
def test_issue_5114_6611():
# See that it doesn't hang; this solves in about 2 seconds.
# Also check that the solution is relatively small.
# Note: the system in issue 6611 solves in about 5 seconds and has
# an op-count of 138336 (with simplify=False).
b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('b:r')
eqs = Matrix([
[b - c/d + r/d], [c*(1/g + 1/e + 1/d) - f/g - r/d],
[-c/g + f*(1/j + 1/i + 1/g) - h/i], [-f/i + h*(1/m + 1/l + 1/i) - k/m],
[-h/m + k*(1/p + 1/o + 1/m) - n/p], [-k/p + n*(1/q + 1/p)]])
v = Matrix([f, h, k, n, b, c])
ans = solve(list(eqs), list(v), simplify=False)
# If time is taken to simplify then then 2617 below becomes
# 1168 and the time is about 50 seconds instead of 2.
assert sum([s.count_ops() for s in ans.values()]) <= 3270
def test_det_quick():
m = Matrix(3, 3, symbols('a:9'))
assert m.det() == det_quick(m) # calls det_perm
m[0, 0] = 1
assert m.det() == det_quick(m) # calls det_minor
m = Matrix(3, 3, list(range(9)))
assert m.det() == det_quick(m) # defaults to .det()
# make sure they work with Sparse
s = SparseMatrix(2, 2, (1, 2, 1, 4))
assert det_perm(s) == det_minor(s) == s.det()
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == \
[-sqrt(-b**2 + 9), sqrt(-b**2 + 9)]
a, b = symbols('a b', imaginary=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == []
def test_issue_7110():
y = -2*x**3 + 4*x**2 - 2*x + 5
assert any(ask(Q.real(i)) for i in solve(y))
def test_units():
assert solve(1/x - 1/(2*cm)) == [2*cm]
def test_issue_7547():
A, B, V = symbols('A,B,V')
eq1 = Eq(630.26*(V - 39.0)*V*(V + 39) - A + B, 0)
eq2 = Eq(B, 1.36*10**8*(V - 39))
eq3 = Eq(A, 5.75*10**5*V*(V + 39.0))
sol = Matrix(nsolve(Tuple(eq1, eq2, eq3), [A, B, V], (0, 0, 0)))
assert str(sol) == str(Matrix(
[['4442890172.68209'],
['4289299466.1432'],
['70.5389666628177']]))
def test_issue_7895():
r = symbols('r', real=True)
assert solve(sqrt(r) - 2) == [4]
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = Rational(191, 20), 3*sqrt(391)/20
ans = [(a, -b), (a, b)]
assert solve((e1, e2), (x, y)) == ans
assert solve((e1, e2/(x - a)), (x, y)) == []
# make the 2nd circle's radius be -3
e2 += 6
assert solve((e1, e2), (x, y)) == []
assert solve((e1, e2), (x, y), check=False) == ans
def test_issue_7322():
number = 5.62527e-35
assert solve(x - number, x)[0] == number
def test_nsolve():
raises(ValueError, lambda: nsolve(x, (-1, 1), method='bisect'))
raises(TypeError, lambda: nsolve((x - y + 3,x + y,z - y),(x,y,z),(-50,50)))
raises(TypeError, lambda: nsolve((x + y, x - y), (0, 1)))
@slow
def test_high_order_multivariate():
assert len(solve(a*x**3 - x + 1, x)) == 3
assert len(solve(a*x**4 - x + 1, x)) == 4
assert solve(a*x**5 - x + 1, x) == [] # incomplete solution allowed
raises(NotImplementedError, lambda:
solve(a*x**5 - x + 1, x, incomplete=False))
# result checking must always consider the denominator and CRootOf
# must be checked, too
d = x**5 - x + 1
assert solve(d*(1 + 1/d)) == [CRootOf(d + 1, i) for i in range(5)]
d = x - 1
assert solve(d*(2 + 1/d)) == [S.Half]
def test_base_0_exp_0():
assert solve(0**x - 1) == [0]
assert solve(0**(x - 2) - 1) == [2]
assert solve(S('x*(1/x**0 - x)', evaluate=False)) == \
[0, 1]
def test__simple_dens():
assert _simple_dens(1/x**0, [x]) == set()
assert _simple_dens(1/x**y, [x]) == {x**y}
assert _simple_dens(1/root(x, 3), [x]) == {x}
def test_issue_8755():
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests the use of
# keyword `composite`.
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 -
1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3
@slow
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = x, y, z
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = f1,f2,f3
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = g1,g2,g3
A = solve(F, v)
B = solve(G, v)
C = solve(G, v, manual=True)
p, q, r = [{tuple(i.evalf(2) for i in j) for j in R} for R in [A, B, C]]
assert p == q == r
@slow
def test_issue_2840_8155():
assert solve(sin(3*x) + sin(6*x)) == [
0, pi*Rational(-5, 3), pi*Rational(-4, 3), -pi, pi*Rational(-2, 3),
pi*Rational(-4, 9), -pi/3, pi*Rational(-2, 9), pi*Rational(2, 9),
pi/3, pi*Rational(4, 9), pi*Rational(2, 3), pi, pi*Rational(4, 3),
pi*Rational(14, 9), pi*Rational(5, 3), pi*Rational(16, 9), 2*pi,
-2*I*log(-(-1)**Rational(1, 9)), -2*I*log(-(-1)**Rational(2, 9)),
-2*I*log(-sin(pi/18) - I*cos(pi/18)),
-2*I*log(-sin(pi/18) + I*cos(pi/18)),
-2*I*log(sin(pi/18) - I*cos(pi/18)),
-2*I*log(sin(pi/18) + I*cos(pi/18))]
assert solve(2*sin(x) - 2*sin(2*x)) == [
0, pi*Rational(-5, 3), -pi, -pi/3, pi/3, pi, pi*Rational(5, 3)]
def test_issue_9567():
assert solve(1 + 1/(x - 1)) == [0]
def test_issue_11538():
assert solve(x + E) == [-E]
assert solve(x**2 + E) == [-I*sqrt(E), I*sqrt(E)]
assert solve(x**3 + 2*E) == [
-cbrt(2 * E),
cbrt(2)*cbrt(E)/2 - cbrt(2)*sqrt(3)*I*cbrt(E)/2,
cbrt(2)*cbrt(E)/2 + cbrt(2)*sqrt(3)*I*cbrt(E)/2]
assert solve([x + 4, y + E], x, y) == {x: -4, y: -E}
assert solve([x**2 + 4, y + E], x, y) == [
(-2*I, -E), (2*I, -E)]
e1 = x - y**3 + 4
e2 = x + y + 4 + 4 * E
assert len(solve([e1, e2], x, y)) == 3
@slow
def test_issue_12114():
a, b, c, d, e, f, g = symbols('a,b,c,d,e,f,g')
terms = [1 + a*b + d*e, 1 + a*c + d*f, 1 + b*c + e*f,
g - a**2 - d**2, g - b**2 - e**2, g - c**2 - f**2]
s = solve(terms, [a, b, c, d, e, f, g], dict=True)
assert s == [{a: -sqrt(-f**2 - 1), b: -sqrt(-f**2 - 1),
c: -sqrt(-f**2 - 1), d: f, e: f, g: -1},
{a: sqrt(-f**2 - 1), b: sqrt(-f**2 - 1),
c: sqrt(-f**2 - 1), d: f, e: f, g: -1},
{a: -sqrt(3)*f/2 - sqrt(-f**2 + 2)/2,
b: sqrt(3)*f/2 - sqrt(-f**2 + 2)/2, c: sqrt(-f**2 + 2),
d: -f/2 + sqrt(-3*f**2 + 6)/2,
e: -f/2 - sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: -sqrt(3)*f/2 + sqrt(-f**2 + 2)/2,
b: sqrt(3)*f/2 + sqrt(-f**2 + 2)/2, c: -sqrt(-f**2 + 2),
d: -f/2 - sqrt(-3*f**2 + 6)/2,
e: -f/2 + sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: sqrt(3)*f/2 - sqrt(-f**2 + 2)/2,
b: -sqrt(3)*f/2 - sqrt(-f**2 + 2)/2, c: sqrt(-f**2 + 2),
d: -f/2 - sqrt(-3*f**2 + 6)/2,
e: -f/2 + sqrt(3)*sqrt(-f**2 + 2)/2, g: 2},
{a: sqrt(3)*f/2 + sqrt(-f**2 + 2)/2,
b: -sqrt(3)*f/2 + sqrt(-f**2 + 2)/2, c: -sqrt(-f**2 + 2),
d: -f/2 + sqrt(-3*f**2 + 6)/2,
e: -f/2 - sqrt(3)*sqrt(-f**2 + 2)/2, g: 2}]
def test_inf():
assert solve(1 - oo*x) == []
assert solve(oo*x, x) == []
assert solve(oo*x - oo, x) == []
def test_issue_12448():
f = Function('f')
fun = [f(i) for i in range(15)]
sym = symbols('x:15')
reps = dict(zip(fun, sym))
(x, y, z), c = sym[:3], sym[3:]
ssym = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
(x, y, z), c = fun[:3], fun[3:]
sfun = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
assert sfun[fun[0]].xreplace(reps).count_ops() == \
ssym[sym[0]].count_ops()
def test_denoms():
assert denoms(x/2 + 1/y) == {2, y}
assert denoms(x/2 + 1/y, y) == {y}
assert denoms(x/2 + 1/y, [y]) == {y}
assert denoms(1/x + 1/y + 1/z, [x, y]) == {x, y}
assert denoms(1/x + 1/y + 1/z, x, y) == {x, y}
assert denoms(1/x + 1/y + 1/z, {x, y}) == {x, y}
def test_issue_12476():
x0, x1, x2, x3, x4, x5 = symbols('x0 x1 x2 x3 x4 x5')
eqns = [x0**2 - x0, x0*x1 - x1, x0*x2 - x2, x0*x3 - x3, x0*x4 - x4, x0*x5 - x5,
x0*x1 - x1, -x0/3 + x1**2 - 2*x2/3, x1*x2 - x1/3 - x2/3 - x3/3,
x1*x3 - x2/3 - x3/3 - x4/3, x1*x4 - 2*x3/3 - x5/3, x1*x5 - x4, x0*x2 - x2,
x1*x2 - x1/3 - x2/3 - x3/3, -x0/6 - x1/6 + x2**2 - x2/6 - x3/3 - x4/6,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6, x2*x4 - x2/3 - x3/3 - x4/3,
x2*x5 - x3, x0*x3 - x3, x1*x3 - x2/3 - x3/3 - x4/3,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6,
-x0/6 - x1/6 - x2/6 + x3**2 - x3/3 - x4/6, -x1/3 - x2/3 + x3*x4 - x3/3,
-x2 + x3*x5, x0*x4 - x4, x1*x4 - 2*x3/3 - x5/3, x2*x4 - x2/3 - x3/3 - x4/3,
-x1/3 - x2/3 + x3*x4 - x3/3, -x0/3 - 2*x2/3 + x4**2, -x1 + x4*x5, x0*x5 - x5,
x1*x5 - x4, x2*x5 - x3, -x2 + x3*x5, -x1 + x4*x5, -x0 + x5**2, x0 - 1]
sols = [{x0: 1, x3: Rational(1, 6), x2: Rational(1, 6), x4: Rational(-2, 3), x1: Rational(-2, 3), x5: 1},
{x0: 1, x3: S.Half, x2: Rational(-1, 2), x4: 0, x1: 0, x5: -1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(-1, 3), x4: Rational(1, 3), x1: Rational(1, 3), x5: 1},
{x0: 1, x3: 1, x2: 1, x4: 1, x1: 1, x5: 1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: sqrt(5)/3, x1: -sqrt(5)/3, x5: -1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: -sqrt(5)/3, x1: sqrt(5)/3, x5: -1}]
assert solve(eqns) == sols
def test_issue_13849():
t = symbols('t')
assert solve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == []
def test_issue_14860():
from sympy.physics.units import newton, kilo
assert solve(8*kilo*newton + x + y, x) == [-8000*newton - y]
def test_issue_14721():
k, h, a, b = symbols(':4')
assert solve([
-1 + (-k + 1)**2/b**2 + (-h - 1)**2/a**2,
-1 + (-k + 1)**2/b**2 + (-h + 1)**2/a**2,
h, k + 2], h, k, a, b) == [
(0, -2, -b*sqrt(1/(b**2 - 9)), b),
(0, -2, b*sqrt(1/(b**2 - 9)), b)]
assert solve([
h, h/a + 1/b**2 - 2, -h/2 + 1/b**2 - 2], a, h, b) == [
(a, 0, -sqrt(2)/2), (a, 0, sqrt(2)/2)]
assert solve((a + b**2 - 1, a + b**2 - 2)) == []
def test_issue_14779():
x = symbols('x', real=True)
assert solve(sqrt(x**4 - 130*x**2 + 1089) + sqrt(x**4 - 130*x**2
+ 3969) - 96*Abs(x)/x,x) == [sqrt(130)]
def test_issue_15307():
assert solve((y - 2, Mul(x + 3,x - 2, evaluate=False))) == \
[{x: -3, y: 2}, {x: 2, y: 2}]
assert solve((y - 2, Mul(3, x - 2, evaluate=False))) == \
{x: 2, y: 2}
assert solve((y - 2, Add(x + 4, x - 2, evaluate=False))) == \
{x: -1, y: 2}
eq1 = Eq(12513*x + 2*y - 219093, -5726*x - y)
eq2 = Eq(-2*x + 8, 2*x - 40)
assert solve([eq1, eq2]) == {x:12, y:75}
def test_issue_15415():
assert solve(x - 3, x) == [3]
assert solve([x - 3], x) == {x:3}
assert solve(Eq(y + 3*x**2/2, y + 3*x), y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x)], y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x), Eq(x, 1)], y) == []
@slow
def test_issue_15731():
# f(x)**g(x)=c
assert solve(Eq((x**2 - 7*x + 11)**(x**2 - 13*x + 42), 1)) == [2, 3, 4, 5, 6, 7]
assert solve((x)**(x + 4) - 4) == [-2]
assert solve((-x)**(-x + 4) - 4) == [2]
assert solve((x**2 - 6)**(x**2 - 2) - 4) == [-2, 2]
assert solve((x**2 - 2*x - 1)**(x**2 - 3) - 1/(1 - 2*sqrt(2))) == [sqrt(2)]
assert solve(x**(x + S.Half) - 4*sqrt(2)) == [S(2)]
assert solve((x**2 + 1)**x - 25) == [2]
assert solve(x**(2/x) - 2) == [2, 4]
assert solve((x/2)**(2/x) - sqrt(2)) == [4, 8]
assert solve(x**(x + S.Half) - Rational(9, 4)) == [Rational(3, 2)]
# a**g(x)=c
assert solve((-sqrt(sqrt(2)))**x - 2) == [4, log(2)/(log(2**Rational(1, 4)) + I*pi)]
assert solve((sqrt(2))**x - sqrt(sqrt(2))) == [S.Half]
assert solve((-sqrt(2))**x + 2*(sqrt(2))) == [3,
(3*log(2)**2 + 4*pi**2 - 4*I*pi*log(2))/(log(2)**2 + 4*pi**2)]
assert solve((sqrt(2))**x - 2*(sqrt(2))) == [3]
assert solve(I**x + 1) == [2]
assert solve((1 + I)**x - 2*I) == [2]
assert solve((sqrt(2) + sqrt(3))**x - (2*sqrt(6) + 5)**Rational(1, 3)) == [Rational(2, 3)]
# bases of both sides are equal
b = Symbol('b')
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
assert solve(b**x - b, x) == [1]
b = Symbol('b', positive=True)
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
def test_issue_10933():
assert solve(x**4 + y*(x + 0.1), x) # doesn't fail
assert solve(I*x**4 + x**3 + x**2 + 1.) # doesn't fail
def test_Abs_handling():
x = symbols('x', real=True)
assert solve(abs(x/y), x) == [0]
def test_issue_7982():
x = Symbol('x')
# Test that no exception happens
assert solve([2*x**2 + 5*x + 20 <= 0, x >= 1.5], x) is S.false
# From #8040
assert solve([x**3 - 8.08*x**2 - 56.48*x/5 - 106 >= 0, x - 1 <= 0], [x]) is S.false
def test_issue_14645():
x, y = symbols('x y')
assert solve([x*y - x - y, x*y - x - y], [x, y]) == [(y/(y - 1), y)]
def test_issue_12024():
x, y = symbols('x y')
assert solve(Piecewise((0.0, x < 0.1), (x, x >= 0.1)) - y) == \
[{y: Piecewise((0.0, x < 0.1), (x, True))}]
def test_issue_17452():
assert solve((7**x)**x + pi, x) == [-sqrt(log(pi) + I*pi)/sqrt(log(7)),
sqrt(log(pi) + I*pi)/sqrt(log(7))]
assert solve(x**(x/11) + pi/11, x) == [exp(LambertW(-11*log(11) + 11*log(pi) + 11*I*pi))]
def test_issue_17799():
assert solve(-erf(x**(S(1)/3))**pi + I, x) == []
def test_issue_17650():
x = Symbol('x', real=True)
assert solve(abs(abs(x**2 - 1) - x) - x) == [1, -1 + sqrt(2), 1 + sqrt(2)]
def test_issue_17882():
eq = -8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3))
assert unrad(eq) is None
def test_issue_17949():
assert solve(exp(+x+x**2), x) == []
assert solve(exp(-x+x**2), x) == []
assert solve(exp(+x-x**2), x) == []
assert solve(exp(-x-x**2), x) == []
def test_issue_10993():
assert solve(Eq(binomial(x, 2), 3)) == [-2, 3]
assert solve(Eq(pow(x, 2) + binomial(x, 3), x)) == [-4, 0, 1]
assert solve(Eq(binomial(x, 2), 0)) == [0, 1]
assert solve(a+binomial(x, 3), a) == [-binomial(x, 3)]
assert solve(x-binomial(a, 3) + binomial(y, 2) + sin(a), x) == [-sin(a) + binomial(a, 3) - binomial(y, 2)]
assert solve((x+1)-binomial(x+1, 3), x) == [-2, -1, 3]
def test_issue_11553():
eq1 = x + y + 1
eq2 = x + GoldenRatio
assert solve([eq1, eq2], x, y) == {x: -GoldenRatio, y: -1 + GoldenRatio}
eq3 = x + 2 + TribonacciConstant
assert solve([eq1, eq3], x, y) == {x: -2 - TribonacciConstant, y: 1 + TribonacciConstant}
def test_issue_19113_19102():
t = S(1)/3
solve(cos(x)**5-sin(x)**5)
assert solve(4*cos(x)**3 - 2*sin(x)**3) == [
atan(2**(t)), -atan(2**(t)*(1 - sqrt(3)*I)/2),
-atan(2**(t)*(1 + sqrt(3)*I)/2)]
h = S.Half
assert solve(cos(x)**2 + sin(x)) == [
2*atan(-h + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2),
-2*atan(h + sqrt(5)/2 + sqrt(2)*sqrt(1 + sqrt(5))/2),
-2*atan(-sqrt(5)/2 + h + sqrt(2)*sqrt(1 - sqrt(5))/2),
-2*atan(-sqrt(2)*sqrt(1 + sqrt(5))/2 + h + sqrt(5)/2)]
assert solve(3*cos(x) - sin(x)) == [atan(3)]
def test_issue_19509():
a = S(3)/4
b = S(5)/8
c = sqrt(5)/8
d = sqrt(5)/4
assert solve(1/(x -1)**5 - 1) == [2,
-d + a - sqrt(-b + c),
-d + a + sqrt(-b + c),
d + a - sqrt(-b - c),
d + a + sqrt(-b - c)]
def test_issue_20747():
THT, HT, DBH, dib, c0, c1, c2, c3, c4 = symbols('THT HT DBH dib c0 c1 c2 c3 c4')
f = DBH*c3 + THT*c4 + c2
rhs = 1 - ((HT - 1)/(THT - 1))**c1*(1 - exp(c0/f))
eq = dib - DBH*(c0 - f*log(rhs))
term = ((1 - exp((DBH*c0 - dib)/(DBH*(DBH*c3 + THT*c4 + c2))))
/ (1 - exp(c0/(DBH*c3 + THT*c4 + c2))))
sol = [THT*term**(1/c1) - term**(1/c1) + 1]
assert solve(eq, HT) == sol
def test_issue_20902():
f = (t / ((1 + t) ** 2))
assert solve(f.subs({t: 3 * x + 2}).diff(x) > 0, x) == (S(-1) < x) & (x < S(-1)/3)
assert solve(f.subs({t: 3 * x + 3}).diff(x) > 0, x) == (S(-4)/3 < x) & (x < S(-2)/3)
assert solve(f.subs({t: 3 * x + 4}).diff(x) > 0, x) == (S(-5)/3 < x) & (x < S(-1))
assert solve(f.subs({t: 3 * x + 2}).diff(x) > 0, x) == (S(-1) < x) & (x < S(-1)/3)
def test_issue_21034():
a = symbols('a', real=True)
system = [x - cosh(cos(4)), y - sinh(cos(a)), z - tanh(x)]
assert solve(system, x, y, z) == {x: cosh(cos(4)), z: tanh(cosh(cos(4))),
y: sinh(cos(a))}
#Constants inside hyperbolic functions should not be rewritten in terms of exp
newsystem = [(exp(x) - exp(-x)) - tanh(x)*(exp(x) + exp(-x)) + x - 5]
assert solve(newsystem, x) == {x: 5}
#If the variable of interest is present in hyperbolic function, only then
# it shouuld be rewritten in terms of exp and solved further
def test_issue_4886():
z = a*sqrt(R**2*a**2 + R**2*b**2 - c**2)/(a**2 + b**2)
t = b*c/(a**2 + b**2)
sol = [((b*(t - z) - c)/(-a), t - z), ((b*(t + z) - c)/(-a), t + z)]
assert solve([x**2 + y**2 - R**2, a*x + b*y - c], x, y) == sol
def test_issue_6819():
a, b, c, d = symbols('a b c d', positive=True)
assert solve(a*b**x - c*d**x, x) == [log(c/a)/log(b/d)]
def test_issue_17454():
x = Symbol('x')
assert solve((1 - x - I)**4, x) == [1 - I]
def test_issue_21852():
solution = [21 - 21*sqrt(2)/2]
assert solve(2*x + sqrt(2*x**2) - 21) == solution
def test_issue_21942():
eq = -d + (a*c**(1 - e) + b**(1 - e)*(1 - a))**(1/(1 - e))
sol = solve(eq, c, simplify=False, check=False)
assert sol == [(b/b**e - b/(a*b**e) + d**(1 - e)/a)**(1/(1 - e))]
def test_solver_flags():
root = solve(x**5 + x**2 - x - 1, cubics=False)
rad = solve(x**5 + x**2 - x - 1, cubics=True)
assert root != rad
def test_issue_22717():
assert solve((-y**2 + log(y**2/x) + 2, -2*x*y + 2*x/y)) == [
{y: -1, x: E}, {y: 1, x: E}]
|
ec652c0d0419b586d920ad29d14c6c3ae34427112de081de392b1e9bf93e9345 | from sympy.solvers.decompogen import decompogen, compogen
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.testing.pytest import XFAIL, raises
x, y = symbols('x y')
def test_decompogen():
assert decompogen(sin(cos(x)), x) == [sin(x), cos(x)]
assert decompogen(sin(x)**2 + sin(x) + 1, x) == [x**2 + x + 1, sin(x)]
assert decompogen(sqrt(6*x**2 - 5), x) == [sqrt(x), 6*x**2 - 5]
assert decompogen(sin(sqrt(cos(x**2 + 1))), x) == [sin(x), sqrt(x), cos(x), x**2 + 1]
assert decompogen(Abs(cos(x)**2 + 3*cos(x) - 4), x) == [Abs(x), x**2 + 3*x - 4, cos(x)]
assert decompogen(sin(x)**2 + sin(x) - sqrt(3)/2, x) == [x**2 + x - sqrt(3)/2, sin(x)]
assert decompogen(Abs(cos(y)**2 + 3*cos(x) - 4), x) == [Abs(x), 3*x + cos(y)**2 - 4, cos(x)]
assert decompogen(x, y) == [x]
assert decompogen(1, x) == [1]
assert decompogen(Max(3, x), x) == [Max(3, x)]
raises(TypeError, lambda: decompogen(x < 5, x))
u = 2*x + 3
assert decompogen(Max(sqrt(u),(u)**2), x) == [Max(sqrt(x), x**2), u]
assert decompogen(Max(u, u**2, y), x) == [Max(x, x**2, y), u]
assert decompogen(Max(sin(x), u), x) == [Max(2*x + 3, sin(x))]
def test_decompogen_poly():
assert decompogen(x**4 + 2*x**2 + 1, x) == [x**2 + 2*x + 1, x**2]
assert decompogen(x**4 + 2*x**3 - x - 1, x) == [x**2 - x - 1, x**2 + x]
@XFAIL
def test_decompogen_fails():
A = lambda x: x**2 + 2*x + 3
B = lambda x: 4*x**2 + 5*x + 6
assert decompogen(A(x*exp(x)), x) == [x**2 + 2*x + 3, x*exp(x)]
assert decompogen(A(B(x)), x) == [x**2 + 2*x + 3, 4*x**2 + 5*x + 6]
assert decompogen(A(1/x + 1/x**2), x) == [x**2 + 2*x + 3, 1/x + 1/x**2]
assert decompogen(A(1/x + 2/(x + 1)), x) == [x**2 + 2*x + 3, 1/x + 2/(x + 1)]
def test_compogen():
assert compogen([sin(x), cos(x)], x) == sin(cos(x))
assert compogen([x**2 + x + 1, sin(x)], x) == sin(x)**2 + sin(x) + 1
assert compogen([sqrt(x), 6*x**2 - 5], x) == sqrt(6*x**2 - 5)
assert compogen([sin(x), sqrt(x), cos(x), x**2 + 1], x) == sin(sqrt(
cos(x**2 + 1)))
assert compogen([Abs(x), x**2 + 3*x - 4, cos(x)], x) == Abs(cos(x)**2 +
3*cos(x) - 4)
assert compogen([x**2 + x - sqrt(3)/2, sin(x)], x) == (sin(x)**2 + sin(x) -
sqrt(3)/2)
assert compogen([Abs(x), 3*x + cos(y)**2 - 4, cos(x)], x) == \
Abs(3*cos(x) + cos(y)**2 - 4)
assert compogen([x**2 + 2*x + 1, x**2], x) == x**4 + 2*x**2 + 1
# the result is in unsimplified form
assert compogen([x**2 - x - 1, x**2 + x], x) == -x**2 - x + (x**2 + x)**2 - 1
|
f2d002b72696277ab1fbea30b71276899edf64b09c1734db161b8781713ba202 | #
# The main tests for the code in single.py are currently located in
# sympy/solvers/tests/test_ode.py
#
r"""
This File contains test functions for the individual hints used for solving ODEs.
Examples of each solver will be returned by _get_examples_ode_sol_name_of_solver.
Examples should have a key 'XFAIL' which stores the list of hints if they are
expected to fail for that hint.
Functions that are for internal use:
1) _ode_solver_test(ode_examples) - It takes a dictionary of examples returned by
_get_examples method and tests them with their respective hints.
2) _test_particular_example(our_hint, example_name) - It tests the ODE example corresponding
to the hint provided.
3) _test_all_hints(runxfail=False) - It is used to test all the examples with all the hints
currently implemented. It calls _test_all_examples_for_one_hint() which outputs whether the
given hint functions properly if it classifies the ODE example.
If runxfail flag is set to True then it will only test the examples which are expected to fail.
Everytime the ODE of a particular solver is added, _test_all_hints() is to be executed to find
the possible failures of different solver hints.
4) _test_all_examples_for_one_hint(our_hint, all_examples) - It takes hint as argument and checks
this hint against all the ODE examples and gives output as the number of ODEs matched, number
of ODEs which were solved correctly, list of ODEs which gives incorrect solution and list of
ODEs which raises exception.
"""
from sympy.core.function import (Derivative, diff)
from sympy.core.mul import Mul
from sympy.core.numbers import (E, I, Rational, pi)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, cosh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import (cbrt, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sec, sin, tan)
from sympy.functions.special.error_functions import (Ei, erfi)
from sympy.functions.special.hyper import hyper
from sympy.integrals.integrals import (Integral, integrate)
from sympy.polys.rootoftools import rootof
from sympy.core import Function, Symbol
from sympy.functions import airyai, airybi, besselj, bessely, lowergamma
from sympy.integrals.risch import NonElementaryIntegral
from sympy.solvers.ode import classify_ode, dsolve
from sympy.solvers.ode.ode import allhints, _remove_redundant_solutions
from sympy.solvers.ode.single import (FirstLinear, ODEMatchError,
SingleODEProblem, SingleODESolver, NthOrderReducible)
from sympy.solvers.ode.subscheck import checkodesol
from sympy.testing.pytest import raises, slow, ON_TRAVIS
import traceback
x = Symbol('x')
u = Symbol('u')
_u = Dummy('u')
y = Symbol('y')
f = Function('f')
g = Function('g')
C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C1:11')
hint_message = """\
Hint did not match the example {example}.
The ODE is:
{eq}.
The expected hint was
{our_hint}\
"""
expected_sol_message = """\
Different solution found from dsolve for example {example}.
The ODE is:
{eq}
The expected solution was
{sol}
What dsolve returned is:
{dsolve_sol}\
"""
checkodesol_msg = """\
solution found is not correct for example {example}.
The ODE is:
{eq}\
"""
dsol_incorrect_msg = """\
solution returned by dsolve is incorrect when using {hint}.
The ODE is:
{eq}
The expected solution was
{sol}
what dsolve returned is:
{dsolve_sol}
You can test this with:
eq = {eq}
sol = dsolve(eq, hint='{hint}')
print(sol)
print(checkodesol(eq, sol))
"""
exception_msg = """\
dsolve raised exception : {e}
when using {hint} for the example {example}
You can test this with:
from sympy.solvers.ode.tests.test_single import _test_an_example
_test_an_example('{hint}', example_name = '{example}')
The ODE is:
{eq}
\
"""
check_hint_msg = """\
Tested hint was : {hint}
Total of {matched} examples matched with this hint.
Out of which {solve} gave correct results.
Examples which gave incorrect results are {unsolve}.
Examples which raised exceptions are {exceptions}
\
"""
def _add_example_keys(func):
def inner():
solver=func()
examples=[]
for example in solver['examples']:
temp={
'eq': solver['examples'][example]['eq'],
'sol': solver['examples'][example]['sol'],
'XFAIL': solver['examples'][example].get('XFAIL', []),
'func': solver['examples'][example].get('func',solver['func']),
'example_name': example,
'slow': solver['examples'][example].get('slow', False),
'simplify_flag':solver['examples'][example].get('simplify_flag',True),
'checkodesol_XFAIL': solver['examples'][example].get('checkodesol_XFAIL', False),
'dsolve_too_slow':solver['examples'][example].get('dsolve_too_slow',False),
'checkodesol_too_slow':solver['examples'][example].get('checkodesol_too_slow',False),
'hint': solver['hint']
}
examples.append(temp)
return examples
return inner()
def _ode_solver_test(ode_examples, run_slow_test=False):
for example in ode_examples:
if ((not run_slow_test) and example['slow']) or (run_slow_test and (not example['slow'])):
continue
result = _test_particular_example(example['hint'], example, solver_flag=True)
if result['xpass_msg'] != "":
print(result['xpass_msg'])
def _test_all_hints(runxfail=False):
all_hints = list(allhints)+["default"]
all_examples = _get_all_examples()
for our_hint in all_hints:
if our_hint.endswith('_Integral') or 'series' in our_hint:
continue
_test_all_examples_for_one_hint(our_hint, all_examples, runxfail)
def _test_dummy_sol(expected_sol,dsolve_sol):
if type(dsolve_sol)==list:
return any(expected_sol.dummy_eq(sub_dsol) for sub_dsol in dsolve_sol)
else:
return expected_sol.dummy_eq(dsolve_sol)
def _test_an_example(our_hint, example_name):
all_examples = _get_all_examples()
for example in all_examples:
if example['example_name'] == example_name:
_test_particular_example(our_hint, example)
def _test_particular_example(our_hint, ode_example, solver_flag=False):
eq = ode_example['eq']
expected_sol = ode_example['sol']
example = ode_example['example_name']
xfail = our_hint in ode_example['XFAIL']
func = ode_example['func']
result = {'msg': '', 'xpass_msg': ''}
simplify_flag=ode_example['simplify_flag']
checkodesol_XFAIL = ode_example['checkodesol_XFAIL']
dsolve_too_slow = ode_example['dsolve_too_slow']
checkodesol_too_slow = ode_example['checkodesol_too_slow']
xpass = True
if solver_flag:
if our_hint not in classify_ode(eq, func):
message = hint_message.format(example=example, eq=eq, our_hint=our_hint)
raise AssertionError(message)
if our_hint in classify_ode(eq, func):
result['match_list'] = example
try:
if not (dsolve_too_slow):
dsolve_sol = dsolve(eq, func, simplify=simplify_flag,hint=our_hint)
else:
if len(expected_sol)==1:
dsolve_sol = expected_sol[0]
else:
dsolve_sol = expected_sol
except Exception as e:
dsolve_sol = []
result['exception_list'] = example
if not solver_flag:
traceback.print_exc()
result['msg'] = exception_msg.format(e=str(e), hint=our_hint, example=example, eq=eq)
if solver_flag and not xfail:
print(result['msg'])
raise
xpass = False
if solver_flag and dsolve_sol!=[]:
expect_sol_check = False
if type(dsolve_sol)==list:
for sub_sol in expected_sol:
if sub_sol.has(Dummy):
expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol)
else:
expect_sol_check = sub_sol not in dsolve_sol
if expect_sol_check:
break
else:
expect_sol_check = dsolve_sol not in expected_sol
for sub_sol in expected_sol:
if sub_sol.has(Dummy):
expect_sol_check = not _test_dummy_sol(sub_sol, dsolve_sol)
if expect_sol_check:
message = expected_sol_message.format(example=example, eq=eq, sol=expected_sol, dsolve_sol=dsolve_sol)
raise AssertionError(message)
expected_checkodesol = [(True, 0) for i in range(len(expected_sol))]
if len(expected_sol) == 1:
expected_checkodesol = (True, 0)
if not (checkodesol_too_slow and ON_TRAVIS):
if not checkodesol_XFAIL:
if checkodesol(eq, dsolve_sol, func, solve_for_func=False) != expected_checkodesol:
result['unsolve_list'] = example
xpass = False
message = dsol_incorrect_msg.format(hint=our_hint, eq=eq, sol=expected_sol,dsolve_sol=dsolve_sol)
if solver_flag:
message = checkodesol_msg.format(example=example, eq=eq)
raise AssertionError(message)
else:
result['msg'] = 'AssertionError: ' + message
if xpass and xfail:
result['xpass_msg'] = example + "is now passing for the hint" + our_hint
return result
def _test_all_examples_for_one_hint(our_hint, all_examples=[], runxfail=None):
if all_examples == []:
all_examples = _get_all_examples()
match_list, unsolve_list, exception_list = [], [], []
for ode_example in all_examples:
xfail = our_hint in ode_example['XFAIL']
if runxfail and not xfail:
continue
if xfail:
continue
result = _test_particular_example(our_hint, ode_example)
match_list += result.get('match_list',[])
unsolve_list += result.get('unsolve_list',[])
exception_list += result.get('exception_list',[])
if runxfail is not None:
msg = result['msg']
if msg!='':
print(result['msg'])
# print(result.get('xpass_msg',''))
if runxfail is None:
match_count = len(match_list)
solved = len(match_list)-len(unsolve_list)-len(exception_list)
msg = check_hint_msg.format(hint=our_hint, matched=match_count, solve=solved, unsolve=unsolve_list, exceptions=exception_list)
print(msg)
def test_SingleODESolver():
# Test that not implemented methods give NotImplementedError
# Subclasses should override these methods.
problem = SingleODEProblem(f(x).diff(x), f(x), x)
solver = SingleODESolver(problem)
raises(NotImplementedError, lambda: solver.matches())
raises(NotImplementedError, lambda: solver.get_general_solution())
raises(NotImplementedError, lambda: solver._matches())
raises(NotImplementedError, lambda: solver._get_general_solution())
# This ODE can not be solved by the FirstLinear solver. Here we test that
# it does not match and the asking for a general solution gives
# ODEMatchError
problem = SingleODEProblem(f(x).diff(x) + f(x)*f(x), f(x), x)
solver = FirstLinear(problem)
raises(ODEMatchError, lambda: solver.get_general_solution())
solver = FirstLinear(problem)
assert solver.matches() is False
#These are just test for order of ODE
problem = SingleODEProblem(f(x).diff(x) + f(x), f(x), x)
assert problem.order == 1
problem = SingleODEProblem(f(x).diff(x,4) + f(x).diff(x,2) - f(x).diff(x,3), f(x), x)
assert problem.order == 4
problem = SingleODEProblem(f(x).diff(x, 3) + f(x).diff(x, 2) - f(x)**2, f(x), x)
assert problem.is_autonomous == True
problem = SingleODEProblem(f(x).diff(x, 3) + x*f(x).diff(x, 2) - f(x)**2, f(x), x)
assert problem.is_autonomous == False
def test_linear_coefficients():
_ode_solver_test(_get_examples_ode_sol_linear_coefficients)
@slow
def test_1st_homogeneous_coeff_ode():
#These were marked as test_1st_homogeneous_coeff_corner_case
eq1 = f(x).diff(x) - f(x)/x
c1 = classify_ode(eq1, f(x))
eq2 = x*f(x).diff(x) - f(x)
c2 = classify_ode(eq2, f(x))
sdi = "1st_homogeneous_coeff_subs_dep_div_indep"
sid = "1st_homogeneous_coeff_subs_indep_div_dep"
assert sid not in c1 and sdi not in c1
assert sid not in c2 and sdi not in c2
_ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep)
_ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_best)
@slow
def test_slow_examples_1st_homogeneous_coeff_ode():
_ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep, run_slow_test=True)
_ode_solver_test(_get_examples_ode_sol_1st_homogeneous_coeff_best, run_slow_test=True)
@slow
def test_nth_linear_constant_coeff_homogeneous():
_ode_solver_test(_get_examples_ode_sol_nth_linear_constant_coeff_homogeneous)
@slow
def test_slow_examples_nth_linear_constant_coeff_homogeneous():
_ode_solver_test(_get_examples_ode_sol_nth_linear_constant_coeff_homogeneous, run_slow_test=True)
def test_Airy_equation():
_ode_solver_test(_get_examples_ode_sol_2nd_linear_airy)
@slow
def test_lie_group():
_ode_solver_test(_get_examples_ode_sol_lie_group)
@slow
def test_separable_reduced():
df = f(x).diff(x)
eq = (x / f(x))*df + tan(x**2*f(x) / (x**2*f(x) - 1))
assert classify_ode(eq) == ('factorable', 'separable_reduced', 'lie_group',
'separable_reduced_Integral')
_ode_solver_test(_get_examples_ode_sol_separable_reduced)
@slow
def test_slow_examples_separable_reduced():
_ode_solver_test(_get_examples_ode_sol_separable_reduced, run_slow_test=True)
@slow
def test_2nd_2F1_hypergeometric():
_ode_solver_test(_get_examples_ode_sol_2nd_2F1_hypergeometric)
def test_2nd_2F1_hypergeometric_integral():
eq = x*(x-1)*f(x).diff(x, 2) + (-1+ S(7)/2*x)*f(x).diff(x) + f(x)
sol = Eq(f(x), (C1 + C2*Integral(exp(Integral((1 - x/2)/(x*(x - 1)), x))/(1 -
x/2)**2, x))*exp(Integral(1/(x - 1), x)/4)*exp(-Integral(7/(x -
1), x)/4)*hyper((S(1)/2, -1), (1,), x))
assert sol == dsolve(eq, hint='2nd_hypergeometric_Integral')
assert checkodesol(eq, sol) == (True, 0)
@slow
def test_2nd_nonlinear_autonomous_conserved():
_ode_solver_test(_get_examples_ode_sol_2nd_nonlinear_autonomous_conserved)
def test_2nd_nonlinear_autonomous_conserved_integral():
eq = f(x).diff(x, 2) + asin(f(x))
actual = [Eq(Integral(1/sqrt(C1 - 2*Integral(asin(_u), _u)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(C1 - 2*Integral(asin(_u), _u)), (_u, f(x))), C2 - x)]
solved = dsolve(eq, hint='2nd_nonlinear_autonomous_conserved_Integral', simplify=False)
for a,s in zip(actual, solved):
assert a.dummy_eq(s)
# checkodesol unable to simplify solutions with f(x) in an integral equation
assert checkodesol(eq, [s.doit() for s in solved]) == [(True, 0), (True, 0)]
@slow
def test_2nd_linear_bessel_equation():
_ode_solver_test(_get_examples_ode_sol_2nd_linear_bessel)
@slow
def test_nth_algebraic():
eqn = f(x) + f(x)*f(x).diff(x)
solns = [Eq(f(x), exp(x)),
Eq(f(x), C1*exp(C2*x))]
solns_final = _remove_redundant_solutions(eqn, solns, 2, x)
assert solns_final == [Eq(f(x), C1*exp(C2*x))]
_ode_solver_test(_get_examples_ode_sol_nth_algebraic)
@slow
def test_slow_examples_nth_linear_constant_coeff_var_of_parameters():
_ode_solver_test(_get_examples_ode_sol_nth_linear_var_of_parameters, run_slow_test=True)
def test_nth_linear_constant_coeff_var_of_parameters():
_ode_solver_test(_get_examples_ode_sol_nth_linear_var_of_parameters)
@slow
def test_nth_linear_constant_coeff_variation_of_parameters__integral():
# solve_variation_of_parameters shouldn't attempt to simplify the
# Wronskian if simplify=False. If wronskian() ever gets good enough
# to simplify the result itself, this test might fail.
our_hint = 'nth_linear_constant_coeff_variation_of_parameters_Integral'
eq = f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x)
sol_simp = dsolve(eq, f(x), hint=our_hint, simplify=True)
sol_nsimp = dsolve(eq, f(x), hint=our_hint, simplify=False)
assert sol_simp != sol_nsimp
assert checkodesol(eq, sol_simp, order=5, solve_for_func=False) == (True, 0)
assert checkodesol(eq, sol_simp, order=5, solve_for_func=False) == (True, 0)
@slow
def test_slow_examples_1st_exact():
_ode_solver_test(_get_examples_ode_sol_1st_exact, run_slow_test=True)
@slow
def test_1st_exact():
_ode_solver_test(_get_examples_ode_sol_1st_exact)
def test_1st_exact_integral():
eq = cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x)
sol_1 = dsolve(eq, f(x), simplify=False, hint='1st_exact_Integral')
assert checkodesol(eq, sol_1, order=1, solve_for_func=False)
@slow
def test_slow_examples_nth_order_reducible():
_ode_solver_test(_get_examples_ode_sol_nth_order_reducible, run_slow_test=True)
@slow
def test_slow_examples_nth_linear_constant_coeff_undetermined_coefficients():
_ode_solver_test(_get_examples_ode_sol_nth_linear_undetermined_coefficients, run_slow_test=True)
@slow
def test_slow_examples_separable():
_ode_solver_test(_get_examples_ode_sol_separable, run_slow_test=True)
@slow
def test_nth_linear_constant_coeff_undetermined_coefficients():
#issue-https://github.com/sympy/sympy/issues/5787
# This test case is to show the classification of imaginary constants under
# nth_linear_constant_coeff_undetermined_coefficients
eq = Eq(diff(f(x), x), I*f(x) + S.Half - I)
our_hint = 'nth_linear_constant_coeff_undetermined_coefficients'
assert our_hint in classify_ode(eq)
_ode_solver_test(_get_examples_ode_sol_nth_linear_undetermined_coefficients)
def test_nth_order_reducible():
F = lambda eq: NthOrderReducible(SingleODEProblem(eq, f(x), x))._matches()
D = Derivative
assert F(D(y*f(x), x, y) + D(f(x), x)) == False
assert F(D(y*f(y), y, y) + D(f(y), y)) == False
assert F(f(x)*D(f(x), x) + D(f(x), x, 2))== False
assert F(D(x*f(y), y, 2) + D(u*y*f(x), x, 3)) == False # no simplification by design
assert F(D(f(y), y, 2) + D(f(y), y, 3) + D(f(x), x, 4)) == False
assert F(D(f(x), x, 2) + D(f(x), x, 3)) == True
_ode_solver_test(_get_examples_ode_sol_nth_order_reducible)
@slow
def test_separable():
_ode_solver_test(_get_examples_ode_sol_separable)
@slow
def test_factorable():
assert integrate(-asin(f(2*x)+pi), x) == -Integral(asin(pi + f(2*x)), x)
_ode_solver_test(_get_examples_ode_sol_factorable)
@slow
def test_slow_examples_factorable():
_ode_solver_test(_get_examples_ode_sol_factorable, run_slow_test=True)
def test_Riccati_special_minus2():
_ode_solver_test(_get_examples_ode_sol_riccati)
@slow
def test_1st_rational_riccati():
_ode_solver_test(_get_examples_ode_sol_1st_rational_riccati)
def test_Bernoulli():
_ode_solver_test(_get_examples_ode_sol_bernoulli)
def test_1st_linear():
_ode_solver_test(_get_examples_ode_sol_1st_linear)
def test_almost_linear():
_ode_solver_test(_get_examples_ode_sol_almost_linear)
@slow
def test_Liouville_ODE():
hint = 'Liouville'
not_Liouville1 = classify_ode(diff(f(x), x)/x + f(x)*diff(f(x), x, x)/2 -
diff(f(x), x)**2/2, f(x))
not_Liouville2 = classify_ode(diff(f(x), x)/x + diff(f(x), x, x)/2 -
x*diff(f(x), x)**2/2, f(x))
assert hint not in not_Liouville1
assert hint not in not_Liouville2
assert hint + '_Integral' not in not_Liouville1
assert hint + '_Integral' not in not_Liouville2
_ode_solver_test(_get_examples_ode_sol_liouville)
def test_nth_order_linear_euler_eq_homogeneous():
x, t, a, b, c = symbols('x t a b c')
y = Function('y')
our_hint = "nth_linear_euler_eq_homogeneous"
eq = diff(f(t), t, 4)*t**4 - 13*diff(f(t), t, 2)*t**2 + 36*f(t)
assert our_hint in classify_ode(eq)
eq = a*y(t) + b*t*diff(y(t), t) + c*t**2*diff(y(t), t, 2)
assert our_hint in classify_ode(eq)
_ode_solver_test(_get_examples_ode_sol_euler_homogeneous)
def test_nth_order_linear_euler_eq_nonhomogeneous_undetermined_coefficients():
x, t = symbols('x t')
a, b, c, d = symbols('a b c d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients"
eq = x**4*diff(f(x), x, 4) - 13*x**2*diff(f(x), x, 2) + 36*f(x) + x
assert our_hint in classify_ode(eq, f(x))
eq = a*x**2*diff(f(x), x, 2) + b*x*diff(f(x), x) + c*f(x) + d*log(x)
assert our_hint in classify_ode(eq, f(x))
_ode_solver_test(_get_examples_ode_sol_euler_undetermined_coeff)
@slow
def test_nth_order_linear_euler_eq_nonhomogeneous_variation_of_parameters():
x, t = symbols('x, t')
a, b, c, d = symbols('a, b, c, d', integer=True)
our_hint = "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters"
eq = Eq(x**2*diff(f(x),x,2) - 8*x*diff(f(x),x) + 12*f(x), x**2)
assert our_hint in classify_ode(eq, f(x))
eq = Eq(a*x**3*diff(f(x),x,3) + b*x**2*diff(f(x),x,2) + c*x*diff(f(x),x) + d*f(x), x*log(x))
assert our_hint in classify_ode(eq, f(x))
_ode_solver_test(_get_examples_ode_sol_euler_var_para)
@_add_example_keys
def _get_examples_ode_sol_euler_homogeneous():
r1, r2, r3, r4, r5 = [rootof(x**5 - 14*x**4 + 71*x**3 - 154*x**2 + 120*x - 1, n) for n in range(5)]
return {
'hint': "nth_linear_euler_eq_homogeneous",
'func': f(x),
'examples':{
'euler_hom_01': {
'eq': Eq(-3*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0),
'sol': [Eq(f(x), C1 + C2*x**Rational(5, 2))],
},
'euler_hom_02': {
'eq': Eq(3*f(x) - 5*diff(f(x), x)*x + 2*x**2*diff(f(x), x, x), 0),
'sol': [Eq(f(x), C1*sqrt(x) + C2*x**3)]
},
'euler_hom_03': {
'eq': Eq(4*f(x) + 5*diff(f(x), x)*x + x**2*diff(f(x), x, x), 0),
'sol': [Eq(f(x), (C1 + C2*log(x))/x**2)]
},
'euler_hom_04': {
'eq': Eq(6*f(x) - 6*diff(f(x), x)*x + 1*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0),
'sol': [Eq(f(x), C1/x**2 + C2*x + C3*x**3)]
},
'euler_hom_05': {
'eq': Eq(-125*f(x) + 61*diff(f(x), x)*x - 12*x**2*diff(f(x), x, x) + x**3*diff(f(x), x, x, x), 0),
'sol': [Eq(f(x), x**5*(C1 + C2*log(x) + C3*log(x)**2))]
},
'euler_hom_06': {
'eq': x**2*diff(f(x), x, 2) + x*diff(f(x), x) - 9*f(x),
'sol': [Eq(f(x), C1*x**-3 + C2*x**3)]
},
'euler_hom_07': {
'eq': sin(x)*x**2*f(x).diff(x, 2) + sin(x)*x*f(x).diff(x) + sin(x)*f(x),
'sol': [Eq(f(x), C1*sin(log(x)) + C2*cos(log(x)))],
'XFAIL': ['2nd_power_series_regular','nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients']
},
'euler_hom_08': {
'eq': x**6 * f(x).diff(x, 6) - x*f(x).diff(x) + f(x),
'sol': [Eq(f(x), C1*x + C2*x**r1 + C3*x**r2 + C4*x**r3 + C5*x**r4 + C6*x**r5)],
'checkodesol_XFAIL':True
},
#This example is from issue: https://github.com/sympy/sympy/issues/15237 #This example is from issue:
# https://github.com/sympy/sympy/issues/15237
'euler_hom_09': {
'eq': Derivative(x*f(x), x, x, x),
'sol': [Eq(f(x), C1 + C2/x + C3*x)],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_euler_undetermined_coeff():
return {
'hint': "nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients",
'func': f(x),
'examples':{
'euler_undet_01': {
'eq': Eq(x**2*diff(f(x), x, x) + x*diff(f(x), x), 1),
'sol': [Eq(f(x), C1 + C2*log(x) + log(x)**2/2)]
},
'euler_undet_02': {
'eq': Eq(x**2*diff(f(x), x, x) - 2*x*diff(f(x), x) + 2*f(x), x**3),
'sol': [Eq(f(x), x*(C1 + C2*x + Rational(1, 2)*x**2))]
},
'euler_undet_03': {
'eq': Eq(x**2*diff(f(x), x, x) - x*diff(f(x), x) - 3*f(x), log(x)/x),
'sol': [Eq(f(x), (C1 + C2*x**4 - log(x)**2/8 - log(x)/16)/x)]
},
'euler_undet_04': {
'eq': Eq(x**2*diff(f(x), x, x) + 3*x*diff(f(x), x) - 8*f(x), log(x)**3 - log(x)),
'sol': [Eq(f(x), C1/x**4 + C2*x**2 - Rational(1,8)*log(x)**3 - Rational(3,32)*log(x)**2 - Rational(1,64)*log(x) - Rational(7, 256))]
},
'euler_undet_05': {
'eq': Eq(x**3*diff(f(x), x, x, x) - 3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), log(x)),
'sol': [Eq(f(x), C1*x + C2*x**2 + C3*x**3 - Rational(1, 6)*log(x) - Rational(11, 36))]
},
#Below examples were added for the issue: https://github.com/sympy/sympy/issues/5096
'euler_undet_06': {
'eq': 2*x**2*f(x).diff(x, 2) + f(x) + sqrt(2*x)*sin(log(2*x)/2),
'sol': [Eq(f(x), sqrt(x)*(C1*sin(log(x)/2) + C2*cos(log(x)/2) + sqrt(2)*log(x)*cos(log(2*x)/2)/2))]
},
'euler_undet_07': {
'eq': 2*x**2*f(x).diff(x, 2) + f(x) + sin(log(2*x)/2),
'sol': [Eq(f(x), C1*sqrt(x)*sin(log(x)/2) + C2*sqrt(x)*cos(log(x)/2) - 2*sin(log(2*x)/2)/5 - 4*cos(log(2*x)/2)/5)]
},
}
}
@_add_example_keys
def _get_examples_ode_sol_euler_var_para():
return {
'hint': "nth_linear_euler_eq_nonhomogeneous_variation_of_parameters",
'func': f(x),
'examples':{
'euler_var_01': {
'eq': Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4),
'sol': [Eq(f(x), x*(C1 + C2*x + x**3/6))]
},
'euler_var_02': {
'eq': Eq(3*x**2*diff(f(x), x, x) + 6*x*diff(f(x), x) - 6*f(x), x**3*exp(x)),
'sol': [Eq(f(x), C1/x**2 + C2*x + x*exp(x)/3 - 4*exp(x)/3 + 8*exp(x)/(3*x) - 8*exp(x)/(3*x**2))]
},
'euler_var_03': {
'eq': Eq(x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x), x**4*exp(x)),
'sol': [Eq(f(x), x*(C1 + C2*x + x*exp(x) - 2*exp(x)))]
},
'euler_var_04': {
'eq': x**2*Derivative(f(x), x, x) - 2*x*Derivative(f(x), x) + 2*f(x) - log(x),
'sol': [Eq(f(x), C1*x + C2*x**2 + log(x)/2 + Rational(3, 4))]
},
'euler_var_05': {
'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x,
'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))]
},
'euler_var_06': {
'eq': x**2 * f(x).diff(x, 2) + x * f(x).diff(x) + 4 * f(x) - 1/x,
'sol': [Eq(f(x), C1*sin(2*log(x)) + C2*cos(2*log(x)) + 1/(5*x))]
},
}
}
@_add_example_keys
def _get_examples_ode_sol_bernoulli():
# Type: Bernoulli, f'(x) + p(x)*f(x) == q(x)*f(x)**n
return {
'hint': "Bernoulli",
'func': f(x),
'examples':{
'bernoulli_01': {
'eq': Eq(x*f(x).diff(x) + f(x) - f(x)**2, 0),
'sol': [Eq(f(x), 1/(C1*x + 1))],
'XFAIL': ['separable_reduced']
},
'bernoulli_02': {
'eq': f(x).diff(x) - y*f(x),
'sol': [Eq(f(x), C1*exp(x*y))]
},
'bernoulli_03': {
'eq': f(x)*f(x).diff(x) - 1,
'sol': [Eq(f(x), -sqrt(C1 + 2*x)), Eq(f(x), sqrt(C1 + 2*x))]
},
}
}
@_add_example_keys
def _get_examples_ode_sol_riccati():
# Type: Riccati special alpha = -2, a*dy/dx + b*y**2 + c*y/x +d/x**2
return {
'hint': "Riccati_special_minus2",
'func': f(x),
'examples':{
'riccati_01': {
'eq': 2*f(x).diff(x) + f(x)**2 - f(x)/x + 3*x**(-2),
'sol': [Eq(f(x), (-sqrt(3)*tan(C1 + sqrt(3)*log(x)/4) + 3)/(2*x))],
},
},
}
@_add_example_keys
def _get_examples_ode_sol_1st_rational_riccati():
# Type: 1st Order Rational Riccati, dy/dx = a + b*y + c*y**2,
# a, b, c are rational functions of x
return {
'hint': "1st_rational_riccati",
'func': f(x),
'examples':{
# a(x) is a constant
"rational_riccati_01": {
"eq": Eq(f(x).diff(x) + f(x)**2 - 2, 0),
"sol": [Eq(f(x), sqrt(2)*(-C1 - exp(2*sqrt(2)*x))/(C1 - exp(2*sqrt(2)*x)))]
},
# a(x) is a constant
"rational_riccati_02": {
"eq": f(x)**2 + Derivative(f(x), x) + 4*f(x)/x + 2/x**2,
"sol": [Eq(f(x), (-2*C1 - x)/(x*(C1 + x)))]
},
# a(x) is a constant
"rational_riccati_03": {
"eq": 2*x**2*Derivative(f(x), x) - x*(4*f(x) + Derivative(f(x), x) - 4) + (f(x) - 1)*f(x),
"sol": [Eq(f(x), (C1 + 2*x**2)/(C1 + x))]
},
# Constant coefficients
"rational_riccati_04": {
"eq": f(x).diff(x) - 6 - 5*f(x) - f(x)**2,
"sol": [Eq(f(x), (-2*C1 + 3*exp(x))/(C1 - exp(x)))]
},
# One pole of multiplicity 2
"rational_riccati_05": {
"eq": x**2 - (2*x + 1/x)*f(x) + f(x)**2 + Derivative(f(x), x),
"sol": [Eq(f(x), x*(C1 + x**2 + 1)/(C1 + x**2 - 1))]
},
# One pole of multiplicity 2
"rational_riccati_06": {
"eq": x**4*Derivative(f(x), x) + x**2 - x*(2*f(x)**2 + Derivative(f(x), x)) + f(x),
"sol": [Eq(f(x), x*(C1*x - x + 1)/(C1 + x**2 - 1))]
},
# Multiple poles of multiplicity 2
"rational_riccati_07": {
"eq": -f(x)**2 + Derivative(f(x), x) + (15*x**2 - 20*x + 7)/((x - 1)**2*(2*x \
- 1)**2),
"sol": [Eq(f(x), (9*C1*x - 6*C1 - 15*x**5 + 60*x**4 - 94*x**3 + 72*x**2 - \
33*x + 8)/(6*C1*x**2 - 9*C1*x + 3*C1 + 6*x**6 - 29*x**5 + 57*x**4 - \
58*x**3 + 28*x**2 - 3*x - 1))]
},
# Imaginary poles
"rational_riccati_08": {
"eq": Derivative(f(x), x) + (3*x**2 + 1)*f(x)**2/x + (6*x**2 - x + 3)*f(x)/(x*(x \
- 1)) + (3*x**2 - 2*x + 2)/(x*(x - 1)**2),
"sol": [Eq(f(x), (-C1 - x**3 + x**2 - 2*x + 1)/(C1*x - C1 + x**4 - x**3 + x**2 - \
2*x + 1))],
},
# Imaginary coefficients in equation
"rational_riccati_09": {
"eq": Derivative(f(x), x) - 2*I*(f(x)**2 + 1)/x,
"sol": [Eq(f(x), (-I*C1 + I*x**4 + I)/(C1 + x**4 - 1))]
},
# Regression: linsolve returning empty solution
# Large value of m (> 10)
"rational_riccati_10": {
"eq": Eq(Derivative(f(x), x), x*f(x)/(S(3)/2 - 2*x) + (x/2 - S(1)/3)*f(x)**2/\
(2*x/3 - S(1)/2) - S(5)/4 + (281*x**2 - 1260*x + 756)/(16*x**3 - 12*x**2)),
"sol": [Eq(f(x), (40*C1*x**14 + 28*C1*x**13 + 420*C1*x**12 + 2940*C1*x**11 + \
18480*C1*x**10 + 103950*C1*x**9 + 519750*C1*x**8 + 2286900*C1*x**7 + \
8731800*C1*x**6 + 28378350*C1*x**5 + 76403250*C1*x**4 + 163721250*C1*x**3 \
+ 261954000*C1*x**2 + 278326125*C1*x + 147349125*C1 + x*exp(2*x) - 9*exp(2*x) \
)/(x*(24*C1*x**13 + 140*C1*x**12 + 840*C1*x**11 + 4620*C1*x**10 + 23100*C1*x**9 \
+ 103950*C1*x**8 + 415800*C1*x**7 + 1455300*C1*x**6 + 4365900*C1*x**5 + \
10914750*C1*x**4 + 21829500*C1*x**3 + 32744250*C1*x**2 + 32744250*C1*x + \
16372125*C1 - exp(2*x))))]
}
}
}
@_add_example_keys
def _get_examples_ode_sol_1st_linear():
# Type: first order linear form f'(x)+p(x)f(x)=q(x)
return {
'hint': "1st_linear",
'func': f(x),
'examples':{
'linear_01': {
'eq': Eq(f(x).diff(x) + x*f(x), x**2),
'sol': [Eq(f(x), (C1 + x*exp(x**2/2)- sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2)/2)*exp(-x**2/2))],
},
},
}
@_add_example_keys
def _get_examples_ode_sol_factorable():
""" some hints are marked as xfail for examples because they missed additional algebraic solution
which could be found by Factorable hint. Fact_01 raise exception for
nth_linear_constant_coeff_undetermined_coefficients"""
y = Dummy('y')
a0,a1,a2,a3,a4 = symbols('a0, a1, a2, a3, a4')
return {
'hint': "factorable",
'func': f(x),
'examples':{
'fact_01': {
'eq': f(x) + f(x)*f(x).diff(x),
'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)],
'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep',
'lie_group', 'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters',
'nth_linear_constant_coeff_undetermined_coefficients']
},
'fact_02': {
'eq': f(x)*(f(x).diff(x)+f(x)*x+2),
'sol': [Eq(f(x), (C1 - sqrt(2)*sqrt(pi)*erfi(sqrt(2)*x/2))*exp(-x**2/2)), Eq(f(x), 0)],
'XFAIL': ['Bernoulli', '1st_linear', 'lie_group']
},
'fact_03': {
'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + x*f(x)),
'sol': [Eq(f(x), C1*airyai(-x) + C2*airybi(-x)),Eq(f(x), C1*exp(-x**3/3))]
},
'fact_04': {
'eq': (f(x).diff(x)+f(x)*x**2)*(f(x).diff(x, 2) + f(x)),
'sol': [Eq(f(x), C1*exp(-x**3/3)), Eq(f(x), C1*sin(x) + C2*cos(x))]
},
'fact_05': {
'eq': (f(x).diff(x)**2-1)*(f(x).diff(x)**2-4),
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x), Eq(f(x), C1 + 2*x), Eq(f(x), C1 - 2*x)]
},
'fact_06': {
'eq': (f(x).diff(x, 2)-exp(f(x)))*f(x).diff(x),
'sol': [
Eq(f(x), log(C1/(cos(C1*sqrt(-1/C1)*(C2 + x)) - 1))),
Eq(f(x), log(C1/(cos(C1*sqrt(-1/C1)*(C2 - x)) - 1))),
Eq(f(x), C1)
],
'slow': True,
},
'fact_07': {
'eq': (f(x).diff(x)**2-1)*(f(x)*f(x).diff(x)-1),
'sol': [Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)]
},
'fact_08': {
'eq': Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1,
'sol': [Eq(f(x), C1 - x), Eq(f(x), C1 + x)]
},
'fact_09': {
'eq': f(x)**2*Derivative(f(x), x)**6 - 2*f(x)**2*Derivative(f(x),
x)**4 + f(x)**2*Derivative(f(x), x)**2 - 2*f(x)*Derivative(f(x),
x)**5 + 4*f(x)*Derivative(f(x), x)**3 - 2*f(x)*Derivative(f(x),
x) + Derivative(f(x), x)**4 - 2*Derivative(f(x), x)**2 + 1,
'sol': [
Eq(f(x), C1 - x), Eq(f(x), -sqrt(C1 + 2*x)),
Eq(f(x), sqrt(C1 + 2*x)), Eq(f(x), C1 + x)
]
},
'fact_10': {
'eq': x**4*f(x)**2 + 2*x**4*f(x)*Derivative(f(x), (x, 2)) + x**4*Derivative(f(x),
(x, 2))**2 + 2*x**3*f(x)*Derivative(f(x), x) + 2*x**3*Derivative(f(x),
x)*Derivative(f(x), (x, 2)) - 7*x**2*f(x)**2 - 7*x**2*f(x)*Derivative(f(x),
(x, 2)) + x**2*Derivative(f(x), x)**2 - 7*x*f(x)*Derivative(f(x), x) + 12*f(x)**2,
'sol': [
Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x)),
Eq(f(x), C1*besselj(sqrt(3), x) + C2*bessely(sqrt(3), x))
],
'slow': True,
},
'fact_11': {
'eq': (f(x).diff(x, 2)-exp(f(x)))*(f(x).diff(x, 2)+exp(f(x))),
'sol': [
Eq(f(x), log(C1/(cos(C1*sqrt(-1/C1)*(C2 + x)) - 1))),
Eq(f(x), log(C1/(cos(C1*sqrt(-1/C1)*(C2 - x)) - 1))),
Eq(f(x), log(C1/(1 - cos(C1*sqrt(-1/C1)*(C2 + x))))),
Eq(f(x), log(C1/(1 - cos(C1*sqrt(-1/C1)*(C2 - x)))))
],
'dsolve_too_slow': True,
},
#Below examples were added for the issue: https://github.com/sympy/sympy/issues/15889
'fact_12': {
'eq': exp(f(x).diff(x))-f(x)**2,
'sol': [Eq(NonElementaryIntegral(1/log(y**2), (y, f(x))), C1 + x)],
'XFAIL': ['lie_group'] #It shows not implemented error for lie_group.
},
'fact_13': {
'eq': f(x).diff(x)**2 - f(x)**3,
'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))],
'XFAIL': ['lie_group'] #It shows not implemented error for lie_group.
},
'fact_14': {
'eq': f(x).diff(x)**2 - f(x),
'sol': [Eq(f(x), C1**2/4 - C1*x/2 + x**2/4)]
},
'fact_15': {
'eq': f(x).diff(x)**2 - f(x)**2,
'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))]
},
'fact_16': {
'eq': f(x).diff(x)**2 - f(x)**3,
'sol': [Eq(f(x), 4/(C1**2 - 2*C1*x + x**2))],
},
# kamke ode 1.1
'fact_17': {
'eq': f(x).diff(x)-(a4*x**4 + a3*x**3 + a2*x**2 + a1*x + a0)**(-1/2),
'sol': [Eq(f(x), C1 + Integral(1/sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4), x))],
'slow': True
},
# This is from issue: https://github.com/sympy/sympy/issues/9446
'fact_18':{
'eq': Eq(f(2 * x), sin(Derivative(f(x)))),
'sol': [Eq(f(x), C1 + Integral(pi - asin(f(2*x)), x)), Eq(f(x), C1 + Integral(asin(f(2*x)), x))],
'checkodesol_XFAIL':True
},
# This is from issue: https://github.com/sympy/sympy/issues/7093
'fact_19': {
'eq': Derivative(f(x), x)**2 - x**3,
'sol': [Eq(f(x), C1 - 2*x**Rational(5,2)/5), Eq(f(x), C1 + 2*x**Rational(5,2)/5)],
},
'fact_20': {
'eq': x*f(x).diff(x, 2) - x*f(x),
'sol': [Eq(f(x), C1*exp(-x) + C2*exp(x))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_almost_linear():
from sympy.functions.special.error_functions import Ei
A = Symbol('A', positive=True)
f = Function('f')
d = f(x).diff(x)
return {
'hint': "almost_linear",
'func': f(x),
'examples':{
'almost_lin_01': {
'eq': x**2*f(x)**2*d + f(x)**3 + 1,
'sol': [Eq(f(x), (C1*exp(3/x) - 1)**Rational(1, 3)),
Eq(f(x), (-1 - sqrt(3)*I)*(C1*exp(3/x) - 1)**Rational(1, 3)/2),
Eq(f(x), (-1 + sqrt(3)*I)*(C1*exp(3/x) - 1)**Rational(1, 3)/2)],
},
'almost_lin_02': {
'eq': x*f(x)*d + 2*x*f(x)**2 + 1,
'sol': [Eq(f(x), -sqrt((C1 - 2*Ei(4*x))*exp(-4*x))), Eq(f(x), sqrt((C1 - 2*Ei(4*x))*exp(-4*x)))]
},
'almost_lin_03': {
'eq': x*d + x*f(x) + 1,
'sol': [Eq(f(x), (C1 - Ei(x))*exp(-x))]
},
'almost_lin_04': {
'eq': x*exp(f(x))*d + exp(f(x)) + 3*x,
'sol': [Eq(f(x), log(C1/x - x*Rational(3, 2)))],
},
'almost_lin_05': {
'eq': x + A*(x + diff(f(x), x) + f(x)) + diff(f(x), x) + f(x) + 2,
'sol': [Eq(f(x), (C1 + Piecewise(
(x, Eq(A + 1, 0)), ((-A*x + A - x - 1)*exp(x)/(A + 1), True)))*exp(-x))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_liouville():
n = Symbol('n')
_y = Dummy('y')
return {
'hint': "Liouville",
'func': f(x),
'examples':{
'liouville_01': {
'eq': diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2,
'sol': [Eq(f(x), log(x/(C1 + C2*x)))],
},
'liouville_02': {
'eq': diff(x*exp(-f(x)), x, x),
'sol': [Eq(f(x), log(x/(C1 + C2*x)))]
},
'liouville_03': {
'eq': ((diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2)*exp(-f(x))/exp(f(x))).expand(),
'sol': [Eq(f(x), log(x/(C1 + C2*x)))]
},
'liouville_04': {
'eq': diff(f(x), x, x) + 1/f(x)*(diff(f(x), x))**2 + 1/x*diff(f(x), x),
'sol': [Eq(f(x), -sqrt(C1 + C2*log(x))), Eq(f(x), sqrt(C1 + C2*log(x)))],
},
'liouville_05': {
'eq': x*diff(f(x), x, x) + x/f(x)*diff(f(x), x)**2 + x*diff(f(x), x),
'sol': [Eq(f(x), -sqrt(C1 + C2*exp(-x))), Eq(f(x), sqrt(C1 + C2*exp(-x)))],
},
'liouville_06': {
'eq': Eq((x*exp(f(x))).diff(x, x), 0),
'sol': [Eq(f(x), log(C1 + C2/x))],
},
'liouville_07': {
'eq': (diff(f(x), x)/x + diff(f(x), x, x)/2 - diff(f(x), x)**2/2)*exp(-f(x))/exp(f(x)),
'sol': [Eq(f(x), log(x/(C1 + C2*x)))],
},
'liouville_08': {
'eq': x**2*diff(f(x),x) + (n*f(x) + f(x)**2)*diff(f(x),x)**2 + diff(f(x), (x, 2)),
'sol': [Eq(C1 + C2*lowergamma(Rational(1,3), x**3/3) + NonElementaryIntegral(exp(_y**3/3)*exp(_y**2*n/2), (_y, f(x))), 0)],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_nth_algebraic():
M, m, r, t = symbols('M m r t')
phi = Function('phi')
k = Symbol('k')
# This one needs a substitution f' = g.
# 'algeb_12': {
# 'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x,
# 'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))],
# },
return {
'hint': "nth_algebraic",
'func': f(x),
'examples':{
'algeb_01': {
'eq': f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1) * (f(x).diff(x) - x),
'sol': [Eq(f(x), C1 + x**2/2), Eq(f(x), C1 + C2*x)]
},
'algeb_02': {
'eq': f(x) * f(x).diff(x) * f(x).diff(x, x) * (f(x) - 1),
'sol': [Eq(f(x), C1 + C2*x)]
},
'algeb_03': {
'eq': f(x) * f(x).diff(x) * f(x).diff(x, x),
'sol': [Eq(f(x), C1 + C2*x)]
},
'algeb_04': {
'eq': Eq(-M * phi(t).diff(t),
Rational(3, 2) * m * r**2 * phi(t).diff(t) * phi(t).diff(t,t)),
'sol': [Eq(phi(t), C1), Eq(phi(t), C1 + C2*t - M*t**2/(3*m*r**2))],
'func': phi(t)
},
'algeb_05': {
'eq': (1 - sin(f(x))) * f(x).diff(x),
'sol': [Eq(f(x), C1)],
'XFAIL': ['separable'] #It raised exception.
},
'algeb_06': {
'eq': (diff(f(x)) - x)*(diff(f(x)) + x),
'sol': [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)]
},
'algeb_07': {
'eq': Eq(Derivative(f(x), x), Derivative(g(x), x)),
'sol': [Eq(f(x), C1 + g(x))],
},
'algeb_08': {
'eq': f(x).diff(x) - C1, #this example is from issue 15999
'sol': [Eq(f(x), C1*x + C2)],
},
'algeb_09': {
'eq': f(x)*f(x).diff(x),
'sol': [Eq(f(x), C1)],
},
'algeb_10': {
'eq': (diff(f(x)) - x)*(diff(f(x)) + x),
'sol': [Eq(f(x), C1 - x**2/2), Eq(f(x), C1 + x**2/2)],
},
'algeb_11': {
'eq': f(x) + f(x)*f(x).diff(x),
'sol': [Eq(f(x), 0), Eq(f(x), C1 - x)],
'XFAIL': ['separable', '1st_exact', '1st_linear', 'Bernoulli', '1st_homogeneous_coeff_best',
'1st_homogeneous_coeff_subs_indep_div_dep', '1st_homogeneous_coeff_subs_dep_div_indep',
'lie_group', 'nth_linear_constant_coeff_undetermined_coefficients',
'nth_linear_euler_eq_nonhomogeneous_undetermined_coefficients',
'nth_linear_constant_coeff_variation_of_parameters',
'nth_linear_euler_eq_nonhomogeneous_variation_of_parameters']
#nth_linear_constant_coeff_undetermined_coefficients raises exception rest all of them misses a solution.
},
'algeb_12': {
'eq': Derivative(x*f(x), x, x, x),
'sol': [Eq(f(x), (C1 + C2*x + C3*x**2) / x)],
'XFAIL': ['nth_algebraic'] # It passes only when prep=False is set in dsolve.
},
'algeb_13': {
'eq': Eq(Derivative(x*Derivative(f(x), x), x)/x, exp(x)),
'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))],
'XFAIL': ['nth_algebraic'] # It passes only when prep=False is set in dsolve.
},
# These are simple tests from the old ode module example 14-18
'algeb_14': {
'eq': Eq(f(x).diff(x), 0),
'sol': [Eq(f(x), C1)],
},
'algeb_15': {
'eq': Eq(3*f(x).diff(x) - 5, 0),
'sol': [Eq(f(x), C1 + x*Rational(5, 3))],
},
'algeb_16': {
'eq': Eq(3*f(x).diff(x), 5),
'sol': [Eq(f(x), C1 + x*Rational(5, 3))],
},
# Type: 2nd order, constant coefficients (two complex roots)
'algeb_17': {
'eq': Eq(3*f(x).diff(x) - 1, 0),
'sol': [Eq(f(x), C1 + x/3)],
},
'algeb_18': {
'eq': Eq(x*f(x).diff(x) - 1, 0),
'sol': [Eq(f(x), C1 + log(x))],
},
# https://github.com/sympy/sympy/issues/6989
'algeb_19': {
'eq': f(x).diff(x) - x*exp(-k*x),
'sol': [Eq(f(x), C1 + Piecewise(((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),(x**2/2, True)))],
},
'algeb_20': {
'eq': -f(x).diff(x) + x*exp(-k*x),
'sol': [Eq(f(x), C1 + Piecewise(((-k*x - 1)*exp(-k*x)/k**2, Ne(k**2, 0)),(x**2/2, True)))],
},
# https://github.com/sympy/sympy/issues/10867
'algeb_21': {
'eq': Eq(g(x).diff(x).diff(x), (x-2)**2 + (x-3)**3),
'sol': [Eq(g(x), C1 + C2*x + x**5/20 - 2*x**4/3 + 23*x**3/6 - 23*x**2/2)],
'func': g(x),
},
# https://github.com/sympy/sympy/issues/13691
'algeb_22': {
'eq': f(x).diff(x) - C1*g(x).diff(x),
'sol': [Eq(f(x), C2 + C1*g(x))],
'func': f(x),
},
# https://github.com/sympy/sympy/issues/4838
'algeb_23': {
'eq': f(x).diff(x) - 3*C1 - 3*x**2,
'sol': [Eq(f(x), C2 + 3*C1*x + x**3)],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_nth_order_reducible():
return {
'hint': "nth_order_reducible",
'func': f(x),
'examples':{
'reducible_01': {
'eq': Eq(x*Derivative(f(x), x)**2 + Derivative(f(x), x, 2), 0),
'sol': [Eq(f(x),C1 - sqrt(-1/C2)*log(-C2*sqrt(-1/C2) + x) +
sqrt(-1/C2)*log(C2*sqrt(-1/C2) + x))],
'slow': True,
},
'reducible_02': {
'eq': -exp(x) + (x*Derivative(f(x), (x, 2)) + Derivative(f(x), x))/x,
'sol': [Eq(f(x), C1 + C2*log(x) + exp(x) - Ei(x))],
'slow': True,
},
'reducible_03': {
'eq': Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0),
'sol': [Eq(f(x), C1 + C2*sin(2**Rational(3, 4)*x/2) + C3*cos(2**Rational(3, 4)*x/2))],
'slow': True,
},
'reducible_04': {
'eq': f(x).diff(x, 2) + 2*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-2*x))],
},
'reducible_05': {
'eq': f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))],
'slow': True,
},
'reducible_06': {
'eq': f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-2*x) + C3*exp(x) + C4*exp(2*x))],
'slow': True,
},
'reducible_07': {
'eq': f(x).diff(x, 4) + 3*f(x).diff(x, 3),
'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))],
'slow': True,
},
'reducible_08': {
'eq': f(x).diff(x, 4) - 2*f(x).diff(x, 2),
'sol': [Eq(f(x), C1 + C2*x + C3*exp(-sqrt(2)*x) + C4*exp(sqrt(2)*x))],
'slow': True,
},
'reducible_09': {
'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2),
'sol': [Eq(f(x), C1 + C2*x + C3*sin(2*x) + C4*cos(2*x))],
'slow': True,
},
'reducible_10': {
'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*x*sin(x) + C2*cos(x) - C3*x*cos(x) + C3*sin(x) + C4*sin(x) + C5*cos(x))],
'slow': True,
},
'reducible_11': {
'eq': f(x).diff(x, 2) - f(x).diff(x)**3,
'sol': [Eq(f(x), C1 - sqrt(2)*sqrt(-1/(C2 + x))*(C2 + x)),
Eq(f(x), C1 + sqrt(2)*sqrt(-1/(C2 + x))*(C2 + x))],
'slow': True,
},
# Needs to be a way to know how to combine derivatives in the expression
'reducible_12': {
'eq': Derivative(x*f(x), x, x, x) + Derivative(f(x), x, x, x),
'sol': [Eq(f(x), C1 + C3/Mul(2, (x**2 + 2*x + 1), evaluate=False) +
x*(C2 + C3/Mul(2, (x**2 + 2*x + 1), evaluate=False)))], # 2-arg Mul!
'slow': True,
},
}
}
@_add_example_keys
def _get_examples_ode_sol_nth_linear_undetermined_coefficients():
# examples 3-27 below are from Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 231
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
t = symbols("t")
u = symbols("u",cls=Function)
R, L, C, E_0, alpha = symbols("R L C E_0 alpha",positive=True)
omega = Symbol('omega')
return {
'hint': "nth_linear_constant_coeff_undetermined_coefficients",
'func': f(x),
'examples':{
'undet_01': {
'eq': c - x*g,
'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x**2/24 - 3*x/32))*exp(-x) - 1)],
'slow': True,
},
'undet_02': {
'eq': c - g,
'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x/8))*exp(-x) - 1)],
'slow': True,
},
'undet_03': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 4,
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2)],
'slow': True,
},
'undet_04': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2*exp(x))],
'slow': True,
},
'undet_05': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - exp(I*x),
'sol': [Eq(f(x), (S(3)/10 + I/10)*(C1*exp(-2*x) + C2*exp(-x) - I*exp(I*x)))],
'slow': True,
},
'undet_06': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - sin(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + sin(x)/10 - 3*cos(x)/10)],
'slow': True,
},
'undet_07': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - cos(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 3*sin(x)/10 + cos(x)/10)],
'slow': True,
},
'undet_08': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - (8 + 6*exp(x) + 2*sin(x)),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + exp(x) + sin(x)/5 - 3*cos(x)/5 + 4)],
'slow': True,
},
'undet_09': {
'eq': f2 + f(x).diff(x) + f(x) - x**2,
'sol': [Eq(f(x), -2*x + x**2 + (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(-x/2))],
'slow': True,
},
'undet_10': {
'eq': f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x),
'sol': [Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))],
'slow': True,
},
'undet_11': {
'eq': f2 - 3*f(x).diff(x) - 2*exp(2*x)*sin(x),
'sol': [Eq(f(x), C1 + C2*exp(3*x) - 3*exp(2*x)*sin(x)/5 - exp(2*x)*cos(x)/5)],
'slow': True,
},
'undet_12': {
'eq': f(x).diff(x, 4) - 2*f2 + f(x) - x + sin(x),
'sol': [Eq(f(x), x - sin(x)/4 + (C1 + C2*x)*exp(-x) + (C3 + C4*x)*exp(x))],
'slow': True,
},
'undet_13': {
'eq': f2 + f(x).diff(x) - x**2 - 2*x,
'sol': [Eq(f(x), C1 + x**3/3 + C2*exp(-x))],
'slow': True,
},
'undet_14': {
'eq': f2 + f(x).diff(x) - x - sin(2*x),
'sol': [Eq(f(x), C1 - x - sin(2*x)/5 - cos(2*x)/10 + x**2/2 + C2*exp(-x))],
'slow': True,
},
'undet_15': {
'eq': f2 + f(x) - 4*x*sin(x),
'sol': [Eq(f(x), (C1 - x**2)*cos(x) + (C2 + x)*sin(x))],
'slow': True,
},
'undet_16': {
'eq': f2 + 4*f(x) - x*sin(2*x),
'sol': [Eq(f(x), (C1 - x**2/8)*cos(2*x) + (C2 + x/16)*sin(2*x))],
'slow': True,
},
'undet_17': {
'eq': f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x),
'sol': [Eq(f(x), (C1 + x*(C2 + x**3/12))*exp(-x))],
'slow': True,
},
'undet_18': {
'eq': f(x).diff(x, 3) + 3*f2 + 3*f(x).diff(x) + f(x) - 2*exp(-x) + \
x**2*exp(-x),
'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 - x**3/60 + x/3)))*exp(-x))],
'slow': True,
},
'undet_19': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - exp(-2*x) - x**2,
'sol': [Eq(f(x), C2*exp(-x) + x**2/2 - x*Rational(3,2) + (C1 - x)*exp(-2*x) + Rational(7,4))],
'slow': True,
},
'undet_20': {
'eq': f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x),
'sol': [Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)],
'slow': True,
},
'undet_21': {
'eq': f2 + f(x).diff(x) - 6*f(x) - x - exp(2*x),
'sol': [Eq(f(x), Rational(-1, 36) - x/6 + C2*exp(-3*x) + (C1 + x/5)*exp(2*x))],
'slow': True,
},
'undet_22': {
'eq': f2 + f(x) - sin(x) - exp(-x),
'sol': [Eq(f(x), C2*sin(x) + (C1 - x/2)*cos(x) + exp(-x)/2)],
'slow': True,
},
'undet_23': {
'eq': f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x),
'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 + x/6)))*exp(x))],
'slow': True,
},
'undet_24': {
'eq': f2 + f(x) - S.Half - cos(2*x)/2,
'sol': [Eq(f(x), S.Half - cos(2*x)/6 + C1*sin(x) + C2*cos(x))],
'slow': True,
},
'undet_25': {
'eq': f(x).diff(x, 3) - f(x).diff(x) - exp(2*x)*(S.Half - cos(2*x)/2),
'sol': [Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + (-21*sin(2*x) + 27*cos(2*x) + 130)*exp(2*x)/1560)],
'slow': True,
},
#Note: 'undet_26' is referred in 'undet_37'
'undet_26': {
'eq': (f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x -
sin(x) - cos(x)),
'sol': [Eq(f(x), C1 + x**2 + (C2 + x*(C3 - x/8))*sin(x) + (C4 + x*(C5 + x/8))*cos(x))],
'slow': True,
},
'undet_27': {
'eq': f2 + f(x) - cos(x)/2 + cos(3*x)/2,
'sol': [Eq(f(x), cos(3*x)/16 + C2*cos(x) + (C1 + x/4)*sin(x))],
'slow': True,
},
'undet_28': {
'eq': f(x).diff(x) - 1,
'sol': [Eq(f(x), C1 + x)],
'slow': True,
},
# https://github.com/sympy/sympy/issues/19358
'undet_29': {
'eq': f2 + f(x).diff(x) + exp(x-C1),
'sol': [Eq(f(x), C2 + C3*exp(-x) - exp(-C1 + x)/2)],
'slow': True,
},
# https://github.com/sympy/sympy/issues/18408
'undet_30': {
'eq': f(x).diff(x, 3) - f(x).diff(x) - sinh(x),
'sol': [Eq(f(x), C1 + C2*exp(-x) + C3*exp(x) + x*sinh(x)/2)],
},
'undet_31': {
'eq': f(x).diff(x, 2) - 49*f(x) - sinh(3*x),
'sol': [Eq(f(x), C1*exp(-7*x) + C2*exp(7*x) - sinh(3*x)/40)],
},
'undet_32': {
'eq': f(x).diff(x, 3) - f(x).diff(x) - sinh(x) - exp(x),
'sol': [Eq(f(x), C1 + C3*exp(-x) + x*sinh(x)/2 + (C2 + x/2)*exp(x))],
},
# https://github.com/sympy/sympy/issues/5096
'undet_33': {
'eq': f(x).diff(x, x) + f(x) - x*sin(x - 2),
'sol': [Eq(f(x), C1*sin(x) + C2*cos(x) - x**2*cos(x - 2)/4 + x*sin(x - 2)/4)],
},
'undet_34': {
'eq': f(x).diff(x, 2) + f(x) - x**4*sin(x-1),
'sol': [ Eq(f(x), C1*sin(x) + C2*cos(x) - x**5*cos(x - 1)/10 + x**4*sin(x - 1)/4 + x**3*cos(x - 1)/2 - 3*x**2*sin(x - 1)/4 - 3*x*cos(x - 1)/4)],
},
'undet_35': {
'eq': f(x).diff(x, 2) - f(x) - exp(x - 1),
'sol': [Eq(f(x), C2*exp(-x) + (C1 + x*exp(-1)/2)*exp(x))],
},
'undet_36': {
'eq': f(x).diff(x, 2)+f(x)-(sin(x-2)+1),
'sol': [Eq(f(x), C1*sin(x) + C2*cos(x) - x*cos(x - 2)/2 + 1)],
},
# Equivalent to example_name 'undet_26'.
# This previously failed because the algorithm for undetermined coefficients
# didn't know to multiply exp(I*x) by sufficient x because it is linearly
# dependent on sin(x) and cos(x).
'undet_37': {
'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x),
'sol': [Eq(f(x), C1 + x**2*(I*exp(I*x)/8 + 1) + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))],
},
# https://github.com/sympy/sympy/issues/12623
'undet_38': {
'eq': Eq( u(t).diff(t,t) + R /L*u(t).diff(t) + 1/(L*C)*u(t), alpha),
'sol': [Eq(u(t), C*L*alpha + C2*exp(-t*(R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L))
+ C1*exp(t*(-R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L)))],
'func': u(t)
},
'undet_39': {
'eq': Eq( L*C*u(t).diff(t,t) + R*C*u(t).diff(t) + u(t), E_0*exp(I*omega*t) ),
'sol': [Eq(u(t), C2*exp(-t*(R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L))
+ C1*exp(t*(-R + sqrt(C*R**2 - 4*L)/sqrt(C))/(2*L))
- E_0*exp(I*omega*t)/(C*L*omega**2 - I*C*R*omega - 1))],
'func': u(t),
},
# https://github.com/sympy/sympy/issues/6879
'undet_40': {
'eq': Eq(Derivative(f(x), x, 2) - 2*Derivative(f(x), x) + f(x), sin(x)),
'sol': [Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_separable():
# test_separable1-5 are from Ordinary Differential Equations, Tenenbaum and
# Pollard, pg. 55
t,a = symbols('a,t')
m = 96
g = 9.8
k = .2
f1 = g * m
v = Function('v')
return {
'hint': "separable",
'func': f(x),
'examples':{
'separable_01': {
'eq': f(x).diff(x) - f(x),
'sol': [Eq(f(x), C1*exp(x))],
},
'separable_02': {
'eq': x*f(x).diff(x) - f(x),
'sol': [Eq(f(x), C1*x)],
},
'separable_03': {
'eq': f(x).diff(x) + sin(x),
'sol': [Eq(f(x), C1 + cos(x))],
},
'separable_04': {
'eq': f(x)**2 + 1 - (x**2 + 1)*f(x).diff(x),
'sol': [Eq(f(x), tan(C1 + atan(x)))],
},
'separable_05': {
'eq': f(x).diff(x)/tan(x) - f(x) - 2,
'sol': [Eq(f(x), C1/cos(x) - 2)],
},
'separable_06': {
'eq': f(x).diff(x) * (1 - sin(f(x))) - 1,
'sol': [Eq(-x + f(x) + cos(f(x)), C1)],
},
'separable_07': {
'eq': f(x)*x**2*f(x).diff(x) - f(x)**3 - 2*x**2*f(x).diff(x),
'sol': [Eq(f(x), (-x - sqrt(x*(4*C1*x + x - 4)))/(C1*x - 1)/2),
Eq(f(x), (-x + sqrt(x*(4*C1*x + x - 4)))/(C1*x - 1)/2)],
'slow': True,
},
'separable_08': {
'eq': f(x)**2 - 1 - (2*f(x) + x*f(x))*f(x).diff(x),
'sol': [Eq(f(x), -sqrt(C1*x**2 + 4*C1*x + 4*C1 + 1)),
Eq(f(x), sqrt(C1*x**2 + 4*C1*x + 4*C1 + 1))],
'slow': True,
},
'separable_09': {
'eq': x*log(x)*f(x).diff(x) + sqrt(1 + f(x)**2),
'sol': [Eq(f(x), sinh(C1 - log(log(x))))], #One more solution is f(x)=I
'slow': True,
'checkodesol_XFAIL': True,
},
'separable_10': {
'eq': exp(x + 1)*tan(f(x)) + cos(f(x))*f(x).diff(x),
'sol': [Eq(E*exp(x) + log(cos(f(x)) - 1)/2 - log(cos(f(x)) + 1)/2 + cos(f(x)), C1)],
'slow': True,
},
'separable_11': {
'eq': (x*cos(f(x)) + x**2*sin(f(x))*f(x).diff(x) - a**2*sin(f(x))*f(x).diff(x)),
'sol': [
Eq(f(x), -acos(C1*sqrt(-a**2 + x**2)) + 2*pi),
Eq(f(x), acos(C1*sqrt(-a**2 + x**2)))
],
'slow': True,
},
'separable_12': {
'eq': f(x).diff(x) - f(x)*tan(x),
'sol': [Eq(f(x), C1/cos(x))],
},
'separable_13': {
'eq': (x - 1)*cos(f(x))*f(x).diff(x) - 2*x*sin(f(x)),
'sol': [
Eq(f(x), pi - asin(C1*(x**2 - 2*x + 1)*exp(2*x))),
Eq(f(x), asin(C1*(x**2 - 2*x + 1)*exp(2*x)))
],
},
'separable_14': {
'eq': f(x).diff(x) - f(x)*log(f(x))/tan(x),
'sol': [Eq(f(x), exp(C1*sin(x)))],
},
'separable_15': {
'eq': x*f(x).diff(x) + (1 + f(x)**2)*atan(f(x)),
'sol': [Eq(f(x), tan(C1/x))], #Two more solutions are f(x)=0 and f(x)=I
'slow': True,
'checkodesol_XFAIL': True,
},
'separable_16': {
'eq': f(x).diff(x) + x*(f(x) + 1),
'sol': [Eq(f(x), -1 + C1*exp(-x**2/2))],
},
'separable_17': {
'eq': exp(f(x)**2)*(x**2 + 2*x + 1) + (x*f(x) + f(x))*f(x).diff(x),
'sol': [
Eq(f(x), -sqrt(log(1/(C1 + x**2 + 2*x)))),
Eq(f(x), sqrt(log(1/(C1 + x**2 + 2*x))))
],
},
'separable_18': {
'eq': f(x).diff(x) + f(x),
'sol': [Eq(f(x), C1*exp(-x))],
},
'separable_19': {
'eq': sin(x)*cos(2*f(x)) + cos(x)*sin(2*f(x))*f(x).diff(x),
'sol': [Eq(f(x), pi - acos(C1/cos(x)**2)/2), Eq(f(x), acos(C1/cos(x)**2)/2)],
},
'separable_20': {
'eq': (1 - x)*f(x).diff(x) - x*(f(x) + 1),
'sol': [Eq(f(x), (C1*exp(-x) - x + 1)/(x - 1))],
},
'separable_21': {
'eq': f(x)*diff(f(x), x) + x - 3*x*f(x)**2,
'sol': [Eq(f(x), -sqrt(3)*sqrt(C1*exp(3*x**2) + 1)/3),
Eq(f(x), sqrt(3)*sqrt(C1*exp(3*x**2) + 1)/3)],
},
'separable_22': {
'eq': f(x).diff(x) - exp(x + f(x)),
'sol': [Eq(f(x), log(-1/(C1 + exp(x))))],
'XFAIL': ['lie_group'] #It shows 'NoneType' object is not subscriptable for lie_group.
},
# https://github.com/sympy/sympy/issues/7081
'separable_23': {
'eq': x*(f(x).diff(x)) + 1 - f(x)**2,
'sol': [Eq(f(x), (-C1 - x**2)/(-C1 + x**2))],
},
# https://github.com/sympy/sympy/issues/10379
'separable_24': {
'eq': f(t).diff(t)-(1-51.05*y*f(t)),
'sol': [Eq(f(t), (0.019588638589618023*exp(y*(C1 - 51.049999999999997*t)) + 0.019588638589618023)/y)],
'func': f(t),
},
# https://github.com/sympy/sympy/issues/15999
'separable_25': {
'eq': f(x).diff(x) - C1*f(x),
'sol': [Eq(f(x), C2*exp(C1*x))],
},
'separable_26': {
'eq': f1 - k * (v(t) ** 2) - m * Derivative(v(t)),
'sol': [Eq(v(t), -68.585712797928991/tanh(C1 - 0.14288690166235204*t))],
'func': v(t),
'checkodesol_XFAIL': True,
},
#https://github.com/sympy/sympy/issues/22155
'separable_27': {
'eq': f(x).diff(x) - exp(f(x) - x),
'sol': [Eq(f(x), log(-exp(x)/(C1*exp(x) - 1)))],
}
}
}
@_add_example_keys
def _get_examples_ode_sol_1st_exact():
# Type: Exact differential equation, p(x,f) + q(x,f)*f' == 0,
# where dp/df == dq/dx
'''
Example 7 is an exact equation that fails under the exact engine. It is caught
by first order homogeneous albeit with a much contorted solution. The
exact engine fails because of a poorly simplified integral of q(0,y)dy,
where q is the function multiplying f'. The solutions should be
Eq(sqrt(x**2+f(x)**2)**3+y**3, C1). The equation below is
equivalent, but it is so complex that checkodesol fails, and takes a long
time to do so.
'''
return {
'hint': "1st_exact",
'func': f(x),
'examples':{
'1st_exact_01': {
'eq': sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x),
'sol': [Eq(f(x), -acos(C1/cos(x)) + 2*pi), Eq(f(x), acos(C1/cos(x)))],
'slow': True,
},
'1st_exact_02': {
'eq': (2*x*f(x) + 1)/f(x) + (f(x) - x)/f(x)**2*f(x).diff(x),
'sol': [Eq(f(x), exp(C1 - x**2 + LambertW(-x*exp(-C1 + x**2))))],
'XFAIL': ['lie_group'], #It shows dsolve raises an exception: List index out of range for lie_group
'slow': True,
'checkodesol_XFAIL':True
},
'1st_exact_03': {
'eq': 2*x + f(x)*cos(x) + (2*f(x) + sin(x) - sin(f(x)))*f(x).diff(x),
'sol': [Eq(f(x)*sin(x) + cos(f(x)) + x**2 + f(x)**2, C1)],
'XFAIL': ['lie_group'], #It goes into infinite loop for lie_group.
'slow': True,
},
'1st_exact_04': {
'eq': cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x),
'sol': [Eq(x*cos(f(x)) + f(x)**3/3, C1)],
'slow': True,
},
'1st_exact_05': {
'eq': 2*x*f(x) + (x**2 + f(x)**2)*f(x).diff(x),
'sol': [Eq(x**2*f(x) + f(x)**3/3, C1)],
'slow': True,
'simplify_flag':False
},
# This was from issue: https://github.com/sympy/sympy/issues/11290
'1st_exact_06': {
'eq': cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x),
'sol': [Eq(x*cos(f(x)) + f(x)**3/3, C1)],
'simplify_flag':False
},
'1st_exact_07': {
'eq': x*sqrt(x**2 + f(x)**2) - (x**2*f(x)/(f(x) - sqrt(x**2 + f(x)**2)))*f(x).diff(x),
'sol': [Eq(log(x),
C1 - 9*sqrt(1 + f(x)**2/x**2)*asinh(f(x)/x)/(-27*f(x)/x +
27*sqrt(1 + f(x)**2/x**2)) - 9*sqrt(1 + f(x)**2/x**2)*
log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2)) +
9*asinh(f(x)/x)*f(x)/(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))) +
9*f(x)*log(1 - sqrt(1 + f(x)**2/x**2)*f(x)/x + 2*f(x)**2/x**2)/
(x*(-27*f(x)/x + 27*sqrt(1 + f(x)**2/x**2))))],
'slow': True,
'dsolve_too_slow':True
},
# Type: a(x)f'(x)+b(x)*f(x)+c(x)=0
'1st_exact_08': {
'eq': Eq(x**2*f(x).diff(x) + 3*x*f(x) - sin(x)/x, 0),
'sol': [Eq(f(x), (C1 - cos(x))/x**3)],
},
# these examples are from test_exact_enhancement
'1st_exact_09': {
'eq': f(x)/x**2 + ((f(x)*x - 1)/x)*f(x).diff(x),
'sol': [Eq(f(x), (i*sqrt(C1*x**2 + 1) + 1)/x) for i in (-1, 1)],
},
'1st_exact_10': {
'eq': (x*f(x) - 1) + f(x).diff(x)*(x**2 - x*f(x)),
'sol': [Eq(f(x), x - sqrt(C1 + x**2 - 2*log(x))), Eq(f(x), x + sqrt(C1 + x**2 - 2*log(x)))],
},
'1st_exact_11': {
'eq': (x + 2)*sin(f(x)) + f(x).diff(x)*x*cos(f(x)),
'sol': [Eq(f(x), -asin(C1*exp(-x)/x**2) + pi), Eq(f(x), asin(C1*exp(-x)/x**2))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_nth_linear_var_of_parameters():
g = exp(-x)
f2 = f(x).diff(x, 2)
c = 3*f(x).diff(x, 3) + 5*f2 + f(x).diff(x) - f(x) - x
return {
'hint': "nth_linear_constant_coeff_variation_of_parameters",
'func': f(x),
'examples':{
'var_of_parameters_01': {
'eq': c - x*g,
'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x**2/24 - 3*x/32))*exp(-x) - 1)],
'slow': True,
},
'var_of_parameters_02': {
'eq': c - g,
'sol': [Eq(f(x), C3*exp(x/3) - x + (C1 + x*(C2 - x/8))*exp(-x) - 1)],
'slow': True,
},
'var_of_parameters_03': {
'eq': f(x).diff(x) - 1,
'sol': [Eq(f(x), C1 + x)],
'slow': True,
},
'var_of_parameters_04': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 4,
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2)],
'slow': True,
},
'var_of_parameters_05': {
'eq': f2 + 3*f(x).diff(x) + 2*f(x) - 12*exp(x),
'sol': [Eq(f(x), C1*exp(-2*x) + C2*exp(-x) + 2*exp(x))],
'slow': True,
},
'var_of_parameters_06': {
'eq': f2 - 2*f(x).diff(x) - 8*f(x) - 9*x*exp(x) - 10*exp(-x),
'sol': [Eq(f(x), -x*exp(x) - 2*exp(-x) + C1*exp(-2*x) + C2*exp(4*x))],
'slow': True,
},
'var_of_parameters_07': {
'eq': f2 + 2*f(x).diff(x) + f(x) - x**2*exp(-x),
'sol': [Eq(f(x), (C1 + x*(C2 + x**3/12))*exp(-x))],
'slow': True,
},
'var_of_parameters_08': {
'eq': f2 - 3*f(x).diff(x) + 2*f(x) - x*exp(-x),
'sol': [Eq(f(x), C1*exp(x) + C2*exp(2*x) + (6*x + 5)*exp(-x)/36)],
'slow': True,
},
'var_of_parameters_09': {
'eq': f(x).diff(x, 3) - 3*f2 + 3*f(x).diff(x) - f(x) - exp(x),
'sol': [Eq(f(x), (C1 + x*(C2 + x*(C3 + x/6)))*exp(x))],
'slow': True,
},
'var_of_parameters_10': {
'eq': f2 + 2*f(x).diff(x) + f(x) - exp(-x)/x,
'sol': [Eq(f(x), (C1 + x*(C2 + log(x)))*exp(-x))],
'slow': True,
},
'var_of_parameters_11': {
'eq': f2 + f(x) - 1/sin(x)*1/cos(x),
'sol': [Eq(f(x), (C1 + log(sin(x) - 1)/2 - log(sin(x) + 1)/2
)*cos(x) + (C2 + log(cos(x) - 1)/2 - log(cos(x) + 1)/2)*sin(x))],
'slow': True,
},
'var_of_parameters_12': {
'eq': f(x).diff(x, 4) - 1/x,
'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + x**3*(C4 + log(x)/6))],
'slow': True,
},
# These were from issue: https://github.com/sympy/sympy/issues/15996
'var_of_parameters_13': {
'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - 2*x - exp(I*x),
'sol': [Eq(f(x), C1 + x**2 + (C2 + x*(C3 - x/8 + 3*exp(I*x)/2 + 3*exp(-I*x)/2) + 5*exp(2*I*x)/16 + 2*I*exp(I*x) - 2*I*exp(-I*x))*sin(x) + (C4 + x*(C5 + I*x/8 + 3*I*exp(I*x)/2 - 3*I*exp(-I*x)/2)
+ 5*I*exp(2*I*x)/16 - 2*exp(I*x) - 2*exp(-I*x))*cos(x) - I*exp(I*x))],
},
'var_of_parameters_14': {
'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x) - exp(I*x),
'sol': [Eq(f(x), C1 + (C2 + x*(C3 - x/8) + 5*exp(2*I*x)/16)*sin(x) + (C4 + x*(C5 + I*x/8) + 5*I*exp(2*I*x)/16)*cos(x) - I*exp(I*x))],
},
# https://github.com/sympy/sympy/issues/14395
'var_of_parameters_15': {
'eq': Derivative(f(x), x, x) + 9*f(x) - sec(x),
'sol': [Eq(f(x), (C1 - x/3 + sin(2*x)/3)*sin(3*x) + (C2 + log(cos(x))
- 2*log(cos(x)**2)/3 + 2*cos(x)**2/3)*cos(3*x))],
'slow': True,
},
}
}
@_add_example_keys
def _get_examples_ode_sol_2nd_linear_bessel():
return {
'hint': "2nd_linear_bessel",
'func': f(x),
'examples':{
'2nd_lin_bessel_01': {
'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - 4)*f(x),
'sol': [Eq(f(x), C1*besselj(2, x) + C2*bessely(2, x))],
},
'2nd_lin_bessel_02': {
'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 +25)*f(x),
'sol': [Eq(f(x), C1*besselj(5*I, x) + C2*bessely(5*I, x))],
},
'2nd_lin_bessel_03': {
'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2)*f(x),
'sol': [Eq(f(x), C1*besselj(0, x) + C2*bessely(0, x))],
},
'2nd_lin_bessel_04': {
'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (81*x**2 -S(1)/9)*f(x),
'sol': [Eq(f(x), C1*besselj(S(1)/3, 9*x) + C2*bessely(S(1)/3, 9*x))],
},
'2nd_lin_bessel_05': {
'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**4 - 4)*f(x),
'sol': [Eq(f(x), C1*besselj(1, x**2/2) + C2*bessely(1, x**2/2))],
},
'2nd_lin_bessel_06': {
'eq': x**2*(f(x).diff(x, 2)) + 2*x*(f(x).diff(x)) + (x**4 - 4)*f(x),
'sol': [Eq(f(x), (C1*besselj(sqrt(17)/4, x**2/2) + C2*bessely(sqrt(17)/4, x**2/2))/sqrt(x))],
},
'2nd_lin_bessel_07': {
'eq': x**2*(f(x).diff(x, 2)) + x*(f(x).diff(x)) + (x**2 - S(1)/4)*f(x),
'sol': [Eq(f(x), C1*besselj(S(1)/2, x) + C2*bessely(S(1)/2, x))],
},
'2nd_lin_bessel_08': {
'eq': x**2*(f(x).diff(x, 2)) - 3*x*(f(x).diff(x)) + (4*x + 4)*f(x),
'sol': [Eq(f(x), x**2*(C1*besselj(0, 4*sqrt(x)) + C2*bessely(0, 4*sqrt(x))))],
},
'2nd_lin_bessel_09': {
'eq': x*(f(x).diff(x, 2)) - f(x).diff(x) + 4*x**3*f(x),
'sol': [Eq(f(x), x*(C1*besselj(S(1)/2, x**2) + C2*bessely(S(1)/2, x**2)))],
},
'2nd_lin_bessel_10': {
'eq': (x-2)**2*(f(x).diff(x, 2)) - (x-2)*f(x).diff(x) + 4*(x-2)**2*f(x),
'sol': [Eq(f(x), (x - 2)*(C1*besselj(1, 2*x - 4) + C2*bessely(1, 2*x - 4)))],
},
# https://github.com/sympy/sympy/issues/4414
'2nd_lin_bessel_11': {
'eq': f(x).diff(x, x) + 2/x*f(x).diff(x) + f(x),
'sol': [Eq(f(x), (C1*besselj(S(1)/2, x) + C2*bessely(S(1)/2, x))/sqrt(x))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_2nd_2F1_hypergeometric():
return {
'hint': "2nd_hypergeometric",
'func': f(x),
'examples':{
'2nd_2F1_hyper_01': {
'eq': x*(x-1)*f(x).diff(x, 2) + (S(3)/2 -2*x)*f(x).diff(x) + 2*f(x),
'sol': [Eq(f(x), C1*x**(S(5)/2)*hyper((S(3)/2, S(1)/2), (S(7)/2,), x) + C2*hyper((-1, -2), (-S(3)/2,), x))],
},
'2nd_2F1_hyper_02': {
'eq': x*(x-1)*f(x).diff(x, 2) + (S(7)/2*x)*f(x).diff(x) + f(x),
'sol': [Eq(f(x), (C1*(1 - x)**(S(5)/2)*hyper((S(1)/2, 2), (S(7)/2,), 1 - x) +
C2*hyper((-S(1)/2, -2), (-S(3)/2,), 1 - x))/(x - 1)**(S(5)/2))],
},
'2nd_2F1_hyper_03': {
'eq': x*(x-1)*f(x).diff(x, 2) + (S(3)+ S(7)/2*x)*f(x).diff(x) + f(x),
'sol': [Eq(f(x), (C1*(1 - x)**(S(11)/2)*hyper((S(1)/2, 2), (S(13)/2,), 1 - x) +
C2*hyper((-S(7)/2, -5), (-S(9)/2,), 1 - x))/(x - 1)**(S(11)/2))],
},
'2nd_2F1_hyper_04': {
'eq': -x**(S(5)/7)*(-416*x**(S(9)/7)/9 - 2385*x**(S(5)/7)/49 + S(298)*x/3)*f(x)/(196*(-x**(S(6)/7) +
x)**2*(x**(S(6)/7) + x)**2) + Derivative(f(x), (x, 2)),
'sol': [Eq(f(x), x**(S(45)/98)*(C1*x**(S(4)/49)*hyper((S(1)/3, -S(1)/2), (S(9)/7,), x**(S(2)/7)) +
C2*hyper((S(1)/21, -S(11)/14), (S(5)/7,), x**(S(2)/7)))/(x**(S(2)/7) - 1)**(S(19)/84))],
'checkodesol_XFAIL':True,
},
}
}
@_add_example_keys
def _get_examples_ode_sol_2nd_nonlinear_autonomous_conserved():
return {
'hint': "2nd_nonlinear_autonomous_conserved",
'func': f(x),
'examples': {
'2nd_nonlinear_autonomous_conserved_01': {
'eq': f(x).diff(x, 2) + exp(f(x)) + log(f(x)),
'sol': [
Eq(Integral(1/sqrt(C1 - 2*_u*log(_u) + 2*_u - 2*exp(_u)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(C1 - 2*_u*log(_u) + 2*_u - 2*exp(_u)), (_u, f(x))), C2 - x)
],
'simplify_flag': False,
},
'2nd_nonlinear_autonomous_conserved_02': {
'eq': f(x).diff(x, 2) + cbrt(f(x)) + 1/f(x),
'sol': [
Eq(sqrt(2)*Integral(1/sqrt(2*C1 - 3*_u**Rational(4, 3) - 4*log(_u)), (_u, f(x))), C2 + x),
Eq(sqrt(2)*Integral(1/sqrt(2*C1 - 3*_u**Rational(4, 3) - 4*log(_u)), (_u, f(x))), C2 - x)
],
'simplify_flag': False,
},
'2nd_nonlinear_autonomous_conserved_03': {
'eq': f(x).diff(x, 2) + sin(f(x)),
'sol': [
Eq(Integral(1/sqrt(C1 + 2*cos(_u)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(C1 + 2*cos(_u)), (_u, f(x))), C2 - x)
],
'simplify_flag': False,
},
'2nd_nonlinear_autonomous_conserved_04': {
'eq': f(x).diff(x, 2) + cosh(f(x)),
'sol': [
Eq(Integral(1/sqrt(C1 - 2*sinh(_u)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(C1 - 2*sinh(_u)), (_u, f(x))), C2 - x)
],
'simplify_flag': False,
},
'2nd_nonlinear_autonomous_conserved_05': {
'eq': f(x).diff(x, 2) + asin(f(x)),
'sol': [
Eq(Integral(1/sqrt(C1 - 2*_u*asin(_u) - 2*sqrt(1 - _u**2)), (_u, f(x))), C2 + x),
Eq(Integral(1/sqrt(C1 - 2*_u*asin(_u) - 2*sqrt(1 - _u**2)), (_u, f(x))), C2 - x)
],
'simplify_flag': False,
'XFAIL': ['2nd_nonlinear_autonomous_conserved_Integral']
}
}
}
@_add_example_keys
def _get_examples_ode_sol_separable_reduced():
df = f(x).diff(x)
return {
'hint': "separable_reduced",
'func': f(x),
'examples':{
'separable_reduced_01': {
'eq': x* df + f(x)* (1 / (x**2*f(x) - 1)),
'sol': [Eq(log(x**2*f(x))/3 + log(x**2*f(x) - Rational(3, 2))/6, C1 + log(x))],
'simplify_flag': False,
'XFAIL': ['lie_group'], #It hangs.
},
#Note: 'separable_reduced_02' is referred in 'separable_reduced_11'
'separable_reduced_02': {
'eq': f(x).diff(x) + (f(x) / (x**4*f(x) - x)),
'sol': [Eq(log(x**3*f(x))/4 + log(x**3*f(x) - Rational(4,3))/12, C1 + log(x))],
'simplify_flag': False,
'checkodesol_XFAIL':True, #It hangs for this.
},
'separable_reduced_03': {
'eq': x*df + f(x)*(x**2*f(x)),
'sol': [Eq(log(x**2*f(x))/2 - log(x**2*f(x) - 2)/2, C1 + log(x))],
'simplify_flag': False,
},
'separable_reduced_04': {
'eq': Eq(f(x).diff(x) + f(x)/x * (1 + (x**(S(2)/3)*f(x))**2), 0),
'sol': [Eq(-3*log(x**(S(2)/3)*f(x)) + 3*log(3*x**(S(4)/3)*f(x)**2 + 1)/2, C1 + log(x))],
'simplify_flag': False,
},
'separable_reduced_05': {
'eq': Eq(f(x).diff(x) + f(x)/x * (1 + (x*f(x))**2), 0),
'sol': [Eq(f(x), -sqrt(2)*sqrt(1/(C1 + log(x)))/(2*x)),\
Eq(f(x), sqrt(2)*sqrt(1/(C1 + log(x)))/(2*x))],
},
'separable_reduced_06': {
'eq': Eq(f(x).diff(x) + (x**4*f(x)**2 + x**2*f(x))*f(x)/(x*(x**6*f(x)**3 + x**4*f(x)**2)), 0),
'sol': [Eq(f(x), C1 + 1/(2*x**2))],
},
'separable_reduced_07': {
'eq': Eq(f(x).diff(x) + (f(x)**2)*f(x)/(x), 0),
'sol': [
Eq(f(x), -sqrt(2)*sqrt(1/(C1 + log(x)))/2),
Eq(f(x), sqrt(2)*sqrt(1/(C1 + log(x)))/2)
],
},
'separable_reduced_08': {
'eq': Eq(f(x).diff(x) + (f(x)+3)*f(x)/(x*(f(x)+2)), 0),
'sol': [Eq(-log(f(x) + 3)/3 - 2*log(f(x))/3, C1 + log(x))],
'simplify_flag': False,
'XFAIL': ['lie_group'], #It hangs.
},
'separable_reduced_09': {
'eq': Eq(f(x).diff(x) + (f(x)+3)*f(x)/x, 0),
'sol': [Eq(f(x), 3/(C1*x**3 - 1))],
},
'separable_reduced_10': {
'eq': Eq(f(x).diff(x) + (f(x)**2+f(x))*f(x)/(x), 0),
'sol': [Eq(- log(x) - log(f(x) + 1) + log(f(x)) + 1/f(x), C1)],
'XFAIL': ['lie_group'],#No algorithms are implemented to solve equation -C1 + x*(_y + 1)*exp(-1/_y)/_y
},
# Equivalent to example_name 'separable_reduced_02'. Only difference is testing with simplify=True
'separable_reduced_11': {
'eq': f(x).diff(x) + (f(x) / (x**4*f(x) - x)),
'sol': [Eq(f(x), -sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6
- sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
+ 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 4/x**6
- 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3)),
Eq(f(x), -sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6
+ sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
+ 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 4/x**6
- 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3)),
Eq(f(x), sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6
- sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
+ 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
+ 4/x**6 + 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3)),
Eq(f(x), sqrt(2)*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3)
- 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)/6
+ sqrt(2)*sqrt(-3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1)
+ x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 4/x**6 + 4*sqrt(2)/(x**9*sqrt(3*3**Rational(1,3)*(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1))
- exp(12*C1)/x**6)**Rational(1,3) - 3*3**Rational(2,3)*exp(12*C1)/(sqrt((3*exp(12*C1) + x**(-12))*exp(24*C1)) - exp(12*C1)/x**6)**Rational(1,3) + 2/x**6)))/6 + 1/(3*x**3))],
'checkodesol_XFAIL':True, #It hangs for this.
'slow': True,
},
#These were from issue: https://github.com/sympy/sympy/issues/6247
'separable_reduced_12': {
'eq': x**2*f(x)**2 + x*Derivative(f(x), x),
'sol': [Eq(f(x), 2*C1/(C1*x**2 - 1))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_lie_group():
a, b, c = symbols("a b c")
return {
'hint': "lie_group",
'func': f(x),
'examples':{
#Example 1-4 and 19-20 were from issue: https://github.com/sympy/sympy/issues/17322
'lie_group_01': {
'eq': x*f(x).diff(x)*(f(x)+4) + (f(x)**2) -2*f(x)-2*x,
'sol': [],
'dsolve_too_slow': True,
'checkodesol_too_slow': True,
},
'lie_group_02': {
'eq': x*f(x).diff(x)*(f(x)+4) + (f(x)**2) -2*f(x)-2*x,
'sol': [],
'dsolve_too_slow': True,
},
'lie_group_03': {
'eq': Eq(x**7*Derivative(f(x), x) + 5*x**3*f(x)**2 - (2*x**2 + 2)*f(x)**3, 0),
'sol': [],
'dsolve_too_slow': True,
},
'lie_group_04': {
'eq': f(x).diff(x) - (f(x) - x*log(x))**2/x**2 + log(x),
'sol': [],
'XFAIL': ['lie_group'],
},
'lie_group_05': {
'eq': f(x).diff(x)**2,
'sol': [Eq(f(x), C1)],
'XFAIL': ['factorable'], #It raises Not Implemented error
},
'lie_group_06': {
'eq': Eq(f(x).diff(x), x**2*f(x)),
'sol': [Eq(f(x), C1*exp(x**3)**Rational(1, 3))],
},
'lie_group_07': {
'eq': f(x).diff(x) + a*f(x) - c*exp(b*x),
'sol': [Eq(f(x), Piecewise(((-C1*(a + b) + c*exp(x*(a + b)))*exp(-a*x)/(a + b),\
Ne(a, -b)), ((-C1 + c*x)*exp(-a*x), True)))],
},
'lie_group_08': {
'eq': f(x).diff(x) + 2*x*f(x) - x*exp(-x**2),
'sol': [Eq(f(x), (C1 + x**2/2)*exp(-x**2))],
},
'lie_group_09': {
'eq': (1 + 2*x)*(f(x).diff(x)) + 2 - 4*exp(-f(x)),
'sol': [Eq(f(x), log(C1/(2*x + 1) + 2))],
},
'lie_group_10': {
'eq': x**2*(f(x).diff(x)) - f(x) + x**2*exp(x - (1/x)),
'sol': [Eq(f(x), (C1 - exp(x))*exp(-1/x))],
'XFAIL': ['factorable'], #It raises Recursion Error (maixmum depth exceeded)
},
'lie_group_11': {
'eq': x**2*f(x)**2 + x*Derivative(f(x), x),
'sol': [Eq(f(x), 2/(C1 + x**2))],
},
'lie_group_12': {
'eq': diff(f(x),x) + 2*x*f(x) - x*exp(-x**2),
'sol': [Eq(f(x), exp(-x**2)*(C1 + x**2/2))],
},
'lie_group_13': {
'eq': diff(f(x),x) + f(x)*cos(x) - exp(2*x),
'sol': [Eq(f(x), exp(-sin(x))*(C1 + Integral(exp(2*x)*exp(sin(x)), x)))],
},
'lie_group_14': {
'eq': diff(f(x),x) + f(x)*cos(x) - sin(2*x)/2,
'sol': [Eq(f(x), C1*exp(-sin(x)) + sin(x) - 1)],
},
'lie_group_15': {
'eq': x*diff(f(x),x) + f(x) - x*sin(x),
'sol': [Eq(f(x), (C1 - x*cos(x) + sin(x))/x)],
},
'lie_group_16': {
'eq': x*diff(f(x),x) - f(x) - x/log(x),
'sol': [Eq(f(x), x*(C1 + log(log(x))))],
},
'lie_group_17': {
'eq': (f(x).diff(x)-f(x)) * (f(x).diff(x)+f(x)),
'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1*exp(-x))],
},
'lie_group_18': {
'eq': f(x).diff(x) * (f(x).diff(x) - f(x)),
'sol': [Eq(f(x), C1*exp(x)), Eq(f(x), C1)],
},
'lie_group_19': {
'eq': (f(x).diff(x)-f(x)) * (f(x).diff(x)+f(x)),
'sol': [Eq(f(x), C1*exp(-x)), Eq(f(x), C1*exp(x))],
},
'lie_group_20': {
'eq': f(x).diff(x)*(f(x).diff(x)+f(x)),
'sol': [Eq(f(x), C1), Eq(f(x), C1*exp(-x))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_2nd_linear_airy():
return {
'hint': "2nd_linear_airy",
'func': f(x),
'examples':{
'2nd_lin_airy_01': {
'eq': f(x).diff(x, 2) - x*f(x),
'sol': [Eq(f(x), C1*airyai(x) + C2*airybi(x))],
},
'2nd_lin_airy_02': {
'eq': f(x).diff(x, 2) + 2*x*f(x),
'sol': [Eq(f(x), C1*airyai(-2**(S(1)/3)*x) + C2*airybi(-2**(S(1)/3)*x))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_nth_linear_constant_coeff_homogeneous():
# From Exercise 20, in Ordinary Differential Equations,
# Tenenbaum and Pollard, pg. 220
a = Symbol('a', positive=True)
k = Symbol('k', real=True)
r1, r2, r3, r4, r5 = [rootof(x**5 + 11*x - 2, n) for n in range(5)]
r6, r7, r8, r9, r10 = [rootof(x**5 - 3*x + 1, n) for n in range(5)]
r11, r12, r13, r14, r15 = [rootof(x**5 - 100*x**3 + 1000*x + 1, n) for n in range(5)]
r16, r17, r18, r19, r20 = [rootof(x**5 - x**4 + 10, n) for n in range(5)]
r21, r22, r23, r24, r25 = [rootof(x**5 - x + 1, n) for n in range(5)]
E = exp(1)
return {
'hint': "nth_linear_constant_coeff_homogeneous",
'func': f(x),
'examples':{
'lin_const_coeff_hom_01': {
'eq': f(x).diff(x, 2) + 2*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-2*x))],
},
'lin_const_coeff_hom_02': {
'eq': f(x).diff(x, 2) - 3*f(x).diff(x) + 2*f(x),
'sol': [Eq(f(x), (C1 + C2*exp(x))*exp(x))],
},
'lin_const_coeff_hom_03': {
'eq': f(x).diff(x, 2) - f(x),
'sol': [Eq(f(x), C1*exp(-x) + C2*exp(x))],
},
'lin_const_coeff_hom_04': {
'eq': f(x).diff(x, 3) + f(x).diff(x, 2) - 6*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-3*x) + C3*exp(2*x))],
'slow': True,
},
'lin_const_coeff_hom_05': {
'eq': 6*f(x).diff(x, 2) - 11*f(x).diff(x) + 4*f(x),
'sol': [Eq(f(x), C1*exp(x/2) + C2*exp(x*Rational(4, 3)))],
'slow': True,
},
'lin_const_coeff_hom_06': {
'eq': Eq(f(x).diff(x, 2) + 2*f(x).diff(x) - f(x), 0),
'sol': [Eq(f(x), C1*exp(x*(-1 + sqrt(2))) + C2*exp(-x*(sqrt(2) + 1)))],
'slow': True,
},
'lin_const_coeff_hom_07': {
'eq': diff(f(x), x, 3) + diff(f(x), x, 2) - 10*diff(f(x), x) - 6*f(x),
'sol': [Eq(f(x), C1*exp(3*x) + C3*exp(-x*(2 + sqrt(2))) + C2*exp(x*(-2 + sqrt(2))))],
'slow': True,
},
'lin_const_coeff_hom_08': {
'eq': f(x).diff(x, 4) - f(x).diff(x, 3) - 4*f(x).diff(x, 2) + \
4*f(x).diff(x),
'sol': [Eq(f(x), C1 + C2*exp(-2*x) + C3*exp(x) + C4*exp(2*x))],
'slow': True,
},
'lin_const_coeff_hom_09': {
'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 3) + f(x).diff(x, 2) - \
4*f(x).diff(x) - 2*f(x),
'sol': [Eq(f(x), C3*exp(-x) + C4*exp(x) + (C1*exp(-sqrt(2)*x) + C2*exp(sqrt(2)*x))*exp(-2*x))],
'slow': True,
},
'lin_const_coeff_hom_10': {
'eq': f(x).diff(x, 4) - a**2*f(x),
'sol': [Eq(f(x), C1*exp(-sqrt(a)*x) + C2*exp(sqrt(a)*x) + C3*sin(sqrt(a)*x) + C4*cos(sqrt(a)*x))],
'slow': True,
},
'lin_const_coeff_hom_11': {
'eq': f(x).diff(x, 2) - 2*k*f(x).diff(x) - 2*f(x),
'sol': [Eq(f(x), C1*exp(x*(k - sqrt(k**2 + 2))) + C2*exp(x*(k + sqrt(k**2 + 2))))],
'slow': True,
},
'lin_const_coeff_hom_12': {
'eq': f(x).diff(x, 2) + 4*k*f(x).diff(x) - 12*k**2*f(x),
'sol': [Eq(f(x), C1*exp(-6*k*x) + C2*exp(2*k*x))],
'slow': True,
},
'lin_const_coeff_hom_13': {
'eq': f(x).diff(x, 4),
'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*x**3)],
'slow': True,
},
'lin_const_coeff_hom_14': {
'eq': f(x).diff(x, 2) + 4*f(x).diff(x) + 4*f(x),
'sol': [Eq(f(x), (C1 + C2*x)*exp(-2*x))],
'slow': True,
},
'lin_const_coeff_hom_15': {
'eq': 3*f(x).diff(x, 3) + 5*f(x).diff(x, 2) + f(x).diff(x) - f(x),
'sol': [Eq(f(x), (C1 + C2*x)*exp(-x) + C3*exp(x/3))],
'slow': True,
},
'lin_const_coeff_hom_16': {
'eq': f(x).diff(x, 3) - 6*f(x).diff(x, 2) + 12*f(x).diff(x) - 8*f(x),
'sol': [Eq(f(x), (C1 + x*(C2 + C3*x))*exp(2*x))],
'slow': True,
},
'lin_const_coeff_hom_17': {
'eq': f(x).diff(x, 2) - 2*a*f(x).diff(x) + a**2*f(x),
'sol': [Eq(f(x), (C1 + C2*x)*exp(a*x))],
'slow': True,
},
'lin_const_coeff_hom_18': {
'eq': f(x).diff(x, 4) + 3*f(x).diff(x, 3),
'sol': [Eq(f(x), C1 + C2*x + C3*x**2 + C4*exp(-3*x))],
'slow': True,
},
'lin_const_coeff_hom_19': {
'eq': f(x).diff(x, 4) - 2*f(x).diff(x, 2),
'sol': [Eq(f(x), C1 + C2*x + C3*exp(-sqrt(2)*x) + C4*exp(sqrt(2)*x))],
'slow': True,
},
'lin_const_coeff_hom_20': {
'eq': f(x).diff(x, 4) + 2*f(x).diff(x, 3) - 11*f(x).diff(x, 2) - \
12*f(x).diff(x) + 36*f(x),
'sol': [Eq(f(x), (C1 + C2*x)*exp(-3*x) + (C3 + C4*x)*exp(2*x))],
'slow': True,
},
'lin_const_coeff_hom_21': {
'eq': 36*f(x).diff(x, 4) - 37*f(x).diff(x, 2) + 4*f(x).diff(x) + 5*f(x),
'sol': [Eq(f(x), C1*exp(-x) + C2*exp(-x/3) + C3*exp(x/2) + C4*exp(x*Rational(5, 6)))],
'slow': True,
},
'lin_const_coeff_hom_22': {
'eq': f(x).diff(x, 4) - 8*f(x).diff(x, 2) + 16*f(x),
'sol': [Eq(f(x), (C1 + C2*x)*exp(-2*x) + (C3 + C4*x)*exp(2*x))],
'slow': True,
},
'lin_const_coeff_hom_23': {
'eq': f(x).diff(x, 2) - 2*f(x).diff(x) + 5*f(x),
'sol': [Eq(f(x), (C1*sin(2*x) + C2*cos(2*x))*exp(x))],
'slow': True,
},
'lin_const_coeff_hom_24': {
'eq': f(x).diff(x, 2) - f(x).diff(x) + f(x),
'sol': [Eq(f(x), (C1*sin(x*sqrt(3)/2) + C2*cos(x*sqrt(3)/2))*exp(x/2))],
'slow': True,
},
'lin_const_coeff_hom_25': {
'eq': f(x).diff(x, 4) + 5*f(x).diff(x, 2) + 6*f(x),
'sol': [Eq(f(x),
C1*sin(sqrt(2)*x) + C2*sin(sqrt(3)*x) + C3*cos(sqrt(2)*x) + C4*cos(sqrt(3)*x))],
'slow': True,
},
'lin_const_coeff_hom_26': {
'eq': f(x).diff(x, 2) - 4*f(x).diff(x) + 20*f(x),
'sol': [Eq(f(x), (C1*sin(4*x) + C2*cos(4*x))*exp(2*x))],
'slow': True,
},
'lin_const_coeff_hom_27': {
'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2) + 4*f(x),
'sol': [Eq(f(x), (C1 + C2*x)*sin(x*sqrt(2)) + (C3 + C4*x)*cos(x*sqrt(2)))],
'slow': True,
},
'lin_const_coeff_hom_28': {
'eq': f(x).diff(x, 3) + 8*f(x),
'sol': [Eq(f(x), (C1*sin(x*sqrt(3)) + C2*cos(x*sqrt(3)))*exp(x) + C3*exp(-2*x))],
'slow': True,
},
'lin_const_coeff_hom_29': {
'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2),
'sol': [Eq(f(x), C1 + C2*x + C3*sin(2*x) + C4*cos(2*x))],
'slow': True,
},
'lin_const_coeff_hom_30': {
'eq': f(x).diff(x, 5) + 2*f(x).diff(x, 3) + f(x).diff(x),
'sol': [Eq(f(x), C1 + (C2 + C3*x)*sin(x) + (C4 + C5*x)*cos(x))],
'slow': True,
},
'lin_const_coeff_hom_31': {
'eq': f(x).diff(x, 4) + f(x).diff(x, 2) + f(x),
'sol': [Eq(f(x), (C1*sin(sqrt(3)*x/2) + C2*cos(sqrt(3)*x/2))*exp(-x/2)
+ (C3*sin(sqrt(3)*x/2) + C4*cos(sqrt(3)*x/2))*exp(x/2))],
'slow': True,
},
'lin_const_coeff_hom_32': {
'eq': f(x).diff(x, 4) + 4*f(x).diff(x, 2) + f(x),
'sol': [Eq(f(x), C1*sin(x*sqrt(-sqrt(3) + 2)) + C2*sin(x*sqrt(sqrt(3) + 2))
+ C3*cos(x*sqrt(-sqrt(3) + 2)) + C4*cos(x*sqrt(sqrt(3) + 2)))],
'slow': True,
},
# One real root, two complex conjugate pairs
'lin_const_coeff_hom_33': {
'eq': f(x).diff(x, 5) + 11*f(x).diff(x) - 2*f(x),
'sol': [Eq(f(x),
C5*exp(r1*x) + exp(re(r2)*x) * (C1*sin(im(r2)*x) + C2*cos(im(r2)*x))
+ exp(re(r4)*x) * (C3*sin(im(r4)*x) + C4*cos(im(r4)*x)))],
'checkodesol_XFAIL':True, #It Hangs
},
# Three real roots, one complex conjugate pair
'lin_const_coeff_hom_34': {
'eq': f(x).diff(x,5) - 3*f(x).diff(x) + f(x),
'sol': [Eq(f(x),
C3*exp(r6*x) + C4*exp(r7*x) + C5*exp(r8*x)
+ exp(re(r9)*x) * (C1*sin(im(r9)*x) + C2*cos(im(r9)*x)))],
'checkodesol_XFAIL':True, #It Hangs
},
# Five distinct real roots
'lin_const_coeff_hom_35': {
'eq': f(x).diff(x,5) - 100*f(x).diff(x,3) + 1000*f(x).diff(x) + f(x),
'sol': [Eq(f(x), C1*exp(r11*x) + C2*exp(r12*x) + C3*exp(r13*x) + C4*exp(r14*x) + C5*exp(r15*x))],
'checkodesol_XFAIL':True, #It Hangs
},
# Rational root and unsolvable quintic
'lin_const_coeff_hom_36': {
'eq': f(x).diff(x, 6) - 6*f(x).diff(x, 5) + 5*f(x).diff(x, 4) + 10*f(x).diff(x) - 50 * f(x),
'sol': [Eq(f(x),
C5*exp(5*x)
+ C6*exp(x*r16)
+ exp(re(r17)*x) * (C1*sin(im(r17)*x) + C2*cos(im(r17)*x))
+ exp(re(r19)*x) * (C3*sin(im(r19)*x) + C4*cos(im(r19)*x)))],
'checkodesol_XFAIL':True, #It Hangs
},
# Five double roots (this is (x**5 - x + 1)**2)
'lin_const_coeff_hom_37': {
'eq': f(x).diff(x, 10) - 2*f(x).diff(x, 6) + 2*f(x).diff(x, 5)
+ f(x).diff(x, 2) - 2*f(x).diff(x, 1) + f(x),
'sol': [Eq(f(x), (C1 + C2*x)*exp(x*r21) + (-((C3 + C4*x)*sin(x*im(r22)))
+ (C5 + C6*x)*cos(x*im(r22)))*exp(x*re(r22)) + (-((C7 + C8*x)*sin(x*im(r24)))
+ (C10*x + C9)*cos(x*im(r24)))*exp(x*re(r24)))],
'checkodesol_XFAIL':True, #It Hangs
},
'lin_const_coeff_hom_38': {
'eq': Eq(sqrt(2) * f(x).diff(x,x,x) + f(x).diff(x), 0),
'sol': [Eq(f(x), C1 + C2*sin(2**Rational(3, 4)*x/2) + C3*cos(2**Rational(3, 4)*x/2))],
},
'lin_const_coeff_hom_39': {
'eq': Eq(E * f(x).diff(x,x,x) + f(x).diff(x), 0),
'sol': [Eq(f(x), C1 + C2*sin(x/sqrt(E)) + C3*cos(x/sqrt(E)))],
},
'lin_const_coeff_hom_40': {
'eq': Eq(pi * f(x).diff(x,x,x) + f(x).diff(x), 0),
'sol': [Eq(f(x), C1 + C2*sin(x/sqrt(pi)) + C3*cos(x/sqrt(pi)))],
},
'lin_const_coeff_hom_41': {
'eq': Eq(I * f(x).diff(x,x,x) + f(x).diff(x), 0),
'sol': [Eq(f(x), C1 + C2*exp(-sqrt(I)*x) + C3*exp(sqrt(I)*x))],
},
'lin_const_coeff_hom_42': {
'eq': f(x).diff(x, x) + y*f(x),
'sol': [Eq(f(x), C1*exp(-x*sqrt(-y)) + C2*exp(x*sqrt(-y)))],
},
'lin_const_coeff_hom_43': {
'eq': Eq(9*f(x).diff(x, x) + f(x), 0),
'sol': [Eq(f(x), C1*sin(x/3) + C2*cos(x/3))],
},
'lin_const_coeff_hom_44': {
'eq': Eq(9*f(x).diff(x, x), f(x)),
'sol': [Eq(f(x), C1*exp(-x/3) + C2*exp(x/3))],
},
'lin_const_coeff_hom_45': {
'eq': Eq(f(x).diff(x, x) - 3*diff(f(x), x) + 2*f(x), 0),
'sol': [Eq(f(x), (C1 + C2*exp(x))*exp(x))],
},
'lin_const_coeff_hom_46': {
'eq': Eq(f(x).diff(x, x) - 4*diff(f(x), x) + 4*f(x), 0),
'sol': [Eq(f(x), (C1 + C2*x)*exp(2*x))],
},
# Type: 2nd order, constant coefficients (two real equal roots)
'lin_const_coeff_hom_47': {
'eq': Eq(f(x).diff(x, x) + 2*diff(f(x), x) + 3*f(x), 0),
'sol': [Eq(f(x), (C1*sin(x*sqrt(2)) + C2*cos(x*sqrt(2)))*exp(-x))],
},
#These were from issue: https://github.com/sympy/sympy/issues/6247
'lin_const_coeff_hom_48': {
'eq': f(x).diff(x, x) + 4*f(x),
'sol': [Eq(f(x), C1*sin(2*x) + C2*cos(2*x))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep():
return {
'hint': "1st_homogeneous_coeff_subs_dep_div_indep",
'func': f(x),
'examples':{
'dep_div_indep_01': {
'eq': f(x)/x*cos(f(x)/x) - (x/f(x)*sin(f(x)/x) + cos(f(x)/x))*f(x).diff(x),
'sol': [Eq(log(x), C1 - log(f(x)*sin(f(x)/x)/x))],
'slow': True
},
#indep_div_dep actually has a simpler solution for example 2 but it runs too slow.
'dep_div_indep_02': {
'eq': x*f(x).diff(x) - f(x) - x*sin(f(x)/x),
'sol': [Eq(log(x), log(C1) + log(cos(f(x)/x) - 1)/2 - log(cos(f(x)/x) + 1)/2)],
'simplify_flag':False,
},
'dep_div_indep_03': {
'eq': x*exp(f(x)/x) - f(x)*sin(f(x)/x) + x*sin(f(x)/x)*f(x).diff(x),
'sol': [Eq(log(x), C1 + exp(-f(x)/x)*sin(f(x)/x)/2 + exp(-f(x)/x)*cos(f(x)/x)/2)],
'slow': True
},
'dep_div_indep_04': {
'eq': f(x).diff(x) - f(x)/x + 1/sin(f(x)/x),
'sol': [Eq(f(x), x*(-acos(C1 + log(x)) + 2*pi)), Eq(f(x), x*acos(C1 + log(x)))],
'slow': True
},
# previous code was testing with these other solution:
# example5_solb = Eq(f(x), log(log(C1/x)**(-x)))
'dep_div_indep_05': {
'eq': x*exp(f(x)/x) + f(x) - x*f(x).diff(x),
'sol': [Eq(f(x), log((1/(C1 - log(x)))**x))],
'checkodesol_XFAIL':True, #(because of **x?)
},
}
}
@_add_example_keys
def _get_examples_ode_sol_linear_coefficients():
return {
'hint': "linear_coefficients",
'func': f(x),
'examples':{
'linear_coeff_01': {
'eq': f(x).diff(x) + (3 + 2*f(x))/(x + 3),
'sol': [Eq(f(x), C1/(x**2 + 6*x + 9) - Rational(3, 2))],
},
}
}
@_add_example_keys
def _get_examples_ode_sol_1st_homogeneous_coeff_best():
return {
'hint': "1st_homogeneous_coeff_best",
'func': f(x),
'examples':{
# previous code was testing this with other solution:
# example1_solb = Eq(-f(x)/(1 + log(x/f(x))), C1)
'1st_homogeneous_coeff_best_01': {
'eq': f(x) + (x*log(f(x)/x) - 2*x)*diff(f(x), x),
'sol': [Eq(f(x), -exp(C1)*LambertW(-x*exp(-C1 + 1)))],
'checkodesol_XFAIL':True, #(because of LambertW?)
},
'1st_homogeneous_coeff_best_02': {
'eq': 2*f(x)*exp(x/f(x)) + f(x)*f(x).diff(x) - 2*x*exp(x/f(x))*f(x).diff(x),
'sol': [Eq(log(f(x)), C1 - 2*exp(x/f(x)))],
},
# previous code was testing this with other solution:
# example3_solb = Eq(log(C1*x*sqrt(1/x)*sqrt(f(x))) + x**2/(2*f(x)**2), 0)
'1st_homogeneous_coeff_best_03': {
'eq': 2*x**2*f(x) + f(x)**3 + (x*f(x)**2 - 2*x**3)*f(x).diff(x),
'sol': [Eq(f(x), exp(2*C1 + LambertW(-2*x**4*exp(-4*C1))/2)/x)],
'checkodesol_XFAIL':True, #(because of LambertW?)
},
'1st_homogeneous_coeff_best_04': {
'eq': (x + sqrt(f(x)**2 - x*f(x)))*f(x).diff(x) - f(x),
'sol': [Eq(log(f(x)), C1 - 2*sqrt(-x/f(x) + 1))],
'slow': True,
},
'1st_homogeneous_coeff_best_05': {
'eq': x + f(x) - (x - f(x))*f(x).diff(x),
'sol': [Eq(log(x), C1 - log(sqrt(1 + f(x)**2/x**2)) + atan(f(x)/x))],
},
'1st_homogeneous_coeff_best_06': {
'eq': x*f(x).diff(x) - f(x) - x*sin(f(x)/x),
'sol': [Eq(f(x), 2*x*atan(C1*x))],
},
'1st_homogeneous_coeff_best_07': {
'eq': x**2 + f(x)**2 - 2*x*f(x)*f(x).diff(x),
'sol': [Eq(f(x), -sqrt(x*(C1 + x))), Eq(f(x), sqrt(x*(C1 + x)))],
},
'1st_homogeneous_coeff_best_08': {
'eq': f(x)**2 + (x*sqrt(f(x)**2 - x**2) - x*f(x))*f(x).diff(x),
'sol': [Eq(log(x), C1 - log(f(x)/x) + acosh(f(x)/x))],
},
}
}
def _get_all_examples():
all_examples = _get_examples_ode_sol_euler_homogeneous + \
_get_examples_ode_sol_euler_undetermined_coeff + \
_get_examples_ode_sol_euler_var_para + \
_get_examples_ode_sol_factorable + \
_get_examples_ode_sol_bernoulli + \
_get_examples_ode_sol_nth_algebraic + \
_get_examples_ode_sol_riccati + \
_get_examples_ode_sol_1st_linear + \
_get_examples_ode_sol_1st_exact + \
_get_examples_ode_sol_almost_linear + \
_get_examples_ode_sol_nth_order_reducible + \
_get_examples_ode_sol_nth_linear_undetermined_coefficients + \
_get_examples_ode_sol_liouville + \
_get_examples_ode_sol_separable + \
_get_examples_ode_sol_1st_rational_riccati + \
_get_examples_ode_sol_nth_linear_var_of_parameters + \
_get_examples_ode_sol_2nd_linear_bessel + \
_get_examples_ode_sol_2nd_2F1_hypergeometric + \
_get_examples_ode_sol_2nd_nonlinear_autonomous_conserved + \
_get_examples_ode_sol_separable_reduced + \
_get_examples_ode_sol_lie_group + \
_get_examples_ode_sol_2nd_linear_airy + \
_get_examples_ode_sol_nth_linear_constant_coeff_homogeneous +\
_get_examples_ode_sol_1st_homogeneous_coeff_best +\
_get_examples_ode_sol_1st_homogeneous_coeff_subs_dep_div_indep +\
_get_examples_ode_sol_linear_coefficients
return all_examples
|
4b9fd81065b4cfbc73ced85286b4ac836c009b313d991e2ee0d773bb3c3f1b1f | from sympy.core.numbers import (E, I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (Abs, re)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import expint
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.simplify.simplify import simplify
from sympy.calculus.util import (function_range, continuous_domain, not_empty_in,
periodicity, lcim, is_convex,
stationary_points, minimum, maximum)
from sympy.sets.sets import (Interval, FiniteSet, Complement, Union)
from sympy.testing.pytest import raises, _both_exp_pow
from sympy.abc import x
a = Symbol('a', real=True)
def test_function_range():
x, y, a, b = symbols('x y a b')
assert function_range(sin(x), x, Interval(-pi/2, pi/2)
) == Interval(-1, 1)
assert function_range(sin(x), x, Interval(0, pi)
) == Interval(0, 1)
assert function_range(tan(x), x, Interval(0, pi)
) == Interval(-oo, oo)
assert function_range(tan(x), x, Interval(pi/2, pi)
) == Interval(-oo, 0)
assert function_range((x + 3)/(x - 2), x, Interval(-5, 5)
) == Union(Interval(-oo, Rational(2, 7)), Interval(Rational(8, 3), oo))
assert function_range(1/(x**2), x, Interval(-1, 1)
) == Interval(1, oo)
assert function_range(exp(x), x, Interval(-1, 1)
) == Interval(exp(-1), exp(1))
assert function_range(log(x) - x, x, S.Reals
) == Interval(-oo, -1)
assert function_range(sqrt(3*x - 1), x, Interval(0, 2)
) == Interval(0, sqrt(5))
assert function_range(x*(x - 1) - (x**2 - x), x, S.Reals
) == FiniteSet(0)
assert function_range(x*(x - 1) - (x**2 - x) + y, x, S.Reals
) == FiniteSet(y)
assert function_range(sin(x), x, Union(Interval(-5, -3), FiniteSet(4))
) == Union(Interval(-sin(3), 1), FiniteSet(sin(4)))
assert function_range(cos(x), x, Interval(-oo, -4)
) == Interval(-1, 1)
assert function_range(cos(x), x, S.EmptySet) == S.EmptySet
assert function_range(x/sqrt(x**2+1), x, S.Reals) == Interval.open(-1,1)
raises(NotImplementedError, lambda : function_range(
exp(x)*(sin(x) - cos(x))/2 - x, x, S.Reals))
raises(NotImplementedError, lambda : function_range(
sin(x) + x, x, S.Reals)) # issue 13273
raises(NotImplementedError, lambda : function_range(
log(x), x, S.Integers))
raises(NotImplementedError, lambda : function_range(
sin(x)/2, x, S.Naturals))
def test_continuous_domain():
x = Symbol('x')
assert continuous_domain(sin(x), x, Interval(0, 2*pi)) == Interval(0, 2*pi)
assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \
Union(Interval(0, pi/2, False, True), Interval(pi/2, pi*Rational(3, 2), True, True),
Interval(pi*Rational(3, 2), 2*pi, True, False))
assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \
Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True))
assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \
Interval(Rational(1, 4), oo, True, True)
assert continuous_domain(1/sqrt(x - 3), x, S.Reals) == Interval(3, oo, True, True)
assert continuous_domain(1/x - 2, x, S.Reals) == \
Union(Interval.open(-oo, 0), Interval.open(0, oo))
assert continuous_domain(1/(x**2 - 4) + 2, x, S.Reals) == \
Union(Interval.open(-oo, -2), Interval.open(-2, 2), Interval.open(2, oo))
domain = continuous_domain(log(tan(x)**2 + 1), x, S.Reals)
assert not domain.contains(3*pi/2)
assert domain.contains(5)
d = Symbol('d', even=True, zero=False)
assert continuous_domain(x**(1/d), x, S.Reals) == Interval(0, oo)
def test_not_empty_in():
assert not_empty_in(FiniteSet(x, 2*x).intersect(Interval(1, 2, True, False)), x) == \
Interval(S.Half, 2, True, False)
assert not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x) == \
Union(Interval(-sqrt(2), -1), Interval(1, 2))
assert not_empty_in(FiniteSet(x**2 + x, x).intersect(Interval(2, 4)), x) == \
Union(Interval(-sqrt(17)/2 - S.Half, -2),
Interval(1, Rational(-1, 2) + sqrt(17)/2), Interval(2, 4))
assert not_empty_in(FiniteSet(x/(x - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet(a/(a - 1)).intersect(S.Reals), a) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet((x**2 - 3*x + 2)/(x - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(1))
assert not_empty_in(FiniteSet(3, 4, x/(x - 1)).intersect(Interval(2, 3)), x) == \
Interval(-oo, oo)
assert not_empty_in(FiniteSet(4, x/(x - 1)).intersect(Interval(2, 3)), x) == \
Interval(S(3)/2, 2)
assert not_empty_in(FiniteSet(x/(x**2 - 1)).intersect(S.Reals), x) == \
Complement(S.Reals, FiniteSet(-1, 1))
assert not_empty_in(FiniteSet(x, x**2).intersect(Union(Interval(1, 3, True, True),
Interval(4, 5))), x) == \
Union(Interval(-sqrt(5), -2), Interval(-sqrt(3), -1, True, True),
Interval(1, 3, True, True), Interval(4, 5))
assert not_empty_in(FiniteSet(1).intersect(Interval(3, 4)), x) == S.EmptySet
assert not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x) == \
Union(Interval(-2, -1, True, False), Interval(2, oo))
raises(ValueError, lambda: not_empty_in(x))
raises(ValueError, lambda: not_empty_in(Interval(0, 1), x))
raises(NotImplementedError,
lambda: not_empty_in(FiniteSet(x).intersect(S.Reals), x, a))
@_both_exp_pow
def test_periodicity():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', real=True)
assert periodicity(sin(2*x), x) == pi
assert periodicity((-2)*tan(4*x), x) == pi/4
assert periodicity(sin(x)**2, x) == 2*pi
assert periodicity(3**tan(3*x), x) == pi/3
assert periodicity(tan(x)*cos(x), x) == 2*pi
assert periodicity(sin(x)**(tan(x)), x) == 2*pi
assert periodicity(tan(x)*sec(x), x) == 2*pi
assert periodicity(sin(2*x)*cos(2*x) - y, x) == pi/2
assert periodicity(tan(x) + cot(x), x) == pi
assert periodicity(sin(x) - cos(2*x), x) == 2*pi
assert periodicity(sin(x) - 1, x) == 2*pi
assert periodicity(sin(4*x) + sin(x)*cos(x), x) == pi
assert periodicity(exp(sin(x)), x) == 2*pi
assert periodicity(log(cot(2*x)) - sin(cos(2*x)), x) == pi
assert periodicity(sin(2*x)*exp(tan(x) - csc(2*x)), x) == pi
assert periodicity(cos(sec(x) - csc(2*x)), x) == 2*pi
assert periodicity(tan(sin(2*x)), x) == pi
assert periodicity(2*tan(x)**2, x) == pi
assert periodicity(sin(x%4), x) == 4
assert periodicity(sin(x)%4, x) == 2*pi
assert periodicity(tan((3*x-2)%4), x) == Rational(4, 3)
assert periodicity((sqrt(2)*(x+1)+x) % 3, x) == 3 / (sqrt(2)+1)
assert periodicity((x**2+1) % x, x) is None
assert periodicity(sin(re(x)), x) == 2*pi
assert periodicity(sin(x)**2 + cos(x)**2, x) is S.Zero
assert periodicity(tan(x), y) is S.Zero
assert periodicity(sin(x) + I*cos(x), x) == 2*pi
assert periodicity(x - sin(2*y), y) == pi
assert periodicity(exp(x), x) is None
assert periodicity(exp(I*x), x) == 2*pi
assert periodicity(exp(I*z), z) == 2*pi
assert periodicity(exp(z), z) is None
assert periodicity(exp(log(sin(z) + I*cos(2*z)), evaluate=False), z) == 2*pi
assert periodicity(exp(log(sin(2*z) + I*cos(z)), evaluate=False), z) == 2*pi
assert periodicity(exp(sin(z)), z) == 2*pi
assert periodicity(exp(2*I*z), z) == pi
assert periodicity(exp(z + I*sin(z)), z) is None
assert periodicity(exp(cos(z/2) + sin(z)), z) == 4*pi
assert periodicity(log(x), x) is None
assert periodicity(exp(x)**sin(x), x) is None
assert periodicity(sin(x)**y, y) is None
assert periodicity(Abs(sin(Abs(sin(x)))), x) == pi
assert all(periodicity(Abs(f(x)), x) == pi for f in (
cos, sin, sec, csc, tan, cot))
assert periodicity(Abs(sin(tan(x))), x) == pi
assert periodicity(Abs(sin(sin(x) + tan(x))), x) == 2*pi
assert periodicity(sin(x) > S.Half, x) == 2*pi
assert periodicity(x > 2, x) is None
assert periodicity(x**3 - x**2 + 1, x) is None
assert periodicity(Abs(x), x) is None
assert periodicity(Abs(x**2 - 1), x) is None
assert periodicity((x**2 + 4)%2, x) is None
assert periodicity((E**x)%3, x) is None
assert periodicity(sin(expint(1, x))/expint(1, x), x) is None
# returning `None` for any Piecewise
p = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True))
assert periodicity(p, x) is None
m = MatrixSymbol('m', 3, 3)
raises(NotImplementedError, lambda: periodicity(sin(m), m))
raises(NotImplementedError, lambda: periodicity(sin(m[0, 0]), m))
raises(NotImplementedError, lambda: periodicity(sin(m), m[0, 0]))
raises(NotImplementedError, lambda: periodicity(sin(m[0, 0]), m[0, 0]))
def test_periodicity_check():
x = Symbol('x')
y = Symbol('y')
assert periodicity(tan(x), x, check=True) == pi
assert periodicity(sin(x) + cos(x), x, check=True) == 2*pi
assert periodicity(sec(x), x) == 2*pi
assert periodicity(sin(x*y), x) == 2*pi/abs(y)
assert periodicity(Abs(sec(sec(x))), x) == pi
def test_lcim():
assert lcim([S.Half, S(2), S(3)]) == 6
assert lcim([pi/2, pi/4, pi]) == pi
assert lcim([2*pi, pi/2]) == 2*pi
assert lcim([S.One, 2*pi]) is None
assert lcim([S(2) + 2*E, E/3 + Rational(1, 3), S.One + E]) == S(2) + 2*E
def test_is_convex():
assert is_convex(1/x, x, domain=Interval.open(0, oo)) == True
assert is_convex(1/x, x, domain=Interval(-oo, 0)) == False
assert is_convex(x**2, x, domain=Interval(0, oo)) == True
assert is_convex(1/x**3, x, domain=Interval.Lopen(0, oo)) == True
assert is_convex(-1/x**3, x, domain=Interval.Ropen(-oo, 0)) == True
assert is_convex(log(x), x) == False
raises(NotImplementedError, lambda: is_convex(log(x), x, a))
def test_stationary_points():
x, y = symbols('x y')
assert stationary_points(sin(x), x, Interval(-pi/2, pi/2)
) == {-pi/2, pi/2}
assert stationary_points(sin(x), x, Interval.Ropen(0, pi/4)
) is S.EmptySet
assert stationary_points(tan(x), x,
) is S.EmptySet
assert stationary_points(sin(x)*cos(x), x, Interval(0, pi)
) == {pi/4, pi*Rational(3, 4)}
assert stationary_points(sec(x), x, Interval(0, pi)
) == {0, pi}
assert stationary_points((x+3)*(x-2), x
) == FiniteSet(Rational(-1, 2))
assert stationary_points((x + 3)/(x - 2), x, Interval(-5, 5)
) is S.EmptySet
assert stationary_points((x**2+3)/(x-2), x
) == {2 - sqrt(7), 2 + sqrt(7)}
assert stationary_points((x**2+3)/(x-2), x, Interval(0, 5)
) == {2 + sqrt(7)}
assert stationary_points(x**4 + x**3 - 5*x**2, x, S.Reals
) == FiniteSet(-2, 0, Rational(5, 4))
assert stationary_points(exp(x), x
) is S.EmptySet
assert stationary_points(log(x) - x, x, S.Reals
) == {1}
assert stationary_points(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
) == {0, -pi, pi}
assert stationary_points(y, x, S.Reals
) == S.Reals
assert stationary_points(y, x, S.EmptySet) == S.EmptySet
def test_maximum():
x, y = symbols('x y')
assert maximum(sin(x), x) is S.One
assert maximum(sin(x), x, Interval(0, 1)) == sin(1)
assert maximum(tan(x), x) is oo
assert maximum(tan(x), x, Interval(-pi/4, pi/4)) is S.One
assert maximum(sin(x)*cos(x), x, S.Reals) == S.Half
assert simplify(maximum(sin(x)*cos(x), x, Interval(pi*Rational(3, 8), pi*Rational(5, 8)))
) == sqrt(2)/4
assert maximum((x+3)*(x-2), x) is oo
assert maximum((x+3)*(x-2), x, Interval(-5, 0)) == S(14)
assert maximum((x+3)/(x-2), x, Interval(-5, 0)) == Rational(2, 7)
assert simplify(maximum(-x**4-x**3+x**2+10, x)
) == 41*sqrt(41)/512 + Rational(5419, 512)
assert maximum(exp(x), x, Interval(-oo, 2)) == exp(2)
assert maximum(log(x) - x, x, S.Reals) is S.NegativeOne
assert maximum(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
) is S.One
assert maximum(cos(x)-sin(x), x, S.Reals) == sqrt(2)
assert maximum(y, x, S.Reals) == y
assert maximum(abs(a**3 + a), a, Interval(0, 2)) == 10
assert maximum(abs(60*a**3 + 24*a), a, Interval(0, 2)) == 528
assert maximum(abs(12*a*(5*a**2 + 2)), a, Interval(0, 2)) == 528
assert maximum(x/sqrt(x**2+1), x, S.Reals) == 1
raises(ValueError, lambda : maximum(sin(x), x, S.EmptySet))
raises(ValueError, lambda : maximum(log(cos(x)), x, S.EmptySet))
raises(ValueError, lambda : maximum(1/(x**2 + y**2 + 1), x, S.EmptySet))
raises(ValueError, lambda : maximum(sin(x), sin(x)))
raises(ValueError, lambda : maximum(sin(x), x*y, S.EmptySet))
raises(ValueError, lambda : maximum(sin(x), S.One))
def test_minimum():
x, y = symbols('x y')
assert minimum(sin(x), x) is S.NegativeOne
assert minimum(sin(x), x, Interval(1, 4)) == sin(4)
assert minimum(tan(x), x) is -oo
assert minimum(tan(x), x, Interval(-pi/4, pi/4)) is S.NegativeOne
assert minimum(sin(x)*cos(x), x, S.Reals) == Rational(-1, 2)
assert simplify(minimum(sin(x)*cos(x), x, Interval(pi*Rational(3, 8), pi*Rational(5, 8)))
) == -sqrt(2)/4
assert minimum((x+3)*(x-2), x) == Rational(-25, 4)
assert minimum((x+3)/(x-2), x, Interval(-5, 0)) == Rational(-3, 2)
assert minimum(x**4-x**3+x**2+10, x) == S(10)
assert minimum(exp(x), x, Interval(-2, oo)) == exp(-2)
assert minimum(log(x) - x, x, S.Reals) is -oo
assert minimum(cos(x), x, Union(Interval(0, 5), Interval(-6, -3))
) is S.NegativeOne
assert minimum(cos(x)-sin(x), x, S.Reals) == -sqrt(2)
assert minimum(y, x, S.Reals) == y
assert minimum(x/sqrt(x**2+1), x, S.Reals) == -1
raises(ValueError, lambda : minimum(sin(x), x, S.EmptySet))
raises(ValueError, lambda : minimum(log(cos(x)), x, S.EmptySet))
raises(ValueError, lambda : minimum(1/(x**2 + y**2 + 1), x, S.EmptySet))
raises(ValueError, lambda : minimum(sin(x), sin(x)))
raises(ValueError, lambda : minimum(sin(x), x*y, S.EmptySet))
raises(ValueError, lambda : minimum(sin(x), S.One))
def test_issue_19869():
t = symbols('t')
assert (maximum(sqrt(3)*(t - 1)/(3*sqrt(t**2 + 1)), t)
) == sqrt(3)/3
def test_issue_16469():
x = Symbol("x", real=True)
f = abs(x)
assert function_range(f, x, S.Reals) == Interval(0, oo, False, True)
@_both_exp_pow
def test_issue_18747():
assert periodicity(exp(pi*I*(x/4+S.Half/2)), x) == 8
|
579dae34336f6ab70afc55e75ab3d98e27c5311593cbfd87e25b77166bc0e16e | from itertools import product
from sympy.core.function import (Function, diff)
from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import exp
from sympy.calculus.finite_diff import (
apply_finite_diff, differentiate_finite, finite_diff_weights,
_as_finite_diff
)
from sympy.testing.pytest import raises, warns_deprecated_sympy
def test_apply_finite_diff():
x, h = symbols('x h')
f = Function('f')
assert (apply_finite_diff(1, [x-h, x+h], [f(x-h), f(x+h)], x) -
(f(x+h)-f(x-h))/(2*h)).simplify() == 0
assert (apply_finite_diff(1, [5, 6, 7], [f(5), f(6), f(7)], 5) -
(Rational(-3, 2)*f(5) + 2*f(6) - S.Half*f(7))).simplify() == 0
raises(ValueError, lambda: apply_finite_diff(1, [x, h], [f(x)]))
def test_finite_diff_weights():
d = finite_diff_weights(1, [5, 6, 7], 5)
assert d[1][2] == [Rational(-3, 2), 2, Rational(-1, 2)]
# Table 1, p. 702 in doi:10.1090/S0025-5718-1988-0935077-0
# --------------------------------------------------------
xl = [0, 1, -1, 2, -2, 3, -3, 4, -4]
# d holds all coefficients
d = finite_diff_weights(4, xl, S.Zero)
# Zeroeth derivative
for i in range(5):
assert d[0][i] == [S.One] + [S.Zero]*8
# First derivative
assert d[1][0] == [S.Zero]*9
assert d[1][2] == [S.Zero, S.Half, Rational(-1, 2)] + [S.Zero]*6
assert d[1][4] == [S.Zero, Rational(2, 3), Rational(-2, 3), Rational(-1, 12), Rational(1, 12)] + [S.Zero]*4
assert d[1][6] == [S.Zero, Rational(3, 4), Rational(-3, 4), Rational(-3, 20), Rational(3, 20),
Rational(1, 60), Rational(-1, 60)] + [S.Zero]*2
assert d[1][8] == [S.Zero, Rational(4, 5), Rational(-4, 5), Rational(-1, 5), Rational(1, 5),
Rational(4, 105), Rational(-4, 105), Rational(-1, 280), Rational(1, 280)]
# Second derivative
for i in range(2):
assert d[2][i] == [S.Zero]*9
assert d[2][2] == [-S(2), S.One, S.One] + [S.Zero]*6
assert d[2][4] == [Rational(-5, 2), Rational(4, 3), Rational(4, 3), Rational(-1, 12), Rational(-1, 12)] + [S.Zero]*4
assert d[2][6] == [Rational(-49, 18), Rational(3, 2), Rational(3, 2), Rational(-3, 20), Rational(-3, 20),
Rational(1, 90), Rational(1, 90)] + [S.Zero]*2
assert d[2][8] == [Rational(-205, 72), Rational(8, 5), Rational(8, 5), Rational(-1, 5), Rational(-1, 5),
Rational(8, 315), Rational(8, 315), Rational(-1, 560), Rational(-1, 560)]
# Third derivative
for i in range(3):
assert d[3][i] == [S.Zero]*9
assert d[3][4] == [S.Zero, -S.One, S.One, S.Half, Rational(-1, 2)] + [S.Zero]*4
assert d[3][6] == [S.Zero, Rational(-13, 8), Rational(13, 8), S.One, -S.One,
Rational(-1, 8), Rational(1, 8)] + [S.Zero]*2
assert d[3][8] == [S.Zero, Rational(-61, 30), Rational(61, 30), Rational(169, 120), Rational(-169, 120),
Rational(-3, 10), Rational(3, 10), Rational(7, 240), Rational(-7, 240)]
# Fourth derivative
for i in range(4):
assert d[4][i] == [S.Zero]*9
assert d[4][4] == [S(6), -S(4), -S(4), S.One, S.One] + [S.Zero]*4
assert d[4][6] == [Rational(28, 3), Rational(-13, 2), Rational(-13, 2), S(2), S(2),
Rational(-1, 6), Rational(-1, 6)] + [S.Zero]*2
assert d[4][8] == [Rational(91, 8), Rational(-122, 15), Rational(-122, 15), Rational(169, 60), Rational(169, 60),
Rational(-2, 5), Rational(-2, 5), Rational(7, 240), Rational(7, 240)]
# Table 2, p. 703 in doi:10.1090/S0025-5718-1988-0935077-0
# --------------------------------------------------------
xl = [[j/S(2) for j in list(range(-i*2+1, 0, 2))+list(range(1, i*2+1, 2))]
for i in range(1, 5)]
# d holds all coefficients
d = [finite_diff_weights({0: 1, 1: 2, 2: 4, 3: 4}[i], xl[i], 0) for
i in range(4)]
# Zeroth derivative
assert d[0][0][1] == [S.Half, S.Half]
assert d[1][0][3] == [Rational(-1, 16), Rational(9, 16), Rational(9, 16), Rational(-1, 16)]
assert d[2][0][5] == [Rational(3, 256), Rational(-25, 256), Rational(75, 128), Rational(75, 128),
Rational(-25, 256), Rational(3, 256)]
assert d[3][0][7] == [Rational(-5, 2048), Rational(49, 2048), Rational(-245, 2048), Rational(1225, 2048),
Rational(1225, 2048), Rational(-245, 2048), Rational(49, 2048), Rational(-5, 2048)]
# First derivative
assert d[0][1][1] == [-S.One, S.One]
assert d[1][1][3] == [Rational(1, 24), Rational(-9, 8), Rational(9, 8), Rational(-1, 24)]
assert d[2][1][5] == [Rational(-3, 640), Rational(25, 384), Rational(-75, 64),
Rational(75, 64), Rational(-25, 384), Rational(3, 640)]
assert d[3][1][7] == [Rational(5, 7168), Rational(-49, 5120),
Rational(245, 3072), Rational(-1225, 1024),
Rational(1225, 1024), Rational(-245, 3072),
Rational(49, 5120), Rational(-5, 7168)]
# Reasonably the rest of the table is also correct... (testing of that
# deemed excessive at the moment)
raises(ValueError, lambda: finite_diff_weights(-1, [1, 2]))
raises(ValueError, lambda: finite_diff_weights(1.2, [1, 2]))
x = symbols('x')
raises(ValueError, lambda: finite_diff_weights(x, [1, 2]))
def test_as_finite_diff():
x = symbols('x')
f = Function('f')
dx = Function('dx')
_as_finite_diff(f(x).diff(x), [x-2, x-1, x, x+1, x+2])
# Use of undefined functions in ``points``
df_true = -f(x+dx(x)/2-dx(x+dx(x)/2)/2) / dx(x+dx(x)/2) \
+ f(x+dx(x)/2+dx(x+dx(x)/2)/2) / dx(x+dx(x)/2)
df_test = diff(f(x), x).as_finite_difference(points=dx(x), x0=x+dx(x)/2)
assert (df_test - df_true).simplify() == 0
def test_differentiate_finite():
x, y, h = symbols('x y h')
f = Function('f')
with warns_deprecated_sympy():
res0 = differentiate_finite(f(x, y) + exp(42), x, y, evaluate=True)
xm, xp, ym, yp = [v + sign*S.Half for v, sign in product([x, y], [-1, 1])]
ref0 = f(xm, ym) + f(xp, yp) - f(xm, yp) - f(xp, ym)
assert (res0 - ref0).simplify() == 0
g = Function('g')
with warns_deprecated_sympy():
res1 = differentiate_finite(f(x)*g(x) + 42, x, evaluate=True)
ref1 = (-f(x - S.Half) + f(x + S.Half))*g(x) + \
(-g(x - S.Half) + g(x + S.Half))*f(x)
assert (res1 - ref1).simplify() == 0
res2 = differentiate_finite(f(x) + x**3 + 42, x, points=[x-1, x+1])
ref2 = (f(x + 1) + (x + 1)**3 - f(x - 1) - (x - 1)**3)/2
assert (res2 - ref2).simplify() == 0
raises(TypeError, lambda: differentiate_finite(f(x)*g(x), x,
pints=[x-1, x+1]))
res3 = differentiate_finite(f(x)*g(x).diff(x), x)
ref3 = (-g(x) + g(x + 1))*f(x + S.Half) - (g(x) - g(x - 1))*f(x - S.Half)
assert res3 == ref3
res4 = differentiate_finite(f(x)*g(x).diff(x).diff(x), x)
ref4 = -((g(x - Rational(3, 2)) - 2*g(x - S.Half) + g(x + S.Half))*f(x - S.Half)) \
+ (g(x - S.Half) - 2*g(x + S.Half) + g(x + Rational(3, 2)))*f(x + S.Half)
assert res4 == ref4
res5_expr = f(x).diff(x)*g(x).diff(x)
res5 = differentiate_finite(res5_expr, points=[x-h, x, x+h])
ref5 = (-2*f(x)/h + f(-h + x)/(2*h) + 3*f(h + x)/(2*h))*(-2*g(x)/h + g(-h + x)/(2*h) \
+ 3*g(h + x)/(2*h))/(2*h) - (2*f(x)/h - 3*f(-h + x)/(2*h) - \
f(h + x)/(2*h))*(2*g(x)/h - 3*g(-h + x)/(2*h) - g(h + x)/(2*h))/(2*h)
assert res5 == ref5
res6 = res5.limit(h, 0).doit()
ref6 = diff(res5_expr, x)
assert res6 == ref6
|
f82efcf5efb93a72e924abd093ebd8eefebc3841bbc21a2b7d7dbc03c8ec0230 | from textwrap import dedent
import sys
from subprocess import Popen, PIPE
import os
from sympy.core.singleton import S
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.utilities.misc import (translate, replace, ordinal, rawlines,
strlines, as_int, find_executable)
def test_translate():
abc = 'abc'
assert translate(abc, None, 'a') == 'bc'
assert translate(abc, None, '') == 'abc'
assert translate(abc, {'a': 'x'}, 'c') == 'xb'
assert translate(abc, {'a': 'bc'}, 'c') == 'bcb'
assert translate(abc, {'ab': 'x'}, 'c') == 'x'
assert translate(abc, {'ab': ''}, 'c') == ''
assert translate(abc, {'bc': 'x'}, 'c') == 'ab'
assert translate(abc, {'abc': 'x', 'a': 'y'}) == 'x'
u = chr(4096)
assert translate(abc, 'a', 'x', u) == 'xbc'
assert (u in translate(abc, 'a', u, u)) is True
def test_replace():
assert replace('abc', ('a', 'b')) == 'bbc'
assert replace('abc', {'a': 'Aa'}) == 'Aabc'
assert replace('abc', ('a', 'b'), ('c', 'C')) == 'bbC'
def test_ordinal():
assert ordinal(-1) == '-1st'
assert ordinal(0) == '0th'
assert ordinal(1) == '1st'
assert ordinal(2) == '2nd'
assert ordinal(3) == '3rd'
assert all(ordinal(i).endswith('th') for i in range(4, 21))
assert ordinal(100) == '100th'
assert ordinal(101) == '101st'
assert ordinal(102) == '102nd'
assert ordinal(103) == '103rd'
assert ordinal(104) == '104th'
assert ordinal(200) == '200th'
assert all(ordinal(i) == str(i) + 'th' for i in range(-220, -203))
def test_rawlines():
assert rawlines('a a\na') == "dedent('''\\\n a a\n a''')"
assert rawlines('a a') == "'a a'"
assert rawlines(strlines('\\le"ft')) == (
'(\n'
" '(\\n'\n"
' \'r\\\'\\\\le"ft\\\'\\n\'\n'
" ')'\n"
')')
def test_strlines():
q = 'this quote (") is in the middle'
# the following assert rhs was prepared with
# print(rawlines(strlines(q, 10)))
assert strlines(q, 10) == dedent('''\
(
'this quo'
'te (") i'
's in the'
' middle'
)''')
assert q == (
'this quo'
'te (") i'
's in the'
' middle'
)
q = "this quote (') is in the middle"
assert strlines(q, 20) == dedent('''\
(
"this quote (') is "
"in the middle"
)''')
assert strlines('\\left') == (
'(\n'
"r'\\left'\n"
')')
assert strlines('\\left', short=True) == r"r'\left'"
assert strlines('\\le"ft') == (
'(\n'
'r\'\\le"ft\'\n'
')')
q = 'this\nother line'
assert strlines(q) == rawlines(q)
def test_translate_args():
try:
translate(None, None, None, 'not_none')
except ValueError:
pass # Exception raised successfully
else:
assert False
assert translate('s', None, None, None) == 's'
try:
translate('s', 'a', 'bc')
except ValueError:
pass # Exception raised successfully
else:
assert False
def test_debug_output():
env = os.environ.copy()
env['SYMPY_DEBUG'] = 'True'
cmd = 'from sympy import *; x = Symbol("x"); print(integrate((1-cos(x))/x, x))'
cmdline = [sys.executable, '-c', cmd]
proc = Popen(cmdline, env=env, stdout=PIPE, stderr=PIPE)
out, err = proc.communicate()
out = out.decode('ascii') # utf-8?
err = err.decode('ascii')
expected = 'substituted: -x*(1 - cos(x)), u: 1/x, u_var: _u'
assert expected in err, err
def test_as_int():
raises(ValueError, lambda : as_int(True))
raises(ValueError, lambda : as_int(1.1))
raises(ValueError, lambda : as_int([]))
raises(ValueError, lambda : as_int(S.NaN))
raises(ValueError, lambda : as_int(S.Infinity))
raises(ValueError, lambda : as_int(S.NegativeInfinity))
raises(ValueError, lambda : as_int(S.ComplexInfinity))
# for the following, limited precision makes int(arg) == arg
# but the int value is not necessarily what a user might have
# expected; Q.prime is more nuanced in its response for
# expressions which might be complex representations of an
# integer. This is not -- by design -- as_ints role.
raises(ValueError, lambda : as_int(1e23))
raises(ValueError, lambda : as_int(S('1.'+'0'*20+'1')))
assert as_int(True, strict=False) == 1
def test_deprecated_find_executable():
with warns_deprecated_sympy():
find_executable('python')
|
7547c37c87687ca62e7f0c36f6357c9954268255350271b6b84c91129358ed5d | import pickle
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.external import import_module
from sympy.testing.pytest import skip
from sympy.utilities.matchpy_connector import WildDot, WildPlus, WildStar, Replacer
matchpy = import_module("matchpy")
x, y, z = symbols("x y z")
def _get_first_match(expr, pattern):
from matchpy import ManyToOneMatcher, Pattern
matcher = ManyToOneMatcher()
matcher.add(Pattern(pattern))
return next(iter(matcher.match(expr)))
def test_matchpy_connector():
if matchpy is None:
skip("matchpy not installed")
from multiset import Multiset
from matchpy import Pattern, Substitution
w_ = WildDot("w_")
w__ = WildPlus("w__")
w___ = WildStar("w___")
expr = x + y
pattern = x + w_
p, subst = _get_first_match(expr, pattern)
assert p == Pattern(pattern)
assert subst == Substitution({'w_': y})
expr = x + y + z
pattern = x + w__
p, subst = _get_first_match(expr, pattern)
assert p == Pattern(pattern)
assert subst == Substitution({'w__': Multiset([y, z])})
expr = x + y + z
pattern = x + y + z + w___
p, subst = _get_first_match(expr, pattern)
assert p == Pattern(pattern)
assert subst == Substitution({'w___': Multiset()})
def test_matchpy_optional():
if matchpy is None:
skip("matchpy not installed")
from matchpy import Pattern, Substitution
from matchpy import ManyToOneReplacer, ReplacementRule
p = WildDot("p", optional=1)
q = WildDot("q", optional=0)
pattern = p*x + q
expr1 = 2*x
pa, subst = _get_first_match(expr1, pattern)
assert pa == Pattern(pattern)
assert subst == Substitution({'p': 2, 'q': 0})
expr2 = x + 3
pa, subst = _get_first_match(expr2, pattern)
assert pa == Pattern(pattern)
assert subst == Substitution({'p': 1, 'q': 3})
expr3 = x
pa, subst = _get_first_match(expr3, pattern)
assert pa == Pattern(pattern)
assert subst == Substitution({'p': 1, 'q': 0})
expr4 = x*y + z
pa, subst = _get_first_match(expr4, pattern)
assert pa == Pattern(pattern)
assert subst == Substitution({'p': y, 'q': z})
replacer = ManyToOneReplacer()
replacer.add(ReplacementRule(Pattern(pattern), lambda p, q: sin(p)*cos(q)))
assert replacer.replace(expr1) == sin(2)*cos(0)
assert replacer.replace(expr2) == sin(1)*cos(3)
assert replacer.replace(expr3) == sin(1)*cos(0)
assert replacer.replace(expr4) == sin(y)*cos(z)
def test_replacer():
if matchpy is None:
skip("matchpy not installed")
x1_ = WildDot("x1_")
x2_ = WildDot("x2_")
a_ = WildDot("a_", optional=S.One)
b_ = WildDot("b_", optional=S.One)
c_ = WildDot("c_", optional=S.Zero)
replacer = Replacer(common_constraints=[
matchpy.CustomConstraint(lambda a_: not a_.has(x)),
matchpy.CustomConstraint(lambda b_: not b_.has(x)),
matchpy.CustomConstraint(lambda c_: not c_.has(x)),
])
# Rewrite the equation into implicit form, unless it's already solved:
replacer.add(Eq(x1_, x2_), Eq(x1_ - x2_, 0), conditions_nonfalse=[Ne(x2_, 0), Ne(x1_, 0), Ne(x1_, x), Ne(x2_, x)])
# Simple equation solver for real numbers:
replacer.add(Eq(a_*x + b_, 0), Eq(x, -b_/a_))
disc = b_**2 - 4*a_*c_
replacer.add(
Eq(a_*x**2 + b_*x + c_, 0),
Eq(x, (-b_ - sqrt(disc))/(2*a_)) | Eq(x, (-b_ + sqrt(disc))/(2*a_)),
conditions_nonfalse=[disc >= 0]
)
replacer.add(
Eq(a_*x**2 + c_, 0),
Eq(x, sqrt(-c_/a_)) | Eq(x, -sqrt(-c_/a_)),
conditions_nonfalse=[-c_*a_ > 0]
)
assert replacer.replace(Eq(3*x, y)) == Eq(x, y/3)
assert replacer.replace(Eq(x**2 + 1, 0)) == Eq(x**2 + 1, 0)
assert replacer.replace(Eq(x**2, 4)) == (Eq(x, 2) | Eq(x, -2))
assert replacer.replace(Eq(x**2 + 4*y*x + 4*y**2, 0)) == Eq(x, -2*y)
def test_matchpy_object_pickle():
if matchpy is None:
return
a1 = WildDot("a")
a2 = pickle.loads(pickle.dumps(a1))
assert a1 == a2
a1 = WildDot("a", S(1))
a2 = pickle.loads(pickle.dumps(a1))
assert a1 == a2
a1 = WildPlus("a", S(1))
a2 = pickle.loads(pickle.dumps(a1))
assert a1 == a2
a1 = WildStar("a", S(1))
a2 = pickle.loads(pickle.dumps(a1))
assert a1 == a2
|
f3815f789ce843d25bc8c593b60ce598e635f4f3517395c29cfaf04e211036ca | import inspect
import copy
import pickle
from sympy.physics.units import meter
from sympy.testing.pytest import XFAIL, raises
from sympy.core.basic import Atom, Basic
from sympy.core.core import BasicMeta
from sympy.core.singleton import SingletonRegistry
from sympy.core.symbol import Str, Dummy, Symbol, Wild
from sympy.core.numbers import (E, I, pi, oo, zoo, nan, Integer,
Rational, Float, AlgebraicNumber)
from sympy.core.relational import (Equality, GreaterThan, LessThan, Relational,
StrictGreaterThan, StrictLessThan, Unequality)
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.function import Derivative, Function, FunctionClass, Lambda, \
WildFunction
from sympy.sets.sets import Interval
from sympy.core.multidimensional import vectorize
from sympy.external.gmpy import HAS_GMPY
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.external import import_module
cloudpickle = import_module('cloudpickle')
excluded_attrs = {
'_assumptions', # This is a local cache that isn't automatically filled on creation
'_mhash', # Cached after __hash__ is called but set to None after creation
'is_EmptySet', # Deprecated from SymPy 1.5. This can be removed when is_EmptySet is removed.
'expr_free_symbols', # Deprecated from SymPy 1.9. This can be removed when exr_free_symbols is removed.
'_mat', # Deprecated from SymPy 1.9. This can be removed when Matrix._mat is removed
'_smat', # Deprecated from SymPy 1.9. This can be removed when SparseMatrix._smat is removed
}
def check(a, exclude=[], check_attr=True):
""" Check that pickling and copying round-trips.
"""
# Pickling with protocols 0 and 1 is disabled for Basic instances:
if isinstance(a, Basic):
for protocol in [0, 1]:
raises(NotImplementedError, lambda: pickle.dumps(a, protocol))
protocols = [2, copy.copy, copy.deepcopy, 3, 4]
if cloudpickle:
protocols.extend([cloudpickle])
for protocol in protocols:
if protocol in exclude:
continue
if callable(protocol):
if isinstance(a, BasicMeta):
# Classes can't be copied, but that's okay.
continue
b = protocol(a)
elif inspect.ismodule(protocol):
b = protocol.loads(protocol.dumps(a))
else:
b = pickle.loads(pickle.dumps(a, protocol))
d1 = dir(a)
d2 = dir(b)
assert set(d1) == set(d2)
if not check_attr:
continue
def c(a, b, d):
for i in d:
if i in excluded_attrs:
continue
if not hasattr(a, i):
continue
attr = getattr(a, i)
if not hasattr(attr, "__call__"):
assert hasattr(b, i), i
assert getattr(b, i) == attr, "%s != %s, protocol: %s" % (getattr(b, i), attr, protocol)
c(a, b, d1)
c(b, a, d2)
#================== core =========================
def test_core_basic():
for c in (Atom, Atom(),
Basic, Basic(),
# XXX: dynamically created types are not picklable
# BasicMeta, BasicMeta("test", (), {}),
SingletonRegistry, S):
check(c)
def test_core_Str():
check(Str('x'))
def test_core_symbol():
# make the Symbol a unique name that doesn't class with any other
# testing variable in this file since after this test the symbol
# having the same name will be cached as noncommutative
for c in (Dummy, Dummy("x", commutative=False), Symbol,
Symbol("_issue_3130", commutative=False), Wild, Wild("x")):
check(c)
def test_core_numbers():
for c in (Integer(2), Rational(2, 3), Float("1.2")):
check(c)
for c in (AlgebraicNumber, AlgebraicNumber(sqrt(3))):
check(c, check_attr=False)
def test_core_float_copy():
# See gh-7457
y = Symbol("x") + 1.0
check(y) # does not raise TypeError ("argument is not an mpz")
def test_core_relational():
x = Symbol("x")
y = Symbol("y")
for c in (Equality, Equality(x, y), GreaterThan, GreaterThan(x, y),
LessThan, LessThan(x, y), Relational, Relational(x, y),
StrictGreaterThan, StrictGreaterThan(x, y), StrictLessThan,
StrictLessThan(x, y), Unequality, Unequality(x, y)):
check(c)
def test_core_add():
x = Symbol("x")
for c in (Add, Add(x, 4)):
check(c)
def test_core_mul():
x = Symbol("x")
for c in (Mul, Mul(x, 4)):
check(c)
def test_core_power():
x = Symbol("x")
for c in (Pow, Pow(x, 4)):
check(c)
def test_core_function():
x = Symbol("x")
for f in (Derivative, Derivative(x), Function, FunctionClass, Lambda,
WildFunction):
check(f)
def test_core_undefinedfunctions():
f = Function("f")
# Full XFAILed test below
exclude = list(range(5))
# https://github.com/cloudpipe/cloudpickle/issues/65
# https://github.com/cloudpipe/cloudpickle/issues/190
exclude.append(cloudpickle)
check(f, exclude=exclude)
@XFAIL
def test_core_undefinedfunctions_fail():
# This fails because f is assumed to be a class at sympy.basic.function.f
f = Function("f")
check(f)
def test_core_interval():
for c in (Interval, Interval(0, 2)):
check(c)
def test_core_multidimensional():
for c in (vectorize, vectorize(0)):
check(c)
def test_Singletons():
protocols = [0, 1, 2, 3, 4]
copiers = [copy.copy, copy.deepcopy]
copiers += [lambda x: pickle.loads(pickle.dumps(x, proto))
for proto in protocols]
if cloudpickle:
copiers += [lambda x: cloudpickle.loads(cloudpickle.dumps(x))]
for obj in (Integer(-1), Integer(0), Integer(1), Rational(1, 2), pi, E, I,
oo, -oo, zoo, nan, S.GoldenRatio, S.TribonacciConstant,
S.EulerGamma, S.Catalan, S.EmptySet, S.IdentityFunction):
for func in copiers:
assert func(obj) is obj
#================== functions ===================
from sympy.functions import (Piecewise, lowergamma, acosh, chebyshevu,
chebyshevt, ln, chebyshevt_root, legendre, Heaviside, bernoulli, coth,
tanh, assoc_legendre, sign, arg, asin, DiracDelta, re, rf, Abs,
uppergamma, binomial, sinh, cos, cot, acos, acot, gamma, bell,
hermite, harmonic, LambertW, zeta, log, factorial, asinh, acoth, cosh,
dirichlet_eta, Eijk, loggamma, erf, ceiling, im, fibonacci,
tribonacci, conjugate, tan, chebyshevu_root, floor, atanh, sqrt, sin,
atan, ff, lucas, atan2, polygamma, exp)
def test_functions():
one_var = (acosh, ln, Heaviside, factorial, bernoulli, coth, tanh,
sign, arg, asin, DiracDelta, re, Abs, sinh, cos, cot, acos, acot,
gamma, bell, harmonic, LambertW, zeta, log, factorial, asinh,
acoth, cosh, dirichlet_eta, loggamma, erf, ceiling, im, fibonacci,
tribonacci, conjugate, tan, floor, atanh, sin, atan, lucas, exp)
two_var = (rf, ff, lowergamma, chebyshevu, chebyshevt, binomial,
atan2, polygamma, hermite, legendre, uppergamma)
x, y, z = symbols("x,y,z")
others = (chebyshevt_root, chebyshevu_root, Eijk(x, y, z),
Piecewise( (0, x < -1), (x**2, x <= 1), (x**3, True)),
assoc_legendre)
for cls in one_var:
check(cls)
c = cls(x)
check(c)
for cls in two_var:
check(cls)
c = cls(x, y)
check(c)
for cls in others:
check(cls)
#================== geometry ====================
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
from sympy.geometry.ellipse import Circle, Ellipse
from sympy.geometry.line import Line, LinearEntity, Ray, Segment
from sympy.geometry.polygon import Polygon, RegularPolygon, Triangle
def test_geometry():
p1 = Point(1, 2)
p2 = Point(2, 3)
p3 = Point(0, 0)
p4 = Point(0, 1)
for c in (
GeometryEntity, GeometryEntity(), Point, p1, Circle, Circle(p1, 2),
Ellipse, Ellipse(p1, 3, 4), Line, Line(p1, p2), LinearEntity,
LinearEntity(p1, p2), Ray, Ray(p1, p2), Segment, Segment(p1, p2),
Polygon, Polygon(p1, p2, p3, p4), RegularPolygon,
RegularPolygon(p1, 4, 5), Triangle, Triangle(p1, p2, p3)):
check(c, check_attr=False)
#================== integrals ====================
from sympy.integrals.integrals import Integral
def test_integrals():
x = Symbol("x")
for c in (Integral, Integral(x)):
check(c)
#==================== logic =====================
from sympy.core.logic import Logic
def test_logic():
for c in (Logic, Logic(1)):
check(c)
#================== matrices ====================
from sympy.matrices import Matrix, SparseMatrix
def test_matrices():
for c in (Matrix, Matrix([1, 2, 3]), SparseMatrix, SparseMatrix([[1, 2], [3, 4]])):
check(c)
#================== ntheory =====================
from sympy.ntheory.generate import Sieve
def test_ntheory():
for c in (Sieve, Sieve()):
check(c)
#================== physics =====================
from sympy.physics.paulialgebra import Pauli
from sympy.physics.units import Unit
def test_physics():
for c in (Unit, meter, Pauli, Pauli(1)):
check(c)
#================== plotting ====================
# XXX: These tests are not complete, so XFAIL them
@XFAIL
def test_plotting():
from sympy.plotting.pygletplot.color_scheme import ColorGradient, ColorScheme
from sympy.plotting.pygletplot.managed_window import ManagedWindow
from sympy.plotting.plot import Plot, ScreenShot
from sympy.plotting.pygletplot.plot_axes import PlotAxes, PlotAxesBase, PlotAxesFrame, PlotAxesOrdinate
from sympy.plotting.pygletplot.plot_camera import PlotCamera
from sympy.plotting.pygletplot.plot_controller import PlotController
from sympy.plotting.pygletplot.plot_curve import PlotCurve
from sympy.plotting.pygletplot.plot_interval import PlotInterval
from sympy.plotting.pygletplot.plot_mode import PlotMode
from sympy.plotting.pygletplot.plot_modes import Cartesian2D, Cartesian3D, Cylindrical, \
ParametricCurve2D, ParametricCurve3D, ParametricSurface, Polar, Spherical
from sympy.plotting.pygletplot.plot_object import PlotObject
from sympy.plotting.pygletplot.plot_surface import PlotSurface
from sympy.plotting.pygletplot.plot_window import PlotWindow
for c in (
ColorGradient, ColorGradient(0.2, 0.4), ColorScheme, ManagedWindow,
ManagedWindow, Plot, ScreenShot, PlotAxes, PlotAxesBase,
PlotAxesFrame, PlotAxesOrdinate, PlotCamera, PlotController,
PlotCurve, PlotInterval, PlotMode, Cartesian2D, Cartesian3D,
Cylindrical, ParametricCurve2D, ParametricCurve3D,
ParametricSurface, Polar, Spherical, PlotObject, PlotSurface,
PlotWindow):
check(c)
@XFAIL
def test_plotting2():
#from sympy.plotting.color_scheme import ColorGradient
from sympy.plotting.pygletplot.color_scheme import ColorScheme
#from sympy.plotting.managed_window import ManagedWindow
from sympy.plotting.plot import Plot
#from sympy.plotting.plot import ScreenShot
from sympy.plotting.pygletplot.plot_axes import PlotAxes
#from sympy.plotting.plot_axes import PlotAxesBase, PlotAxesFrame, PlotAxesOrdinate
#from sympy.plotting.plot_camera import PlotCamera
#from sympy.plotting.plot_controller import PlotController
#from sympy.plotting.plot_curve import PlotCurve
#from sympy.plotting.plot_interval import PlotInterval
#from sympy.plotting.plot_mode import PlotMode
#from sympy.plotting.plot_modes import Cartesian2D, Cartesian3D, Cylindrical, \
# ParametricCurve2D, ParametricCurve3D, ParametricSurface, Polar, Spherical
#from sympy.plotting.plot_object import PlotObject
#from sympy.plotting.plot_surface import PlotSurface
# from sympy.plotting.plot_window import PlotWindow
check(ColorScheme("rainbow"))
check(Plot(1, visible=False))
check(PlotAxes())
#================== polys =======================
from sympy.polys.domains.integerring import ZZ
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.orderings import lex
from sympy.polys.polytools import Poly
def test_pickling_polys_polytools():
from sympy.polys.polytools import PurePoly
# from sympy.polys.polytools import GroebnerBasis
x = Symbol('x')
for c in (Poly, Poly(x, x)):
check(c)
for c in (PurePoly, PurePoly(x)):
check(c)
# TODO: fix pickling of Options class (see GroebnerBasis._options)
# for c in (GroebnerBasis, GroebnerBasis([x**2 - 1], x, order=lex)):
# check(c)
def test_pickling_polys_polyclasses():
from sympy.polys.polyclasses import DMP, DMF, ANP
for c in (DMP, DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)]], ZZ)):
check(c)
for c in (DMF, DMF(([ZZ(1), ZZ(2)], [ZZ(1), ZZ(3)]), ZZ)):
check(c)
for c in (ANP, ANP([QQ(1), QQ(2)], [QQ(1), QQ(2), QQ(3)], QQ)):
check(c)
@XFAIL
def test_pickling_polys_rings():
# NOTE: can't use protocols < 2 because we have to execute __new__ to
# make sure caching of rings works properly.
from sympy.polys.rings import PolyRing
ring = PolyRing("x,y,z", ZZ, lex)
for c in (PolyRing, ring):
check(c, exclude=[0, 1])
for c in (ring.dtype, ring.one):
check(c, exclude=[0, 1], check_attr=False) # TODO: Py3k
def test_pickling_polys_fields():
pass
# NOTE: can't use protocols < 2 because we have to execute __new__ to
# make sure caching of fields works properly.
# from sympy.polys.fields import FracField
# field = FracField("x,y,z", ZZ, lex)
# TODO: AssertionError: assert id(obj) not in self.memo
# for c in (FracField, field):
# check(c, exclude=[0, 1])
# TODO: AssertionError: assert id(obj) not in self.memo
# for c in (field.dtype, field.one):
# check(c, exclude=[0, 1])
def test_pickling_polys_elements():
from sympy.polys.domains.pythonrational import PythonRational
#from sympy.polys.domains.pythonfinitefield import PythonFiniteField
#from sympy.polys.domains.mpelements import MPContext
for c in (PythonRational, PythonRational(1, 7)):
check(c)
#gf = PythonFiniteField(17)
# TODO: fix pickling of ModularInteger
# for c in (gf.dtype, gf(5)):
# check(c)
#mp = MPContext()
# TODO: fix pickling of RealElement
# for c in (mp.mpf, mp.mpf(1.0)):
# check(c)
# TODO: fix pickling of ComplexElement
# for c in (mp.mpc, mp.mpc(1.0, -1.5)):
# check(c)
def test_pickling_polys_domains():
# from sympy.polys.domains.pythonfinitefield import PythonFiniteField
from sympy.polys.domains.pythonintegerring import PythonIntegerRing
from sympy.polys.domains.pythonrationalfield import PythonRationalField
# TODO: fix pickling of ModularInteger
# for c in (PythonFiniteField, PythonFiniteField(17)):
# check(c)
for c in (PythonIntegerRing, PythonIntegerRing()):
check(c, check_attr=False)
for c in (PythonRationalField, PythonRationalField()):
check(c, check_attr=False)
if HAS_GMPY:
# from sympy.polys.domains.gmpyfinitefield import GMPYFiniteField
from sympy.polys.domains.gmpyintegerring import GMPYIntegerRing
from sympy.polys.domains.gmpyrationalfield import GMPYRationalField
# TODO: fix pickling of ModularInteger
# for c in (GMPYFiniteField, GMPYFiniteField(17)):
# check(c)
for c in (GMPYIntegerRing, GMPYIntegerRing()):
check(c, check_attr=False)
for c in (GMPYRationalField, GMPYRationalField()):
check(c, check_attr=False)
#from sympy.polys.domains.realfield import RealField
#from sympy.polys.domains.complexfield import ComplexField
from sympy.polys.domains.algebraicfield import AlgebraicField
#from sympy.polys.domains.polynomialring import PolynomialRing
#from sympy.polys.domains.fractionfield import FractionField
from sympy.polys.domains.expressiondomain import ExpressionDomain
# TODO: fix pickling of RealElement
# for c in (RealField, RealField(100)):
# check(c)
# TODO: fix pickling of ComplexElement
# for c in (ComplexField, ComplexField(100)):
# check(c)
for c in (AlgebraicField, AlgebraicField(QQ, sqrt(3))):
check(c, check_attr=False)
# TODO: AssertionError
# for c in (PolynomialRing, PolynomialRing(ZZ, "x,y,z")):
# check(c)
# TODO: AttributeError: 'PolyElement' object has no attribute 'ring'
# for c in (FractionField, FractionField(ZZ, "x,y,z")):
# check(c)
for c in (ExpressionDomain, ExpressionDomain()):
check(c, check_attr=False)
def test_pickling_polys_orderings():
from sympy.polys.orderings import (LexOrder, GradedLexOrder,
ReversedGradedLexOrder, InverseOrder)
# from sympy.polys.orderings import ProductOrder
for c in (LexOrder, LexOrder()):
check(c)
for c in (GradedLexOrder, GradedLexOrder()):
check(c)
for c in (ReversedGradedLexOrder, ReversedGradedLexOrder()):
check(c)
# TODO: Argh, Python is so naive. No lambdas nor inner function support in
# pickling module. Maybe someone could figure out what to do with this.
#
# for c in (ProductOrder, ProductOrder((LexOrder(), lambda m: m[:2]),
# (GradedLexOrder(), lambda m: m[2:]))):
# check(c)
for c in (InverseOrder, InverseOrder(LexOrder())):
check(c)
def test_pickling_polys_monomials():
from sympy.polys.monomials import MonomialOps, Monomial
x, y, z = symbols("x,y,z")
for c in (MonomialOps, MonomialOps(3)):
check(c)
for c in (Monomial, Monomial((1, 2, 3), (x, y, z))):
check(c)
def test_pickling_polys_errors():
from sympy.polys.polyerrors import (HeuristicGCDFailed,
HomomorphismFailed, IsomorphismFailed, ExtraneousFactors,
EvaluationFailed, RefinementFailed, CoercionFailed, NotInvertible,
NotReversible, NotAlgebraic, DomainError, PolynomialError,
UnificationFailed, GeneratorsError, GeneratorsNeeded,
UnivariatePolynomialError, MultivariatePolynomialError, OptionError,
FlagError)
# from sympy.polys.polyerrors import (ExactQuotientFailed,
# OperationNotSupported, ComputationFailed, PolificationFailed)
# x = Symbol('x')
# TODO: TypeError: __init__() takes at least 3 arguments (1 given)
# for c in (ExactQuotientFailed, ExactQuotientFailed(x, 3*x, ZZ)):
# check(c)
# TODO: TypeError: can't pickle instancemethod objects
# for c in (OperationNotSupported, OperationNotSupported(Poly(x), Poly.gcd)):
# check(c)
for c in (HeuristicGCDFailed, HeuristicGCDFailed()):
check(c)
for c in (HomomorphismFailed, HomomorphismFailed()):
check(c)
for c in (IsomorphismFailed, IsomorphismFailed()):
check(c)
for c in (ExtraneousFactors, ExtraneousFactors()):
check(c)
for c in (EvaluationFailed, EvaluationFailed()):
check(c)
for c in (RefinementFailed, RefinementFailed()):
check(c)
for c in (CoercionFailed, CoercionFailed()):
check(c)
for c in (NotInvertible, NotInvertible()):
check(c)
for c in (NotReversible, NotReversible()):
check(c)
for c in (NotAlgebraic, NotAlgebraic()):
check(c)
for c in (DomainError, DomainError()):
check(c)
for c in (PolynomialError, PolynomialError()):
check(c)
for c in (UnificationFailed, UnificationFailed()):
check(c)
for c in (GeneratorsError, GeneratorsError()):
check(c)
for c in (GeneratorsNeeded, GeneratorsNeeded()):
check(c)
# TODO: PicklingError: Can't pickle <function <lambda> at 0x38578c0>: it's not found as __main__.<lambda>
# for c in (ComputationFailed, ComputationFailed(lambda t: t, 3, None)):
# check(c)
for c in (UnivariatePolynomialError, UnivariatePolynomialError()):
check(c)
for c in (MultivariatePolynomialError, MultivariatePolynomialError()):
check(c)
# TODO: TypeError: __init__() takes at least 3 arguments (1 given)
# for c in (PolificationFailed, PolificationFailed({}, x, x, False)):
# check(c)
for c in (OptionError, OptionError()):
check(c)
for c in (FlagError, FlagError()):
check(c)
#def test_pickling_polys_options():
#from sympy.polys.polyoptions import Options
# TODO: fix pickling of `symbols' flag
# for c in (Options, Options((), dict(domain='ZZ', polys=False))):
# check(c)
# TODO: def test_pickling_polys_rootisolation():
# RealInterval
# ComplexInterval
def test_pickling_polys_rootoftools():
from sympy.polys.rootoftools import CRootOf, RootSum
x = Symbol('x')
f = x**3 + x + 3
for c in (CRootOf, CRootOf(f, 0)):
check(c)
for c in (RootSum, RootSum(f, exp)):
check(c)
#================== printing ====================
from sympy.printing.latex import LatexPrinter
from sympy.printing.mathml import MathMLContentPrinter, MathMLPresentationPrinter
from sympy.printing.pretty.pretty import PrettyPrinter
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.printer import Printer
from sympy.printing.python import PythonPrinter
def test_printing():
for c in (LatexPrinter, LatexPrinter(), MathMLContentPrinter,
MathMLPresentationPrinter, PrettyPrinter, prettyForm, stringPict,
stringPict("a"), Printer, Printer(), PythonPrinter,
PythonPrinter()):
check(c)
@XFAIL
def test_printing1():
check(MathMLContentPrinter())
@XFAIL
def test_printing2():
check(MathMLPresentationPrinter())
@XFAIL
def test_printing3():
check(PrettyPrinter())
#================== series ======================
from sympy.series.limits import Limit
from sympy.series.order import Order
def test_series():
e = Symbol("e")
x = Symbol("x")
for c in (Limit, Limit(e, x, 1), Order, Order(e)):
check(c)
#================== concrete ==================
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
def test_concrete():
x = Symbol("x")
for c in (Product, Product(x, (x, 2, 4)), Sum, Sum(x, (x, 2, 4))):
check(c)
def test_deprecation_warning():
w = SymPyDeprecationWarning("message", deprecated_since_version='1.0', active_deprecations_target="active-deprecations")
check(w)
def test_issue_18438():
assert pickle.loads(pickle.dumps(S.Half)) == 1/2
#================= old pickles =================
def test_unpickle_from_older_versions():
data = (
b'\x80\x04\x95^\x00\x00\x00\x00\x00\x00\x00\x8c\x10sympy.core.power'
b'\x94\x8c\x03Pow\x94\x93\x94\x8c\x12sympy.core.numbers\x94\x8c'
b'\x07Integer\x94\x93\x94K\x02\x85\x94R\x94}\x94bh\x03\x8c\x04Half'
b'\x94\x93\x94)R\x94}\x94b\x86\x94R\x94}\x94b.'
)
assert pickle.loads(data) == sqrt(2)
|
e4feeab8ca11ad8fcc60ceda14563ba566faa746d63a0596dc08aff69a9efbb5 | from itertools import product
import math
import inspect
import mpmath
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.concrete.summations import Sum
from sympy.core.function import (Function, Lambda, diff)
from sympy.core.numbers import (E, Float, I, Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.hyperbolic import acosh
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, cos, sin, sinc, tan)
from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely)
from sympy.functions.special.beta_functions import (beta, betainc, betainc_regularized)
from sympy.functions.special.delta_functions import (Heaviside)
from sympy.functions.special.error_functions import (erf, erfc, fresnelc, fresnels)
from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma)
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (And, false, ITE, Not, Or, true)
from sympy.matrices.expressions.dotproduct import DotProduct
from sympy.tensor.array import derive_by_array, Array
from sympy.tensor.indexed import IndexedBase
from sympy.utilities.lambdify import lambdify
from sympy.core.expr import UnevaluatedExpr
from sympy.codegen.cfunctions import expm1, log1p, exp2, log2, log10, hypot
from sympy.codegen.numpy_nodes import logaddexp, logaddexp2
from sympy.codegen.scipy_nodes import cosm1
from sympy.functions.elementary.complexes import re, im, arg
from sympy.functions.special.polynomials import \
chebyshevt, chebyshevu, legendre, hermite, laguerre, gegenbauer, \
assoc_legendre, assoc_laguerre, jacobi
from sympy.matrices import Matrix, MatrixSymbol, SparseMatrix
from sympy.printing.lambdarepr import LambdaPrinter
from sympy.printing.numpy import NumPyPrinter
from sympy.utilities.lambdify import implemented_function, lambdastr
from sympy.testing.pytest import skip
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.external import import_module
from sympy.functions.special.gamma_functions import uppergamma, lowergamma
import sympy
MutableDenseMatrix = Matrix
numpy = import_module('numpy')
scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
numexpr = import_module('numexpr')
tensorflow = import_module('tensorflow')
cupy = import_module('cupy')
numba = import_module('numba')
if tensorflow:
# Hide Tensorflow warnings
import os
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'
w, x, y, z = symbols('w,x,y,z')
#================== Test different arguments =======================
def test_no_args():
f = lambdify([], 1)
raises(TypeError, lambda: f(-1))
assert f() == 1
def test_single_arg():
f = lambdify(x, 2*x)
assert f(1) == 2
def test_list_args():
f = lambdify([x, y], x + y)
assert f(1, 2) == 3
def test_nested_args():
f1 = lambdify([[w, x]], [w, x])
assert f1([91, 2]) == [91, 2]
raises(TypeError, lambda: f1(1, 2))
f2 = lambdify([(w, x), (y, z)], [w, x, y, z])
assert f2((18, 12), (73, 4)) == [18, 12, 73, 4]
raises(TypeError, lambda: f2(3, 4))
f3 = lambdify([w, [[[x]], y], z], [w, x, y, z])
assert f3(10, [[[52]], 31], 44) == [10, 52, 31, 44]
def test_str_args():
f = lambdify('x,y,z', 'z,y,x')
assert f(3, 2, 1) == (1, 2, 3)
assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0)
# make sure correct number of args required
raises(TypeError, lambda: f(0))
def test_own_namespace_1():
myfunc = lambda x: 1
f = lambdify(x, sin(x), {"sin": myfunc})
assert f(0.1) == 1
assert f(100) == 1
def test_own_namespace_2():
def myfunc(x):
return 1
f = lambdify(x, sin(x), {'sin': myfunc})
assert f(0.1) == 1
assert f(100) == 1
def test_own_module():
f = lambdify(x, sin(x), math)
assert f(0) == 0.0
p, q, r = symbols("p q r", real=True)
ae = abs(exp(p+UnevaluatedExpr(q+r)))
f = lambdify([p, q, r], [ae, ae], modules=math)
results = f(1.0, 1e18, -1e18)
refvals = [math.exp(1.0)]*2
for res, ref in zip(results, refvals):
assert abs((res-ref)/ref) < 1e-15
def test_bad_args():
# no vargs given
raises(TypeError, lambda: lambdify(1))
# same with vector exprs
raises(TypeError, lambda: lambdify([1, 2]))
def test_atoms():
# Non-Symbol atoms should not be pulled out from the expression namespace
f = lambdify(x, pi + x, {"pi": 3.14})
assert f(0) == 3.14
f = lambdify(x, I + x, {"I": 1j})
assert f(1) == 1 + 1j
#================== Test different modules =========================
# high precision output of sin(0.2*pi) is used to detect if precision is lost unwanted
@conserve_mpmath_dps
def test_sympy_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "sympy")
assert f(x) == sin(x)
prec = 1e-15
assert -prec < f(Rational(1, 5)).evalf() - Float(str(sin02)) < prec
# arctan is in numpy module and should not be available
# The arctan below gives NameError. What is this supposed to test?
# raises(NameError, lambda: lambdify(x, arctan(x), "sympy"))
@conserve_mpmath_dps
def test_math_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "math")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
raises(TypeError, lambda: f(x))
# if this succeeds, it can't be a Python math function
@conserve_mpmath_dps
def test_mpmath_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
raises(TypeError, lambda: f(x))
# if this succeeds, it can't be a mpmath function
@conserve_mpmath_dps
def test_number_precision():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin02, "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(0) - sin02 < prec
@conserve_mpmath_dps
def test_mpmath_precision():
mpmath.mp.dps = 100
assert str(lambdify((), pi.evalf(100), 'mpmath')()) == str(pi.evalf(100))
#================== Test Translations ==============================
# We can only check if all translated functions are valid. It has to be checked
# by hand if they are complete.
def test_math_transl():
from sympy.utilities.lambdify import MATH_TRANSLATIONS
for sym, mat in MATH_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert mat in math.__dict__
def test_mpmath_transl():
from sympy.utilities.lambdify import MPMATH_TRANSLATIONS
for sym, mat in MPMATH_TRANSLATIONS.items():
assert sym in sympy.__dict__ or sym == 'Matrix'
assert mat in mpmath.__dict__
def test_numpy_transl():
if not numpy:
skip("numpy not installed.")
from sympy.utilities.lambdify import NUMPY_TRANSLATIONS
for sym, nump in NUMPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert nump in numpy.__dict__
def test_scipy_transl():
if not scipy:
skip("scipy not installed.")
from sympy.utilities.lambdify import SCIPY_TRANSLATIONS
for sym, scip in SCIPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert scip in scipy.__dict__ or scip in scipy.special.__dict__
def test_numpy_translation_abs():
if not numpy:
skip("numpy not installed.")
f = lambdify(x, Abs(x), "numpy")
assert f(-1) == 1
assert f(1) == 1
def test_numexpr_printer():
if not numexpr:
skip("numexpr not installed.")
# if translation/printing is done incorrectly then evaluating
# a lambdified numexpr expression will throw an exception
from sympy.printing.lambdarepr import NumExprPrinter
blacklist = ('where', 'complex', 'contains')
arg_tuple = (x, y, z) # some functions take more than one argument
for sym in NumExprPrinter._numexpr_functions.keys():
if sym in blacklist:
continue
ssym = S(sym)
if hasattr(ssym, '_nargs'):
nargs = ssym._nargs[0]
else:
nargs = 1
args = arg_tuple[:nargs]
f = lambdify(args, ssym(*args), modules='numexpr')
assert f(*(1, )*nargs) is not None
def test_issue_9334():
if not numexpr:
skip("numexpr not installed.")
if not numpy:
skip("numpy not installed.")
expr = S('b*a - sqrt(a**2)')
a, b = sorted(expr.free_symbols, key=lambda s: s.name)
func_numexpr = lambdify((a,b), expr, modules=[numexpr], dummify=False)
foo, bar = numpy.random.random((2, 4))
func_numexpr(foo, bar)
def test_issue_12984():
import warnings
if not numexpr:
skip("numexpr not installed.")
func_numexpr = lambdify((x,y,z), Piecewise((y, x >= 0), (z, x > -1)), numexpr)
assert func_numexpr(1, 24, 42) == 24
with warnings.catch_warnings():
warnings.simplefilter("ignore", RuntimeWarning)
assert str(func_numexpr(-1, 24, 42)) == 'nan'
def test_empty_modules():
x, y = symbols('x y')
expr = -(x % y)
no_modules = lambdify([x, y], expr)
empty_modules = lambdify([x, y], expr, modules=[])
assert no_modules(3, 7) == empty_modules(3, 7)
assert no_modules(3, 7) == -3
def test_exponentiation():
f = lambdify(x, x**2)
assert f(-1) == 1
assert f(0) == 0
assert f(1) == 1
assert f(-2) == 4
assert f(2) == 4
assert f(2.5) == 6.25
def test_sqrt():
f = lambdify(x, sqrt(x))
assert f(0) == 0.0
assert f(1) == 1.0
assert f(4) == 2.0
assert abs(f(2) - 1.414) < 0.001
assert f(6.25) == 2.5
def test_trig():
f = lambdify([x], [cos(x), sin(x)], 'math')
d = f(pi)
prec = 1e-11
assert -prec < d[0] + 1 < prec
assert -prec < d[1] < prec
d = f(3.14159)
prec = 1e-5
assert -prec < d[0] + 1 < prec
assert -prec < d[1] < prec
def test_integral():
f = Lambda(x, exp(-x**2))
l = lambdify(y, Integral(f(x), (x, y, oo)))
d = l(-oo)
assert 1.77245385 < d < 1.772453851
def test_double_integral():
# example from http://mpmath.org/doc/current/calculus/integration.html
i = Integral(1/(1 - x**2*y**2), (x, 0, 1), (y, 0, z))
l = lambdify([z], i)
d = l(1)
assert 1.23370055 < d < 1.233700551
#================== Test vectors ===================================
def test_vector_simple():
f = lambdify((x, y, z), (z, y, x))
assert f(3, 2, 1) == (1, 2, 3)
assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0)
# make sure correct number of args required
raises(TypeError, lambda: f(0))
def test_vector_discontinuous():
f = lambdify(x, (-1/x, 1/x))
raises(ZeroDivisionError, lambda: f(0))
assert f(1) == (-1.0, 1.0)
assert f(2) == (-0.5, 0.5)
assert f(-2) == (0.5, -0.5)
def test_trig_symbolic():
f = lambdify([x], [cos(x), sin(x)], 'math')
d = f(pi)
assert abs(d[0] + 1) < 0.0001
assert abs(d[1] - 0) < 0.0001
def test_trig_float():
f = lambdify([x], [cos(x), sin(x)])
d = f(3.14159)
assert abs(d[0] + 1) < 0.0001
assert abs(d[1] - 0) < 0.0001
def test_docs():
f = lambdify(x, x**2)
assert f(2) == 4
f = lambdify([x, y, z], [z, y, x])
assert f(1, 2, 3) == [3, 2, 1]
f = lambdify(x, sqrt(x))
assert f(4) == 2.0
f = lambdify((x, y), sin(x*y)**2)
assert f(0, 5) == 0
def test_math():
f = lambdify((x, y), sin(x), modules="math")
assert f(0, 5) == 0
def test_sin():
f = lambdify(x, sin(x)**2)
assert isinstance(f(2), float)
f = lambdify(x, sin(x)**2, modules="math")
assert isinstance(f(2), float)
def test_matrix():
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol = Matrix([[1, 2], [sin(3) + 4, 1]])
f = lambdify((x, y, z), A, modules="sympy")
assert f(1, 2, 3) == sol
f = lambdify((x, y, z), (A, [A]), modules="sympy")
assert f(1, 2, 3) == (sol, [sol])
J = Matrix((x, x + y)).jacobian((x, y))
v = Matrix((x, y))
sol = Matrix([[1, 0], [1, 1]])
assert lambdify(v, J, modules='sympy')(1, 2) == sol
assert lambdify(v.T, J, modules='sympy')(1, 2) == sol
def test_numpy_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
#Lambdify array first, to ensure return to array as default
f = lambdify((x, y, z), A, ['numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
#Check that the types are arrays and matrices
assert isinstance(f(1, 2, 3), numpy.ndarray)
# gh-15071
class dot(Function):
pass
x_dot_mtx = dot(x, Matrix([[2], [1], [0]]))
f_dot1 = lambdify(x, x_dot_mtx)
inp = numpy.zeros((17, 3))
assert numpy.all(f_dot1(inp) == 0)
strict_kw = dict(allow_unknown_functions=False, inline=True, fully_qualified_modules=False)
p2 = NumPyPrinter(dict(user_functions={'dot': 'dot'}, **strict_kw))
f_dot2 = lambdify(x, x_dot_mtx, printer=p2)
assert numpy.all(f_dot2(inp) == 0)
p3 = NumPyPrinter(strict_kw)
# The line below should probably fail upon construction (before calling with "(inp)"):
raises(Exception, lambda: lambdify(x, x_dot_mtx, printer=p3)(inp))
def test_numpy_transpose():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A.T, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, 0], [2, 1]]))
def test_numpy_dotproduct():
if not numpy:
skip("numpy not installed")
A = Matrix([x, y, z])
f1 = lambdify([x, y, z], DotProduct(A, A), modules='numpy')
f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy')
f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='numpy')
f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy')
assert f1(1, 2, 3) == \
f2(1, 2, 3) == \
f3(1, 2, 3) == \
f4(1, 2, 3) == \
numpy.array([14])
def test_numpy_inverse():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A**-1, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, -2], [0, 1]]))
def test_numpy_old_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
f = lambdify((x, y, z), A, [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
assert isinstance(f(1, 2, 3), numpy.matrix)
def test_scipy_sparse_matrix():
if not scipy:
skip("scipy not installed.")
A = SparseMatrix([[x, 0], [0, y]])
f = lambdify((x, y), A, modules="scipy")
B = f(1, 2)
assert isinstance(B, scipy.sparse.coo_matrix)
def test_python_div_zero_issue_11306():
if not numpy:
skip("numpy not installed.")
p = Piecewise((1 / x, y < -1), (x, y < 1), (1 / x, True))
f = lambdify([x, y], p, modules='numpy')
numpy.seterr(divide='ignore')
assert float(f(numpy.array([0]),numpy.array([0.5]))) == 0
assert str(float(f(numpy.array([0]),numpy.array([1])))) == 'inf'
numpy.seterr(divide='warn')
def test_issue9474():
mods = [None, 'math']
if numpy:
mods.append('numpy')
if mpmath:
mods.append('mpmath')
for mod in mods:
f = lambdify(x, S.One/x, modules=mod)
assert f(2) == 0.5
f = lambdify(x, floor(S.One/x), modules=mod)
assert f(2) == 0
for absfunc, modules in product([Abs, abs], mods):
f = lambdify(x, absfunc(x), modules=modules)
assert f(-1) == 1
assert f(1) == 1
assert f(3+4j) == 5
def test_issue_9871():
if not numexpr:
skip("numexpr not installed.")
if not numpy:
skip("numpy not installed.")
r = sqrt(x**2 + y**2)
expr = diff(1/r, x)
xn = yn = numpy.linspace(1, 10, 16)
# expr(xn, xn) = -xn/(sqrt(2)*xn)^3
fv_exact = -numpy.sqrt(2.)**-3 * xn**-2
fv_numpy = lambdify((x, y), expr, modules='numpy')(xn, yn)
fv_numexpr = lambdify((x, y), expr, modules='numexpr')(xn, yn)
numpy.testing.assert_allclose(fv_numpy, fv_exact, rtol=1e-10)
numpy.testing.assert_allclose(fv_numexpr, fv_exact, rtol=1e-10)
def test_numpy_piecewise():
if not numpy:
skip("numpy not installed.")
pieces = Piecewise((x, x < 3), (x**2, x > 5), (0, True))
f = lambdify(x, pieces, modules="numpy")
numpy.testing.assert_array_equal(f(numpy.arange(10)),
numpy.array([0, 1, 2, 0, 0, 0, 36, 49, 64, 81]))
# If we evaluate somewhere all conditions are False, we should get back NaN
nodef_func = lambdify(x, Piecewise((x, x > 0), (-x, x < 0)))
numpy.testing.assert_array_equal(nodef_func(numpy.array([-1, 0, 1])),
numpy.array([1, numpy.nan, 1]))
def test_numpy_logical_ops():
if not numpy:
skip("numpy not installed.")
and_func = lambdify((x, y), And(x, y), modules="numpy")
and_func_3 = lambdify((x, y, z), And(x, y, z), modules="numpy")
or_func = lambdify((x, y), Or(x, y), modules="numpy")
or_func_3 = lambdify((x, y, z), Or(x, y, z), modules="numpy")
not_func = lambdify((x), Not(x), modules="numpy")
arr1 = numpy.array([True, True])
arr2 = numpy.array([False, True])
arr3 = numpy.array([True, False])
numpy.testing.assert_array_equal(and_func(arr1, arr2), numpy.array([False, True]))
numpy.testing.assert_array_equal(and_func_3(arr1, arr2, arr3), numpy.array([False, False]))
numpy.testing.assert_array_equal(or_func(arr1, arr2), numpy.array([True, True]))
numpy.testing.assert_array_equal(or_func_3(arr1, arr2, arr3), numpy.array([True, True]))
numpy.testing.assert_array_equal(not_func(arr2), numpy.array([True, False]))
def test_numpy_matmul():
if not numpy:
skip("numpy not installed.")
xmat = Matrix([[x, y], [z, 1+z]])
ymat = Matrix([[x**2], [Abs(x)]])
mat_func = lambdify((x, y, z), xmat*ymat, modules="numpy")
numpy.testing.assert_array_equal(mat_func(0.5, 3, 4), numpy.array([[1.625], [3.5]]))
numpy.testing.assert_array_equal(mat_func(-0.5, 3, 4), numpy.array([[1.375], [3.5]]))
# Multiple matrices chained together in multiplication
f = lambdify((x, y, z), xmat*xmat*xmat, modules="numpy")
numpy.testing.assert_array_equal(f(0.5, 3, 4), numpy.array([[72.125, 119.25],
[159, 251]]))
def test_numpy_numexpr():
if not numpy:
skip("numpy not installed.")
if not numexpr:
skip("numexpr not installed.")
a, b, c = numpy.random.randn(3, 128, 128)
# ensure that numpy and numexpr return same value for complicated expression
expr = sin(x) + cos(y) + tan(z)**2 + Abs(z-y)*acos(sin(y*z)) + \
Abs(y-z)*acosh(2+exp(y-x))- sqrt(x**2+I*y**2)
npfunc = lambdify((x, y, z), expr, modules='numpy')
nefunc = lambdify((x, y, z), expr, modules='numexpr')
assert numpy.allclose(npfunc(a, b, c), nefunc(a, b, c))
def test_numexpr_userfunctions():
if not numpy:
skip("numpy not installed.")
if not numexpr:
skip("numexpr not installed.")
a, b = numpy.random.randn(2, 10)
uf = type('uf', (Function, ),
{'eval' : classmethod(lambda x, y : y**2+1)})
func = lambdify(x, 1-uf(x), modules='numexpr')
assert numpy.allclose(func(a), -(a**2))
uf = implemented_function(Function('uf'), lambda x, y : 2*x*y+1)
func = lambdify((x, y), uf(x, y), modules='numexpr')
assert numpy.allclose(func(a, b), 2*a*b+1)
def test_tensorflow_basic_math():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
a = tensorflow.constant(0, dtype=tensorflow.float32)
assert func(a).eval(session=s) == 0.5
def test_tensorflow_placeholders():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
a = tensorflow.compat.v1.placeholder(dtype=tensorflow.float32)
assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5
def test_tensorflow_variables():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
a = tensorflow.Variable(0, dtype=tensorflow.float32)
s.run(a.initializer)
assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5
def test_tensorflow_logical_operations():
if not tensorflow:
skip("tensorflow not installed.")
expr = Not(And(Or(x, y), y))
func = lambdify([x, y], expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(False, True).eval(session=s) == False
def test_tensorflow_piecewise():
if not tensorflow:
skip("tensorflow not installed.")
expr = Piecewise((0, Eq(x,0)), (-1, x < 0), (1, x > 0))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(-1).eval(session=s) == -1
assert func(0).eval(session=s) == 0
assert func(1).eval(session=s) == 1
def test_tensorflow_multi_max():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(-2).eval(session=s) == 4
def test_tensorflow_multi_min():
if not tensorflow:
skip("tensorflow not installed.")
expr = Min(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(-2).eval(session=s) == -2
def test_tensorflow_relational():
if not tensorflow:
skip("tensorflow not installed.")
expr = x >= 0
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(1).eval(session=s) == True
def test_tensorflow_complexes():
if not tensorflow:
skip("tensorflow not installed")
func1 = lambdify(x, re(x), modules="tensorflow")
func2 = lambdify(x, im(x), modules="tensorflow")
func3 = lambdify(x, Abs(x), modules="tensorflow")
func4 = lambdify(x, arg(x), modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
# For versions before
# https://github.com/tensorflow/tensorflow/issues/30029
# resolved, using Python numeric types may not work
a = tensorflow.constant(1+2j)
assert func1(a).eval(session=s) == 1
assert func2(a).eval(session=s) == 2
tensorflow_result = func3(a).eval(session=s)
sympy_result = Abs(1 + 2j).evalf()
assert abs(tensorflow_result-sympy_result) < 10**-6
tensorflow_result = func4(a).eval(session=s)
sympy_result = arg(1 + 2j).evalf()
assert abs(tensorflow_result-sympy_result) < 10**-6
def test_tensorflow_array_arg():
# Test for issue 14655 (tensorflow part)
if not tensorflow:
skip("tensorflow not installed.")
f = lambdify([[x, y]], x*x + y, 'tensorflow')
with tensorflow.compat.v1.Session() as s:
fcall = f(tensorflow.constant([2.0, 1.0]))
assert fcall.eval(session=s) == 5.0
#================== Test symbolic ==================================
def test_sym_single_arg():
f = lambdify(x, x * y)
assert f(z) == z * y
def test_sym_list_args():
f = lambdify([x, y], x + y + z)
assert f(1, 2) == 3 + z
def test_sym_integral():
f = Lambda(x, exp(-x**2))
l = lambdify(x, Integral(f(x), (x, -oo, oo)), modules="sympy")
assert l(y) == Integral(exp(-y**2), (y, -oo, oo))
assert l(y).doit() == sqrt(pi)
def test_namespace_order():
# lambdify had a bug, such that module dictionaries or cached module
# dictionaries would pull earlier namespaces into themselves.
# Because the module dictionaries form the namespace of the
# generated lambda, this meant that the behavior of a previously
# generated lambda function could change as a result of later calls
# to lambdify.
n1 = {'f': lambda x: 'first f'}
n2 = {'f': lambda x: 'second f',
'g': lambda x: 'function g'}
f = sympy.Function('f')
g = sympy.Function('g')
if1 = lambdify(x, f(x), modules=(n1, "sympy"))
assert if1(1) == 'first f'
if2 = lambdify(x, g(x), modules=(n2, "sympy"))
# previously gave 'second f'
assert if1(1) == 'first f'
assert if2(1) == 'function g'
def test_imps():
# Here we check if the default returned functions are anonymous - in
# the sense that we can have more than one function with the same name
f = implemented_function('f', lambda x: 2*x)
g = implemented_function('f', lambda x: math.sqrt(x))
l1 = lambdify(x, f(x))
l2 = lambdify(x, g(x))
assert str(f(x)) == str(g(x))
assert l1(3) == 6
assert l2(3) == math.sqrt(3)
# check that we can pass in a Function as input
func = sympy.Function('myfunc')
assert not hasattr(func, '_imp_')
my_f = implemented_function(func, lambda x: 2*x)
assert hasattr(my_f, '_imp_')
# Error for functions with same name and different implementation
f2 = implemented_function("f", lambda x: x + 101)
raises(ValueError, lambda: lambdify(x, f(f2(x))))
def test_imps_errors():
# Test errors that implemented functions can return, and still be able to
# form expressions.
# See: https://github.com/sympy/sympy/issues/10810
#
# XXX: Removed AttributeError here. This test was added due to issue 10810
# but that issue was about ValueError. It doesn't seem reasonable to
# "support" catching AttributeError in the same context...
for val, error_class in product((0, 0., 2, 2.0), (TypeError, ValueError)):
def myfunc(a):
if a == 0:
raise error_class
return 1
f = implemented_function('f', myfunc)
expr = f(val)
assert expr == f(val)
def test_imps_wrong_args():
raises(ValueError, lambda: implemented_function(sin, lambda x: x))
def test_lambdify_imps():
# Test lambdify with implemented functions
# first test basic (sympy) lambdify
f = sympy.cos
assert lambdify(x, f(x))(0) == 1
assert lambdify(x, 1 + f(x))(0) == 2
assert lambdify((x, y), y + f(x))(0, 1) == 2
# make an implemented function and test
f = implemented_function("f", lambda x: x + 100)
assert lambdify(x, f(x))(0) == 100
assert lambdify(x, 1 + f(x))(0) == 101
assert lambdify((x, y), y + f(x))(0, 1) == 101
# Can also handle tuples, lists, dicts as expressions
lam = lambdify(x, (f(x), x))
assert lam(3) == (103, 3)
lam = lambdify(x, [f(x), x])
assert lam(3) == [103, 3]
lam = lambdify(x, [f(x), (f(x), x)])
assert lam(3) == [103, (103, 3)]
lam = lambdify(x, {f(x): x})
assert lam(3) == {103: 3}
lam = lambdify(x, {f(x): x})
assert lam(3) == {103: 3}
lam = lambdify(x, {x: f(x)})
assert lam(3) == {3: 103}
# Check that imp preferred to other namespaces by default
d = {'f': lambda x: x + 99}
lam = lambdify(x, f(x), d)
assert lam(3) == 103
# Unless flag passed
lam = lambdify(x, f(x), d, use_imps=False)
assert lam(3) == 102
def test_dummification():
t = symbols('t')
F = Function('F')
G = Function('G')
#"\alpha" is not a valid Python variable name
#lambdify should sub in a dummy for it, and return
#without a syntax error
alpha = symbols(r'\alpha')
some_expr = 2 * F(t)**2 / G(t)
lam = lambdify((F(t), G(t)), some_expr)
assert lam(3, 9) == 2
lam = lambdify(sin(t), 2 * sin(t)**2)
assert lam(F(t)) == 2 * F(t)**2
#Test that \alpha was properly dummified
lam = lambdify((alpha, t), 2*alpha + t)
assert lam(2, 1) == 5
raises(SyntaxError, lambda: lambdify(F(t) * G(t), F(t) * G(t) + 5))
raises(SyntaxError, lambda: lambdify(2 * F(t), 2 * F(t) + 5))
raises(SyntaxError, lambda: lambdify(2 * F(t), 4 * F(t) + 5))
def test_curly_matrix_symbol():
# Issue #15009
curlyv = sympy.MatrixSymbol("{v}", 2, 1)
lam = lambdify(curlyv, curlyv)
assert lam(1)==1
lam = lambdify(curlyv, curlyv, dummify=True)
assert lam(1)==1
def test_python_keywords():
# Test for issue 7452. The automatic dummification should ensure use of
# Python reserved keywords as symbol names will create valid lambda
# functions. This is an additional regression test.
python_if = symbols('if')
expr = python_if / 2
f = lambdify(python_if, expr)
assert f(4.0) == 2.0
def test_lambdify_docstring():
func = lambdify((w, x, y, z), w + x + y + z)
ref = (
"Created with lambdify. Signature:\n\n"
"func(w, x, y, z)\n\n"
"Expression:\n\n"
"w + x + y + z"
).splitlines()
assert func.__doc__.splitlines()[:len(ref)] == ref
syms = symbols('a1:26')
func = lambdify(syms, sum(syms))
ref = (
"Created with lambdify. Signature:\n\n"
"func(a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15,\n"
" a16, a17, a18, a19, a20, a21, a22, a23, a24, a25)\n\n"
"Expression:\n\n"
"a1 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18 + a19 + a2 + a20 +..."
).splitlines()
assert func.__doc__.splitlines()[:len(ref)] == ref
#================== Test special printers ==========================
def test_special_printers():
from sympy.printing.lambdarepr import IntervalPrinter
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sqrt(sqrt(2) + sqrt(3)) + S.Half
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
# To check Is lambdify loggamma works for mpmath or not
exp1 = lambdify(x, loggamma(x), 'mpmath')(5)
exp2 = lambdify(x, loggamma(x), 'mpmath')(1.8)
exp3 = lambdify(x, loggamma(x), 'mpmath')(15)
exp_ls = [exp1, exp2, exp3]
sol1 = mpmath.loggamma(5)
sol2 = mpmath.loggamma(1.8)
sol3 = mpmath.loggamma(15)
sol_ls = [sol1, sol2, sol3]
assert exp_ls == sol_ls
def test_true_false():
# We want exact is comparison here, not just ==
assert lambdify([], true)() is True
assert lambdify([], false)() is False
def test_issue_2790():
assert lambdify((x, (y, z)), x + y)(1, (2, 4)) == 3
assert lambdify((x, (y, (w, z))), w + x + y + z)(1, (2, (3, 4))) == 10
assert lambdify(x, x + 1, dummify=False)(1) == 2
def test_issue_12092():
f = implemented_function('f', lambda x: x**2)
assert f(f(2)).evalf() == Float(16)
def test_issue_14911():
class Variable(sympy.Symbol):
def _sympystr(self, printer):
return printer.doprint(self.name)
_lambdacode = _sympystr
_numpycode = _sympystr
x = Variable('x')
y = 2 * x
code = LambdaPrinter().doprint(y)
assert code.replace(' ', '') == '2*x'
def test_ITE():
assert lambdify((x, y, z), ITE(x, y, z))(True, 5, 3) == 5
assert lambdify((x, y, z), ITE(x, y, z))(False, 5, 3) == 3
def test_Min_Max():
# see gh-10375
assert lambdify((x, y, z), Min(x, y, z))(1, 2, 3) == 1
assert lambdify((x, y, z), Max(x, y, z))(1, 2, 3) == 3
def test_Indexed():
# Issue #10934
if not numpy:
skip("numpy not installed")
a = IndexedBase('a')
i, j = symbols('i j')
b = numpy.array([[1, 2], [3, 4]])
assert lambdify(a, Sum(a[x, y], (x, 0, 1), (y, 0, 1)))(b) == 10
def test_issue_12173():
#test for issue 12173
expr1 = lambdify((x, y), uppergamma(x, y),"mpmath")(1, 2)
expr2 = lambdify((x, y), lowergamma(x, y),"mpmath")(1, 2)
assert expr1 == uppergamma(1, 2).evalf()
assert expr2 == lowergamma(1, 2).evalf()
def test_issue_13642():
if not numpy:
skip("numpy not installed")
f = lambdify(x, sinc(x))
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_sinc_mpmath():
f = lambdify(x, sinc(x), "mpmath")
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_lambdify_dummy_arg():
d1 = Dummy()
f1 = lambdify(d1, d1 + 1, dummify=False)
assert f1(2) == 3
f1b = lambdify(d1, d1 + 1)
assert f1b(2) == 3
d2 = Dummy('x')
f2 = lambdify(d2, d2 + 1)
assert f2(2) == 3
f3 = lambdify([[d2]], d2 + 1)
assert f3([2]) == 3
def test_lambdify_mixed_symbol_dummy_args():
d = Dummy()
# Contrived example of name clash
dsym = symbols(str(d))
f = lambdify([d, dsym], d - dsym)
assert f(4, 1) == 3
def test_numpy_array_arg():
# Test for issue 14655 (numpy part)
if not numpy:
skip("numpy not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
assert f(numpy.array([2.0, 1.0])) == 5
def test_scipy_fns():
if not scipy:
skip("scipy not installed")
single_arg_sympy_fns = [erf, erfc, factorial, gamma, loggamma, digamma]
single_arg_scipy_fns = [scipy.special.erf, scipy.special.erfc,
scipy.special.factorial, scipy.special.gamma, scipy.special.gammaln,
scipy.special.psi]
numpy.random.seed(0)
for (sympy_fn, scipy_fn) in zip(single_arg_sympy_fns, single_arg_scipy_fns):
f = lambdify(x, sympy_fn(x), modules="scipy")
for i in range(20):
tv = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy thinks that factorial(z) is 0 when re(z) < 0 and
# does not support complex numbers.
# SymPy does not think so.
if sympy_fn == factorial:
tv = numpy.abs(tv)
# SciPy supports gammaln for real arguments only,
# and there is also a branch cut along the negative real axis
if sympy_fn == loggamma:
tv = numpy.abs(tv)
# SymPy's digamma evaluates as polygamma(0, z)
# which SciPy supports for real arguments only
if sympy_fn == digamma:
tv = numpy.real(tv)
sympy_result = sympy_fn(tv).evalf()
assert abs(f(tv) - sympy_result) < 1e-13*(1 + abs(sympy_result))
assert abs(f(tv) - scipy_fn(tv)) < 1e-13*(1 + abs(sympy_result))
double_arg_sympy_fns = [RisingFactorial, besselj, bessely, besseli,
besselk]
double_arg_scipy_fns = [scipy.special.poch, scipy.special.jv,
scipy.special.yv, scipy.special.iv, scipy.special.kv]
for (sympy_fn, scipy_fn) in zip(double_arg_sympy_fns, double_arg_scipy_fns):
f = lambdify((x, y), sympy_fn(x, y), modules="scipy")
for i in range(20):
# SciPy supports only real orders of Bessel functions
tv1 = numpy.random.uniform(-10, 10)
tv2 = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy supports poch for real arguments only
if sympy_fn == RisingFactorial:
tv2 = numpy.real(tv2)
sympy_result = sympy_fn(tv1, tv2).evalf()
assert abs(f(tv1, tv2) - sympy_result) < 1e-13*(1 + abs(sympy_result))
assert abs(f(tv1, tv2) - scipy_fn(tv1, tv2)) < 1e-13*(1 + abs(sympy_result))
def test_scipy_polys():
if not scipy:
skip("scipy not installed")
numpy.random.seed(0)
params = symbols('n k a b')
# list polynomials with the number of parameters
polys = [
(chebyshevt, 1),
(chebyshevu, 1),
(legendre, 1),
(hermite, 1),
(laguerre, 1),
(gegenbauer, 2),
(assoc_legendre, 2),
(assoc_laguerre, 2),
(jacobi, 3)
]
msg = \
"The random test of the function {func} with the arguments " \
"{args} had failed because the SymPy result {sympy_result} " \
"and SciPy result {scipy_result} had failed to converge " \
"within the tolerance {tol} " \
"(Actual absolute difference : {diff})"
for sympy_fn, num_params in polys:
args = params[:num_params] + (x,)
f = lambdify(args, sympy_fn(*args))
for _ in range(10):
tn = numpy.random.randint(3, 10)
tparams = tuple(numpy.random.uniform(0, 5, size=num_params-1))
tv = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy supports hermite for real arguments only
if sympy_fn == hermite:
tv = numpy.real(tv)
# assoc_legendre needs x in (-1, 1) and integer param at most n
if sympy_fn == assoc_legendre:
tv = numpy.random.uniform(-1, 1)
tparams = tuple(numpy.random.randint(1, tn, size=1))
vals = (tn,) + tparams + (tv,)
scipy_result = f(*vals)
sympy_result = sympy_fn(*vals).evalf()
atol = 1e-9*(1 + abs(sympy_result))
diff = abs(scipy_result - sympy_result)
try:
assert diff < atol
except TypeError:
raise AssertionError(
msg.format(
func=repr(sympy_fn),
args=repr(vals),
sympy_result=repr(sympy_result),
scipy_result=repr(scipy_result),
diff=diff,
tol=atol)
)
def test_lambdify_inspect():
f = lambdify(x, x**2)
# Test that inspect.getsource works but don't hard-code implementation
# details
assert 'x**2' in inspect.getsource(f)
def test_issue_14941():
x, y = Dummy(), Dummy()
# test dict
f1 = lambdify([x, y], {x: 3, y: 3}, 'sympy')
assert f1(2, 3) == {2: 3, 3: 3}
# test tuple
f2 = lambdify([x, y], (y, x), 'sympy')
assert f2(2, 3) == (3, 2)
# test list
f3 = lambdify([x, y], [y, x], 'sympy')
assert f3(2, 3) == [3, 2]
def test_lambdify_Derivative_arg_issue_16468():
f = Function('f')(x)
fx = f.diff()
assert lambdify((f, fx), f + fx)(10, 5) == 15
assert eval(lambdastr((f, fx), f/fx))(10, 5) == 2
raises(SyntaxError, lambda:
eval(lambdastr((f, fx), f/fx, dummify=False)))
assert eval(lambdastr((f, fx), f/fx, dummify=True))(10, 5) == 2
assert eval(lambdastr((fx, f), f/fx, dummify=True))(S(10), 5) == S.Half
assert lambdify(fx, 1 + fx)(41) == 42
assert eval(lambdastr(fx, 1 + fx, dummify=True))(41) == 42
def test_imag_real():
f_re = lambdify([z], sympy.re(z))
val = 3+2j
assert f_re(val) == val.real
f_im = lambdify([z], sympy.im(z)) # see #15400
assert f_im(val) == val.imag
def test_MatrixSymbol_issue_15578():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol('A', 2, 2)
A0 = numpy.array([[1, 2], [3, 4]])
f = lambdify(A, A**(-1))
assert numpy.allclose(f(A0), numpy.array([[-2., 1.], [1.5, -0.5]]))
g = lambdify(A, A**3)
assert numpy.allclose(g(A0), numpy.array([[37, 54], [81, 118]]))
def test_issue_15654():
if not scipy:
skip("scipy not installed")
from sympy.abc import n, l, r, Z
from sympy.physics import hydrogen
nv, lv, rv, Zv = 1, 0, 3, 1
sympy_value = hydrogen.R_nl(nv, lv, rv, Zv).evalf()
f = lambdify((n, l, r, Z), hydrogen.R_nl(n, l, r, Z))
scipy_value = f(nv, lv, rv, Zv)
assert abs(sympy_value - scipy_value) < 1e-15
def test_issue_15827():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 2, 3)
C = MatrixSymbol("C", 3, 4)
D = MatrixSymbol("D", 4, 5)
k=symbols("k")
f = lambdify(A, (2*k)*A)
g = lambdify(A, (2+k)*A)
h = lambdify(A, 2*A)
i = lambdify((B, C, D), 2*B*C*D)
assert numpy.array_equal(f(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2*k, 4*k, 6*k], [2*k, 4*k, 6*k], [2*k, 4*k, 6*k]], dtype=object))
assert numpy.array_equal(g(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[k + 2, 2*k + 4, 3*k + 6], [k + 2, 2*k + 4, 3*k + 6], \
[k + 2, 2*k + 4, 3*k + 6]], dtype=object))
assert numpy.array_equal(h(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2, 4, 6], [2, 4, 6], [2, 4, 6]]))
assert numpy.array_equal(i(numpy.array([[1, 2, 3], [1, 2, 3]]), numpy.array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]), \
numpy.array([[1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]])), numpy.array([[ 120, 240, 360, 480, 600], \
[ 120, 240, 360, 480, 600]]))
def test_issue_16930():
if not scipy:
skip("scipy not installed")
x = symbols("x")
f = lambda x: S.GoldenRatio * x**2
f_ = lambdify(x, f(x), modules='scipy')
assert f_(1) == scipy.constants.golden_ratio
def test_issue_17898():
if not scipy:
skip("scipy not installed")
x = symbols("x")
f_ = lambdify([x], sympy.LambertW(x,-1), modules='scipy')
assert f_(0.1) == mpmath.lambertw(0.1, -1)
def test_issue_13167_21411():
if not numpy:
skip("numpy not installed")
f1 = lambdify(x, sympy.Heaviside(x))
f2 = lambdify(x, sympy.Heaviside(x, 1))
res1 = f1([-1, 0, 1])
res2 = f2([-1, 0, 1])
assert Abs(res1[0]).n() < 1e-15 # First functionality: only one argument passed
assert Abs(res1[1] - 1/2).n() < 1e-15
assert Abs(res1[2] - 1).n() < 1e-15
assert Abs(res2[0]).n() < 1e-15 # Second functionality: two arguments passed
assert Abs(res2[1] - 1).n() < 1e-15
assert Abs(res2[2] - 1).n() < 1e-15
def test_single_e():
f = lambdify(x, E)
assert f(23) == exp(1.0)
def test_issue_16536():
if not scipy:
skip("scipy not installed")
a = symbols('a')
f1 = lowergamma(a, x)
F = lambdify((a, x), f1, modules='scipy')
assert abs(lowergamma(1, 3) - F(1, 3)) <= 1e-10
f2 = uppergamma(a, x)
F = lambdify((a, x), f2, modules='scipy')
assert abs(uppergamma(1, 3) - F(1, 3)) <= 1e-10
def test_issue_22726():
if not numpy:
skip("numpy not installed")
x1, x2 = symbols('x1 x2')
f = Max(S.Zero, Min(x1, x2))
g = derive_by_array(f, (x1, x2))
G = lambdify((x1, x2), g, modules='numpy')
point = {x1: 1, x2: 2}
assert (abs(g.subs(point) - G(*point.values())) <= 1e-10).all()
def test_issue_22739():
if not numpy:
skip("numpy not installed")
x1, x2 = symbols('x1 x2')
f = Heaviside(Min(x1, x2))
F = lambdify((x1, x2), f, modules='numpy')
point = {x1: 1, x2: 2}
assert abs(f.subs(point) - F(*point.values())) <= 1e-10
def test_issue_19764():
if not numpy:
skip("numpy not installed")
expr = Array([x, x**2])
f = lambdify(x, expr, 'numpy')
assert f(1).__class__ == numpy.ndarray
def test_issue_20070():
if not numba:
skip("numba not installed")
f = lambdify(x, sin(x), 'numpy')
assert numba.jit(f)(1)==0.8414709848078965
def test_fresnel_integrals_scipy():
if not scipy:
skip("scipy not installed")
f1 = fresnelc(x)
f2 = fresnels(x)
F1 = lambdify(x, f1, modules='scipy')
F2 = lambdify(x, f2, modules='scipy')
assert abs(fresnelc(1.3) - F1(1.3)) <= 1e-10
assert abs(fresnels(1.3) - F2(1.3)) <= 1e-10
def test_beta_scipy():
if not scipy:
skip("scipy not installed")
f = beta(x, y)
F = lambdify((x, y), f, modules='scipy')
assert abs(beta(1.3, 2.3) - F(1.3, 2.3)) <= 1e-10
def test_beta_math():
f = beta(x, y)
F = lambdify((x, y), f, modules='math')
assert abs(beta(1.3, 2.3) - F(1.3, 2.3)) <= 1e-10
def test_betainc_scipy():
if not scipy:
skip("scipy not installed")
f = betainc(w, x, y, z)
F = lambdify((w, x, y, z), f, modules='scipy')
assert abs(betainc(1.4, 3.1, 0.1, 0.5) - F(1.4, 3.1, 0.1, 0.5)) <= 1e-10
def test_betainc_regularized_scipy():
if not scipy:
skip("scipy not installed")
f = betainc_regularized(w, x, y, z)
F = lambdify((w, x, y, z), f, modules='scipy')
assert abs(betainc_regularized(0.2, 3.5, 0.1, 1) - F(0.2, 3.5, 0.1, 1)) <= 1e-10
def test_numpy_special_math():
if not numpy:
skip("numpy not installed")
funcs = [expm1, log1p, exp2, log2, log10, hypot, logaddexp, logaddexp2]
for func in funcs:
if 2 in func.nargs:
expr = func(x, y)
args = (x, y)
num_args = (0.3, 0.4)
elif 1 in func.nargs:
expr = func(x)
args = (x,)
num_args = (0.3,)
else:
raise NotImplementedError("Need to handle other than unary & binary functions in test")
f = lambdify(args, expr)
result = f(*num_args)
reference = expr.subs(dict(zip(args, num_args))).evalf()
assert numpy.allclose(result, float(reference))
lae2 = lambdify((x, y), logaddexp2(log2(x), log2(y)))
assert abs(2.0**lae2(1e-50, 2.5e-50) - 3.5e-50) < 1e-62 # from NumPy's docstring
def test_scipy_special_math():
if not scipy:
skip("scipy not installed")
cm1 = lambdify((x,), cosm1(x), modules='scipy')
assert abs(cm1(1e-20) + 5e-41) < 1e-200
def test_cupy_array_arg():
if not cupy:
skip("CuPy not installed")
f = lambdify([[x, y]], x*x + y, 'cupy')
result = f(cupy.array([2.0, 1.0]))
assert result == 5
assert "cupy" in str(type(result))
def test_cupy_array_arg_using_numpy():
# numpy functions can be run on cupy arrays
# unclear if we can "officialy" support this,
# depends on numpy __array_function__ support
if not cupy:
skip("CuPy not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
result = f(cupy.array([2.0, 1.0]))
assert result == 5
assert "cupy" in str(type(result))
def test_cupy_dotproduct():
if not cupy:
skip("CuPy not installed")
A = Matrix([x, y, z])
f1 = lambdify([x, y, z], DotProduct(A, A), modules='cupy')
f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='cupy')
f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='cupy')
f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='cupy')
assert f1(1, 2, 3) == \
f2(1, 2, 3) == \
f3(1, 2, 3) == \
f4(1, 2, 3) == \
cupy.array([14])
def test_lambdify_cse():
def dummy_cse(exprs):
return (), exprs
def minmem(exprs):
from sympy.simplify.cse_main import cse_release_variables, cse
return cse(exprs, postprocess=cse_release_variables)
class Case:
def __init__(self, *, args, exprs, num_args, requires_numpy=False):
self.args = args
self.exprs = exprs
self.num_args = num_args
subs_dict = dict(zip(self.args, self.num_args))
self.ref = [e.subs(subs_dict).evalf() for e in exprs]
self.requires_numpy = requires_numpy
def lambdify(self, *, cse):
return lambdify(self.args, self.exprs, cse=cse)
def assertAllClose(self, result, *, abstol=1e-15, reltol=1e-15):
if self.requires_numpy:
assert all(numpy.allclose(result[i], numpy.asarray(r, dtype=float),
rtol=reltol, atol=abstol)
for i, r in enumerate(self.ref))
return
for i, r in enumerate(self.ref):
abs_err = abs(result[i] - r)
if r == 0:
assert abs_err < abstol
else:
assert abs_err/abs(r) < reltol
cases = [
Case(
args=(x, y, z),
exprs=[
x + y + z,
x + y - z,
2*x + 2*y - z,
(x+y)**2 + (y+z)**2,
],
num_args=(2., 3., 4.)
),
Case(
args=(x, y, z),
exprs=[
x + sympy.Heaviside(x),
y + sympy.Heaviside(x),
z + sympy.Heaviside(x, 1),
z/sympy.Heaviside(x, 1)
],
num_args=(0., 3., 4.)
),
Case(
args=(x, y, z),
exprs=[
x + sinc(y),
y + sinc(y),
z - sinc(y)
],
num_args=(0.1, 0.2, 0.3)
),
Case(
args=(x, y, z),
exprs=[
Matrix([[x, x*y], [sin(z) + 4, x**z]]),
x*y+sin(z)-x**z,
Matrix([x*x, sin(z), x**z])
],
num_args=(1.,2.,3.),
requires_numpy=True
),
Case(
args=(x, y),
exprs=[(x + y - 1)**2, x, x + y,
(x + y)/(2*x + 1) + (x + y - 1)**2, (2*x + 1)**(x + y)],
num_args=(1,2)
)
]
for case in cases:
if not numpy and case.requires_numpy:
continue
for cse in [False, True, minmem, dummy_cse]:
f = case.lambdify(cse=cse)
result = f(*case.num_args)
case.assertAllClose(result)
def test_deprecated_set():
with warns_deprecated_sympy():
lambdify({x, y}, x + y)
|
c59f976908b7dd8b4ca0e95ec56899690b32debd4e2a053da7cca9f3adb6f9e1 | from textwrap import dedent
from itertools import islice, product
from sympy.core.basic import Basic
from sympy.core.numbers import Integer
from sympy.core.sorting import ordered
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions.combinatorial.factorials import factorial
from sympy.matrices.dense import Matrix
from sympy.combinatorics import RGS_enum, RGS_unrank, Permutation
from sympy.utilities.iterables import (
_partition, _set_partitions, binary_partitions, bracelets, capture,
cartes, common_prefix, common_suffix, connected_components, dict_merge,
filter_symbols, flatten, generate_bell, generate_derangements,
generate_involutions, generate_oriented_forest, group, has_dups, ibin,
iproduct, kbins, minlex, multiset, multiset_combinations,
multiset_partitions, multiset_permutations, necklaces, numbered_symbols,
partitions, permutations, postfixes,
prefixes, reshape, rotate_left, rotate_right, runs, sift,
strongly_connected_components, subsets, take, topological_sort, unflatten,
uniq, variations, ordered_partitions, rotations, is_palindromic, iterable,
NotIterable, multiset_derangements)
from sympy.utilities.enumerative import (
factoring_visitor, multiset_partitions_taocp )
from sympy.core.singleton import S
from sympy.testing.pytest import raises, warns_deprecated_sympy
w, x, y, z = symbols('w,x,y,z')
def test_deprecated_iterables():
from sympy.utilities.iterables import default_sort_key, ordered
with warns_deprecated_sympy():
assert list(ordered([y, x])) == [x, y]
with warns_deprecated_sympy():
assert sorted([y, x], key=default_sort_key) == [x, y]
def test_is_palindromic():
assert is_palindromic('')
assert is_palindromic('x')
assert is_palindromic('xx')
assert is_palindromic('xyx')
assert not is_palindromic('xy')
assert not is_palindromic('xyzx')
assert is_palindromic('xxyzzyx', 1)
assert not is_palindromic('xxyzzyx', 2)
assert is_palindromic('xxyzzyx', 2, -1)
assert is_palindromic('xxyzzyx', 2, 6)
assert is_palindromic('xxyzyx', 1)
assert not is_palindromic('xxyzyx', 2)
assert is_palindromic('xxyzyx', 2, 2 + 3)
def test_flatten():
assert flatten((1, (1,))) == [1, 1]
assert flatten((x, (x,))) == [x, x]
ls = [[(-2, -1), (1, 2)], [(0, 0)]]
assert flatten(ls, levels=0) == ls
assert flatten(ls, levels=1) == [(-2, -1), (1, 2), (0, 0)]
assert flatten(ls, levels=2) == [-2, -1, 1, 2, 0, 0]
assert flatten(ls, levels=3) == [-2, -1, 1, 2, 0, 0]
raises(ValueError, lambda: flatten(ls, levels=-1))
class MyOp(Basic):
pass
assert flatten([MyOp(x, y), z]) == [MyOp(x, y), z]
assert flatten([MyOp(x, y), z], cls=MyOp) == [x, y, z]
assert flatten({1, 11, 2}) == list({1, 11, 2})
def test_iproduct():
assert list(iproduct()) == [()]
assert list(iproduct([])) == []
assert list(iproduct([1,2,3])) == [(1,),(2,),(3,)]
assert sorted(iproduct([1, 2], [3, 4, 5])) == [
(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)]
assert sorted(iproduct([0,1],[0,1],[0,1])) == [
(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),(1,1,1)]
assert iterable(iproduct(S.Integers)) is True
assert iterable(iproduct(S.Integers, S.Integers)) is True
assert (3,) in iproduct(S.Integers)
assert (4, 5) in iproduct(S.Integers, S.Integers)
assert (1, 2, 3) in iproduct(S.Integers, S.Integers, S.Integers)
triples = set(islice(iproduct(S.Integers, S.Integers, S.Integers), 1000))
for n1, n2, n3 in triples:
assert isinstance(n1, Integer)
assert isinstance(n2, Integer)
assert isinstance(n3, Integer)
for t in set(product(*([range(-2, 3)]*3))):
assert t in iproduct(S.Integers, S.Integers, S.Integers)
def test_group():
assert group([]) == []
assert group([], multiple=False) == []
assert group([1]) == [[1]]
assert group([1], multiple=False) == [(1, 1)]
assert group([1, 1]) == [[1, 1]]
assert group([1, 1], multiple=False) == [(1, 2)]
assert group([1, 1, 1]) == [[1, 1, 1]]
assert group([1, 1, 1], multiple=False) == [(1, 3)]
assert group([1, 2, 1]) == [[1], [2], [1]]
assert group([1, 2, 1], multiple=False) == [(1, 1), (2, 1), (1, 1)]
assert group([1, 1, 2, 2, 2, 1, 3, 3]) == [[1, 1], [2, 2, 2], [1], [3, 3]]
assert group([1, 1, 2, 2, 2, 1, 3, 3], multiple=False) == [(1, 2),
(2, 3), (1, 1), (3, 2)]
def test_subsets():
# combinations
assert list(subsets([1, 2, 3], 0)) == [()]
assert list(subsets([1, 2, 3], 1)) == [(1,), (2,), (3,)]
assert list(subsets([1, 2, 3], 2)) == [(1, 2), (1, 3), (2, 3)]
assert list(subsets([1, 2, 3], 3)) == [(1, 2, 3)]
l = list(range(4))
assert list(subsets(l, 0, repetition=True)) == [()]
assert list(subsets(l, 1, repetition=True)) == [(0,), (1,), (2,), (3,)]
assert list(subsets(l, 2, repetition=True)) == [(0, 0), (0, 1), (0, 2),
(0, 3), (1, 1), (1, 2),
(1, 3), (2, 2), (2, 3),
(3, 3)]
assert list(subsets(l, 3, repetition=True)) == [(0, 0, 0), (0, 0, 1),
(0, 0, 2), (0, 0, 3),
(0, 1, 1), (0, 1, 2),
(0, 1, 3), (0, 2, 2),
(0, 2, 3), (0, 3, 3),
(1, 1, 1), (1, 1, 2),
(1, 1, 3), (1, 2, 2),
(1, 2, 3), (1, 3, 3),
(2, 2, 2), (2, 2, 3),
(2, 3, 3), (3, 3, 3)]
assert len(list(subsets(l, 4, repetition=True))) == 35
assert list(subsets(l[:2], 3, repetition=False)) == []
assert list(subsets(l[:2], 3, repetition=True)) == [(0, 0, 0),
(0, 0, 1),
(0, 1, 1),
(1, 1, 1)]
assert list(subsets([1, 2], repetition=True)) == \
[(), (1,), (2,), (1, 1), (1, 2), (2, 2)]
assert list(subsets([1, 2], repetition=False)) == \
[(), (1,), (2,), (1, 2)]
assert list(subsets([1, 2, 3], 2)) == \
[(1, 2), (1, 3), (2, 3)]
assert list(subsets([1, 2, 3], 2, repetition=True)) == \
[(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)]
def test_variations():
# permutations
l = list(range(4))
assert list(variations(l, 0, repetition=False)) == [()]
assert list(variations(l, 1, repetition=False)) == [(0,), (1,), (2,), (3,)]
assert list(variations(l, 2, repetition=False)) == [(0, 1), (0, 2), (0, 3), (1, 0), (1, 2), (1, 3), (2, 0), (2, 1), (2, 3), (3, 0), (3, 1), (3, 2)]
assert list(variations(l, 3, repetition=False)) == [(0, 1, 2), (0, 1, 3), (0, 2, 1), (0, 2, 3), (0, 3, 1), (0, 3, 2), (1, 0, 2), (1, 0, 3), (1, 2, 0), (1, 2, 3), (1, 3, 0), (1, 3, 2), (2, 0, 1), (2, 0, 3), (2, 1, 0), (2, 1, 3), (2, 3, 0), (2, 3, 1), (3, 0, 1), (3, 0, 2), (3, 1, 0), (3, 1, 2), (3, 2, 0), (3, 2, 1)]
assert list(variations(l, 0, repetition=True)) == [()]
assert list(variations(l, 1, repetition=True)) == [(0,), (1,), (2,), (3,)]
assert list(variations(l, 2, repetition=True)) == [(0, 0), (0, 1), (0, 2),
(0, 3), (1, 0), (1, 1),
(1, 2), (1, 3), (2, 0),
(2, 1), (2, 2), (2, 3),
(3, 0), (3, 1), (3, 2),
(3, 3)]
assert len(list(variations(l, 3, repetition=True))) == 64
assert len(list(variations(l, 4, repetition=True))) == 256
assert list(variations(l[:2], 3, repetition=False)) == []
assert list(variations(l[:2], 3, repetition=True)) == [
(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1),
(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)
]
def test_cartes():
assert list(cartes([1, 2], [3, 4, 5])) == \
[(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)]
assert list(cartes()) == [()]
assert list(cartes('a')) == [('a',)]
assert list(cartes('a', repeat=2)) == [('a', 'a')]
assert list(cartes(list(range(2)))) == [(0,), (1,)]
def test_filter_symbols():
s = numbered_symbols()
filtered = filter_symbols(s, symbols("x0 x2 x3"))
assert take(filtered, 3) == list(symbols("x1 x4 x5"))
def test_numbered_symbols():
s = numbered_symbols(cls=Dummy)
assert isinstance(next(s), Dummy)
assert next(numbered_symbols('C', start=1, exclude=[symbols('C1')])) == \
symbols('C2')
def test_sift():
assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([S.One], lambda _: _.has(x)) == {False: [1]}
assert sift([0, 1, 2, 3], lambda x: x % 2, binary=True) == (
[1, 3], [0, 2])
assert sift([0, 1, 2, 3], lambda x: x % 3 == 1, binary=True) == (
[1], [0, 2, 3])
raises(ValueError, lambda:
sift([0, 1, 2, 3], lambda x: x % 3, binary=True))
def test_take():
X = numbered_symbols()
assert take(X, 5) == list(symbols('x0:5'))
assert take(X, 5) == list(symbols('x5:10'))
assert take([1, 2, 3, 4, 5], 5) == [1, 2, 3, 4, 5]
def test_dict_merge():
assert dict_merge({}, {1: x, y: z}) == {1: x, y: z}
assert dict_merge({1: x, y: z}, {}) == {1: x, y: z}
assert dict_merge({2: z}, {1: x, y: z}) == {1: x, 2: z, y: z}
assert dict_merge({1: x, y: z}, {2: z}) == {1: x, 2: z, y: z}
assert dict_merge({1: y, 2: z}, {1: x, y: z}) == {1: x, 2: z, y: z}
assert dict_merge({1: x, y: z}, {1: y, 2: z}) == {1: y, 2: z, y: z}
def test_prefixes():
assert list(prefixes([])) == []
assert list(prefixes([1])) == [[1]]
assert list(prefixes([1, 2])) == [[1], [1, 2]]
assert list(prefixes([1, 2, 3, 4, 5])) == \
[[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 3, 4, 5]]
def test_postfixes():
assert list(postfixes([])) == []
assert list(postfixes([1])) == [[1]]
assert list(postfixes([1, 2])) == [[2], [1, 2]]
assert list(postfixes([1, 2, 3, 4, 5])) == \
[[5], [4, 5], [3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5]]
def test_topological_sort():
V = [2, 3, 5, 7, 8, 9, 10, 11]
E = [(7, 11), (7, 8), (5, 11),
(3, 8), (3, 10), (11, 2),
(11, 9), (11, 10), (8, 9)]
assert topological_sort((V, E)) == [3, 5, 7, 8, 11, 2, 9, 10]
assert topological_sort((V, E), key=lambda v: -v) == \
[7, 5, 11, 3, 10, 8, 9, 2]
raises(ValueError, lambda: topological_sort((V, E + [(10, 7)])))
def test_strongly_connected_components():
assert strongly_connected_components(([], [])) == []
assert strongly_connected_components(([1, 2, 3], [])) == [[1], [2], [3]]
V = [1, 2, 3]
E = [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1)]
assert strongly_connected_components((V, E)) == [[1, 2, 3]]
V = [1, 2, 3, 4]
E = [(1, 2), (2, 3), (3, 2), (3, 4)]
assert strongly_connected_components((V, E)) == [[4], [2, 3], [1]]
V = [1, 2, 3, 4]
E = [(1, 2), (2, 1), (3, 4), (4, 3)]
assert strongly_connected_components((V, E)) == [[1, 2], [3, 4]]
def test_connected_components():
assert connected_components(([], [])) == []
assert connected_components(([1, 2, 3], [])) == [[1], [2], [3]]
V = [1, 2, 3]
E = [(1, 2), (1, 3), (2, 1), (2, 3), (3, 1)]
assert connected_components((V, E)) == [[1, 2, 3]]
V = [1, 2, 3, 4]
E = [(1, 2), (2, 3), (3, 2), (3, 4)]
assert connected_components((V, E)) == [[1, 2, 3, 4]]
V = [1, 2, 3, 4]
E = [(1, 2), (3, 4)]
assert connected_components((V, E)) == [[1, 2], [3, 4]]
def test_rotate():
A = [0, 1, 2, 3, 4]
assert rotate_left(A, 2) == [2, 3, 4, 0, 1]
assert rotate_right(A, 1) == [4, 0, 1, 2, 3]
A = []
B = rotate_right(A, 1)
assert B == []
B.append(1)
assert A == []
B = rotate_left(A, 1)
assert B == []
B.append(1)
assert A == []
def test_multiset_partitions():
A = [0, 1, 2, 3, 4]
assert list(multiset_partitions(A, 5)) == [[[0], [1], [2], [3], [4]]]
assert len(list(multiset_partitions(A, 4))) == 10
assert len(list(multiset_partitions(A, 3))) == 25
assert list(multiset_partitions([1, 1, 1, 2, 2], 2)) == [
[[1, 1, 1, 2], [2]], [[1, 1, 1], [2, 2]], [[1, 1, 2, 2], [1]],
[[1, 1, 2], [1, 2]], [[1, 1], [1, 2, 2]]]
assert list(multiset_partitions([1, 1, 2, 2], 2)) == [
[[1, 1, 2], [2]], [[1, 1], [2, 2]], [[1, 2, 2], [1]],
[[1, 2], [1, 2]]]
assert list(multiset_partitions([1, 2, 3, 4], 2)) == [
[[1, 2, 3], [4]], [[1, 2, 4], [3]], [[1, 2], [3, 4]],
[[1, 3, 4], [2]], [[1, 3], [2, 4]], [[1, 4], [2, 3]],
[[1], [2, 3, 4]]]
assert list(multiset_partitions([1, 2, 2], 2)) == [
[[1, 2], [2]], [[1], [2, 2]]]
assert list(multiset_partitions(3)) == [
[[0, 1, 2]], [[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]],
[[0], [1], [2]]]
assert list(multiset_partitions(3, 2)) == [
[[0, 1], [2]], [[0, 2], [1]], [[0], [1, 2]]]
assert list(multiset_partitions([1] * 3, 2)) == [[[1], [1, 1]]]
assert list(multiset_partitions([1] * 3)) == [
[[1, 1, 1]], [[1], [1, 1]], [[1], [1], [1]]]
a = [3, 2, 1]
assert list(multiset_partitions(a)) == \
list(multiset_partitions(sorted(a)))
assert list(multiset_partitions(a, 5)) == []
assert list(multiset_partitions(a, 1)) == [[[1, 2, 3]]]
assert list(multiset_partitions(a + [4], 5)) == []
assert list(multiset_partitions(a + [4], 1)) == [[[1, 2, 3, 4]]]
assert list(multiset_partitions(2, 5)) == []
assert list(multiset_partitions(2, 1)) == [[[0, 1]]]
assert list(multiset_partitions('a')) == [[['a']]]
assert list(multiset_partitions('a', 2)) == []
assert list(multiset_partitions('ab')) == [[['a', 'b']], [['a'], ['b']]]
assert list(multiset_partitions('ab', 1)) == [[['a', 'b']]]
assert list(multiset_partitions('aaa', 1)) == [['aaa']]
assert list(multiset_partitions([1, 1], 1)) == [[[1, 1]]]
ans = [('mpsyy',), ('mpsy', 'y'), ('mps', 'yy'), ('mps', 'y', 'y'),
('mpyy', 's'), ('mpy', 'sy'), ('mpy', 's', 'y'), ('mp', 'syy'),
('mp', 'sy', 'y'), ('mp', 's', 'yy'), ('mp', 's', 'y', 'y'),
('msyy', 'p'), ('msy', 'py'), ('msy', 'p', 'y'), ('ms', 'pyy'),
('ms', 'py', 'y'), ('ms', 'p', 'yy'), ('ms', 'p', 'y', 'y'),
('myy', 'ps'), ('myy', 'p', 's'), ('my', 'psy'), ('my', 'ps', 'y'),
('my', 'py', 's'), ('my', 'p', 'sy'), ('my', 'p', 's', 'y'),
('m', 'psyy'), ('m', 'psy', 'y'), ('m', 'ps', 'yy'),
('m', 'ps', 'y', 'y'), ('m', 'pyy', 's'), ('m', 'py', 'sy'),
('m', 'py', 's', 'y'), ('m', 'p', 'syy'),
('m', 'p', 'sy', 'y'), ('m', 'p', 's', 'yy'),
('m', 'p', 's', 'y', 'y')]
assert list(tuple("".join(part) for part in p)
for p in multiset_partitions('sympy')) == ans
factorings = [[24], [8, 3], [12, 2], [4, 6], [4, 2, 3],
[6, 2, 2], [2, 2, 2, 3]]
assert list(factoring_visitor(p, [2,3]) for
p in multiset_partitions_taocp([3, 1])) == factorings
def test_multiset_combinations():
ans = ['iii', 'iim', 'iip', 'iis', 'imp', 'ims', 'ipp', 'ips',
'iss', 'mpp', 'mps', 'mss', 'pps', 'pss', 'sss']
assert [''.join(i) for i in
list(multiset_combinations('mississippi', 3))] == ans
M = multiset('mississippi')
assert [''.join(i) for i in
list(multiset_combinations(M, 3))] == ans
assert [''.join(i) for i in multiset_combinations(M, 30)] == []
assert list(multiset_combinations([[1], [2, 3]], 2)) == [[[1], [2, 3]]]
assert len(list(multiset_combinations('a', 3))) == 0
assert len(list(multiset_combinations('a', 0))) == 1
assert list(multiset_combinations('abc', 1)) == [['a'], ['b'], ['c']]
raises(ValueError, lambda: list(multiset_combinations({0: 3, 1: -1}, 2)))
def test_multiset_permutations():
ans = ['abby', 'abyb', 'aybb', 'baby', 'bayb', 'bbay', 'bbya', 'byab',
'byba', 'yabb', 'ybab', 'ybba']
assert [''.join(i) for i in multiset_permutations('baby')] == ans
assert [''.join(i) for i in multiset_permutations(multiset('baby'))] == ans
assert list(multiset_permutations([0, 0, 0], 2)) == [[0, 0]]
assert list(multiset_permutations([0, 2, 1], 2)) == [
[0, 1], [0, 2], [1, 0], [1, 2], [2, 0], [2, 1]]
assert len(list(multiset_permutations('a', 0))) == 1
assert len(list(multiset_permutations('a', 3))) == 0
for nul in ([], {}, ''):
assert list(multiset_permutations(nul)) == [[]]
assert list(multiset_permutations(nul, 0)) == [[]]
# impossible requests give no result
assert list(multiset_permutations(nul, 1)) == []
assert list(multiset_permutations(nul, -1)) == []
def test():
for i in range(1, 7):
print(i)
for p in multiset_permutations([0, 0, 1, 0, 1], i):
print(p)
assert capture(lambda: test()) == dedent('''\
1
[0]
[1]
2
[0, 0]
[0, 1]
[1, 0]
[1, 1]
3
[0, 0, 0]
[0, 0, 1]
[0, 1, 0]
[0, 1, 1]
[1, 0, 0]
[1, 0, 1]
[1, 1, 0]
4
[0, 0, 0, 1]
[0, 0, 1, 0]
[0, 0, 1, 1]
[0, 1, 0, 0]
[0, 1, 0, 1]
[0, 1, 1, 0]
[1, 0, 0, 0]
[1, 0, 0, 1]
[1, 0, 1, 0]
[1, 1, 0, 0]
5
[0, 0, 0, 1, 1]
[0, 0, 1, 0, 1]
[0, 0, 1, 1, 0]
[0, 1, 0, 0, 1]
[0, 1, 0, 1, 0]
[0, 1, 1, 0, 0]
[1, 0, 0, 0, 1]
[1, 0, 0, 1, 0]
[1, 0, 1, 0, 0]
[1, 1, 0, 0, 0]
6\n''')
raises(ValueError, lambda: list(multiset_permutations({0: 3, 1: -1})))
def test_partitions():
ans = [[{}], [(0, {})]]
for i in range(2):
assert list(partitions(0, size=i)) == ans[i]
assert list(partitions(1, 0, size=i)) == ans[i]
assert list(partitions(6, 2, 2, size=i)) == ans[i]
assert list(partitions(6, 2, None, size=i)) != ans[i]
assert list(partitions(6, None, 2, size=i)) != ans[i]
assert list(partitions(6, 2, 0, size=i)) == ans[i]
assert [p for p in partitions(6, k=2)] == [
{2: 3}, {1: 2, 2: 2}, {1: 4, 2: 1}, {1: 6}]
assert [p for p in partitions(6, k=3)] == [
{3: 2}, {1: 1, 2: 1, 3: 1}, {1: 3, 3: 1}, {2: 3}, {1: 2, 2: 2},
{1: 4, 2: 1}, {1: 6}]
assert [p for p in partitions(8, k=4, m=3)] == [
{4: 2}, {1: 1, 3: 1, 4: 1}, {2: 2, 4: 1}, {2: 1, 3: 2}] == [
i for i in partitions(8, k=4, m=3) if all(k <= 4 for k in i)
and sum(i.values()) <=3]
assert [p for p in partitions(S(3), m=2)] == [
{3: 1}, {1: 1, 2: 1}]
assert [i for i in partitions(4, k=3)] == [
{1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}] == [
i for i in partitions(4) if all(k <= 3 for k in i)]
# Consistency check on output of _partitions and RGS_unrank.
# This provides a sanity test on both routines. Also verifies that
# the total number of partitions is the same in each case.
# (from pkrathmann2)
for n in range(2, 6):
i = 0
for m, q in _set_partitions(n):
assert q == RGS_unrank(i, n)
i += 1
assert i == RGS_enum(n)
def test_binary_partitions():
assert [i[:] for i in binary_partitions(10)] == [[8, 2], [8, 1, 1],
[4, 4, 2], [4, 4, 1, 1], [4, 2, 2, 2], [4, 2, 2, 1, 1],
[4, 2, 1, 1, 1, 1], [4, 1, 1, 1, 1, 1, 1], [2, 2, 2, 2, 2],
[2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1, 1],
[2, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
assert len([j[:] for j in binary_partitions(16)]) == 36
def test_bell_perm():
assert [len(set(generate_bell(i))) for i in range(1, 7)] == [
factorial(i) for i in range(1, 7)]
assert list(generate_bell(3)) == [
(0, 1, 2), (0, 2, 1), (2, 0, 1), (2, 1, 0), (1, 2, 0), (1, 0, 2)]
# generate_bell and trotterjohnson are advertised to return the same
# permutations; this is not technically necessary so this test could
# be removed
for n in range(1, 5):
p = Permutation(range(n))
b = generate_bell(n)
for bi in b:
assert bi == tuple(p.array_form)
p = p.next_trotterjohnson()
raises(ValueError, lambda: list(generate_bell(0))) # XXX is this consistent with other permutation algorithms?
def test_involutions():
lengths = [1, 2, 4, 10, 26, 76]
for n, N in enumerate(lengths):
i = list(generate_involutions(n + 1))
assert len(i) == N
assert len({Permutation(j)**2 for j in i}) == 1
def test_derangements():
assert len(list(generate_derangements(list(range(6))))) == 265
assert ''.join(''.join(i) for i in generate_derangements('abcde')) == (
'badecbaecdbcaedbcdeabceadbdaecbdeacbdecabeacdbedacbedcacabedcadebcaebd'
'cdaebcdbeacdeabcdebaceabdcebadcedabcedbadabecdaebcdaecbdcaebdcbeadceab'
'dcebadeabcdeacbdebacdebcaeabcdeadbceadcbecabdecbadecdabecdbaedabcedacb'
'edbacedbca')
assert list(generate_derangements([0, 1, 2, 3])) == [
[1, 0, 3, 2], [1, 2, 3, 0], [1, 3, 0, 2], [2, 0, 3, 1],
[2, 3, 0, 1], [2, 3, 1, 0], [3, 0, 1, 2], [3, 2, 0, 1], [3, 2, 1, 0]]
assert list(generate_derangements([0, 1, 2, 2])) == [
[2, 2, 0, 1], [2, 2, 1, 0]]
assert list(generate_derangements('ba')) == [list('ab')]
# multiset_derangements
D = multiset_derangements
assert list(D('abb')) == []
assert [''.join(i) for i in D('ab')] == ['ba']
assert [''.join(i) for i in D('abc')] == ['bca', 'cab']
assert [''.join(i) for i in D('aabb')] == ['bbaa']
assert [''.join(i) for i in D('aabbcccc')] == [
'ccccaabb', 'ccccabab', 'ccccabba', 'ccccbaab', 'ccccbaba',
'ccccbbaa']
assert [''.join(i) for i in D('aabbccc')] == [
'cccabba', 'cccabab', 'cccaabb', 'ccacbba', 'ccacbab',
'ccacabb', 'cbccbaa', 'cbccaba', 'cbccaab', 'bcccbaa',
'bcccaba', 'bcccaab']
assert [''.join(i) for i in D('books')] == ['kbsoo', 'ksboo',
'sbkoo', 'skboo', 'oksbo', 'oskbo', 'okbso', 'obkso', 'oskob',
'oksob', 'osbok', 'obsok']
assert list(generate_derangements([[3], [2], [2], [1]])) == [
[[2], [1], [3], [2]], [[2], [3], [1], [2]]]
def test_necklaces():
def count(n, k, f):
return len(list(necklaces(n, k, f)))
m = []
for i in range(1, 8):
m.append((
i, count(i, 2, 0), count(i, 2, 1), count(i, 3, 1)))
assert Matrix(m) == Matrix([
[1, 2, 2, 3],
[2, 3, 3, 6],
[3, 4, 4, 10],
[4, 6, 6, 21],
[5, 8, 8, 39],
[6, 14, 13, 92],
[7, 20, 18, 198]])
def test_bracelets():
bc = [i for i in bracelets(2, 4)]
assert Matrix(bc) == Matrix([
[0, 0],
[0, 1],
[0, 2],
[0, 3],
[1, 1],
[1, 2],
[1, 3],
[2, 2],
[2, 3],
[3, 3]
])
bc = [i for i in bracelets(4, 2)]
assert Matrix(bc) == Matrix([
[0, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 1],
[0, 1, 0, 1],
[0, 1, 1, 1],
[1, 1, 1, 1]
])
def test_generate_oriented_forest():
assert list(generate_oriented_forest(5)) == [[0, 1, 2, 3, 4],
[0, 1, 2, 3, 3], [0, 1, 2, 3, 2], [0, 1, 2, 3, 1], [0, 1, 2, 3, 0],
[0, 1, 2, 2, 2], [0, 1, 2, 2, 1], [0, 1, 2, 2, 0], [0, 1, 2, 1, 2],
[0, 1, 2, 1, 1], [0, 1, 2, 1, 0], [0, 1, 2, 0, 1], [0, 1, 2, 0, 0],
[0, 1, 1, 1, 1], [0, 1, 1, 1, 0], [0, 1, 1, 0, 1], [0, 1, 1, 0, 0],
[0, 1, 0, 1, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 0]]
assert len(list(generate_oriented_forest(10))) == 1842
def test_unflatten():
r = list(range(10))
assert unflatten(r) == list(zip(r[::2], r[1::2]))
assert unflatten(r, 5) == [tuple(r[:5]), tuple(r[5:])]
raises(ValueError, lambda: unflatten(list(range(10)), 3))
raises(ValueError, lambda: unflatten(list(range(10)), -2))
def test_common_prefix_suffix():
assert common_prefix([], [1]) == []
assert common_prefix(list(range(3))) == [0, 1, 2]
assert common_prefix(list(range(3)), list(range(4))) == [0, 1, 2]
assert common_prefix([1, 2, 3], [1, 2, 5]) == [1, 2]
assert common_prefix([1, 2, 3], [1, 3, 5]) == [1]
assert common_suffix([], [1]) == []
assert common_suffix(list(range(3))) == [0, 1, 2]
assert common_suffix(list(range(3)), list(range(3))) == [0, 1, 2]
assert common_suffix(list(range(3)), list(range(4))) == []
assert common_suffix([1, 2, 3], [9, 2, 3]) == [2, 3]
assert common_suffix([1, 2, 3], [9, 7, 3]) == [3]
def test_minlex():
assert minlex([1, 2, 0]) == (0, 1, 2)
assert minlex((1, 2, 0)) == (0, 1, 2)
assert minlex((1, 0, 2)) == (0, 2, 1)
assert minlex((1, 0, 2), directed=False) == (0, 1, 2)
assert minlex('aba') == 'aab'
assert minlex(('bb', 'aaa', 'c', 'a'), key=len) == ('c', 'a', 'bb', 'aaa')
def test_ordered():
assert list(ordered((x, y), hash, default=False)) in [[x, y], [y, x]]
assert list(ordered((x, y), hash, default=False)) == \
list(ordered((y, x), hash, default=False))
assert list(ordered((x, y))) == [x, y]
seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]],
(lambda x: len(x), lambda x: sum(x))]
assert list(ordered(seq, keys, default=False, warn=False)) == \
[[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]
raises(ValueError, lambda:
list(ordered(seq, keys, default=False, warn=True)))
def test_runs():
assert runs([]) == []
assert runs([1]) == [[1]]
assert runs([1, 1]) == [[1], [1]]
assert runs([1, 1, 2]) == [[1], [1, 2]]
assert runs([1, 2, 1]) == [[1, 2], [1]]
assert runs([2, 1, 1]) == [[2], [1], [1]]
from operator import lt
assert runs([2, 1, 1], lt) == [[2, 1], [1]]
def test_reshape():
seq = list(range(1, 9))
assert reshape(seq, [4]) == \
[[1, 2, 3, 4], [5, 6, 7, 8]]
assert reshape(seq, (4,)) == \
[(1, 2, 3, 4), (5, 6, 7, 8)]
assert reshape(seq, (2, 2)) == \
[(1, 2, 3, 4), (5, 6, 7, 8)]
assert reshape(seq, (2, [2])) == \
[(1, 2, [3, 4]), (5, 6, [7, 8])]
assert reshape(seq, ((2,), [2])) == \
[((1, 2), [3, 4]), ((5, 6), [7, 8])]
assert reshape(seq, (1, [2], 1)) == \
[(1, [2, 3], 4), (5, [6, 7], 8)]
assert reshape(tuple(seq), ([[1], 1, (2,)],)) == \
(([[1], 2, (3, 4)],), ([[5], 6, (7, 8)],))
assert reshape(tuple(seq), ([1], 1, (2,))) == \
(([1], 2, (3, 4)), ([5], 6, (7, 8)))
assert reshape(list(range(12)), [2, [3], {2}, (1, (3,), 1)]) == \
[[0, 1, [2, 3, 4], {5, 6}, (7, (8, 9, 10), 11)]]
raises(ValueError, lambda: reshape([0, 1], [-1]))
raises(ValueError, lambda: reshape([0, 1], [3]))
def test_uniq():
assert list(uniq(p for p in partitions(4))) == \
[{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2: 1}, {1: 4}]
assert list(uniq(x % 2 for x in range(5))) == [0, 1]
assert list(uniq('a')) == ['a']
assert list(uniq('ababc')) == list('abc')
assert list(uniq([[1], [2, 1], [1]])) == [[1], [2, 1]]
assert list(uniq(permutations(i for i in [[1], 2, 2]))) == \
[([1], 2, 2), (2, [1], 2), (2, 2, [1])]
assert list(uniq([2, 3, 2, 4, [2], [1], [2], [3], [1]])) == \
[2, 3, 4, [2], [1], [3]]
f = [1]
raises(RuntimeError, lambda: [f.remove(i) for i in uniq(f)])
f = [[1]]
raises(RuntimeError, lambda: [f.remove(i) for i in uniq(f)])
def test_kbins():
assert len(list(kbins('1123', 2, ordered=1))) == 24
assert len(list(kbins('1123', 2, ordered=11))) == 36
assert len(list(kbins('1123', 2, ordered=10))) == 10
assert len(list(kbins('1123', 2, ordered=0))) == 5
assert len(list(kbins('1123', 2, ordered=None))) == 3
def test1():
for orderedval in [None, 0, 1, 10, 11]:
print('ordered =', orderedval)
for p in kbins([0, 0, 1], 2, ordered=orderedval):
print(' ', p)
assert capture(lambda : test1()) == dedent('''\
ordered = None
[[0], [0, 1]]
[[0, 0], [1]]
ordered = 0
[[0, 0], [1]]
[[0, 1], [0]]
ordered = 1
[[0], [0, 1]]
[[0], [1, 0]]
[[1], [0, 0]]
ordered = 10
[[0, 0], [1]]
[[1], [0, 0]]
[[0, 1], [0]]
[[0], [0, 1]]
ordered = 11
[[0], [0, 1]]
[[0, 0], [1]]
[[0], [1, 0]]
[[0, 1], [0]]
[[1], [0, 0]]
[[1, 0], [0]]\n''')
def test2():
for orderedval in [None, 0, 1, 10, 11]:
print('ordered =', orderedval)
for p in kbins(list(range(3)), 2, ordered=orderedval):
print(' ', p)
assert capture(lambda : test2()) == dedent('''\
ordered = None
[[0], [1, 2]]
[[0, 1], [2]]
ordered = 0
[[0, 1], [2]]
[[0, 2], [1]]
[[0], [1, 2]]
ordered = 1
[[0], [1, 2]]
[[0], [2, 1]]
[[1], [0, 2]]
[[1], [2, 0]]
[[2], [0, 1]]
[[2], [1, 0]]
ordered = 10
[[0, 1], [2]]
[[2], [0, 1]]
[[0, 2], [1]]
[[1], [0, 2]]
[[0], [1, 2]]
[[1, 2], [0]]
ordered = 11
[[0], [1, 2]]
[[0, 1], [2]]
[[0], [2, 1]]
[[0, 2], [1]]
[[1], [0, 2]]
[[1, 0], [2]]
[[1], [2, 0]]
[[1, 2], [0]]
[[2], [0, 1]]
[[2, 0], [1]]
[[2], [1, 0]]
[[2, 1], [0]]\n''')
def test_has_dups():
assert has_dups(set()) is False
assert has_dups(list(range(3))) is False
assert has_dups([1, 2, 1]) is True
assert has_dups([[1], [1]]) is True
assert has_dups([[1], [2]]) is False
def test__partition():
assert _partition('abcde', [1, 0, 1, 2, 0]) == [
['b', 'e'], ['a', 'c'], ['d']]
assert _partition('abcde', [1, 0, 1, 2, 0], 3) == [
['b', 'e'], ['a', 'c'], ['d']]
output = (3, [1, 0, 1, 2, 0])
assert _partition('abcde', *output) == [['b', 'e'], ['a', 'c'], ['d']]
def test_ordered_partitions():
from sympy.functions.combinatorial.numbers import nT
f = ordered_partitions
assert list(f(0, 1)) == [[]]
assert list(f(1, 0)) == [[]]
for i in range(1, 7):
for j in [None] + list(range(1, i)):
assert (
sum(1 for p in f(i, j, 1)) ==
sum(1 for p in f(i, j, 0)) ==
nT(i, j))
def test_rotations():
assert list(rotations('ab')) == [['a', 'b'], ['b', 'a']]
assert list(rotations(range(3))) == [[0, 1, 2], [1, 2, 0], [2, 0, 1]]
assert list(rotations(range(3), dir=-1)) == [[0, 1, 2], [2, 0, 1], [1, 2, 0]]
def test_ibin():
assert ibin(3) == [1, 1]
assert ibin(3, 3) == [0, 1, 1]
assert ibin(3, str=True) == '11'
assert ibin(3, 3, str=True) == '011'
assert list(ibin(2, 'all')) == [(0, 0), (0, 1), (1, 0), (1, 1)]
assert list(ibin(2, '', str=True)) == ['00', '01', '10', '11']
raises(ValueError, lambda: ibin(-.5))
raises(ValueError, lambda: ibin(2, 1))
def test_iterable():
assert iterable(0) is False
assert iterable(1) is False
assert iterable(None) is False
class Test1(NotIterable):
pass
assert iterable(Test1()) is False
class Test2(NotIterable):
_iterable = True
assert iterable(Test2()) is True
class Test3:
pass
assert iterable(Test3()) is False
class Test4:
_iterable = True
assert iterable(Test4()) is True
class Test5:
def __iter__(self):
yield 1
assert iterable(Test5()) is True
class Test6(Test5):
_iterable = False
assert iterable(Test6()) is False
|
38a45adb2b8732d9d57278acbbec5c69e762d912ac1fd42b814f5ec60a0e2b7d | from sympy.testing.pytest import raises
from sympy.utilities.exceptions import sympy_deprecation_warning
# Only test exceptions here because the other cases are tested in the
# warns_deprecated_sympy tests
def test_sympy_deprecation_warning():
raises(TypeError, lambda: sympy_deprecation_warning('test',
deprecated_since_version=1.10,
active_deprecations_target='active-deprecations'))
raises(ValueError, lambda: sympy_deprecation_warning('test',
deprecated_since_version="1.10", active_deprecations_target='(active-deprecations)='))
|
55852fcd4ad523b23b30ad186712cb12612ee2957976a681d422c427c39c6afe | from functools import wraps
from sympy.utilities.decorator import threaded, xthreaded, memoize_property, deprecated
from sympy.testing.pytest import warns_deprecated_sympy
from sympy.core.basic import Basic
from sympy.core.relational import Eq
from sympy.matrices.dense import Matrix
from sympy.abc import x, y
def test_threaded():
@threaded
def function(expr, *args):
return 2*expr + sum(args)
assert function(Matrix([[x, y], [1, x]]), 1, 2) == \
Matrix([[2*x + 3, 2*y + 3], [5, 2*x + 3]])
assert function(Eq(x, y), 1, 2) == Eq(2*x + 3, 2*y + 3)
assert function([x, y], 1, 2) == [2*x + 3, 2*y + 3]
assert function((x, y), 1, 2) == (2*x + 3, 2*y + 3)
assert function({x, y}, 1, 2) == {2*x + 3, 2*y + 3}
@threaded
def function(expr, n):
return expr**n
assert function(x + y, 2) == x**2 + y**2
assert function(x, 2) == x**2
def test_xthreaded():
@xthreaded
def function(expr, n):
return expr**n
assert function(x + y, 2) == (x + y)**2
def test_wraps():
def my_func(x):
"""My function. """
my_func.is_my_func = True
new_my_func = threaded(my_func)
new_my_func = wraps(my_func)(new_my_func)
assert new_my_func.__name__ == 'my_func'
assert new_my_func.__doc__ == 'My function. '
assert hasattr(new_my_func, 'is_my_func')
assert new_my_func.is_my_func is True
def test_memoize_property():
class TestMemoize(Basic):
@memoize_property
def prop(self):
return Basic()
member = TestMemoize()
obj1 = member.prop
obj2 = member.prop
assert obj1 is obj2
def test_deprecated():
@deprecated('deprecated_function is deprecated',
deprecated_since_version='1.10',
# This is the target at the top of the file, which will never
# go away.
active_deprecations_target='active-deprecations')
def deprecated_function(x):
return x
with warns_deprecated_sympy():
assert deprecated_function(1) == 1
@deprecated('deprecated_class is deprecated',
deprecated_since_version='1.10',
active_deprecations_target='active-deprecations')
class deprecated_class:
pass
with warns_deprecated_sympy():
assert isinstance(deprecated_class(), deprecated_class)
# Ensure the class decorator works even when the class never returns
# itself
@deprecated('deprecated_class_new is deprecated',
deprecated_since_version='1.10',
active_deprecations_target='active-deprecations')
class deprecated_class_new:
def __new__(cls, arg):
return arg
with warns_deprecated_sympy():
assert deprecated_class_new(1) == 1
@deprecated('deprecated_class_init is deprecated',
deprecated_since_version='1.10',
active_deprecations_target='active-deprecations')
class deprecated_class_init:
def __init__(self, arg):
self.arg = 1
with warns_deprecated_sympy():
assert deprecated_class_init(1).arg == 1
@deprecated('deprecated_class_new_init is deprecated',
deprecated_since_version='1.10',
active_deprecations_target='active-deprecations')
class deprecated_class_new_init:
def __new__(cls, arg):
if arg == 0:
return arg
return object.__new__(cls)
def __init__(self, arg):
self.arg = 1
with warns_deprecated_sympy():
assert deprecated_class_new_init(0) == 0
with warns_deprecated_sympy():
assert deprecated_class_new_init(1).arg == 1
|
632884d4db230ce62b222a12b0840ff95ad6a7c452550cf0dde5d935efe665b2 | import itertools
from sympy.core import S
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import Function
from sympy.core.mul import Mul
from sympy.core.numbers import Number, Rational
from sympy.core.power import Pow
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Symbol
from sympy.core.sympify import SympifyError
from sympy.printing.conventions import requires_partial
from sympy.printing.precedence import PRECEDENCE, precedence, precedence_traditional
from sympy.printing.printer import Printer, print_function
from sympy.printing.str import sstr
from sympy.utilities.iterables import has_variety
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.printing.pretty.pretty_symbology import hobj, vobj, xobj, \
xsym, pretty_symbol, pretty_atom, pretty_use_unicode, greek_unicode, U, \
pretty_try_use_unicode, annotated
# rename for usage from outside
pprint_use_unicode = pretty_use_unicode
pprint_try_use_unicode = pretty_try_use_unicode
class PrettyPrinter(Printer):
"""Printer, which converts an expression into 2D ASCII-art figure."""
printmethod = "_pretty"
_default_settings = {
"order": None,
"full_prec": "auto",
"use_unicode": None,
"wrap_line": True,
"num_columns": None,
"use_unicode_sqrt_char": True,
"root_notation": True,
"mat_symbol_style": "plain",
"imaginary_unit": "i",
"perm_cyclic": True
}
def __init__(self, settings=None):
Printer.__init__(self, settings)
if not isinstance(self._settings['imaginary_unit'], str):
raise TypeError("'imaginary_unit' must a string, not {}".format(self._settings['imaginary_unit']))
elif self._settings['imaginary_unit'] not in ("i", "j"):
raise ValueError("'imaginary_unit' must be either 'i' or 'j', not '{}'".format(self._settings['imaginary_unit']))
def emptyPrinter(self, expr):
return prettyForm(str(expr))
@property
def _use_unicode(self):
if self._settings['use_unicode']:
return True
else:
return pretty_use_unicode()
def doprint(self, expr):
return self._print(expr).render(**self._settings)
# empty op so _print(stringPict) returns the same
def _print_stringPict(self, e):
return e
def _print_basestring(self, e):
return prettyForm(e)
def _print_atan2(self, e):
pform = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(*pform.left('atan2'))
return pform
def _print_Symbol(self, e, bold_name=False):
symb = pretty_symbol(e.name, bold_name)
return prettyForm(symb)
_print_RandomSymbol = _print_Symbol
def _print_MatrixSymbol(self, e):
return self._print_Symbol(e, self._settings['mat_symbol_style'] == "bold")
def _print_Float(self, e):
# we will use StrPrinter's Float printer, but we need to handle the
# full_prec ourselves, according to the self._print_level
full_prec = self._settings["full_prec"]
if full_prec == "auto":
full_prec = self._print_level == 1
return prettyForm(sstr(e, full_prec=full_prec))
def _print_Cross(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Curl(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('MULTIPLICATION SIGN'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Divergence(self, e):
vec = e._expr
pform = self._print(vec)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Dot(self, e):
vec1 = e._expr1
vec2 = e._expr2
pform = self._print(vec2)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('DOT OPERATOR'))))
pform = prettyForm(*pform.left(')'))
pform = prettyForm(*pform.left(self._print(vec1)))
pform = prettyForm(*pform.left('('))
return pform
def _print_Gradient(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('NABLA'))))
return pform
def _print_Laplacian(self, e):
func = e._expr
pform = self._print(func)
pform = prettyForm(*pform.left('('))
pform = prettyForm(*pform.right(')'))
pform = prettyForm(*pform.left(self._print(U('INCREMENT'))))
return pform
def _print_Atom(self, e):
try:
# print atoms like Exp1 or Pi
return prettyForm(pretty_atom(e.__class__.__name__, printer=self))
except KeyError:
return self.emptyPrinter(e)
# Infinity inherits from Number, so we have to override _print_XXX order
_print_Infinity = _print_Atom
_print_NegativeInfinity = _print_Atom
_print_EmptySet = _print_Atom
_print_Naturals = _print_Atom
_print_Naturals0 = _print_Atom
_print_Integers = _print_Atom
_print_Rationals = _print_Atom
_print_Complexes = _print_Atom
_print_EmptySequence = _print_Atom
def _print_Reals(self, e):
if self._use_unicode:
return self._print_Atom(e)
else:
inf_list = ['-oo', 'oo']
return self._print_seq(inf_list, '(', ')')
def _print_subfactorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('!'))
return pform
def _print_factorial(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!'))
return pform
def _print_factorial2(self, e):
x = e.args[0]
pform = self._print(x)
# Add parentheses if needed
if not ((x.is_Integer and x.is_nonnegative) or x.is_Symbol):
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right('!!'))
return pform
def _print_binomial(self, e):
n, k = e.args
n_pform = self._print(n)
k_pform = self._print(k)
bar = ' '*max(n_pform.width(), k_pform.width())
pform = prettyForm(*k_pform.above(bar))
pform = prettyForm(*pform.above(n_pform))
pform = prettyForm(*pform.parens('(', ')'))
pform.baseline = (pform.baseline + 1)//2
return pform
def _print_Relational(self, e):
op = prettyForm(' ' + xsym(e.rel_op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r), binding=prettyForm.OPEN)
return pform
def _print_Not(self, e):
from sympy.logic.boolalg import (Equivalent, Implies)
if self._use_unicode:
arg = e.args[0]
pform = self._print(arg)
if isinstance(arg, Equivalent):
return self._print_Equivalent(arg, altchar="\N{LEFT RIGHT DOUBLE ARROW WITH STROKE}")
if isinstance(arg, Implies):
return self._print_Implies(arg, altchar="\N{RIGHTWARDS ARROW WITH STROKE}")
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left("\N{NOT SIGN}"))
else:
return self._print_Function(e)
def __print_Boolean(self, e, char, sort=True):
args = e.args
if sort:
args = sorted(e.args, key=default_sort_key)
arg = args[0]
pform = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform = prettyForm(*pform.parens())
for arg in args[1:]:
pform_arg = self._print(arg)
if arg.is_Boolean and not arg.is_Not:
pform_arg = prettyForm(*pform_arg.parens())
pform = prettyForm(*pform.right(' %s ' % char))
pform = prettyForm(*pform.right(pform_arg))
return pform
def _print_And(self, e):
if self._use_unicode:
return self.__print_Boolean(e, "\N{LOGICAL AND}")
else:
return self._print_Function(e, sort=True)
def _print_Or(self, e):
if self._use_unicode:
return self.__print_Boolean(e, "\N{LOGICAL OR}")
else:
return self._print_Function(e, sort=True)
def _print_Xor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, "\N{XOR}")
else:
return self._print_Function(e, sort=True)
def _print_Nand(self, e):
if self._use_unicode:
return self.__print_Boolean(e, "\N{NAND}")
else:
return self._print_Function(e, sort=True)
def _print_Nor(self, e):
if self._use_unicode:
return self.__print_Boolean(e, "\N{NOR}")
else:
return self._print_Function(e, sort=True)
def _print_Implies(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or "\N{RIGHTWARDS ARROW}", sort=False)
else:
return self._print_Function(e)
def _print_Equivalent(self, e, altchar=None):
if self._use_unicode:
return self.__print_Boolean(e, altchar or "\N{LEFT RIGHT DOUBLE ARROW}")
else:
return self._print_Function(e, sort=True)
def _print_conjugate(self, e):
pform = self._print(e.args[0])
return prettyForm( *pform.above( hobj('_', pform.width())) )
def _print_Abs(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('|', '|'))
return pform
_print_Determinant = _print_Abs
def _print_floor(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lfloor', 'rfloor'))
return pform
else:
return self._print_Function(e)
def _print_ceiling(self, e):
if self._use_unicode:
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens('lceil', 'rceil'))
return pform
else:
return self._print_Function(e)
def _print_Derivative(self, deriv):
if requires_partial(deriv.expr) and self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
count_total_deriv = 0
for sym, num in reversed(deriv.variable_count):
s = self._print(sym)
ds = prettyForm(*s.left(deriv_symbol))
count_total_deriv += num
if (not num.is_Integer) or (num > 1):
ds = ds**prettyForm(str(num))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
if (count_total_deriv > 1) != False:
pform = pform**prettyForm(str(count_total_deriv))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Cycle(self, dc):
from sympy.combinatorics.permutations import Permutation, Cycle
# for Empty Cycle
if dc == Cycle():
cyc = stringPict('')
return prettyForm(*cyc.parens())
dc_list = Permutation(dc.list()).cyclic_form
# for Identity Cycle
if dc_list == []:
cyc = self._print(dc.size - 1)
return prettyForm(*cyc.parens())
cyc = stringPict('')
for i in dc_list:
l = self._print(str(tuple(i)).replace(',', ''))
cyc = prettyForm(*cyc.right(l))
return cyc
def _print_Permutation(self, expr):
from sympy.combinatorics.permutations import Permutation, Cycle
perm_cyclic = Permutation.print_cyclic
if perm_cyclic is not None:
sympy_deprecation_warning(
f"""
Setting Permutation.print_cyclic is deprecated. Instead use
init_printing(perm_cyclic={perm_cyclic}).
""",
deprecated_since_version="1.6",
active_deprecations_target="deprecated-permutation-print_cyclic",
stacklevel=7,
)
else:
perm_cyclic = self._settings.get("perm_cyclic", True)
if perm_cyclic:
return self._print_Cycle(Cycle(expr))
lower = expr.array_form
upper = list(range(len(lower)))
result = stringPict('')
first = True
for u, l in zip(upper, lower):
s1 = self._print(u)
s2 = self._print(l)
col = prettyForm(*s1.below(s2))
if first:
first = False
else:
col = prettyForm(*col.left(" "))
result = prettyForm(*result.right(col))
return prettyForm(*result.parens())
def _print_Integral(self, integral):
f = integral.function
# Add parentheses if arg involves addition of terms and
# create a pretty form for the argument
prettyF = self._print(f)
# XXX generalize parens
if f.is_Add:
prettyF = prettyForm(*prettyF.parens())
# dx dy dz ...
arg = prettyF
for x in integral.limits:
prettyArg = self._print(x[0])
# XXX qparens (parens if needs-parens)
if prettyArg.width() > 1:
prettyArg = prettyForm(*prettyArg.parens())
arg = prettyForm(*arg.right(' d', prettyArg))
# \int \int \int ...
firstterm = True
s = None
for lim in integral.limits:
# Create bar based on the height of the argument
h = arg.height()
H = h + 2
# XXX hack!
ascii_mode = not self._use_unicode
if ascii_mode:
H += 2
vint = vobj('int', H)
# Construct the pretty form with the integral sign and the argument
pform = prettyForm(vint)
pform.baseline = arg.baseline + (
H - h)//2 # covering the whole argument
if len(lim) > 1:
# Create pretty forms for endpoints, if definite integral.
# Do not print empty endpoints.
if len(lim) == 2:
prettyA = prettyForm("")
prettyB = self._print(lim[1])
if len(lim) == 3:
prettyA = self._print(lim[1])
prettyB = self._print(lim[2])
if ascii_mode: # XXX hack
# Add spacing so that endpoint can more easily be
# identified with the correct integral sign
spc = max(1, 3 - prettyB.width())
prettyB = prettyForm(*prettyB.left(' ' * spc))
spc = max(1, 4 - prettyA.width())
prettyA = prettyForm(*prettyA.right(' ' * spc))
pform = prettyForm(*pform.above(prettyB))
pform = prettyForm(*pform.below(prettyA))
if not ascii_mode: # XXX hack
pform = prettyForm(*pform.right(' '))
if firstterm:
s = pform # first term
firstterm = False
else:
s = prettyForm(*s.left(pform))
pform = prettyForm(*arg.left(s))
pform.binding = prettyForm.MUL
return pform
def _print_Product(self, expr):
func = expr.term
pretty_func = self._print(func)
horizontal_chr = xobj('_', 1)
corner_chr = xobj('_', 1)
vertical_chr = xobj('|', 1)
if self._use_unicode:
# use unicode corners
horizontal_chr = xobj('-', 1)
corner_chr = '\N{BOX DRAWINGS LIGHT DOWN AND HORIZONTAL}'
func_height = pretty_func.height()
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
pretty_lower, pretty_upper = self.__print_SumProduct_Limits(lim)
width = (func_height + 2) * 5 // 3 - 2
sign_lines = [horizontal_chr + corner_chr + (horizontal_chr * (width-2)) + corner_chr + horizontal_chr]
for _ in range(func_height + 1):
sign_lines.append(' ' + vertical_chr + (' ' * (width-2)) + vertical_chr + ' ')
pretty_sign = stringPict('')
pretty_sign = prettyForm(*pretty_sign.stack(*sign_lines))
max_upper = max(max_upper, pretty_upper.height())
if first:
sign_height = pretty_sign.height()
pretty_sign = prettyForm(*pretty_sign.above(pretty_upper))
pretty_sign = prettyForm(*pretty_sign.below(pretty_lower))
if first:
pretty_func.baseline = 0
first = False
height = pretty_sign.height()
padding = stringPict('')
padding = prettyForm(*padding.stack(*[' ']*(height - 1)))
pretty_sign = prettyForm(*pretty_sign.right(padding))
pretty_func = prettyForm(*pretty_sign.right(pretty_func))
pretty_func.baseline = max_upper + sign_height//2
pretty_func.binding = prettyForm.MUL
return pretty_func
def __print_SumProduct_Limits(self, lim):
def print_start(lhs, rhs):
op = prettyForm(' ' + xsym("==") + ' ')
l = self._print(lhs)
r = self._print(rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
prettyUpper = self._print(lim[2])
prettyLower = print_start(lim[0], lim[1])
return prettyLower, prettyUpper
def _print_Sum(self, expr):
ascii_mode = not self._use_unicode
def asum(hrequired, lower, upper, use_ascii):
def adjust(s, wid=None, how='<^>'):
if not wid or len(s) > wid:
return s
need = wid - len(s)
if how in ('<^>', "<") or how not in list('<^>'):
return s + ' '*need
half = need//2
lead = ' '*half
if how == ">":
return " "*need + s
return lead + s + ' '*(need - len(lead))
h = max(hrequired, 2)
d = h//2
w = d + 1
more = hrequired % 2
lines = []
if use_ascii:
lines.append("_"*(w) + ' ')
lines.append(r"\%s`" % (' '*(w - 1)))
for i in range(1, d):
lines.append('%s\\%s' % (' '*i, ' '*(w - i)))
if more:
lines.append('%s)%s' % (' '*(d), ' '*(w - d)))
for i in reversed(range(1, d)):
lines.append('%s/%s' % (' '*i, ' '*(w - i)))
lines.append("/" + "_"*(w - 1) + ',')
return d, h + more, lines, more
else:
w = w + more
d = d + more
vsum = vobj('sum', 4)
lines.append("_"*(w))
for i in range(0, d):
lines.append('%s%s%s' % (' '*i, vsum[2], ' '*(w - i - 1)))
for i in reversed(range(0, d)):
lines.append('%s%s%s' % (' '*i, vsum[4], ' '*(w - i - 1)))
lines.append(vsum[8]*(w))
return d, h + 2*more, lines, more
f = expr.function
prettyF = self._print(f)
if f.is_Add: # add parens
prettyF = prettyForm(*prettyF.parens())
H = prettyF.height() + 2
# \sum \sum \sum ...
first = True
max_upper = 0
sign_height = 0
for lim in expr.limits:
prettyLower, prettyUpper = self.__print_SumProduct_Limits(lim)
max_upper = max(max_upper, prettyUpper.height())
# Create sum sign based on the height of the argument
d, h, slines, adjustment = asum(
H, prettyLower.width(), prettyUpper.width(), ascii_mode)
prettySign = stringPict('')
prettySign = prettyForm(*prettySign.stack(*slines))
if first:
sign_height = prettySign.height()
prettySign = prettyForm(*prettySign.above(prettyUpper))
prettySign = prettyForm(*prettySign.below(prettyLower))
if first:
# change F baseline so it centers on the sign
prettyF.baseline -= d - (prettyF.height()//2 -
prettyF.baseline)
first = False
# put padding to the right
pad = stringPict('')
pad = prettyForm(*pad.stack(*[' ']*h))
prettySign = prettyForm(*prettySign.right(pad))
# put the present prettyF to the right
prettyF = prettyForm(*prettySign.right(prettyF))
# adjust baseline of ascii mode sigma with an odd height so that it is
# exactly through the center
ascii_adjustment = ascii_mode if not adjustment else 0
prettyF.baseline = max_upper + sign_height//2 + ascii_adjustment
prettyF.binding = prettyForm.MUL
return prettyF
def _print_Limit(self, l):
e, z, z0, dir = l.args
E = self._print(e)
if precedence(e) <= PRECEDENCE["Mul"]:
E = prettyForm(*E.parens('(', ')'))
Lim = prettyForm('lim')
LimArg = self._print(z)
if self._use_unicode:
LimArg = prettyForm(*LimArg.right('\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{RIGHTWARDS ARROW}'))
else:
LimArg = prettyForm(*LimArg.right('->'))
LimArg = prettyForm(*LimArg.right(self._print(z0)))
if str(dir) == '+-' or z0 in (S.Infinity, S.NegativeInfinity):
dir = ""
else:
if self._use_unicode:
dir = '\N{SUPERSCRIPT PLUS SIGN}' if str(dir) == "+" else '\N{SUPERSCRIPT MINUS}'
LimArg = prettyForm(*LimArg.right(self._print(dir)))
Lim = prettyForm(*Lim.below(LimArg))
Lim = prettyForm(*Lim.right(E), binding=prettyForm.MUL)
return Lim
def _print_matrix_contents(self, e):
"""
This method factors out what is essentially grid printing.
"""
M = e # matrix
Ms = {} # i,j -> pretty(M[i,j])
for i in range(M.rows):
for j in range(M.cols):
Ms[i, j] = self._print(M[i, j])
# h- and v- spacers
hsep = 2
vsep = 1
# max width for columns
maxw = [-1] * M.cols
for j in range(M.cols):
maxw[j] = max([Ms[i, j].width() for i in range(M.rows)] or [0])
# drawing result
D = None
for i in range(M.rows):
D_row = None
for j in range(M.cols):
s = Ms[i, j]
# reshape s to maxw
# XXX this should be generalized, and go to stringPict.reshape ?
assert s.width() <= maxw[j]
# hcenter it, +0.5 to the right 2
# ( it's better to align formula starts for say 0 and r )
# XXX this is not good in all cases -- maybe introduce vbaseline?
wdelta = maxw[j] - s.width()
wleft = wdelta // 2
wright = wdelta - wleft
s = prettyForm(*s.right(' '*wright))
s = prettyForm(*s.left(' '*wleft))
# we don't need vcenter cells -- this is automatically done in
# a pretty way because when their baselines are taking into
# account in .right()
if D_row is None:
D_row = s # first box in a row
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(s))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
if D is None:
D = prettyForm('') # Empty Matrix
return D
def _print_MatrixBase(self, e):
D = self._print_matrix_contents(e)
D.baseline = D.height()//2
D = prettyForm(*D.parens('[', ']'))
return D
def _print_TensorProduct(self, expr):
# This should somehow share the code with _print_WedgeProduct:
if self._use_unicode:
circled_times = "\u2297"
else:
circled_times = ".*"
return self._print_seq(expr.args, None, None, circled_times,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_WedgeProduct(self, expr):
# This should somehow share the code with _print_TensorProduct:
if self._use_unicode:
wedge_symbol = "\u2227"
else:
wedge_symbol = '/\\'
return self._print_seq(expr.args, None, None, wedge_symbol,
parenthesize=lambda x: precedence_traditional(x) <= PRECEDENCE["Mul"])
def _print_Trace(self, e):
D = self._print(e.arg)
D = prettyForm(*D.parens('(',')'))
D.baseline = D.height()//2
D = prettyForm(*D.left('\n'*(0) + 'tr'))
return D
def _print_MatrixElement(self, expr):
from sympy.matrices import MatrixSymbol
if (isinstance(expr.parent, MatrixSymbol)
and expr.i.is_number and expr.j.is_number):
return self._print(
Symbol(expr.parent.name + '_%d%d' % (expr.i, expr.j)))
else:
prettyFunc = self._print(expr.parent)
prettyFunc = prettyForm(*prettyFunc.parens())
prettyIndices = self._print_seq((expr.i, expr.j), delimiter=', '
).parens(left='[', right=']')[0]
pform = prettyForm(binding=prettyForm.FUNC,
*stringPict.next(prettyFunc, prettyIndices))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyIndices
return pform
def _print_MatrixSlice(self, m):
# XXX works only for applied functions
from sympy.matrices import MatrixSymbol
prettyFunc = self._print(m.parent)
if not isinstance(m.parent, MatrixSymbol):
prettyFunc = prettyForm(*prettyFunc.parens())
def ppslice(x, dim):
x = list(x)
if x[2] == 1:
del x[2]
if x[0] == 0:
x[0] = ''
if x[1] == dim:
x[1] = ''
return prettyForm(*self._print_seq(x, delimiter=':'))
prettyArgs = self._print_seq((ppslice(m.rowslice, m.parent.rows),
ppslice(m.colslice, m.parent.cols)), delimiter=', ').parens(left='[', right=']')[0]
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_Transpose(self, expr):
pform = self._print(expr.arg)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol) and expr.arg.is_MatrixExpr:
pform = prettyForm(*pform.parens())
pform = pform**(prettyForm('T'))
return pform
def _print_Adjoint(self, expr):
pform = self._print(expr.arg)
if self._use_unicode:
dag = prettyForm('\N{DAGGER}')
else:
dag = prettyForm('+')
from sympy.matrices import MatrixSymbol
if not isinstance(expr.arg, MatrixSymbol) and expr.arg.is_MatrixExpr:
pform = prettyForm(*pform.parens())
pform = pform**dag
return pform
def _print_BlockMatrix(self, B):
if B.blocks.shape == (1, 1):
return self._print(B.blocks[0, 0])
return self._print(B.blocks)
def _print_MatAdd(self, expr):
s = None
for item in expr.args:
pform = self._print(item)
if s is None:
s = pform # First element
else:
coeff = item.as_coeff_mmul()[0]
if S(coeff).could_extract_minus_sign():
s = prettyForm(*stringPict.next(s, ' '))
pform = self._print(item)
else:
s = prettyForm(*stringPict.next(s, ' + '))
s = prettyForm(*stringPict.next(s, pform))
return s
def _print_MatMul(self, expr):
args = list(expr.args)
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions.matadd import MatAdd
for i, a in enumerate(args):
if (isinstance(a, (Add, MatAdd, HadamardProduct, KroneckerProduct))
and len(expr.args) > 1):
args[i] = prettyForm(*self._print(a).parens())
else:
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_Identity(self, expr):
if self._use_unicode:
return prettyForm('\N{MATHEMATICAL DOUBLE-STRUCK CAPITAL I}')
else:
return prettyForm('I')
def _print_ZeroMatrix(self, expr):
if self._use_unicode:
return prettyForm('\N{MATHEMATICAL DOUBLE-STRUCK DIGIT ZERO}')
else:
return prettyForm('0')
def _print_OneMatrix(self, expr):
if self._use_unicode:
return prettyForm('\N{MATHEMATICAL DOUBLE-STRUCK DIGIT ONE}')
else:
return prettyForm('1')
def _print_DotProduct(self, expr):
args = list(expr.args)
for i, a in enumerate(args):
args[i] = self._print(a)
return prettyForm.__mul__(*args)
def _print_MatPow(self, expr):
pform = self._print(expr.base)
from sympy.matrices import MatrixSymbol
if not isinstance(expr.base, MatrixSymbol):
pform = prettyForm(*pform.parens())
pform = pform**(self._print(expr.exp))
return pform
def _print_HadamardProduct(self, expr):
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matmul import MatMul
if self._use_unicode:
delim = pretty_atom('Ring')
else:
delim = '.*'
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul, HadamardProduct)))
def _print_HadamardPower(self, expr):
# from sympy import MatAdd, MatMul
if self._use_unicode:
circ = pretty_atom('Ring')
else:
circ = self._print('.')
pretty_base = self._print(expr.base)
pretty_exp = self._print(expr.exp)
if precedence(expr.exp) < PRECEDENCE["Mul"]:
pretty_exp = prettyForm(*pretty_exp.parens())
pretty_circ_exp = prettyForm(
binding=prettyForm.LINE,
*stringPict.next(circ, pretty_exp)
)
return pretty_base**pretty_circ_exp
def _print_KroneckerProduct(self, expr):
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matmul import MatMul
if self._use_unicode:
delim = ' \N{N-ARY CIRCLED TIMES OPERATOR} '
else:
delim = ' x '
return self._print_seq(expr.args, None, None, delim,
parenthesize=lambda x: isinstance(x, (MatAdd, MatMul)))
def _print_FunctionMatrix(self, X):
D = self._print(X.lamda.expr)
D = prettyForm(*D.parens('[', ']'))
return D
def _print_TransferFunction(self, expr):
if not expr.num == 1:
num, den = expr.num, expr.den
res = Mul(num, Pow(den, -1, evaluate=False), evaluate=False)
return self._print_Mul(res)
else:
return self._print(1)/self._print(expr.den)
def _print_Series(self, expr):
args = list(expr.args)
for i, a in enumerate(expr.args):
args[i] = prettyForm(*self._print(a).parens())
return prettyForm.__mul__(*args)
def _print_MIMOSeries(self, expr):
from sympy.physics.control.lti import MIMOParallel
args = list(expr.args)
pretty_args = []
for i, a in enumerate(reversed(args)):
if (isinstance(a, MIMOParallel) and len(expr.args) > 1):
expression = self._print(a)
expression.baseline = expression.height()//2
pretty_args.append(prettyForm(*expression.parens()))
else:
expression = self._print(a)
expression.baseline = expression.height()//2
pretty_args.append(expression)
return prettyForm.__mul__(*pretty_args)
def _print_Parallel(self, expr):
s = None
for item in expr.args:
pform = self._print(item)
if s is None:
s = pform # First element
else:
s = prettyForm(*stringPict.next(s))
s.baseline = s.height()//2
s = prettyForm(*stringPict.next(s, ' + '))
s = prettyForm(*stringPict.next(s, pform))
return s
def _print_MIMOParallel(self, expr):
from sympy.physics.control.lti import TransferFunctionMatrix
s = None
for item in expr.args:
pform = self._print(item)
if s is None:
s = pform # First element
else:
s = prettyForm(*stringPict.next(s))
s.baseline = s.height()//2
s = prettyForm(*stringPict.next(s, ' + '))
if isinstance(item, TransferFunctionMatrix):
s.baseline = s.height() - 1
s = prettyForm(*stringPict.next(s, pform))
# s.baseline = s.height()//2
return s
def _print_Feedback(self, expr):
from sympy.physics.control import TransferFunction, Series
num, tf = expr.sys1, TransferFunction(1, 1, expr.var)
num_arg_list = list(num.args) if isinstance(num, Series) else [num]
den_arg_list = list(expr.sys2.args) if \
isinstance(expr.sys2, Series) else [expr.sys2]
if isinstance(num, Series) and isinstance(expr.sys2, Series):
den = Series(*num_arg_list, *den_arg_list)
elif isinstance(num, Series) and isinstance(expr.sys2, TransferFunction):
if expr.sys2 == tf:
den = Series(*num_arg_list)
else:
den = Series(*num_arg_list, expr.sys2)
elif isinstance(num, TransferFunction) and isinstance(expr.sys2, Series):
if num == tf:
den = Series(*den_arg_list)
else:
den = Series(num, *den_arg_list)
else:
if num == tf:
den = Series(*den_arg_list)
elif expr.sys2 == tf:
den = Series(*num_arg_list)
else:
den = Series(*num_arg_list, *den_arg_list)
denom = prettyForm(*stringPict.next(self._print(tf)))
denom.baseline = denom.height()//2
denom = prettyForm(*stringPict.next(denom, ' + ')) if expr.sign == -1 \
else prettyForm(*stringPict.next(denom, ' - '))
denom = prettyForm(*stringPict.next(denom, self._print(den)))
return self._print(num)/denom
def _print_MIMOFeedback(self, expr):
from sympy.physics.control import MIMOSeries, TransferFunctionMatrix
inv_mat = self._print(MIMOSeries(expr.sys2, expr.sys1))
plant = self._print(expr.sys1)
_feedback = prettyForm(*stringPict.next(inv_mat))
_feedback = prettyForm(*stringPict.right("I + ", _feedback)) if expr.sign == -1 \
else prettyForm(*stringPict.right("I - ", _feedback))
_feedback = prettyForm(*stringPict.parens(_feedback))
_feedback.baseline = 0
_feedback = prettyForm(*stringPict.right(_feedback, '-1 '))
_feedback.baseline = _feedback.height()//2
_feedback = prettyForm.__mul__(_feedback, prettyForm(" "))
if isinstance(expr.sys1, TransferFunctionMatrix):
_feedback.baseline = _feedback.height() - 1
_feedback = prettyForm(*stringPict.next(_feedback, plant))
return _feedback
def _print_TransferFunctionMatrix(self, expr):
mat = self._print(expr._expr_mat)
mat.baseline = mat.height() - 1
subscript = greek_unicode['tau'] if self._use_unicode else r'{t}'
mat = prettyForm(*mat.right(subscript))
return mat
def _print_BasisDependent(self, expr):
from sympy.vector import Vector
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of BasisDependent is not implemented")
if expr == expr.zero:
return prettyForm(expr.zero._pretty_form)
o1 = []
vectstrs = []
if isinstance(expr, Vector):
items = expr.separate().items()
else:
items = [(0, expr)]
for system, vect in items:
inneritems = list(vect.components.items())
inneritems.sort(key = lambda x: x[0].__str__())
for k, v in inneritems:
#if the coef of the basis vector is 1
#we skip the 1
if v == 1:
o1.append("" +
k._pretty_form)
#Same for -1
elif v == -1:
o1.append("(-1) " +
k._pretty_form)
#For a general expr
else:
#We always wrap the measure numbers in
#parentheses
arg_str = self._print(
v).parens()[0]
o1.append(arg_str + ' ' + k._pretty_form)
vectstrs.append(k._pretty_form)
#outstr = u("").join(o1)
if o1[0].startswith(" + "):
o1[0] = o1[0][3:]
elif o1[0].startswith(" "):
o1[0] = o1[0][1:]
#Fixing the newlines
lengths = []
strs = ['']
flag = []
for i, partstr in enumerate(o1):
flag.append(0)
# XXX: What is this hack?
if '\n' in partstr:
tempstr = partstr
tempstr = tempstr.replace(vectstrs[i], '')
if '\N{RIGHT PARENTHESIS EXTENSION}' in tempstr: # If scalar is a fraction
for paren in range(len(tempstr)):
flag[i] = 1
if tempstr[paren] == '\N{RIGHT PARENTHESIS EXTENSION}' and tempstr[paren + 1] == '\n':
# We want to place the vector string after all the right parentheses, because
# otherwise, the vector will be in the middle of the string
tempstr = tempstr[:paren] + '\N{RIGHT PARENTHESIS EXTENSION}'\
+ ' ' + vectstrs[i] + tempstr[paren + 1:]
break
elif '\N{RIGHT PARENTHESIS LOWER HOOK}' in tempstr:
# We want to place the vector string after all the right parentheses, because
# otherwise, the vector will be in the middle of the string. For this reason,
# we insert the vector string at the rightmost index.
index = tempstr.rfind('\N{RIGHT PARENTHESIS LOWER HOOK}')
if index != -1: # then this character was found in this string
flag[i] = 1
tempstr = tempstr[:index] + '\N{RIGHT PARENTHESIS LOWER HOOK}'\
+ ' ' + vectstrs[i] + tempstr[index + 1:]
o1[i] = tempstr
o1 = [x.split('\n') for x in o1]
n_newlines = max([len(x) for x in o1]) # Width of part in its pretty form
if 1 in flag: # If there was a fractional scalar
for i, parts in enumerate(o1):
if len(parts) == 1: # If part has no newline
parts.insert(0, ' ' * (len(parts[0])))
flag[i] = 1
for i, parts in enumerate(o1):
lengths.append(len(parts[flag[i]]))
for j in range(n_newlines):
if j+1 <= len(parts):
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
if j == flag[i]:
strs[flag[i]] += parts[flag[i]] + ' + '
else:
strs[j] += parts[j] + ' '*(lengths[-1] -
len(parts[j])+
3)
else:
if j >= len(strs):
strs.append(' ' * (sum(lengths[:-1]) +
3*(len(lengths)-1)))
strs[j] += ' '*(lengths[-1]+3)
return prettyForm('\n'.join([s[:-3] for s in strs]))
def _print_NDimArray(self, expr):
from sympy.matrices.immutable import ImmutableMatrix
if expr.rank() == 0:
return self._print(expr[()])
level_str = [[]] + [[] for i in range(expr.rank())]
shape_ranges = [list(range(i)) for i in expr.shape]
# leave eventual matrix elements unflattened
mat = lambda x: ImmutableMatrix(x, evaluate=False)
for outer_i in itertools.product(*shape_ranges):
level_str[-1].append(expr[outer_i])
even = True
for back_outer_i in range(expr.rank()-1, -1, -1):
if len(level_str[back_outer_i+1]) < expr.shape[back_outer_i]:
break
if even:
level_str[back_outer_i].append(level_str[back_outer_i+1])
else:
level_str[back_outer_i].append(mat(
level_str[back_outer_i+1]))
if len(level_str[back_outer_i + 1]) == 1:
level_str[back_outer_i][-1] = mat(
[[level_str[back_outer_i][-1]]])
even = not even
level_str[back_outer_i+1] = []
out_expr = level_str[0][0]
if expr.rank() % 2 == 1:
out_expr = mat([out_expr])
return self._print(out_expr)
def _printer_tensor_indices(self, name, indices, index_map={}):
center = stringPict(name)
top = stringPict(" "*center.width())
bot = stringPict(" "*center.width())
last_valence = None
prev_map = None
for i, index in enumerate(indices):
indpic = self._print(index.args[0])
if ((index in index_map) or prev_map) and last_valence == index.is_up:
if index.is_up:
top = prettyForm(*stringPict.next(top, ","))
else:
bot = prettyForm(*stringPict.next(bot, ","))
if index in index_map:
indpic = prettyForm(*stringPict.next(indpic, "="))
indpic = prettyForm(*stringPict.next(indpic, self._print(index_map[index])))
prev_map = True
else:
prev_map = False
if index.is_up:
top = stringPict(*top.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
bot = stringPict(*bot.right(" "*indpic.width()))
else:
bot = stringPict(*bot.right(indpic))
center = stringPict(*center.right(" "*indpic.width()))
top = stringPict(*top.right(" "*indpic.width()))
last_valence = index.is_up
pict = prettyForm(*center.above(top))
pict = prettyForm(*pict.below(bot))
return pict
def _print_Tensor(self, expr):
name = expr.args[0].name
indices = expr.get_indices()
return self._printer_tensor_indices(name, indices)
def _print_TensorElement(self, expr):
name = expr.expr.args[0].name
indices = expr.expr.get_indices()
index_map = expr.index_map
return self._printer_tensor_indices(name, indices, index_map)
def _print_TensMul(self, expr):
sign, args = expr._get_args_for_traditional_printer()
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in args
]
pform = prettyForm.__mul__(*args)
if sign:
return prettyForm(*pform.left(sign))
else:
return pform
def _print_TensAdd(self, expr):
args = [
prettyForm(*self._print(i).parens()) if
precedence_traditional(i) < PRECEDENCE["Mul"] else self._print(i)
for i in expr.args
]
return prettyForm.__add__(*args)
def _print_TensorIndex(self, expr):
sym = expr.args[0]
if not expr.is_up:
sym = -sym
return self._print(sym)
def _print_PartialDerivative(self, deriv):
if self._use_unicode:
deriv_symbol = U('PARTIAL DIFFERENTIAL')
else:
deriv_symbol = r'd'
x = None
for variable in reversed(deriv.variables):
s = self._print(variable)
ds = prettyForm(*s.left(deriv_symbol))
if x is None:
x = ds
else:
x = prettyForm(*x.right(' '))
x = prettyForm(*x.right(ds))
f = prettyForm(
binding=prettyForm.FUNC, *self._print(deriv.expr).parens())
pform = prettyForm(deriv_symbol)
if len(deriv.variables) > 1:
pform = pform**self._print(len(deriv.variables))
pform = prettyForm(*pform.below(stringPict.LINE, x))
pform.baseline = pform.baseline + 1
pform = prettyForm(*stringPict.next(pform, f))
pform.binding = prettyForm.MUL
return pform
def _print_Piecewise(self, pexpr):
P = {}
for n, ec in enumerate(pexpr.args):
P[n, 0] = self._print(ec.expr)
if ec.cond == True:
P[n, 1] = prettyForm('otherwise')
else:
P[n, 1] = prettyForm(
*prettyForm('for ').right(self._print(ec.cond)))
hsep = 2
vsep = 1
len_args = len(pexpr.args)
# max widths
maxw = [max([P[i, j].width() for i in range(len_args)])
for j in range(2)]
# FIXME: Refactor this code and matrix into some tabular environment.
# drawing result
D = None
for i in range(len_args):
D_row = None
for j in range(2):
p = P[i, j]
assert p.width() <= maxw[j]
wdelta = maxw[j] - p.width()
wleft = wdelta // 2
wright = wdelta - wleft
p = prettyForm(*p.right(' '*wright))
p = prettyForm(*p.left(' '*wleft))
if D_row is None:
D_row = p
continue
D_row = prettyForm(*D_row.right(' '*hsep)) # h-spacer
D_row = prettyForm(*D_row.right(p))
if D is None:
D = D_row # first row in a picture
continue
# v-spacer
for _ in range(vsep):
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
D = prettyForm(*D.parens('{', ''))
D.baseline = D.height()//2
D.binding = prettyForm.OPEN
return D
def _print_ITE(self, ite):
from sympy.functions.elementary.piecewise import Piecewise
return self._print(ite.rewrite(Piecewise))
def _hprint_vec(self, v):
D = None
for a in v:
p = a
if D is None:
D = p
else:
D = prettyForm(*D.right(', '))
D = prettyForm(*D.right(p))
if D is None:
D = stringPict(' ')
return D
def _hprint_vseparator(self, p1, p2, left=None, right=None, delimiter='', ifascii_nougly=False):
if ifascii_nougly and not self._use_unicode:
return self._print_seq((p1, '|', p2), left=left, right=right,
delimiter=delimiter, ifascii_nougly=True)
tmp = self._print_seq((p1, p2,), left=left, right=right, delimiter=delimiter)
sep = stringPict(vobj('|', tmp.height()), baseline=tmp.baseline)
return self._print_seq((p1, sep, p2), left=left, right=right,
delimiter=delimiter)
def _print_hyper(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
ap = [self._print(a) for a in e.ap]
bq = [self._print(b) for b in e.bq]
P = self._print(e.argument)
P.baseline = P.height()//2
# Drawing result - first create the ap, bq vectors
D = None
for v in [ap, bq]:
D_row = self._hprint_vec(v)
if D is None:
D = D_row # first row in a picture
else:
D = prettyForm(*D.below(' '))
D = prettyForm(*D.below(D_row))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the F symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('F')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
add = (sz + 1)//2
F = prettyForm(*F.left(self._print(len(e.ap))))
F = prettyForm(*F.right(self._print(len(e.bq))))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_meijerg(self, e):
# FIXME refactor Matrix, Piecewise, and this into a tabular environment
v = {}
v[(0, 0)] = [self._print(a) for a in e.an]
v[(0, 1)] = [self._print(a) for a in e.aother]
v[(1, 0)] = [self._print(b) for b in e.bm]
v[(1, 1)] = [self._print(b) for b in e.bother]
P = self._print(e.argument)
P.baseline = P.height()//2
vp = {}
for idx in v:
vp[idx] = self._hprint_vec(v[idx])
for i in range(2):
maxw = max(vp[(0, i)].width(), vp[(1, i)].width())
for j in range(2):
s = vp[(j, i)]
left = (maxw - s.width()) // 2
right = maxw - left - s.width()
s = prettyForm(*s.left(' ' * left))
s = prettyForm(*s.right(' ' * right))
vp[(j, i)] = s
D1 = prettyForm(*vp[(0, 0)].right(' ', vp[(0, 1)]))
D1 = prettyForm(*D1.below(' '))
D2 = prettyForm(*vp[(1, 0)].right(' ', vp[(1, 1)]))
D = prettyForm(*D1.below(D2))
# make sure that the argument `z' is centred vertically
D.baseline = D.height()//2
# insert horizontal separator
P = prettyForm(*P.left(' '))
D = prettyForm(*D.right(' '))
# insert separating `|`
D = self._hprint_vseparator(D, P)
# add parens
D = prettyForm(*D.parens('(', ')'))
# create the G symbol
above = D.height()//2 - 1
below = D.height() - above - 1
sz, t, b, add, img = annotated('G')
F = prettyForm('\n' * (above - t) + img + '\n' * (below - b),
baseline=above + sz)
pp = self._print(len(e.ap))
pq = self._print(len(e.bq))
pm = self._print(len(e.bm))
pn = self._print(len(e.an))
def adjust(p1, p2):
diff = p1.width() - p2.width()
if diff == 0:
return p1, p2
elif diff > 0:
return p1, prettyForm(*p2.left(' '*diff))
else:
return prettyForm(*p1.left(' '*-diff)), p2
pp, pm = adjust(pp, pm)
pq, pn = adjust(pq, pn)
pu = prettyForm(*pm.right(', ', pn))
pl = prettyForm(*pp.right(', ', pq))
ht = F.baseline - above - 2
if ht > 0:
pu = prettyForm(*pu.below('\n'*ht))
p = prettyForm(*pu.below(pl))
F.baseline = above
F = prettyForm(*F.right(p))
F.baseline = above + add
D = prettyForm(*F.right(' ', D))
return D
def _print_ExpBase(self, e):
# TODO should exp_polar be printed differently?
# what about exp_polar(0), exp_polar(1)?
base = prettyForm(pretty_atom('Exp1', 'e'))
return base ** self._print(e.args[0])
def _print_Exp1(self, e):
return prettyForm(pretty_atom('Exp1', 'e'))
def _print_Function(self, e, sort=False, func_name=None, left='(',
right=')'):
# optional argument func_name for supplying custom names
# XXX works only for applied functions
return self._helper_print_function(e.func, e.args, sort=sort, func_name=func_name, left=left, right=right)
def _print_mathieuc(self, e):
return self._print_Function(e, func_name='C')
def _print_mathieus(self, e):
return self._print_Function(e, func_name='S')
def _print_mathieucprime(self, e):
return self._print_Function(e, func_name="C'")
def _print_mathieusprime(self, e):
return self._print_Function(e, func_name="S'")
def _helper_print_function(self, func, args, sort=False, func_name=None,
delimiter=', ', elementwise=False, left='(',
right=')'):
if sort:
args = sorted(args, key=default_sort_key)
if not func_name and hasattr(func, "__name__"):
func_name = func.__name__
if func_name:
prettyFunc = self._print(Symbol(func_name))
else:
prettyFunc = prettyForm(*self._print(func).parens())
if elementwise:
if self._use_unicode:
circ = pretty_atom('Modifier Letter Low Ring')
else:
circ = '.'
circ = self._print(circ)
prettyFunc = prettyForm(
binding=prettyForm.LINE,
*stringPict.next(prettyFunc, circ)
)
prettyArgs = prettyForm(*self._print_seq(args, delimiter=delimiter).parens(
left=left, right=right))
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_ElementwiseApplyFunction(self, e):
func = e.function
arg = e.expr
args = [arg]
return self._helper_print_function(func, args, delimiter="", elementwise=True)
@property
def _special_function_classes(self):
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.gamma_functions import gamma, lowergamma
from sympy.functions.special.zeta_functions import lerchphi
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import Chi
return {KroneckerDelta: [greek_unicode['delta'], 'delta'],
gamma: [greek_unicode['Gamma'], 'Gamma'],
lerchphi: [greek_unicode['Phi'], 'lerchphi'],
lowergamma: [greek_unicode['gamma'], 'gamma'],
beta: [greek_unicode['Beta'], 'B'],
DiracDelta: [greek_unicode['delta'], 'delta'],
Chi: ['Chi', 'Chi']}
def _print_FunctionClass(self, expr):
for cls in self._special_function_classes:
if issubclass(expr, cls) and expr.__name__ == cls.__name__:
if self._use_unicode:
return prettyForm(self._special_function_classes[cls][0])
else:
return prettyForm(self._special_function_classes[cls][1])
func_name = expr.__name__
return prettyForm(pretty_symbol(func_name))
def _print_GeometryEntity(self, expr):
# GeometryEntity is based on Tuple but should not print like a Tuple
return self.emptyPrinter(expr)
def _print_lerchphi(self, e):
func_name = greek_unicode['Phi'] if self._use_unicode else 'lerchphi'
return self._print_Function(e, func_name=func_name)
def _print_dirichlet_eta(self, e):
func_name = greek_unicode['eta'] if self._use_unicode else 'dirichlet_eta'
return self._print_Function(e, func_name=func_name)
def _print_Heaviside(self, e):
func_name = greek_unicode['theta'] if self._use_unicode else 'Heaviside'
if e.args[1]==1/2:
pform = prettyForm(*self._print(e.args[0]).parens())
pform = prettyForm(*pform.left(func_name))
return pform
else:
return self._print_Function(e, func_name=func_name)
def _print_fresnels(self, e):
return self._print_Function(e, func_name="S")
def _print_fresnelc(self, e):
return self._print_Function(e, func_name="C")
def _print_airyai(self, e):
return self._print_Function(e, func_name="Ai")
def _print_airybi(self, e):
return self._print_Function(e, func_name="Bi")
def _print_airyaiprime(self, e):
return self._print_Function(e, func_name="Ai'")
def _print_airybiprime(self, e):
return self._print_Function(e, func_name="Bi'")
def _print_LambertW(self, e):
return self._print_Function(e, func_name="W")
def _print_Covariance(self, e):
return self._print_Function(e, func_name="Cov")
def _print_Variance(self, e):
return self._print_Function(e, func_name="Var")
def _print_Probability(self, e):
return self._print_Function(e, func_name="P")
def _print_Expectation(self, e):
return self._print_Function(e, func_name="E", left='[', right=']')
def _print_Lambda(self, e):
expr = e.expr
sig = e.signature
if self._use_unicode:
arrow = " \N{RIGHTWARDS ARROW FROM BAR} "
else:
arrow = " -> "
if len(sig) == 1 and sig[0].is_symbol:
sig = sig[0]
var_form = self._print(sig)
return prettyForm(*stringPict.next(var_form, arrow, self._print(expr)), binding=8)
def _print_Order(self, expr):
pform = self._print(expr.expr)
if (expr.point and any(p != S.Zero for p in expr.point)) or \
len(expr.variables) > 1:
pform = prettyForm(*pform.right("; "))
if len(expr.variables) > 1:
pform = prettyForm(*pform.right(self._print(expr.variables)))
elif len(expr.variables):
pform = prettyForm(*pform.right(self._print(expr.variables[0])))
if self._use_unicode:
pform = prettyForm(*pform.right(" \N{RIGHTWARDS ARROW} "))
else:
pform = prettyForm(*pform.right(" -> "))
if len(expr.point) > 1:
pform = prettyForm(*pform.right(self._print(expr.point)))
else:
pform = prettyForm(*pform.right(self._print(expr.point[0])))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left("O"))
return pform
def _print_SingularityFunction(self, e):
if self._use_unicode:
shift = self._print(e.args[0]-e.args[1])
n = self._print(e.args[2])
base = prettyForm("<")
base = prettyForm(*base.right(shift))
base = prettyForm(*base.right(">"))
pform = base**n
return pform
else:
n = self._print(e.args[2])
shift = self._print(e.args[0]-e.args[1])
base = self._print_seq(shift, "<", ">", ' ')
return base**n
def _print_beta(self, e):
func_name = greek_unicode['Beta'] if self._use_unicode else 'B'
return self._print_Function(e, func_name=func_name)
def _print_betainc(self, e):
func_name = "B'"
return self._print_Function(e, func_name=func_name)
def _print_betainc_regularized(self, e):
func_name = 'I'
return self._print_Function(e, func_name=func_name)
def _print_gamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_uppergamma(self, e):
func_name = greek_unicode['Gamma'] if self._use_unicode else 'Gamma'
return self._print_Function(e, func_name=func_name)
def _print_lowergamma(self, e):
func_name = greek_unicode['gamma'] if self._use_unicode else 'lowergamma'
return self._print_Function(e, func_name=func_name)
def _print_DiracDelta(self, e):
if self._use_unicode:
if len(e.args) == 2:
a = prettyForm(greek_unicode['delta'])
b = self._print(e.args[1])
b = prettyForm(*b.parens())
c = self._print(e.args[0])
c = prettyForm(*c.parens())
pform = a**b
pform = prettyForm(*pform.right(' '))
pform = prettyForm(*pform.right(c))
return pform
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(greek_unicode['delta']))
return pform
else:
return self._print_Function(e)
def _print_expint(self, e):
if e.args[0].is_Integer and self._use_unicode:
return self._print_Function(Function('E_%s' % e.args[0])(e.args[1]))
return self._print_Function(e)
def _print_Chi(self, e):
# This needs a special case since otherwise it comes out as greek
# letter chi...
prettyFunc = prettyForm("Chi")
prettyArgs = prettyForm(*self._print_seq(e.args).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
# store pform parts so it can be reassembled e.g. when powered
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_elliptic_e(self, e):
pforma0 = self._print(e.args[0])
if len(e.args) == 1:
pform = pforma0
else:
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('E'))
return pform
def _print_elliptic_k(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('K'))
return pform
def _print_elliptic_f(self, e):
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
pform = self._hprint_vseparator(pforma0, pforma1)
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left('F'))
return pform
def _print_elliptic_pi(self, e):
name = greek_unicode['Pi'] if self._use_unicode else 'Pi'
pforma0 = self._print(e.args[0])
pforma1 = self._print(e.args[1])
if len(e.args) == 2:
pform = self._hprint_vseparator(pforma0, pforma1)
else:
pforma2 = self._print(e.args[2])
pforma = self._hprint_vseparator(pforma1, pforma2, ifascii_nougly=False)
pforma = prettyForm(*pforma.left('; '))
pform = prettyForm(*pforma.left(pforma0))
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(name))
return pform
def _print_GoldenRatio(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('phi'))
return self._print(Symbol("GoldenRatio"))
def _print_EulerGamma(self, expr):
if self._use_unicode:
return prettyForm(pretty_symbol('gamma'))
return self._print(Symbol("EulerGamma"))
def _print_Catalan(self, expr):
return self._print(Symbol("G"))
def _print_Mod(self, expr):
pform = self._print(expr.args[0])
if pform.binding > prettyForm.MUL:
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.right(' mod '))
pform = prettyForm(*pform.right(self._print(expr.args[1])))
pform.binding = prettyForm.OPEN
return pform
def _print_Add(self, expr, order=None):
terms = self._as_ordered_terms(expr, order=order)
pforms, indices = [], []
def pretty_negative(pform, index):
"""Prepend a minus sign to a pretty form. """
#TODO: Move this code to prettyForm
if index == 0:
if pform.height() > 1:
pform_neg = '- '
else:
pform_neg = '-'
else:
pform_neg = ' - '
if (pform.binding > prettyForm.NEG
or pform.binding == prettyForm.ADD):
p = stringPict(*pform.parens())
else:
p = pform
p = stringPict.next(pform_neg, p)
# Lower the binding to NEG, even if it was higher. Otherwise, it
# will print as a + ( - (b)), instead of a - (b).
return prettyForm(binding=prettyForm.NEG, *p)
for i, term in enumerate(terms):
if term.is_Mul and term.could_extract_minus_sign():
coeff, other = term.as_coeff_mul(rational=False)
if coeff == -1:
negterm = Mul(*other, evaluate=False)
else:
negterm = Mul(-coeff, *other, evaluate=False)
pform = self._print(negterm)
pforms.append(pretty_negative(pform, i))
elif term.is_Rational and term.q > 1:
pforms.append(None)
indices.append(i)
elif term.is_Number and term < 0:
pform = self._print(-term)
pforms.append(pretty_negative(pform, i))
elif term.is_Relational:
pforms.append(prettyForm(*self._print(term).parens()))
else:
pforms.append(self._print(term))
if indices:
large = True
for pform in pforms:
if pform is not None and pform.height() > 1:
break
else:
large = False
for i in indices:
term, negative = terms[i], False
if term < 0:
term, negative = -term, True
if large:
pform = prettyForm(str(term.p))/prettyForm(str(term.q))
else:
pform = self._print(term)
if negative:
pform = pretty_negative(pform, i)
pforms[i] = pform
return prettyForm.__add__(*pforms)
def _print_Mul(self, product):
from sympy.physics.units import Quantity
# Check for unevaluated Mul. In this case we need to make sure the
# identities are visible, multiple Rational factors are not combined
# etc so we display in a straight-forward form that fully preserves all
# args and their order.
args = product.args
if args[0] is S.One or any(isinstance(arg, Number) for arg in args[1:]):
strargs = list(map(self._print, args))
# XXX: This is a hack to work around the fact that
# prettyForm.__mul__ absorbs a leading -1 in the args. Probably it
# would be better to fix this in prettyForm.__mul__ instead.
negone = strargs[0] == '-1'
if negone:
strargs[0] = prettyForm('1', 0, 0)
obj = prettyForm.__mul__(*strargs)
if negone:
obj = prettyForm('-' + obj.s, obj.baseline, obj.binding)
return obj
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
if self.order not in ('old', 'none'):
args = product.as_ordered_factors()
else:
args = list(product.args)
# If quantities are present append them at the back
args = sorted(args, key=lambda x: isinstance(x, Quantity) or
(isinstance(x, Pow) and isinstance(x.base, Quantity)))
# Gather terms for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
b.append(Pow(item.base, -item.exp))
elif item.is_Rational and item is not S.Infinity:
if item.p != 1:
a.append( Rational(item.p) )
if item.q != 1:
b.append( Rational(item.q) )
else:
a.append(item)
# Convert to pretty forms. Parentheses are added by `__mul__`.
a = [self._print(ai) for ai in a]
b = [self._print(bi) for bi in b]
# Construct a pretty form
if len(b) == 0:
return prettyForm.__mul__(*a)
else:
if len(a) == 0:
a.append( self._print(S.One) )
return prettyForm.__mul__(*a)/prettyForm.__mul__(*b)
# A helper function for _print_Pow to print x**(1/n)
def _print_nth_root(self, base, root):
bpretty = self._print(base)
# In very simple cases, use a single-char root sign
if (self._settings['use_unicode_sqrt_char'] and self._use_unicode
and root == 2 and bpretty.height() == 1
and (bpretty.width() == 1
or (base.is_Integer and base.is_nonnegative))):
return prettyForm(*bpretty.left('\N{SQUARE ROOT}'))
# Construct root sign, start with the \/ shape
_zZ = xobj('/', 1)
rootsign = xobj('\\', 1) + _zZ
# Constructing the number to put on root
rpretty = self._print(root)
# roots look bad if they are not a single line
if rpretty.height() != 1:
return self._print(base)**self._print(1/root)
# If power is half, no number should appear on top of root sign
exp = '' if root == 2 else str(rpretty).ljust(2)
if len(exp) > 2:
rootsign = ' '*(len(exp) - 2) + rootsign
# Stack the exponent
rootsign = stringPict(exp + '\n' + rootsign)
rootsign.baseline = 0
# Diagonal: length is one less than height of base
linelength = bpretty.height() - 1
diagonal = stringPict('\n'.join(
' '*(linelength - i - 1) + _zZ + ' '*i
for i in range(linelength)
))
# Put baseline just below lowest line: next to exp
diagonal.baseline = linelength - 1
# Make the root symbol
rootsign = prettyForm(*rootsign.right(diagonal))
# Det the baseline to match contents to fix the height
# but if the height of bpretty is one, the rootsign must be one higher
rootsign.baseline = max(1, bpretty.baseline)
#build result
s = prettyForm(hobj('_', 2 + bpretty.width()))
s = prettyForm(*bpretty.above(s))
s = prettyForm(*s.left(rootsign))
return s
def _print_Pow(self, power):
from sympy.simplify.simplify import fraction
b, e = power.as_base_exp()
if power.is_commutative:
if e is S.NegativeOne:
return prettyForm("1")/self._print(b)
n, d = fraction(e)
if n is S.One and d.is_Atom and not e.is_Integer and (e.is_Rational or d.is_Symbol) \
and self._settings['root_notation']:
return self._print_nth_root(b, d)
if e.is_Rational and e < 0:
return prettyForm("1")/self._print(Pow(b, -e, evaluate=False))
if b.is_Relational:
return prettyForm(*self._print(b).parens()).__pow__(self._print(e))
return self._print(b)**self._print(e)
def _print_UnevaluatedExpr(self, expr):
return self._print(expr.args[0])
def __print_numer_denom(self, p, q):
if q == 1:
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)
else:
return prettyForm(str(p))
elif abs(p) >= 10 and abs(q) >= 10:
# If more than one digit in numer and denom, print larger fraction
if p < 0:
return prettyForm(str(p), binding=prettyForm.NEG)/prettyForm(str(q))
# Old printing method:
#pform = prettyForm(str(-p))/prettyForm(str(q))
#return prettyForm(binding=prettyForm.NEG, *pform.left('- '))
else:
return prettyForm(str(p))/prettyForm(str(q))
else:
return None
def _print_Rational(self, expr):
result = self.__print_numer_denom(expr.p, expr.q)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_Fraction(self, expr):
result = self.__print_numer_denom(expr.numerator, expr.denominator)
if result is not None:
return result
else:
return self.emptyPrinter(expr)
def _print_ProductSet(self, p):
if len(p.sets) >= 1 and not has_variety(p.sets):
return self._print(p.sets[0]) ** self._print(len(p.sets))
else:
prod_char = "\N{MULTIPLICATION SIGN}" if self._use_unicode else 'x'
return self._print_seq(p.sets, None, None, ' %s ' % prod_char,
parenthesize=lambda set: set.is_Union or
set.is_Intersection or set.is_ProductSet)
def _print_FiniteSet(self, s):
items = sorted(s.args, key=default_sort_key)
return self._print_seq(items, '{', '}', ', ' )
def _print_Range(self, s):
if self._use_unicode:
dots = "\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if s.start.is_infinite and s.stop.is_infinite:
if s.step.is_positive:
printset = dots, -1, 0, 1, dots
else:
printset = dots, 1, 0, -1, dots
elif s.start.is_infinite:
printset = dots, s[-1] - s.step, s[-1]
elif s.stop.is_infinite:
it = iter(s)
printset = next(it), next(it), dots
elif len(s) > 4:
it = iter(s)
printset = next(it), next(it), dots, s[-1]
else:
printset = tuple(s)
return self._print_seq(printset, '{', '}', ', ' )
def _print_Interval(self, i):
if i.start == i.end:
return self._print_seq(i.args[:1], '{', '}')
else:
if i.left_open:
left = '('
else:
left = '['
if i.right_open:
right = ')'
else:
right = ']'
return self._print_seq(i.args[:2], left, right)
def _print_AccumulationBounds(self, i):
left = '<'
right = '>'
return self._print_seq(i.args[:2], left, right)
def _print_Intersection(self, u):
delimiter = ' %s ' % pretty_atom('Intersection', 'n')
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Union or set.is_Complement)
def _print_Union(self, u):
union_delimiter = ' %s ' % pretty_atom('Union', 'U')
return self._print_seq(u.args, None, None, union_delimiter,
parenthesize=lambda set: set.is_ProductSet or
set.is_Intersection or set.is_Complement)
def _print_SymmetricDifference(self, u):
if not self._use_unicode:
raise NotImplementedError("ASCII pretty printing of SymmetricDifference is not implemented")
sym_delimeter = ' %s ' % pretty_atom('SymmetricDifference')
return self._print_seq(u.args, None, None, sym_delimeter)
def _print_Complement(self, u):
delimiter = r' \ '
return self._print_seq(u.args, None, None, delimiter,
parenthesize=lambda set: set.is_ProductSet or set.is_Intersection
or set.is_Union)
def _print_ImageSet(self, ts):
if self._use_unicode:
inn = "\N{SMALL ELEMENT OF}"
else:
inn = 'in'
fun = ts.lamda
sets = ts.base_sets
signature = fun.signature
expr = self._print(fun.expr)
# TODO: the stuff to the left of the | and the stuff to the right of
# the | should have independent baselines, that way something like
# ImageSet(Lambda(x, 1/x**2), S.Naturals) prints the "x in N" part
# centered on the right instead of aligned with the fraction bar on
# the left. The same also applies to ConditionSet and ComplexRegion
if len(signature) == 1:
S = self._print_seq((signature[0], inn, sets[0]),
delimiter=' ')
return self._hprint_vseparator(expr, S,
left='{', right='}',
ifascii_nougly=True, delimiter=' ')
else:
pargs = tuple(j for var, setv in zip(signature, sets) for j in
(var, ' ', inn, ' ', setv, ", "))
S = self._print_seq(pargs[:-1], delimiter='')
return self._hprint_vseparator(expr, S,
left='{', right='}',
ifascii_nougly=True, delimiter=' ')
def _print_ConditionSet(self, ts):
if self._use_unicode:
inn = "\N{SMALL ELEMENT OF}"
# using _and because and is a keyword and it is bad practice to
# overwrite them
_and = "\N{LOGICAL AND}"
else:
inn = 'in'
_and = 'and'
variables = self._print_seq(Tuple(ts.sym))
as_expr = getattr(ts.condition, 'as_expr', None)
if as_expr is not None:
cond = self._print(ts.condition.as_expr())
else:
cond = self._print(ts.condition)
if self._use_unicode:
cond = self._print(cond)
cond = prettyForm(*cond.parens())
if ts.base_set is S.UniversalSet:
return self._hprint_vseparator(variables, cond, left="{",
right="}", ifascii_nougly=True,
delimiter=' ')
base = self._print(ts.base_set)
C = self._print_seq((variables, inn, base, _and, cond),
delimiter=' ')
return self._hprint_vseparator(variables, C, left="{", right="}",
ifascii_nougly=True, delimiter=' ')
def _print_ComplexRegion(self, ts):
if self._use_unicode:
inn = "\N{SMALL ELEMENT OF}"
else:
inn = 'in'
variables = self._print_seq(ts.variables)
expr = self._print(ts.expr)
prodsets = self._print(ts.sets)
C = self._print_seq((variables, inn, prodsets),
delimiter=' ')
return self._hprint_vseparator(expr, C, left="{", right="}",
ifascii_nougly=True, delimiter=' ')
def _print_Contains(self, e):
var, set = e.args
if self._use_unicode:
el = " \N{ELEMENT OF} "
return prettyForm(*stringPict.next(self._print(var),
el, self._print(set)), binding=8)
else:
return prettyForm(sstr(e))
def _print_FourierSeries(self, s):
if s.an.formula is S.Zero and s.bn.formula is S.Zero:
return self._print(s.a0)
if self._use_unicode:
dots = "\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
return self._print_Add(s.truncate()) + self._print(dots)
def _print_FormalPowerSeries(self, s):
return self._print_Add(s.infinite)
def _print_SetExpr(self, se):
pretty_set = prettyForm(*self._print(se.set).parens())
pretty_name = self._print(Symbol("SetExpr"))
return prettyForm(*pretty_name.right(pretty_set))
def _print_SeqFormula(self, s):
if self._use_unicode:
dots = "\N{HORIZONTAL ELLIPSIS}"
else:
dots = '...'
if len(s.start.free_symbols) > 0 or len(s.stop.free_symbols) > 0:
raise NotImplementedError("Pretty printing of sequences with symbolic bound not implemented")
if s.start is S.NegativeInfinity:
stop = s.stop
printset = (dots, s.coeff(stop - 3), s.coeff(stop - 2),
s.coeff(stop - 1), s.coeff(stop))
elif s.stop is S.Infinity or s.length > 4:
printset = s[:4]
printset.append(dots)
printset = tuple(printset)
else:
printset = tuple(s)
return self._print_list(printset)
_print_SeqPer = _print_SeqFormula
_print_SeqAdd = _print_SeqFormula
_print_SeqMul = _print_SeqFormula
def _print_seq(self, seq, left=None, right=None, delimiter=', ',
parenthesize=lambda x: False, ifascii_nougly=True):
try:
pforms = []
for item in seq:
pform = self._print(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if pforms:
pforms.append(delimiter)
pforms.append(pform)
if not pforms:
s = stringPict('')
else:
s = prettyForm(*stringPict.next(*pforms))
# XXX: Under the tests from #15686 the above raises:
# AttributeError: 'Fake' object has no attribute 'baseline'
# This is caught below but that is not the right way to
# fix it.
except AttributeError:
s = None
for item in seq:
pform = self.doprint(item)
if parenthesize(item):
pform = prettyForm(*pform.parens())
if s is None:
# first element
s = pform
else :
s = prettyForm(*stringPict.next(s, delimiter))
s = prettyForm(*stringPict.next(s, pform))
if s is None:
s = stringPict('')
s = prettyForm(*s.parens(left, right, ifascii_nougly=ifascii_nougly))
return s
def join(self, delimiter, args):
pform = None
for arg in args:
if pform is None:
pform = arg
else:
pform = prettyForm(*pform.right(delimiter))
pform = prettyForm(*pform.right(arg))
if pform is None:
return prettyForm("")
else:
return pform
def _print_list(self, l):
return self._print_seq(l, '[', ']')
def _print_tuple(self, t):
if len(t) == 1:
ptuple = prettyForm(*stringPict.next(self._print(t[0]), ','))
return prettyForm(*ptuple.parens('(', ')', ifascii_nougly=True))
else:
return self._print_seq(t, '(', ')')
def _print_Tuple(self, expr):
return self._print_tuple(expr)
def _print_dict(self, d):
keys = sorted(d.keys(), key=default_sort_key)
items = []
for k in keys:
K = self._print(k)
V = self._print(d[k])
s = prettyForm(*stringPict.next(K, ': ', V))
items.append(s)
return self._print_seq(items, '{', '}')
def _print_Dict(self, d):
return self._print_dict(d)
def _print_set(self, s):
if not s:
return prettyForm('set()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
return pretty
def _print_frozenset(self, s):
if not s:
return prettyForm('frozenset()')
items = sorted(s, key=default_sort_key)
pretty = self._print_seq(items)
pretty = prettyForm(*pretty.parens('{', '}', ifascii_nougly=True))
pretty = prettyForm(*pretty.parens('(', ')', ifascii_nougly=True))
pretty = prettyForm(*stringPict.next(type(s).__name__, pretty))
return pretty
def _print_UniversalSet(self, s):
if self._use_unicode:
return prettyForm("\N{MATHEMATICAL DOUBLE-STRUCK CAPITAL U}")
else:
return prettyForm('UniversalSet')
def _print_PolyRing(self, ring):
return prettyForm(sstr(ring))
def _print_FracField(self, field):
return prettyForm(sstr(field))
def _print_FreeGroupElement(self, elm):
return prettyForm(str(elm))
def _print_PolyElement(self, poly):
return prettyForm(sstr(poly))
def _print_FracElement(self, frac):
return prettyForm(sstr(frac))
def _print_AlgebraicNumber(self, expr):
if expr.is_aliased:
return self._print(expr.as_poly().as_expr())
else:
return self._print(expr.as_expr())
def _print_ComplexRootOf(self, expr):
args = [self._print_Add(expr.expr, order='lex'), expr.index]
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('CRootOf'))
return pform
def _print_RootSum(self, expr):
args = [self._print_Add(expr.expr, order='lex')]
if expr.fun is not S.IdentityFunction:
args.append(self._print(expr.fun))
pform = prettyForm(*self._print_seq(args).parens())
pform = prettyForm(*pform.left('RootSum'))
return pform
def _print_FiniteField(self, expr):
if self._use_unicode:
form = '\N{DOUBLE-STRUCK CAPITAL Z}_%d'
else:
form = 'GF(%d)'
return prettyForm(pretty_symbol(form % expr.mod))
def _print_IntegerRing(self, expr):
if self._use_unicode:
return prettyForm('\N{DOUBLE-STRUCK CAPITAL Z}')
else:
return prettyForm('ZZ')
def _print_RationalField(self, expr):
if self._use_unicode:
return prettyForm('\N{DOUBLE-STRUCK CAPITAL Q}')
else:
return prettyForm('QQ')
def _print_RealField(self, domain):
if self._use_unicode:
prefix = '\N{DOUBLE-STRUCK CAPITAL R}'
else:
prefix = 'RR'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_ComplexField(self, domain):
if self._use_unicode:
prefix = '\N{DOUBLE-STRUCK CAPITAL C}'
else:
prefix = 'CC'
if domain.has_default_precision:
return prettyForm(prefix)
else:
return self._print(pretty_symbol(prefix + "_" + str(domain.precision)))
def _print_PolynomialRing(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_FractionField(self, expr):
args = list(expr.symbols)
if not expr.order.is_default:
order = prettyForm(*prettyForm("order=").right(self._print(expr.order)))
args.append(order)
pform = self._print_seq(args, '(', ')')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_PolynomialRingBase(self, expr):
g = expr.symbols
if str(expr.order) != str(expr.default_order):
g = g + ("order=" + str(expr.order),)
pform = self._print_seq(g, '[', ']')
pform = prettyForm(*pform.left(self._print(expr.domain)))
return pform
def _print_GroebnerBasis(self, basis):
exprs = [ self._print_Add(arg, order=basis.order)
for arg in basis.exprs ]
exprs = prettyForm(*self.join(", ", exprs).parens(left="[", right="]"))
gens = [ self._print(gen) for gen in basis.gens ]
domain = prettyForm(
*prettyForm("domain=").right(self._print(basis.domain)))
order = prettyForm(
*prettyForm("order=").right(self._print(basis.order)))
pform = self.join(", ", [exprs] + gens + [domain, order])
pform = prettyForm(*pform.parens())
pform = prettyForm(*pform.left(basis.__class__.__name__))
return pform
def _print_Subs(self, e):
pform = self._print(e.expr)
pform = prettyForm(*pform.parens())
h = pform.height() if pform.height() > 1 else 2
rvert = stringPict(vobj('|', h), baseline=pform.baseline)
pform = prettyForm(*pform.right(rvert))
b = pform.baseline
pform.baseline = pform.height() - 1
pform = prettyForm(*pform.right(self._print_seq([
self._print_seq((self._print(v[0]), xsym('=='), self._print(v[1])),
delimiter='') for v in zip(e.variables, e.point) ])))
pform.baseline = b
return pform
def _print_number_function(self, e, name):
# Print name_arg[0] for one argument or name_arg[0](arg[1])
# for more than one argument
pform = prettyForm(name)
arg = self._print(e.args[0])
pform_arg = prettyForm(" "*arg.width())
pform_arg = prettyForm(*pform_arg.below(arg))
pform = prettyForm(*pform.right(pform_arg))
if len(e.args) == 1:
return pform
m, x = e.args
# TODO: copy-pasted from _print_Function: can we do better?
prettyFunc = pform
prettyArgs = prettyForm(*self._print_seq([x]).parens())
pform = prettyForm(
binding=prettyForm.FUNC, *stringPict.next(prettyFunc, prettyArgs))
pform.prettyFunc = prettyFunc
pform.prettyArgs = prettyArgs
return pform
def _print_euler(self, e):
return self._print_number_function(e, "E")
def _print_catalan(self, e):
return self._print_number_function(e, "C")
def _print_bernoulli(self, e):
return self._print_number_function(e, "B")
_print_bell = _print_bernoulli
def _print_lucas(self, e):
return self._print_number_function(e, "L")
def _print_fibonacci(self, e):
return self._print_number_function(e, "F")
def _print_tribonacci(self, e):
return self._print_number_function(e, "T")
def _print_stieltjes(self, e):
if self._use_unicode:
return self._print_number_function(e, '\N{GREEK SMALL LETTER GAMMA}')
else:
return self._print_number_function(e, "stieltjes")
def _print_KroneckerDelta(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.right(prettyForm(',')))
pform = prettyForm(*pform.right(self._print(e.args[1])))
if self._use_unicode:
a = stringPict(pretty_symbol('delta'))
else:
a = stringPict('d')
b = pform
top = stringPict(*b.left(' '*a.width()))
bot = stringPict(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.below(top))
def _print_RandomDomain(self, d):
if hasattr(d, 'as_boolean'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.as_boolean())))
return pform
elif hasattr(d, 'set'):
pform = self._print('Domain: ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
pform = prettyForm(*pform.right(self._print(' in ')))
pform = prettyForm(*pform.right(self._print(d.set)))
return pform
elif hasattr(d, 'symbols'):
pform = self._print('Domain on ')
pform = prettyForm(*pform.right(self._print(d.symbols)))
return pform
else:
return self._print(None)
def _print_DMP(self, p):
try:
if p.ring is not None:
# TODO incorporate order
return self._print(p.ring.to_sympy(p))
except SympifyError:
pass
return self._print(repr(p))
def _print_DMF(self, p):
return self._print_DMP(p)
def _print_Object(self, object):
return self._print(pretty_symbol(object.name))
def _print_Morphism(self, morphism):
arrow = xsym("-->")
domain = self._print(morphism.domain)
codomain = self._print(morphism.codomain)
tail = domain.right(arrow, codomain)[0]
return prettyForm(tail)
def _print_NamedMorphism(self, morphism):
pretty_name = self._print(pretty_symbol(morphism.name))
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(":", pretty_morphism)[0])
def _print_IdentityMorphism(self, morphism):
from sympy.categories import NamedMorphism
return self._print_NamedMorphism(
NamedMorphism(morphism.domain, morphism.codomain, "id"))
def _print_CompositeMorphism(self, morphism):
circle = xsym(".")
# All components of the morphism have names and it is thus
# possible to build the name of the composite.
component_names_list = [pretty_symbol(component.name) for
component in morphism.components]
component_names_list.reverse()
component_names = circle.join(component_names_list) + ":"
pretty_name = self._print(component_names)
pretty_morphism = self._print_Morphism(morphism)
return prettyForm(pretty_name.right(pretty_morphism)[0])
def _print_Category(self, category):
return self._print(pretty_symbol(category.name))
def _print_Diagram(self, diagram):
if not diagram.premises:
# This is an empty diagram.
return self._print(S.EmptySet)
pretty_result = self._print(diagram.premises)
if diagram.conclusions:
results_arrow = " %s " % xsym("==>")
pretty_conclusions = self._print(diagram.conclusions)[0]
pretty_result = pretty_result.right(
results_arrow, pretty_conclusions)
return prettyForm(pretty_result[0])
def _print_DiagramGrid(self, grid):
from sympy.matrices import Matrix
matrix = Matrix([[grid[i, j] if grid[i, j] else Symbol(" ")
for j in range(grid.width)]
for i in range(grid.height)])
return self._print_matrix_contents(matrix)
def _print_FreeModuleElement(self, m):
# Print as row vector for convenience, for now.
return self._print_seq(m, '[', ']')
def _print_SubModule(self, M):
return self._print_seq(M.gens, '<', '>')
def _print_FreeModule(self, M):
return self._print(M.ring)**self._print(M.rank)
def _print_ModuleImplementedIdeal(self, M):
return self._print_seq([x for [x] in M._module.gens], '<', '>')
def _print_QuotientRing(self, R):
return self._print(R.ring) / self._print(R.base_ideal)
def _print_QuotientRingElement(self, R):
return self._print(R.data) + self._print(R.ring.base_ideal)
def _print_QuotientModuleElement(self, m):
return self._print(m.data) + self._print(m.module.killed_module)
def _print_QuotientModule(self, M):
return self._print(M.base) / self._print(M.killed_module)
def _print_MatrixHomomorphism(self, h):
matrix = self._print(h._sympy_matrix())
matrix.baseline = matrix.height() // 2
pform = prettyForm(*matrix.right(' : ', self._print(h.domain),
' %s> ' % hobj('-', 2), self._print(h.codomain)))
return pform
def _print_Manifold(self, manifold):
return self._print(manifold.name)
def _print_Patch(self, patch):
return self._print(patch.name)
def _print_CoordSystem(self, coords):
return self._print(coords.name)
def _print_BaseScalarField(self, field):
string = field._coord_sys.symbols[field._index].name
return self._print(pretty_symbol(string))
def _print_BaseVectorField(self, field):
s = U('PARTIAL DIFFERENTIAL') + '_' + field._coord_sys.symbols[field._index].name
return self._print(pretty_symbol(s))
def _print_Differential(self, diff):
if self._use_unicode:
d = '\N{DOUBLE-STRUCK ITALIC SMALL D}'
else:
d = 'd'
field = diff._form_field
if hasattr(field, '_coord_sys'):
string = field._coord_sys.symbols[field._index].name
return self._print(d + ' ' + pretty_symbol(string))
else:
pform = self._print(field)
pform = prettyForm(*pform.parens())
return prettyForm(*pform.left(d))
def _print_Tr(self, p):
#TODO: Handle indices
pform = self._print(p.args[0])
pform = prettyForm(*pform.left('%s(' % (p.__class__.__name__)))
pform = prettyForm(*pform.right(')'))
return pform
def _print_primenu(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['nu']))
else:
pform = prettyForm(*pform.left('nu'))
return pform
def _print_primeomega(self, e):
pform = self._print(e.args[0])
pform = prettyForm(*pform.parens())
if self._use_unicode:
pform = prettyForm(*pform.left(greek_unicode['Omega']))
else:
pform = prettyForm(*pform.left('Omega'))
return pform
def _print_Quantity(self, e):
if e.name.name == 'degree':
pform = self._print("\N{DEGREE SIGN}")
return pform
else:
return self.emptyPrinter(e)
def _print_AssignmentBase(self, e):
op = prettyForm(' ' + xsym(e.op) + ' ')
l = self._print(e.lhs)
r = self._print(e.rhs)
pform = prettyForm(*stringPict.next(l, op, r))
return pform
def _print_Str(self, s):
return self._print(s.name)
@print_function(PrettyPrinter)
def pretty(expr, **settings):
"""Returns a string containing the prettified form of expr.
For information on keyword arguments see pretty_print function.
"""
pp = PrettyPrinter(settings)
# XXX: this is an ugly hack, but at least it works
use_unicode = pp._settings['use_unicode']
uflag = pretty_use_unicode(use_unicode)
try:
return pp.doprint(expr)
finally:
pretty_use_unicode(uflag)
def pretty_print(expr, **kwargs):
"""Prints expr in pretty form.
pprint is just a shortcut for this function.
Parameters
==========
expr : expression
The expression to print.
wrap_line : bool, optional (default=True)
Line wrapping enabled/disabled.
num_columns : int or None, optional (default=None)
Number of columns before line breaking (default to None which reads
the terminal width), useful when using SymPy without terminal.
use_unicode : bool or None, optional (default=None)
Use unicode characters, such as the Greek letter pi instead of
the string pi.
full_prec : bool or string, optional (default="auto")
Use full precision.
order : bool or string, optional (default=None)
Set to 'none' for long expressions if slow; default is None.
use_unicode_sqrt_char : bool, optional (default=True)
Use compact single-character square root symbol (when unambiguous).
root_notation : bool, optional (default=True)
Set to 'False' for printing exponents of the form 1/n in fractional form.
By default exponent is printed in root form.
mat_symbol_style : string, optional (default="plain")
Set to "bold" for printing MatrixSymbols using a bold mathematical symbol face.
By default the standard face is used.
imaginary_unit : string, optional (default="i")
Letter to use for imaginary unit when use_unicode is True.
Can be "i" (default) or "j".
"""
print(pretty(expr, **kwargs))
pprint = pretty_print
def pager_print(expr, **settings):
"""Prints expr using the pager, in pretty form.
This invokes a pager command using pydoc. Lines are not wrapped
automatically. This routine is meant to be used with a pager that allows
sideways scrolling, like ``less -S``.
Parameters are the same as for ``pretty_print``. If you wish to wrap lines,
pass ``num_columns=None`` to auto-detect the width of the terminal.
"""
from pydoc import pager
from locale import getpreferredencoding
if 'num_columns' not in settings:
settings['num_columns'] = 500000 # disable line wrap
pager(pretty(expr, **settings).encode(getpreferredencoding()))
|
9df048f9c2d82c13fd56e6716fb0d25ad148837be00e984d7d5fae06d1433081 | """Symbolic primitives + unicode/ASCII abstraction for pretty.py"""
import sys
import warnings
from string import ascii_lowercase, ascii_uppercase
import unicodedata
unicode_warnings = ''
def U(name):
"""
Get a unicode character by name or, None if not found.
This exists because older versions of Python use older unicode databases.
"""
try:
return unicodedata.lookup(name)
except KeyError:
global unicode_warnings
unicode_warnings += 'No \'%s\' in unicodedata\n' % name
return None
from sympy.printing.conventions import split_super_sub
from sympy.core.alphabets import greeks
from sympy.utilities.exceptions import sympy_deprecation_warning
# prefix conventions when constructing tables
# L - LATIN i
# G - GREEK beta
# D - DIGIT 0
# S - SYMBOL +
__all__ = ['greek_unicode', 'sub', 'sup', 'xsym', 'vobj', 'hobj', 'pretty_symbol',
'annotated']
_use_unicode = False
def pretty_use_unicode(flag=None):
"""Set whether pretty-printer should use unicode by default"""
global _use_unicode
global unicode_warnings
if flag is None:
return _use_unicode
if flag and unicode_warnings:
# print warnings (if any) on first unicode usage
warnings.warn(unicode_warnings)
unicode_warnings = ''
use_unicode_prev = _use_unicode
_use_unicode = flag
return use_unicode_prev
def pretty_try_use_unicode():
"""See if unicode output is available and leverage it if possible"""
encoding = getattr(sys.stdout, 'encoding', None)
# this happens when e.g. stdout is redirected through a pipe, or is
# e.g. a cStringIO.StringO
if encoding is None:
return # sys.stdout has no encoding
symbols = []
# see if we can represent greek alphabet
symbols += greek_unicode.values()
# and atoms
symbols += atoms_table.values()
for s in symbols:
if s is None:
return # common symbols not present!
try:
s.encode(encoding)
except UnicodeEncodeError:
return
# all the characters were present and encodable
pretty_use_unicode(True)
def xstr(*args):
sympy_deprecation_warning(
"""
The sympy.printing.pretty.pretty_symbology.xstr() function is
deprecated. Use str() instead.
""",
deprecated_since_version="1.7",
active_deprecations_target="deprecated-pretty-printing-functions"
)
return str(*args)
# GREEK
g = lambda l: U('GREEK SMALL LETTER %s' % l.upper())
G = lambda l: U('GREEK CAPITAL LETTER %s' % l.upper())
greek_letters = list(greeks) # make a copy
# deal with Unicode's funny spelling of lambda
greek_letters[greek_letters.index('lambda')] = 'lamda'
# {} greek letter -> (g,G)
greek_unicode = {L: g(L) for L in greek_letters}
greek_unicode.update((L[0].upper() + L[1:], G(L)) for L in greek_letters)
# aliases
greek_unicode['lambda'] = greek_unicode['lamda']
greek_unicode['Lambda'] = greek_unicode['Lamda']
greek_unicode['varsigma'] = '\N{GREEK SMALL LETTER FINAL SIGMA}'
# BOLD
b = lambda l: U('MATHEMATICAL BOLD SMALL %s' % l.upper())
B = lambda l: U('MATHEMATICAL BOLD CAPITAL %s' % l.upper())
bold_unicode = {l: b(l) for l in ascii_lowercase}
bold_unicode.update((L, B(L)) for L in ascii_uppercase)
# GREEK BOLD
gb = lambda l: U('MATHEMATICAL BOLD SMALL %s' % l.upper())
GB = lambda l: U('MATHEMATICAL BOLD CAPITAL %s' % l.upper())
greek_bold_letters = list(greeks) # make a copy, not strictly required here
# deal with Unicode's funny spelling of lambda
greek_bold_letters[greek_bold_letters.index('lambda')] = 'lamda'
# {} greek letter -> (g,G)
greek_bold_unicode = {L: g(L) for L in greek_bold_letters}
greek_bold_unicode.update((L[0].upper() + L[1:], G(L)) for L in greek_bold_letters)
greek_bold_unicode['lambda'] = greek_unicode['lamda']
greek_bold_unicode['Lambda'] = greek_unicode['Lamda']
greek_bold_unicode['varsigma'] = '\N{MATHEMATICAL BOLD SMALL FINAL SIGMA}'
digit_2txt = {
'0': 'ZERO',
'1': 'ONE',
'2': 'TWO',
'3': 'THREE',
'4': 'FOUR',
'5': 'FIVE',
'6': 'SIX',
'7': 'SEVEN',
'8': 'EIGHT',
'9': 'NINE',
}
symb_2txt = {
'+': 'PLUS SIGN',
'-': 'MINUS',
'=': 'EQUALS SIGN',
'(': 'LEFT PARENTHESIS',
')': 'RIGHT PARENTHESIS',
'[': 'LEFT SQUARE BRACKET',
']': 'RIGHT SQUARE BRACKET',
'{': 'LEFT CURLY BRACKET',
'}': 'RIGHT CURLY BRACKET',
# non-std
'{}': 'CURLY BRACKET',
'sum': 'SUMMATION',
'int': 'INTEGRAL',
}
# SUBSCRIPT & SUPERSCRIPT
LSUB = lambda letter: U('LATIN SUBSCRIPT SMALL LETTER %s' % letter.upper())
GSUB = lambda letter: U('GREEK SUBSCRIPT SMALL LETTER %s' % letter.upper())
DSUB = lambda digit: U('SUBSCRIPT %s' % digit_2txt[digit])
SSUB = lambda symb: U('SUBSCRIPT %s' % symb_2txt[symb])
LSUP = lambda letter: U('SUPERSCRIPT LATIN SMALL LETTER %s' % letter.upper())
DSUP = lambda digit: U('SUPERSCRIPT %s' % digit_2txt[digit])
SSUP = lambda symb: U('SUPERSCRIPT %s' % symb_2txt[symb])
sub = {} # symb -> subscript symbol
sup = {} # symb -> superscript symbol
# latin subscripts
for l in 'aeioruvxhklmnpst':
sub[l] = LSUB(l)
for l in 'in':
sup[l] = LSUP(l)
for gl in ['beta', 'gamma', 'rho', 'phi', 'chi']:
sub[gl] = GSUB(gl)
for d in [str(i) for i in range(10)]:
sub[d] = DSUB(d)
sup[d] = DSUP(d)
for s in '+-=()':
sub[s] = SSUB(s)
sup[s] = SSUP(s)
# Variable modifiers
# TODO: Make brackets adjust to height of contents
modifier_dict = {
# Accents
'mathring': lambda s: center_accent(s, '\N{COMBINING RING ABOVE}'),
'ddddot': lambda s: center_accent(s, '\N{COMBINING FOUR DOTS ABOVE}'),
'dddot': lambda s: center_accent(s, '\N{COMBINING THREE DOTS ABOVE}'),
'ddot': lambda s: center_accent(s, '\N{COMBINING DIAERESIS}'),
'dot': lambda s: center_accent(s, '\N{COMBINING DOT ABOVE}'),
'check': lambda s: center_accent(s, '\N{COMBINING CARON}'),
'breve': lambda s: center_accent(s, '\N{COMBINING BREVE}'),
'acute': lambda s: center_accent(s, '\N{COMBINING ACUTE ACCENT}'),
'grave': lambda s: center_accent(s, '\N{COMBINING GRAVE ACCENT}'),
'tilde': lambda s: center_accent(s, '\N{COMBINING TILDE}'),
'hat': lambda s: center_accent(s, '\N{COMBINING CIRCUMFLEX ACCENT}'),
'bar': lambda s: center_accent(s, '\N{COMBINING OVERLINE}'),
'vec': lambda s: center_accent(s, '\N{COMBINING RIGHT ARROW ABOVE}'),
'prime': lambda s: s+'\N{PRIME}',
'prm': lambda s: s+'\N{PRIME}',
# # Faces -- these are here for some compatibility with latex printing
# 'bold': lambda s: s,
# 'bm': lambda s: s,
# 'cal': lambda s: s,
# 'scr': lambda s: s,
# 'frak': lambda s: s,
# Brackets
'norm': lambda s: '\N{DOUBLE VERTICAL LINE}'+s+'\N{DOUBLE VERTICAL LINE}',
'avg': lambda s: '\N{MATHEMATICAL LEFT ANGLE BRACKET}'+s+'\N{MATHEMATICAL RIGHT ANGLE BRACKET}',
'abs': lambda s: '\N{VERTICAL LINE}'+s+'\N{VERTICAL LINE}',
'mag': lambda s: '\N{VERTICAL LINE}'+s+'\N{VERTICAL LINE}',
}
# VERTICAL OBJECTS
HUP = lambda symb: U('%s UPPER HOOK' % symb_2txt[symb])
CUP = lambda symb: U('%s UPPER CORNER' % symb_2txt[symb])
MID = lambda symb: U('%s MIDDLE PIECE' % symb_2txt[symb])
EXT = lambda symb: U('%s EXTENSION' % symb_2txt[symb])
HLO = lambda symb: U('%s LOWER HOOK' % symb_2txt[symb])
CLO = lambda symb: U('%s LOWER CORNER' % symb_2txt[symb])
TOP = lambda symb: U('%s TOP' % symb_2txt[symb])
BOT = lambda symb: U('%s BOTTOM' % symb_2txt[symb])
# {} '(' -> (extension, start, end, middle) 1-character
_xobj_unicode = {
# vertical symbols
# (( ext, top, bot, mid ), c1)
'(': (( EXT('('), HUP('('), HLO('(') ), '('),
')': (( EXT(')'), HUP(')'), HLO(')') ), ')'),
'[': (( EXT('['), CUP('['), CLO('[') ), '['),
']': (( EXT(']'), CUP(']'), CLO(']') ), ']'),
'{': (( EXT('{}'), HUP('{'), HLO('{'), MID('{') ), '{'),
'}': (( EXT('{}'), HUP('}'), HLO('}'), MID('}') ), '}'),
'|': U('BOX DRAWINGS LIGHT VERTICAL'),
'<': ((U('BOX DRAWINGS LIGHT VERTICAL'),
U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT'),
U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT')), '<'),
'>': ((U('BOX DRAWINGS LIGHT VERTICAL'),
U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT'),
U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT')), '>'),
'lfloor': (( EXT('['), EXT('['), CLO('[') ), U('LEFT FLOOR')),
'rfloor': (( EXT(']'), EXT(']'), CLO(']') ), U('RIGHT FLOOR')),
'lceil': (( EXT('['), CUP('['), EXT('[') ), U('LEFT CEILING')),
'rceil': (( EXT(']'), CUP(']'), EXT(']') ), U('RIGHT CEILING')),
'int': (( EXT('int'), U('TOP HALF INTEGRAL'), U('BOTTOM HALF INTEGRAL') ), U('INTEGRAL')),
'sum': (( U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT'), '_', U('OVERLINE'), U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT')), U('N-ARY SUMMATION')),
# horizontal objects
#'-': '-',
'-': U('BOX DRAWINGS LIGHT HORIZONTAL'),
'_': U('LOW LINE'),
# We used to use this, but LOW LINE looks better for roots, as it's a
# little lower (i.e., it lines up with the / perfectly. But perhaps this
# one would still be wanted for some cases?
# '_': U('HORIZONTAL SCAN LINE-9'),
# diagonal objects '\' & '/' ?
'/': U('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT'),
'\\': U('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT'),
}
_xobj_ascii = {
# vertical symbols
# (( ext, top, bot, mid ), c1)
'(': (( '|', '/', '\\' ), '('),
')': (( '|', '\\', '/' ), ')'),
# XXX this looks ugly
# '[': (( '|', '-', '-' ), '['),
# ']': (( '|', '-', '-' ), ']'),
# XXX not so ugly :(
'[': (( '[', '[', '[' ), '['),
']': (( ']', ']', ']' ), ']'),
'{': (( '|', '/', '\\', '<' ), '{'),
'}': (( '|', '\\', '/', '>' ), '}'),
'|': '|',
'<': (( '|', '/', '\\' ), '<'),
'>': (( '|', '\\', '/' ), '>'),
'int': ( ' | ', ' /', '/ ' ),
# horizontal objects
'-': '-',
'_': '_',
# diagonal objects '\' & '/' ?
'/': '/',
'\\': '\\',
}
def xobj(symb, length):
"""Construct spatial object of given length.
return: [] of equal-length strings
"""
if length <= 0:
raise ValueError("Length should be greater than 0")
# TODO robustify when no unicodedat available
if _use_unicode:
_xobj = _xobj_unicode
else:
_xobj = _xobj_ascii
vinfo = _xobj[symb]
c1 = top = bot = mid = None
if not isinstance(vinfo, tuple): # 1 entry
ext = vinfo
else:
if isinstance(vinfo[0], tuple): # (vlong), c1
vlong = vinfo[0]
c1 = vinfo[1]
else: # (vlong), c1
vlong = vinfo
ext = vlong[0]
try:
top = vlong[1]
bot = vlong[2]
mid = vlong[3]
except IndexError:
pass
if c1 is None:
c1 = ext
if top is None:
top = ext
if bot is None:
bot = ext
if mid is not None:
if (length % 2) == 0:
# even height, but we have to print it somehow anyway...
# XXX is it ok?
length += 1
else:
mid = ext
if length == 1:
return c1
res = []
next = (length - 2)//2
nmid = (length - 2) - next*2
res += [top]
res += [ext]*next
res += [mid]*nmid
res += [ext]*next
res += [bot]
return res
def vobj(symb, height):
"""Construct vertical object of a given height
see: xobj
"""
return '\n'.join( xobj(symb, height) )
def hobj(symb, width):
"""Construct horizontal object of a given width
see: xobj
"""
return ''.join( xobj(symb, width) )
# RADICAL
# n -> symbol
root = {
2: U('SQUARE ROOT'), # U('RADICAL SYMBOL BOTTOM')
3: U('CUBE ROOT'),
4: U('FOURTH ROOT'),
}
# RATIONAL
VF = lambda txt: U('VULGAR FRACTION %s' % txt)
# (p,q) -> symbol
frac = {
(1, 2): VF('ONE HALF'),
(1, 3): VF('ONE THIRD'),
(2, 3): VF('TWO THIRDS'),
(1, 4): VF('ONE QUARTER'),
(3, 4): VF('THREE QUARTERS'),
(1, 5): VF('ONE FIFTH'),
(2, 5): VF('TWO FIFTHS'),
(3, 5): VF('THREE FIFTHS'),
(4, 5): VF('FOUR FIFTHS'),
(1, 6): VF('ONE SIXTH'),
(5, 6): VF('FIVE SIXTHS'),
(1, 8): VF('ONE EIGHTH'),
(3, 8): VF('THREE EIGHTHS'),
(5, 8): VF('FIVE EIGHTHS'),
(7, 8): VF('SEVEN EIGHTHS'),
}
# atom symbols
_xsym = {
'==': ('=', '='),
'<': ('<', '<'),
'>': ('>', '>'),
'<=': ('<=', U('LESS-THAN OR EQUAL TO')),
'>=': ('>=', U('GREATER-THAN OR EQUAL TO')),
'!=': ('!=', U('NOT EQUAL TO')),
':=': (':=', ':='),
'+=': ('+=', '+='),
'-=': ('-=', '-='),
'*=': ('*=', '*='),
'/=': ('/=', '/='),
'%=': ('%=', '%='),
'*': ('*', U('DOT OPERATOR')),
'-->': ('-->', U('EM DASH') + U('EM DASH') +
U('BLACK RIGHT-POINTING TRIANGLE') if U('EM DASH')
and U('BLACK RIGHT-POINTING TRIANGLE') else None),
'==>': ('==>', U('BOX DRAWINGS DOUBLE HORIZONTAL') +
U('BOX DRAWINGS DOUBLE HORIZONTAL') +
U('BLACK RIGHT-POINTING TRIANGLE') if
U('BOX DRAWINGS DOUBLE HORIZONTAL') and
U('BOX DRAWINGS DOUBLE HORIZONTAL') and
U('BLACK RIGHT-POINTING TRIANGLE') else None),
'.': ('*', U('RING OPERATOR')),
}
def xsym(sym):
"""get symbology for a 'character'"""
op = _xsym[sym]
if _use_unicode:
return op[1]
else:
return op[0]
# SYMBOLS
atoms_table = {
# class how-to-display
'Exp1': U('SCRIPT SMALL E'),
'Pi': U('GREEK SMALL LETTER PI'),
'Infinity': U('INFINITY'),
'NegativeInfinity': U('INFINITY') and ('-' + U('INFINITY')), # XXX what to do here
#'ImaginaryUnit': U('GREEK SMALL LETTER IOTA'),
#'ImaginaryUnit': U('MATHEMATICAL ITALIC SMALL I'),
'ImaginaryUnit': U('DOUBLE-STRUCK ITALIC SMALL I'),
'EmptySet': U('EMPTY SET'),
'Naturals': U('DOUBLE-STRUCK CAPITAL N'),
'Naturals0': (U('DOUBLE-STRUCK CAPITAL N') and
(U('DOUBLE-STRUCK CAPITAL N') +
U('SUBSCRIPT ZERO'))),
'Integers': U('DOUBLE-STRUCK CAPITAL Z'),
'Rationals': U('DOUBLE-STRUCK CAPITAL Q'),
'Reals': U('DOUBLE-STRUCK CAPITAL R'),
'Complexes': U('DOUBLE-STRUCK CAPITAL C'),
'Union': U('UNION'),
'SymmetricDifference': U('INCREMENT'),
'Intersection': U('INTERSECTION'),
'Ring': U('RING OPERATOR'),
'Modifier Letter Low Ring':U('Modifier Letter Low Ring'),
'EmptySequence': 'EmptySequence',
}
def pretty_atom(atom_name, default=None, printer=None):
"""return pretty representation of an atom"""
if _use_unicode:
if printer is not None and atom_name == 'ImaginaryUnit' and printer._settings['imaginary_unit'] == 'j':
return U('DOUBLE-STRUCK ITALIC SMALL J')
else:
return atoms_table[atom_name]
else:
if default is not None:
return default
raise KeyError('only unicode') # send it default printer
def pretty_symbol(symb_name, bold_name=False):
"""return pretty representation of a symbol"""
# let's split symb_name into symbol + index
# UC: beta1
# UC: f_beta
if not _use_unicode:
return symb_name
name, sups, subs = split_super_sub(symb_name)
def translate(s, bold_name) :
if bold_name:
gG = greek_bold_unicode.get(s)
else:
gG = greek_unicode.get(s)
if gG is not None:
return gG
for key in sorted(modifier_dict.keys(), key=lambda k:len(k), reverse=True) :
if s.lower().endswith(key) and len(s)>len(key):
return modifier_dict[key](translate(s[:-len(key)], bold_name))
if bold_name:
return ''.join([bold_unicode[c] for c in s])
return s
name = translate(name, bold_name)
# Let's prettify sups/subs. If it fails at one of them, pretty sups/subs are
# not used at all.
def pretty_list(l, mapping):
result = []
for s in l:
pretty = mapping.get(s)
if pretty is None:
try: # match by separate characters
pretty = ''.join([mapping[c] for c in s])
except (TypeError, KeyError):
return None
result.append(pretty)
return result
pretty_sups = pretty_list(sups, sup)
if pretty_sups is not None:
pretty_subs = pretty_list(subs, sub)
else:
pretty_subs = None
# glue the results into one string
if pretty_subs is None: # nice formatting of sups/subs did not work
if subs:
name += '_'+'_'.join([translate(s, bold_name) for s in subs])
if sups:
name += '__'+'__'.join([translate(s, bold_name) for s in sups])
return name
else:
sups_result = ' '.join(pretty_sups)
subs_result = ' '.join(pretty_subs)
return ''.join([name, sups_result, subs_result])
def annotated(letter):
"""
Return a stylised drawing of the letter ``letter``, together with
information on how to put annotations (super- and subscripts to the
left and to the right) on it.
See pretty.py functions _print_meijerg, _print_hyper on how to use this
information.
"""
ucode_pics = {
'F': (2, 0, 2, 0, '\N{BOX DRAWINGS LIGHT DOWN AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\n'
'\N{BOX DRAWINGS LIGHT VERTICAL AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\n'
'\N{BOX DRAWINGS LIGHT UP}'),
'G': (3, 0, 3, 1, '\N{BOX DRAWINGS LIGHT ARC DOWN AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{BOX DRAWINGS LIGHT ARC DOWN AND LEFT}\n'
'\N{BOX DRAWINGS LIGHT VERTICAL}\N{BOX DRAWINGS LIGHT RIGHT}\N{BOX DRAWINGS LIGHT DOWN AND LEFT}\n'
'\N{BOX DRAWINGS LIGHT ARC UP AND RIGHT}\N{BOX DRAWINGS LIGHT HORIZONTAL}\N{BOX DRAWINGS LIGHT ARC UP AND LEFT}')
}
ascii_pics = {
'F': (3, 0, 3, 0, ' _\n|_\n|\n'),
'G': (3, 0, 3, 1, ' __\n/__\n\\_|')
}
if _use_unicode:
return ucode_pics[letter]
else:
return ascii_pics[letter]
_remove_combining = dict.fromkeys(list(range(ord('\N{COMBINING GRAVE ACCENT}'), ord('\N{COMBINING LATIN SMALL LETTER X}')))
+ list(range(ord('\N{COMBINING LEFT HARPOON ABOVE}'), ord('\N{COMBINING ASTERISK ABOVE}'))))
def is_combining(sym):
"""Check whether symbol is a unicode modifier. """
return ord(sym) in _remove_combining
def center_accent(string, accent):
"""
Returns a string with accent inserted on the middle character. Useful to
put combining accents on symbol names, including multi-character names.
Parameters
==========
string : string
The string to place the accent in.
accent : string
The combining accent to insert
References
==========
.. [1] https://en.wikipedia.org/wiki/Combining_character
.. [2] https://en.wikipedia.org/wiki/Combining_Diacritical_Marks
"""
# Accent is placed on the previous character, although it may not always look
# like that depending on console
midpoint = len(string) // 2 + 1
firstpart = string[:midpoint]
secondpart = string[midpoint:]
return firstpart + accent + secondpart
def line_width(line):
"""Unicode combining symbols (modifiers) are not ever displayed as
separate symbols and thus should not be counted
"""
return len(line.translate(_remove_combining))
|
d14504476e9c94f25760706a01cb9195744e1778d78ccf321da7669b944eae4e | """Prettyprinter by Jurjen Bos.
(I hate spammers: mail me at pietjepuk314 at the reverse of ku.oc.oohay).
All objects have a method that create a "stringPict",
that can be used in the str method for pretty printing.
Updates by Jason Gedge (email <my last name> at cs mun ca)
- terminal_string() method
- minor fixes and changes (mostly to prettyForm)
TODO:
- Allow left/center/right alignment options for above/below and
top/center/bottom alignment options for left/right
"""
from .pretty_symbology import hobj, vobj, xsym, xobj, pretty_use_unicode, line_width
from sympy.utilities.exceptions import sympy_deprecation_warning
class stringPict:
"""An ASCII picture.
The pictures are represented as a list of equal length strings.
"""
#special value for stringPict.below
LINE = 'line'
def __init__(self, s, baseline=0):
"""Initialize from string.
Multiline strings are centered.
"""
self.s = s
#picture is a string that just can be printed
self.picture = stringPict.equalLengths(s.splitlines())
#baseline is the line number of the "base line"
self.baseline = baseline
self.binding = None
@staticmethod
def equalLengths(lines):
# empty lines
if not lines:
return ['']
width = max(line_width(line) for line in lines)
return [line.center(width) for line in lines]
def height(self):
"""The height of the picture in characters."""
return len(self.picture)
def width(self):
"""The width of the picture in characters."""
return line_width(self.picture[0])
@staticmethod
def next(*args):
"""Put a string of stringPicts next to each other.
Returns string, baseline arguments for stringPict.
"""
#convert everything to stringPicts
objects = []
for arg in args:
if isinstance(arg, str):
arg = stringPict(arg)
objects.append(arg)
#make a list of pictures, with equal height and baseline
newBaseline = max(obj.baseline for obj in objects)
newHeightBelowBaseline = max(
obj.height() - obj.baseline
for obj in objects)
newHeight = newBaseline + newHeightBelowBaseline
pictures = []
for obj in objects:
oneEmptyLine = [' '*obj.width()]
basePadding = newBaseline - obj.baseline
totalPadding = newHeight - obj.height()
pictures.append(
oneEmptyLine * basePadding +
obj.picture +
oneEmptyLine * (totalPadding - basePadding))
result = [''.join(lines) for lines in zip(*pictures)]
return '\n'.join(result), newBaseline
def right(self, *args):
r"""Put pictures next to this one.
Returns string, baseline arguments for stringPict.
(Multiline) strings are allowed, and are given a baseline of 0.
Examples
========
>>> from sympy.printing.pretty.stringpict import stringPict
>>> print(stringPict("10").right(" + ",stringPict("1\r-\r2",1))[0])
1
10 + -
2
"""
return stringPict.next(self, *args)
def left(self, *args):
"""Put pictures (left to right) at left.
Returns string, baseline arguments for stringPict.
"""
return stringPict.next(*(args + (self,)))
@staticmethod
def stack(*args):
"""Put pictures on top of each other,
from top to bottom.
Returns string, baseline arguments for stringPict.
The baseline is the baseline of the second picture.
Everything is centered.
Baseline is the baseline of the second picture.
Strings are allowed.
The special value stringPict.LINE is a row of '-' extended to the width.
"""
#convert everything to stringPicts; keep LINE
objects = []
for arg in args:
if arg is not stringPict.LINE and isinstance(arg, str):
arg = stringPict(arg)
objects.append(arg)
#compute new width
newWidth = max(
obj.width()
for obj in objects
if obj is not stringPict.LINE)
lineObj = stringPict(hobj('-', newWidth))
#replace LINE with proper lines
for i, obj in enumerate(objects):
if obj is stringPict.LINE:
objects[i] = lineObj
#stack the pictures, and center the result
newPicture = []
for obj in objects:
newPicture.extend(obj.picture)
newPicture = [line.center(newWidth) for line in newPicture]
newBaseline = objects[0].height() + objects[1].baseline
return '\n'.join(newPicture), newBaseline
def below(self, *args):
"""Put pictures under this picture.
Returns string, baseline arguments for stringPict.
Baseline is baseline of top picture
Examples
========
>>> from sympy.printing.pretty.stringpict import stringPict
>>> print(stringPict("x+3").below(
... stringPict.LINE, '3')[0]) #doctest: +NORMALIZE_WHITESPACE
x+3
---
3
"""
s, baseline = stringPict.stack(self, *args)
return s, self.baseline
def above(self, *args):
"""Put pictures above this picture.
Returns string, baseline arguments for stringPict.
Baseline is baseline of bottom picture.
"""
string, baseline = stringPict.stack(*(args + (self,)))
baseline = len(string.splitlines()) - self.height() + self.baseline
return string, baseline
def parens(self, left='(', right=')', ifascii_nougly=False):
"""Put parentheses around self.
Returns string, baseline arguments for stringPict.
left or right can be None or empty string which means 'no paren from
that side'
"""
h = self.height()
b = self.baseline
# XXX this is a hack -- ascii parens are ugly!
if ifascii_nougly and not pretty_use_unicode():
h = 1
b = 0
res = self
if left:
lparen = stringPict(vobj(left, h), baseline=b)
res = stringPict(*lparen.right(self))
if right:
rparen = stringPict(vobj(right, h), baseline=b)
res = stringPict(*res.right(rparen))
return ('\n'.join(res.picture), res.baseline)
def leftslash(self):
"""Precede object by a slash of the proper size.
"""
# XXX not used anywhere ?
height = max(
self.baseline,
self.height() - 1 - self.baseline)*2 + 1
slash = '\n'.join(
' '*(height - i - 1) + xobj('/', 1) + ' '*i
for i in range(height)
)
return self.left(stringPict(slash, height//2))
def root(self, n=None):
"""Produce a nice root symbol.
Produces ugly results for big n inserts.
"""
# XXX not used anywhere
# XXX duplicate of root drawing in pretty.py
#put line over expression
result = self.above('_'*self.width())
#construct right half of root symbol
height = self.height()
slash = '\n'.join(
' ' * (height - i - 1) + '/' + ' ' * i
for i in range(height)
)
slash = stringPict(slash, height - 1)
#left half of root symbol
if height > 2:
downline = stringPict('\\ \n \\', 1)
else:
downline = stringPict('\\')
#put n on top, as low as possible
if n is not None and n.width() > downline.width():
downline = downline.left(' '*(n.width() - downline.width()))
downline = downline.above(n)
#build root symbol
root = downline.right(slash)
#glue it on at the proper height
#normally, the root symbel is as high as self
#which is one less than result
#this moves the root symbol one down
#if the root became higher, the baseline has to grow too
root.baseline = result.baseline - result.height() + root.height()
return result.left(root)
def render(self, * args, **kwargs):
"""Return the string form of self.
Unless the argument line_break is set to False, it will
break the expression in a form that can be printed
on the terminal without being broken up.
"""
if kwargs["wrap_line"] is False:
return "\n".join(self.picture)
if kwargs["num_columns"] is not None:
# Read the argument num_columns if it is not None
ncols = kwargs["num_columns"]
else:
# Attempt to get a terminal width
ncols = self.terminal_width()
ncols -= 2
if ncols <= 0:
ncols = 78
# If smaller than the terminal width, no need to correct
if self.width() <= ncols:
return type(self.picture[0])(self)
# for one-line pictures we don't need v-spacers. on the other hand, for
# multiline-pictures, we need v-spacers between blocks, compare:
#
# 2 2 3 | a*c*e + a*c*f + a*d | a*c*e + a*c*f + a*d | 3.14159265358979323
# 6*x *y + 4*x*y + | | *e + a*d*f + b*c*e | 84626433832795
# | *e + a*d*f + b*c*e | + b*c*f + b*d*e + b |
# 3 4 4 | | *d*f |
# 4*y*x + x + y | + b*c*f + b*d*e + b | |
# | | |
# | *d*f
i = 0
svals = []
do_vspacers = (self.height() > 1)
while i < self.width():
svals.extend([ sval[i:i + ncols] for sval in self.picture ])
if do_vspacers:
svals.append("") # a vertical spacer
i += ncols
if svals[-1] == '':
del svals[-1] # Get rid of the last spacer
return "\n".join(svals)
def terminal_width(self):
"""Return the terminal width if possible, otherwise return 0.
"""
ncols = 0
try:
import curses
import io
try:
curses.setupterm()
ncols = curses.tigetnum('cols')
except AttributeError:
# windows curses doesn't implement setupterm or tigetnum
# code below from
# http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/440694
from ctypes import windll, create_string_buffer
# stdin handle is -10
# stdout handle is -11
# stderr handle is -12
h = windll.kernel32.GetStdHandle(-12)
csbi = create_string_buffer(22)
res = windll.kernel32.GetConsoleScreenBufferInfo(h, csbi)
if res:
import struct
(bufx, bufy, curx, cury, wattr,
left, top, right, bottom, maxx, maxy) = struct.unpack("hhhhHhhhhhh", csbi.raw)
ncols = right - left + 1
except curses.error:
pass
except io.UnsupportedOperation:
pass
except (ImportError, TypeError):
pass
return ncols
def __eq__(self, o):
if isinstance(o, str):
return '\n'.join(self.picture) == o
elif isinstance(o, stringPict):
return o.picture == self.picture
return False
def __hash__(self):
return super().__hash__()
def __str__(self):
return '\n'.join(self.picture)
def __repr__(self):
return "stringPict(%r,%d)" % ('\n'.join(self.picture), self.baseline)
def __getitem__(self, index):
return self.picture[index]
def __len__(self):
return len(self.s)
class prettyForm(stringPict):
"""
Extension of the stringPict class that knows about basic math applications,
optimizing double minus signs.
"Binding" is interpreted as follows::
ATOM this is an atom: never needs to be parenthesized
FUNC this is a function application: parenthesize if added (?)
DIV this is a division: make wider division if divided
POW this is a power: only parenthesize if exponent
MUL this is a multiplication: parenthesize if powered
ADD this is an addition: parenthesize if multiplied or powered
NEG this is a negative number: optimize if added, parenthesize if
multiplied or powered
OPEN this is an open object: parenthesize if added, multiplied, or
powered (example: Piecewise)
"""
ATOM, FUNC, DIV, POW, MUL, ADD, NEG, OPEN = range(8)
def __init__(self, s, baseline=0, binding=0, unicode=None):
"""Initialize from stringPict and binding power."""
stringPict.__init__(self, s, baseline)
self.binding = binding
if unicode is not None:
sympy_deprecation_warning(
"""
The unicode argument to prettyForm is deprecated. Only the s
argument (the first positional argument) should be passed.
""",
deprecated_since_version="1.7",
active_deprecations_target="deprecated-pretty-printing-functions")
self._unicode = unicode or s
@property
def unicode(self):
sympy_deprecation_warning(
"""
The prettyForm.unicode attribute is deprecated. Use the
prettyForm.s attribute instead.
""",
deprecated_since_version="1.7",
active_deprecations_target="deprecated-pretty-printing-functions")
return self._unicode
# Note: code to handle subtraction is in _print_Add
def __add__(self, *others):
"""Make a pretty addition.
Addition of negative numbers is simplified.
"""
arg = self
if arg.binding > prettyForm.NEG:
arg = stringPict(*arg.parens())
result = [arg]
for arg in others:
#add parentheses for weak binders
if arg.binding > prettyForm.NEG:
arg = stringPict(*arg.parens())
#use existing minus sign if available
if arg.binding != prettyForm.NEG:
result.append(' + ')
result.append(arg)
return prettyForm(binding=prettyForm.ADD, *stringPict.next(*result))
def __truediv__(self, den, slashed=False):
"""Make a pretty division; stacked or slashed.
"""
if slashed:
raise NotImplementedError("Can't do slashed fraction yet")
num = self
if num.binding == prettyForm.DIV:
num = stringPict(*num.parens())
if den.binding == prettyForm.DIV:
den = stringPict(*den.parens())
if num.binding==prettyForm.NEG:
num = num.right(" ")[0]
return prettyForm(binding=prettyForm.DIV, *stringPict.stack(
num,
stringPict.LINE,
den))
def __mul__(self, *others):
"""Make a pretty multiplication.
Parentheses are needed around +, - and neg.
"""
quantity = {
'degree': "\N{DEGREE SIGN}"
}
if len(others) == 0:
return self # We aren't actually multiplying... So nothing to do here.
# add parens on args that need them
arg = self
if arg.binding > prettyForm.MUL and arg.binding != prettyForm.NEG:
arg = stringPict(*arg.parens())
result = [arg]
for arg in others:
if arg.picture[0] not in quantity.values():
result.append(xsym('*'))
#add parentheses for weak binders
if arg.binding > prettyForm.MUL and arg.binding != prettyForm.NEG:
arg = stringPict(*arg.parens())
result.append(arg)
len_res = len(result)
for i in range(len_res):
if i < len_res - 1 and result[i] == '-1' and result[i + 1] == xsym('*'):
# substitute -1 by -, like in -1*x -> -x
result.pop(i)
result.pop(i)
result.insert(i, '-')
if result[0][0] == '-':
# if there is a - sign in front of all
# This test was failing to catch a prettyForm.__mul__(prettyForm("-1", 0, 6)) being negative
bin = prettyForm.NEG
if result[0] == '-':
right = result[1]
if right.picture[right.baseline][0] == '-':
result[0] = '- '
else:
bin = prettyForm.MUL
return prettyForm(binding=bin, *stringPict.next(*result))
def __repr__(self):
return "prettyForm(%r,%d,%d)" % (
'\n'.join(self.picture),
self.baseline,
self.binding)
def __pow__(self, b):
"""Make a pretty power.
"""
a = self
use_inline_func_form = False
if b.binding == prettyForm.POW:
b = stringPict(*b.parens())
if a.binding > prettyForm.FUNC:
a = stringPict(*a.parens())
elif a.binding == prettyForm.FUNC:
# heuristic for when to use inline power
if b.height() > 1:
a = stringPict(*a.parens())
else:
use_inline_func_form = True
if use_inline_func_form:
# 2
# sin + + (x)
b.baseline = a.prettyFunc.baseline + b.height()
func = stringPict(*a.prettyFunc.right(b))
return prettyForm(*func.right(a.prettyArgs))
else:
# 2 <-- top
# (x+y) <-- bot
top = stringPict(*b.left(' '*a.width()))
bot = stringPict(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.above(top))
simpleFunctions = ["sin", "cos", "tan"]
@staticmethod
def apply(function, *args):
"""Functions of one or more variables.
"""
if function in prettyForm.simpleFunctions:
#simple function: use only space if possible
assert len(
args) == 1, "Simple function %s must have 1 argument" % function
arg = args[0].__pretty__()
if arg.binding <= prettyForm.DIV:
#optimization: no parentheses necessary
return prettyForm(binding=prettyForm.FUNC, *arg.left(function + ' '))
argumentList = []
for arg in args:
argumentList.append(',')
argumentList.append(arg.__pretty__())
argumentList = stringPict(*stringPict.next(*argumentList[1:]))
argumentList = stringPict(*argumentList.parens())
return prettyForm(binding=prettyForm.ATOM, *argumentList.left(function))
|
a3b9d0b64505607d9078de18fc88614effb9123b3288e149fcc4a8c164c2b8e5 | from typing import Any, Dict as tDict
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.assumptions.ask import Q
from sympy.core.function import (Function, WildFunction)
from sympy.core.numbers import (AlgebraicNumber, Float, Integer, Rational)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, Wild, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.miscellaneous import (root, sqrt)
from sympy.functions.elementary.trigonometric import sin
from sympy.functions.special.delta_functions import Heaviside
from sympy.logic.boolalg import (false, true)
from sympy.matrices.dense import (Matrix, ones)
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.combinatorics import Cycle, Permutation
from sympy.core.symbol import Str
from sympy.geometry import Point, Ellipse
from sympy.printing import srepr
from sympy.polys import ring, field, ZZ, QQ, lex, grlex, Poly
from sympy.polys.polyclasses import DMP
from sympy.polys.agca.extensions import FiniteExtension
x, y = symbols('x,y')
# eval(srepr(expr)) == expr has to succeed in the right environment. The right
# environment is the scope of "from sympy import *" for most cases.
ENV = {"Str": Str} # type: tDict[str, Any]
exec("from sympy import *", ENV)
def sT(expr, string, import_stmt=None, **kwargs):
"""
sT := sreprTest
Tests that srepr delivers the expected string and that
the condition eval(srepr(expr))==expr holds.
"""
if import_stmt is None:
ENV2 = ENV
else:
ENV2 = ENV.copy()
exec(import_stmt, ENV2)
assert srepr(expr, **kwargs) == string
assert eval(string, ENV2) == expr
def test_printmethod():
class R(Abs):
def _sympyrepr(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert srepr(R(x)) == "foo(Symbol('x'))"
def test_Add():
sT(x + y, "Add(Symbol('x'), Symbol('y'))")
assert srepr(x**2 + 1, order='lex') == "Add(Pow(Symbol('x'), Integer(2)), Integer(1))"
assert srepr(x**2 + 1, order='old') == "Add(Integer(1), Pow(Symbol('x'), Integer(2)))"
assert srepr(sympify('x + 3 - 2', evaluate=False), order='none') == "Add(Symbol('x'), Integer(3), Mul(Integer(-1), Integer(2)))"
def test_more_than_255_args_issue_10259():
from sympy.core.add import Add
from sympy.core.mul import Mul
for op in (Add, Mul):
expr = op(*symbols('x:256'))
assert eval(srepr(expr)) == expr
def test_Function():
sT(Function("f")(x), "Function('f')(Symbol('x'))")
# test unapplied Function
sT(Function('f'), "Function('f')")
sT(sin(x), "sin(Symbol('x'))")
sT(sin, "sin")
def test_Heaviside():
sT(Heaviside(x), "Heaviside(Symbol('x'))")
sT(Heaviside(x, 1), "Heaviside(Symbol('x'), Integer(1))")
def test_Geometry():
sT(Point(0, 0), "Point2D(Integer(0), Integer(0))")
sT(Ellipse(Point(0, 0), 5, 1),
"Ellipse(Point2D(Integer(0), Integer(0)), Integer(5), Integer(1))")
# TODO more tests
def test_Singletons():
sT(S.Catalan, 'Catalan')
sT(S.ComplexInfinity, 'zoo')
sT(S.EulerGamma, 'EulerGamma')
sT(S.Exp1, 'E')
sT(S.GoldenRatio, 'GoldenRatio')
sT(S.TribonacciConstant, 'TribonacciConstant')
sT(S.Half, 'Rational(1, 2)')
sT(S.ImaginaryUnit, 'I')
sT(S.Infinity, 'oo')
sT(S.NaN, 'nan')
sT(S.NegativeInfinity, '-oo')
sT(S.NegativeOne, 'Integer(-1)')
sT(S.One, 'Integer(1)')
sT(S.Pi, 'pi')
sT(S.Zero, 'Integer(0)')
sT(S.Complexes, 'Complexes')
sT(S.EmptySequence, 'EmptySequence')
sT(S.EmptySet, 'EmptySet')
# sT(S.IdentityFunction, 'Lambda(_x, _x)')
sT(S.Naturals, 'Naturals')
sT(S.Naturals0, 'Naturals0')
sT(S.Rationals, 'Rationals')
sT(S.Reals, 'Reals')
sT(S.UniversalSet, 'UniversalSet')
def test_Integer():
sT(Integer(4), "Integer(4)")
def test_list():
sT([x, Integer(4)], "[Symbol('x'), Integer(4)]")
def test_Matrix():
for cls, name in [(Matrix, "MutableDenseMatrix"), (ImmutableDenseMatrix, "ImmutableDenseMatrix")]:
sT(cls([[x**+1, 1], [y, x + y]]),
"%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name)
sT(cls(), "%s([])" % name)
sT(cls([[x**+1, 1], [y, x + y]]), "%s([[Symbol('x'), Integer(1)], [Symbol('y'), Add(Symbol('x'), Symbol('y'))]])" % name)
def test_empty_Matrix():
sT(ones(0, 3), "MutableDenseMatrix(0, 3, [])")
sT(ones(4, 0), "MutableDenseMatrix(4, 0, [])")
sT(ones(0, 0), "MutableDenseMatrix([])")
def test_Rational():
sT(Rational(1, 3), "Rational(1, 3)")
sT(Rational(-1, 3), "Rational(-1, 3)")
def test_Float():
sT(Float('1.23', dps=3), "Float('1.22998', precision=13)")
sT(Float('1.23456789', dps=9), "Float('1.23456788994', precision=33)")
sT(Float('1.234567890123456789', dps=19),
"Float('1.234567890123456789013', precision=66)")
sT(Float('0.60038617995049726', dps=15),
"Float('0.60038617995049726', precision=53)")
sT(Float('1.23', precision=13), "Float('1.22998', precision=13)")
sT(Float('1.23456789', precision=33),
"Float('1.23456788994', precision=33)")
sT(Float('1.234567890123456789', precision=66),
"Float('1.234567890123456789013', precision=66)")
sT(Float('0.60038617995049726', precision=53),
"Float('0.60038617995049726', precision=53)")
sT(Float('0.60038617995049726', 15),
"Float('0.60038617995049726', precision=53)")
def test_Symbol():
sT(x, "Symbol('x')")
sT(y, "Symbol('y')")
sT(Symbol('x', negative=True), "Symbol('x', negative=True)")
def test_Symbol_two_assumptions():
x = Symbol('x', negative=0, integer=1)
# order could vary
s1 = "Symbol('x', integer=True, negative=False)"
s2 = "Symbol('x', negative=False, integer=True)"
assert srepr(x) in (s1, s2)
assert eval(srepr(x), ENV) == x
def test_Symbol_no_special_commutative_treatment():
sT(Symbol('x'), "Symbol('x')")
sT(Symbol('x', commutative=False), "Symbol('x', commutative=False)")
sT(Symbol('x', commutative=0), "Symbol('x', commutative=False)")
sT(Symbol('x', commutative=True), "Symbol('x', commutative=True)")
sT(Symbol('x', commutative=1), "Symbol('x', commutative=True)")
def test_Wild():
sT(Wild('x', even=True), "Wild('x', even=True)")
def test_Dummy():
d = Dummy('d')
sT(d, "Dummy('d', dummy_index=%s)" % str(d.dummy_index))
def test_Dummy_assumption():
d = Dummy('d', nonzero=True)
assert d == eval(srepr(d))
s1 = "Dummy('d', dummy_index=%s, nonzero=True)" % str(d.dummy_index)
s2 = "Dummy('d', nonzero=True, dummy_index=%s)" % str(d.dummy_index)
assert srepr(d) in (s1, s2)
def test_Dummy_from_Symbol():
# should not get the full dictionary of assumptions
n = Symbol('n', integer=True)
d = n.as_dummy()
assert srepr(d
) == "Dummy('n', dummy_index=%s)" % str(d.dummy_index)
def test_tuple():
sT((x,), "(Symbol('x'),)")
sT((x, y), "(Symbol('x'), Symbol('y'))")
def test_WildFunction():
sT(WildFunction('w'), "WildFunction('w')")
def test_settins():
raises(TypeError, lambda: srepr(x, method="garbage"))
def test_Mul():
sT(3*x**3*y, "Mul(Integer(3), Pow(Symbol('x'), Integer(3)), Symbol('y'))")
assert srepr(3*x**3*y, order='old') == "Mul(Integer(3), Symbol('y'), Pow(Symbol('x'), Integer(3)))"
assert srepr(sympify('(x+4)*2*x*7', evaluate=False), order='none') == "Mul(Add(Symbol('x'), Integer(4)), Integer(2), Symbol('x'), Integer(7))"
def test_AlgebraicNumber():
a = AlgebraicNumber(sqrt(2))
sT(a, "AlgebraicNumber(Pow(Integer(2), Rational(1, 2)), [Integer(1), Integer(0)])")
a = AlgebraicNumber(root(-2, 3))
sT(a, "AlgebraicNumber(Pow(Integer(-2), Rational(1, 3)), [Integer(1), Integer(0)])")
def test_PolyRing():
assert srepr(ring("x", ZZ, lex)[0]) == "PolyRing((Symbol('x'),), ZZ, lex)"
assert srepr(ring("x,y", QQ, grlex)[0]) == "PolyRing((Symbol('x'), Symbol('y')), QQ, grlex)"
assert srepr(ring("x,y,z", ZZ["t"], lex)[0]) == "PolyRing((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)"
def test_FracField():
assert srepr(field("x", ZZ, lex)[0]) == "FracField((Symbol('x'),), ZZ, lex)"
assert srepr(field("x,y", QQ, grlex)[0]) == "FracField((Symbol('x'), Symbol('y')), QQ, grlex)"
assert srepr(field("x,y,z", ZZ["t"], lex)[0]) == "FracField((Symbol('x'), Symbol('y'), Symbol('z')), ZZ[t], lex)"
def test_PolyElement():
R, x, y = ring("x,y", ZZ)
assert srepr(3*x**2*y + 1) == "PolyElement(PolyRing((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)])"
def test_FracElement():
F, x, y = field("x,y", ZZ)
assert srepr((3*x**2*y + 1)/(x - y**2)) == "FracElement(FracField((Symbol('x'), Symbol('y')), ZZ, lex), [((2, 1), 3), ((0, 0), 1)], [((1, 0), 1), ((0, 2), -1)])"
def test_FractionField():
assert srepr(QQ.frac_field(x)) == \
"FractionField(FracField((Symbol('x'),), QQ, lex))"
assert srepr(QQ.frac_field(x, y, order=grlex)) == \
"FractionField(FracField((Symbol('x'), Symbol('y')), QQ, grlex))"
def test_PolynomialRingBase():
assert srepr(ZZ.old_poly_ring(x)) == \
"GlobalPolynomialRing(ZZ, Symbol('x'))"
assert srepr(ZZ[x].old_poly_ring(y)) == \
"GlobalPolynomialRing(ZZ[x], Symbol('y'))"
assert srepr(QQ.frac_field(x).old_poly_ring(y)) == \
"GlobalPolynomialRing(FractionField(FracField((Symbol('x'),), QQ, lex)), Symbol('y'))"
def test_DMP():
assert srepr(DMP([1, 2], ZZ)) == 'DMP([1, 2], ZZ)'
assert srepr(ZZ.old_poly_ring(x)([1, 2])) == \
"DMP([1, 2], ZZ, ring=GlobalPolynomialRing(ZZ, Symbol('x')))"
def test_FiniteExtension():
assert srepr(FiniteExtension(Poly(x**2 + 1, x))) == \
"FiniteExtension(Poly(x**2 + 1, x, domain='ZZ'))"
def test_ExtensionElement():
A = FiniteExtension(Poly(x**2 + 1, x))
assert srepr(A.generator) == \
"ExtElem(DMP([1, 0], ZZ, ring=GlobalPolynomialRing(ZZ, Symbol('x'))), FiniteExtension(Poly(x**2 + 1, x, domain='ZZ')))"
def test_BooleanAtom():
assert srepr(true) == "true"
assert srepr(false) == "false"
def test_Integers():
sT(S.Integers, "Integers")
def test_Naturals():
sT(S.Naturals, "Naturals")
def test_Naturals0():
sT(S.Naturals0, "Naturals0")
def test_Reals():
sT(S.Reals, "Reals")
def test_matrix_expressions():
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, n)
sT(A, "MatrixSymbol(Str('A'), Symbol('n', integer=True), Symbol('n', integer=True))")
sT(A*B, "MatMul(MatrixSymbol(Str('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Str('B'), Symbol('n', integer=True), Symbol('n', integer=True)))")
sT(A + B, "MatAdd(MatrixSymbol(Str('A'), Symbol('n', integer=True), Symbol('n', integer=True)), MatrixSymbol(Str('B'), Symbol('n', integer=True), Symbol('n', integer=True)))")
def test_Cycle():
# FIXME: sT fails because Cycle is not immutable and calling srepr(Cycle(1, 2))
# adds keys to the Cycle dict (GH-17661)
#import_stmt = "from sympy.combinatorics import Cycle"
#sT(Cycle(1, 2), "Cycle(1, 2)", import_stmt)
assert srepr(Cycle(1, 2)) == "Cycle(1, 2)"
def test_Permutation():
import_stmt = "from sympy.combinatorics import Permutation"
sT(Permutation(1, 2)(3, 4), "Permutation([0, 2, 1, 4, 3])", import_stmt, perm_cyclic=False)
sT(Permutation(1, 2)(3, 4), "Permutation(1, 2)(3, 4)", import_stmt, perm_cyclic=True)
with warns_deprecated_sympy():
old_print_cyclic = Permutation.print_cyclic
Permutation.print_cyclic = False
sT(Permutation(1, 2)(3, 4), "Permutation([0, 2, 1, 4, 3])", import_stmt)
Permutation.print_cyclic = old_print_cyclic
def test_dict():
from sympy.abc import x, y, z
d = {}
assert srepr(d) == "{}"
d = {x: y}
assert srepr(d) == "{Symbol('x'): Symbol('y')}"
d = {x: y, y: z}
assert srepr(d) in (
"{Symbol('x'): Symbol('y'), Symbol('y'): Symbol('z')}",
"{Symbol('y'): Symbol('z'), Symbol('x'): Symbol('y')}",
)
d = {x: {y: z}}
assert srepr(d) == "{Symbol('x'): {Symbol('y'): Symbol('z')}}"
def test_set():
from sympy.abc import x, y
s = set()
assert srepr(s) == "set()"
s = {x, y}
assert srepr(s) in ("{Symbol('x'), Symbol('y')}", "{Symbol('y'), Symbol('x')}")
def test_Predicate():
sT(Q.even, "Q.even")
def test_AppliedPredicate():
sT(Q.even(Symbol('z')), "AppliedPredicate(Q.even, Symbol('z'))")
|
99e15ed8479bae29d4a82f4c794189de0b2244019d503a01139bae7386c97126 | from sympy.codegen import Assignment
from sympy.codegen.ast import none
from sympy.codegen.cfunctions import expm1, log1p
from sympy.codegen.scipy_nodes import cosm1
from sympy.codegen.matrix_nodes import MatrixSolve
from sympy.core import Expr, Mod, symbols, Eq, Le, Gt, zoo, oo, Rational, Pow
from sympy.core.numbers import pi
from sympy.core.singleton import S
from sympy.functions import acos, KroneckerDelta, Piecewise, sign, sqrt, Min, Max
from sympy.logic import And, Or
from sympy.matrices import SparseMatrix, MatrixSymbol, Identity
from sympy.printing.pycode import (
MpmathPrinter, PythonCodePrinter, pycode, SymPyPrinter
)
from sympy.printing.tensorflow import TensorflowPrinter
from sympy.printing.numpy import NumPyPrinter, SciPyPrinter
from sympy.testing.pytest import raises, skip
from sympy.tensor import IndexedBase, Idx
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayDiagonal, ArrayContraction, ZeroArray, OneArray
from sympy.external import import_module
from sympy.functions.special.gamma_functions import loggamma
from sympy.parsing.latex import parse_latex
x, y, z = symbols('x y z')
p = IndexedBase("p")
def test_PythonCodePrinter():
prntr = PythonCodePrinter()
assert not prntr.module_imports
assert prntr.doprint(x**y) == 'x**y'
assert prntr.doprint(Mod(x, 2)) == 'x % 2'
assert prntr.doprint(-Mod(x, y)) == '-(x % y)'
assert prntr.doprint(Mod(-x, y)) == '(-x) % y'
assert prntr.doprint(And(x, y)) == 'x and y'
assert prntr.doprint(Or(x, y)) == 'x or y'
assert not prntr.module_imports
assert prntr.doprint(pi) == 'math.pi'
assert prntr.module_imports == {'math': {'pi'}}
assert prntr.doprint(x**Rational(1, 2)) == 'math.sqrt(x)'
assert prntr.doprint(sqrt(x)) == 'math.sqrt(x)'
assert prntr.module_imports == {'math': {'pi', 'sqrt'}}
assert prntr.doprint(acos(x)) == 'math.acos(x)'
assert prntr.doprint(Assignment(x, 2)) == 'x = 2'
assert prntr.doprint(Piecewise((1, Eq(x, 0)),
(2, x>6))) == '((1) if (x == 0) else (2) if (x > 6) else None)'
assert prntr.doprint(Piecewise((2, Le(x, 0)),
(3, Gt(x, 0)), evaluate=False)) == '((2) if (x <= 0) else'\
' (3) if (x > 0) else None)'
assert prntr.doprint(sign(x)) == '(0.0 if x == 0 else math.copysign(1, x))'
assert prntr.doprint(p[0, 1]) == 'p[0, 1]'
assert prntr.doprint(KroneckerDelta(x,y)) == '(1 if x == y else 0)'
assert prntr.doprint((2,3)) == "(2, 3)"
assert prntr.doprint([2,3]) == "[2, 3]"
assert prntr.doprint(Min(x, y)) == "min(x, y)"
assert prntr.doprint(Max(x, y)) == "max(x, y)"
def test_PythonCodePrinter_standard():
prntr = PythonCodePrinter()
assert prntr.standard == 'python3'
raises(ValueError, lambda: PythonCodePrinter({'standard':'python4'}))
def test_MpmathPrinter():
p = MpmathPrinter()
assert p.doprint(sign(x)) == 'mpmath.sign(x)'
assert p.doprint(Rational(1, 2)) == 'mpmath.mpf(1)/mpmath.mpf(2)'
assert p.doprint(S.Exp1) == 'mpmath.e'
assert p.doprint(S.Pi) == 'mpmath.pi'
assert p.doprint(S.GoldenRatio) == 'mpmath.phi'
assert p.doprint(S.EulerGamma) == 'mpmath.euler'
assert p.doprint(S.NaN) == 'mpmath.nan'
assert p.doprint(S.Infinity) == 'mpmath.inf'
assert p.doprint(S.NegativeInfinity) == 'mpmath.ninf'
assert p.doprint(loggamma(x)) == 'mpmath.loggamma(x)'
def test_NumPyPrinter():
from sympy.core.function import Lambda
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions.diagonal import (DiagMatrix, DiagonalMatrix, DiagonalOf)
from sympy.matrices.expressions.funcmatrix import FunctionMatrix
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions.special import (OneMatrix, ZeroMatrix)
from sympy.abc import a, b
p = NumPyPrinter()
assert p.doprint(sign(x)) == 'numpy.sign(x)'
A = MatrixSymbol("A", 2, 2)
B = MatrixSymbol("B", 2, 2)
C = MatrixSymbol("C", 1, 5)
D = MatrixSymbol("D", 3, 4)
assert p.doprint(A**(-1)) == "numpy.linalg.inv(A)"
assert p.doprint(A**5) == "numpy.linalg.matrix_power(A, 5)"
assert p.doprint(Identity(3)) == "numpy.eye(3)"
u = MatrixSymbol('x', 2, 1)
v = MatrixSymbol('y', 2, 1)
assert p.doprint(MatrixSolve(A, u)) == 'numpy.linalg.solve(A, x)'
assert p.doprint(MatrixSolve(A, u) + v) == 'numpy.linalg.solve(A, x) + y'
assert p.doprint(ZeroMatrix(2, 3)) == "numpy.zeros((2, 3))"
assert p.doprint(OneMatrix(2, 3)) == "numpy.ones((2, 3))"
assert p.doprint(FunctionMatrix(4, 5, Lambda((a, b), a + b))) == \
"numpy.fromfunction(lambda a, b: a + b, (4, 5))"
assert p.doprint(HadamardProduct(A, B)) == "numpy.multiply(A, B)"
assert p.doprint(KroneckerProduct(A, B)) == "numpy.kron(A, B)"
assert p.doprint(Adjoint(A)) == "numpy.conjugate(numpy.transpose(A))"
assert p.doprint(DiagonalOf(A)) == "numpy.reshape(numpy.diag(A), (-1, 1))"
assert p.doprint(DiagMatrix(C)) == "numpy.diagflat(C)"
assert p.doprint(DiagonalMatrix(D)) == "numpy.multiply(D, numpy.eye(3, 4))"
# Workaround for numpy negative integer power errors
assert p.doprint(x**-1) == 'x**(-1.0)'
assert p.doprint(x**-2) == 'x**(-2.0)'
expr = Pow(2, -1, evaluate=False)
assert p.doprint(expr) == "2**(-1.0)"
assert p.doprint(S.Exp1) == 'numpy.e'
assert p.doprint(S.Pi) == 'numpy.pi'
assert p.doprint(S.EulerGamma) == 'numpy.euler_gamma'
assert p.doprint(S.NaN) == 'numpy.nan'
assert p.doprint(S.Infinity) == 'numpy.PINF'
assert p.doprint(S.NegativeInfinity) == 'numpy.NINF'
def test_issue_18770():
numpy = import_module('numpy')
if not numpy:
skip("numpy not installed.")
from sympy.functions.elementary.miscellaneous import (Max, Min)
from sympy.utilities.lambdify import lambdify
expr1 = Min(0.1*x + 3, x + 1, 0.5*x + 1)
func = lambdify(x, expr1, "numpy")
assert (func(numpy.linspace(0, 3, 3)) == [1.0, 1.75, 2.5 ]).all()
assert func(4) == 3
expr1 = Max(x**2, x**3)
func = lambdify(x,expr1, "numpy")
assert (func(numpy.linspace(-1, 2, 4)) == [1, 0, 1, 8] ).all()
assert func(4) == 64
def test_SciPyPrinter():
p = SciPyPrinter()
expr = acos(x)
assert 'numpy' not in p.module_imports
assert p.doprint(expr) == 'numpy.arccos(x)'
assert 'numpy' in p.module_imports
assert not any(m.startswith('scipy') for m in p.module_imports)
smat = SparseMatrix(2, 5, {(0, 1): 3})
assert p.doprint(smat) == \
'scipy.sparse.coo_matrix(([3], ([0], [1])), shape=(2, 5))'
assert 'scipy.sparse' in p.module_imports
assert p.doprint(S.GoldenRatio) == 'scipy.constants.golden_ratio'
assert p.doprint(S.Pi) == 'scipy.constants.pi'
assert p.doprint(S.Exp1) == 'numpy.e'
def test_pycode_reserved_words():
s1, s2 = symbols('if else')
raises(ValueError, lambda: pycode(s1 + s2, error_on_reserved=True))
py_str = pycode(s1 + s2)
assert py_str in ('else_ + if_', 'if_ + else_')
def test_issue_20762():
antlr4 = import_module("antlr4")
if not antlr4:
skip('antlr not installed.')
# Make sure pycode removes curly braces from subscripted variables
expr = parse_latex(r'a_b \cdot b')
assert pycode(expr) == 'a_b*b'
expr = parse_latex(r'a_{11} \cdot b')
assert pycode(expr) == 'a_11*b'
def test_sqrt():
prntr = PythonCodePrinter()
assert prntr._print_Pow(sqrt(x), rational=False) == 'math.sqrt(x)'
assert prntr._print_Pow(1/sqrt(x), rational=False) == '1/math.sqrt(x)'
prntr = PythonCodePrinter({'standard' : 'python3'})
assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)'
assert prntr._print_Pow(1/sqrt(x), rational=True) == 'x**(-1/2)'
prntr = MpmathPrinter()
assert prntr._print_Pow(sqrt(x), rational=False) == 'mpmath.sqrt(x)'
assert prntr._print_Pow(sqrt(x), rational=True) == \
"x**(mpmath.mpf(1)/mpmath.mpf(2))"
prntr = NumPyPrinter()
assert prntr._print_Pow(sqrt(x), rational=False) == 'numpy.sqrt(x)'
assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)'
prntr = SciPyPrinter()
assert prntr._print_Pow(sqrt(x), rational=False) == 'numpy.sqrt(x)'
assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)'
prntr = SymPyPrinter()
assert prntr._print_Pow(sqrt(x), rational=False) == 'sympy.sqrt(x)'
assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)'
def test_frac():
from sympy.functions.elementary.integers import frac
expr = frac(x)
prntr = NumPyPrinter()
assert prntr.doprint(expr) == 'numpy.mod(x, 1)'
prntr = SciPyPrinter()
assert prntr.doprint(expr) == 'numpy.mod(x, 1)'
prntr = PythonCodePrinter()
assert prntr.doprint(expr) == 'x % 1'
prntr = MpmathPrinter()
assert prntr.doprint(expr) == 'mpmath.frac(x)'
prntr = SymPyPrinter()
assert prntr.doprint(expr) == 'sympy.functions.elementary.integers.frac(x)'
class CustomPrintedObject(Expr):
def _numpycode(self, printer):
return 'numpy'
def _mpmathcode(self, printer):
return 'mpmath'
def test_printmethod():
obj = CustomPrintedObject()
assert NumPyPrinter().doprint(obj) == 'numpy'
assert MpmathPrinter().doprint(obj) == 'mpmath'
def test_codegen_ast_nodes():
assert pycode(none) == 'None'
def test_issue_14283():
prntr = PythonCodePrinter()
assert prntr.doprint(zoo) == "math.nan"
assert prntr.doprint(-oo) == "float('-inf')"
def test_NumPyPrinter_print_seq():
n = NumPyPrinter()
assert n._print_seq(range(2)) == '(0, 1,)'
def test_issue_16535_16536():
from sympy.functions.special.gamma_functions import (lowergamma, uppergamma)
a = symbols('a')
expr1 = lowergamma(a, x)
expr2 = uppergamma(a, x)
prntr = SciPyPrinter()
assert prntr.doprint(expr1) == 'scipy.special.gamma(a)*scipy.special.gammainc(a, x)'
assert prntr.doprint(expr2) == 'scipy.special.gamma(a)*scipy.special.gammaincc(a, x)'
prntr = NumPyPrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
prntr = PythonCodePrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
def test_Integral():
from sympy.functions.elementary.exponential import exp
from sympy.integrals.integrals import Integral
single = Integral(exp(-x), (x, 0, oo))
double = Integral(x**2*exp(x*y), (x, -z, z), (y, 0, z))
indefinite = Integral(x**2, x)
evaluateat = Integral(x**2, (x, 1))
prntr = SciPyPrinter()
assert prntr.doprint(single) == 'scipy.integrate.quad(lambda x: numpy.exp(-x), 0, numpy.PINF)[0]'
assert prntr.doprint(double) == 'scipy.integrate.nquad(lambda x, y: x**2*numpy.exp(x*y), ((-z, z), (0, z)))[0]'
raises(NotImplementedError, lambda: prntr.doprint(indefinite))
raises(NotImplementedError, lambda: prntr.doprint(evaluateat))
prntr = MpmathPrinter()
assert prntr.doprint(single) == 'mpmath.quad(lambda x: mpmath.exp(-x), (0, mpmath.inf))'
assert prntr.doprint(double) == 'mpmath.quad(lambda x, y: x**2*mpmath.exp(x*y), (-z, z), (0, z))'
raises(NotImplementedError, lambda: prntr.doprint(indefinite))
raises(NotImplementedError, lambda: prntr.doprint(evaluateat))
def test_fresnel_integrals():
from sympy.functions.special.error_functions import (fresnelc, fresnels)
expr1 = fresnelc(x)
expr2 = fresnels(x)
prntr = SciPyPrinter()
assert prntr.doprint(expr1) == 'scipy.special.fresnel(x)[1]'
assert prntr.doprint(expr2) == 'scipy.special.fresnel(x)[0]'
prntr = NumPyPrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
prntr = PythonCodePrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
prntr = MpmathPrinter()
assert prntr.doprint(expr1) == 'mpmath.fresnelc(x)'
assert prntr.doprint(expr2) == 'mpmath.fresnels(x)'
def test_beta():
from sympy.functions.special.beta_functions import beta
expr = beta(x, y)
prntr = SciPyPrinter()
assert prntr.doprint(expr) == 'scipy.special.beta(x, y)'
prntr = NumPyPrinter()
assert prntr.doprint(expr) == 'math.gamma(x)*math.gamma(y)/math.gamma(x + y)'
prntr = PythonCodePrinter()
assert prntr.doprint(expr) == 'math.gamma(x)*math.gamma(y)/math.gamma(x + y)'
prntr = PythonCodePrinter({'allow_unknown_functions': True})
assert prntr.doprint(expr) == 'math.gamma(x)*math.gamma(y)/math.gamma(x + y)'
prntr = MpmathPrinter()
assert prntr.doprint(expr) == 'mpmath.beta(x, y)'
def test_airy():
from sympy.functions.special.bessel import (airyai, airybi)
expr1 = airyai(x)
expr2 = airybi(x)
prntr = SciPyPrinter()
assert prntr.doprint(expr1) == 'scipy.special.airy(x)[0]'
assert prntr.doprint(expr2) == 'scipy.special.airy(x)[2]'
prntr = NumPyPrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
prntr = PythonCodePrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
def test_airy_prime():
from sympy.functions.special.bessel import (airyaiprime, airybiprime)
expr1 = airyaiprime(x)
expr2 = airybiprime(x)
prntr = SciPyPrinter()
assert prntr.doprint(expr1) == 'scipy.special.airy(x)[1]'
assert prntr.doprint(expr2) == 'scipy.special.airy(x)[3]'
prntr = NumPyPrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
prntr = PythonCodePrinter()
assert "Not supported" in prntr.doprint(expr1)
assert "Not supported" in prntr.doprint(expr2)
def test_numerical_accuracy_functions():
prntr = SciPyPrinter()
assert prntr.doprint(expm1(x)) == 'numpy.expm1(x)'
assert prntr.doprint(log1p(x)) == 'numpy.log1p(x)'
assert prntr.doprint(cosm1(x)) == 'scipy.special.cosm1(x)'
def test_array_printer():
A = ArraySymbol('A', (4,4,6,6,6))
I = IndexedBase('I')
i,j,k = Idx('i', (0,1)), Idx('j', (2,3)), Idx('k', (4,5))
prntr = NumPyPrinter()
assert prntr.doprint(ZeroArray(5)) == 'numpy.zeros((5,))'
assert prntr.doprint(OneArray(5)) == 'numpy.ones((5,))'
assert prntr.doprint(ArrayContraction(A, [2,3])) == 'numpy.einsum("abccd->abd", A)'
assert prntr.doprint(I) == 'I'
assert prntr.doprint(ArrayDiagonal(A, [2,3,4])) == 'numpy.einsum("abccc->abc", A)'
assert prntr.doprint(ArrayDiagonal(A, [0,1], [2,3])) == 'numpy.einsum("aabbc->cab", A)'
assert prntr.doprint(ArrayContraction(A, [2], [3])) == 'numpy.einsum("abcde->abe", A)'
assert prntr.doprint(Assignment(I[i,j,k], I[i,j,k])) == 'I = I'
prntr = TensorflowPrinter()
assert prntr.doprint(ZeroArray(5)) == 'tensorflow.zeros((5,))'
assert prntr.doprint(OneArray(5)) == 'tensorflow.ones((5,))'
assert prntr.doprint(ArrayContraction(A, [2,3])) == 'tensorflow.linalg.einsum("abccd->abd", A)'
assert prntr.doprint(I) == 'I'
assert prntr.doprint(ArrayDiagonal(A, [2,3,4])) == 'tensorflow.linalg.einsum("abccc->abc", A)'
assert prntr.doprint(ArrayDiagonal(A, [0,1], [2,3])) == 'tensorflow.linalg.einsum("aabbc->cab", A)'
assert prntr.doprint(ArrayContraction(A, [2], [3])) == 'tensorflow.linalg.einsum("abcde->abe", A)'
assert prntr.doprint(Assignment(I[i,j,k], I[i,j,k])) == 'I = I'
|
bd5046fc736db61c4e1e2fe407225eb4b1a5999b456a99a68edae647513dba75 | from sympy.algebras.quaternion import Quaternion
from sympy.assumptions.ask import Q
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.combinatorics.partitions import Partition
from sympy.concrete.summations import (Sum, summation)
from sympy.core.add import Add
from sympy.core.containers import (Dict, Tuple)
from sympy.core.expr import UnevaluatedExpr, Expr
from sympy.core.function import (Derivative, Function, Lambda, Subs, WildFunction)
from sympy.core.mul import Mul
from sympy.core import (Catalan, EulerGamma, GoldenRatio, TribonacciConstant)
from sympy.core.numbers import (E, Float, I, Integer, Rational, nan, oo, pi, zoo)
from sympy.core.parameters import _exp_is_pow
from sympy.core.power import Pow
from sympy.core.relational import (Eq, Rel, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, Wild, symbols)
from sympy.functions.combinatorial.factorials import (factorial, factorial2, subfactorial)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.zeta_functions import zeta
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (Equivalent, false, true, Xor)
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices import SparseMatrix
from sympy.polys.polytools import factor
from sympy.series.limits import Limit
from sympy.series.order import O
from sympy.sets.sets import (Complement, FiniteSet, Interval, SymmetricDifference)
from sympy.external import import_module
from sympy.physics.control.lti import TransferFunction, Series, Parallel, \
Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback
from sympy.physics.units import second, joule
from sympy.polys import (Poly, rootof, RootSum, groebner, ring, field, ZZ, QQ,
ZZ_I, QQ_I, lex, grlex)
from sympy.geometry import Point, Circle, Polygon, Ellipse, Triangle
from sympy.tensor import NDimArray
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElement
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.printing import sstr, sstrrepr, StrPrinter
from sympy.physics.quantum.trace import Tr
x, y, z, w, t = symbols('x,y,z,w,t')
d = Dummy('d')
def test_printmethod():
class R(Abs):
def _sympystr(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert sstr(R(x)) == "foo(x)"
class R(Abs):
def _sympystr(self, printer):
return "foo"
assert sstr(R(x)) == "foo"
def test_Abs():
assert str(Abs(x)) == "Abs(x)"
assert str(Abs(Rational(1, 6))) == "1/6"
assert str(Abs(Rational(-1, 6))) == "1/6"
def test_Add():
assert str(x + y) == "x + y"
assert str(x + 1) == "x + 1"
assert str(x + x**2) == "x**2 + x"
assert str(Add(0, 1, evaluate=False)) == "0 + 1"
assert str(Add(0, 0, 1, evaluate=False)) == "0 + 0 + 1"
assert str(1.0*x) == "1.0*x"
assert str(5 + x + y + x*y + x**2 + y**2) == "x**2 + x*y + x + y**2 + y + 5"
assert str(1 + x + x**2/2 + x**3/3) == "x**3/3 + x**2/2 + x + 1"
assert str(2*x - 7*x**2 + 2 + 3*y) == "-7*x**2 + 2*x + 3*y + 2"
assert str(x - y) == "x - y"
assert str(2 - x) == "2 - x"
assert str(x - 2) == "x - 2"
assert str(x - y - z - w) == "-w + x - y - z"
assert str(x - z*y**2*z*w) == "-w*y**2*z**2 + x"
assert str(x - 1*y*x*y) == "-x*y**2 + x"
assert str(sin(x).series(x, 0, 15)) == "x - x**3/6 + x**5/120 - x**7/5040 + x**9/362880 - x**11/39916800 + x**13/6227020800 + O(x**15)"
def test_Catalan():
assert str(Catalan) == "Catalan"
def test_ComplexInfinity():
assert str(zoo) == "zoo"
def test_Derivative():
assert str(Derivative(x, y)) == "Derivative(x, y)"
assert str(Derivative(x**2, x, evaluate=False)) == "Derivative(x**2, x)"
assert str(Derivative(
x**2/y, x, y, evaluate=False)) == "Derivative(x**2/y, x, y)"
def test_dict():
assert str({1: 1 + x}) == sstr({1: 1 + x}) == "{1: x + 1}"
assert str({1: x**2, 2: y*x}) in ("{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}")
assert sstr({1: x**2, 2: y*x}) == "{1: x**2, 2: x*y}"
def test_Dict():
assert str(Dict({1: 1 + x})) == sstr({1: 1 + x}) == "{1: x + 1}"
assert str(Dict({1: x**2, 2: y*x})) in (
"{1: x**2, 2: x*y}", "{2: x*y, 1: x**2}")
assert sstr(Dict({1: x**2, 2: y*x})) == "{1: x**2, 2: x*y}"
def test_Dummy():
assert str(d) == "_d"
assert str(d + x) == "_d + x"
def test_EulerGamma():
assert str(EulerGamma) == "EulerGamma"
def test_Exp():
assert str(E) == "E"
with _exp_is_pow(True):
assert str(exp(x)) == "E**x"
def test_factorial():
n = Symbol('n', integer=True)
assert str(factorial(-2)) == "zoo"
assert str(factorial(0)) == "1"
assert str(factorial(7)) == "5040"
assert str(factorial(n)) == "factorial(n)"
assert str(factorial(2*n)) == "factorial(2*n)"
assert str(factorial(factorial(n))) == 'factorial(factorial(n))'
assert str(factorial(factorial2(n))) == 'factorial(factorial2(n))'
assert str(factorial2(factorial(n))) == 'factorial2(factorial(n))'
assert str(factorial2(factorial2(n))) == 'factorial2(factorial2(n))'
assert str(subfactorial(3)) == "2"
assert str(subfactorial(n)) == "subfactorial(n)"
assert str(subfactorial(2*n)) == "subfactorial(2*n)"
def test_Function():
f = Function('f')
fx = f(x)
w = WildFunction('w')
assert str(f) == "f"
assert str(fx) == "f(x)"
assert str(w) == "w_"
def test_Geometry():
assert sstr(Point(0, 0)) == 'Point2D(0, 0)'
assert sstr(Circle(Point(0, 0), 3)) == 'Circle(Point2D(0, 0), 3)'
assert sstr(Ellipse(Point(1, 2), 3, 4)) == 'Ellipse(Point2D(1, 2), 3, 4)'
assert sstr(Triangle(Point(1, 1), Point(7, 8), Point(0, -1))) == \
'Triangle(Point2D(1, 1), Point2D(7, 8), Point2D(0, -1))'
assert sstr(Polygon(Point(5, 6), Point(-2, -3), Point(0, 0), Point(4, 7))) == \
'Polygon(Point2D(5, 6), Point2D(-2, -3), Point2D(0, 0), Point2D(4, 7))'
assert sstr(Triangle(Point(0, 0), Point(1, 0), Point(0, 1)), sympy_integers=True) == \
'Triangle(Point2D(S(0), S(0)), Point2D(S(1), S(0)), Point2D(S(0), S(1)))'
assert sstr(Ellipse(Point(1, 2), 3, 4), sympy_integers=True) == \
'Ellipse(Point2D(S(1), S(2)), S(3), S(4))'
def test_GoldenRatio():
assert str(GoldenRatio) == "GoldenRatio"
def test_Heaviside():
assert str(Heaviside(x)) == str(Heaviside(x, S.Half)) == "Heaviside(x)"
assert str(Heaviside(x, 1)) == "Heaviside(x, 1)"
def test_TribonacciConstant():
assert str(TribonacciConstant) == "TribonacciConstant"
def test_ImaginaryUnit():
assert str(I) == "I"
def test_Infinity():
assert str(oo) == "oo"
assert str(oo*I) == "oo*I"
def test_Integer():
assert str(Integer(-1)) == "-1"
assert str(Integer(1)) == "1"
assert str(Integer(-3)) == "-3"
assert str(Integer(0)) == "0"
assert str(Integer(25)) == "25"
def test_Integral():
assert str(Integral(sin(x), y)) == "Integral(sin(x), y)"
assert str(Integral(sin(x), (y, 0, 1))) == "Integral(sin(x), (y, 0, 1))"
def test_Interval():
n = (S.NegativeInfinity, 1, 2, S.Infinity)
for i in range(len(n)):
for j in range(i + 1, len(n)):
for l in (True, False):
for r in (True, False):
ival = Interval(n[i], n[j], l, r)
assert S(str(ival)) == ival
def test_AccumBounds():
a = Symbol('a', real=True)
assert str(AccumBounds(0, a)) == "AccumBounds(0, a)"
assert str(AccumBounds(0, 1)) == "AccumBounds(0, 1)"
def test_Lambda():
assert str(Lambda(d, d**2)) == "Lambda(_d, _d**2)"
# issue 2908
assert str(Lambda((), 1)) == "Lambda((), 1)"
assert str(Lambda((), x)) == "Lambda((), x)"
assert str(Lambda((x, y), x+y)) == "Lambda((x, y), x + y)"
assert str(Lambda(((x, y),), x+y)) == "Lambda(((x, y),), x + y)"
def test_Limit():
assert str(Limit(sin(x)/x, x, y)) == "Limit(sin(x)/x, x, y)"
assert str(Limit(1/x, x, 0)) == "Limit(1/x, x, 0)"
assert str(
Limit(sin(x)/x, x, y, dir="-")) == "Limit(sin(x)/x, x, y, dir='-')"
def test_list():
assert str([x]) == sstr([x]) == "[x]"
assert str([x**2, x*y + 1]) == sstr([x**2, x*y + 1]) == "[x**2, x*y + 1]"
assert str([x**2, [y + x]]) == sstr([x**2, [y + x]]) == "[x**2, [x + y]]"
def test_Matrix_str():
M = Matrix([[x**+1, 1], [y, x + y]])
assert str(M) == "Matrix([[x, 1], [y, x + y]])"
assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])"
M = Matrix([[1]])
assert str(M) == sstr(M) == "Matrix([[1]])"
M = Matrix([[1, 2]])
assert str(M) == sstr(M) == "Matrix([[1, 2]])"
M = Matrix()
assert str(M) == sstr(M) == "Matrix(0, 0, [])"
M = Matrix(0, 1, lambda i, j: 0)
assert str(M) == sstr(M) == "Matrix(0, 1, [])"
def test_Mul():
assert str(x/y) == "x/y"
assert str(y/x) == "y/x"
assert str(x/y/z) == "x/(y*z)"
assert str((x + 1)/(y + 2)) == "(x + 1)/(y + 2)"
assert str(2*x/3) == '2*x/3'
assert str(-2*x/3) == '-2*x/3'
assert str(-1.0*x) == '-1.0*x'
assert str(1.0*x) == '1.0*x'
assert str(Mul(0, 1, evaluate=False)) == '0*1'
assert str(Mul(1, 0, evaluate=False)) == '1*0'
assert str(Mul(1, 1, evaluate=False)) == '1*1'
assert str(Mul(1, 1, 1, evaluate=False)) == '1*1*1'
assert str(Mul(1, 2, evaluate=False)) == '1*2'
assert str(Mul(1, S.Half, evaluate=False)) == '1*(1/2)'
assert str(Mul(1, 1, S.Half, evaluate=False)) == '1*1*(1/2)'
assert str(Mul(1, 1, 2, 3, x, evaluate=False)) == '1*1*2*3*x'
assert str(Mul(1, -1, evaluate=False)) == '1*(-1)'
assert str(Mul(-1, 1, evaluate=False)) == '-1*1'
assert str(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == '4*3*2*1*0*y*x'
assert str(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == '4*3*2*(z + 1)*0*y*x'
assert str(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == '(2/3)*(5/7)'
# For issue 14160
assert str(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x/(y*y)'
# issue 21537
assert str(Mul(x, Pow(1/y, -1, evaluate=False), evaluate=False)) == 'x/(1/y)'
class CustomClass1(Expr):
is_commutative = True
class CustomClass2(Expr):
is_commutative = True
cc1 = CustomClass1()
cc2 = CustomClass2()
assert str(Rational(2)*cc1) == '2*CustomClass1()'
assert str(cc1*Rational(2)) == '2*CustomClass1()'
assert str(cc1*Float("1.5")) == '1.5*CustomClass1()'
assert str(cc2*Rational(2)) == '2*CustomClass2()'
assert str(cc2*Rational(2)*cc1) == '2*CustomClass1()*CustomClass2()'
assert str(cc1*Rational(2)*cc2) == '2*CustomClass1()*CustomClass2()'
def test_NaN():
assert str(nan) == "nan"
def test_NegativeInfinity():
assert str(-oo) == "-oo"
def test_Order():
assert str(O(x)) == "O(x)"
assert str(O(x**2)) == "O(x**2)"
assert str(O(x*y)) == "O(x*y, x, y)"
assert str(O(x, x)) == "O(x)"
assert str(O(x, (x, 0))) == "O(x)"
assert str(O(x, (x, oo))) == "O(x, (x, oo))"
assert str(O(x, x, y)) == "O(x, x, y)"
assert str(O(x, x, y)) == "O(x, x, y)"
assert str(O(x, (x, oo), (y, oo))) == "O(x, (x, oo), (y, oo))"
def test_Permutation_Cycle():
from sympy.combinatorics import Permutation, Cycle
# general principle: economically, canonically show all moved elements
# and the size of the permutation.
for p, s in [
(Cycle(),
'()'),
(Cycle(2),
'(2)'),
(Cycle(2, 1),
'(1 2)'),
(Cycle(1, 2)(5)(6, 7)(10),
'(1 2)(6 7)(10)'),
(Cycle(3, 4)(1, 2)(3, 4),
'(1 2)(4)'),
]:
assert sstr(p) == s
for p, s in [
(Permutation([]),
'Permutation([])'),
(Permutation([], size=1),
'Permutation([0])'),
(Permutation([], size=2),
'Permutation([0, 1])'),
(Permutation([], size=10),
'Permutation([], size=10)'),
(Permutation([1, 0, 2]),
'Permutation([1, 0, 2])'),
(Permutation([1, 0, 2, 3, 4, 5]),
'Permutation([1, 0], size=6)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10),
'Permutation([1, 0], size=10)'),
]:
assert sstr(p, perm_cyclic=False) == s
for p, s in [
(Permutation([]),
'()'),
(Permutation([], size=1),
'(0)'),
(Permutation([], size=2),
'(1)'),
(Permutation([], size=10),
'(9)'),
(Permutation([1, 0, 2]),
'(2)(0 1)'),
(Permutation([1, 0, 2, 3, 4, 5]),
'(5)(0 1)'),
(Permutation([1, 0, 2, 3, 4, 5], size=10),
'(9)(0 1)'),
(Permutation([0, 1, 3, 2, 4, 5], size=10),
'(9)(2 3)'),
]:
assert sstr(p) == s
with warns_deprecated_sympy():
old_print_cyclic = Permutation.print_cyclic
Permutation.print_cyclic = False
assert sstr(Permutation([1, 0, 2])) == 'Permutation([1, 0, 2])'
Permutation.print_cyclic = old_print_cyclic
def test_Pi():
assert str(pi) == "pi"
def test_Poly():
assert str(Poly(0, x)) == "Poly(0, x, domain='ZZ')"
assert str(Poly(1, x)) == "Poly(1, x, domain='ZZ')"
assert str(Poly(x, x)) == "Poly(x, x, domain='ZZ')"
assert str(Poly(2*x + 1, x)) == "Poly(2*x + 1, x, domain='ZZ')"
assert str(Poly(2*x - 1, x)) == "Poly(2*x - 1, x, domain='ZZ')"
assert str(Poly(-1, x)) == "Poly(-1, x, domain='ZZ')"
assert str(Poly(-x, x)) == "Poly(-x, x, domain='ZZ')"
assert str(Poly(-2*x + 1, x)) == "Poly(-2*x + 1, x, domain='ZZ')"
assert str(Poly(-2*x - 1, x)) == "Poly(-2*x - 1, x, domain='ZZ')"
assert str(Poly(x - 1, x)) == "Poly(x - 1, x, domain='ZZ')"
assert str(Poly(2*x + x**5, x)) == "Poly(x**5 + 2*x, x, domain='ZZ')"
assert str(Poly(3**(2*x), 3**x)) == "Poly((3**x)**2, 3**x, domain='ZZ')"
assert str(Poly((x**2)**x)) == "Poly(((x**2)**x), (x**2)**x, domain='ZZ')"
assert str(Poly((x + y)**3, (x + y), expand=False)
) == "Poly((x + y)**3, x + y, domain='ZZ')"
assert str(Poly((x - 1)**2, (x - 1), expand=False)
) == "Poly((x - 1)**2, x - 1, domain='ZZ')"
assert str(
Poly(x**2 + 1 + y, x)) == "Poly(x**2 + y + 1, x, domain='ZZ[y]')"
assert str(
Poly(x**2 - 1 + y, x)) == "Poly(x**2 + y - 1, x, domain='ZZ[y]')"
assert str(Poly(x**2 + I*x, x)) == "Poly(x**2 + I*x, x, domain='ZZ_I')"
assert str(Poly(x**2 - I*x, x)) == "Poly(x**2 - I*x, x, domain='ZZ_I')"
assert str(Poly(-x*y*z + x*y - 1, x, y, z)
) == "Poly(-x*y*z + x*y - 1, x, y, z, domain='ZZ')"
assert str(Poly(-w*x**21*y**7*z + (1 + w)*z**3 - 2*x*z + 1, x, y, z)) == \
"Poly(-w*x**21*y**7*z - 2*x*z + (w + 1)*z**3 + 1, x, y, z, domain='ZZ[w]')"
assert str(Poly(x**2 + 1, x, modulus=2)) == "Poly(x**2 + 1, x, modulus=2)"
assert str(Poly(2*x**2 + 3*x + 4, x, modulus=17)) == "Poly(2*x**2 + 3*x + 4, x, modulus=17)"
def test_PolyRing():
assert str(ring("x", ZZ, lex)[0]) == "Polynomial ring in x over ZZ with lex order"
assert str(ring("x,y", QQ, grlex)[0]) == "Polynomial ring in x, y over QQ with grlex order"
assert str(ring("x,y,z", ZZ["t"], lex)[0]) == "Polynomial ring in x, y, z over ZZ[t] with lex order"
def test_FracField():
assert str(field("x", ZZ, lex)[0]) == "Rational function field in x over ZZ with lex order"
assert str(field("x,y", QQ, grlex)[0]) == "Rational function field in x, y over QQ with grlex order"
assert str(field("x,y,z", ZZ["t"], lex)[0]) == "Rational function field in x, y, z over ZZ[t] with lex order"
def test_PolyElement():
Ruv, u,v = ring("u,v", ZZ)
Rxyz, x,y,z = ring("x,y,z", Ruv)
Rx_zzi, xz = ring("x", ZZ_I)
assert str(x - x) == "0"
assert str(x - 1) == "x - 1"
assert str(x + 1) == "x + 1"
assert str(x**2) == "x**2"
assert str(x**(-2)) == "x**(-2)"
assert str(x**QQ(1, 2)) == "x**(1/2)"
assert str((u**2 + 3*u*v + 1)*x**2*y + u + 1) == "(u**2 + 3*u*v + 1)*x**2*y + u + 1"
assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x"
assert str((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == "(u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1"
assert str((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == "-(u**2 - 3*u*v + 1)*x**2*y - (u + 1)*x - 1"
assert str(-(v**2 + v + 1)*x + 3*u*v + 1) == "-(v**2 + v + 1)*x + 3*u*v + 1"
assert str(-(v**2 + v + 1)*x - 3*u*v + 1) == "-(v**2 + v + 1)*x - 3*u*v + 1"
assert str((1+I)*xz + 2) == "(1 + 1*I)*x + (2 + 0*I)"
def test_FracElement():
Fuv, u,v = field("u,v", ZZ)
Fxyzt, x,y,z,t = field("x,y,z,t", Fuv)
Rx_zzi, xz = field("x", QQ_I)
i = QQ_I(0, 1)
assert str(x - x) == "0"
assert str(x - 1) == "x - 1"
assert str(x + 1) == "x + 1"
assert str(x/3) == "x/3"
assert str(x/z) == "x/z"
assert str(x*y/z) == "x*y/z"
assert str(x/(z*t)) == "x/(z*t)"
assert str(x*y/(z*t)) == "x*y/(z*t)"
assert str((x - 1)/y) == "(x - 1)/y"
assert str((x + 1)/y) == "(x + 1)/y"
assert str((-x - 1)/y) == "(-x - 1)/y"
assert str((x + 1)/(y*z)) == "(x + 1)/(y*z)"
assert str(-y/(x + 1)) == "-y/(x + 1)"
assert str(y*z/(x + 1)) == "y*z/(x + 1)"
assert str(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - 1)"
assert str(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == "((u + 1)*x*y + 1)/((v - 1)*z - u*v*t - 1)"
assert str((1+i)/xz) == "(1 + 1*I)/x"
assert str(((1+i)*xz - i)/xz) == "((1 + 1*I)*x + (0 + -1*I))/x"
def test_GaussianInteger():
assert str(ZZ_I(1, 0)) == "1"
assert str(ZZ_I(-1, 0)) == "-1"
assert str(ZZ_I(0, 1)) == "I"
assert str(ZZ_I(0, -1)) == "-I"
assert str(ZZ_I(0, 2)) == "2*I"
assert str(ZZ_I(0, -2)) == "-2*I"
assert str(ZZ_I(1, 1)) == "1 + I"
assert str(ZZ_I(-1, -1)) == "-1 - I"
assert str(ZZ_I(-1, -2)) == "-1 - 2*I"
def test_GaussianRational():
assert str(QQ_I(1, 0)) == "1"
assert str(QQ_I(QQ(2, 3), 0)) == "2/3"
assert str(QQ_I(0, QQ(2, 3))) == "2*I/3"
assert str(QQ_I(QQ(1, 2), QQ(-2, 3))) == "1/2 - 2*I/3"
def test_Pow():
assert str(x**-1) == "1/x"
assert str(x**-2) == "x**(-2)"
assert str(x**2) == "x**2"
assert str((x + y)**-1) == "1/(x + y)"
assert str((x + y)**-2) == "(x + y)**(-2)"
assert str((x + y)**2) == "(x + y)**2"
assert str((x + y)**(1 + x)) == "(x + y)**(x + 1)"
assert str(x**Rational(1, 3)) == "x**(1/3)"
assert str(1/x**Rational(1, 3)) == "x**(-1/3)"
assert str(sqrt(sqrt(x))) == "x**(1/4)"
# not the same as x**-1
assert str(x**-1.0) == 'x**(-1.0)'
# see issue #2860
assert str(Pow(S(2), -1.0, evaluate=False)) == '2**(-1.0)'
def test_sqrt():
assert str(sqrt(x)) == "sqrt(x)"
assert str(sqrt(x**2)) == "sqrt(x**2)"
assert str(1/sqrt(x)) == "1/sqrt(x)"
assert str(1/sqrt(x**2)) == "1/sqrt(x**2)"
assert str(y/sqrt(x)) == "y/sqrt(x)"
assert str(x**0.5) == "x**0.5"
assert str(1/x**0.5) == "x**(-0.5)"
def test_Rational():
n1 = Rational(1, 4)
n2 = Rational(1, 3)
n3 = Rational(2, 4)
n4 = Rational(2, -4)
n5 = Rational(0)
n7 = Rational(3)
n8 = Rational(-3)
assert str(n1*n2) == "1/12"
assert str(n1*n2) == "1/12"
assert str(n3) == "1/2"
assert str(n1*n3) == "1/8"
assert str(n1 + n3) == "3/4"
assert str(n1 + n2) == "7/12"
assert str(n1 + n4) == "-1/4"
assert str(n4*n4) == "1/4"
assert str(n4 + n2) == "-1/6"
assert str(n4 + n5) == "-1/2"
assert str(n4*n5) == "0"
assert str(n3 + n4) == "0"
assert str(n1**n7) == "1/64"
assert str(n2**n7) == "1/27"
assert str(n2**n8) == "27"
assert str(n7**n8) == "1/27"
assert str(Rational("-25")) == "-25"
assert str(Rational("1.25")) == "5/4"
assert str(Rational("-2.6e-2")) == "-13/500"
assert str(S("25/7")) == "25/7"
assert str(S("-123/569")) == "-123/569"
assert str(S("0.1[23]", rational=1)) == "61/495"
assert str(S("5.1[666]", rational=1)) == "31/6"
assert str(S("-5.1[666]", rational=1)) == "-31/6"
assert str(S("0.[9]", rational=1)) == "1"
assert str(S("-0.[9]", rational=1)) == "-1"
assert str(sqrt(Rational(1, 4))) == "1/2"
assert str(sqrt(Rational(1, 36))) == "1/6"
assert str((123**25) ** Rational(1, 25)) == "123"
assert str((123**25 + 1)**Rational(1, 25)) != "123"
assert str((123**25 - 1)**Rational(1, 25)) != "123"
assert str((123**25 - 1)**Rational(1, 25)) != "122"
assert str(sqrt(Rational(81, 36))**3) == "27/8"
assert str(1/sqrt(Rational(81, 36))**3) == "8/27"
assert str(sqrt(-4)) == str(2*I)
assert str(2**Rational(1, 10**10)) == "2**(1/10000000000)"
assert sstr(Rational(2, 3), sympy_integers=True) == "S(2)/3"
x = Symbol("x")
assert sstr(x**Rational(2, 3), sympy_integers=True) == "x**(S(2)/3)"
assert sstr(Eq(x, Rational(2, 3)), sympy_integers=True) == "Eq(x, S(2)/3)"
assert sstr(Limit(x, x, Rational(7, 2)), sympy_integers=True) == \
"Limit(x, x, S(7)/2)"
def test_Float():
# NOTE dps is the whole number of decimal digits
assert str(Float('1.23', dps=1 + 2)) == '1.23'
assert str(Float('1.23456789', dps=1 + 8)) == '1.23456789'
assert str(
Float('1.234567890123456789', dps=1 + 18)) == '1.234567890123456789'
assert str(pi.evalf(1 + 2)) == '3.14'
assert str(pi.evalf(1 + 14)) == '3.14159265358979'
assert str(pi.evalf(1 + 64)) == ('3.141592653589793238462643383279'
'5028841971693993751058209749445923')
assert str(pi.round(-1)) == '0.0'
assert str((pi**400 - (pi**400).round(1)).n(2)) == '-0.e+88'
assert sstr(Float("100"), full_prec=False, min=-2, max=2) == '1.0e+2'
assert sstr(Float("100"), full_prec=False, min=-2, max=3) == '100.0'
assert sstr(Float("0.1"), full_prec=False, min=-2, max=3) == '0.1'
assert sstr(Float("0.099"), min=-2, max=3) == '9.90000000000000e-2'
def test_Relational():
assert str(Rel(x, y, "<")) == "x < y"
assert str(Rel(x + y, y, "==")) == "Eq(x + y, y)"
assert str(Rel(x, y, "!=")) == "Ne(x, y)"
assert str(Eq(x, 1) | Eq(x, 2)) == "Eq(x, 1) | Eq(x, 2)"
assert str(Ne(x, 1) & Ne(x, 2)) == "Ne(x, 1) & Ne(x, 2)"
def test_AppliedBinaryRelation():
assert str(Q.eq(x, y)) == "Q.eq(x, y)"
assert str(Q.ne(x, y)) == "Q.ne(x, y)"
def test_CRootOf():
assert str(rootof(x**5 + 2*x - 1, 0)) == "CRootOf(x**5 + 2*x - 1, 0)"
def test_RootSum():
f = x**5 + 2*x - 1
assert str(
RootSum(f, Lambda(z, z), auto=False)) == "RootSum(x**5 + 2*x - 1)"
assert str(RootSum(f, Lambda(
z, z**2), auto=False)) == "RootSum(x**5 + 2*x - 1, Lambda(z, z**2))"
def test_GroebnerBasis():
assert str(groebner(
[], x, y)) == "GroebnerBasis([], x, y, domain='ZZ', order='lex')"
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
assert str(groebner(F, order='grlex')) == \
"GroebnerBasis([x**2 - x - 3*y + 1, y**2 - 2*x + y - 1], x, y, domain='ZZ', order='grlex')"
assert str(groebner(F, order='lex')) == \
"GroebnerBasis([2*x - y**2 - y + 1, y**4 + 2*y**3 - 3*y**2 - 16*y + 7], x, y, domain='ZZ', order='lex')"
def test_set():
assert sstr(set()) == 'set()'
assert sstr(frozenset()) == 'frozenset()'
assert sstr({1}) == '{1}'
assert sstr(frozenset([1])) == 'frozenset({1})'
assert sstr({1, 2, 3}) == '{1, 2, 3}'
assert sstr(frozenset([1, 2, 3])) == 'frozenset({1, 2, 3})'
assert sstr(
{1, x, x**2, x**3, x**4}) == '{1, x, x**2, x**3, x**4}'
assert sstr(
frozenset([1, x, x**2, x**3, x**4])) == 'frozenset({1, x, x**2, x**3, x**4})'
def test_SparseMatrix():
M = SparseMatrix([[x**+1, 1], [y, x + y]])
assert str(M) == "Matrix([[x, 1], [y, x + y]])"
assert sstr(M) == "Matrix([\n[x, 1],\n[y, x + y]])"
def test_Sum():
assert str(summation(cos(3*z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
"Sum(x*y**2, (x, -2, 2), (y, -5, 5))"
def test_Symbol():
assert str(y) == "y"
assert str(x) == "x"
e = x
assert str(e) == "x"
def test_tuple():
assert str((x,)) == sstr((x,)) == "(x,)"
assert str((x + y, 1 + x)) == sstr((x + y, 1 + x)) == "(x + y, x + 1)"
assert str((x + y, (
1 + x, x**2))) == sstr((x + y, (1 + x, x**2))) == "(x + y, (x + 1, x**2))"
def test_Series_str():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y)
assert str(Series(tf1, tf2)) == \
"Series(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y))"
assert str(Series(tf1, tf2, tf3)) == \
"Series(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y), TransferFunction(t*x**2 - t**w*x + w, t - y, y))"
assert str(Series(-tf2, tf1)) == \
"Series(TransferFunction(-x + y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y))"
def test_MIMOSeries_str():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
assert str(MIMOSeries(tfm_1, tfm_2)) == \
"MIMOSeries(TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), "\
"(TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)))), "\
"TransferFunctionMatrix(((TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)), "\
"(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)))))"
def test_TransferFunction_str():
tf1 = TransferFunction(x - 1, x + 1, x)
assert str(tf1) == "TransferFunction(x - 1, x + 1, x)"
tf2 = TransferFunction(x + 1, 2 - y, x)
assert str(tf2) == "TransferFunction(x + 1, 2 - y, x)"
tf3 = TransferFunction(y, y**2 + 2*y + 3, y)
assert str(tf3) == "TransferFunction(y, y**2 + 2*y + 3, y)"
def test_Parallel_str():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y)
assert str(Parallel(tf1, tf2)) == \
"Parallel(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y))"
assert str(Parallel(tf1, tf2, tf3)) == \
"Parallel(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y), TransferFunction(t*x**2 - t**w*x + w, t - y, y))"
assert str(Parallel(-tf2, tf1)) == \
"Parallel(TransferFunction(-x + y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y))"
def test_MIMOParallel_str():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
assert str(MIMOParallel(tfm_1, tfm_2)) == \
"MIMOParallel(TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), "\
"(TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)))), "\
"TransferFunctionMatrix(((TransferFunction(x - y, x + y, y), TransferFunction(x*y**2 - z, -t**3 + y**3, y)), "\
"(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)))))"
def test_Feedback_str():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y)
assert str(Feedback(tf1*tf2, tf3)) == \
"Feedback(Series(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), " \
"TransferFunction(t*x**2 - t**w*x + w, t - y, y), -1)"
assert str(Feedback(tf1, TransferFunction(1, 1, y), 1)) == \
"Feedback(TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(1, 1, y), 1)"
def test_MIMOFeedback_str():
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
tfm_1 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
tfm_2 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
assert (str(MIMOFeedback(tfm_1, tfm_2)) \
== "MIMOFeedback(TransferFunctionMatrix(((TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x))," \
" (TransferFunction(x**2 - y**3, y - z, x), TransferFunction(-x + y, y + z, x)))), " \
"TransferFunctionMatrix(((TransferFunction(x**2 - y**3, y - z, x), " \
"TransferFunction(-x + y, y + z, x)), (TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x)))), -1)")
assert (str(MIMOFeedback(tfm_1, tfm_2, 1)) \
== "MIMOFeedback(TransferFunctionMatrix(((TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x)), " \
"(TransferFunction(x**2 - y**3, y - z, x), TransferFunction(-x + y, y + z, x)))), " \
"TransferFunctionMatrix(((TransferFunction(x**2 - y**3, y - z, x), TransferFunction(-x + y, y + z, x)), "\
"(TransferFunction(-x + y, y + z, x), TransferFunction(x**2 - y**3, y - z, x)))), 1)")
def test_TransferFunctionMatrix_str():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y)
assert str(TransferFunctionMatrix([[tf1], [tf2]])) == \
"TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y),), (TransferFunction(x - y, x + y, y),)))"
assert str(TransferFunctionMatrix([[tf1, tf2], [tf3, tf2]])) == \
"TransferFunctionMatrix(((TransferFunction(x*y**2 - z, -t**3 + y**3, y), TransferFunction(x - y, x + y, y)), (TransferFunction(t*x**2 - t**w*x + w, t - y, y), TransferFunction(x - y, x + y, y))))"
def test_Quaternion_str_printer():
q = Quaternion(x, y, z, t)
assert str(q) == "x + y*i + z*j + t*k"
q = Quaternion(x,y,z,x*t)
assert str(q) == "x + y*i + z*j + t*x*k"
q = Quaternion(x,y,z,x+t)
assert str(q) == "x + y*i + z*j + (t + x)*k"
def test_Quantity_str():
assert sstr(second, abbrev=True) == "s"
assert sstr(joule, abbrev=True) == "J"
assert str(second) == "second"
assert str(joule) == "joule"
def test_wild_str():
# Check expressions containing Wild not causing infinite recursion
w = Wild('x')
assert str(w + 1) == 'x_ + 1'
assert str(exp(2**w) + 5) == 'exp(2**x_) + 5'
assert str(3*w + 1) == '3*x_ + 1'
assert str(1/w + 1) == '1 + 1/x_'
assert str(w**2 + 1) == 'x_**2 + 1'
assert str(1/(1 - w)) == '1/(1 - x_)'
def test_wild_matchpy():
from sympy.utilities.matchpy_connector import WildDot, WildPlus, WildStar
matchpy = import_module("matchpy")
if matchpy is None:
return
wd = WildDot('w_')
wp = WildPlus('w__')
ws = WildStar('w___')
assert str(wd) == 'w_'
assert str(wp) == 'w__'
assert str(ws) == 'w___'
assert str(wp/ws + 2**wd) == '2**w_ + w__/w___'
assert str(sin(wd)*cos(wp)*sqrt(ws)) == 'sqrt(w___)*sin(w_)*cos(w__)'
def test_zeta():
assert str(zeta(3)) == "zeta(3)"
def test_issue_3101():
e = x - y
a = str(e)
b = str(e)
assert a == b
def test_issue_3103():
e = -2*sqrt(x) - y/sqrt(x)/2
assert str(e) not in ["(-2)*x**1/2(-1/2)*x**(-1/2)*y",
"-2*x**1/2(-1/2)*x**(-1/2)*y", "-2*x**1/2-1/2*x**-1/2*w"]
assert str(e) == "-2*sqrt(x) - y/(2*sqrt(x))"
def test_issue_4021():
e = Integral(x, x) + 1
assert str(e) == 'Integral(x, x) + 1'
def test_sstrrepr():
assert sstr('abc') == 'abc'
assert sstrrepr('abc') == "'abc'"
e = ['a', 'b', 'c', x]
assert sstr(e) == "[a, b, c, x]"
assert sstrrepr(e) == "['a', 'b', 'c', x]"
def test_infinity():
assert sstr(oo*I) == "oo*I"
def test_full_prec():
assert sstr(S("0.3"), full_prec=True) == "0.300000000000000"
assert sstr(S("0.3"), full_prec="auto") == "0.300000000000000"
assert sstr(S("0.3"), full_prec=False) == "0.3"
assert sstr(S("0.3")*x, full_prec=True) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert sstr(S("0.3")*x, full_prec="auto") in [
"0.3*x",
"x*0.3"
]
assert sstr(S("0.3")*x, full_prec=False) in [
"0.3*x",
"x*0.3"
]
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert sstr(A*B*C**-1) == "A*B*C**(-1)"
assert sstr(C**-1*A*B) == "C**(-1)*A*B"
assert sstr(A*C**-1*B) == "A*C**(-1)*B"
assert sstr(sqrt(A)) == "sqrt(A)"
assert sstr(1/sqrt(A)) == "A**(-1/2)"
def test_empty_printer():
str_printer = StrPrinter()
assert str_printer.emptyPrinter("foo") == "foo"
assert str_printer.emptyPrinter(x*y) == "x*y"
assert str_printer.emptyPrinter(32) == "32"
def test_settings():
raises(TypeError, lambda: sstr(S(4), method="garbage"))
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert str(where(X > 0)) == "Domain: (0 < x1) & (x1 < oo)"
D = Die('d1', 6)
assert str(where(D > 4)) == "Domain: Eq(d1, 5) | Eq(d1, 6)"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert str(pspace(Tuple(A, B)).domain) == "Domain: (0 <= a) & (0 <= b) & (a < oo) & (b < oo)"
def test_FiniteSet():
assert str(FiniteSet(*range(1, 51))) == (
'{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,'
' 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,'
' 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50}'
)
assert str(FiniteSet(*range(1, 6))) == '{1, 2, 3, 4, 5}'
assert str(FiniteSet(*[x*y, x**2])) == '{x**2, x*y}'
assert str(FiniteSet(FiniteSet(FiniteSet(x, y), 5), FiniteSet(x,y), 5)
) == 'FiniteSet(5, FiniteSet(5, {x, y}), {x, y})'
def test_Partition():
assert str(Partition(FiniteSet(x, y), {z})) == 'Partition({z}, {x, y})'
def test_UniversalSet():
assert str(S.UniversalSet) == 'UniversalSet'
def test_PrettyPoly():
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert sstr(F.convert(x/(x + y))) == sstr(x/(x + y))
assert sstr(R.convert(x + y)) == sstr(x + y)
def test_categories():
from sympy.categories import (Object, NamedMorphism,
IdentityMorphism, Category)
A = Object("A")
B = Object("B")
f = NamedMorphism(A, B, "f")
id_A = IdentityMorphism(A)
K = Category("K")
assert str(A) == 'Object("A")'
assert str(f) == 'NamedMorphism(Object("A"), Object("B"), "f")'
assert str(id_A) == 'IdentityMorphism(Object("A"))'
assert str(K) == 'Category("K")'
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert str(t) == 'Tr(A*B)'
def test_issue_6387():
assert str(factor(-3.0*z + 3)) == '-3.0*(1.0*z - 1.0)'
def test_MatMul_MatAdd():
X, Y = MatrixSymbol("X", 2, 2), MatrixSymbol("Y", 2, 2)
assert str(2*(X + Y)) == "2*X + 2*Y"
assert str(I*X) == "I*X"
assert str(-I*X) == "-I*X"
assert str((1 + I)*X) == '(1 + I)*X'
assert str(-(1 + I)*X) == '(-1 - I)*X'
def test_MatrixSlice():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
assert str(MatrixSlice(X, (None, None, None), (None, None, None))) == 'X[:, :]'
assert str(X[x:x + 1, y:y + 1]) == 'X[x:x + 1, y:y + 1]'
assert str(X[x:x + 1:2, y:y + 1:2]) == 'X[x:x + 1:2, y:y + 1:2]'
assert str(X[:x, y:]) == 'X[:x, y:]'
assert str(X[:x, y:]) == 'X[:x, y:]'
assert str(X[x:, :y]) == 'X[x:, :y]'
assert str(X[x:y, z:w]) == 'X[x:y, z:w]'
assert str(X[x:y:t, w:t:x]) == 'X[x:y:t, w:t:x]'
assert str(X[x::y, t::w]) == 'X[x::y, t::w]'
assert str(X[:x:y, :t:w]) == 'X[:x:y, :t:w]'
assert str(X[::x, ::y]) == 'X[::x, ::y]'
assert str(MatrixSlice(X, (0, None, None), (0, None, None))) == 'X[:, :]'
assert str(MatrixSlice(X, (None, n, None), (None, n, None))) == 'X[:, :]'
assert str(MatrixSlice(X, (0, n, None), (0, n, None))) == 'X[:, :]'
assert str(MatrixSlice(X, (0, n, 2), (0, n, 2))) == 'X[::2, ::2]'
assert str(X[1:2:3, 4:5:6]) == 'X[1:2:3, 4:5:6]'
assert str(X[1:3:5, 4:6:8]) == 'X[1:3:5, 4:6:8]'
assert str(X[1:10:2]) == 'X[1:10:2, :]'
assert str(Y[:5, 1:9:2]) == 'Y[:5, 1:9:2]'
assert str(Y[:5, 1:10:2]) == 'Y[:5, 1::2]'
assert str(Y[5, :5:2]) == 'Y[5:6, :5:2]'
assert str(X[0:1, 0:1]) == 'X[:1, :1]'
assert str(X[0:1:2, 0:1:2]) == 'X[:1:2, :1:2]'
assert str((Y + Z)[2:, 2:]) == '(Y + Z)[2:, 2:]'
def test_true_false():
assert str(true) == repr(true) == sstr(true) == "True"
assert str(false) == repr(false) == sstr(false) == "False"
def test_Equivalent():
assert str(Equivalent(y, x)) == "Equivalent(x, y)"
def test_Xor():
assert str(Xor(y, x, evaluate=False)) == "x ^ y"
def test_Complement():
assert str(Complement(S.Reals, S.Naturals)) == 'Complement(Reals, Naturals)'
def test_SymmetricDifference():
assert str(SymmetricDifference(Interval(2, 3), Interval(3, 4),evaluate=False)) == \
'SymmetricDifference(Interval(2, 3), Interval(3, 4))'
def test_UnevaluatedExpr():
a, b = symbols("a b")
expr1 = 2*UnevaluatedExpr(a+b)
assert str(expr1) == "2*(a + b)"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert(str(A[0, 0]) == "A[0, 0]")
assert(str(3 * A[0, 0]) == "3*A[0, 0]")
F = C[0, 0].subs(C, A - B)
assert str(F) == "(A - B)[0, 0]"
def test_MatrixSymbol_printing():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert str(A - A*B - B) == "A - A*B - B"
assert str(A*B - (A+B)) == "-A + A*B - B"
assert str(A**(-1)) == "A**(-1)"
assert str(A**3) == "A**3"
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert str(X) == "X"
# Apply function elementwise (`ElementwiseApplyFunc`):
expr = (X.T*X).applyfunc(sin)
assert str(expr) == 'Lambda(_d, sin(_d)).(X.T*X)'
lamda = Lambda(x, 1/x)
expr = (n*X).applyfunc(lamda)
assert str(expr) == 'Lambda(x, 1/x).(n*X)'
def test_Subs_printing():
assert str(Subs(x, (x,), (1,))) == 'Subs(x, x, 1)'
assert str(Subs(x + y, (x, y), (1, 2))) == 'Subs(x + y, (x, y), (1, 2))'
def test_issue_15716():
e = Integral(factorial(x), (x, -oo, oo))
assert e.as_terms() == ([(e, ((1.0, 0.0), (1,), ()))], [e])
def test_str_special_matrices():
from sympy.matrices import Identity, ZeroMatrix, OneMatrix
assert str(Identity(4)) == 'I'
assert str(ZeroMatrix(2, 2)) == '0'
assert str(OneMatrix(2, 2)) == '1'
def test_issue_14567():
assert factorial(Sum(-1, (x, 0, 0))) + y # doesn't raise an error
def test_issue_21823():
assert str(Partition([1, 2])) == 'Partition({1, 2})'
assert str(Partition({1, 2})) == 'Partition({1, 2})'
def test_issue_22689():
assert str(Mul(Pow(x,-2, evaluate=False), Pow(3,-1,evaluate=False), evaluate=False)) == "1/(x**2*3)"
def test_issue_21119_21460():
ss = lambda x: str(S(x, evaluate=False))
assert ss('4/2') == '4/2'
assert ss('4/-2') == '4/(-2)'
assert ss('-4/2') == '-4/2'
assert ss('-4/-2') == '-4/(-2)'
assert ss('-2*3/-1') == '-2*3/(-1)'
assert ss('-2*3/-1/2') == '-2*3/(-1*2)'
assert ss('4/2/1') == '4/(2*1)'
assert ss('-2/-1/2') == '-2/(-1*2)'
assert ss('2*3*4**(-2*3)') == '2*3/4**(2*3)'
assert ss('2*3*1*4**(-2*3)') == '2*3*1/4**(2*3)'
def test_Str():
from sympy.core.symbol import Str
assert str(Str('x')) == 'x'
assert sstrrepr(Str('x')) == "Str('x')"
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
x,y = symbols('x y', real=True)
m = Manifold('M', 2)
assert str(m) == "M"
p = Patch('P', m)
assert str(p) == "P"
rect = CoordSystem('rect', p, [x, y])
assert str(rect) == "rect"
b = BaseScalarField(rect, 0)
assert str(b) == "x"
def test_NDimArray():
assert sstr(NDimArray(1.0), full_prec=True) == '1.00000000000000'
assert sstr(NDimArray(1.0), full_prec=False) == '1.0'
assert sstr(NDimArray([1.0, 2.0]), full_prec=True) == '[1.00000000000000, 2.00000000000000]'
assert sstr(NDimArray([1.0, 2.0]), full_prec=False) == '[1.0, 2.0]'
def test_Predicate():
assert sstr(Q.even) == 'Q.even'
def test_AppliedPredicate():
assert sstr(Q.even(x)) == 'Q.even(x)'
def test_printing_str_array_expressions():
assert sstr(ArraySymbol("A", (2, 3, 4))) == "A"
assert sstr(ArrayElement("A", (2, 1/(1-x), 0))) == "A[2, 1/(1 - x), 0]"
M = MatrixSymbol("M", 3, 3)
N = MatrixSymbol("N", 3, 3)
assert sstr(ArrayElement(M*N, [x, 0])) == "(M*N)[x, 0]"
|
041b9869382141ed48afce3e536abd2e8272d5e076187863386aed3949d3a2dc | from sympy.concrete.summations import Sum
from sympy.core.expr import Expr
from sympy.core.symbol import symbols
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import sin
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.sets.sets import Interval
from sympy.utilities.lambdify import lambdify
from sympy.testing.pytest import raises
from sympy.printing.tensorflow import TensorflowPrinter
from sympy.printing.lambdarepr import lambdarepr, LambdaPrinter, NumExprPrinter
x, y, z = symbols("x,y,z")
i, a, b = symbols("i,a,b")
j, c, d = symbols("j,c,d")
def test_basic():
assert lambdarepr(x*y) == "x*y"
assert lambdarepr(x + y) in ["y + x", "x + y"]
assert lambdarepr(x**y) == "x**y"
def test_matrix():
# Test printing a Matrix that has an element that is printed differently
# with the LambdaPrinter than with the StrPrinter.
e = x % 2
assert lambdarepr(e) != str(e)
assert lambdarepr(Matrix([e])) == 'ImmutableDenseMatrix([[x % 2]])'
def test_piecewise():
# In each case, test eval() the lambdarepr() to make sure there are a
# correct number of parentheses. It will give a SyntaxError if there aren't.
h = "lambda x: "
p = Piecewise((x, x < 0))
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (x < 0) else None)"
p = Piecewise(
(1, x < 1),
(2, x < 2),
(0, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (2) if (x < 2) else (0))"
p = Piecewise(
(1, x < 1),
(2, x < 2),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (2) if (x < 2) else None)"
p = Piecewise(
(x, x < 1),
(x**2, Interval(3, 4, True, False).contains(x)),
(0, True),
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x) if (x < 1) else (x**2) if (((x <= 4)) and ((x > 3))) else (0))"
p = Piecewise(
(x**2, x < 0),
(x, x < 1),
(2 - x, x >= 1),
(0, True), evaluate=False
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x**2) if (x < 0) else (x) if (x < 1)"\
" else (2 - x) if (x >= 1) else (0))"
p = Piecewise(
(x**2, x < 0),
(x, x < 1),
(2 - x, x >= 1), evaluate=False
)
l = lambdarepr(p)
eval(h + l)
assert l == "((x**2) if (x < 0) else (x) if (x < 1)"\
" else (2 - x) if (x >= 1) else None)"
p = Piecewise(
(1, x >= 1),
(2, x >= 2),
(3, x >= 3),
(4, x >= 4),
(5, x >= 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x >= 1) else (2) if (x >= 2) else (3) if (x >= 3)"\
" else (4) if (x >= 4) else (5) if (x >= 5) else (6))"
p = Piecewise(
(1, x <= 1),
(2, x <= 2),
(3, x <= 3),
(4, x <= 4),
(5, x <= 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x <= 1) else (2) if (x <= 2) else (3) if (x <= 3)"\
" else (4) if (x <= 4) else (5) if (x <= 5) else (6))"
p = Piecewise(
(1, x > 1),
(2, x > 2),
(3, x > 3),
(4, x > 4),
(5, x > 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l =="((1) if (x > 1) else (2) if (x > 2) else (3) if (x > 3)"\
" else (4) if (x > 4) else (5) if (x > 5) else (6))"
p = Piecewise(
(1, x < 1),
(2, x < 2),
(3, x < 3),
(4, x < 4),
(5, x < 5),
(6, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((1) if (x < 1) else (2) if (x < 2) else (3) if (x < 3)"\
" else (4) if (x < 4) else (5) if (x < 5) else (6))"
p = Piecewise(
(Piecewise(
(1, x > 0),
(2, True)
), y > 0),
(3, True)
)
l = lambdarepr(p)
eval(h + l)
assert l == "((((1) if (x > 0) else (2))) if (y > 0) else (3))"
def test_sum__1():
# In each case, test eval() the lambdarepr() to make sure that
# it evaluates to the same results as the symbolic expression
s = Sum(x ** i, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(x**i for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_sum__2():
s = Sum(i * x, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_multiple_sums():
s = Sum(i * x + j, (i, a, b), (j, c, d))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"
args = x, a, b, c, d
f = lambdify(args, s)
vals = 2, 3, 4, 5, 6
f_ref = s.subs(zip(args, vals)).doit()
f_res = f(*vals)
assert f_res == f_ref
def test_sqrt():
prntr = LambdaPrinter({'standard' : 'python3'})
assert prntr._print_Pow(sqrt(x), rational=False) == 'sqrt(x)'
assert prntr._print_Pow(sqrt(x), rational=True) == 'x**(1/2)'
def test_settings():
raises(TypeError, lambda: lambdarepr(sin(x), method="garbage"))
def test_numexpr():
# test ITE rewrite as Piecewise
from sympy.logic.boolalg import ITE
expr = ITE(x > 0, True, False, evaluate=False)
assert NumExprPrinter().doprint(expr) == \
"numexpr.evaluate('where((x > 0), True, False)', truediv=True)"
from sympy.codegen.ast import Return, FunctionDefinition, Variable, Assignment
func_def = FunctionDefinition(None, 'foo', [Variable(x)], [Assignment(y,x), Return(y**2)])
print("")
print(NumExprPrinter().doprint(func_def))
expected = "def foo(x):\n"\
" y = numexpr.evaluate('x', truediv=True)\n"\
" return numexpr.evaluate('y**2', truediv=True)"
print(expected)
assert NumExprPrinter().doprint(func_def) == expected
class CustomPrintedObject(Expr):
def _lambdacode(self, printer):
return 'lambda'
def _tensorflowcode(self, printer):
return 'tensorflow'
def _numpycode(self, printer):
return 'numpy'
def _numexprcode(self, printer):
return 'numexpr'
def _mpmathcode(self, printer):
return 'mpmath'
def test_printmethod():
# In each case, printmethod is called to test
# its working
obj = CustomPrintedObject()
assert LambdaPrinter().doprint(obj) == 'lambda'
assert TensorflowPrinter().doprint(obj) == 'tensorflow'
assert NumExprPrinter().doprint(obj) == "numexpr.evaluate('numexpr', truediv=True)"
assert NumExprPrinter().doprint(Piecewise((y, x >= 0), (z, x < 0))) == \
"numexpr.evaluate('where((x >= 0), y, z)', truediv=True)"
|
411e6bf211f0e2e74e74cec6c19863bc543158611204d79fb8ee3e673f2f43a3 | from sympy.algebras.quaternion import Quaternion
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.combinatorics.permutations import Cycle, Permutation, AppliedPermutation
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.containers import Tuple, Dict
from sympy.core.expr import UnevaluatedExpr
from sympy.core.function import (Derivative, Function, Lambda, Subs, diff)
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core.numbers import (AlgebraicNumber, Float, I, Integer, Rational, oo, pi)
from sympy.core.power import Pow
from sympy.core.relational import Eq, Ne
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, Wild, symbols)
from sympy.functions.combinatorial.factorials import (FallingFactorial, RisingFactorial, binomial, factorial, factorial2, subfactorial)
from sympy.functions.combinatorial.numbers import bernoulli, bell, catalan, euler, lucas, fibonacci, tribonacci
from sympy.functions.elementary.complexes import (Abs, arg, conjugate, im, polar_lift, re)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (asinh, coth)
from sympy.functions.elementary.integers import (ceiling, floor, frac)
from sympy.functions.elementary.miscellaneous import (Max, Min, root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acsc, asin, cos, cot, sin, tan)
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
from sympy.functions.special.elliptic_integrals import (elliptic_e, elliptic_f, elliptic_k, elliptic_pi)
from sympy.functions.special.error_functions import (Chi, Ci, Ei, Shi, Si, expint)
from sympy.functions.special.gamma_functions import (gamma, uppergamma)
from sympy.functions.special.hyper import (hyper, meijerg)
from sympy.functions.special.mathieu_functions import (mathieuc, mathieucprime, mathieus, mathieusprime)
from sympy.functions.special.polynomials import (assoc_laguerre, assoc_legendre, chebyshevt, chebyshevu, gegenbauer, hermite, jacobi, laguerre, legendre)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.spherical_harmonics import (Ynm, Znm)
from sympy.functions.special.tensor_functions import (KroneckerDelta, LeviCivita)
from sympy.functions.special.zeta_functions import (dirichlet_eta, lerchphi, polylog, stieltjes, zeta)
from sympy.integrals.integrals import Integral
from sympy.integrals.transforms import (CosineTransform, FourierTransform, InverseCosineTransform, InverseFourierTransform, InverseLaplaceTransform, InverseMellinTransform, InverseSineTransform, LaplaceTransform, MellinTransform, SineTransform)
from sympy.logic import Implies
from sympy.logic.boolalg import (And, Or, Xor, Equivalent, false, Not, true)
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.permutation import PermutationMatrix
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.physics.control.lti import TransferFunction, Series, Parallel, Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback
from sympy.ntheory.factor_ import (divisor_sigma, primenu, primeomega, reduced_totient, totient, udivisor_sigma)
from sympy.physics.quantum import Commutator, Operator
from sympy.physics.quantum.trace import Tr
from sympy.physics.units import meter, gibibyte, microgram, second
from sympy.polys.domains.integerring import ZZ
from sympy.polys.fields import field
from sympy.polys.polytools import Poly
from sympy.polys.rings import ring
from sympy.polys.rootoftools import (RootSum, rootof)
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
from sympy.series.limits import Limit
from sympy.series.order import Order
from sympy.series.sequences import (SeqAdd, SeqFormula, SeqMul, SeqPer)
from sympy.sets.conditionset import ConditionSet
from sympy.sets.contains import Contains
from sympy.sets.fancysets import (ComplexRegion, ImageSet, Range)
from sympy.sets.ordinals import Ordinal, OrdinalOmega, OmegaPower
from sympy.sets.powerset import PowerSet
from sympy.sets.sets import (FiniteSet, Interval, Union, Intersection, Complement, SymmetricDifference, ProductSet)
from sympy.sets.setexpr import SetExpr
from sympy.stats.crv_types import Normal
from sympy.stats.symbolic_probability import (Covariance, Expectation,
Probability, Variance)
from sympy.tensor.array import (ImmutableDenseNDimArray,
ImmutableSparseNDimArray,
MutableSparseNDimArray,
MutableDenseNDimArray,
tensorproduct)
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElement
from sympy.tensor.indexed import (Idx, Indexed, IndexedBase)
from sympy.tensor.toperators import PartialDerivative
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
from sympy.testing.pytest import (XFAIL, raises, _both_exp_pow,
warns_deprecated_sympy)
from sympy.printing.latex import (latex, translate, greek_letters_set,
tex_greek_dictionary, multiline_latex,
latex_escape, LatexPrinter)
import sympy as sym
from sympy.abc import mu, tau
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
x, y, z, t, w, a, b, c, s, p = symbols('x y z t w a b c s p')
k, m, n = symbols('k m n', integer=True)
def test_printmethod():
class R(Abs):
def _latex(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert latex(R(x)) == r"foo(x)"
class R(Abs):
def _latex(self, printer):
return "foo"
assert latex(R(x)) == r"foo"
def test_latex_basic():
assert latex(1 + x) == r"x + 1"
assert latex(x**2) == r"x^{2}"
assert latex(x**(1 + x)) == r"x^{x + 1}"
assert latex(x**3 + x + 1 + x**2) == r"x^{3} + x^{2} + x + 1"
assert latex(2*x*y) == r"2 x y"
assert latex(2*x*y, mul_symbol='dot') == r"2 \cdot x \cdot y"
assert latex(3*x**2*y, mul_symbol='\\,') == r"3\,x^{2}\,y"
assert latex(1.5*3**x, mul_symbol='\\,') == r"1.5 \cdot 3^{x}"
assert latex(x**S.Half**5) == r"\sqrt[32]{x}"
assert latex(Mul(S.Half, x**2, -5, evaluate=False)) == r"\frac{1}{2} x^{2} \left(-5\right)"
assert latex(Mul(S.Half, x**2, 5, evaluate=False)) == r"\frac{1}{2} x^{2} \cdot 5"
assert latex(Mul(-5, -5, evaluate=False)) == r"\left(-5\right) \left(-5\right)"
assert latex(Mul(5, -5, evaluate=False)) == r"5 \left(-5\right)"
assert latex(Mul(S.Half, -5, S.Half, evaluate=False)) == r"\frac{1}{2} \left(-5\right) \frac{1}{2}"
assert latex(Mul(5, I, 5, evaluate=False)) == r"5 i 5"
assert latex(Mul(5, I, -5, evaluate=False)) == r"5 i \left(-5\right)"
assert latex(Mul(0, 1, evaluate=False)) == r'0 \cdot 1'
assert latex(Mul(1, 0, evaluate=False)) == r'1 \cdot 0'
assert latex(Mul(1, 1, evaluate=False)) == r'1 \cdot 1'
assert latex(Mul(-1, 1, evaluate=False)) == r'\left(-1\right) 1'
assert latex(Mul(1, 1, 1, evaluate=False)) == r'1 \cdot 1 \cdot 1'
assert latex(Mul(1, 2, evaluate=False)) == r'1 \cdot 2'
assert latex(Mul(1, S.Half, evaluate=False)) == r'1 \cdot \frac{1}{2}'
assert latex(Mul(1, 1, S.Half, evaluate=False)) == \
r'1 \cdot 1 \cdot \frac{1}{2}'
assert latex(Mul(1, 1, 2, 3, x, evaluate=False)) == \
r'1 \cdot 1 \cdot 2 \cdot 3 x'
assert latex(Mul(1, -1, evaluate=False)) == r'1 \left(-1\right)'
assert latex(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == \
r'4 \cdot 3 \cdot 2 \cdot 1 \cdot 0 y x'
assert latex(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == \
r'4 \cdot 3 \cdot 2 \left(z + 1\right) 0 y x'
assert latex(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == \
r'\frac{2}{3} \cdot \frac{5}{7}'
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/x, fold_short_frac=True) == r"1 / x"
assert latex(-S(3)/2) == r"- \frac{3}{2}"
assert latex(-S(3)/2, fold_short_frac=True) == r"- 3 / 2"
assert latex(1/x**2) == r"\frac{1}{x^{2}}"
assert latex(1/(x + y)/2) == r"\frac{1}{2 \left(x + y\right)}"
assert latex(x/2) == r"\frac{x}{2}"
assert latex(x/2, fold_short_frac=True) == r"x / 2"
assert latex((x + y)/(2*x)) == r"\frac{x + y}{2 x}"
assert latex((x + y)/(2*x), fold_short_frac=True) == \
r"\left(x + y\right) / 2 x"
assert latex((x + y)/(2*x), long_frac_ratio=0) == \
r"\frac{1}{2 x} \left(x + y\right)"
assert latex((x + y)/x) == r"\frac{x + y}{x}"
assert latex((x + y)/x, long_frac_ratio=3) == r"\frac{x + y}{x}"
assert latex((2*sqrt(2)*x)/3) == r"\frac{2 \sqrt{2} x}{3}"
assert latex((2*sqrt(2)*x)/3, long_frac_ratio=2) == \
r"\frac{2 x}{3} \sqrt{2}"
assert latex(binomial(x, y)) == r"{\binom{x}{y}}"
x_star = Symbol('x^*')
f = Function('f')
assert latex(x_star**2) == r"\left(x^{*}\right)^{2}"
assert latex(x_star**2, parenthesize_super=False) == r"{x^{*}}^{2}"
assert latex(Derivative(f(x_star), x_star,2)) == r"\frac{d^{2}}{d \left(x^{*}\right)^{2}} f{\left(x^{*} \right)}"
assert latex(Derivative(f(x_star), x_star,2), parenthesize_super=False) == r"\frac{d^{2}}{d {x^{*}}^{2}} f{\left(x^{*} \right)}"
assert latex(2*Integral(x, x)/3) == r"\frac{2 \int x\, dx}{3}"
assert latex(2*Integral(x, x)/3, fold_short_frac=True) == \
r"\left(2 \int x\, dx\right) / 3"
assert latex(sqrt(x)) == r"\sqrt{x}"
assert latex(x**Rational(1, 3)) == r"\sqrt[3]{x}"
assert latex(x**Rational(1, 3), root_notation=False) == r"x^{\frac{1}{3}}"
assert latex(sqrt(x)**3) == r"x^{\frac{3}{2}}"
assert latex(sqrt(x), itex=True) == r"\sqrt{x}"
assert latex(x**Rational(1, 3), itex=True) == r"\root{3}{x}"
assert latex(sqrt(x)**3, itex=True) == r"x^{\frac{3}{2}}"
assert latex(x**Rational(3, 4)) == r"x^{\frac{3}{4}}"
assert latex(x**Rational(3, 4), fold_frac_powers=True) == r"x^{3/4}"
assert latex((x + 1)**Rational(3, 4)) == \
r"\left(x + 1\right)^{\frac{3}{4}}"
assert latex((x + 1)**Rational(3, 4), fold_frac_powers=True) == \
r"\left(x + 1\right)^{3/4}"
assert latex(AlgebraicNumber(sqrt(2))) == r"\sqrt{2}"
assert latex(AlgebraicNumber(sqrt(2), [3, -7])) == r"-7 + 3 \sqrt{2}"
assert latex(AlgebraicNumber(sqrt(2), alias='alpha')) == r"\alpha"
assert latex(AlgebraicNumber(sqrt(2), [3, -7], alias='alpha')) == \
r"3 \alpha - 7"
assert latex(AlgebraicNumber(2**(S(1)/3), [1, 3, -7], alias='beta')) == \
r"\beta^{2} + 3 \beta - 7"
assert latex(1.5e20*x) == r"1.5 \cdot 10^{20} x"
assert latex(1.5e20*x, mul_symbol='dot') == r"1.5 \cdot 10^{20} \cdot x"
assert latex(1.5e20*x, mul_symbol='times') == \
r"1.5 \times 10^{20} \times x"
assert latex(1/sin(x)) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**-1) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**Rational(3, 2)) == \
r"\sin^{\frac{3}{2}}{\left(x \right)}"
assert latex(sin(x)**Rational(3, 2), fold_frac_powers=True) == \
r"\sin^{3/2}{\left(x \right)}"
assert latex(~x) == r"\neg x"
assert latex(x & y) == r"x \wedge y"
assert latex(x & y & z) == r"x \wedge y \wedge z"
assert latex(x | y) == r"x \vee y"
assert latex(x | y | z) == r"x \vee y \vee z"
assert latex((x & y) | z) == r"z \vee \left(x \wedge y\right)"
assert latex(Implies(x, y)) == r"x \Rightarrow y"
assert latex(~(x >> ~y)) == r"x \not\Rightarrow \neg y"
assert latex(Implies(Or(x,y), z)) == r"\left(x \vee y\right) \Rightarrow z"
assert latex(Implies(z, Or(x,y))) == r"z \Rightarrow \left(x \vee y\right)"
assert latex(~(x & y)) == r"\neg \left(x \wedge y\right)"
assert latex(~x, symbol_names={x: "x_i"}) == r"\neg x_i"
assert latex(x & y, symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \wedge y_i"
assert latex(x & y & z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \wedge y_i \wedge z_i"
assert latex(x | y, symbol_names={x: "x_i", y: "y_i"}) == r"x_i \vee y_i"
assert latex(x | y | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \vee y_i \vee z_i"
assert latex((x & y) | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"z_i \vee \left(x_i \wedge y_i\right)"
assert latex(Implies(x, y), symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \Rightarrow y_i"
assert latex(Pow(Rational(1, 3), -1, evaluate=False)) == r"\frac{1}{\frac{1}{3}}"
assert latex(Pow(Rational(1, 3), -2, evaluate=False)) == r"\frac{1}{(\frac{1}{3})^{2}}"
assert latex(Pow(Integer(1)/100, -1, evaluate=False)) == r"\frac{1}{\frac{1}{100}}"
p = Symbol('p', positive=True)
assert latex(exp(-p)*log(p)) == r"e^{- p} \log{\left(p \right)}"
def test_latex_builtins():
assert latex(True) == r"\text{True}"
assert latex(False) == r"\text{False}"
assert latex(None) == r"\text{None}"
assert latex(true) == r"\text{True}"
assert latex(false) == r'\text{False}'
def test_latex_SingularityFunction():
assert latex(SingularityFunction(x, 4, 5)) == \
r"{\left\langle x - 4 \right\rangle}^{5}"
assert latex(SingularityFunction(x, -3, 4)) == \
r"{\left\langle x + 3 \right\rangle}^{4}"
assert latex(SingularityFunction(x, 0, 4)) == \
r"{\left\langle x \right\rangle}^{4}"
assert latex(SingularityFunction(x, a, n)) == \
r"{\left\langle - a + x \right\rangle}^{n}"
assert latex(SingularityFunction(x, 4, -2)) == \
r"{\left\langle x - 4 \right\rangle}^{-2}"
assert latex(SingularityFunction(x, 4, -1)) == \
r"{\left\langle x - 4 \right\rangle}^{-1}"
assert latex(SingularityFunction(x, 4, 5)**3) == \
r"{\left({\langle x - 4 \rangle}^{5}\right)}^{3}"
assert latex(SingularityFunction(x, -3, 4)**3) == \
r"{\left({\langle x + 3 \rangle}^{4}\right)}^{3}"
assert latex(SingularityFunction(x, 0, 4)**3) == \
r"{\left({\langle x \rangle}^{4}\right)}^{3}"
assert latex(SingularityFunction(x, a, n)**3) == \
r"{\left({\langle - a + x \rangle}^{n}\right)}^{3}"
assert latex(SingularityFunction(x, 4, -2)**3) == \
r"{\left({\langle x - 4 \rangle}^{-2}\right)}^{3}"
assert latex((SingularityFunction(x, 4, -1)**3)**3) == \
r"{\left({\langle x - 4 \rangle}^{-1}\right)}^{9}"
def test_latex_cycle():
assert latex(Cycle(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Cycle(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Cycle()) == r"\left( \right)"
def test_latex_permutation():
assert latex(Permutation(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Permutation(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Permutation()) == r"\left( \right)"
assert latex(Permutation(2, 4)*Permutation(5)) == \
r"\left( 2\; 4\right)\left( 5\right)"
assert latex(Permutation(5)) == r"\left( 5\right)"
assert latex(Permutation(0, 1), perm_cyclic=False) == \
r"\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}"
assert latex(Permutation(0, 1)(2, 3), perm_cyclic=False) == \
r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}"
assert latex(Permutation(), perm_cyclic=False) == \
r"\left( \right)"
with warns_deprecated_sympy():
old_print_cyclic = Permutation.print_cyclic
Permutation.print_cyclic = False
assert latex(Permutation(0, 1)(2, 3)) == \
r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}"
Permutation.print_cyclic = old_print_cyclic
def test_latex_Float():
assert latex(Float(1.0e100)) == r"1.0 \cdot 10^{100}"
assert latex(Float(1.0e-100)) == r"1.0 \cdot 10^{-100}"
assert latex(Float(1.0e-100), mul_symbol="times") == \
r"1.0 \times 10^{-100}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=2) == \
r"1.0 \cdot 10^{4}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=4) == \
r"1.0 \cdot 10^{4}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=5) == \
r"10000.0"
assert latex(Float('0.099999'), full_prec=True, min=-2, max=5) == \
r"9.99990000000000 \cdot 10^{-2}"
def test_latex_vector_expressions():
A = CoordSys3D('A')
assert latex(Cross(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Cross(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}"
assert latex(x*Cross(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}\right)"
assert latex(Cross(x*A.i, A.j)) == \
r'- \mathbf{\hat{j}_{A}} \times \left(\left(x\right)\mathbf{\hat{i}_{A}}\right)'
assert latex(Curl(3*A.x*A.j)) == \
r"\nabla\times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*A.x*A.j+A.i)) == \
r"\nabla\times \left(\mathbf{\hat{i}_{A}} + \left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*x*A.x*A.j)) == \
r"\nabla\times \left(\left(3 \mathbf{{x}_{A}} x\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Curl(3*A.x*A.j)) == \
r"x \left(\nabla\times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Divergence(3*A.x*A.j+A.i)) == \
r"\nabla\cdot \left(\mathbf{\hat{i}_{A}} + \left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(Divergence(3*A.x*A.j)) == \
r"\nabla\cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Divergence(3*A.x*A.j)) == \
r"x \left(\nabla\cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Dot(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Dot(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}"
assert latex(Dot(x*A.i, A.j)) == \
r"\mathbf{\hat{j}_{A}} \cdot \left(\left(x\right)\mathbf{\hat{i}_{A}}\right)"
assert latex(x*Dot(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}\right)"
assert latex(Gradient(A.x)) == r"\nabla \mathbf{{x}_{A}}"
assert latex(Gradient(A.x + 3*A.y)) == \
r"\nabla \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Gradient(A.x)) == r"x \left(\nabla \mathbf{{x}_{A}}\right)"
assert latex(Gradient(x*A.x)) == r"\nabla \left(\mathbf{{x}_{A}} x\right)"
assert latex(Laplacian(A.x)) == r"\Delta \mathbf{{x}_{A}}"
assert latex(Laplacian(A.x + 3*A.y)) == \
r"\Delta \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Laplacian(A.x)) == r"x \left(\Delta \mathbf{{x}_{A}}\right)"
assert latex(Laplacian(x*A.x)) == r"\Delta \left(\mathbf{{x}_{A}} x\right)"
def test_latex_symbols():
Gamma, lmbda, rho = symbols('Gamma, lambda, rho')
tau, Tau, TAU, taU = symbols('tau, Tau, TAU, taU')
assert latex(tau) == r"\tau"
assert latex(Tau) == r"T"
assert latex(TAU) == r"\tau"
assert latex(taU) == r"\tau"
# Check that all capitalized greek letters are handled explicitly
capitalized_letters = {l.capitalize() for l in greek_letters_set}
assert len(capitalized_letters - set(tex_greek_dictionary.keys())) == 0
assert latex(Gamma + lmbda) == r"\Gamma + \lambda"
assert latex(Gamma * lmbda) == r"\Gamma \lambda"
assert latex(Symbol('q1')) == r"q_{1}"
assert latex(Symbol('q21')) == r"q_{21}"
assert latex(Symbol('epsilon0')) == r"\epsilon_{0}"
assert latex(Symbol('omega1')) == r"\omega_{1}"
assert latex(Symbol('91')) == r"91"
assert latex(Symbol('alpha_new')) == r"\alpha_{new}"
assert latex(Symbol('C^orig')) == r"C^{orig}"
assert latex(Symbol('x^alpha')) == r"x^{\alpha}"
assert latex(Symbol('beta^alpha')) == r"\beta^{\alpha}"
assert latex(Symbol('e^Alpha')) == r"e^{A}"
assert latex(Symbol('omega_alpha^beta')) == r"\omega^{\beta}_{\alpha}"
assert latex(Symbol('omega') ** Symbol('beta')) == r"\omega^{\beta}"
@XFAIL
def test_latex_symbols_failing():
rho, mass, volume = symbols('rho, mass, volume')
assert latex(
volume * rho == mass) == r"\rho \mathrm{volume} = \mathrm{mass}"
assert latex(volume / mass * rho == 1) == \
r"\rho \mathrm{volume} {\mathrm{mass}}^{(-1)} = 1"
assert latex(mass**3 * volume**3) == \
r"{\mathrm{mass}}^{3} \cdot {\mathrm{volume}}^{3}"
@_both_exp_pow
def test_latex_functions():
assert latex(exp(x)) == r"e^{x}"
assert latex(exp(1) + exp(2)) == r"e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left(x \right)}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left(x,y \right)}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left(x,y,z \right)}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left(x \right)}'
mybeta = Function('beta')
# not to be confused with the beta function
assert latex(mybeta(x, y, z)) == r"\beta{\left(x,y,z \right)}"
assert latex(beta(x, y)) == r'\operatorname{B}\left(x, y\right)'
assert latex(beta(x, y)**2) == r'\operatorname{B}^{2}\left(x, y\right)'
assert latex(mybeta(x)) == r"\beta{\left(x \right)}"
assert latex(mybeta) == r"\beta"
g = Function('gamma')
# not to be confused with the gamma function
assert latex(g(x, y, z)) == r"\gamma{\left(x,y,z \right)}"
assert latex(g(x)) == r"\gamma{\left(x \right)}"
assert latex(g) == r"\gamma"
a1 = Function('a_1')
assert latex(a1) == r"\operatorname{a_{1}}"
assert latex(a1(x)) == r"\operatorname{a_{1}}{\left(x \right)}"
# issue 5868
omega1 = Function('omega1')
assert latex(omega1) == r"\omega_{1}"
assert latex(omega1(x)) == r"\omega_{1}{\left(x \right)}"
assert latex(sin(x)) == r"\sin{\left(x \right)}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left(x \right)}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(acsc(x), inv_trig_style="full") == \
r"\operatorname{arccsc}{\left(x \right)}"
assert latex(asinh(x), inv_trig_style="full") == \
r"\operatorname{arsinh}{\left(x \right)}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial(k)**2) == r"k!^{2}"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(subfactorial(k)**2) == r"\left(!k\right)^{2}"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(factorial2(k)**2) == r"k!!^{2}"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(binomial(2, k)**2) == r"{\binom{2}{k}}^{2}"
assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{k}"
assert latex(RisingFactorial(3, k)) == r"{3}^{\left(k\right)}"
assert latex(floor(x)) == r"\left\lfloor{x}\right\rfloor"
assert latex(ceiling(x)) == r"\left\lceil{x}\right\rceil"
assert latex(frac(x)) == r"\operatorname{frac}{\left(x\right)}"
assert latex(floor(x)**2) == r"\left\lfloor{x}\right\rfloor^{2}"
assert latex(ceiling(x)**2) == r"\left\lceil{x}\right\rceil^{2}"
assert latex(frac(x)**2) == r"\operatorname{frac}{\left(x\right)}^{2}"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left|{x}\right|"
assert latex(Abs(x)**2) == r"\left|{x}\right|^{2}"
assert latex(re(x)) == r"\operatorname{re}{\left(x\right)}"
assert latex(re(x + y)) == \
r"\operatorname{re}{\left(x\right)} + \operatorname{re}{\left(y\right)}"
assert latex(im(x)) == r"\operatorname{im}{\left(x\right)}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(conjugate(x)**2) == r"\overline{x}^{2}"
assert latex(conjugate(x**2)) == r"\overline{x}^{2}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma\left(w\right)"
assert latex(Order(x)) == r"O\left(x\right)"
assert latex(Order(x, x)) == r"O\left(x\right)"
assert latex(Order(x, (x, 0))) == r"O\left(x\right)"
assert latex(Order(x, (x, oo))) == r"O\left(x; x\rightarrow \infty\right)"
assert latex(Order(x - y, (x, y))) == \
r"O\left(x - y; x\rightarrow y\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, (x, oo), (y, oo))) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( \infty, \ \infty\right)\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(lowergamma(x, y)**2) == r'\gamma^{2}\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(uppergamma(x, y)**2) == r'\Gamma^{2}\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left(x \right)}'
assert latex(coth(x)) == r'\coth{\left(x \right)}'
assert latex(re(x)) == r'\operatorname{re}{\left(x\right)}'
assert latex(im(x)) == r'\operatorname{im}{\left(x\right)}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left(x \right)}'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(stieltjes(x)) == r"\gamma_{x}"
assert latex(stieltjes(x)**2) == r"\gamma_{x}^{2}"
assert latex(stieltjes(x, y)) == r"\gamma_{x}\left(y\right)"
assert latex(stieltjes(x, y)**2) == r"\gamma_{x}\left(y\right)^{2}"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left(x \right)}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left(x \right)}'
assert latex(expint(x, y)) == r'\operatorname{E}_{x}\left(y\right)'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left(x \right)}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left(x \right)}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left(x \right)}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}\left(x\right)'
assert latex(Chi(x)) == r'\operatorname{Chi}\left(x\right)'
assert latex(jacobi(n, a, b, x)) == \
r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == \
r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a, x)) == \
r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == \
r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n, x)**2) == \
r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n, x)**2) == \
r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a, x)) == \
r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == \
r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a, x)) == \
r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == \
r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(Ynm(n, m, theta, phi)**3) == \
r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(Znm(n, m, theta, phi)**3) == \
r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(polar_lift(0)) == \
r"\operatorname{polar\_lift}{\left(0 \right)}"
assert latex(polar_lift(0)**3) == \
r"\operatorname{polar\_lift}^{3}{\left(0 \right)}"
assert latex(totient(n)) == r'\phi\left(n\right)'
assert latex(totient(n) ** 2) == r'\left(\phi\left(n\right)\right)^{2}'
assert latex(reduced_totient(n)) == r'\lambda\left(n\right)'
assert latex(reduced_totient(n) ** 2) == \
r'\left(\lambda\left(n\right)\right)^{2}'
assert latex(divisor_sigma(x)) == r"\sigma\left(x\right)"
assert latex(divisor_sigma(x)**2) == r"\sigma^{2}\left(x\right)"
assert latex(divisor_sigma(x, y)) == r"\sigma_y\left(x\right)"
assert latex(divisor_sigma(x, y)**2) == r"\sigma^{2}_y\left(x\right)"
assert latex(udivisor_sigma(x)) == r"\sigma^*\left(x\right)"
assert latex(udivisor_sigma(x)**2) == r"\sigma^*^{2}\left(x\right)"
assert latex(udivisor_sigma(x, y)) == r"\sigma^*_y\left(x\right)"
assert latex(udivisor_sigma(x, y)**2) == r"\sigma^*^{2}_y\left(x\right)"
assert latex(primenu(n)) == r'\nu\left(n\right)'
assert latex(primenu(n) ** 2) == r'\left(\nu\left(n\right)\right)^{2}'
assert latex(primeomega(n)) == r'\Omega\left(n\right)'
assert latex(primeomega(n) ** 2) == \
r'\left(\Omega\left(n\right)\right)^{2}'
assert latex(LambertW(n)) == r'W\left(n\right)'
assert latex(LambertW(n, -1)) == r'W_{-1}\left(n\right)'
assert latex(LambertW(n, k)) == r'W_{k}\left(n\right)'
assert latex(LambertW(n) * LambertW(n)) == r"W^{2}\left(n\right)"
assert latex(Pow(LambertW(n), 2)) == r"W^{2}\left(n\right)"
assert latex(LambertW(n)**k) == r"W^{k}\left(n\right)"
assert latex(LambertW(n, k)**p) == r"W^{p}_{k}\left(n\right)"
assert latex(Mod(x, 7)) == r'x \bmod 7'
assert latex(Mod(x + 1, 7)) == r'\left(x + 1\right) \bmod 7'
assert latex(Mod(7, x + 1)) == r'7 \bmod \left(x + 1\right)'
assert latex(Mod(2 * x, 7)) == r'2 x \bmod 7'
assert latex(Mod(7, 2 * x)) == r'7 \bmod 2 x'
assert latex(Mod(x, 7) + 1) == r'\left(x \bmod 7\right) + 1'
assert latex(2 * Mod(x, 7)) == r'2 \left(x \bmod 7\right)'
assert latex(Mod(7, 2 * x)**n) == r'\left(7 \bmod 2 x\right)^{n}'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left(x \right)}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert latex(mygamma) == r"\operatorname{mygamma}"
assert latex(mygamma(x)) == r"\operatorname{mygamma}{\left(x \right)}"
def test_hyper_printing():
from sympy.abc import x, z
assert latex(meijerg(Tuple(pi, pi, x), Tuple(1),
(0, 1), Tuple(1, 2, 3/pi), z)) == \
r'{G_{4, 5}^{2, 3}\left(\begin{matrix} \pi, \pi, x & 1 \\0, 1 & 1, 2, '\
r'\frac{3}{\pi} \end{matrix} \middle| {z} \right)}'
assert latex(meijerg(Tuple(), Tuple(1), (0,), Tuple(), z)) == \
r'{G_{1, 1}^{1, 0}\left(\begin{matrix} & 1 \\0 & \end{matrix} \middle| {z} \right)}'
assert latex(hyper((x, 2), (3,), z)) == \
r'{{}_{2}F_{1}\left(\begin{matrix} x, 2 ' \
r'\\ 3 \end{matrix}\middle| {z} \right)}'
assert latex(hyper(Tuple(), Tuple(1), z)) == \
r'{{}_{0}F_{1}\left(\begin{matrix} ' \
r'\\ 1 \end{matrix}\middle| {z} \right)}'
def test_latex_bessel():
from sympy.functions.special.bessel import (besselj, bessely, besseli,
besselk, hankel1, hankel2,
jn, yn, hn1, hn2)
from sympy.abc import z
assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)'
assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)'
assert latex(besseli(n, z)) == r'I_{n}\left(z\right)'
assert latex(besselk(n, z)) == r'K_{n}\left(z\right)'
assert latex(hankel1(n, z**2)**2) == \
r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}'
assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)'
assert latex(jn(n, z)) == r'j_{n}\left(z\right)'
assert latex(yn(n, z)) == r'y_{n}\left(z\right)'
assert latex(hn1(n, z)) == r'h^{(1)}_{n}\left(z\right)'
assert latex(hn2(n, z)) == r'h^{(2)}_{n}\left(z\right)'
def test_latex_fresnel():
from sympy.functions.special.error_functions import (fresnels, fresnelc)
from sympy.abc import z
assert latex(fresnels(z)) == r'S\left(z\right)'
assert latex(fresnelc(z)) == r'C\left(z\right)'
assert latex(fresnels(z)**2) == r'S^{2}\left(z\right)'
assert latex(fresnelc(z)**2) == r'C^{2}\left(z\right)'
def test_latex_brackets():
assert latex((-1)**x) == r"\left(-1\right)^{x}"
def test_latex_indexed():
Psi_symbol = Symbol('Psi_0', complex=True, real=False)
Psi_indexed = IndexedBase(Symbol('Psi', complex=True, real=False))
symbol_latex = latex(Psi_symbol * conjugate(Psi_symbol))
indexed_latex = latex(Psi_indexed[0] * conjugate(Psi_indexed[0]))
# \\overline{{\\Psi}_{0}} {\\Psi}_{0} vs. \\Psi_{0} \\overline{\\Psi_{0}}
assert symbol_latex == r'\Psi_{0} \overline{\Psi_{0}}'
assert indexed_latex == r'\overline{{\Psi}_{0}} {\Psi}_{0}'
# Symbol('gamma') gives r'\gamma'
interval = '\\mathrel{..}\\nobreak'
assert latex(Indexed('x1', Symbol('i'))) == r'{x_{1}}_{i}'
assert latex(Indexed('x2', Idx('i'))) == r'{x_{2}}_{i}'
assert latex(Indexed('x3', Idx('i', Symbol('N')))) == r'{x_{3}}_{{i}_{0'+interval+'N - 1}}'
assert latex(Indexed('x3', Idx('i', Symbol('N')+1))) == r'{x_{3}}_{{i}_{0'+interval+'N}}'
assert latex(Indexed('x4', Idx('i', (Symbol('a'),Symbol('b'))))) == r'{x_{4}}_{{i}_{a'+interval+'b}}'
assert latex(IndexedBase('gamma')) == r'\gamma'
assert latex(IndexedBase('a b')) == r'a b'
assert latex(IndexedBase('a_b')) == r'a_{b}'
def test_latex_derivatives():
# regular "d" for ordinary derivatives
assert latex(diff(x**3, x, evaluate=False)) == \
r"\frac{d}{d x} x^{3}"
assert latex(diff(sin(x) + x**2, x, evaluate=False)) == \
r"\frac{d}{d x} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False))\
== \
r"\frac{d^{2}}{d x^{2}} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False), evaluate=False)) == \
r"\frac{d^{3}}{d x^{3}} \left(x^{2} + \sin{\left(x \right)}\right)"
# \partial for partial derivatives
assert latex(diff(sin(x * y), x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \sin{\left(x y \right)}"
assert latex(diff(sin(x * y) + x**2, x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial x^{3}} \left(x^{2} + \sin{\left(x y \right)}\right)"
# mixed partial derivatives
f = Function("f")
assert latex(diff(diff(f(x, y), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial y\partial x} " + latex(f(x, y))
assert latex(diff(diff(diff(f(x, y), x, evaluate=False), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial y\partial x^{2}} " + latex(f(x, y))
# for negative nested Derivative
assert latex(diff(-diff(y**2,x,evaluate=False),x,evaluate=False)) == r'\frac{d}{d x} \left(- \frac{d}{d x} y^{2}\right)'
assert latex(diff(diff(-diff(diff(y,x,evaluate=False),x,evaluate=False),x,evaluate=False),x,evaluate=False)) == \
r'\frac{d^{2}}{d x^{2}} \left(- \frac{d^{2}}{d x^{2}} y\right)'
# use ordinary d when one of the variables has been integrated out
assert latex(diff(Integral(exp(-x*y), (x, 0, oo)), y, evaluate=False)) == \
r"\frac{d}{d y} \int\limits_{0}^{\infty} e^{- x y}\, dx"
# Derivative wrapped in power:
assert latex(diff(x, x, evaluate=False)**2) == \
r"\left(\frac{d}{d x} x\right)^{2}"
assert latex(diff(f(x), x)**2) == \
r"\left(\frac{d}{d x} f{\left(x \right)}\right)^{2}"
assert latex(diff(f(x), (x, n))) == \
r"\frac{d^{n}}{d x^{n}} f{\left(x \right)}"
x1 = Symbol('x1')
x2 = Symbol('x2')
assert latex(diff(f(x1, x2), x1)) == r'\frac{\partial}{\partial x_{1}} f{\left(x_{1},x_{2} \right)}'
n1 = Symbol('n1')
assert latex(diff(f(x), (x, n1))) == r'\frac{d^{n_{1}}}{d x^{n_{1}}} f{\left(x \right)}'
n2 = Symbol('n2')
assert latex(diff(f(x), (x, Max(n1, n2)))) == \
r'\frac{d^{\max\left(n_{1}, n_{2}\right)}}{d x^{\max\left(n_{1}, n_{2}\right)}} f{\left(x \right)}'
# set diff operator
assert latex(diff(f(x), x), diff_operator="rd") == r'\frac{\mathrm{d}}{\mathrm{d} x} f{\left(x \right)}'
def test_latex_subs():
assert latex(Subs(x*y, (x, y), (1, 2))) == r'\left. x y \right|_{\substack{ x=1\\ y=2 }}'
def test_latex_integrals():
assert latex(Integral(log(x), x)) == r"\int \log{\left(x \right)}\, dx"
assert latex(Integral(x**2, (x, 0, 1))) == \
r"\int\limits_{0}^{1} x^{2}\, dx"
assert latex(Integral(x**2, (x, 10, 20))) == \
r"\int\limits_{10}^{20} x^{2}\, dx"
assert latex(Integral(y*x**2, (x, 0, 1), y)) == \
r"\int\int\limits_{0}^{1} x^{2} y\, dx\, dy"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*') == \
r"\begin{equation*}\int\int\limits_{0}^{1} x^{2} y\, dx\, dy\end{equation*}"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*', itex=True) \
== r"$$\int\int_{0}^{1} x^{2} y\, dx\, dy$$"
assert latex(Integral(x, (x, 0))) == r"\int\limits^{0} x\, dx"
assert latex(Integral(x*y, x, y)) == r"\iint x y\, dx\, dy"
assert latex(Integral(x*y*z, x, y, z)) == r"\iiint x y z\, dx\, dy\, dz"
assert latex(Integral(x*y*z*t, x, y, z, t)) == \
r"\iiiint t x y z\, dx\, dy\, dz\, dt"
assert latex(Integral(x, x, x, x, x, x, x)) == \
r"\int\int\int\int\int\int x\, dx\, dx\, dx\, dx\, dx\, dx"
assert latex(Integral(x, x, y, (z, 0, 1))) == \
r"\int\limits_{0}^{1}\int\int x\, dx\, dy\, dz"
# for negative nested Integral
assert latex(Integral(-Integral(y**2,x),x)) == \
r'\int \left(- \int y^{2}\, dx\right)\, dx'
assert latex(Integral(-Integral(-Integral(y,x),x),x)) == \
r'\int \left(- \int \left(- \int y\, dx\right)\, dx\right)\, dx'
# fix issue #10806
assert latex(Integral(z, z)**2) == r"\left(\int z\, dz\right)^{2}"
assert latex(Integral(x + z, z)) == r"\int \left(x + z\right)\, dz"
assert latex(Integral(x+z/2, z)) == \
r"\int \left(x + \frac{z}{2}\right)\, dz"
assert latex(Integral(x**y, z)) == r"\int x^{y}\, dz"
# set diff operator
assert latex(Integral(x, x), diff_operator="rd") == r'\int x\, \mathrm{d}x'
assert latex(Integral(x, (x, 0, 1)), diff_operator="rd") == r'\int\limits_{0}^{1} x\, \mathrm{d}x'
def test_latex_sets():
for s in (frozenset, set):
assert latex(s([x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
s = FiniteSet
assert latex(s(*[x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(*range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(*range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
def test_latex_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
assert latex(se) == r"SetExpr\left(\left[1, 3\right]\right)"
def test_latex_Range():
assert latex(Range(1, 51)) == r'\left\{1, 2, \ldots, 50\right\}'
assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}'
assert latex(Range(0, 3, 1)) == r'\left\{0, 1, 2\right\}'
assert latex(Range(0, 30, 1)) == r'\left\{0, 1, \ldots, 29\right\}'
assert latex(Range(30, 1, -1)) == r'\left\{30, 29, \ldots, 2\right\}'
assert latex(Range(0, oo, 2)) == r'\left\{0, 2, \ldots\right\}'
assert latex(Range(oo, -2, -2)) == r'\left\{\ldots, 2, 0\right\}'
assert latex(Range(-2, -oo, -1)) == r'\left\{-2, -3, \ldots\right\}'
assert latex(Range(-oo, oo)) == r'\left\{\ldots, -1, 0, 1, \ldots\right\}'
assert latex(Range(oo, -oo, -1)) == r'\left\{\ldots, 1, 0, -1, \ldots\right\}'
a, b, c = symbols('a:c')
assert latex(Range(a, b, c)) == r'\text{Range}\left(a, b, c\right)'
assert latex(Range(a, 10, 1)) == r'\text{Range}\left(a, 10\right)'
assert latex(Range(0, b, 1)) == r'\text{Range}\left(b\right)'
assert latex(Range(0, 10, c)) == r'\text{Range}\left(0, 10, c\right)'
i = Symbol('i', integer=True)
n = Symbol('n', negative=True, integer=True)
p = Symbol('p', positive=True, integer=True)
assert latex(Range(i, i + 3)) == r'\left\{i, i + 1, i + 2\right\}'
assert latex(Range(-oo, n, 2)) == r'\left\{\ldots, n - 4, n - 2\right\}'
assert latex(Range(p, oo)) == r'\left\{p, p + 1, \ldots\right\}'
# The following will work if __iter__ is improved
# assert latex(Range(-3, p + 7)) == r'\left\{-3, -2, \ldots, p + 6\right\}'
# Must have integer assumptions
assert latex(Range(a, a + 3)) == r'\text{Range}\left(a, a + 3\right)'
def test_latex_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
latex_str = r'\left[0, 1, 4, 9, \ldots\right]'
assert latex(s1) == latex_str
latex_str = r'\left[1, 2, 1, 2, \ldots\right]'
assert latex(s2) == latex_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
latex_str = r'\left[0, 1, 4\right]'
assert latex(s3) == latex_str
latex_str = r'\left[1, 2, 1\right]'
assert latex(s4) == latex_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
latex_str = r'\left[\ldots, 9, 4, 1, 0\right]'
assert latex(s5) == latex_str
latex_str = r'\left[\ldots, 2, 1, 2, 1\right]'
assert latex(s6) == latex_str
latex_str = r'\left[1, 3, 5, 11, \ldots\right]'
assert latex(SeqAdd(s1, s2)) == latex_str
latex_str = r'\left[1, 3, 5\right]'
assert latex(SeqAdd(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 11, 5, 3, 1\right]'
assert latex(SeqAdd(s5, s6)) == latex_str
latex_str = r'\left[0, 2, 4, 18, \ldots\right]'
assert latex(SeqMul(s1, s2)) == latex_str
latex_str = r'\left[0, 2, 4\right]'
assert latex(SeqMul(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 18, 4, 2, 0\right]'
assert latex(SeqMul(s5, s6)) == latex_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
latex_str = r'\left\{a^{2}\right\}_{a=0}^{x}'
assert latex(s7) == latex_str
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
latex_str = r'\left[0, b, 4 b\right]'
assert latex(s8) == latex_str
def test_latex_FourierSeries():
latex_str = \
r'2 \sin{\left(x \right)} - \sin{\left(2 x \right)} + \frac{2 \sin{\left(3 x \right)}}{3} + \ldots'
assert latex(fourier_series(x, (x, -pi, pi))) == latex_str
def test_latex_FormalPowerSeries():
latex_str = r'\sum_{k=1}^{\infty} - \frac{\left(-1\right)^{- k} x^{k}}{k}'
assert latex(fps(log(1 + x))) == latex_str
def test_latex_intervals():
a = Symbol('a', real=True)
assert latex(Interval(0, 0)) == r"\left\{0\right\}"
assert latex(Interval(0, a)) == r"\left[0, a\right]"
assert latex(Interval(0, a, False, False)) == r"\left[0, a\right]"
assert latex(Interval(0, a, True, False)) == r"\left(0, a\right]"
assert latex(Interval(0, a, False, True)) == r"\left[0, a\right)"
assert latex(Interval(0, a, True, True)) == r"\left(0, a\right)"
def test_latex_AccumuBounds():
a = Symbol('a', real=True)
assert latex(AccumBounds(0, 1)) == r"\left\langle 0, 1\right\rangle"
assert latex(AccumBounds(0, a)) == r"\left\langle 0, a\right\rangle"
assert latex(AccumBounds(a + 1, a + 2)) == \
r"\left\langle a + 1, a + 2\right\rangle"
def test_latex_emptyset():
assert latex(S.EmptySet) == r"\emptyset"
def test_latex_universalset():
assert latex(S.UniversalSet) == r"\mathbb{U}"
def test_latex_commutator():
A = Operator('A')
B = Operator('B')
comm = Commutator(B, A)
assert latex(comm.doit()) == r"- (A B - B A)"
def test_latex_union():
assert latex(Union(Interval(0, 1), Interval(2, 3))) == \
r"\left[0, 1\right] \cup \left[2, 3\right]"
assert latex(Union(Interval(1, 1), Interval(2, 2), Interval(3, 4))) == \
r"\left\{1, 2\right\} \cup \left[3, 4\right]"
def test_latex_intersection():
assert latex(Intersection(Interval(0, 1), Interval(x, y))) == \
r"\left[0, 1\right] \cap \left[x, y\right]"
def test_latex_symmetric_difference():
assert latex(SymmetricDifference(Interval(2, 5), Interval(4, 7),
evaluate=False)) == \
r'\left[2, 5\right] \triangle \left[4, 7\right]'
def test_latex_Complement():
assert latex(Complement(S.Reals, S.Naturals)) == \
r"\mathbb{R} \setminus \mathbb{N}"
def test_latex_productset():
line = Interval(0, 1)
bigline = Interval(0, 10)
fset = FiniteSet(1, 2, 3)
assert latex(line**2) == r"%s^{2}" % latex(line)
assert latex(line**10) == r"%s^{10}" % latex(line)
assert latex((line * bigline * fset).flatten()) == r"%s \times %s \times %s" % (
latex(line), latex(bigline), latex(fset))
def test_latex_powerset():
fset = FiniteSet(1, 2, 3)
assert latex(PowerSet(fset)) == r'\mathcal{P}\left(\left\{1, 2, 3\right\}\right)'
def test_latex_ordinals():
w = OrdinalOmega()
assert latex(w) == r"\omega"
wp = OmegaPower(2, 3)
assert latex(wp) == r'3 \omega^{2}'
assert latex(Ordinal(wp, OmegaPower(1, 1))) == r'3 \omega^{2} + \omega'
assert latex(Ordinal(OmegaPower(2, 1), OmegaPower(1, 2))) == r'\omega^{2} + 2 \omega'
def test_set_operators_parenthesis():
a, b, c, d = symbols('a:d')
A = FiniteSet(a)
B = FiniteSet(b)
C = FiniteSet(c)
D = FiniteSet(d)
U1 = Union(A, B, evaluate=False)
U2 = Union(C, D, evaluate=False)
I1 = Intersection(A, B, evaluate=False)
I2 = Intersection(C, D, evaluate=False)
C1 = Complement(A, B, evaluate=False)
C2 = Complement(C, D, evaluate=False)
D1 = SymmetricDifference(A, B, evaluate=False)
D2 = SymmetricDifference(C, D, evaluate=False)
# XXX ProductSet does not support evaluate keyword
P1 = ProductSet(A, B)
P2 = ProductSet(C, D)
assert latex(Intersection(A, U2, evaluate=False)) == \
r'\left\{a\right\} \cap ' \
r'\left(\left\{c\right\} \cup \left\{d\right\}\right)'
assert latex(Intersection(U1, U2, evaluate=False)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\cap \left(\left\{c\right\} \cup \left\{d\right\}\right)'
assert latex(Intersection(C1, C2, evaluate=False)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \cap \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(Intersection(D1, D2, evaluate=False)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \cap \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
assert latex(Intersection(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \
r'\cap \left(\left\{c\right\} \times ' \
r'\left\{d\right\}\right)'
assert latex(Union(A, I2, evaluate=False)) == \
r'\left\{a\right\} \cup ' \
r'\left(\left\{c\right\} \cap \left\{d\right\}\right)'
assert latex(Union(I1, I2, evaluate=False)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\cup \left(\left\{c\right\} \cap \left\{d\right\}\right)'
assert latex(Union(C1, C2, evaluate=False)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \cup \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(Union(D1, D2, evaluate=False)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \cup \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
assert latex(Union(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \
r'\cup \left(\left\{c\right\} \times ' \
r'\left\{d\right\}\right)'
assert latex(Complement(A, C2, evaluate=False)) == \
r'\left\{a\right\} \setminus \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(Complement(U1, U2, evaluate=False)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\setminus \left(\left\{c\right\} \cup ' \
r'\left\{d\right\}\right)'
assert latex(Complement(I1, I2, evaluate=False)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\setminus \left(\left\{c\right\} \cap ' \
r'\left\{d\right\}\right)'
assert latex(Complement(D1, D2, evaluate=False)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \setminus ' \
r'\left(\left\{c\right\} \triangle \left\{d\right\}\right)'
assert latex(Complement(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) '\
r'\setminus \left(\left\{c\right\} \times '\
r'\left\{d\right\}\right)'
assert latex(SymmetricDifference(A, D2, evaluate=False)) == \
r'\left\{a\right\} \triangle \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
assert latex(SymmetricDifference(U1, U2, evaluate=False)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\triangle \left(\left\{c\right\} \cup ' \
r'\left\{d\right\}\right)'
assert latex(SymmetricDifference(I1, I2, evaluate=False)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\triangle \left(\left\{c\right\} \cap ' \
r'\left\{d\right\}\right)'
assert latex(SymmetricDifference(C1, C2, evaluate=False)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \triangle ' \
r'\left(\left\{c\right\} \setminus \left\{d\right\}\right)'
assert latex(SymmetricDifference(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \
r'\triangle \left(\left\{c\right\} \times ' \
r'\left\{d\right\}\right)'
# XXX This can be incorrect since cartesian product is not associative
assert latex(ProductSet(A, P2).flatten()) == \
r'\left\{a\right\} \times \left\{c\right\} \times ' \
r'\left\{d\right\}'
assert latex(ProductSet(U1, U2)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\times \left(\left\{c\right\} \cup ' \
r'\left\{d\right\}\right)'
assert latex(ProductSet(I1, I2)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\times \left(\left\{c\right\} \cap ' \
r'\left\{d\right\}\right)'
assert latex(ProductSet(C1, C2)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \times \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(ProductSet(D1, D2)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \times \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
def test_latex_Complexes():
assert latex(S.Complexes) == r"\mathbb{C}"
def test_latex_Naturals():
assert latex(S.Naturals) == r"\mathbb{N}"
def test_latex_Naturals0():
assert latex(S.Naturals0) == r"\mathbb{N}_0"
def test_latex_Integers():
assert latex(S.Integers) == r"\mathbb{Z}"
def test_latex_ImageSet():
x = Symbol('x')
assert latex(ImageSet(Lambda(x, x**2), S.Naturals)) == \
r"\left\{x^{2}\; \middle|\; x \in \mathbb{N}\right\}"
y = Symbol('y')
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
assert latex(imgset) == \
r"\left\{x + y\; \middle|\; x \in \left\{1, 2, 3\right\}, y \in \left\{3, 4\right\}\right\}"
imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4}))
assert latex(imgset) == \
r"\left\{x + y\; \middle|\; \left( x, \ y\right) \in \left\{1, 2, 3\right\} \times \left\{3, 4\right\}\right\}"
def test_latex_ConditionSet():
x = Symbol('x')
assert latex(ConditionSet(x, Eq(x**2, 1), S.Reals)) == \
r"\left\{x\; \middle|\; x \in \mathbb{R} \wedge x^{2} = 1 \right\}"
assert latex(ConditionSet(x, Eq(x**2, 1), S.UniversalSet)) == \
r"\left\{x\; \middle|\; x^{2} = 1 \right\}"
def test_latex_ComplexRegion():
assert latex(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == \
r"\left\{x + y i\; \middle|\; x, y \in \left[3, 5\right] \times \left[4, 6\right] \right\}"
assert latex(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == \
r"\left\{r \left(i \sin{\left(\theta \right)} + \cos{\left(\theta "\
r"\right)}\right)\; \middle|\; r, \theta \in \left[0, 1\right] \times \left[0, 2 \pi\right) \right\}"
def test_latex_Contains():
x = Symbol('x')
assert latex(Contains(x, S.Naturals)) == r"x \in \mathbb{N}"
def test_latex_sum():
assert latex(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\sum_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Sum(x**2, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} x^{2}"
assert latex(Sum(x**2 + y, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Sum(x**2 + y, (x, -2, 2))**2) == \
r"\left(\sum_{x=-2}^{2} \left(x^{2} + y\right)\right)^{2}"
def test_latex_product():
assert latex(Product(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\prod_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Product(x**2, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} x^{2}"
assert latex(Product(x**2 + y, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Product(x, (x, -2, 2))**2) == \
r"\left(\prod_{x=-2}^{2} x\right)^{2}"
def test_latex_limits():
assert latex(Limit(x, x, oo)) == r"\lim_{x \to \infty} x"
# issue 8175
f = Function('f')
assert latex(Limit(f(x), x, 0)) == r"\lim_{x \to 0^+} f{\left(x \right)}"
assert latex(Limit(f(x), x, 0, "-")) == \
r"\lim_{x \to 0^-} f{\left(x \right)}"
# issue #10806
assert latex(Limit(f(x), x, 0)**2) == \
r"\left(\lim_{x \to 0^+} f{\left(x \right)}\right)^{2}"
# bi-directional limit
assert latex(Limit(f(x), x, 0, dir='+-')) == \
r"\lim_{x \to 0} f{\left(x \right)}"
def test_latex_log():
assert latex(log(x)) == r"\log{\left(x \right)}"
assert latex(log(x), ln_notation=True) == r"\ln{\left(x \right)}"
assert latex(log(x) + log(y)) == \
r"\log{\left(x \right)} + \log{\left(y \right)}"
assert latex(log(x) + log(y), ln_notation=True) == \
r"\ln{\left(x \right)} + \ln{\left(y \right)}"
assert latex(pow(log(x), x)) == r"\log{\left(x \right)}^{x}"
assert latex(pow(log(x), x), ln_notation=True) == \
r"\ln{\left(x \right)}^{x}"
def test_issue_3568():
beta = Symbol(r'\beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
beta = Symbol(r'beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
def test_latex():
assert latex((2*tau)**Rational(7, 2)) == r"8 \sqrt{2} \tau^{\frac{7}{2}}"
assert latex((2*mu)**Rational(7, 2), mode='equation*') == \
r"\begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*}"
assert latex((2*mu)**Rational(7, 2), mode='equation', itex=True) == \
r"$$8 \sqrt{2} \mu^{\frac{7}{2}}$$"
assert latex([2/x, y]) == r"\left[ \frac{2}{x}, \ y\right]"
def test_latex_dict():
d = {Rational(1): 1, x**2: 2, x: 3, x**3: 4}
assert latex(d) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
D = Dict(d)
assert latex(D) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
def test_latex_list():
ll = [Symbol('omega1'), Symbol('a'), Symbol('alpha')]
assert latex(ll) == r'\left[ \omega_{1}, \ a, \ \alpha\right]'
def test_latex_NumberSymbols():
assert latex(S.Catalan) == "G"
assert latex(S.EulerGamma) == r"\gamma"
assert latex(S.Exp1) == "e"
assert latex(S.GoldenRatio) == r"\phi"
assert latex(S.Pi) == r"\pi"
assert latex(S.TribonacciConstant) == r"\text{TribonacciConstant}"
def test_latex_rational():
# tests issue 3973
assert latex(-Rational(1, 2)) == r"- \frac{1}{2}"
assert latex(Rational(-1, 2)) == r"- \frac{1}{2}"
assert latex(Rational(1, -2)) == r"- \frac{1}{2}"
assert latex(-Rational(-1, 2)) == r"\frac{1}{2}"
assert latex(-Rational(1, 2)*x) == r"- \frac{x}{2}"
assert latex(-Rational(1, 2)*x + Rational(-2, 3)*y) == \
r"- \frac{x}{2} - \frac{2 y}{3}"
def test_latex_inverse():
# tests issue 4129
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/(x + y)) == r"\frac{1}{x + y}"
def test_latex_DiracDelta():
assert latex(DiracDelta(x)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x)**2) == r"\left(\delta\left(x\right)\right)^{2}"
assert latex(DiracDelta(x, 0)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x, 5)) == \
r"\delta^{\left( 5 \right)}\left( x \right)"
assert latex(DiracDelta(x, 5)**2) == \
r"\left(\delta^{\left( 5 \right)}\left( x \right)\right)^{2}"
def test_latex_Heaviside():
assert latex(Heaviside(x)) == r"\theta\left(x\right)"
assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}"
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 6578
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
assert latex(Pow(KroneckerDelta(x, y), 2, evaluate=False)) == \
r"\left(\delta_{x y}\right)^{2}"
def test_latex_LeviCivita():
assert latex(LeviCivita(x, y, z)) == r"\varepsilon_{x y z}"
assert latex(LeviCivita(x, y, z)**2) == \
r"\left(\varepsilon_{x y z}\right)^{2}"
assert latex(LeviCivita(x, y, z + 1)) == r"\varepsilon_{x, y, z + 1}"
assert latex(LeviCivita(x, y + 1, z)) == r"\varepsilon_{x, y + 1, z}"
assert latex(LeviCivita(x + 1, y, z)) == r"\varepsilon_{x + 1, y, z}"
def test_mode():
expr = x + y
assert latex(expr) == r'x + y'
assert latex(expr, mode='plain') == r'x + y'
assert latex(expr, mode='inline') == r'$x + y$'
assert latex(
expr, mode='equation*') == r'\begin{equation*}x + y\end{equation*}'
assert latex(
expr, mode='equation') == r'\begin{equation}x + y\end{equation}'
raises(ValueError, lambda: latex(expr, mode='foo'))
def test_latex_mathieu():
assert latex(mathieuc(x, y, z)) == r"C\left(x, y, z\right)"
assert latex(mathieus(x, y, z)) == r"S\left(x, y, z\right)"
assert latex(mathieuc(x, y, z)**2) == r"C\left(x, y, z\right)^{2}"
assert latex(mathieus(x, y, z)**2) == r"S\left(x, y, z\right)^{2}"
assert latex(mathieucprime(x, y, z)) == r"C^{\prime}\left(x, y, z\right)"
assert latex(mathieusprime(x, y, z)) == r"S^{\prime}\left(x, y, z\right)"
assert latex(mathieucprime(x, y, z)**2) == r"C^{\prime}\left(x, y, z\right)^{2}"
assert latex(mathieusprime(x, y, z)**2) == r"S^{\prime}\left(x, y, z\right)^{2}"
def test_latex_Piecewise():
p = Piecewise((x, x < 1), (x**2, True))
assert latex(p) == r"\begin{cases} x & \text{for}\: x < 1 \\x^{2} &" \
r" \text{otherwise} \end{cases}"
assert latex(p, itex=True) == \
r"\begin{cases} x & \text{for}\: x \lt 1 \\x^{2} &" \
r" \text{otherwise} \end{cases}"
p = Piecewise((x, x < 0), (0, x >= 0))
assert latex(p) == r'\begin{cases} x & \text{for}\: x < 0 \\0 &' \
r' \text{otherwise} \end{cases}'
A, B = symbols("A B", commutative=False)
p = Piecewise((A**2, Eq(A, B)), (A*B, True))
s = r"\begin{cases} A^{2} & \text{for}\: A = B \\A B & \text{otherwise} \end{cases}"
assert latex(p) == s
assert latex(A*p) == r"A \left(%s\right)" % s
assert latex(p*A) == r"\left(%s\right) A" % s
assert latex(Piecewise((x, x < 1), (x**2, x < 2))) == \
r'\begin{cases} x & ' \
r'\text{for}\: x < 1 \\x^{2} & \text{for}\: x < 2 \end{cases}'
def test_latex_Matrix():
M = Matrix([[1 + x, y], [y, x - 1]])
assert latex(M) == \
r'\left[\begin{matrix}x + 1 & y\\y & x - 1\end{matrix}\right]'
assert latex(M, mode='inline') == \
r'$\left[\begin{smallmatrix}x + 1 & y\\' \
r'y & x - 1\end{smallmatrix}\right]$'
assert latex(M, mat_str='array') == \
r'\left[\begin{array}{cc}x + 1 & y\\y & x - 1\end{array}\right]'
assert latex(M, mat_str='bmatrix') == \
r'\left[\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}\right]'
assert latex(M, mat_delim=None, mat_str='bmatrix') == \
r'\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}'
M2 = Matrix(1, 11, range(11))
assert latex(M2) == \
r'\left[\begin{array}{ccccccccccc}' \
r'0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]'
def test_latex_matrix_with_functions():
t = symbols('t')
theta1 = symbols('theta1', cls=Function)
M = Matrix([[sin(theta1(t)), cos(theta1(t))],
[cos(theta1(t).diff(t)), sin(theta1(t).diff(t))]])
expected = (r'\left[\begin{matrix}\sin{\left('
r'\theta_{1}{\left(t \right)} \right)} & '
r'\cos{\left(\theta_{1}{\left(t \right)} \right)'
r'}\\\cos{\left(\frac{d}{d t} \theta_{1}{\left(t '
r'\right)} \right)} & \sin{\left(\frac{d}{d t} '
r'\theta_{1}{\left(t \right)} \right'
r')}\end{matrix}\right]')
assert latex(M) == expected
def test_latex_NDimArray():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert latex(M) == r"x"
M = ArrayType([[1 / x, y], [z, w]])
M1 = ArrayType([1 / x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
assert latex(M) == \
r'\left[\begin{matrix}\frac{1}{x} & y\\z & w\end{matrix}\right]'
assert latex(M1) == \
r"\left[\begin{matrix}\frac{1}{x} & y & z\end{matrix}\right]"
assert latex(M2) == \
r"\left[\begin{matrix}" \
r"\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right]" \
r"\end{matrix}\right]"
assert latex(M3) == \
r"""\left[\begin{matrix}"""\
r"""\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right]\\"""\
r"""\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{w}{x} & w y\\w z & w^{2}\end{matrix}\right]"""\
r"""\end{matrix}\right]"""
Mrow = ArrayType([[x, y, 1/z]])
Mcolumn = ArrayType([[x], [y], [1/z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
assert latex(Mrow) == \
r"\left[\left[\begin{matrix}x & y & \frac{1}{z}\end{matrix}\right]\right]"
assert latex(Mcolumn) == \
r"\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]"
assert latex(Mcol2) == \
r'\left[\begin{matrix}\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]\end{matrix}\right]'
def test_latex_mul_symbol():
assert latex(4*4**x, mul_symbol='times') == r"4 \times 4^{x}"
assert latex(4*4**x, mul_symbol='dot') == r"4 \cdot 4^{x}"
assert latex(4*4**x, mul_symbol='ldot') == r"4 \,.\, 4^{x}"
assert latex(4*x, mul_symbol='times') == r"4 \times x"
assert latex(4*x, mul_symbol='dot') == r"4 \cdot x"
assert latex(4*x, mul_symbol='ldot') == r"4 \,.\, x"
def test_latex_issue_4381():
y = 4*4**log(2)
assert latex(y) == r'4 \cdot 4^{\log{\left(2 \right)}}'
assert latex(1/y) == r'\frac{1}{4 \cdot 4^{\log{\left(2 \right)}}}'
def test_latex_issue_4576():
assert latex(Symbol("beta_13_2")) == r"\beta_{13 2}"
assert latex(Symbol("beta_132_20")) == r"\beta_{132 20}"
assert latex(Symbol("beta_13")) == r"\beta_{13}"
assert latex(Symbol("x_a_b")) == r"x_{a b}"
assert latex(Symbol("x_1_2_3")) == r"x_{1 2 3}"
assert latex(Symbol("x_a_b1")) == r"x_{a b1}"
assert latex(Symbol("x_a_1")) == r"x_{a 1}"
assert latex(Symbol("x_1_a")) == r"x_{1 a}"
assert latex(Symbol("x_1^aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_1__aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_11^a")) == r"x^{a}_{11}"
assert latex(Symbol("x_11__a")) == r"x^{a}_{11}"
assert latex(Symbol("x_a_a_a_a")) == r"x_{a a a a}"
assert latex(Symbol("x_a_a^a^a")) == r"x^{a a}_{a a}"
assert latex(Symbol("x_a_a__a__a")) == r"x^{a a}_{a a}"
assert latex(Symbol("alpha_11")) == r"\alpha_{11}"
assert latex(Symbol("alpha_11_11")) == r"\alpha_{11 11}"
assert latex(Symbol("alpha_alpha")) == r"\alpha_{\alpha}"
assert latex(Symbol("alpha^aleph")) == r"\alpha^{\aleph}"
assert latex(Symbol("alpha__aleph")) == r"\alpha^{\aleph}"
def test_latex_pow_fraction():
x = Symbol('x')
# Testing exp
assert r'e^{-x}' in latex(exp(-x)/2).replace(' ', '') # Remove Whitespace
# Testing e^{-x} in case future changes alter behavior of muls or fracs
# In particular current output is \frac{1}{2}e^{- x} but perhaps this will
# change to \frac{e^{-x}}{2}
# Testing general, non-exp, power
assert r'3^{-x}' in latex(3**-x/2).replace(' ', '')
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert latex(A*B*C**-1) == r"A B C^{-1}"
assert latex(C**-1*A*B) == r"C^{-1} A B"
assert latex(A*C**-1*B) == r"A C^{-1} B"
def test_latex_order():
expr = x**3 + x**2*y + y**4 + 3*x*y**3
assert latex(expr, order='lex') == r"x^{3} + x^{2} y + 3 x y^{3} + y^{4}"
assert latex(
expr, order='rev-lex') == r"y^{4} + 3 x y^{3} + x^{2} y + x^{3}"
assert latex(expr, order='none') == r"x^{3} + y^{4} + y x^{2} + 3 x y^{3}"
def test_latex_Lambda():
assert latex(Lambda(x, x + 1)) == r"\left( x \mapsto x + 1 \right)"
assert latex(Lambda((x, y), x + 1)) == r"\left( \left( x, \ y\right) \mapsto x + 1 \right)"
assert latex(Lambda(x, x)) == r"\left( x \mapsto x \right)"
def test_latex_PolyElement():
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == \
r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
assert latex(-(v**2 + v + 1)*x + 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x + 3 u v + 1"
assert latex(-(v**2 + v + 1)*x - 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x - 3 u v + 1"
def test_latex_FracElement():
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex(x/3) == r"\frac{x}{3}"
assert latex(x/z) == r"\frac{x}{z}"
assert latex(x*y/z) == r"\frac{x y}{z}"
assert latex(x/(z*t)) == r"\frac{x}{z t}"
assert latex(x*y/(z*t)) == r"\frac{x y}{z t}"
assert latex((x - 1)/y) == r"\frac{x - 1}{y}"
assert latex((x + 1)/y) == r"\frac{x + 1}{y}"
assert latex((-x - 1)/y) == r"\frac{-x - 1}{y}"
assert latex((x + 1)/(y*z)) == r"\frac{x + 1}{y z}"
assert latex(-y/(x + 1)) == r"\frac{-y}{x + 1}"
assert latex(y*z/(x + 1)) == r"\frac{y z}{x + 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - u v t - 1}"
def test_latex_Poly():
assert latex(Poly(x**2 + 2 * x, x)) == \
r"\operatorname{Poly}{\left( x^{2} + 2 x, x, domain=\mathbb{Z} \right)}"
assert latex(Poly(x/y, x)) == \
r"\operatorname{Poly}{\left( \frac{1}{y} x, x, domain=\mathbb{Z}\left(y\right) \right)}"
assert latex(Poly(2.0*x + y)) == \
r"\operatorname{Poly}{\left( 2.0 x + 1.0 y, x, y, domain=\mathbb{R} \right)}"
def test_latex_Poly_order():
assert latex(Poly([a, 1, b, 2, c, 3], x)) == \
r'\operatorname{Poly}{\left( a x^{5} + x^{4} + b x^{3} + 2 x^{2} + c'\
r' x + 3, x, domain=\mathbb{Z}\left[a, b, c\right] \right)}'
assert latex(Poly([a, 1, b+c, 2, 3], x)) == \
r'\operatorname{Poly}{\left( a x^{4} + x^{3} + \left(b + c\right) '\
r'x^{2} + 2 x + 3, x, domain=\mathbb{Z}\left[a, b, c\right] \right)}'
assert latex(Poly(a*x**3 + x**2*y - x*y - c*y**3 - b*x*y**2 + y - a*x + b,
(x, y))) == \
r'\operatorname{Poly}{\left( a x^{3} + x^{2}y - b xy^{2} - xy - '\
r'a x - c y^{3} + y + b, x, y, domain=\mathbb{Z}\left[a, b, c\right] \right)}'
def test_latex_ComplexRootOf():
assert latex(rootof(x**5 + x + 3, 0)) == \
r"\operatorname{CRootOf} {\left(x^{5} + x + 3, 0\right)}"
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum} {\left(x^{5} + x + 3, \left( x \mapsto \sin{\left(x \right)} \right)\right)}"
def test_settings():
raises(TypeError, lambda: latex(x*y, method="garbage"))
def test_latex_numbers():
assert latex(catalan(n)) == r"C_{n}"
assert latex(catalan(n)**2) == r"C_{n}^{2}"
assert latex(bernoulli(n)) == r"B_{n}"
assert latex(bernoulli(n, x)) == r"B_{n}\left(x\right)"
assert latex(bernoulli(n)**2) == r"B_{n}^{2}"
assert latex(bernoulli(n, x)**2) == r"B_{n}^{2}\left(x\right)"
assert latex(bell(n)) == r"B_{n}"
assert latex(bell(n, x)) == r"B_{n}\left(x\right)"
assert latex(bell(n, m, (x, y))) == r"B_{n, m}\left(x, y\right)"
assert latex(bell(n)**2) == r"B_{n}^{2}"
assert latex(bell(n, x)**2) == r"B_{n}^{2}\left(x\right)"
assert latex(bell(n, m, (x, y))**2) == r"B_{n, m}^{2}\left(x, y\right)"
assert latex(fibonacci(n)) == r"F_{n}"
assert latex(fibonacci(n, x)) == r"F_{n}\left(x\right)"
assert latex(fibonacci(n)**2) == r"F_{n}^{2}"
assert latex(fibonacci(n, x)**2) == r"F_{n}^{2}\left(x\right)"
assert latex(lucas(n)) == r"L_{n}"
assert latex(lucas(n)**2) == r"L_{n}^{2}"
assert latex(tribonacci(n)) == r"T_{n}"
assert latex(tribonacci(n, x)) == r"T_{n}\left(x\right)"
assert latex(tribonacci(n)**2) == r"T_{n}^{2}"
assert latex(tribonacci(n, x)**2) == r"T_{n}^{2}\left(x\right)"
def test_latex_euler():
assert latex(euler(n)) == r"E_{n}"
assert latex(euler(n, x)) == r"E_{n}\left(x\right)"
assert latex(euler(n, x)**2) == r"E_{n}^{2}\left(x\right)"
def test_lamda():
assert latex(Symbol('lamda')) == r"\lambda"
assert latex(Symbol('Lamda')) == r"\Lambda"
def test_custom_symbol_names():
x = Symbol('x')
y = Symbol('y')
assert latex(x) == r"x"
assert latex(x, symbol_names={x: "x_i"}) == r"x_i"
assert latex(x + y, symbol_names={x: "x_i"}) == r"x_i + y"
assert latex(x**2, symbol_names={x: "x_i"}) == r"x_i^{2}"
assert latex(x + y, symbol_names={x: "x_i", y: "y_j"}) == r"x_i + y_j"
def test_matAdd():
C = MatrixSymbol('C', 5, 5)
B = MatrixSymbol('B', 5, 5)
l = LatexPrinter()
assert l._print(C - 2*B) in [r'- 2 B + C', r'C -2 B']
assert l._print(C + 2*B) in [r'2 B + C', r'C + 2 B']
assert l._print(B - 2*C) in [r'B - 2 C', r'- 2 C + B']
assert l._print(B + 2*C) in [r'B + 2 C', r'2 C + B']
def test_matMul():
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
x = Symbol('x')
lp = LatexPrinter()
assert lp._print_MatMul(2*A) == r'2 A'
assert lp._print_MatMul(2*x*A) == r'2 x A'
assert lp._print_MatMul(-2*A) == r'- 2 A'
assert lp._print_MatMul(1.5*A) == r'1.5 A'
assert lp._print_MatMul(sqrt(2)*A) == r'\sqrt{2} A'
assert lp._print_MatMul(-sqrt(2)*A) == r'- \sqrt{2} A'
assert lp._print_MatMul(2*sqrt(2)*x*A) == r'2 \sqrt{2} x A'
assert lp._print_MatMul(-2*A*(A + 2*B)) in [r'- 2 A \left(A + 2 B\right)',
r'- 2 A \left(2 B + A\right)']
def test_latex_MatrixSlice():
n = Symbol('n', integer=True)
x, y, z, w, t, = symbols('x y z w t')
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
assert latex(MatrixSlice(X, (None, None, None), (None, None, None))) == r'X\left[:, :\right]'
assert latex(X[x:x + 1, y:y + 1]) == r'X\left[x:x + 1, y:y + 1\right]'
assert latex(X[x:x + 1:2, y:y + 1:2]) == r'X\left[x:x + 1:2, y:y + 1:2\right]'
assert latex(X[:x, y:]) == r'X\left[:x, y:\right]'
assert latex(X[:x, y:]) == r'X\left[:x, y:\right]'
assert latex(X[x:, :y]) == r'X\left[x:, :y\right]'
assert latex(X[x:y, z:w]) == r'X\left[x:y, z:w\right]'
assert latex(X[x:y:t, w:t:x]) == r'X\left[x:y:t, w:t:x\right]'
assert latex(X[x::y, t::w]) == r'X\left[x::y, t::w\right]'
assert latex(X[:x:y, :t:w]) == r'X\left[:x:y, :t:w\right]'
assert latex(X[::x, ::y]) == r'X\left[::x, ::y\right]'
assert latex(MatrixSlice(X, (0, None, None), (0, None, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (None, n, None), (None, n, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (0, n, None), (0, n, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (0, n, 2), (0, n, 2))) == r'X\left[::2, ::2\right]'
assert latex(X[1:2:3, 4:5:6]) == r'X\left[1:2:3, 4:5:6\right]'
assert latex(X[1:3:5, 4:6:8]) == r'X\left[1:3:5, 4:6:8\right]'
assert latex(X[1:10:2]) == r'X\left[1:10:2, :\right]'
assert latex(Y[:5, 1:9:2]) == r'Y\left[:5, 1:9:2\right]'
assert latex(Y[:5, 1:10:2]) == r'Y\left[:5, 1::2\right]'
assert latex(Y[5, :5:2]) == r'Y\left[5:6, :5:2\right]'
assert latex(X[0:1, 0:1]) == r'X\left[:1, :1\right]'
assert latex(X[0:1:2, 0:1:2]) == r'X\left[:1:2, :1:2\right]'
assert latex((Y + Z)[2:, 2:]) == r'\left(Y + Z\right)\left[2:, 2:\right]'
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
from sympy.stats.rv import RandomDomain
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == r"\text{Domain: }0 < x_{1} \wedge x_{1} < \infty"
D = Die('d1', 6)
assert latex(where(D > 4)) == r"\text{Domain: }d_{1} = 5 \vee d_{1} = 6"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert latex(
pspace(Tuple(A, B)).domain) == \
r"\text{Domain: }0 \leq a \wedge 0 \leq b \wedge a < \infty \wedge b < \infty"
assert latex(RandomDomain(FiniteSet(x), FiniteSet(1, 2))) == \
r'\text{Domain: }\left\{x\right\} \in \left\{1, 2\right\}'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert latex(F.convert(x/(x + y))) == latex(x/(x + y))
assert latex(R.convert(x + y)) == latex(x + y)
def test_integral_transforms():
x = Symbol("x")
k = Symbol("k")
f = Function("f")
a = Symbol("a")
b = Symbol("b")
assert latex(MellinTransform(f(x), x, k)) == \
r"\mathcal{M}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseMellinTransform(f(k), k, x, a, b)) == \
r"\mathcal{M}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(LaplaceTransform(f(x), x, k)) == \
r"\mathcal{L}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseLaplaceTransform(f(k), k, x, (a, b))) == \
r"\mathcal{L}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(FourierTransform(f(x), x, k)) == \
r"\mathcal{F}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseFourierTransform(f(k), k, x)) == \
r"\mathcal{F}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(CosineTransform(f(x), x, k)) == \
r"\mathcal{COS}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseCosineTransform(f(k), k, x)) == \
r"\mathcal{COS}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(SineTransform(f(x), x, k)) == \
r"\mathcal{SIN}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseSineTransform(f(k), k, x)) == \
r"\mathcal{SIN}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
def test_PolynomialRingBase():
from sympy.polys.domains import QQ
assert latex(QQ.old_poly_ring(x, y)) == r"\mathbb{Q}\left[x, y\right]"
assert latex(QQ.old_poly_ring(x, y, order="ilex")) == \
r"S_<^{-1}\mathbb{Q}\left[x, y\right]"
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram,
DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert latex(A1) == r"A_{1}"
assert latex(f1) == r"f_{1}:A_{1}\rightarrow A_{2}"
assert latex(id_A1) == r"id:A_{1}\rightarrow A_{1}"
assert latex(f2*f1) == r"f_{2}\circ f_{1}:A_{1}\rightarrow A_{3}"
assert latex(K1) == r"\mathbf{K_{1}}"
d = Diagram()
assert latex(d) == r"\emptyset"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}, " \
r"\ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}," \
r" \ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" \
r"\Longrightarrow \left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \left\{unique\right\}\right\}"
# A linear diagram.
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert latex(grid) == r"\begin{array}{cc}" + "\n" \
r"A & B \\" + "\n" \
r" & C " + "\n" \
r"\end{array}" + "\n"
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == \
r"\left\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle"
I = R.ideal(x**2, y)
assert latex(I) == r"\left\langle {x^{2}},{y} \right\rangle"
Q = F / M
assert latex(Q) == \
r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left\langle {\left[ {x},"\
r"{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}"
assert latex(Q.submodule([1, x**3/2], [2, y])) == \
r"\left\langle {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left"\
r"\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} "\
r"\right\rangle}},{{\left[ {2},{y} \right]} + {\left\langle {\left[ "\
r"{x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}} \right\rangle"
h = homomorphism(QQ.old_poly_ring(x).free_module(2),
QQ.old_poly_ring(x).free_module(2), [0, 0])
assert latex(h) == \
r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : "\
r"{{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
def test_QuotientRing():
from sympy.polys.domains import QQ
R = QQ.old_poly_ring(x)/[x**2 + 1]
assert latex(R) == \
r"\frac{\mathbb{Q}\left[x\right]}{\left\langle {x^{2} + 1} \right\rangle}"
assert latex(R.one) == r"{1} + {\left\langle {x^{2} + 1} \right\rangle}"
def test_Tr():
#TODO: Handle indices
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert latex(t) == r'\operatorname{tr}\left(A B\right)'
def test_Adjoint():
from sympy.matrices import Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Adjoint(X)) == r'X^{\dagger}'
assert latex(Adjoint(X + Y)) == r'\left(X + Y\right)^{\dagger}'
assert latex(Adjoint(X) + Adjoint(Y)) == r'X^{\dagger} + Y^{\dagger}'
assert latex(Adjoint(X*Y)) == r'\left(X Y\right)^{\dagger}'
assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}'
assert latex(Adjoint(X**2)) == r'\left(X^{2}\right)^{\dagger}'
assert latex(Adjoint(X)**2) == r'\left(X^{\dagger}\right)^{2}'
assert latex(Adjoint(Inverse(X))) == r'\left(X^{-1}\right)^{\dagger}'
assert latex(Inverse(Adjoint(X))) == r'\left(X^{\dagger}\right)^{-1}'
assert latex(Adjoint(Transpose(X))) == r'\left(X^{T}\right)^{\dagger}'
assert latex(Transpose(Adjoint(X))) == r'\left(X^{\dagger}\right)^{T}'
assert latex(Transpose(Adjoint(X) + Y)) == r'\left(X^{\dagger} + Y\right)^{T}'
m = Matrix(((1, 2), (3, 4)))
assert latex(Adjoint(m)) == '\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{\\dagger}'
assert latex(Adjoint(m+X)) == \
'\\left(\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X\\right)^{\\dagger}'
# Issue 20959
Mx = MatrixSymbol('M^x', 2, 2)
assert latex(Adjoint(Mx)) == r'\left(M^{x}\right)^{\dagger}'
def test_Transpose():
from sympy.matrices import Transpose, MatPow, HadamardPower
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Transpose(X)) == r'X^{T}'
assert latex(Transpose(X + Y)) == r'\left(X + Y\right)^{T}'
assert latex(Transpose(HadamardPower(X, 2))) == r'\left(X^{\circ {2}}\right)^{T}'
assert latex(HadamardPower(Transpose(X), 2)) == r'\left(X^{T}\right)^{\circ {2}}'
assert latex(Transpose(MatPow(X, 2))) == r'\left(X^{2}\right)^{T}'
assert latex(MatPow(Transpose(X), 2)) == r'\left(X^{T}\right)^{2}'
m = Matrix(((1, 2), (3, 4)))
assert latex(Transpose(m)) == '\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{T}'
assert latex(Transpose(m+X)) == \
'\\left(\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X\\right)^{T}'
# Issue 20959
Mx = MatrixSymbol('M^x', 2, 2)
assert latex(Transpose(Mx)) == r'\left(M^{x}\right)^{T}'
def test_Hadamard():
from sympy.matrices import HadamardProduct, HadamardPower
from sympy.matrices.expressions import MatAdd, MatMul, MatPow
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(HadamardProduct(X, Y*Y)) == r'X \circ Y^{2}'
assert latex(HadamardProduct(X, Y)*Y) == r'\left(X \circ Y\right) Y'
assert latex(HadamardPower(X, 2)) == r'X^{\circ {2}}'
assert latex(HadamardPower(X, -1)) == r'X^{\circ \left({-1}\right)}'
assert latex(HadamardPower(MatAdd(X, Y), 2)) == \
r'\left(X + Y\right)^{\circ {2}}'
assert latex(HadamardPower(MatMul(X, Y), 2)) == \
r'\left(X Y\right)^{\circ {2}}'
assert latex(HadamardPower(MatPow(X, -1), -1)) == \
r'\left(X^{-1}\right)^{\circ \left({-1}\right)}'
assert latex(MatPow(HadamardPower(X, -1), -1)) == \
r'\left(X^{\circ \left({-1}\right)}\right)^{-1}'
assert latex(HadamardPower(X, n+1)) == \
r'X^{\circ \left({n + 1}\right)}'
def test_MatPow():
from sympy.matrices.expressions import MatPow
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(MatPow(X, 2)) == 'X^{2}'
assert latex(MatPow(X*X, 2)) == '\\left(X^{2}\\right)^{2}'
assert latex(MatPow(X*Y, 2)) == '\\left(X Y\\right)^{2}'
assert latex(MatPow(X + Y, 2)) == '\\left(X + Y\\right)^{2}'
assert latex(MatPow(X + X, 2)) == '\\left(2 X\\right)^{2}'
# Issue 20959
Mx = MatrixSymbol('M^x', 2, 2)
assert latex(MatPow(Mx, 2)) == r'\left(M^{x}\right)^{2}'
def test_ElementwiseApplyFunction():
X = MatrixSymbol('X', 2, 2)
expr = (X.T*X).applyfunc(sin)
assert latex(expr) == r"{\left( d \mapsto \sin{\left(d \right)} \right)}_{\circ}\left({X^{T} X}\right)"
expr = X.applyfunc(Lambda(x, 1/x))
assert latex(expr) == r'{\left( x \mapsto \frac{1}{x} \right)}_{\circ}\left({X}\right)'
def test_ZeroMatrix():
from sympy.matrices.expressions.special import ZeroMatrix
assert latex(ZeroMatrix(1, 1), mat_symbol_style='plain') == r"0"
assert latex(ZeroMatrix(1, 1), mat_symbol_style='bold') == r"\mathbf{0}"
def test_OneMatrix():
from sympy.matrices.expressions.special import OneMatrix
assert latex(OneMatrix(3, 4), mat_symbol_style='plain') == r"1"
assert latex(OneMatrix(3, 4), mat_symbol_style='bold') == r"\mathbf{1}"
def test_Identity():
from sympy.matrices.expressions.special import Identity
assert latex(Identity(1), mat_symbol_style='plain') == r"\mathbb{I}"
assert latex(Identity(1), mat_symbol_style='bold') == r"\mathbf{I}"
def test_latex_DFT_IDFT():
from sympy.matrices.expressions.fourier import DFT, IDFT
assert latex(DFT(13)) == r"\text{DFT}_{13}"
assert latex(IDFT(x)) == r"\text{IDFT}_{x}"
def test_boolean_args_order():
syms = symbols('a:f')
expr = And(*syms)
assert latex(expr) == r'a \wedge b \wedge c \wedge d \wedge e \wedge f'
expr = Or(*syms)
assert latex(expr) == r'a \vee b \vee c \vee d \vee e \vee f'
expr = Equivalent(*syms)
assert latex(expr) == \
r'a \Leftrightarrow b \Leftrightarrow c \Leftrightarrow d \Leftrightarrow e \Leftrightarrow f'
expr = Xor(*syms)
assert latex(expr) == \
r'a \veebar b \veebar c \veebar d \veebar e \veebar f'
def test_imaginary():
i = sqrt(-1)
assert latex(i) == r'i'
def test_builtins_without_args():
assert latex(sin) == r'\sin'
assert latex(cos) == r'\cos'
assert latex(tan) == r'\tan'
assert latex(log) == r'\log'
assert latex(Ei) == r'\operatorname{Ei}'
assert latex(zeta) == r'\zeta'
def test_latex_greek_functions():
# bug because capital greeks that have roman equivalents should not use
# \Alpha, \Beta, \Eta, etc.
s = Function('Alpha')
assert latex(s) == r'A'
assert latex(s(x)) == r'A{\left(x \right)}'
s = Function('Beta')
assert latex(s) == r'B'
s = Function('Eta')
assert latex(s) == r'H'
assert latex(s(x)) == r'H{\left(x \right)}'
# bug because sympy.core.numbers.Pi is special
p = Function('Pi')
# assert latex(p(x)) == r'\Pi{\left(x \right)}'
assert latex(p) == r'\Pi'
# bug because not all greeks are included
c = Function('chi')
assert latex(c(x)) == r'\chi{\left(x \right)}'
assert latex(c) == r'\chi'
def test_translate():
s = 'Alpha'
assert translate(s) == r'A'
s = 'Beta'
assert translate(s) == r'B'
s = 'Eta'
assert translate(s) == r'H'
s = 'omicron'
assert translate(s) == r'o'
s = 'Pi'
assert translate(s) == r'\Pi'
s = 'pi'
assert translate(s) == r'\pi'
s = 'LamdaHatDOT'
assert translate(s) == r'\dot{\hat{\Lambda}}'
def test_other_symbols():
from sympy.printing.latex import other_symbols
for s in other_symbols:
assert latex(symbols(s)) == r"" "\\" + s
def test_modifiers():
# Test each modifier individually in the simplest case
# (with funny capitalizations)
assert latex(symbols("xMathring")) == r"\mathring{x}"
assert latex(symbols("xCheck")) == r"\check{x}"
assert latex(symbols("xBreve")) == r"\breve{x}"
assert latex(symbols("xAcute")) == r"\acute{x}"
assert latex(symbols("xGrave")) == r"\grave{x}"
assert latex(symbols("xTilde")) == r"\tilde{x}"
assert latex(symbols("xPrime")) == r"{x}'"
assert latex(symbols("xddDDot")) == r"\ddddot{x}"
assert latex(symbols("xDdDot")) == r"\dddot{x}"
assert latex(symbols("xDDot")) == r"\ddot{x}"
assert latex(symbols("xBold")) == r"\boldsymbol{x}"
assert latex(symbols("xnOrM")) == r"\left\|{x}\right\|"
assert latex(symbols("xAVG")) == r"\left\langle{x}\right\rangle"
assert latex(symbols("xHat")) == r"\hat{x}"
assert latex(symbols("xDot")) == r"\dot{x}"
assert latex(symbols("xBar")) == r"\bar{x}"
assert latex(symbols("xVec")) == r"\vec{x}"
assert latex(symbols("xAbs")) == r"\left|{x}\right|"
assert latex(symbols("xMag")) == r"\left|{x}\right|"
assert latex(symbols("xPrM")) == r"{x}'"
assert latex(symbols("xBM")) == r"\boldsymbol{x}"
# Test strings that are *only* the names of modifiers
assert latex(symbols("Mathring")) == r"Mathring"
assert latex(symbols("Check")) == r"Check"
assert latex(symbols("Breve")) == r"Breve"
assert latex(symbols("Acute")) == r"Acute"
assert latex(symbols("Grave")) == r"Grave"
assert latex(symbols("Tilde")) == r"Tilde"
assert latex(symbols("Prime")) == r"Prime"
assert latex(symbols("DDot")) == r"\dot{D}"
assert latex(symbols("Bold")) == r"Bold"
assert latex(symbols("NORm")) == r"NORm"
assert latex(symbols("AVG")) == r"AVG"
assert latex(symbols("Hat")) == r"Hat"
assert latex(symbols("Dot")) == r"Dot"
assert latex(symbols("Bar")) == r"Bar"
assert latex(symbols("Vec")) == r"Vec"
assert latex(symbols("Abs")) == r"Abs"
assert latex(symbols("Mag")) == r"Mag"
assert latex(symbols("PrM")) == r"PrM"
assert latex(symbols("BM")) == r"BM"
assert latex(symbols("hbar")) == r"\hbar"
# Check a few combinations
assert latex(symbols("xvecdot")) == r"\dot{\vec{x}}"
assert latex(symbols("xDotVec")) == r"\vec{\dot{x}}"
assert latex(symbols("xHATNorm")) == r"\left\|{\hat{x}}\right\|"
# Check a couple big, ugly combinations
assert latex(symbols('xMathringBm_yCheckPRM__zbreveAbs')) == \
r"\boldsymbol{\mathring{x}}^{\left|{\breve{z}}\right|}_{{\check{y}}'}"
assert latex(symbols('alphadothat_nVECDOT__tTildePrime')) == \
r"\hat{\dot{\alpha}}^{{\tilde{t}}'}_{\dot{\vec{n}}}"
def test_greek_symbols():
assert latex(Symbol('alpha')) == r'\alpha'
assert latex(Symbol('beta')) == r'\beta'
assert latex(Symbol('gamma')) == r'\gamma'
assert latex(Symbol('delta')) == r'\delta'
assert latex(Symbol('epsilon')) == r'\epsilon'
assert latex(Symbol('zeta')) == r'\zeta'
assert latex(Symbol('eta')) == r'\eta'
assert latex(Symbol('theta')) == r'\theta'
assert latex(Symbol('iota')) == r'\iota'
assert latex(Symbol('kappa')) == r'\kappa'
assert latex(Symbol('lambda')) == r'\lambda'
assert latex(Symbol('mu')) == r'\mu'
assert latex(Symbol('nu')) == r'\nu'
assert latex(Symbol('xi')) == r'\xi'
assert latex(Symbol('omicron')) == r'o'
assert latex(Symbol('pi')) == r'\pi'
assert latex(Symbol('rho')) == r'\rho'
assert latex(Symbol('sigma')) == r'\sigma'
assert latex(Symbol('tau')) == r'\tau'
assert latex(Symbol('upsilon')) == r'\upsilon'
assert latex(Symbol('phi')) == r'\phi'
assert latex(Symbol('chi')) == r'\chi'
assert latex(Symbol('psi')) == r'\psi'
assert latex(Symbol('omega')) == r'\omega'
assert latex(Symbol('Alpha')) == r'A'
assert latex(Symbol('Beta')) == r'B'
assert latex(Symbol('Gamma')) == r'\Gamma'
assert latex(Symbol('Delta')) == r'\Delta'
assert latex(Symbol('Epsilon')) == r'E'
assert latex(Symbol('Zeta')) == r'Z'
assert latex(Symbol('Eta')) == r'H'
assert latex(Symbol('Theta')) == r'\Theta'
assert latex(Symbol('Iota')) == r'I'
assert latex(Symbol('Kappa')) == r'K'
assert latex(Symbol('Lambda')) == r'\Lambda'
assert latex(Symbol('Mu')) == r'M'
assert latex(Symbol('Nu')) == r'N'
assert latex(Symbol('Xi')) == r'\Xi'
assert latex(Symbol('Omicron')) == r'O'
assert latex(Symbol('Pi')) == r'\Pi'
assert latex(Symbol('Rho')) == r'P'
assert latex(Symbol('Sigma')) == r'\Sigma'
assert latex(Symbol('Tau')) == r'T'
assert latex(Symbol('Upsilon')) == r'\Upsilon'
assert latex(Symbol('Phi')) == r'\Phi'
assert latex(Symbol('Chi')) == r'X'
assert latex(Symbol('Psi')) == r'\Psi'
assert latex(Symbol('Omega')) == r'\Omega'
assert latex(Symbol('varepsilon')) == r'\varepsilon'
assert latex(Symbol('varkappa')) == r'\varkappa'
assert latex(Symbol('varphi')) == r'\varphi'
assert latex(Symbol('varpi')) == r'\varpi'
assert latex(Symbol('varrho')) == r'\varrho'
assert latex(Symbol('varsigma')) == r'\varsigma'
assert latex(Symbol('vartheta')) == r'\vartheta'
def test_fancyset_symbols():
assert latex(S.Rationals) == r'\mathbb{Q}'
assert latex(S.Naturals) == r'\mathbb{N}'
assert latex(S.Naturals0) == r'\mathbb{N}_0'
assert latex(S.Integers) == r'\mathbb{Z}'
assert latex(S.Reals) == r'\mathbb{R}'
assert latex(S.Complexes) == r'\mathbb{C}'
@XFAIL
def test_builtin_without_args_mismatched_names():
assert latex(CosineTransform) == r'\mathcal{COS}'
def test_builtin_no_args():
assert latex(Chi) == r'\operatorname{Chi}'
assert latex(beta) == r'\operatorname{B}'
assert latex(gamma) == r'\Gamma'
assert latex(KroneckerDelta) == r'\delta'
assert latex(DiracDelta) == r'\delta'
assert latex(lowergamma) == r'\gamma'
def test_issue_6853():
p = Function('Pi')
assert latex(p(x)) == r"\Pi{\left(x \right)}"
def test_Mul():
e = Mul(-2, x + 1, evaluate=False)
assert latex(e) == r'- 2 \left(x + 1\right)'
e = Mul(2, x + 1, evaluate=False)
assert latex(e) == r'2 \left(x + 1\right)'
e = Mul(S.Half, x + 1, evaluate=False)
assert latex(e) == r'\frac{x + 1}{2}'
e = Mul(y, x + 1, evaluate=False)
assert latex(e) == r'y \left(x + 1\right)'
e = Mul(-y, x + 1, evaluate=False)
assert latex(e) == r'- y \left(x + 1\right)'
e = Mul(-2, x + 1)
assert latex(e) == r'- 2 x - 2'
e = Mul(2, x + 1)
assert latex(e) == r'2 x + 2'
def test_Pow():
e = Pow(2, 2, evaluate=False)
assert latex(e) == r'2^{2}'
assert latex(x**(Rational(-1, 3))) == r'\frac{1}{\sqrt[3]{x}}'
x2 = Symbol(r'x^2')
assert latex(x2**2) == r'\left(x^{2}\right)^{2}'
def test_issue_7180():
assert latex(Equivalent(x, y)) == r"x \Leftrightarrow y"
assert latex(Not(Equivalent(x, y))) == r"x \not\Leftrightarrow y"
def test_issue_8409():
assert latex(S.Half**n) == r"\left(\frac{1}{2}\right)^{n}"
def test_issue_8470():
from sympy.parsing.sympy_parser import parse_expr
e = parse_expr("-B*A", evaluate=False)
assert latex(e) == r"A \left(- B\right)"
def test_issue_15439():
x = MatrixSymbol('x', 2, 2)
y = MatrixSymbol('y', 2, 2)
assert latex((x * y).subs(y, -y)) == r"x \left(- y\right)"
assert latex((x * y).subs(y, -2*y)) == r"x \left(- 2 y\right)"
assert latex((x * y).subs(x, -x)) == r"- x y"
def test_issue_2934():
assert latex(Symbol(r'\frac{a_1}{b_1}')) == r'\frac{a_1}{b_1}'
def test_issue_10489():
latexSymbolWithBrace = r'C_{x_{0}}'
s = Symbol(latexSymbolWithBrace)
assert latex(s) == latexSymbolWithBrace
assert latex(cos(s)) == r'\cos{\left(C_{x_{0}} \right)}'
def test_issue_12886():
m__1, l__1 = symbols('m__1, l__1')
assert latex(m__1**2 + l__1**2) == \
r'\left(l^{1}\right)^{2} + \left(m^{1}\right)^{2}'
def test_issue_13559():
from sympy.parsing.sympy_parser import parse_expr
expr = parse_expr('5/1', evaluate=False)
assert latex(expr) == r"\frac{5}{1}"
def test_issue_13651():
expr = c + Mul(-1, a + b, evaluate=False)
assert latex(expr) == r"c - \left(a + b\right)"
def test_latex_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
assert latex(he) == latex(1/x) == r"\frac{1}{x}"
assert latex(he**2) == r"\left(\frac{1}{x}\right)^{2}"
assert latex(he + 1) == r"1 + \frac{1}{x}"
assert latex(x*he) == r"x \frac{1}{x}"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert latex(A[0, 0]) == r"A_{0, 0}"
assert latex(3 * A[0, 0]) == r"3 A_{0, 0}"
F = C[0, 0].subs(C, A - B)
assert latex(F) == r"\left(A - B\right)_{0, 0}"
i, j, k = symbols("i j k")
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
assert latex((M*N)[i, j]) == \
r'\sum_{i_{1}=0}^{k - 1} M_{i, i_{1}} N_{i_{1}, j}'
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A) == r"- A"
assert latex(A - A*B - B) == r"A - A B - B"
assert latex(-A*B - A*B*C - B) == r"- A B - A B C - B"
def test_KroneckerProduct_printing():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 2, 2)
assert latex(KroneckerProduct(A, B)) == r'A \otimes B'
def test_Series_printing():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y)
assert latex(Series(tf1, tf2)) == \
r'\left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right) \left(\frac{x - y}{x + y}\right)'
assert latex(Series(tf1, tf2, tf3)) == \
r'\left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right) \left(\frac{x - y}{x + y}\right) \left(\frac{t x^{2} - t^{w} x + w}{t - y}\right)'
assert latex(Series(-tf2, tf1)) == \
r'\left(\frac{- x + y}{x + y}\right) \left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right)'
M_1 = Matrix([[5/s], [5/(2*s)]])
T_1 = TransferFunctionMatrix.from_Matrix(M_1, s)
M_2 = Matrix([[5, 6*s**3]])
T_2 = TransferFunctionMatrix.from_Matrix(M_2, s)
# Brackets
assert latex(T_1*(T_2 + T_2)) == \
r'\left[\begin{matrix}\frac{5}{s}\\\frac{5}{2 s}\end{matrix}\right]_\tau\cdot\left(\left[\begin{matrix}\frac{5}{1} &' \
r' \frac{6 s^{3}}{1}\end{matrix}\right]_\tau + \left[\begin{matrix}\frac{5}{1} & \frac{6 s^{3}}{1}\end{matrix}\right]_\tau\right)' \
== latex(MIMOSeries(MIMOParallel(T_2, T_2), T_1))
# No Brackets
M_3 = Matrix([[5, 6], [6, 5/s]])
T_3 = TransferFunctionMatrix.from_Matrix(M_3, s)
assert latex(T_1*T_2 + T_3) == r'\left[\begin{matrix}\frac{5}{s}\\\frac{5}{2 s}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}' \
r'\frac{5}{1} & \frac{6 s^{3}}{1}\end{matrix}\right]_\tau + \left[\begin{matrix}\frac{5}{1} & \frac{6}{1}\\\frac{6}{1} & ' \
r'\frac{5}{s}\end{matrix}\right]_\tau' == latex(MIMOParallel(MIMOSeries(T_2, T_1), T_3))
def test_TransferFunction_printing():
tf1 = TransferFunction(x - 1, x + 1, x)
assert latex(tf1) == r"\frac{x - 1}{x + 1}"
tf2 = TransferFunction(x + 1, 2 - y, x)
assert latex(tf2) == r"\frac{x + 1}{2 - y}"
tf3 = TransferFunction(y, y**2 + 2*y + 3, y)
assert latex(tf3) == r"\frac{y}{y^{2} + 2 y + 3}"
def test_Parallel_printing():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
assert latex(Parallel(tf1, tf2)) == \
r'\frac{x y^{2} - z}{- t^{3} + y^{3}} + \frac{x - y}{x + y}'
assert latex(Parallel(-tf2, tf1)) == \
r'\frac{- x + y}{x + y} + \frac{x y^{2} - z}{- t^{3} + y^{3}}'
M_1 = Matrix([[5, 6], [6, 5/s]])
T_1 = TransferFunctionMatrix.from_Matrix(M_1, s)
M_2 = Matrix([[5/s, 6], [6, 5/(s - 1)]])
T_2 = TransferFunctionMatrix.from_Matrix(M_2, s)
M_3 = Matrix([[6, 5/(s*(s - 1))], [5, 6]])
T_3 = TransferFunctionMatrix.from_Matrix(M_3, s)
assert latex(T_1 + T_2 + T_3) == r'\left[\begin{matrix}\frac{5}{1} & \frac{6}{1}\\\frac{6}{1} & \frac{5}{s}\end{matrix}\right]' \
r'_\tau + \left[\begin{matrix}\frac{5}{s} & \frac{6}{1}\\\frac{6}{1} & \frac{5}{s - 1}\end{matrix}\right]_\tau + \left[\begin{matrix}' \
r'\frac{6}{1} & \frac{5}{s \left(s - 1\right)}\\\frac{5}{1} & \frac{6}{1}\end{matrix}\right]_\tau' \
== latex(MIMOParallel(T_1, T_2, T_3)) == latex(MIMOParallel(T_1, MIMOParallel(T_2, T_3))) == latex(MIMOParallel(MIMOParallel(T_1, T_2), T_3))
def test_TransferFunctionMatrix_printing():
tf1 = TransferFunction(p, p + x, p)
tf2 = TransferFunction(-s + p, p + s, p)
tf3 = TransferFunction(p, y**2 + 2*y + 3, p)
assert latex(TransferFunctionMatrix([[tf1], [tf2]])) == \
r'\left[\begin{matrix}\frac{p}{p + x}\\\frac{p - s}{p + s}\end{matrix}\right]_\tau'
assert latex(TransferFunctionMatrix([[tf1, tf2], [tf3, -tf1]])) == \
r'\left[\begin{matrix}\frac{p}{p + x} & \frac{p - s}{p + s}\\\frac{p}{y^{2} + 2 y + 3} & \frac{\left(-1\right) p}{p + x}\end{matrix}\right]_\tau'
def test_Feedback_printing():
tf1 = TransferFunction(p, p + x, p)
tf2 = TransferFunction(-s + p, p + s, p)
# Negative Feedback (Default)
assert latex(Feedback(tf1, tf2)) == \
r'\frac{\frac{p}{p + x}}{\frac{1}{1} + \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
assert latex(Feedback(tf1*tf2, TransferFunction(1, 1, p))) == \
r'\frac{\left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}{\frac{1}{1} + \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
# Positive Feedback
assert latex(Feedback(tf1, tf2, 1)) == \
r'\frac{\frac{p}{p + x}}{\frac{1}{1} - \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
assert latex(Feedback(tf1*tf2, sign=1)) == \
r'\frac{\left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}{\frac{1}{1} - \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
def test_MIMOFeedback_printing():
tf1 = TransferFunction(1, s, s)
tf2 = TransferFunction(s, s**2 - 1, s)
tf3 = TransferFunction(s, s - 1, s)
tf4 = TransferFunction(s**2, s**2 - 1, s)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
tfm_2 = TransferFunctionMatrix([[tf4, tf3], [tf2, tf1]])
# Negative Feedback (Default)
assert latex(MIMOFeedback(tfm_1, tfm_2)) == \
r'\left(I_{\tau} + \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left[' \
r'\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1} & \frac{1}{s}\end{matrix}\right]_\tau\right)^{-1} \cdot \left[\begin{matrix}' \
r'\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau'
# Positive Feedback
assert latex(MIMOFeedback(tfm_1*tfm_2, tfm_1, 1)) == \
r'\left(I_{\tau} - \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left' \
r'[\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1} & \frac{1}{s}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}' \
r'\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\right)^{-1} \cdot \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}' \
r'\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1}' \
r' & \frac{1}{s}\end{matrix}\right]_\tau'
def test_Quaternion_latex_printing():
q = Quaternion(x, y, z, t)
assert latex(q) == r"x + y i + z j + t k"
q = Quaternion(x, y, z, x*t)
assert latex(q) == r"x + y i + z j + t x k"
q = Quaternion(x, y, z, x + t)
assert latex(q) == r"x + y i + z j + \left(t + x\right) k"
def test_TensorProduct_printing():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert latex(TensorProduct(A, B)) == r"A \otimes B"
def test_WedgeProduct_printing():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert latex(wp) == r"\operatorname{d}x \wedge \operatorname{d}y"
def test_issue_9216():
expr_1 = Pow(1, -1, evaluate=False)
assert latex(expr_1) == r"1^{-1}"
expr_2 = Pow(1, Pow(1, -1, evaluate=False), evaluate=False)
assert latex(expr_2) == r"1^{1^{-1}}"
expr_3 = Pow(3, -2, evaluate=False)
assert latex(expr_3) == r"\frac{1}{9}"
expr_4 = Pow(1, -2, evaluate=False)
assert latex(expr_4) == r"1^{-2}"
def test_latex_printer_tensor():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, tensor_heads
L = TensorIndexType("L")
i, j, k, l = tensor_indices("i j k l", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
K = TensorHead("K", [L, L, L, L])
assert latex(i) == r"{}^{i}"
assert latex(-i) == r"{}_{i}"
expr = A(i)
assert latex(expr) == r"A{}^{i}"
expr = A(i0)
assert latex(expr) == r"A{}^{i_{0}}"
expr = A(-i)
assert latex(expr) == r"A{}_{i}"
expr = -3*A(i)
assert latex(expr) == r"-3A{}^{i}"
expr = K(i, j, -k, -i0)
assert latex(expr) == r"K{}^{ij}{}_{ki_{0}}"
expr = K(i, -j, -k, i0)
assert latex(expr) == r"K{}^{i}{}_{jk}{}^{i_{0}}"
expr = K(i, -j, k, -i0)
assert latex(expr) == r"K{}^{i}{}_{j}{}^{k}{}_{i_{0}}"
expr = H(i, -j)
assert latex(expr) == r"H{}^{i}{}_{j}"
expr = H(i, j)
assert latex(expr) == r"H{}^{ij}"
expr = H(-i, -j)
assert latex(expr) == r"H{}_{ij}"
expr = (1+x)*A(i)
assert latex(expr) == r"\left(x + 1\right)A{}^{i}"
expr = H(i, -i)
assert latex(expr) == r"H{}^{L_{0}}{}_{L_{0}}"
expr = H(i, -j)*A(j)*B(k)
assert latex(expr) == r"H{}^{i}{}_{L_{0}}A{}^{L_{0}}B{}^{k}"
expr = A(i) + 3*B(i)
assert latex(expr) == r"3B{}^{i} + A{}^{i}"
# Test ``TensorElement``:
from sympy.tensor.tensor import TensorElement
expr = TensorElement(K(i, j, k, l), {i: 3, k: 2})
assert latex(expr) == r'K{}^{i=3,j,k=2,l}'
expr = TensorElement(K(i, j, k, l), {i: 3})
assert latex(expr) == r'K{}^{i=3,jkl}'
expr = TensorElement(K(i, -j, k, l), {i: 3, k: 2})
assert latex(expr) == r'K{}^{i=3}{}_{j}{}^{k=2,l}'
expr = TensorElement(K(i, -j, k, -l), {i: 3, k: 2})
assert latex(expr) == r'K{}^{i=3}{}_{j}{}^{k=2}{}_{l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3, -k: 2})
assert latex(expr) == r'K{}^{i=3,j}{}_{k=2,l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3})
assert latex(expr) == r'K{}^{i=3,j}{}_{kl}'
expr = PartialDerivative(A(i), A(i))
assert latex(expr) == r"\frac{\partial}{\partial {A{}^{L_{0}}}}{A{}^{L_{0}}}"
expr = PartialDerivative(A(-i), A(-j))
assert latex(expr) == r"\frac{\partial}{\partial {A{}_{j}}}{A{}_{i}}"
expr = PartialDerivative(K(i, j, -k, -l), A(m), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}^{m}} \partial {A{}_{n}}}{K{}^{ij}{}_{kl}}"
expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(A{}_{i} + B{}_{i}\right)}"
expr = PartialDerivative(3*A(-i), A(-j), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(3A{}_{i}\right)}"
def test_multiline_latex():
a, b, c, d, e, f = symbols('a b c d e f')
expr = -a + 2*b -3*c +4*d -5*e
expected = r"\begin{eqnarray}" + "\n"\
r"f & = &- a \nonumber\\" + "\n"\
r"& & + 2 b \nonumber\\" + "\n"\
r"& & - 3 c \nonumber\\" + "\n"\
r"& & + 4 d \nonumber\\" + "\n"\
r"& & - 5 e " + "\n"\
r"\end{eqnarray}"
assert multiline_latex(f, expr, environment="eqnarray") == expected
expected2 = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b \nonumber\\' + '\n'\
r'& & - 3 c + 4 d \nonumber\\' + '\n'\
r'& & - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 2, environment="eqnarray") == expected2
expected3 = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b - 3 c \nonumber\\'+ '\n'\
r'& & + 4 d - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 3, environment="eqnarray") == expected3
expected3dots = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b - 3 c \dots\nonumber\\'+ '\n'\
r'& & + 4 d - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 3, environment="eqnarray", use_dots=True) == expected3dots
expected3align = r'\begin{align*}' + '\n'\
r'f = &- a + 2 b - 3 c \\'+ '\n'\
r'& + 4 d - 5 e ' + '\n'\
r'\end{align*}'
assert multiline_latex(f, expr, 3) == expected3align
assert multiline_latex(f, expr, 3, environment='align*') == expected3align
expected2ieee = r'\begin{IEEEeqnarray}{rCl}' + '\n'\
r'f & = &- a + 2 b \nonumber\\' + '\n'\
r'& & - 3 c + 4 d \nonumber\\' + '\n'\
r'& & - 5 e ' + '\n'\
r'\end{IEEEeqnarray}'
assert multiline_latex(f, expr, 2, environment="IEEEeqnarray") == expected2ieee
raises(ValueError, lambda: multiline_latex(f, expr, environment="foo"))
def test_issue_15353():
a, x = symbols('a x')
# Obtained from nonlinsolve([(sin(a*x)),cos(a*x)],[x,a])
sol = ConditionSet(
Tuple(x, a), Eq(sin(a*x), 0) & Eq(cos(a*x), 0), S.Complexes**2)
assert latex(sol) == \
r'\left\{\left( x, \ a\right)\; \middle|\; \left( x, \ a\right) \in ' \
r'\mathbb{C}^{2} \wedge \sin{\left(a x \right)} = 0 \wedge ' \
r'\cos{\left(a x \right)} = 0 \right\}'
def test_latex_symbolic_probability():
mu = symbols("mu")
sigma = symbols("sigma", positive=True)
X = Normal("X", mu, sigma)
assert latex(Expectation(X)) == r'\operatorname{E}\left[X\right]'
assert latex(Variance(X)) == r'\operatorname{Var}\left(X\right)'
assert latex(Probability(X > 0)) == r'\operatorname{P}\left(X > 0\right)'
Y = Normal("Y", mu, sigma)
assert latex(Covariance(X, Y)) == r'\operatorname{Cov}\left(X, Y\right)'
def test_trace():
# Issue 15303
from sympy.matrices.expressions.trace import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A)) == r"\operatorname{tr}\left(A \right)"
assert latex(trace(A**2)) == r"\operatorname{tr}\left(A^{2} \right)"
def test_print_basic():
# Issue 15303
from sympy.core.basic import Basic
from sympy.core.expr import Expr
# dummy class for testing printing where the function is not
# implemented in latex.py
class UnimplementedExpr(Expr):
def __new__(cls, e):
return Basic.__new__(cls, e)
# dummy function for testing
def unimplemented_expr(expr):
return UnimplementedExpr(expr).doit()
# override class name to use superscript / subscript
def unimplemented_expr_sup_sub(expr):
result = UnimplementedExpr(expr)
result.__class__.__name__ = 'UnimplementedExpr_x^1'
return result
assert latex(unimplemented_expr(x)) == r'\operatorname{UnimplementedExpr}\left(x\right)'
assert latex(unimplemented_expr(x**2)) == \
r'\operatorname{UnimplementedExpr}\left(x^{2}\right)'
assert latex(unimplemented_expr_sup_sub(x)) == \
r'\operatorname{UnimplementedExpr^{1}_{x}}\left(x\right)'
def test_MatrixSymbol_bold():
# Issue #15871
from sympy.matrices.expressions.trace import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A), mat_symbol_style='bold') == \
r"\operatorname{tr}\left(\mathbf{A} \right)"
assert latex(trace(A), mat_symbol_style='plain') == \
r"\operatorname{tr}\left(A \right)"
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A, mat_symbol_style='bold') == r"- \mathbf{A}"
assert latex(A - A*B - B, mat_symbol_style='bold') == \
r"\mathbf{A} - \mathbf{A} \mathbf{B} - \mathbf{B}"
assert latex(-A*B - A*B*C - B, mat_symbol_style='bold') == \
r"- \mathbf{A} \mathbf{B} - \mathbf{A} \mathbf{B} \mathbf{C} - \mathbf{B}"
A_k = MatrixSymbol("A_k", 3, 3)
assert latex(A_k, mat_symbol_style='bold') == r"\mathbf{A}_{k}"
A = MatrixSymbol(r"\nabla_k", 3, 3)
assert latex(A, mat_symbol_style='bold') == r"\mathbf{\nabla}_{k}"
def test_AppliedPermutation():
p = Permutation(0, 1, 2)
x = Symbol('x')
assert latex(AppliedPermutation(p, x)) == \
r'\sigma_{\left( 0\; 1\; 2\right)}(x)'
def test_PermutationMatrix():
p = Permutation(0, 1, 2)
assert latex(PermutationMatrix(p)) == r'P_{\left( 0\; 1\; 2\right)}'
p = Permutation(0, 3)(1, 2)
assert latex(PermutationMatrix(p)) == \
r'P_{\left( 0\; 3\right)\left( 1\; 2\right)}'
def test_issue_21758():
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.series.fourier import FourierSeries
x = Symbol('x')
k, n = symbols('k n')
fo = FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), SeqFormula(
Piecewise((-2*pi*cos(n*pi)/n + 2*sin(n*pi)/n**2, (n > -oo) & (n < oo) & Ne(n, 0)),
(0, True))*sin(n*x)/pi, (n, 1, oo))))
assert latex(piecewise_fold(fo)) == '\\begin{cases} 2 \\sin{\\left(x \\right)}' \
' - \\sin{\\left(2 x \\right)} + \\frac{2 \\sin{\\left(3 x \\right)}}{3} +' \
' \\ldots & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge ' \
'n \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}'
assert latex(FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)),
SeqFormula(0, (n, 1, oo))))) == '0'
def test_imaginary_unit():
assert latex(1 + I) == r'1 + i'
assert latex(1 + I, imaginary_unit='i') == r'1 + i'
assert latex(1 + I, imaginary_unit='j') == r'1 + j'
assert latex(1 + I, imaginary_unit='foo') == r'1 + foo'
assert latex(I, imaginary_unit="ti") == r'\text{i}'
assert latex(I, imaginary_unit="tj") == r'\text{j}'
def test_text_re_im():
assert latex(im(x), gothic_re_im=True) == r'\Im{\left(x\right)}'
assert latex(im(x), gothic_re_im=False) == r'\operatorname{im}{\left(x\right)}'
assert latex(re(x), gothic_re_im=True) == r'\Re{\left(x\right)}'
assert latex(re(x), gothic_re_im=False) == r'\operatorname{re}{\left(x\right)}'
def test_latex_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
from sympy.diffgeom.rn import R2
x,y = symbols('x y', real=True)
m = Manifold('M', 2)
assert latex(m) == r'\text{M}'
p = Patch('P', m)
assert latex(p) == r'\text{P}_{\text{M}}'
rect = CoordSystem('rect', p, [x, y])
assert latex(rect) == r'\text{rect}^{\text{P}}_{\text{M}}'
b = BaseScalarField(rect, 0)
assert latex(b) == r'\mathbf{x}'
g = Function('g')
s_field = g(R2.x, R2.y)
assert latex(Differential(s_field)) == \
r'\operatorname{d}\left(g{\left(\mathbf{x},\mathbf{y} \right)}\right)'
def test_unit_printing():
assert latex(5*meter) == r'5 \text{m}'
assert latex(3*gibibyte) == r'3 \text{gibibyte}'
assert latex(4*microgram/second) == r'\frac{4 \mu\text{g}}{\text{s}}'
def test_issue_17092():
x_star = Symbol('x^*')
assert latex(Derivative(x_star, x_star,2)) == r'\frac{d^{2}}{d \left(x^{*}\right)^{2}} x^{*}'
def test_latex_decimal_separator():
x, y, z, t = symbols('x y z t')
k, m, n = symbols('k m n', integer=True)
f, g, h = symbols('f g h', cls=Function)
# comma decimal_separator
assert(latex([1, 2.3, 4.5], decimal_separator='comma') == r'\left[ 1; \ 2{,}3; \ 4{,}5\right]')
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='comma') == r'\left\{1; 2{,}3; 4{,}5\right\}')
assert(latex((1, 2.3, 4.6), decimal_separator = 'comma') == r'\left( 1; \ 2{,}3; \ 4{,}6\right)')
assert(latex((1,), decimal_separator='comma') == r'\left( 1;\right)')
# period decimal_separator
assert(latex([1, 2.3, 4.5], decimal_separator='period') == r'\left[ 1, \ 2.3, \ 4.5\right]' )
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}')
assert(latex((1, 2.3, 4.6), decimal_separator = 'period') == r'\left( 1, \ 2.3, \ 4.6\right)')
assert(latex((1,), decimal_separator='period') == r'\left( 1,\right)')
# default decimal_separator
assert(latex([1, 2.3, 4.5]) == r'\left[ 1, \ 2.3, \ 4.5\right]')
assert(latex(FiniteSet(1, 2.3, 4.5)) == r'\left\{1, 2.3, 4.5\right\}')
assert(latex((1, 2.3, 4.6)) == r'\left( 1, \ 2.3, \ 4.6\right)')
assert(latex((1,)) == r'\left( 1,\right)')
assert(latex(Mul(3.4,5.3), decimal_separator = 'comma') == r'18{,}02')
assert(latex(3.4*5.3, decimal_separator = 'comma') == r'18{,}02')
x = symbols('x')
y = symbols('y')
z = symbols('z')
assert(latex(x*5.3 + 2**y**3.4 + 4.5 + z, decimal_separator = 'comma') == r'2^{y^{3{,}4}} + 5{,}3 x + z + 4{,}5')
assert(latex(0.987, decimal_separator='comma') == r'0{,}987')
assert(latex(S(0.987), decimal_separator='comma') == r'0{,}987')
assert(latex(.3, decimal_separator='comma') == r'0{,}3')
assert(latex(S(.3), decimal_separator='comma') == r'0{,}3')
assert(latex(5.8*10**(-7), decimal_separator='comma') == r'5{,}8 \cdot 10^{-7}')
assert(latex(S(5.7)*10**(-7), decimal_separator='comma') == r'5{,}7 \cdot 10^{-7}')
assert(latex(S(5.7*10**(-7)), decimal_separator='comma') == r'5{,}7 \cdot 10^{-7}')
x = symbols('x')
assert(latex(1.2*x+3.4, decimal_separator='comma') == r'1{,}2 x + 3{,}4')
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}')
# Error Handling tests
raises(ValueError, lambda: latex([1,2.3,4.5], decimal_separator='non_existing_decimal_separator_in_list'))
raises(ValueError, lambda: latex(FiniteSet(1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_set'))
raises(ValueError, lambda: latex((1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_tuple'))
def test_Str():
from sympy.core.symbol import Str
assert str(Str('x')) == r'x'
def test_latex_escape():
assert latex_escape(r"~^\&%$#_{}") == "".join([
r'\textasciitilde',
r'\textasciicircum',
r'\textbackslash',
r'\&',
r'\%',
r'\$',
r'\#',
r'\_',
r'\{',
r'\}',
])
def test_emptyPrinter():
class MyObject:
def __repr__(self):
return "<MyObject with {...}>"
# unknown objects are monospaced
assert latex(MyObject()) == r"\mathtt{\text{<MyObject with \{...\}>}}"
# even if they are nested within other objects
assert latex((MyObject(),)) == r"\left( \mathtt{\text{<MyObject with \{...\}>}},\right)"
def test_global_settings():
import inspect
# settings should be visible in the signature of `latex`
assert inspect.signature(latex).parameters['imaginary_unit'].default == r'i'
assert latex(I) == r'i'
try:
# but changing the defaults...
LatexPrinter.set_global_settings(imaginary_unit='j')
# ... should change the signature
assert inspect.signature(latex).parameters['imaginary_unit'].default == r'j'
assert latex(I) == r'j'
finally:
# there's no public API to undo this, but we need to make sure we do
# so as not to impact other tests
del LatexPrinter._global_settings['imaginary_unit']
# check we really did undo it
assert inspect.signature(latex).parameters['imaginary_unit'].default == r'i'
assert latex(I) == r'i'
def test_pickleable():
# this tests that the _PrintFunction instance is pickleable
import pickle
assert pickle.loads(pickle.dumps(latex)) is latex
def test_printing_latex_array_expressions():
assert latex(ArraySymbol("A", (2, 3, 4))) == "A"
assert latex(ArrayElement("A", (2, 1/(1-x), 0))) == "{{A}_{2, \\frac{1}{1 - x}, 0}}"
M = MatrixSymbol("M", 3, 3)
N = MatrixSymbol("N", 3, 3)
assert latex(ArrayElement(M*N, [x, 0])) == "{{\\left(M N\\right)}_{x, 0}}"
|
4d8908cf20bd3144ce59c06cce1a3cecd36b0724222264687c6b8bc4ea37b84d | from sympy.concrete.summations import Sum
from sympy.core.mod import Mod
from sympy.core.relational import (Equality, Unequality)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.special import Identity
from sympy.utilities.lambdify import lambdify
from sympy.abc import x, i, j, a, b, c, d
from sympy.core import Pow
from sympy.codegen.matrix_nodes import MatrixSolve
from sympy.codegen.numpy_nodes import logaddexp, logaddexp2
from sympy.codegen.cfunctions import log1p, expm1, hypot, log10, exp2, log2, Sqrt
from sympy.tensor.array import Array
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayAdd, \
PermuteDims, ArrayDiagonal
from sympy.printing.numpy import NumPyPrinter, SciPyPrinter, _numpy_known_constants, \
_numpy_known_functions, _scipy_known_constants, _scipy_known_functions
from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
from sympy.testing.pytest import skip, raises
from sympy.external import import_module
np = import_module('numpy')
def test_numpy_piecewise_regression():
"""
NumPyPrinter needs to print Piecewise()'s choicelist as a list to avoid
breaking compatibility with numpy 1.8. This is not necessary in numpy 1.9+.
See gh-9747 and gh-9749 for details.
"""
printer = NumPyPrinter()
p = Piecewise((1, x < 0), (0, True))
assert printer.doprint(p) == \
'numpy.select([numpy.less(x, 0),True], [1,0], default=numpy.nan)'
assert printer.module_imports == {'numpy': {'select', 'less', 'nan'}}
def test_numpy_logaddexp():
lae = logaddexp(a, b)
assert NumPyPrinter().doprint(lae) == 'numpy.logaddexp(a, b)'
lae2 = logaddexp2(a, b)
assert NumPyPrinter().doprint(lae2) == 'numpy.logaddexp2(a, b)'
def test_sum():
if not np:
skip("NumPy not installed")
s = Sum(x ** i, (i, a, b))
f = lambdify((a, b, x), s, 'numpy')
a_, b_ = 0, 10
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, x_), sum(x_ ** i_ for i_ in range(a_, b_ + 1)))
s = Sum(i * x, (i, a, b))
f = lambdify((a, b, x), s, 'numpy')
a_, b_ = 0, 10
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, x_), sum(i_ * x_ for i_ in range(a_, b_ + 1)))
def test_multiple_sums():
if not np:
skip("NumPy not installed")
s = Sum((x + j) * i, (i, a, b), (j, c, d))
f = lambdify((a, b, c, d, x), s, 'numpy')
a_, b_ = 0, 10
c_, d_ = 11, 21
x_ = np.linspace(-1, +1, 10)
assert np.allclose(f(a_, b_, c_, d_, x_),
sum((x_ + j_) * i_ for i_ in range(a_, b_ + 1) for j_ in range(c_, d_ + 1)))
def test_codegen_einsum():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
cg = convert_matrix_to_array(M * N)
f = lambdify((M, N), cg, 'numpy')
ma = np.array([[1, 2], [3, 4]])
mb = np.array([[1,-2], [-1, 3]])
assert (f(ma, mb) == np.matmul(ma, mb)).all()
def test_codegen_extra():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
P = MatrixSymbol("P", 2, 2)
Q = MatrixSymbol("Q", 2, 2)
ma = np.array([[1, 2], [3, 4]])
mb = np.array([[1,-2], [-1, 3]])
mc = np.array([[2, 0], [1, 2]])
md = np.array([[1,-1], [4, 7]])
cg = ArrayTensorProduct(M, N)
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.einsum(ma, [0, 1], mb, [2, 3])).all()
cg = ArrayAdd(M, N)
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == ma+mb).all()
cg = ArrayAdd(M, N, P)
f = lambdify((M, N, P), cg, 'numpy')
assert (f(ma, mb, mc) == ma+mb+mc).all()
cg = ArrayAdd(M, N, P, Q)
f = lambdify((M, N, P, Q), cg, 'numpy')
assert (f(ma, mb, mc, md) == ma+mb+mc+md).all()
cg = PermuteDims(M, [1, 0])
f = lambdify((M,), cg, 'numpy')
assert (f(ma) == ma.T).all()
cg = PermuteDims(ArrayTensorProduct(M, N), [1, 2, 3, 0])
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.transpose(np.einsum(ma, [0, 1], mb, [2, 3]), (1, 2, 3, 0))).all()
cg = ArrayDiagonal(ArrayTensorProduct(M, N), (1, 2))
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.diagonal(np.einsum(ma, [0, 1], mb, [2, 3]), axis1=1, axis2=2)).all()
def test_relational():
if not np:
skip("NumPy not installed")
e = Equality(x, 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [False, True, False])
e = Unequality(x, 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [True, False, True])
e = (x < 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [True, False, False])
e = (x <= 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [True, True, False])
e = (x > 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [False, False, True])
e = (x >= 1)
f = lambdify((x,), e)
x_ = np.array([0, 1, 2])
assert np.array_equal(f(x_), [False, True, True])
def test_mod():
if not np:
skip("NumPy not installed")
e = Mod(a, b)
f = lambdify((a, b), e)
a_ = np.array([0, 1, 2, 3])
b_ = 2
assert np.array_equal(f(a_, b_), [0, 1, 0, 1])
a_ = np.array([0, 1, 2, 3])
b_ = np.array([2, 2, 2, 2])
assert np.array_equal(f(a_, b_), [0, 1, 0, 1])
a_ = np.array([2, 3, 4, 5])
b_ = np.array([2, 3, 4, 5])
assert np.array_equal(f(a_, b_), [0, 0, 0, 0])
def test_pow():
if not np:
skip('NumPy not installed')
expr = Pow(2, -1, evaluate=False)
f = lambdify([], expr, 'numpy')
assert f() == 0.5
def test_expm1():
if not np:
skip("NumPy not installed")
f = lambdify((a,), expm1(a), 'numpy')
assert abs(f(1e-10) - 1e-10 - 5e-21) < 1e-22
def test_log1p():
if not np:
skip("NumPy not installed")
f = lambdify((a,), log1p(a), 'numpy')
assert abs(f(1e-99) - 1e-99) < 1e-100
def test_hypot():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a, b), hypot(a, b), 'numpy')(3, 4) - 5) < 1e-16
def test_log10():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), log10(a), 'numpy')(100) - 2) < 1e-16
def test_exp2():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), exp2(a), 'numpy')(5) - 32) < 1e-16
def test_log2():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), log2(a), 'numpy')(256) - 8) < 1e-16
def test_Sqrt():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), Sqrt(a), 'numpy')(4) - 2) < 1e-16
def test_sqrt():
if not np:
skip("NumPy not installed")
assert abs(lambdify((a,), sqrt(a), 'numpy')(4) - 2) < 1e-16
def test_matsolve():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 3, 3)
x = MatrixSymbol("x", 3, 1)
expr = M**(-1) * x + x
matsolve_expr = MatrixSolve(M, x) + x
f = lambdify((M, x), expr)
f_matsolve = lambdify((M, x), matsolve_expr)
m0 = np.array([[1, 2, 3], [3, 2, 5], [5, 6, 7]])
assert np.linalg.matrix_rank(m0) == 3
x0 = np.array([3, 4, 5])
assert np.allclose(f_matsolve(m0, x0), f(m0, x0))
def test_16857():
if not np:
skip("NumPy not installed")
a_1 = MatrixSymbol('a_1', 10, 3)
a_2 = MatrixSymbol('a_2', 10, 3)
a_3 = MatrixSymbol('a_3', 10, 3)
a_4 = MatrixSymbol('a_4', 10, 3)
A = BlockMatrix([[a_1, a_2], [a_3, a_4]])
assert A.shape == (20, 6)
printer = NumPyPrinter()
assert printer.doprint(A) == 'numpy.block([[a_1, a_2], [a_3, a_4]])'
def test_issue_17006():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
f = lambdify(M, M + Identity(2))
ma = np.array([[1, 2], [3, 4]])
mr = np.array([[2, 2], [3, 5]])
assert (f(ma) == mr).all()
from sympy.core.symbol import symbols
n = symbols('n', integer=True)
N = MatrixSymbol("M", n, n)
raises(NotImplementedError, lambda: lambdify(N, N + Identity(n)))
def test_numpy_array():
assert NumPyPrinter().doprint(Array(((1, 2), (3, 5)))) == 'numpy.array([[1, 2], [3, 5]])'
assert NumPyPrinter().doprint(Array((1, 2))) == 'numpy.array((1, 2))'
def test_numpy_known_funcs_consts():
assert _numpy_known_constants['NaN'] == 'numpy.nan'
assert _numpy_known_constants['EulerGamma'] == 'numpy.euler_gamma'
assert _numpy_known_functions['acos'] == 'numpy.arccos'
assert _numpy_known_functions['log'] == 'numpy.log'
def test_scipy_known_funcs_consts():
assert _scipy_known_constants['GoldenRatio'] == 'scipy.constants.golden_ratio'
assert _scipy_known_constants['Pi'] == 'scipy.constants.pi'
assert _scipy_known_functions['erf'] == 'scipy.special.erf'
assert _scipy_known_functions['factorial'] == 'scipy.special.factorial'
def test_numpy_print_methods():
prntr = NumPyPrinter()
assert hasattr(prntr, '_print_acos')
assert hasattr(prntr, '_print_log')
def test_scipy_print_methods():
prntr = SciPyPrinter()
assert hasattr(prntr, '_print_acos')
assert hasattr(prntr, '_print_log')
assert hasattr(prntr, '_print_erf')
assert hasattr(prntr, '_print_factorial')
assert hasattr(prntr, '_print_chebyshevt')
|
7dc4368287cafc8b5cf73b785e7ef78e4b66804fd2c2d4c728bb35e08e556df6 | # -*- coding: utf-8 -*-
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import (Dict, Tuple)
from sympy.core.function import (Derivative, Function, Lambda, Subs)
from sympy.core.mul import Mul
from sympy.core import (EulerGamma, GoldenRatio, Catalan)
from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.power import Pow
from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import conjugate
from sympy.functions.elementary.exponential import LambertW
from sympy.functions.special.bessel import (airyai, airyaiprime, airybi, airybiprime)
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.error_functions import (fresnelc, fresnels)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import dirichlet_eta
from sympy.geometry.line import (Ray, Segment)
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (And, Equivalent, ITE, Implies, Nand, Nor, Not, Or, Xor)
from sympy.matrices.dense import (Matrix, diag)
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions.trace import Trace
from sympy.polys.domains.finitefield import FF
from sympy.polys.domains.integerring import ZZ
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.domains.realfield import RR
from sympy.polys.orderings import (grlex, ilex)
from sympy.polys.polytools import groebner
from sympy.polys.rootoftools import (RootSum, rootof)
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
from sympy.series.limits import Limit
from sympy.series.order import O
from sympy.series.sequences import (SeqAdd, SeqFormula, SeqMul, SeqPer)
from sympy.sets.contains import Contains
from sympy.sets.fancysets import Range
from sympy.sets.sets import (Complement, FiniteSet, Intersection, Interval, Union)
from sympy.codegen.ast import (Assignment, AddAugmentedAssignment,
SubAugmentedAssignment, MulAugmentedAssignment, DivAugmentedAssignment, ModAugmentedAssignment)
from sympy.core.expr import UnevaluatedExpr
from sympy.physics.quantum.trace import Tr
from sympy.functions import (Abs, Chi, Ci, Ei, KroneckerDelta,
Piecewise, Shi, Si, atan2, beta, binomial, catalan, ceiling, cos,
euler, exp, expint, factorial, factorial2, floor, gamma, hyper, log,
meijerg, sin, sqrt, subfactorial, tan, uppergamma, lerchphi,
elliptic_k, elliptic_f, elliptic_e, elliptic_pi, DiracDelta, bell,
bernoulli, fibonacci, tribonacci, lucas, stieltjes, mathieuc, mathieus,
mathieusprime, mathieucprime)
from sympy.matrices import Adjoint, Inverse, MatrixSymbol, Transpose, KroneckerProduct
from sympy.matrices.expressions import hadamard_power
from sympy.physics import mechanics
from sympy.physics.control.lti import (TransferFunction, Feedback, TransferFunctionMatrix,
Series, Parallel, MIMOSeries, MIMOParallel, MIMOFeedback)
from sympy.physics.units import joule, degree
from sympy.printing.pretty import pprint, pretty as xpretty
from sympy.printing.pretty.pretty_symbology import center_accent, is_combining
from sympy.sets.conditionset import ConditionSet
from sympy.sets import ImageSet, ProductSet
from sympy.sets.setexpr import SetExpr
from sympy.stats.crv_types import Normal
from sympy.stats.symbolic_probability import (Covariance, Expectation,
Probability, Variance)
from sympy.tensor.array import (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray, tensorproduct)
from sympy.tensor.functions import TensorProduct
from sympy.tensor.tensor import (TensorIndexType, tensor_indices, TensorHead,
TensorElement, tensor_heads)
from sympy.testing.pytest import raises, _both_exp_pow, warns_deprecated_sympy
from sympy.vector import CoordSys3D, Gradient, Curl, Divergence, Dot, Cross, Laplacian
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
a, b, c, d, x, y, z, k, n, s, p = symbols('a,b,c,d,x,y,z,k,n,s,p')
f = Function("f")
th = Symbol('theta')
ph = Symbol('phi')
"""
Expressions whose pretty-printing is tested here:
(A '#' to the right of an expression indicates that its various acceptable
orderings are accounted for by the tests.)
BASIC EXPRESSIONS:
oo
(x**2)
1/x
y*x**-2
x**Rational(-5,2)
(-2)**x
Pow(3, 1, evaluate=False)
(x**2 + x + 1) #
1-x #
1-2*x #
x/y
-x/y
(x+2)/y #
(1+x)*y #3
-5*x/(x+10) # correct placement of negative sign
1 - Rational(3,2)*(x+1)
-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5) # issue 5524
ORDERING:
x**2 + x + 1
1 - x
1 - 2*x
2*x**4 + y**2 - x**2 + y**3
RELATIONAL:
Eq(x, y)
Lt(x, y)
Gt(x, y)
Le(x, y)
Ge(x, y)
Ne(x/(y+1), y**2) #
RATIONAL NUMBERS:
y*x**-2
y**Rational(3,2) * x**Rational(-5,2)
sin(x)**3/tan(x)**2
FUNCTIONS (ABS, CONJ, EXP, FUNCTION BRACES, FACTORIAL, FLOOR, CEILING):
(2*x + exp(x)) #
Abs(x)
Abs(x/(x**2+1)) #
Abs(1 / (y - Abs(x)))
factorial(n)
factorial(2*n)
subfactorial(n)
subfactorial(2*n)
factorial(factorial(factorial(n)))
factorial(n+1) #
conjugate(x)
conjugate(f(x+1)) #
f(x)
f(x, y)
f(x/(y+1), y) #
f(x**x**x**x**x**x)
sin(x)**2
conjugate(a+b*I)
conjugate(exp(a+b*I))
conjugate( f(1 + conjugate(f(x))) ) #
f(x/(y+1), y) # denom of first arg
floor(1 / (y - floor(x)))
ceiling(1 / (y - ceiling(x)))
SQRT:
sqrt(2)
2**Rational(1,3)
2**Rational(1,1000)
sqrt(x**2 + 1)
(1 + sqrt(5))**Rational(1,3)
2**(1/x)
sqrt(2+pi)
(2+(1+x**2)/(2+x))**Rational(1,4)+(1+x**Rational(1,1000))/sqrt(3+x**2)
DERIVATIVES:
Derivative(log(x), x, evaluate=False)
Derivative(log(x), x, evaluate=False) + x #
Derivative(log(x) + x**2, x, y, evaluate=False)
Derivative(2*x*y, y, x, evaluate=False) + x**2 #
beta(alpha).diff(alpha)
INTEGRALS:
Integral(log(x), x)
Integral(x**2, x)
Integral((sin(x))**2 / (tan(x))**2)
Integral(x**(2**x), x)
Integral(x**2, (x,1,2))
Integral(x**2, (x,Rational(1,2),10))
Integral(x**2*y**2, x,y)
Integral(x**2, (x, None, 1))
Integral(x**2, (x, 1, None))
Integral(sin(th)/cos(ph), (th,0,pi), (ph, 0, 2*pi))
MATRICES:
Matrix([[x**2+1, 1], [y, x+y]]) #
Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
PIECEWISE:
Piecewise((x,x<1),(x**2,True))
ITE:
ITE(x, y, z)
SEQUENCES (TUPLES, LISTS, DICTIONARIES):
()
[]
{}
(1/x,)
[x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
{x: sin(x)}
{1/x: 1/y, x: sin(x)**2} #
[x**2]
(x**2,)
{x**2: 1}
LIMITS:
Limit(x, x, oo)
Limit(x**2, x, 0)
Limit(1/x, x, 0)
Limit(sin(x)/x, x, 0)
UNITS:
joule => kg*m**2/s
SUBS:
Subs(f(x), x, ph**2)
Subs(f(x).diff(x), x, 0)
Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ORDER:
O(1)
O(1/x)
O(x**2 + y**2)
"""
def pretty(expr, order=None):
"""ASCII pretty-printing"""
return xpretty(expr, order=order, use_unicode=False, wrap_line=False)
def upretty(expr, order=None):
"""Unicode pretty-printing"""
return xpretty(expr, order=order, use_unicode=True, wrap_line=False)
def test_pretty_ascii_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( "xxx" ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_pretty_unicode_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( 'xxx' ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_upretty_greek():
assert upretty( oo ) == '∞'
assert upretty( Symbol('alpha^+_1') ) == 'α⁺₁'
assert upretty( Symbol('beta') ) == 'β'
assert upretty(Symbol('lambda')) == 'λ'
def test_upretty_multiindex():
assert upretty( Symbol('beta12') ) == 'β₁₂'
assert upretty( Symbol('Y00') ) == 'Y₀₀'
assert upretty( Symbol('Y_00') ) == 'Y₀₀'
assert upretty( Symbol('F^+-') ) == 'F⁺⁻'
def test_upretty_sub_super():
assert upretty( Symbol('beta_1_2') ) == 'β₁ ₂'
assert upretty( Symbol('beta^1^2') ) == 'β¹ ²'
assert upretty( Symbol('beta_1^2') ) == 'β²₁'
assert upretty( Symbol('beta_10_20') ) == 'β₁₀ ₂₀'
assert upretty( Symbol('beta_ax_gamma^i') ) == 'βⁱₐₓ ᵧ'
assert upretty( Symbol("F^1^2_3_4") ) == 'F¹ ²₃ ₄'
assert upretty( Symbol("F_1_2^3^4") ) == 'F³ ⁴₁ ₂'
assert upretty( Symbol("F_1_2_3_4") ) == 'F₁ ₂ ₃ ₄'
assert upretty( Symbol("F^1^2^3^4") ) == 'F¹ ² ³ ⁴'
def test_upretty_subs_missing_in_24():
assert upretty( Symbol('F_beta') ) == 'Fᵦ'
assert upretty( Symbol('F_gamma') ) == 'Fᵧ'
assert upretty( Symbol('F_rho') ) == 'Fᵨ'
assert upretty( Symbol('F_phi') ) == 'Fᵩ'
assert upretty( Symbol('F_chi') ) == 'Fᵪ'
assert upretty( Symbol('F_a') ) == 'Fₐ'
assert upretty( Symbol('F_e') ) == 'Fₑ'
assert upretty( Symbol('F_i') ) == 'Fᵢ'
assert upretty( Symbol('F_o') ) == 'Fₒ'
assert upretty( Symbol('F_u') ) == 'Fᵤ'
assert upretty( Symbol('F_r') ) == 'Fᵣ'
assert upretty( Symbol('F_v') ) == 'Fᵥ'
assert upretty( Symbol('F_x') ) == 'Fₓ'
def test_missing_in_2X_issue_9047():
assert upretty( Symbol('F_h') ) == 'Fₕ'
assert upretty( Symbol('F_k') ) == 'Fₖ'
assert upretty( Symbol('F_l') ) == 'Fₗ'
assert upretty( Symbol('F_m') ) == 'Fₘ'
assert upretty( Symbol('F_n') ) == 'Fₙ'
assert upretty( Symbol('F_p') ) == 'Fₚ'
assert upretty( Symbol('F_s') ) == 'Fₛ'
assert upretty( Symbol('F_t') ) == 'Fₜ'
def test_upretty_modifiers():
# Accents
assert upretty( Symbol('Fmathring') ) == 'F̊'
assert upretty( Symbol('Fddddot') ) == 'F⃜'
assert upretty( Symbol('Fdddot') ) == 'F⃛'
assert upretty( Symbol('Fddot') ) == 'F̈'
assert upretty( Symbol('Fdot') ) == 'Ḟ'
assert upretty( Symbol('Fcheck') ) == 'F̌'
assert upretty( Symbol('Fbreve') ) == 'F̆'
assert upretty( Symbol('Facute') ) == 'F́'
assert upretty( Symbol('Fgrave') ) == 'F̀'
assert upretty( Symbol('Ftilde') ) == 'F̃'
assert upretty( Symbol('Fhat') ) == 'F̂'
assert upretty( Symbol('Fbar') ) == 'F̅'
assert upretty( Symbol('Fvec') ) == 'F⃗'
assert upretty( Symbol('Fprime') ) == 'F′'
assert upretty( Symbol('Fprm') ) == 'F′'
# No faces are actually implemented, but test to make sure the modifiers are stripped
assert upretty( Symbol('Fbold') ) == 'Fbold'
assert upretty( Symbol('Fbm') ) == 'Fbm'
assert upretty( Symbol('Fcal') ) == 'Fcal'
assert upretty( Symbol('Fscr') ) == 'Fscr'
assert upretty( Symbol('Ffrak') ) == 'Ffrak'
# Brackets
assert upretty( Symbol('Fnorm') ) == '‖F‖'
assert upretty( Symbol('Favg') ) == '⟨F⟩'
assert upretty( Symbol('Fabs') ) == '|F|'
assert upretty( Symbol('Fmag') ) == '|F|'
# Combinations
assert upretty( Symbol('xvecdot') ) == 'x⃗̇'
assert upretty( Symbol('xDotVec') ) == 'ẋ⃗'
assert upretty( Symbol('xHATNorm') ) == '‖x̂‖'
assert upretty( Symbol('xMathring_yCheckPRM__zbreveAbs') ) == 'x̊_y̌′__|z̆|'
assert upretty( Symbol('alphadothat_nVECDOT__tTildePrime') ) == 'α̇̂_n⃗̇__t̃′'
assert upretty( Symbol('x_dot') ) == 'x_dot'
assert upretty( Symbol('x__dot') ) == 'x__dot'
def test_pretty_Cycle():
from sympy.combinatorics.permutations import Cycle
assert pretty(Cycle(1, 2)) == '(1 2)'
assert pretty(Cycle(2)) == '(2)'
assert pretty(Cycle(1, 3)(4, 5)) == '(1 3)(4 5)'
assert pretty(Cycle()) == '()'
def test_pretty_Permutation():
from sympy.combinatorics.permutations import Permutation
p1 = Permutation(1, 2)(3, 4)
assert xpretty(p1, perm_cyclic=True, use_unicode=True) == "(1 2)(3 4)"
assert xpretty(p1, perm_cyclic=True, use_unicode=False) == "(1 2)(3 4)"
assert xpretty(p1, perm_cyclic=False, use_unicode=True) == \
'⎛0 1 2 3 4⎞\n'\
'⎝0 2 1 4 3⎠'
assert xpretty(p1, perm_cyclic=False, use_unicode=False) == \
"/0 1 2 3 4\\\n"\
"\\0 2 1 4 3/"
with warns_deprecated_sympy():
old_print_cyclic = Permutation.print_cyclic
Permutation.print_cyclic = False
assert xpretty(p1, use_unicode=True) == \
'⎛0 1 2 3 4⎞\n'\
'⎝0 2 1 4 3⎠'
assert xpretty(p1, use_unicode=False) == \
"/0 1 2 3 4\\\n"\
"\\0 2 1 4 3/"
Permutation.print_cyclic = old_print_cyclic
def test_pretty_basic():
assert pretty( -Rational(1)/2 ) == '-1/2'
assert pretty( -Rational(13)/22 ) == \
"""\
-13 \n\
----\n\
22 \
"""
expr = oo
ascii_str = \
"""\
oo\
"""
ucode_str = \
"""\
∞\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2)
ascii_str = \
"""\
2\n\
x \
"""
ucode_str = \
"""\
2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 1/x
ascii_str = \
"""\
1\n\
-\n\
x\
"""
ucode_str = \
"""\
1\n\
─\n\
x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# not the same as 1/x
expr = x**-1.0
ascii_str = \
"""\
-1.0\n\
x \
"""
ucode_str = \
"""\
-1.0\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# see issue #2860
expr = Pow(S(2), -1.0, evaluate=False)
ascii_str = \
"""\
-1.0\n\
2 \
"""
ucode_str = \
"""\
-1.0\n\
2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
"""\
y \n\
──\n\
2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
#see issue #14033
expr = x**Rational(1, 3)
ascii_str = \
"""\
1/3\n\
x \
"""
ucode_str = \
"""\
1/3\n\
x \
"""
assert xpretty(expr, use_unicode=False, wrap_line=False,\
root_notation = False) == ascii_str
assert xpretty(expr, use_unicode=True, wrap_line=False,\
root_notation = False) == ucode_str
expr = x**Rational(-5, 2)
ascii_str = \
"""\
1 \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
"""\
1 \n\
────\n\
5/2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (-2)**x
ascii_str = \
"""\
x\n\
(-2) \
"""
ucode_str = \
"""\
x\n\
(-2) \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# See issue 4923
expr = Pow(3, 1, evaluate=False)
ascii_str = \
"""\
1\n\
3 \
"""
ucode_str = \
"""\
1\n\
3 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2 + x + 1)
ascii_str_1 = \
"""\
2\n\
1 + x + x \
"""
ascii_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ascii_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
ucode_str_1 = \
"""\
2\n\
1 + x + x \
"""
ucode_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ucode_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = 1 - x
ascii_str_1 = \
"""\
1 - x\
"""
ascii_str_2 = \
"""\
-x + 1\
"""
ucode_str_1 = \
"""\
1 - x\
"""
ucode_str_2 = \
"""\
-x + 1\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 1 - 2*x
ascii_str_1 = \
"""\
1 - 2*x\
"""
ascii_str_2 = \
"""\
-2*x + 1\
"""
ucode_str_1 = \
"""\
1 - 2⋅x\
"""
ucode_str_2 = \
"""\
-2⋅x + 1\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = x/y
ascii_str = \
"""\
x\n\
-\n\
y\
"""
ucode_str = \
"""\
x\n\
─\n\
y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/y
ascii_str = \
"""\
-x \n\
---\n\
y \
"""
ucode_str = \
"""\
-x \n\
───\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x + 2)/y
ascii_str_1 = \
"""\
2 + x\n\
-----\n\
y \
"""
ascii_str_2 = \
"""\
x + 2\n\
-----\n\
y \
"""
ucode_str_1 = \
"""\
2 + x\n\
─────\n\
y \
"""
ucode_str_2 = \
"""\
x + 2\n\
─────\n\
y \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = (1 + x)*y
ascii_str_1 = \
"""\
y*(1 + x)\
"""
ascii_str_2 = \
"""\
(1 + x)*y\
"""
ascii_str_3 = \
"""\
y*(x + 1)\
"""
ucode_str_1 = \
"""\
y⋅(1 + x)\
"""
ucode_str_2 = \
"""\
(1 + x)⋅y\
"""
ucode_str_3 = \
"""\
y⋅(x + 1)\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
# Test for correct placement of the negative sign
expr = -5*x/(x + 10)
ascii_str_1 = \
"""\
-5*x \n\
------\n\
10 + x\
"""
ascii_str_2 = \
"""\
-5*x \n\
------\n\
x + 10\
"""
ucode_str_1 = \
"""\
-5⋅x \n\
──────\n\
10 + x\
"""
ucode_str_2 = \
"""\
-5⋅x \n\
──────\n\
x + 10\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = -S.Half - 3*x
ascii_str = \
"""\
-3*x - 1/2\
"""
ucode_str = \
"""\
-3⋅x - 1/2\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S.Half - 3*x
ascii_str = \
"""\
1/2 - 3*x\
"""
ucode_str = \
"""\
1/2 - 3⋅x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -S.Half - 3*x/2
ascii_str = \
"""\
3*x 1\n\
- --- - -\n\
2 2\
"""
ucode_str = \
"""\
3⋅x 1\n\
- ─── - ─\n\
2 2\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S.Half - 3*x/2
ascii_str = \
"""\
1 3*x\n\
- - ---\n\
2 2 \
"""
ucode_str = \
"""\
1 3⋅x\n\
─ - ───\n\
2 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_negative_fractions():
expr = -x/y
ascii_str =\
"""\
-x \n\
---\n\
y \
"""
ucode_str =\
"""\
-x \n\
───\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x*z/y
ascii_str =\
"""\
-x*z \n\
-----\n\
y \
"""
ucode_str =\
"""\
-x⋅z \n\
─────\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x**2/y
ascii_str =\
"""\
2\n\
x \n\
--\n\
y \
"""
ucode_str =\
"""\
2\n\
x \n\
──\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x**2/y
ascii_str =\
"""\
2 \n\
-x \n\
----\n\
y \
"""
ucode_str =\
"""\
2 \n\
-x \n\
────\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/(y*z)
ascii_str =\
"""\
-x \n\
---\n\
y*z\
"""
ucode_str =\
"""\
-x \n\
───\n\
y⋅z\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -a/y**2
ascii_str =\
"""\
-a \n\
---\n\
2\n\
y \
"""
ucode_str =\
"""\
-a \n\
───\n\
2\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**(-a/b)
ascii_str =\
"""\
-a \n\
---\n\
b \n\
y \
"""
ucode_str =\
"""\
-a \n\
───\n\
b \n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -1/y**2
ascii_str =\
"""\
-1 \n\
---\n\
2\n\
y \
"""
ucode_str =\
"""\
-1 \n\
───\n\
2\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -10/b**2
ascii_str =\
"""\
-10 \n\
----\n\
2 \n\
b \
"""
ucode_str =\
"""\
-10 \n\
────\n\
2 \n\
b \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Rational(-200, 37)
ascii_str =\
"""\
-200 \n\
-----\n\
37 \
"""
ucode_str =\
"""\
-200 \n\
─────\n\
37 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_Mul():
expr = Mul(0, 1, evaluate=False)
assert pretty(expr) == "0*1"
assert upretty(expr) == "0⋅1"
expr = Mul(1, 0, evaluate=False)
assert pretty(expr) == "1*0"
assert upretty(expr) == "1⋅0"
expr = Mul(1, 1, evaluate=False)
assert pretty(expr) == "1*1"
assert upretty(expr) == "1⋅1"
expr = Mul(1, 1, 1, evaluate=False)
assert pretty(expr) == "1*1*1"
assert upretty(expr) == "1⋅1⋅1"
expr = Mul(1, 2, evaluate=False)
assert pretty(expr) == "1*2"
assert upretty(expr) == "1⋅2"
expr = Add(0, 1, evaluate=False)
assert pretty(expr) == "0 + 1"
assert upretty(expr) == "0 + 1"
expr = Mul(1, 1, 2, evaluate=False)
assert pretty(expr) == "1*1*2"
assert upretty(expr) == "1⋅1⋅2"
expr = Add(0, 0, 1, evaluate=False)
assert pretty(expr) == "0 + 0 + 1"
assert upretty(expr) == "0 + 0 + 1"
expr = Mul(1, -1, evaluate=False)
assert pretty(expr) == "1*-1"
assert upretty(expr) == "1⋅-1"
expr = Mul(1.0, x, evaluate=False)
assert pretty(expr) == "1.0*x"
assert upretty(expr) == "1.0⋅x"
expr = Mul(1, 1, 2, 3, x, evaluate=False)
assert pretty(expr) == "1*1*2*3*x"
assert upretty(expr) == "1⋅1⋅2⋅3⋅x"
expr = Mul(-1, 1, evaluate=False)
assert pretty(expr) == "-1*1"
assert upretty(expr) == "-1⋅1"
expr = Mul(4, 3, 2, 1, 0, y, x, evaluate=False)
assert pretty(expr) == "4*3*2*1*0*y*x"
assert upretty(expr) == "4⋅3⋅2⋅1⋅0⋅y⋅x"
expr = Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)
assert pretty(expr) == "4*3*2*(z + 1)*0*y*x"
assert upretty(expr) == "4⋅3⋅2⋅(z + 1)⋅0⋅y⋅x"
expr = Mul(Rational(2, 3), Rational(5, 7), evaluate=False)
assert pretty(expr) == "2/3*5/7"
assert upretty(expr) == "2/3⋅5/7"
expr = Mul(x + y, Rational(1, 2), evaluate=False)
assert pretty(expr) == "(x + y)*1/2"
assert upretty(expr) == "(x + y)⋅1/2"
expr = Mul(Rational(1, 2), x + y, evaluate=False)
assert pretty(expr) == "x + y\n-----\n 2 "
assert upretty(expr) == "x + y\n─────\n 2 "
expr = Mul(S.One, x + y, evaluate=False)
assert pretty(expr) == "1*(x + y)"
assert upretty(expr) == "1⋅(x + y)"
expr = Mul(x - y, S.One, evaluate=False)
assert pretty(expr) == "(x - y)*1"
assert upretty(expr) == "(x - y)⋅1"
expr = Mul(Rational(1, 2), x - y, S.One, x + y, evaluate=False)
assert pretty(expr) == "1/2*(x - y)*1*(x + y)"
assert upretty(expr) == "1/2⋅(x - y)⋅1⋅(x + y)"
expr = Mul(x + y, Rational(3, 4), S.One, y - z, evaluate=False)
assert pretty(expr) == "(x + y)*3/4*1*(y - z)"
assert upretty(expr) == "(x + y)⋅3/4⋅1⋅(y - z)"
expr = Mul(x + y, Rational(1, 1), Rational(3, 4), Rational(5, 6),evaluate=False)
assert pretty(expr) == "(x + y)*1*3/4*5/6"
assert upretty(expr) == "(x + y)⋅1⋅3/4⋅5/6"
expr = Mul(Rational(3, 4), x + y, S.One, y - z, evaluate=False)
assert pretty(expr) == "3/4*(x + y)*1*(y - z)"
assert upretty(expr) == "3/4⋅(x + y)⋅1⋅(y - z)"
def test_issue_5524():
assert pretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 / ___ \\\n\
- (5 - y) + (x - 5)*\\-x - 2*\\/ 2 + 5/\
"""
assert upretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 \n\
- (5 - y) + (x - 5)⋅(-x - 2⋅√2 + 5)\
"""
def test_pretty_ordering():
assert pretty(x**2 + x + 1, order='lex') == \
"""\
2 \n\
x + x + 1\
"""
assert pretty(x**2 + x + 1, order='rev-lex') == \
"""\
2\n\
1 + x + x \
"""
assert pretty(1 - x, order='lex') == '-x + 1'
assert pretty(1 - x, order='rev-lex') == '1 - x'
assert pretty(1 - 2*x, order='lex') == '-2*x + 1'
assert pretty(1 - 2*x, order='rev-lex') == '1 - 2*x'
f = 2*x**4 + y**2 - x**2 + y**3
assert pretty(f, order=None) == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='lex') == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='rev-lex') == \
"""\
2 3 2 4\n\
y + y - x + 2*x \
"""
expr = x - x**3/6 + x**5/120 + O(x**6)
ascii_str = \
"""\
3 5 \n\
x x / 6\\\n\
x - -- + --- + O\\x /\n\
6 120 \
"""
ucode_str = \
"""\
3 5 \n\
x x ⎛ 6⎞\n\
x - ── + ─── + O⎝x ⎠\n\
6 120 \
"""
assert pretty(expr, order=None) == ascii_str
assert upretty(expr, order=None) == ucode_str
assert pretty(expr, order='lex') == ascii_str
assert upretty(expr, order='lex') == ucode_str
assert pretty(expr, order='rev-lex') == ascii_str
assert upretty(expr, order='rev-lex') == ucode_str
def test_EulerGamma():
assert pretty(EulerGamma) == str(EulerGamma) == "EulerGamma"
assert upretty(EulerGamma) == "γ"
def test_GoldenRatio():
assert pretty(GoldenRatio) == str(GoldenRatio) == "GoldenRatio"
assert upretty(GoldenRatio) == "φ"
def test_Catalan():
assert pretty(Catalan) == upretty(Catalan) == "G"
def test_pretty_relational():
expr = Eq(x, y)
ascii_str = \
"""\
x = y\
"""
ucode_str = \
"""\
x = y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lt(x, y)
ascii_str = \
"""\
x < y\
"""
ucode_str = \
"""\
x < y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Gt(x, y)
ascii_str = \
"""\
x > y\
"""
ucode_str = \
"""\
x > y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Le(x, y)
ascii_str = \
"""\
x <= y\
"""
ucode_str = \
"""\
x ≤ y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ge(x, y)
ascii_str = \
"""\
x >= y\
"""
ucode_str = \
"""\
x ≥ y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ne(x/(y + 1), y**2)
ascii_str_1 = \
"""\
x 2\n\
----- != y \n\
1 + y \
"""
ascii_str_2 = \
"""\
x 2\n\
----- != y \n\
y + 1 \
"""
ucode_str_1 = \
"""\
x 2\n\
───── ≠ y \n\
1 + y \
"""
ucode_str_2 = \
"""\
x 2\n\
───── ≠ y \n\
y + 1 \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
def test_Assignment():
expr = Assignment(x, y)
ascii_str = \
"""\
x := y\
"""
ucode_str = \
"""\
x := y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_AugmentedAssignment():
expr = AddAugmentedAssignment(x, y)
ascii_str = \
"""\
x += y\
"""
ucode_str = \
"""\
x += y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = SubAugmentedAssignment(x, y)
ascii_str = \
"""\
x -= y\
"""
ucode_str = \
"""\
x -= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = MulAugmentedAssignment(x, y)
ascii_str = \
"""\
x *= y\
"""
ucode_str = \
"""\
x *= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = DivAugmentedAssignment(x, y)
ascii_str = \
"""\
x /= y\
"""
ucode_str = \
"""\
x /= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ModAugmentedAssignment(x, y)
ascii_str = \
"""\
x %= y\
"""
ucode_str = \
"""\
x %= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_rational():
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
"""\
y \n\
──\n\
2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**Rational(3, 2) * x**Rational(-5, 2)
ascii_str = \
"""\
3/2\n\
y \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
"""\
3/2\n\
y \n\
────\n\
5/2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**3/tan(x)**2
ascii_str = \
"""\
3 \n\
sin (x)\n\
-------\n\
2 \n\
tan (x)\
"""
ucode_str = \
"""\
3 \n\
sin (x)\n\
───────\n\
2 \n\
tan (x)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
@_both_exp_pow
def test_pretty_functions():
"""Tests for Abs, conjugate, exp, function braces, and factorial."""
expr = (2*x + exp(x))
ascii_str_1 = \
"""\
x\n\
2*x + e \
"""
ascii_str_2 = \
"""\
x \n\
e + 2*x\
"""
ucode_str_1 = \
"""\
x\n\
2⋅x + ℯ \
"""
ucode_str_2 = \
"""\
x \n\
ℯ + 2⋅x\
"""
ucode_str_3 = \
"""\
x \n\
ℯ + 2⋅x\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = Abs(x)
ascii_str = \
"""\
|x|\
"""
ucode_str = \
"""\
│x│\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Abs(x/(x**2 + 1))
ascii_str_1 = \
"""\
| x |\n\
|------|\n\
| 2|\n\
|1 + x |\
"""
ascii_str_2 = \
"""\
| x |\n\
|------|\n\
| 2 |\n\
|x + 1|\
"""
ucode_str_1 = \
"""\
│ x │\n\
│──────│\n\
│ 2│\n\
│1 + x │\
"""
ucode_str_2 = \
"""\
│ x │\n\
│──────│\n\
│ 2 │\n\
│x + 1│\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(1 / (y - Abs(x)))
ascii_str = \
"""\
1 \n\
---------\n\
|y - |x||\
"""
ucode_str = \
"""\
1 \n\
─────────\n\
│y - │x││\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial(n)
ascii_str = \
"""\
n!\
"""
ucode_str = \
"""\
n!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(2*n)
ascii_str = \
"""\
(2*n)!\
"""
ucode_str = \
"""\
(2⋅n)!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(factorial(factorial(n)))
ascii_str = \
"""\
((n!)!)!\
"""
ucode_str = \
"""\
((n!)!)!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(n + 1)
ascii_str_1 = \
"""\
(1 + n)!\
"""
ascii_str_2 = \
"""\
(n + 1)!\
"""
ucode_str_1 = \
"""\
(1 + n)!\
"""
ucode_str_2 = \
"""\
(n + 1)!\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = subfactorial(n)
ascii_str = \
"""\
!n\
"""
ucode_str = \
"""\
!n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = subfactorial(2*n)
ascii_str = \
"""\
!(2*n)\
"""
ucode_str = \
"""\
!(2⋅n)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial2(n)
ascii_str = \
"""\
n!!\
"""
ucode_str = \
"""\
n!!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(2*n)
ascii_str = \
"""\
(2*n)!!\
"""
ucode_str = \
"""\
(2⋅n)!!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(factorial2(factorial2(n)))
ascii_str = \
"""\
((n!!)!!)!!\
"""
ucode_str = \
"""\
((n!!)!!)!!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(n + 1)
ascii_str_1 = \
"""\
(1 + n)!!\
"""
ascii_str_2 = \
"""\
(n + 1)!!\
"""
ucode_str_1 = \
"""\
(1 + n)!!\
"""
ucode_str_2 = \
"""\
(n + 1)!!\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 2*binomial(n, k)
ascii_str = \
"""\
/n\\\n\
2*| |\n\
\\k/\
"""
ucode_str = \
"""\
⎛n⎞\n\
2⋅⎜ ⎟\n\
⎝k⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(2*n, k)
ascii_str = \
"""\
/2*n\\\n\
2*| |\n\
\\ k /\
"""
ucode_str = \
"""\
⎛2⋅n⎞\n\
2⋅⎜ ⎟\n\
⎝ k ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(n**2, k)
ascii_str = \
"""\
/ 2\\\n\
|n |\n\
2*| |\n\
\\k /\
"""
ucode_str = \
"""\
⎛ 2⎞\n\
⎜n ⎟\n\
2⋅⎜ ⎟\n\
⎝k ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
"""\
C \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
"""\
C \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bell(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
"""\
B \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
"""\
B \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n, x)
ascii_str = \
"""\
B (x)\n\
n \
"""
ucode_str = \
"""\
B (x)\n\
n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = fibonacci(n)
ascii_str = \
"""\
F \n\
n\
"""
ucode_str = \
"""\
F \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = lucas(n)
ascii_str = \
"""\
L \n\
n\
"""
ucode_str = \
"""\
L \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = tribonacci(n)
ascii_str = \
"""\
T \n\
n\
"""
ucode_str = \
"""\
T \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = stieltjes(n)
ascii_str = \
"""\
stieltjes \n\
n\
"""
ucode_str = \
"""\
γ \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = stieltjes(n, x)
ascii_str = \
"""\
stieltjes (x)\n\
n \
"""
ucode_str = \
"""\
γ (x)\n\
n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieuc(x, y, z)
ascii_str = 'C(x, y, z)'
ucode_str = 'C(x, y, z)'
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieus(x, y, z)
ascii_str = 'S(x, y, z)'
ucode_str = 'S(x, y, z)'
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieucprime(x, y, z)
ascii_str = "C'(x, y, z)"
ucode_str = "C'(x, y, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieusprime(x, y, z)
ascii_str = "S'(x, y, z)"
ucode_str = "S'(x, y, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(x)
ascii_str = \
"""\
_\n\
x\
"""
ucode_str = \
"""\
_\n\
x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
f = Function('f')
expr = conjugate(f(x + 1))
ascii_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ascii_str_2 = \
"""\
________\n\
f(x + 1)\
"""
ucode_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ucode_str_2 = \
"""\
________\n\
f(x + 1)\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x)
ascii_str = \
"""\
f(x)\
"""
ucode_str = \
"""\
f(x)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x, y)
ascii_str = \
"""\
f(x, y)\
"""
ucode_str = \
"""\
f(x, y)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
"""
ucode_str_2 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x**x**x**x**x**x)
ascii_str = \
"""\
/ / / / / x\\\\\\\\\\
| | | | \\x /||||
| | | \\x /|||
| | \\x /||
| \\x /|
f\\x /\
"""
ucode_str = \
"""\
⎛ ⎛ ⎛ ⎛ ⎛ x⎞⎞⎞⎞⎞
⎜ ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟⎟
⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟
⎜ ⎜ ⎝x ⎠⎟⎟
⎜ ⎝x ⎠⎟
f⎝x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**2
ascii_str = \
"""\
2 \n\
sin (x)\
"""
ucode_str = \
"""\
2 \n\
sin (x)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(a + b*I)
ascii_str = \
"""\
_ _\n\
a - I*b\
"""
ucode_str = \
"""\
_ _\n\
a - ⅈ⋅b\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(exp(a + b*I))
ascii_str = \
"""\
_ _\n\
a - I*b\n\
e \
"""
ucode_str = \
"""\
_ _\n\
a - ⅈ⋅b\n\
ℯ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate( f(1 + conjugate(f(x))) )
ascii_str_1 = \
"""\
___________\n\
/ ____\\\n\
f\\1 + f(x)/\
"""
ascii_str_2 = \
"""\
___________\n\
/____ \\\n\
f\\f(x) + 1/\
"""
ucode_str_1 = \
"""\
___________\n\
⎛ ____⎞\n\
f⎝1 + f(x)⎠\
"""
ucode_str_2 = \
"""\
___________\n\
⎛____ ⎞\n\
f⎝f(x) + 1⎠\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
"""
ucode_str_2 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = floor(1 / (y - floor(x)))
ascii_str = \
"""\
/ 1 \\\n\
floor|------------|\n\
\\y - floor(x)/\
"""
ucode_str = \
"""\
⎢ 1 ⎥\n\
⎢───────⎥\n\
⎣y - ⌊x⌋⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ceiling(1 / (y - ceiling(x)))
ascii_str = \
"""\
/ 1 \\\n\
ceiling|--------------|\n\
\\y - ceiling(x)/\
"""
ucode_str = \
"""\
⎡ 1 ⎤\n\
⎢───────⎥\n\
⎢y - ⌈x⌉⎥\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n)
ascii_str = \
"""\
E \n\
n\
"""
ucode_str = \
"""\
E \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(1/(1 + 1/(1 + 1/n)))
ascii_str = \
"""\
E \n\
1 \n\
---------\n\
1 \n\
1 + -----\n\
1\n\
1 + -\n\
n\
"""
ucode_str = \
"""\
E \n\
1 \n\
─────────\n\
1 \n\
1 + ─────\n\
1\n\
1 + ─\n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x)
ascii_str = \
"""\
E (x)\n\
n \
"""
ucode_str = \
"""\
E (x)\n\
n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x/2)
ascii_str = \
"""\
/x\\\n\
E |-|\n\
n\\2/\
"""
ucode_str = \
"""\
⎛x⎞\n\
E ⎜─⎟\n\
n⎝2⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt():
expr = sqrt(2)
ascii_str = \
"""\
___\n\
\\/ 2 \
"""
ucode_str = \
"√2"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 3)
ascii_str = \
"""\
3 ___\n\
\\/ 2 \
"""
ucode_str = \
"""\
3 ___\n\
╲╱ 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 1000)
ascii_str = \
"""\
1000___\n\
\\/ 2 \
"""
ucode_str = \
"""\
1000___\n\
╲╱ 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(x**2 + 1)
ascii_str = \
"""\
________\n\
/ 2 \n\
\\/ x + 1 \
"""
ucode_str = \
"""\
________\n\
╱ 2 \n\
╲╱ x + 1 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1 + sqrt(5))**Rational(1, 3)
ascii_str = \
"""\
___________\n\
3 / ___ \n\
\\/ 1 + \\/ 5 \
"""
ucode_str = \
"""\
3 ________\n\
╲╱ 1 + √5 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**(1/x)
ascii_str = \
"""\
x ___\n\
\\/ 2 \
"""
ucode_str = \
"""\
x ___\n\
╲╱ 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(2 + pi)
ascii_str = \
"""\
________\n\
\\/ 2 + pi \
"""
ucode_str = \
"""\
_______\n\
╲╱ 2 + π \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (2 + (
1 + x**2)/(2 + x))**Rational(1, 4) + (1 + x**Rational(1, 1000))/sqrt(3 + x**2)
ascii_str = \
"""\
____________ \n\
/ 2 1000___ \n\
/ x + 1 \\/ x + 1\n\
4 / 2 + ------ + -----------\n\
\\/ x + 2 ________\n\
/ 2 \n\
\\/ x + 3 \
"""
ucode_str = \
"""\
____________ \n\
╱ 2 1000___ \n\
╱ x + 1 ╲╱ x + 1\n\
4 ╱ 2 + ────── + ───────────\n\
╲╱ x + 2 ________\n\
╱ 2 \n\
╲╱ x + 3 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt_char_knob():
# See PR #9234.
expr = sqrt(2)
ucode_str1 = \
"""\
___\n\
╲╱ 2 \
"""
ucode_str2 = \
"√2"
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=False) == ucode_str1
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=True) == ucode_str2
def test_pretty_sqrt_longsymbol_no_sqrt_char():
# Do not use unicode sqrt char for long symbols (see PR #9234).
expr = sqrt(Symbol('C1'))
ucode_str = \
"""\
____\n\
╲╱ C₁ \
"""
assert upretty(expr) == ucode_str
def test_pretty_KroneckerDelta():
x, y = symbols("x, y")
expr = KroneckerDelta(x, y)
ascii_str = \
"""\
d \n\
x,y\
"""
ucode_str = \
"""\
δ \n\
x,y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_product():
n, m, k, l = symbols('n m k l')
f = symbols('f', cls=Function)
expr = Product(f((n/3)**2), (n, k**2, l))
unicode_str = \
"""\
l \n\
─┬──────┬─ \n\
│ │ ⎛ 2⎞\n\
│ │ ⎜n ⎟\n\
│ │ f⎜──⎟\n\
│ │ ⎝9 ⎠\n\
│ │ \n\
2 \n\
n = k """
ascii_str = \
"""\
l \n\
__________ \n\
| | / 2\\\n\
| | |n |\n\
| | f|--|\n\
| | \\9 /\n\
| | \n\
2 \n\
n = k """
expr = Product(f((n/3)**2), (n, k**2, l), (l, 1, m))
unicode_str = \
"""\
m l \n\
─┬──────┬─ ─┬──────┬─ \n\
│ │ │ │ ⎛ 2⎞\n\
│ │ │ │ ⎜n ⎟\n\
│ │ │ │ f⎜──⎟\n\
│ │ │ │ ⎝9 ⎠\n\
│ │ │ │ \n\
l = 1 2 \n\
n = k """
ascii_str = \
"""\
m l \n\
__________ __________ \n\
| | | | / 2\\\n\
| | | | |n |\n\
| | | | f|--|\n\
| | | | \\9 /\n\
| | | | \n\
l = 1 2 \n\
n = k """
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_pretty_Lambda():
# S.IdentityFunction is a special case
expr = Lambda(y, y)
assert pretty(expr) == "x -> x"
assert upretty(expr) == "x ↦ x"
expr = Lambda(x, x+1)
assert pretty(expr) == "x -> x + 1"
assert upretty(expr) == "x ↦ x + 1"
expr = Lambda(x, x**2)
ascii_str = \
"""\
2\n\
x -> x \
"""
ucode_str = \
"""\
2\n\
x ↦ x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(x, x**2)**2
ascii_str = \
"""\
2
/ 2\\ \n\
\\x -> x / \
"""
ucode_str = \
"""\
2
⎛ 2⎞ \n\
⎝x ↦ x ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x)
ascii_str = "(x, y) -> x"
ucode_str = "(x, y) ↦ x"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x**2)
ascii_str = \
"""\
2\n\
(x, y) -> x \
"""
ucode_str = \
"""\
2\n\
(x, y) ↦ x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(((x, y),), x**2)
ascii_str = \
"""\
2\n\
((x, y),) -> x \
"""
ucode_str = \
"""\
2\n\
((x, y),) ↦ x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_TransferFunction():
tf1 = TransferFunction(s - 1, s + 1, s)
assert upretty(tf1) == "s - 1\n─────\ns + 1"
tf2 = TransferFunction(2*s + 1, 3 - p, s)
assert upretty(tf2) == "2⋅s + 1\n───────\n 3 - p "
tf3 = TransferFunction(p, p + 1, p)
assert upretty(tf3) == " p \n─────\np + 1"
def test_pretty_Series():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(x**2 + y, y - x, y)
tf4 = TransferFunction(2, 3, y)
tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
tfm2 = TransferFunctionMatrix([[tf3], [-tf4]])
tfm3 = TransferFunctionMatrix([[tf1, -tf2, -tf3], [tf3, -tf4, tf2]])
tfm4 = TransferFunctionMatrix([[tf1, tf2], [tf3, -tf4], [-tf2, -tf1]])
tfm5 = TransferFunctionMatrix([[-tf2, -tf1], [tf4, -tf3], [tf1, tf2]])
expected1 = \
"""\
⎛ 2 ⎞\n\
⎛ x + y ⎞ ⎜x + y⎟\n\
⎜───────⎟⋅⎜──────⎟\n\
⎝x - 2⋅y⎠ ⎝-x + y⎠\
"""
expected2 = \
"""\
⎛-x + y⎞ ⎛ -x - y⎞\n\
⎜──────⎟⋅⎜───────⎟\n\
⎝x + y ⎠ ⎝x - 2⋅y⎠\
"""
expected3 = \
"""\
⎛ 2 ⎞ \n\
⎜x + y⎟ ⎛ x + y ⎞ ⎛ -x - y x - y⎞\n\
⎜──────⎟⋅⎜───────⎟⋅⎜─────── + ─────⎟\n\
⎝-x + y⎠ ⎝x - 2⋅y⎠ ⎝x - 2⋅y x + y⎠\
"""
expected4 = \
"""\
⎛ 2 ⎞\n\
⎛ x + y x - y⎞ ⎜x - y x + y⎟\n\
⎜─────── + ─────⎟⋅⎜───── + ──────⎟\n\
⎝x - 2⋅y x + y⎠ ⎝x + y -x + y⎠\
"""
expected5 = \
"""\
⎡ x + y x - y⎤ ⎡ 2 ⎤ \n\
⎢─────── ─────⎥ ⎢x + y⎥ \n\
⎢x - 2⋅y x + y⎥ ⎢──────⎥ \n\
⎢ ⎥ ⎢-x + y⎥ \n\
⎢ 2 ⎥ ⋅⎢ ⎥ \n\
⎢x + y 2 ⎥ ⎢ -2 ⎥ \n\
⎢────── ─ ⎥ ⎢ ─── ⎥ \n\
⎣-x + y 3 ⎦τ ⎣ 3 ⎦τ\
"""
expected6 = \
"""\
⎛⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎞\n\
⎜⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎟\n\
⎡ x + y x - y⎤ ⎡ 2 ⎤ ⎜⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎟\n\
⎢─────── ─────⎥ ⎢ x + y -x + y - x - y⎥ ⎜⎢ ⎥ ⎢ ⎥ ⎟\n\
⎢x - 2⋅y x + y⎥ ⎢─────── ────── ────────⎥ ⎜⎢ 2 ⎥ ⎢ 2 ⎥ ⎟\n\
⎢ ⎥ ⎢x - 2⋅y x + y -x + y ⎥ ⎜⎢x + y -2 ⎥ ⎢ -2 x + y ⎥ ⎟\n\
⎢ 2 ⎥ ⋅⎢ ⎥ ⋅⎜⎢────── ─── ⎥ + ⎢ ─── ────── ⎥ ⎟\n\
⎢x + y 2 ⎥ ⎢ 2 ⎥ ⎜⎢-x + y 3 ⎥ ⎢ 3 -x + y ⎥ ⎟\n\
⎢────── ─ ⎥ ⎢x + y -2 x - y ⎥ ⎜⎢ ⎥ ⎢ ⎥ ⎟\n\
⎣-x + y 3 ⎦τ ⎢────── ─── ───── ⎥ ⎜⎢-x + y -x - y⎥ ⎢ -x - y -x + y ⎥ ⎟\n\
⎣-x + y 3 x + y ⎦τ ⎜⎢────── ───────⎥ ⎢─────── ────── ⎥ ⎟\n\
⎝⎣x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ⎠\
"""
assert upretty(Series(tf1, tf3)) == expected1
assert upretty(Series(-tf2, -tf1)) == expected2
assert upretty(Series(tf3, tf1, Parallel(-tf1, tf2))) == expected3
assert upretty(Series(Parallel(tf1, tf2), Parallel(tf2, tf3))) == expected4
assert upretty(MIMOSeries(tfm2, tfm1)) == expected5
assert upretty(MIMOSeries(MIMOParallel(tfm4, -tfm5), tfm3, tfm1)) == expected6
def test_pretty_Parallel():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(x**2 + y, y - x, y)
tf4 = TransferFunction(y**2 - x, x**3 + x, y)
tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf3, -tf4], [-tf2, -tf1]])
tfm2 = TransferFunctionMatrix([[-tf2, -tf1], [tf4, -tf3], [tf1, tf2]])
tfm3 = TransferFunctionMatrix([[-tf1, tf2], [-tf3, tf4], [tf2, tf1]])
tfm4 = TransferFunctionMatrix([[-tf1, -tf2], [-tf3, -tf4]])
expected1 = \
"""\
x + y x - y\n\
─────── + ─────\n\
x - 2⋅y x + y\
"""
expected2 = \
"""\
-x + y -x - y\n\
────── + ───────\n\
x + y x - 2⋅y\
"""
expected3 = \
"""\
2 \n\
x + y x + y ⎛ -x - y⎞ ⎛x - y⎞\n\
────── + ─────── + ⎜───────⎟⋅⎜─────⎟\n\
-x + y x - 2⋅y ⎝x - 2⋅y⎠ ⎝x + y⎠\
"""
expected4 = \
"""\
⎛ 2 ⎞\n\
⎛ x + y ⎞ ⎛x - y⎞ ⎛x - y⎞ ⎜x + y⎟\n\
⎜───────⎟⋅⎜─────⎟ + ⎜─────⎟⋅⎜──────⎟\n\
⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x + y⎠ ⎝-x + y⎠\
"""
expected5 = \
"""\
⎡ x + y -x + y ⎤ ⎡ x - y x + y ⎤ ⎡ x + y x - y ⎤ \n\
⎢─────── ────── ⎥ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ \n\
⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎢x - 2⋅y x + y ⎥ \n\
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ \n\
⎢ 2 2 ⎥ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ \n\
⎢x + y x - y ⎥ ⎢x - y x + y ⎥ ⎢x + y x - y ⎥ \n\
⎢────── ────── ⎥ + ⎢────── ────── ⎥ + ⎢────── ────── ⎥ \n\
⎢-x + y 3 ⎥ ⎢ 3 -x + y ⎥ ⎢-x + y 3 ⎥ \n\
⎢ x + x ⎥ ⎢x + x ⎥ ⎢ x + x ⎥ \n\
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ \n\
⎢-x + y -x - y⎥ ⎢ -x - y -x + y ⎥ ⎢-x + y -x - y⎥ \n\
⎢────── ───────⎥ ⎢─────── ────── ⎥ ⎢────── ───────⎥ \n\
⎣x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ ⎣x + y x - 2⋅y⎦τ\
"""
expected6 = \
"""\
⎡ x - y x + y ⎤ ⎡-x + y -x - y ⎤ \n\
⎢ ───── ───────⎥ ⎢────── ─────── ⎥ \n\
⎢ x + y x - 2⋅y⎥ ⎡ -x - y -x + y⎤ ⎢x + y x - 2⋅y ⎥ \n\
⎢ ⎥ ⎢─────── ──────⎥ ⎢ ⎥ \n\
⎢ 2 2 ⎥ ⎢x - 2⋅y x + y ⎥ ⎢ 2 2 ⎥ \n\
⎢x - y x + y ⎥ ⎢ ⎥ ⎢-x + y - x - y⎥ \n\
⎢────── ────── ⎥ ⋅⎢ 2 2⎥ + ⎢─────── ────────⎥ \n\
⎢ 3 -x + y ⎥ ⎢- x - y x - y ⎥ ⎢ 3 -x + y ⎥ \n\
⎢x + x ⎥ ⎢──────── ──────⎥ ⎢ x + x ⎥ \n\
⎢ ⎥ ⎢ -x + y 3 ⎥ ⎢ ⎥ \n\
⎢ -x - y -x + y ⎥ ⎣ x + x⎦τ ⎢ x + y x - y ⎥ \n\
⎢─────── ────── ⎥ ⎢─────── ───── ⎥ \n\
⎣x - 2⋅y x + y ⎦τ ⎣x - 2⋅y x + y ⎦τ\
"""
assert upretty(Parallel(tf1, tf2)) == expected1
assert upretty(Parallel(-tf2, -tf1)) == expected2
assert upretty(Parallel(tf3, tf1, Series(-tf1, tf2))) == expected3
assert upretty(Parallel(Series(tf1, tf2), Series(tf2, tf3))) == expected4
assert upretty(MIMOParallel(-tfm3, -tfm2, tfm1)) == expected5
assert upretty(MIMOParallel(MIMOSeries(tfm4, -tfm2), tfm2)) == expected6
def test_pretty_Feedback():
tf = TransferFunction(1, 1, y)
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(y**2 - 2*y + 1, y + 5, y)
tf4 = TransferFunction(x - 2*y**3, x + y, x)
tf5 = TransferFunction(1 - x, x - y, y)
tf6 = TransferFunction(2, 2, x)
expected1 = \
"""\
⎛1⎞ \n\
⎜─⎟ \n\
⎝1⎠ \n\
─────────────\n\
1 ⎛ x + y ⎞\n\
─ + ⎜───────⎟\n\
1 ⎝x - 2⋅y⎠\
"""
expected2 = \
"""\
⎛1⎞ \n\
⎜─⎟ \n\
⎝1⎠ \n\
────────────────────────────────────\n\
⎛ 2 ⎞\n\
1 ⎛x - y⎞ ⎛ x + y ⎞ ⎜y - 2⋅y + 1⎟\n\
─ + ⎜─────⎟⋅⎜───────⎟⋅⎜────────────⎟\n\
1 ⎝x + y⎠ ⎝x - 2⋅y⎠ ⎝ y + 5 ⎠\
"""
expected3 = \
"""\
⎛ x + y ⎞ \n\
⎜───────⎟ \n\
⎝x - 2⋅y⎠ \n\
────────────────────────────────────────────\n\
⎛ 2 ⎞ \n\
1 ⎛ x + y ⎞ ⎛x - y⎞ ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞\n\
─ + ⎜───────⎟⋅⎜─────⎟⋅⎜────────────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝ y + 5 ⎠ ⎝x - y⎠\
"""
expected4 = \
"""\
⎛ x + y ⎞ ⎛x - y⎞ \n\
⎜───────⎟⋅⎜─────⎟ \n\
⎝x - 2⋅y⎠ ⎝x + y⎠ \n\
─────────────────────\n\
1 ⎛ x + y ⎞ ⎛x - y⎞\n\
─ + ⎜───────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠\
"""
expected5 = \
"""\
⎛ x + y ⎞ ⎛x - y⎞ \n\
⎜───────⎟⋅⎜─────⎟ \n\
⎝x - 2⋅y⎠ ⎝x + y⎠ \n\
─────────────────────────────\n\
1 ⎛ x + y ⎞ ⎛x - y⎞ ⎛1 - x⎞\n\
─ + ⎜───────⎟⋅⎜─────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x - y⎠\
"""
expected6 = \
"""\
⎛ 2 ⎞ \n\
⎜y - 2⋅y + 1⎟ ⎛1 - x⎞ \n\
⎜────────────⎟⋅⎜─────⎟ \n\
⎝ y + 5 ⎠ ⎝x - y⎠ \n\
────────────────────────────────────────────\n\
⎛ 2 ⎞ \n\
1 ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞ ⎛x - y⎞ ⎛ x + y ⎞\n\
─ + ⎜────────────⎟⋅⎜─────⎟⋅⎜─────⎟⋅⎜───────⎟\n\
1 ⎝ y + 5 ⎠ ⎝x - y⎠ ⎝x + y⎠ ⎝x - 2⋅y⎠\
"""
expected7 = \
"""\
⎛ 3⎞ \n\
⎜x - 2⋅y ⎟ \n\
⎜────────⎟ \n\
⎝ x + y ⎠ \n\
──────────────────\n\
⎛ 3⎞ \n\
1 ⎜x - 2⋅y ⎟ ⎛2⎞\n\
─ + ⎜────────⎟⋅⎜─⎟\n\
1 ⎝ x + y ⎠ ⎝2⎠\
"""
expected8 = \
"""\
⎛1 - x⎞ \n\
⎜─────⎟ \n\
⎝x - y⎠ \n\
───────────\n\
1 ⎛1 - x⎞\n\
─ + ⎜─────⎟\n\
1 ⎝x - y⎠\
"""
expected9 = \
"""\
⎛ x + y ⎞ ⎛x - y⎞ \n\
⎜───────⎟⋅⎜─────⎟ \n\
⎝x - 2⋅y⎠ ⎝x + y⎠ \n\
─────────────────────────────\n\
1 ⎛ x + y ⎞ ⎛x - y⎞ ⎛1 - x⎞\n\
─ - ⎜───────⎟⋅⎜─────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x - y⎠\
"""
expected10 = \
"""\
⎛1 - x⎞ \n\
⎜─────⎟ \n\
⎝x - y⎠ \n\
───────────\n\
1 ⎛1 - x⎞\n\
─ - ⎜─────⎟\n\
1 ⎝x - y⎠\
"""
assert upretty(Feedback(tf, tf1)) == expected1
assert upretty(Feedback(tf, tf2*tf1*tf3)) == expected2
assert upretty(Feedback(tf1, tf2*tf3*tf5)) == expected3
assert upretty(Feedback(tf1*tf2, tf)) == expected4
assert upretty(Feedback(tf1*tf2, tf5)) == expected5
assert upretty(Feedback(tf3*tf5, tf2*tf1)) == expected6
assert upretty(Feedback(tf4, tf6)) == expected7
assert upretty(Feedback(tf5, tf)) == expected8
assert upretty(Feedback(tf1*tf2, tf5, 1)) == expected9
assert upretty(Feedback(tf5, tf, 1)) == expected10
def test_pretty_MIMOFeedback():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
tfm_3 = TransferFunctionMatrix([[tf1, tf1], [tf2, tf2]])
expected1 = \
"""\
⎛ ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎞-1 ⎡ x + y x - y ⎤ \n\
⎜ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎟ ⎢─────── ───── ⎥ \n\
⎜ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎟ ⎢x - 2⋅y x + y ⎥ \n\
⎜I - ⎢ ⎥ ⋅⎢ ⎥ ⎟ ⋅ ⎢ ⎥ \n\
⎜ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ ⎟ ⎢ x - y x + y ⎥ \n\
⎜ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ ⎟ ⎢ ───── ───────⎥ \n\
⎝ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ⎠ ⎣ x + y x - 2⋅y⎦τ\
"""
expected2 = \
"""\
⎛ ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎡ x + y x + y ⎤ ⎞-1 ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ \n\
⎜ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎢─────── ───────⎥ ⎟ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ \n\
⎜ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎢x - 2⋅y x - 2⋅y⎥ ⎟ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ \n\
⎜I + ⎢ ⎥ ⋅⎢ ⎥ ⋅⎢ ⎥ ⎟ ⋅ ⎢ ⎥ ⋅⎢ ⎥ \n\
⎜ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ ⎢ x - y x - y ⎥ ⎟ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ \n\
⎜ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ ⎢ ───── ───── ⎥ ⎟ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ \n\
⎝ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ ⎣ x + y x + y ⎦τ⎠ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ\
"""
assert upretty(MIMOFeedback(tfm_1, tfm_2, 1)) == \
expected1 # Positive MIMOFeedback
assert upretty(MIMOFeedback(tfm_1*tfm_2, tfm_3)) == \
expected2 # Negative MIMOFeedback (Default)
def test_pretty_TransferFunctionMatrix():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(y**2 - 2*y + 1, y + 5, y)
tf4 = TransferFunction(y, x**2 + x + 1, y)
tf5 = TransferFunction(1 - x, x - y, y)
tf6 = TransferFunction(2, 2, y)
expected1 = \
"""\
⎡ x + y ⎤ \n\
⎢───────⎥ \n\
⎢x - 2⋅y⎥ \n\
⎢ ⎥ \n\
⎢ x - y ⎥ \n\
⎢ ───── ⎥ \n\
⎣ x + y ⎦τ\
"""
expected2 = \
"""\
⎡ x + y ⎤ \n\
⎢ ─────── ⎥ \n\
⎢ x - 2⋅y ⎥ \n\
⎢ ⎥ \n\
⎢ x - y ⎥ \n\
⎢ ───── ⎥ \n\
⎢ x + y ⎥ \n\
⎢ ⎥ \n\
⎢ 2 ⎥ \n\
⎢- y + 2⋅y - 1⎥ \n\
⎢──────────────⎥ \n\
⎣ y + 5 ⎦τ\
"""
expected3 = \
"""\
⎡ x + y x - y ⎤ \n\
⎢ ─────── ───── ⎥ \n\
⎢ x - 2⋅y x + y ⎥ \n\
⎢ ⎥ \n\
⎢ 2 ⎥ \n\
⎢y - 2⋅y + 1 y ⎥ \n\
⎢──────────── ──────────⎥ \n\
⎢ y + 5 2 ⎥ \n\
⎢ x + x + 1⎥ \n\
⎢ ⎥ \n\
⎢ 1 - x 2 ⎥ \n\
⎢ ───── ─ ⎥ \n\
⎣ x - y 2 ⎦τ\
"""
expected4 = \
"""\
⎡ x - y x + y y ⎤ \n\
⎢ ───── ─────── ──────────⎥ \n\
⎢ x + y x - 2⋅y 2 ⎥ \n\
⎢ x + x + 1⎥ \n\
⎢ ⎥ \n\
⎢ 2 ⎥ \n\
⎢- y + 2⋅y - 1 x - 1 -2 ⎥ \n\
⎢────────────── ───── ─── ⎥ \n\
⎣ y + 5 x - y 2 ⎦τ\
"""
expected5 = \
"""\
⎡ x + y x - y x + y y ⎤ \n\
⎢───────⋅───── ─────── ──────────⎥ \n\
⎢x - 2⋅y x + y x - 2⋅y 2 ⎥ \n\
⎢ x + x + 1⎥ \n\
⎢ ⎥ \n\
⎢ 1 - x 2 x + y -2 ⎥ \n\
⎢ ───── + ─ ─────── ─── ⎥ \n\
⎣ x - y 2 x - 2⋅y 2 ⎦τ\
"""
assert upretty(TransferFunctionMatrix([[tf1], [tf2]])) == expected1
assert upretty(TransferFunctionMatrix([[tf1], [tf2], [-tf3]])) == expected2
assert upretty(TransferFunctionMatrix([[tf1, tf2], [tf3, tf4], [tf5, tf6]])) == expected3
assert upretty(TransferFunctionMatrix([[tf2, tf1, tf4], [-tf3, -tf5, -tf6]])) == expected4
assert upretty(TransferFunctionMatrix([[Series(tf2, tf1), tf1, tf4], [Parallel(tf6, tf5), tf1, -tf6]])) == \
expected5
def test_pretty_order():
expr = O(1)
ascii_str = \
"""\
O(1)\
"""
ucode_str = \
"""\
O(1)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x)
ascii_str = \
"""\
/1\\\n\
O|-|\n\
\\x/\
"""
ucode_str = \
"""\
⎛1⎞\n\
O⎜─⎟\n\
⎝x⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2)
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (0, 0)/\
"""
ucode_str = \
"""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (0, 0)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1, (x, oo))
ascii_str = \
"""\
O(1; x -> oo)\
"""
ucode_str = \
"""\
O(1; x → ∞)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x, (x, oo))
ascii_str = \
"""\
/1 \\\n\
O|-; x -> oo|\n\
\\x /\
"""
ucode_str = \
"""\
⎛1 ⎞\n\
O⎜─; x → ∞⎟\n\
⎝x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2, (x, oo), (y, oo))
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (oo, oo)/\
"""
ucode_str = \
"""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (∞, ∞)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_derivatives():
# Simple
expr = Derivative(log(x), x, evaluate=False)
ascii_str = \
"""\
d \n\
--(log(x))\n\
dx \
"""
ucode_str = \
"""\
d \n\
──(log(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(log(x), x, evaluate=False) + x
ascii_str_1 = \
"""\
d \n\
x + --(log(x))\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(log(x)) + x\n\
dx \
"""
ucode_str_1 = \
"""\
d \n\
x + ──(log(x))\n\
dx \
"""
ucode_str_2 = \
"""\
d \n\
──(log(x)) + x\n\
dx \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
# basic partial derivatives
expr = Derivative(log(x + y) + x, x)
ascii_str_1 = \
"""\
d \n\
--(log(x + y) + x)\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(x + log(x + y))\n\
dx \
"""
ucode_str_1 = \
"""\
∂ \n\
──(log(x + y) + x)\n\
∂x \
"""
ucode_str_2 = \
"""\
∂ \n\
──(x + log(x + y))\n\
∂x \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2], upretty(expr)
# Multiple symbols
expr = Derivative(log(x) + x**2, x, y)
ascii_str_1 = \
"""\
2 \n\
d / 2\\\n\
-----\\log(x) + x /\n\
dy dx \
"""
ascii_str_2 = \
"""\
2 \n\
d / 2 \\\n\
-----\\x + log(x)/\n\
dy dx \
"""
ucode_str_1 = \
"""\
2 \n\
d ⎛ 2⎞\n\
─────⎝log(x) + x ⎠\n\
dy dx \
"""
ucode_str_2 = \
"""\
2 \n\
d ⎛ 2 ⎞\n\
─────⎝x + log(x)⎠\n\
dy dx \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, y, x) + x**2
ascii_str_1 = \
"""\
2 \n\
d 2\n\
-----(2*x*y) + x \n\
dx dy \
"""
ascii_str_2 = \
"""\
2 \n\
2 d \n\
x + -----(2*x*y)\n\
dx dy \
"""
ucode_str_1 = \
"""\
2 \n\
∂ 2\n\
─────(2⋅x⋅y) + x \n\
∂x ∂y \
"""
ucode_str_2 = \
"""\
2 \n\
2 ∂ \n\
x + ─────(2⋅x⋅y)\n\
∂x ∂y \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, x, x)
ascii_str = \
"""\
2 \n\
d \n\
---(2*x*y)\n\
2 \n\
dx \
"""
ucode_str = \
"""\
2 \n\
∂ \n\
───(2⋅x⋅y)\n\
2 \n\
∂x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, 17)
ascii_str = \
"""\
17 \n\
d \n\
----(2*x*y)\n\
17 \n\
dx \
"""
ucode_str = \
"""\
17 \n\
∂ \n\
────(2⋅x⋅y)\n\
17 \n\
∂x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, x, y)
ascii_str = \
"""\
3 \n\
d \n\
------(2*x*y)\n\
2 \n\
dy dx \
"""
ucode_str = \
"""\
3 \n\
∂ \n\
──────(2⋅x⋅y)\n\
2 \n\
∂y ∂x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# Greek letters
alpha = Symbol('alpha')
beta = Function('beta')
expr = beta(alpha).diff(alpha)
ascii_str = \
"""\
d \n\
------(beta(alpha))\n\
dalpha \
"""
ucode_str = \
"""\
d \n\
──(β(α))\n\
dα \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(f(x), (x, n))
ascii_str = \
"""\
n \n\
d \n\
---(f(x))\n\
n \n\
dx \
"""
ucode_str = \
"""\
n \n\
d \n\
───(f(x))\n\
n \n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_integrals():
expr = Integral(log(x), x)
ascii_str = \
"""\
/ \n\
| \n\
| log(x) dx\n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ log(x) dx\n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, x)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral((sin(x))**2 / (tan(x))**2)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| sin (x) \n\
| ------- dx\n\
| 2 \n\
| tan (x) \n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ 2 \n\
⎮ sin (x) \n\
⎮ ─────── dx\n\
⎮ 2 \n\
⎮ tan (x) \n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**(2**x), x)
ascii_str = \
"""\
/ \n\
| \n\
| / x\\ \n\
| \\2 / \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ ⎛ x⎞ \n\
⎮ ⎝2 ⎠ \n\
⎮ x dx\n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, 1, 2))
ascii_str = \
"""\
2 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1 \
"""
ucode_str = \
"""\
2 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, Rational(1, 2), 10))
ascii_str = \
"""\
10 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1/2 \
"""
ucode_str = \
"""\
10 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1/2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2*y**2, x, y)
ascii_str = \
"""\
/ / \n\
| | \n\
| | 2 2 \n\
| | x *y dx dy\n\
| | \n\
/ / \
"""
ucode_str = \
"""\
⌠ ⌠ \n\
⎮ ⎮ 2 2 \n\
⎮ ⎮ x ⋅y dx dy\n\
⌡ ⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(sin(th)/cos(ph), (th, 0, pi), (ph, 0, 2*pi))
ascii_str = \
"""\
2*pi pi \n\
/ / \n\
| | \n\
| | sin(theta) \n\
| | ---------- d(theta) d(phi)\n\
| | cos(phi) \n\
| | \n\
/ / \n\
0 0 \
"""
ucode_str = \
"""\
2⋅π π \n\
⌠ ⌠ \n\
⎮ ⎮ sin(θ) \n\
⎮ ⎮ ────── dθ dφ\n\
⎮ ⎮ cos(φ) \n\
⌡ ⌡ \n\
0 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_matrix():
# Empty Matrix
expr = Matrix()
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(2, 0, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(0, 2, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix([[x**2 + 1, 1], [y, x + y]])
ascii_str_1 = \
"""\
[ 2 ]
[1 + x 1 ]
[ ]
[ y x + y]\
"""
ascii_str_2 = \
"""\
[ 2 ]
[x + 1 1 ]
[ ]
[ y x + y]\
"""
ucode_str_1 = \
"""\
⎡ 2 ⎤
⎢1 + x 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
"""
ucode_str_2 = \
"""\
⎡ 2 ⎤
⎢x + 1 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
ascii_str = \
"""\
[x ]
[- y theta]
[y ]
[ ]
[ I*k*phi ]
[0 e 1 ]\
"""
ucode_str = \
"""\
⎡x ⎤
⎢─ y θ⎥
⎢y ⎥
⎢ ⎥
⎢ ⅈ⋅k⋅φ ⎥
⎣0 ℯ 1⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
unicode_str = \
"""\
⎡v̇_msc_00 0 0 ⎤
⎢ ⎥
⎢ 0 v̇_msc_01 0 ⎥
⎢ ⎥
⎣ 0 0 v̇_msc_02⎦\
"""
expr = diag(*MatrixSymbol('vdot_msc',1,3))
assert upretty(expr) == unicode_str
def test_pretty_ndim_arrays():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert pretty(M) == "x"
assert upretty(M) == "x"
M = ArrayType([[1/x, y], [z, w]])
M1 = ArrayType([1/x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
ascii_str = \
"""\
[1 ]\n\
[- y]\n\
[x ]\n\
[ ]\n\
[z w]\
"""
ucode_str = \
"""\
⎡1 ⎤\n\
⎢─ y⎥\n\
⎢x ⎥\n\
⎢ ⎥\n\
⎣z w⎦\
"""
assert pretty(M) == ascii_str
assert upretty(M) == ucode_str
ascii_str = \
"""\
[1 ]\n\
[- y z]\n\
[x ]\
"""
ucode_str = \
"""\
⎡1 ⎤\n\
⎢─ y z⎥\n\
⎣x ⎦\
"""
assert pretty(M1) == ascii_str
assert upretty(M1) == ucode_str
ascii_str = \
"""\
[[1 y] ]\n\
[[-- -] [z ]]\n\
[[ 2 x] [ y 2 ] [- y*z]]\n\
[[x ] [ - y ] [x ]]\n\
[[ ] [ x ] [ ]]\n\
[[z w] [ ] [ 2 ]]\n\
[[- -] [y*z w*y] [z w*z]]\n\
[[x x] ]\
"""
ucode_str = \
"""\
⎡⎡1 y⎤ ⎤\n\
⎢⎢── ─⎥ ⎡z ⎤⎥\n\
⎢⎢ 2 x⎥ ⎡ y 2 ⎤ ⎢─ y⋅z⎥⎥\n\
⎢⎢x ⎥ ⎢ ─ y ⎥ ⎢x ⎥⎥\n\
⎢⎢ ⎥ ⎢ x ⎥ ⎢ ⎥⎥\n\
⎢⎢z w⎥ ⎢ ⎥ ⎢ 2 ⎥⎥\n\
⎢⎢─ ─⎥ ⎣y⋅z w⋅y⎦ ⎣z w⋅z⎦⎥\n\
⎣⎣x x⎦ ⎦\
"""
assert pretty(M2) == ascii_str
assert upretty(M2) == ucode_str
ascii_str = \
"""\
[ [1 y] ]\n\
[ [-- -] ]\n\
[ [ 2 x] [ y 2 ]]\n\
[ [x ] [ - y ]]\n\
[ [ ] [ x ]]\n\
[ [z w] [ ]]\n\
[ [- -] [y*z w*y]]\n\
[ [x x] ]\n\
[ ]\n\
[[z ] [ w ]]\n\
[[- y*z] [ - w*y]]\n\
[[x ] [ x ]]\n\
[[ ] [ ]]\n\
[[ 2 ] [ 2 ]]\n\
[[z w*z] [w*z w ]]\
"""
ucode_str = \
"""\
⎡ ⎡1 y⎤ ⎤\n\
⎢ ⎢── ─⎥ ⎥\n\
⎢ ⎢ 2 x⎥ ⎡ y 2 ⎤⎥\n\
⎢ ⎢x ⎥ ⎢ ─ y ⎥⎥\n\
⎢ ⎢ ⎥ ⎢ x ⎥⎥\n\
⎢ ⎢z w⎥ ⎢ ⎥⎥\n\
⎢ ⎢─ ─⎥ ⎣y⋅z w⋅y⎦⎥\n\
⎢ ⎣x x⎦ ⎥\n\
⎢ ⎥\n\
⎢⎡z ⎤ ⎡ w ⎤⎥\n\
⎢⎢─ y⋅z⎥ ⎢ ─ w⋅y⎥⎥\n\
⎢⎢x ⎥ ⎢ x ⎥⎥\n\
⎢⎢ ⎥ ⎢ ⎥⎥\n\
⎢⎢ 2 ⎥ ⎢ 2 ⎥⎥\n\
⎣⎣z w⋅z⎦ ⎣w⋅z w ⎦⎦\
"""
assert pretty(M3) == ascii_str
assert upretty(M3) == ucode_str
Mrow = ArrayType([[x, y, 1 / z]])
Mcolumn = ArrayType([[x], [y], [1 / z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
ascii_str = \
"""\
[[ 1]]\n\
[[x y -]]\n\
[[ z]]\
"""
ucode_str = \
"""\
⎡⎡ 1⎤⎤\n\
⎢⎢x y ─⎥⎥\n\
⎣⎣ z⎦⎦\
"""
assert pretty(Mrow) == ascii_str
assert upretty(Mrow) == ucode_str
ascii_str = \
"""\
[x]\n\
[ ]\n\
[y]\n\
[ ]\n\
[1]\n\
[-]\n\
[z]\
"""
ucode_str = \
"""\
⎡x⎤\n\
⎢ ⎥\n\
⎢y⎥\n\
⎢ ⎥\n\
⎢1⎥\n\
⎢─⎥\n\
⎣z⎦\
"""
assert pretty(Mcolumn) == ascii_str
assert upretty(Mcolumn) == ucode_str
ascii_str = \
"""\
[[x]]\n\
[[ ]]\n\
[[y]]\n\
[[ ]]\n\
[[1]]\n\
[[-]]\n\
[[z]]\
"""
ucode_str = \
"""\
⎡⎡x⎤⎤\n\
⎢⎢ ⎥⎥\n\
⎢⎢y⎥⎥\n\
⎢⎢ ⎥⎥\n\
⎢⎢1⎥⎥\n\
⎢⎢─⎥⎥\n\
⎣⎣z⎦⎦\
"""
assert pretty(Mcol2) == ascii_str
assert upretty(Mcol2) == ucode_str
def test_tensor_TensorProduct():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert upretty(TensorProduct(A, B)) == "A\u2297B"
assert upretty(TensorProduct(A, B, A)) == "A\u2297B\u2297A"
def test_diffgeom_print_WedgeProduct():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert upretty(wp) == "ⅆ x∧ⅆ y"
assert pretty(wp) == r"d x/\d y"
def test_Adjoint():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert pretty(Adjoint(X)) == " +\nX "
assert pretty(Adjoint(X + Y)) == " +\n(X + Y) "
assert pretty(Adjoint(X) + Adjoint(Y)) == " + +\nX + Y "
assert pretty(Adjoint(X*Y)) == " +\n(X*Y) "
assert pretty(Adjoint(Y)*Adjoint(X)) == " + +\nY *X "
assert pretty(Adjoint(X**2)) == " +\n/ 2\\ \n\\X / "
assert pretty(Adjoint(X)**2) == " 2\n/ +\\ \n\\X / "
assert pretty(Adjoint(Inverse(X))) == " +\n/ -1\\ \n\\X / "
assert pretty(Inverse(Adjoint(X))) == " -1\n/ +\\ \n\\X / "
assert pretty(Adjoint(Transpose(X))) == " +\n/ T\\ \n\\X / "
assert pretty(Transpose(Adjoint(X))) == " T\n/ +\\ \n\\X / "
assert upretty(Adjoint(X)) == " †\nX "
assert upretty(Adjoint(X + Y)) == " †\n(X + Y) "
assert upretty(Adjoint(X) + Adjoint(Y)) == " † †\nX + Y "
assert upretty(Adjoint(X*Y)) == " †\n(X⋅Y) "
assert upretty(Adjoint(Y)*Adjoint(X)) == " † †\nY ⋅X "
assert upretty(Adjoint(X**2)) == \
" †\n⎛ 2⎞ \n⎝X ⎠ "
assert upretty(Adjoint(X)**2) == \
" 2\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Inverse(X))) == \
" †\n⎛ -1⎞ \n⎝X ⎠ "
assert upretty(Inverse(Adjoint(X))) == \
" -1\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Transpose(X))) == \
" †\n⎛ T⎞ \n⎝X ⎠ "
assert upretty(Transpose(Adjoint(X))) == \
" T\n⎛ †⎞ \n⎝X ⎠ "
m = Matrix(((1, 2), (3, 4)))
assert upretty(Adjoint(m)) == \
' †\n'\
'⎡1 2⎤ \n'\
'⎢ ⎥ \n'\
'⎣3 4⎦ '
assert upretty(Adjoint(m+X)) == \
' †\n'\
'⎛⎡1 2⎤ ⎞ \n'\
'⎜⎢ ⎥ + X⎟ \n'\
'⎝⎣3 4⎦ ⎠ '
def test_pretty_Trace_issue_9044():
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[2, 4], [6, 8]])
ascii_str_1 = \
"""\
/[1 2]\\
tr|[ ]|
\\[3 4]/\
"""
ucode_str_1 = \
"""\
⎛⎡1 2⎤⎞
tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠\
"""
ascii_str_2 = \
"""\
/[1 2]\\ /[2 4]\\
tr|[ ]| + tr|[ ]|
\\[3 4]/ \\[6 8]/\
"""
ucode_str_2 = \
"""\
⎛⎡1 2⎤⎞ ⎛⎡2 4⎤⎞
tr⎜⎢ ⎥⎟ + tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠ ⎝⎣6 8⎦⎠\
"""
assert pretty(Trace(X)) == ascii_str_1
assert upretty(Trace(X)) == ucode_str_1
assert pretty(Trace(X) + Trace(Y)) == ascii_str_2
assert upretty(Trace(X) + Trace(Y)) == ucode_str_2
def test_MatrixSlice():
n = Symbol('n', integer=True)
x, y, z, w, t, = symbols('x y z w t')
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
expr = MatrixSlice(X, (None, None, None), (None, None, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = X[x:x + 1, y:y + 1]
assert pretty(expr) == upretty(expr) == 'X[x:x + 1, y:y + 1]'
expr = X[x:x + 1:2, y:y + 1:2]
assert pretty(expr) == upretty(expr) == 'X[x:x + 1:2, y:y + 1:2]'
expr = X[:x, y:]
assert pretty(expr) == upretty(expr) == 'X[:x, y:]'
expr = X[:x, y:]
assert pretty(expr) == upretty(expr) == 'X[:x, y:]'
expr = X[x:, :y]
assert pretty(expr) == upretty(expr) == 'X[x:, :y]'
expr = X[x:y, z:w]
assert pretty(expr) == upretty(expr) == 'X[x:y, z:w]'
expr = X[x:y:t, w:t:x]
assert pretty(expr) == upretty(expr) == 'X[x:y:t, w:t:x]'
expr = X[x::y, t::w]
assert pretty(expr) == upretty(expr) == 'X[x::y, t::w]'
expr = X[:x:y, :t:w]
assert pretty(expr) == upretty(expr) == 'X[:x:y, :t:w]'
expr = X[::x, ::y]
assert pretty(expr) == upretty(expr) == 'X[::x, ::y]'
expr = MatrixSlice(X, (0, None, None), (0, None, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (None, n, None), (None, n, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (0, n, None), (0, n, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (0, n, 2), (0, n, 2))
assert pretty(expr) == upretty(expr) == 'X[::2, ::2]'
expr = X[1:2:3, 4:5:6]
assert pretty(expr) == upretty(expr) == 'X[1:2:3, 4:5:6]'
expr = X[1:3:5, 4:6:8]
assert pretty(expr) == upretty(expr) == 'X[1:3:5, 4:6:8]'
expr = X[1:10:2]
assert pretty(expr) == upretty(expr) == 'X[1:10:2, :]'
expr = Y[:5, 1:9:2]
assert pretty(expr) == upretty(expr) == 'Y[:5, 1:9:2]'
expr = Y[:5, 1:10:2]
assert pretty(expr) == upretty(expr) == 'Y[:5, 1::2]'
expr = Y[5, :5:2]
assert pretty(expr) == upretty(expr) == 'Y[5:6, :5:2]'
expr = X[0:1, 0:1]
assert pretty(expr) == upretty(expr) == 'X[:1, :1]'
expr = X[0:1:2, 0:1:2]
assert pretty(expr) == upretty(expr) == 'X[:1:2, :1:2]'
expr = (Y + Z)[2:, 2:]
assert pretty(expr) == upretty(expr) == '(Y + Z)[2:, 2:]'
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert pretty(X) == upretty(X) == "X"
# Apply function elementwise (`ElementwiseApplyFunc`):
expr = (X.T*X).applyfunc(sin)
ascii_str = """\
/ T \\\n\
(d -> sin(d)).\\X *X/\
"""
ucode_str = """\
⎛ T ⎞\n\
(d ↦ sin(d))˳⎝X ⋅X⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
lamda = Lambda(x, 1/x)
expr = (n*X).applyfunc(lamda)
ascii_str = """\
/ 1\\ \n\
|x -> -|.(n*X)\n\
\\ x/ \
"""
ucode_str = """\
⎛ 1⎞ \n\
⎜x ↦ ─⎟˳(n⋅X)\n\
⎝ x⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_dotproduct():
from sympy.matrices.expressions.dotproduct import DotProduct
n = symbols("n", integer=True)
A = MatrixSymbol('A', n, 1)
B = MatrixSymbol('B', n, 1)
C = Matrix(1, 3, [1, 2, 3])
D = Matrix(1, 3, [1, 3, 4])
assert pretty(DotProduct(A, B)) == "A*B"
assert pretty(DotProduct(C, D)) == "[1 2 3]*[1 3 4]"
assert upretty(DotProduct(A, B)) == "A⋅B"
assert upretty(DotProduct(C, D)) == "[1 2 3]⋅[1 3 4]"
def test_pretty_piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
/x for x < 1\n\
| \n\
< 2 \n\
|x otherwise\n\
\\ \
"""
ucode_str = \
"""\
⎧x for x < 1\n\
⎪ \n\
⎨ 2 \n\
⎪x otherwise\n\
⎩ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
//x for x < 1\\\n\
|| |\n\
-|< 2 |\n\
||x otherwise|\n\
\\\\ /\
"""
ucode_str = \
"""\
⎛⎧x for x < 1⎞\n\
⎜⎪ ⎟\n\
-⎜⎨ 2 ⎟\n\
⎜⎪x otherwise⎟\n\
⎝⎩ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x + Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x + |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
"""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x + ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x - Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x - |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
"""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x - ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x*Piecewise((x, x > 0), (y, True))
ascii_str = \
"""\
//x for x > 0\\\n\
x*|< |\n\
\\\\y otherwise/\
"""
ucode_str = \
"""\
⎛⎧x for x > 0⎞\n\
x⋅⎜⎨ ⎟\n\
⎝⎩y otherwise⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x >
2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
"""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x
> 2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
-|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
"""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
-⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((0, Abs(1/y) < 1), (1, Abs(y) < 1), (y*meijerg(((2, 1),
()), ((), (1, 0)), 1/y), True))
ascii_str = \
"""\
/ 1 \n\
| 0 for --- < 1\n\
| |y| \n\
| \n\
< 1 for |y| < 1\n\
| \n\
| __0, 2 /2, 1 | 1\\ \n\
|y*/__ | | -| otherwise \n\
\\ \\_|2, 2 \\ 1, 0 | y/ \
"""
ucode_str = \
"""\
⎧ 1 \n\
⎪ 0 for ─── < 1\n\
⎪ │y│ \n\
⎪ \n\
⎨ 1 for │y│ < 1\n\
⎪ \n\
⎪ ╭─╮0, 2 ⎛2, 1 │ 1⎞ \n\
⎪y⋅│╶┐ ⎜ │ ─⎟ otherwise \n\
⎩ ╰─╯2, 2 ⎝ 1, 0 │ y⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# XXX: We have to use evaluate=False here because Piecewise._eval_power
# denests the power.
expr = Pow(Piecewise((x, x > 0), (y, True)), 2, evaluate=False)
ascii_str = \
"""\
2\n\
//x for x > 0\\ \n\
|< | \n\
\\\\y otherwise/ \
"""
ucode_str = \
"""\
2\n\
⎛⎧x for x > 0⎞ \n\
⎜⎨ ⎟ \n\
⎝⎩y otherwise⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ITE():
expr = ITE(x, y, z)
assert pretty(expr) == (
'/y for x \n'
'< \n'
'\\z otherwise'
)
assert upretty(expr) == """\
⎧y for x \n\
⎨ \n\
⎩z otherwise\
"""
def test_pretty_seq():
expr = ()
ascii_str = \
"""\
()\
"""
ucode_str = \
"""\
()\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = []
ascii_str = \
"""\
[]\
"""
ucode_str = \
"""\
[]\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {}
expr_2 = {}
ascii_str = \
"""\
{}\
"""
ucode_str = \
"""\
{}\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = (1/x,)
ascii_str = \
"""\
1 \n\
(-,)\n\
x \
"""
ucode_str = \
"""\
⎛1 ⎞\n\
⎜─,⎟\n\
⎝x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = [x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
[x , -, x, y, -----------]\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
"""\
⎡ 2 ⎤\n\
⎢ 2 1 sin (θ)⎥\n\
⎢x , ─, x, y, ───────⎥\n\
⎢ x 2 ⎥\n\
⎣ cos (φ)⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x: sin(x)}
expr_2 = Dict({x: sin(x)})
ascii_str = \
"""\
{x: sin(x)}\
"""
ucode_str = \
"""\
{x: sin(x)}\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = {1/x: 1/y, x: sin(x)**2}
expr_2 = Dict({1/x: 1/y, x: sin(x)**2})
ascii_str = \
"""\
1 1 2 \n\
{-: -, x: sin (x)}\n\
x y \
"""
ucode_str = \
"""\
⎧1 1 2 ⎫\n\
⎨─: ─, x: sin (x)⎬\n\
⎩x y ⎭\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
# There used to be a bug with pretty-printing sequences of even height.
expr = [x**2]
ascii_str = \
"""\
2 \n\
[x ]\
"""
ucode_str = \
"""\
⎡ 2⎤\n\
⎣x ⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2,)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎝x ,⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎝x ,⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x**2: 1}
expr_2 = Dict({x**2: 1})
ascii_str = \
"""\
2 \n\
{x : 1}\
"""
ucode_str = \
"""\
⎧ 2 ⎫\n\
⎨x : 1⎬\n\
⎩ ⎭\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
def test_any_object_in_sequence():
# Cf. issue 5306
b1 = Basic()
b2 = Basic(Basic())
expr = [b2, b1]
assert pretty(expr) == "[Basic(Basic()), Basic()]"
assert upretty(expr) == "[Basic(Basic()), Basic()]"
expr = {b2, b1}
assert pretty(expr) == "{Basic(), Basic(Basic())}"
assert upretty(expr) == "{Basic(), Basic(Basic())}"
expr = {b2: b1, b1: b2}
expr2 = Dict({b2: b1, b1: b2})
assert pretty(expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert pretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
def test_print_builtin_set():
assert pretty(set()) == 'set()'
assert upretty(set()) == 'set()'
assert pretty(frozenset()) == 'frozenset()'
assert upretty(frozenset()) == 'frozenset()'
s1 = {1/x, x}
s2 = frozenset(s1)
assert pretty(s1) == \
"""\
1 \n\
{-, x}
x \
"""
assert upretty(s1) == \
"""\
⎧1 ⎫
⎨─, x⎬
⎩x ⎭\
"""
assert pretty(s2) == \
"""\
1 \n\
frozenset({-, x})
x \
"""
assert upretty(s2) == \
"""\
⎛⎧1 ⎫⎞
frozenset⎜⎨─, x⎬⎟
⎝⎩x ⎭⎠\
"""
def test_pretty_sets():
s = FiniteSet
assert pretty(s(*[x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(s(*range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(s(*range(1, 13))) == "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty({x*y, x**2}) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(set(range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(set(range(1, 13))) == \
"{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(frozenset([x*y, x**2])) == \
"""\
2 \n\
frozenset({x , x*y})\
"""
assert pretty(frozenset(range(1, 6))) == "frozenset({1, 2, 3, 4, 5})"
assert pretty(frozenset(range(1, 13))) == \
"frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12})"
assert pretty(Range(0, 3, 1)) == '{0, 1, 2}'
ascii_str = '{0, 1, ..., 29}'
ucode_str = '{0, 1, …, 29}'
assert pretty(Range(0, 30, 1)) == ascii_str
assert upretty(Range(0, 30, 1)) == ucode_str
ascii_str = '{30, 29, ..., 2}'
ucode_str = '{30, 29, …, 2}'
assert pretty(Range(30, 1, -1)) == ascii_str
assert upretty(Range(30, 1, -1)) == ucode_str
ascii_str = '{0, 2, ...}'
ucode_str = '{0, 2, …}'
assert pretty(Range(0, oo, 2)) == ascii_str
assert upretty(Range(0, oo, 2)) == ucode_str
ascii_str = '{..., 2, 0}'
ucode_str = '{…, 2, 0}'
assert pretty(Range(oo, -2, -2)) == ascii_str
assert upretty(Range(oo, -2, -2)) == ucode_str
ascii_str = '{-2, -3, ...}'
ucode_str = '{-2, -3, …}'
assert pretty(Range(-2, -oo, -1)) == ascii_str
assert upretty(Range(-2, -oo, -1)) == ucode_str
def test_pretty_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
ascii_str = "SetExpr([1, 3])"
ucode_str = "SetExpr([1, 3])"
assert pretty(se) == ascii_str
assert upretty(se) == ucode_str
def test_pretty_ImageSet():
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
ascii_str = '{x + y | x in {1, 2, 3}, y in {3, 4}}'
ucode_str = '{x + y │ x ∊ {1, 2, 3}, y ∊ {3, 4}}'
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4}))
ascii_str = '{x + y | (x, y) in {1, 2, 3} x {3, 4}}'
ucode_str = '{x + y │ (x, y) ∊ {1, 2, 3} × {3, 4}}'
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(x, x**2), S.Naturals)
ascii_str = '''\
2 \n\
{x | x in Naturals}'''
ucode_str = '''\
⎧ 2 │ ⎫\n\
⎨x │ x ∊ ℕ⎬\n\
⎩ │ ⎭'''
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
# TODO: The "x in N" parts below should be centered independently of the
# 1/x**2 fraction
imgset = ImageSet(Lambda(x, 1/x**2), S.Naturals)
ascii_str = '''\
1 \n\
{-- | x in Naturals}
2 \n\
x '''
ucode_str = '''\
⎧1 │ ⎫\n\
⎪── │ x ∊ ℕ⎪\n\
⎨ 2 │ ⎬\n\
⎪x │ ⎪\n\
⎩ │ ⎭'''
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda((x, y), 1/(x + y)**2), S.Naturals, S.Naturals)
ascii_str = '''\
1 \n\
{-------- | x in Naturals, y in Naturals}
2 \n\
(x + y) '''
ucode_str = '''\
⎧ 1 │ ⎫
⎪──────── │ x ∊ ℕ, y ∊ ℕ⎪
⎨ 2 │ ⎬
⎪(x + y) │ ⎪
⎩ │ ⎭'''
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
def test_pretty_ConditionSet():
ascii_str = '{x | x in (-oo, oo) and sin(x) = 0}'
ucode_str = '{x │ x ∊ ℝ ∧ (sin(x) = 0)}'
assert pretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ascii_str
assert upretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ucode_str
assert pretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert upretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert pretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "EmptySet"
assert upretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "∅"
assert pretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
assert upretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
condset = ConditionSet(x, 1/x**2 > 0)
ascii_str = '''\
1 \n\
{x | -- > 0}
2 \n\
x '''
ucode_str = '''\
⎧ │ ⎛1 ⎞⎫
⎪x │ ⎜── > 0⎟⎪
⎨ │ ⎜ 2 ⎟⎬
⎪ │ ⎝x ⎠⎪
⎩ │ ⎭'''
assert pretty(condset) == ascii_str
assert upretty(condset) == ucode_str
condset = ConditionSet(x, 1/x**2 > 0, S.Reals)
ascii_str = '''\
1 \n\
{x | x in (-oo, oo) and -- > 0}
2 \n\
x '''
ucode_str = '''\
⎧ │ ⎛1 ⎞⎫
⎪x │ x ∊ ℝ ∧ ⎜── > 0⎟⎪
⎨ │ ⎜ 2 ⎟⎬
⎪ │ ⎝x ⎠⎪
⎩ │ ⎭'''
assert pretty(condset) == ascii_str
assert upretty(condset) == ucode_str
def test_pretty_ComplexRegion():
from sympy.sets.fancysets import ComplexRegion
cregion = ComplexRegion(Interval(3, 5)*Interval(4, 6))
ascii_str = '{x + y*I | x, y in [3, 5] x [4, 6]}'
ucode_str = '{x + y⋅ⅈ │ x, y ∊ [3, 5] × [4, 6]}'
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
cregion = ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)
ascii_str = '{r*(I*sin(theta) + cos(theta)) | r, theta in [0, 1] x [0, 2*pi)}'
ucode_str = '{r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ [0, 1] × [0, 2⋅π)}'
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
cregion = ComplexRegion(Interval(3, 1/a**2)*Interval(4, 6))
ascii_str = '''\
1 \n\
{x + y*I | x, y in [3, --] x [4, 6]}
2 \n\
a '''
ucode_str = '''\
⎧ │ ⎡ 1 ⎤ ⎫
⎪x + y⋅ⅈ │ x, y ∊ ⎢3, ──⎥ × [4, 6]⎪
⎨ │ ⎢ 2⎥ ⎬
⎪ │ ⎣ a ⎦ ⎪
⎩ │ ⎭'''
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
cregion = ComplexRegion(Interval(0, 1/a**2)*Interval(0, 2*pi), polar=True)
ascii_str = '''\
1 \n\
{r*(I*sin(theta) + cos(theta)) | r, theta in [0, --] x [0, 2*pi)}
2 \n\
a '''
ucode_str = '''\
⎧ │ ⎡ 1 ⎤ ⎫
⎪r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ ⎢0, ──⎥ × [0, 2⋅π)⎪
⎨ │ ⎢ 2⎥ ⎬
⎪ │ ⎣ a ⎦ ⎪
⎩ │ ⎭'''
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
def test_pretty_Union_issue_10414():
a, b = Interval(2, 3), Interval(4, 7)
ucode_str = '[2, 3] ∪ [4, 7]'
ascii_str = '[2, 3] U [4, 7]'
assert upretty(Union(a, b)) == ucode_str
assert pretty(Union(a, b)) == ascii_str
def test_pretty_Intersection_issue_10414():
x, y, z, w = symbols('x, y, z, w')
a, b = Interval(x, y), Interval(z, w)
ucode_str = '[x, y] ∩ [z, w]'
ascii_str = '[x, y] n [z, w]'
assert upretty(Intersection(a, b)) == ucode_str
assert pretty(Intersection(a, b)) == ascii_str
def test_ProductSet_exponent():
ucode_str = ' 1\n[0, 1] '
assert upretty(Interval(0, 1)**1) == ucode_str
ucode_str = ' 2\n[0, 1] '
assert upretty(Interval(0, 1)**2) == ucode_str
def test_ProductSet_parenthesis():
ucode_str = '([4, 7] × {1, 2}) ∪ ([2, 3] × [4, 7])'
a, b = Interval(2, 3), Interval(4, 7)
assert upretty(Union(a*b, b*FiniteSet(1, 2))) == ucode_str
def test_ProductSet_prod_char_issue_10413():
ascii_str = '[2, 3] x [4, 7]'
ucode_str = '[2, 3] × [4, 7]'
a, b = Interval(2, 3), Interval(4, 7)
assert pretty(a*b) == ascii_str
assert upretty(a*b) == ucode_str
def test_pretty_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
ascii_str = '[0, 1, 4, 9, ...]'
ucode_str = '[0, 1, 4, 9, …]'
assert pretty(s1) == ascii_str
assert upretty(s1) == ucode_str
ascii_str = '[1, 2, 1, 2, ...]'
ucode_str = '[1, 2, 1, 2, …]'
assert pretty(s2) == ascii_str
assert upretty(s2) == ucode_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
ascii_str = '[0, 1, 4]'
ucode_str = '[0, 1, 4]'
assert pretty(s3) == ascii_str
assert upretty(s3) == ucode_str
ascii_str = '[1, 2, 1]'
ucode_str = '[1, 2, 1]'
assert pretty(s4) == ascii_str
assert upretty(s4) == ucode_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
ascii_str = '[..., 9, 4, 1, 0]'
ucode_str = '[…, 9, 4, 1, 0]'
assert pretty(s5) == ascii_str
assert upretty(s5) == ucode_str
ascii_str = '[..., 2, 1, 2, 1]'
ucode_str = '[…, 2, 1, 2, 1]'
assert pretty(s6) == ascii_str
assert upretty(s6) == ucode_str
ascii_str = '[1, 3, 5, 11, ...]'
ucode_str = '[1, 3, 5, 11, …]'
assert pretty(SeqAdd(s1, s2)) == ascii_str
assert upretty(SeqAdd(s1, s2)) == ucode_str
ascii_str = '[1, 3, 5]'
ucode_str = '[1, 3, 5]'
assert pretty(SeqAdd(s3, s4)) == ascii_str
assert upretty(SeqAdd(s3, s4)) == ucode_str
ascii_str = '[..., 11, 5, 3, 1]'
ucode_str = '[…, 11, 5, 3, 1]'
assert pretty(SeqAdd(s5, s6)) == ascii_str
assert upretty(SeqAdd(s5, s6)) == ucode_str
ascii_str = '[0, 2, 4, 18, ...]'
ucode_str = '[0, 2, 4, 18, …]'
assert pretty(SeqMul(s1, s2)) == ascii_str
assert upretty(SeqMul(s1, s2)) == ucode_str
ascii_str = '[0, 2, 4]'
ucode_str = '[0, 2, 4]'
assert pretty(SeqMul(s3, s4)) == ascii_str
assert upretty(SeqMul(s3, s4)) == ucode_str
ascii_str = '[..., 18, 4, 2, 0]'
ucode_str = '[…, 18, 4, 2, 0]'
assert pretty(SeqMul(s5, s6)) == ascii_str
assert upretty(SeqMul(s5, s6)) == ucode_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
raises(NotImplementedError, lambda: pretty(s7))
raises(NotImplementedError, lambda: upretty(s7))
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
ascii_str = '[0, b, 4*b]'
ucode_str = '[0, b, 4⋅b]'
assert pretty(s8) == ascii_str
assert upretty(s8) == ucode_str
def test_pretty_FourierSeries():
f = fourier_series(x, (x, -pi, pi))
ascii_str = \
"""\
2*sin(3*x) \n\
2*sin(x) - sin(2*x) + ---------- + ...\n\
3 \
"""
ucode_str = \
"""\
2⋅sin(3⋅x) \n\
2⋅sin(x) - sin(2⋅x) + ────────── + …\n\
3 \
"""
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_FormalPowerSeries():
f = fps(log(1 + x))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -k k \n\
\\ -(-1) *x \n\
/ -----------\n\
/ k \n\
/___, \n\
k = 1 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ -k k \n\
╲ -(-1) ⋅x \n\
╱ ───────────\n\
╱ k \n\
╱ \n\
‾‾‾‾ \n\
k = 1 \
"""
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_limits():
expr = Limit(x, x, oo)
ascii_str = \
"""\
lim x\n\
x->oo \
"""
ucode_str = \
"""\
lim x\n\
x─→∞ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x**2, x, 0)
ascii_str = \
"""\
2\n\
lim x \n\
x->0+ \
"""
ucode_str = \
"""\
2\n\
lim x \n\
x─→0⁺ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(1/x, x, 0)
ascii_str = \
"""\
1\n\
lim -\n\
x->0+x\
"""
ucode_str = \
"""\
1\n\
lim ─\n\
x─→0⁺x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0)
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0+\\ x /\
"""
ucode_str = \
"""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁺⎝ x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0, "-")
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0-\\ x /\
"""
ucode_str = \
"""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁻⎝ x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x + sin(x), x, 0)
ascii_str = \
"""\
lim (x + sin(x))\n\
x->0+ \
"""
ucode_str = \
"""\
lim (x + sin(x))\n\
x─→0⁺ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x, x, 0)**2
ascii_str = \
"""\
2\n\
/ lim x\\ \n\
\\x->0+ / \
"""
ucode_str = \
"""\
2\n\
⎛ lim x⎞ \n\
⎝x─→0⁺ ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
"""\
⎛ ⎛y⎞⎞\n\
lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
2* lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
"""\
⎛ ⎛y⎞⎞\n\
2⋅ lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x), x, 0, dir='+-')
ascii_str = \
"""\
lim sin(x)\n\
x->0 \
"""
ucode_str = \
"""\
lim sin(x)\n\
x─→0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ComplexRootOf():
expr = rootof(x**5 + 11*x - 2, 0)
ascii_str = \
"""\
/ 5 \\\n\
CRootOf\\x + 11*x - 2, 0/\
"""
ucode_str = \
"""\
⎛ 5 ⎞\n\
CRootOf⎝x + 11⋅x - 2, 0⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_RootSum():
expr = RootSum(x**5 + 11*x - 2, auto=False)
ascii_str = \
"""\
/ 5 \\\n\
RootSum\\x + 11*x - 2/\
"""
ucode_str = \
"""\
⎛ 5 ⎞\n\
RootSum⎝x + 11⋅x - 2⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = RootSum(x**5 + 11*x - 2, Lambda(z, exp(z)))
ascii_str = \
"""\
/ 5 z\\\n\
RootSum\\x + 11*x - 2, z -> e /\
"""
ucode_str = \
"""\
⎛ 5 z⎞\n\
RootSum⎝x + 11⋅x - 2, z ↦ ℯ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_GroebnerBasis():
expr = groebner([], x, y)
ascii_str = \
"""\
GroebnerBasis([], x, y, domain=ZZ, order=lex)\
"""
ucode_str = \
"""\
GroebnerBasis([], x, y, domain=ℤ, order=lex)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
expr = groebner(F, x, y, order='grlex')
ascii_str = \
"""\
/[ 2 2 ] \\\n\
GroebnerBasis\\[x - x - 3*y + 1, y - 2*x + y - 1], x, y, domain=ZZ, order=grlex/\
"""
ucode_str = \
"""\
⎛⎡ 2 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣x - x - 3⋅y + 1, y - 2⋅x + y - 1⎦, x, y, domain=ℤ, order=grlex⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = expr.fglm('lex')
ascii_str = \
"""\
/[ 2 4 3 2 ] \\\n\
GroebnerBasis\\[2*x - y - y + 1, y + 2*y - 3*y - 16*y + 7], x, y, domain=ZZ, order=lex/\
"""
ucode_str = \
"""\
⎛⎡ 2 4 3 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣2⋅x - y - y + 1, y + 2⋅y - 3⋅y - 16⋅y + 7⎦, x, y, domain=ℤ, order=lex⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_UniversalSet():
assert pretty(S.UniversalSet) == "UniversalSet"
assert upretty(S.UniversalSet) == '𝕌'
def test_pretty_Boolean():
expr = Not(x, evaluate=False)
assert pretty(expr) == "Not(x)"
assert upretty(expr) == "¬x"
expr = And(x, y)
assert pretty(expr) == "And(x, y)"
assert upretty(expr) == "x ∧ y"
expr = Or(x, y)
assert pretty(expr) == "Or(x, y)"
assert upretty(expr) == "x ∨ y"
syms = symbols('a:f')
expr = And(*syms)
assert pretty(expr) == "And(a, b, c, d, e, f)"
assert upretty(expr) == "a ∧ b ∧ c ∧ d ∧ e ∧ f"
expr = Or(*syms)
assert pretty(expr) == "Or(a, b, c, d, e, f)"
assert upretty(expr) == "a ∨ b ∨ c ∨ d ∨ e ∨ f"
expr = Xor(x, y, evaluate=False)
assert pretty(expr) == "Xor(x, y)"
assert upretty(expr) == "x ⊻ y"
expr = Nand(x, y, evaluate=False)
assert pretty(expr) == "Nand(x, y)"
assert upretty(expr) == "x ⊼ y"
expr = Nor(x, y, evaluate=False)
assert pretty(expr) == "Nor(x, y)"
assert upretty(expr) == "x ⊽ y"
expr = Implies(x, y, evaluate=False)
assert pretty(expr) == "Implies(x, y)"
assert upretty(expr) == "x → y"
# don't sort args
expr = Implies(y, x, evaluate=False)
assert pretty(expr) == "Implies(y, x)"
assert upretty(expr) == "y → x"
expr = Equivalent(x, y, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == "x ⇔ y"
expr = Equivalent(y, x, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == "x ⇔ y"
def test_pretty_Domain():
expr = FF(23)
assert pretty(expr) == "GF(23)"
assert upretty(expr) == "ℤ₂₃"
expr = ZZ
assert pretty(expr) == "ZZ"
assert upretty(expr) == "ℤ"
expr = QQ
assert pretty(expr) == "QQ"
assert upretty(expr) == "ℚ"
expr = RR
assert pretty(expr) == "RR"
assert upretty(expr) == "ℝ"
expr = QQ[x]
assert pretty(expr) == "QQ[x]"
assert upretty(expr) == "ℚ[x]"
expr = QQ[x, y]
assert pretty(expr) == "QQ[x, y]"
assert upretty(expr) == "ℚ[x, y]"
expr = ZZ.frac_field(x)
assert pretty(expr) == "ZZ(x)"
assert upretty(expr) == "ℤ(x)"
expr = ZZ.frac_field(x, y)
assert pretty(expr) == "ZZ(x, y)"
assert upretty(expr) == "ℤ(x, y)"
expr = QQ.poly_ring(x, y, order=grlex)
assert pretty(expr) == "QQ[x, y, order=grlex]"
assert upretty(expr) == "ℚ[x, y, order=grlex]"
expr = QQ.poly_ring(x, y, order=ilex)
assert pretty(expr) == "QQ[x, y, order=ilex]"
assert upretty(expr) == "ℚ[x, y, order=ilex]"
def test_pretty_prec():
assert xpretty(S("0.3"), full_prec=True, wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec="auto", wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec=False, wrap_line=False) == "0.3"
assert xpretty(S("0.3")*x, full_prec=True, use_unicode=False, wrap_line=False) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert xpretty(S("0.3")*x, full_prec="auto", use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
assert xpretty(S("0.3")*x, full_prec=False, use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
def test_pprint():
import sys
from io import StringIO
fd = StringIO()
sso = sys.stdout
sys.stdout = fd
try:
pprint(pi, use_unicode=False, wrap_line=False)
finally:
sys.stdout = sso
assert fd.getvalue() == 'pi\n'
def test_pretty_class():
"""Test that the printer dispatcher correctly handles classes."""
class C:
pass # C has no .__class__ and this was causing problems
class D:
pass
assert pretty( C ) == str( C )
assert pretty( D ) == str( D )
def test_pretty_no_wrap_line():
huge_expr = 0
for i in range(20):
huge_expr += i*sin(i + x)
assert xpretty(huge_expr ).find('\n') != -1
assert xpretty(huge_expr, wrap_line=False).find('\n') == -1
def test_settings():
raises(TypeError, lambda: pretty(S(4), method="garbage"))
def test_pretty_sum():
from sympy.abc import x, a, b, k, m, n
expr = Sum(k**k, (k, 0, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = 0 \
"""
ucode_str = \
"""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**k, (k, oo, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = oo \
"""
ucode_str = \
"""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = ∞ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (k, 0, n**n))
ascii_str = \
"""\
n \n\
n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
"""\
n \n\
n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
"""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (
k, x + n + x**2 + n**2 + (x/n) + (1/x), Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
2 2 1 x \n\
k = n + n + x + x + - + - \n\
x n \
"""
ucode_str = \
"""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
2 2 1 x \n\
k = n + n + x + x + ─ + ─ \n\
x n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, x + n + x**2 + n**2 + (x/n) + (1/x)))
ascii_str = \
"""\
2 2 1 x \n\
n + n + x + x + - + - \n\
x n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
"""\
2 2 1 x \n\
n + n + x + x + ─ + ─ \n\
x n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x, (x, 0, oo))
ascii_str = \
"""\
oo \n\
__ \n\
\\ ` \n\
) x\n\
/_, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
___ \n\
╲ \n\
╲ \n\
╱ x\n\
╱ \n\
‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ 2\n\
/ x \n\
/__, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
___ \n\
╲ \n\
╲ 2\n\
╱ x \n\
╱ \n\
‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ x\n\
) -\n\
/ 2\n\
/__, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ \n\
╲ x\n\
╱ ─\n\
╱ 2\n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**3/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 3\n\
\\ x \n\
/ --\n\
/ 2 \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ 3\n\
╲ x \n\
╱ ──\n\
╱ 2 \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum((x**3*y**(x/2))**n, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ n\n\
\\ / x\\ \n\
) | -| \n\
/ | 3 2| \n\
/ \\x *y / \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
_____ \n\
╲ \n\
╲ \n\
╲ n\n\
╲ ⎛ x⎞ \n\
╱ ⎜ ─⎟ \n\
╱ ⎜ 3 2⎟ \n\
╱ ⎝x ⋅y ⎠ \n\
╱ \n\
‾‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 1 \n\
\\ --\n\
/ 2\n\
/ x \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ 1 \n\
╲ ──\n\
╱ 2\n\
╱ x \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -a \n\
\\ ---\n\
/ b \n\
/ y \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ -a \n\
╲ ───\n\
╱ b \n\
╱ y \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo), (y, 1, 2))
ascii_str = \
"""\
2 oo \n\
____ ____ \n\
\\ ` \\ ` \n\
\\ \\ -a\n\
\\ \\ --\n\
/ / b \n\
/ / y \n\
/___, /___, \n\
y = 1 x = 0 \
"""
ucode_str = \
"""\
2 ∞ \n\
____ ____ \n\
╲ ╲ \n\
╲ ╲ -a\n\
╲ ╲ ──\n\
╱ ╱ b \n\
╱ ╱ y \n\
╱ ╱ \n\
‾‾‾‾ ‾‾‾‾ \n\
y = 1 x = 0 \
"""
expr = Sum(1/(1 + 1/(
1 + 1/k)) + 1, (k, 111, 1 + 1/n), (k, 1/(1 + m), oo)) + 1/(1 + 1/k)
ascii_str = \
"""\
1 \n\
1 + - \n\
oo n \n\
_____ _____ \n\
\\ ` \\ ` \n\
\\ \\ / 1 \\ \n\
\\ \\ |1 + ---------| \n\
\\ \\ | 1 | 1 \n\
) ) | 1 + -----| + -----\n\
/ / | 1| 1\n\
/ / | 1 + -| 1 + -\n\
/ / \\ k/ k\n\
/____, /____, \n\
1 k = 111 \n\
k = ----- \n\
m + 1 \
"""
ucode_str = \
"""\
1 \n\
1 + ─ \n\
∞ n \n\
______ ______ \n\
╲ ╲ \n\
╲ ╲ \n\
╲ ╲ ⎛ 1 ⎞ \n\
╲ ╲ ⎜1 + ─────────⎟ \n\
╲ ╲ ⎜ 1 ⎟ 1 \n\
╱ ╱ ⎜ 1 + ─────⎟ + ─────\n\
╱ ╱ ⎜ 1⎟ 1\n\
╱ ╱ ⎜ 1 + ─⎟ 1 + ─\n\
╱ ╱ ⎝ k⎠ k\n\
╱ ╱ \n\
‾‾‾‾‾‾ ‾‾‾‾‾‾ \n\
1 k = 111 \n\
k = ───── \n\
m + 1 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_units():
expr = joule
ascii_str1 = \
"""\
2\n\
kilogram*meter \n\
---------------\n\
2 \n\
second \
"""
unicode_str1 = \
"""\
2\n\
kilogram⋅meter \n\
───────────────\n\
2 \n\
second \
"""
ascii_str2 = \
"""\
2\n\
3*x*y*kilogram*meter \n\
---------------------\n\
2 \n\
second \
"""
unicode_str2 = \
"""\
2\n\
3⋅x⋅y⋅kilogram⋅meter \n\
─────────────────────\n\
2 \n\
second \
"""
from sympy.physics.units import kg, m, s
assert upretty(expr) == "joule"
assert pretty(expr) == "joule"
assert upretty(expr.convert_to(kg*m**2/s**2)) == unicode_str1
assert pretty(expr.convert_to(kg*m**2/s**2)) == ascii_str1
assert upretty(3*kg*x*m**2*y/s**2) == unicode_str2
assert pretty(3*kg*x*m**2*y/s**2) == ascii_str2
def test_pretty_Subs():
f = Function('f')
expr = Subs(f(x), x, ph**2)
ascii_str = \
"""\
(f(x))| 2\n\
|x=phi \
"""
unicode_str = \
"""\
(f(x))│ 2\n\
│x=φ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x), x, 0)
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
\\dx /|x=0\
"""
unicode_str = \
"""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎝dx ⎠│x=0\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
|dx || \n\
|--------|| \n\
\\ y /|x=0, y=1/2\
"""
unicode_str = \
"""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎜dx ⎟│ \n\
⎜────────⎟│ \n\
⎝ y ⎠│x=0, y=1/2\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_gammas():
assert upretty(lowergamma(x, y)) == "γ(x, y)"
assert upretty(uppergamma(x, y)) == "Γ(x, y)"
assert xpretty(gamma(x), use_unicode=True) == 'Γ(x)'
assert xpretty(gamma, use_unicode=True) == 'Γ'
assert xpretty(symbols('gamma', cls=Function)(x), use_unicode=True) == 'γ(x)'
assert xpretty(symbols('gamma', cls=Function), use_unicode=True) == 'γ'
def test_beta():
assert xpretty(beta(x,y), use_unicode=True) == 'Β(x, y)'
assert xpretty(beta(x,y), use_unicode=False) == 'B(x, y)'
assert xpretty(beta, use_unicode=True) == 'Β'
assert xpretty(beta, use_unicode=False) == 'B'
mybeta = Function('beta')
assert xpretty(mybeta(x), use_unicode=True) == 'β(x)'
assert xpretty(mybeta(x, y, z), use_unicode=False) == 'beta(x, y, z)'
assert xpretty(mybeta, use_unicode=True) == 'β'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert xpretty(mygamma, use_unicode=True) == r"mygamma"
assert xpretty(mygamma(x), use_unicode=True) == r"mygamma(x)"
def test_SingularityFunction():
assert xpretty(SingularityFunction(x, 0, n), use_unicode=True) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=True) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=True) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=True) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=True) == (
"""\
n\n\
<x - y> \
""")
assert xpretty(SingularityFunction(x, 0, n), use_unicode=False) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=False) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=False) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=False) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=False) == (
"""\
n\n\
<x - y> \
""")
def test_deltas():
assert xpretty(DiracDelta(x), use_unicode=True) == 'δ(x)'
assert xpretty(DiracDelta(x, 1), use_unicode=True) == \
"""\
(1) \n\
δ (x)\
"""
assert xpretty(x*DiracDelta(x, 1), use_unicode=True) == \
"""\
(1) \n\
x⋅δ (x)\
"""
def test_hyper():
expr = hyper((), (), z)
ucode_str = \
"""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ z⎟\n\
0╵ 0 ⎝ │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | z|\n\
0 0 \\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((), (1,), x)
ucode_str = \
"""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ x⎟\n\
0╵ 1 ⎝1 │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | x|\n\
0 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([2], [1], x)
ucode_str = \
"""\
┌─ ⎛2 │ ⎞\n\
├─ ⎜ │ x⎟\n\
1╵ 1 ⎝1 │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ /2 | \\\n\
| | | x|\n\
1 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi/3, -2*k), (3, 4, 5, -3), x)
ucode_str = \
"""\
⎛ π │ ⎞\n\
┌─ ⎜ ─, -2⋅k │ ⎟\n\
├─ ⎜ 3 │ x⎟\n\
2╵ 4 ⎜ │ ⎟\n\
⎝3, 4, 5, -3 │ ⎠\
"""
ascii_str = \
"""\
\n\
_ / pi | \\\n\
|_ | --, -2*k | |\n\
| | 3 | x|\n\
2 4 | | |\n\
\\3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi, S('2/3'), -2*k), (3, 4, 5, -3), x**2)
ucode_str = \
"""\
┌─ ⎛π, 2/3, -2⋅k │ 2⎞\n\
├─ ⎜ │ x ⎟\n\
3╵ 4 ⎝3, 4, 5, -3 │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ /pi, 2/3, -2*k | 2\\\n\
| | | x |\n\
3 4 \\ 3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([1, 2], [3, 4], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
┌─ ⎜1, 2 │ 1 + ─────────⎟\n\
├─ ⎜ │ 1 ⎟\n\
2╵ 2 ⎜3, 4 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
"""
ascii_str = \
"""\
\n\
/ | 1 \\\n\
| | -------------|\n\
_ | | 1 |\n\
|_ |1, 2 | 1 + ---------|\n\
| | | 1 |\n\
2 2 |3, 4 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_meijerg():
expr = meijerg([pi, pi, x], [1], [0, 1], [1, 2, 3], z)
ucode_str = \
"""\
╭─╮2, 3 ⎛π, π, x 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯4, 5 ⎝ 0, 1 1, 2, 3 │ ⎠\
"""
ascii_str = \
"""\
__2, 3 /pi, pi, x 1 | \\\n\
/__ | | z|\n\
\\_|4, 5 \\ 0, 1 1, 2, 3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, pi/7], [2, pi, 5], [], [], z**2)
ucode_str = \
"""\
⎛ π │ ⎞\n\
╭─╮0, 2 ⎜1, ─ 2, π, 5 │ 2⎟\n\
│╶┐ ⎜ 7 │ z ⎟\n\
╰─╯5, 0 ⎜ │ ⎟\n\
⎝ │ ⎠\
"""
ascii_str = \
"""\
/ pi | \\\n\
__0, 2 |1, -- 2, pi, 5 | 2|\n\
/__ | 7 | z |\n\
\\_|5, 0 | | |\n\
\\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ucode_str = \
"""\
╭─╮ 1, 10 ⎛1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯11, 2 ⎝ 1 1 │ ⎠\
"""
ascii_str = \
"""\
__ 1, 10 /1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 | \\\n\
/__ | | z|\n\
\\_|11, 2 \\ 1 1 | /\
"""
expr = meijerg([1]*10, [1], [1], [1], z)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, 2, ], [4, 3], [3], [4, 5], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟\n\
│╶┐ ⎜ │ 1 ⎟\n\
╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
"""
ascii_str = \
"""\
/ | 1 \\\n\
| | -------------|\n\
| | 1 |\n\
__1, 2 |1, 2 4, 3 | 1 + ---------|\n\
/__ | | 1 |\n\
\\_|4, 3 | 3 4, 5 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(expr, x)
ucode_str = \
"""\
⌠ \n\
⎮ ⎛ │ 1 ⎞ \n\
⎮ ⎜ │ ─────────────⎟ \n\
⎮ ⎜ │ 1 ⎟ \n\
⎮ ╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟ \n\
⎮ │╶┐ ⎜ │ 1 ⎟ dx\n\
⎮ ╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟ \n\
⎮ ⎜ │ 1⎟ \n\
⎮ ⎜ │ 1 + ─⎟ \n\
⎮ ⎝ │ x⎠ \n\
⌡ \
"""
ascii_str = \
"""\
/ \n\
| \n\
| / | 1 \\ \n\
| | | -------------| \n\
| | | 1 | \n\
| __1, 2 |1, 2 4, 3 | 1 + ---------| \n\
| /__ | | 1 | dx\n\
| \\_|4, 3 | 3 4, 5 | 1 + -----| \n\
| | | 1| \n\
| | | 1 + -| \n\
| \\ | x/ \n\
| \n\
/ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
expr = A*B*C**-1
ascii_str = \
"""\
-1\n\
A*B*C \
"""
ucode_str = \
"""\
-1\n\
A⋅B⋅C \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = C**-1*A*B
ascii_str = \
"""\
-1 \n\
C *A*B\
"""
ucode_str = \
"""\
-1 \n\
C ⋅A⋅B\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B
ascii_str = \
"""\
-1 \n\
A*C *B\
"""
ucode_str = \
"""\
-1 \n\
A⋅C ⋅B\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B/x
ascii_str = \
"""\
-1 \n\
A*C *B\n\
-------\n\
x \
"""
ucode_str = \
"""\
-1 \n\
A⋅C ⋅B\n\
───────\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_special_functions():
x, y = symbols("x y")
# atan2
expr = atan2(y/sqrt(200), sqrt(x))
ascii_str = \
"""\
/ ___ \\\n\
|\\/ 2 *y ___|\n\
atan2|-------, \\/ x |\n\
\\ 20 /\
"""
ucode_str = \
"""\
⎛√2⋅y ⎞\n\
atan2⎜────, √x⎟\n\
⎝ 20 ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_geometry():
e = Segment((0, 1), (0, 2))
assert pretty(e) == 'Segment2D(Point2D(0, 1), Point2D(0, 2))'
e = Ray((1, 1), angle=4.02*pi)
assert pretty(e) == 'Ray2D(Point2D(1, 1), Point2D(2, tan(pi/50) + 1))'
def test_expint():
expr = Ei(x)
string = 'Ei(x)'
assert pretty(expr) == string
assert upretty(expr) == string
expr = expint(1, z)
ucode_str = "E₁(z)"
ascii_str = "expint(1, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
assert pretty(Shi(x)) == 'Shi(x)'
assert pretty(Si(x)) == 'Si(x)'
assert pretty(Ci(x)) == 'Ci(x)'
assert pretty(Chi(x)) == 'Chi(x)'
assert upretty(Shi(x)) == 'Shi(x)'
assert upretty(Si(x)) == 'Si(x)'
assert upretty(Ci(x)) == 'Ci(x)'
assert upretty(Chi(x)) == 'Chi(x)'
def test_elliptic_functions():
ascii_str = \
"""\
/ 1 \\\n\
K|-----|\n\
\\z + 1/\
"""
ucode_str = \
"""\
⎛ 1 ⎞\n\
K⎜─────⎟\n\
⎝z + 1⎠\
"""
expr = elliptic_k(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
F|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
F⎜1│─────⎟\n\
⎝ │z + 1⎠\
"""
expr = elliptic_f(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 1 \\\n\
E|-----|\n\
\\z + 1/\
"""
ucode_str = \
"""\
⎛ 1 ⎞\n\
E⎜─────⎟\n\
⎝z + 1⎠\
"""
expr = elliptic_e(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
E|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
E⎜1│─────⎟\n\
⎝ │z + 1⎠\
"""
expr = elliptic_e(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ |4\\\n\
Pi|3|-|\n\
\\ |x/\
"""
ucode_str = \
"""\
⎛ │4⎞\n\
Π⎜3│─⎟\n\
⎝ │x⎠\
"""
expr = elliptic_pi(3, 4/x)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 4| \\\n\
Pi|3; -|6|\n\
\\ x| /\
"""
ucode_str = \
"""\
⎛ 4│ ⎞\n\
Π⎜3; ─│6⎟\n\
⎝ x│ ⎠\
"""
expr = elliptic_pi(3, 4/x, 6)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert upretty(where(X > 0)) == "Domain: 0 < x₁ ∧ x₁ < ∞"
D = Die('d1', 6)
assert upretty(where(D > 4)) == 'Domain: d₁ = 5 ∨ d₁ = 6'
A = Exponential('a', 1)
B = Exponential('b', 1)
assert upretty(pspace(Tuple(A, B)).domain) == \
'Domain: 0 ≤ a ∧ 0 ≤ b ∧ a < ∞ ∧ b < ∞'
def test_PrettyPoly():
F = QQ.frac_field(x, y)
R = QQ.poly_ring(x, y)
expr = F.convert(x/(x + y))
assert pretty(expr) == "x/(x + y)"
assert upretty(expr) == "x/(x + y)"
expr = R.convert(x + y)
assert pretty(expr) == "x + y"
assert upretty(expr) == "x + y"
def test_issue_6285():
assert pretty(Pow(2, -5, evaluate=False)) == '1 \n--\n 5\n2 '
assert pretty(Pow(x, (1/pi))) == \
' 1 \n'\
' --\n'\
' pi\n'\
'x '
def test_issue_6359():
assert pretty(Integral(x**2, x)**2) == \
"""\
2
/ / \\ \n\
| | | \n\
| | 2 | \n\
| | x dx| \n\
| | | \n\
\\/ / \
"""
assert upretty(Integral(x**2, x)**2) == \
"""\
2
⎛⌠ ⎞ \n\
⎜⎮ 2 ⎟ \n\
⎜⎮ x dx⎟ \n\
⎝⌡ ⎠ \
"""
assert pretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
/ 1 \\ \n\
| ___ | \n\
| \\ ` | \n\
| \\ 2| \n\
| / x | \n\
| /__, | \n\
\\x = 0 / \
"""
assert upretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
⎛ 1 ⎞ \n\
⎜ ___ ⎟ \n\
⎜ ╲ ⎟ \n\
⎜ ╲ 2⎟ \n\
⎜ ╱ x ⎟ \n\
⎜ ╱ ⎟ \n\
⎜ ‾‾‾ ⎟ \n\
⎝x = 0 ⎠ \
"""
assert pretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
/ 2 \\ \n\
|______ | \n\
| | | 2| \n\
| | | x | \n\
| | | | \n\
\\x = 1 / \
"""
assert upretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
⎛ 2 ⎞ \n\
⎜─┬──┬─ ⎟ \n\
⎜ │ │ 2⎟ \n\
⎜ │ │ x ⎟ \n\
⎜ │ │ ⎟ \n\
⎝x = 1 ⎠ \
"""
f = Function('f')
assert pretty(Derivative(f(x), x)**2) == \
"""\
2
/d \\ \n\
|--(f(x))| \n\
\\dx / \
"""
assert upretty(Derivative(f(x), x)**2) == \
"""\
2
⎛d ⎞ \n\
⎜──(f(x))⎟ \n\
⎝dx ⎠ \
"""
def test_issue_6739():
ascii_str = \
"""\
1 \n\
-----\n\
___\n\
\\/ x \
"""
ucode_str = \
"""\
1 \n\
──\n\
√x\
"""
assert pretty(1/sqrt(x)) == ascii_str
assert upretty(1/sqrt(x)) == ucode_str
def test_complicated_symbol_unchanged():
for symb_name in ["dexpr2_d1tau", "dexpr2^d1tau"]:
assert pretty(Symbol(symb_name)) == symb_name
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram, DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert pretty(A1) == "A1"
assert upretty(A1) == "A₁"
assert pretty(f1) == "f1:A1-->A2"
assert upretty(f1) == "f₁:A₁——▶A₂"
assert pretty(id_A1) == "id:A1-->A1"
assert upretty(id_A1) == "id:A₁——▶A₁"
assert pretty(f2*f1) == "f2*f1:A1-->A3"
assert upretty(f2*f1) == "f₂∘f₁:A₁——▶A₃"
assert pretty(K1) == "K1"
assert upretty(K1) == "K₁"
# Test how diagrams are printed.
d = Diagram()
assert pretty(d) == "EmptySet"
assert upretty(d) == "∅"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \
"EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \
"EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}"
assert upretty(d) == "{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, " \
"id:A₂——▶A₂: ∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \
"EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \
"EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}" \
" ==> {f2*f1:A1-->A3: {unique}}"
assert upretty(d) == "{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, id:A₂——▶A₂: " \
"∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" \
" ══▶ {f₂∘f₁:A₁——▶A₃: {unique}}"
grid = DiagramGrid(d)
assert pretty(grid) == "A1 A2\n \nA3 "
assert upretty(grid) == "A₁ A₂\n \nA₃ "
def test_PrettyModules():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
ucode_str = \
"""\
2\n\
ℚ[x, y] \
"""
ascii_str = \
"""\
2\n\
QQ[x, y] \
"""
assert upretty(F) == ucode_str
assert pretty(F) == ascii_str
ucode_str = \
"""\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
"""
ascii_str = \
"""\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(M) == ucode_str
assert pretty(M) == ascii_str
I = R.ideal(x**2, y)
ucode_str = \
"""\
╱ 2 ╲\n\
╲x , y╱\
"""
ascii_str = \
"""\
2 \n\
<x , y>\
"""
assert upretty(I) == ucode_str
assert pretty(I) == ascii_str
Q = F / M
ucode_str = \
"""\
2 \n\
ℚ[x, y] \n\
─────────────────\n\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
"""
ascii_str = \
"""\
2 \n\
QQ[x, y] \n\
-----------------\n\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(Q) == ucode_str
assert pretty(Q) == ascii_str
ucode_str = \
"""\
╱⎡ 3⎤ ╲\n\
│⎢ x ⎥ ╱ ⎡ 2⎤╲ ╱ ⎡ 2⎤╲│\n\
│⎢1, ──⎥ + ╲[x, y], ⎣1, x ⎦╱, [2, y] + ╲[x, y], ⎣1, x ⎦╱│\n\
╲⎣ 2 ⎦ ╱\
"""
ascii_str = \
"""\
3 \n\
x 2 2 \n\
<[1, --] + <[x, y], [1, x ]>, [2, y] + <[x, y], [1, x ]>>\n\
2 \
"""
def test_QuotientRing():
R = QQ.old_poly_ring(x)/[x**2 + 1]
ucode_str = \
"""\
ℚ[x] \n\
────────\n\
╱ 2 ╲\n\
╲x + 1╱\
"""
ascii_str = \
"""\
QQ[x] \n\
--------\n\
2 \n\
<x + 1>\
"""
assert upretty(R) == ucode_str
assert pretty(R) == ascii_str
ucode_str = \
"""\
╱ 2 ╲\n\
1 + ╲x + 1╱\
"""
ascii_str = \
"""\
2 \n\
1 + <x + 1>\
"""
assert upretty(R.one) == ucode_str
assert pretty(R.one) == ascii_str
def test_Homomorphism():
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x)
expr = homomorphism(R.free_module(1), R.free_module(1), [0])
ucode_str = \
"""\
1 1\n\
[0] : ℚ[x] ──> ℚ[x] \
"""
ascii_str = \
"""\
1 1\n\
[0] : QQ[x] --> QQ[x] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(2), R.free_module(2), [0, 0])
ucode_str = \
"""\
⎡0 0⎤ 2 2\n\
⎢ ⎥ : ℚ[x] ──> ℚ[x] \n\
⎣0 0⎦ \
"""
ascii_str = \
"""\
[0 0] 2 2\n\
[ ] : QQ[x] --> QQ[x] \n\
[0 0] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])
ucode_str = \
"""\
1\n\
1 ℚ[x] \n\
[0] : ℚ[x] ──> ─────\n\
<[x]>\
"""
ascii_str = \
"""\
1\n\
1 QQ[x] \n\
[0] : QQ[x] --> ------\n\
<[x]> \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert pretty(t) == r'Tr(A*B)'
assert upretty(t) == 'Tr(A⋅B)'
def test_pretty_Add():
eq = Mul(-2, x - 2, evaluate=False) + 5
assert pretty(eq) == '5 - 2*(x - 2)'
def test_issue_7179():
assert upretty(Not(Equivalent(x, y))) == 'x ⇎ y'
assert upretty(Not(Implies(x, y))) == 'x ↛ y'
def test_issue_7180():
assert upretty(Equivalent(x, y)) == 'x ⇔ y'
def test_pretty_Complement():
assert pretty(S.Reals - S.Naturals) == '(-oo, oo) \\ Naturals'
assert upretty(S.Reals - S.Naturals) == 'ℝ \\ ℕ'
assert pretty(S.Reals - S.Naturals0) == '(-oo, oo) \\ Naturals0'
assert upretty(S.Reals - S.Naturals0) == 'ℝ \\ ℕ₀'
def test_pretty_SymmetricDifference():
from sympy.sets.sets import SymmetricDifference
assert upretty(SymmetricDifference(Interval(2,3), Interval(3,5), \
evaluate = False)) == '[2, 3] ∆ [3, 5]'
with raises(NotImplementedError):
pretty(SymmetricDifference(Interval(2,3), Interval(3,5), evaluate = False))
def test_pretty_Contains():
assert pretty(Contains(x, S.Integers)) == 'Contains(x, Integers)'
assert upretty(Contains(x, S.Integers)) == 'x ∈ ℤ'
def test_issue_8292():
from sympy.core import sympify
e = sympify('((x+x**4)/(x-1))-(2*(x-1)**4/(x-1)**4)', evaluate=False)
ucode_str = \
"""\
4 4 \n\
2⋅(x - 1) x + x\n\
- ────────── + ──────\n\
4 x - 1 \n\
(x - 1) \
"""
ascii_str = \
"""\
4 4 \n\
2*(x - 1) x + x\n\
- ---------- + ------\n\
4 x - 1 \n\
(x - 1) \
"""
assert pretty(e) == ascii_str
assert upretty(e) == ucode_str
def test_issue_4335():
y = Function('y')
expr = -y(x).diff(x)
ucode_str = \
"""\
d \n\
-──(y(x))\n\
dx \
"""
ascii_str = \
"""\
d \n\
- --(y(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_8344():
from sympy.core import sympify
e = sympify('2*x*y**2/1**2 + 1', evaluate=False)
ucode_str = \
"""\
2 \n\
2⋅x⋅y \n\
────── + 1\n\
2 \n\
1 \
"""
assert upretty(e) == ucode_str
def test_issue_6324():
x = Pow(2, 3, evaluate=False)
y = Pow(10, -2, evaluate=False)
e = Mul(x, y, evaluate=False)
ucode_str = \
"""\
3\n\
2 \n\
───\n\
2\n\
10 \
"""
assert upretty(e) == ucode_str
def test_issue_7927():
e = sin(x/2)**cos(x/2)
ucode_str = \
"""\
⎛x⎞\n\
cos⎜─⎟\n\
⎝2⎠\n\
⎛ ⎛x⎞⎞ \n\
⎜sin⎜─⎟⎟ \n\
⎝ ⎝2⎠⎠ \
"""
assert upretty(e) == ucode_str
e = sin(x)**(S(11)/13)
ucode_str = \
"""\
11\n\
──\n\
13\n\
(sin(x)) \
"""
assert upretty(e) == ucode_str
def test_issue_6134():
from sympy.abc import lamda, t
phi = Function('phi')
e = lamda*x*Integral(phi(t)*pi*sin(pi*t), (t, 0, 1)) + lamda*x**2*Integral(phi(t)*2*pi*sin(2*pi*t), (t, 0, 1))
ucode_str = \
"""\
1 1 \n\
2 ⌠ ⌠ \n\
λ⋅x ⋅⎮ 2⋅π⋅φ(t)⋅sin(2⋅π⋅t) dt + λ⋅x⋅⎮ π⋅φ(t)⋅sin(π⋅t) dt\n\
⌡ ⌡ \n\
0 0 \
"""
assert upretty(e) == ucode_str
def test_issue_9877():
ucode_str1 = '(2, 3) ∪ ([1, 2] \\ {x})'
a, b, c = Interval(2, 3, True, True), Interval(1, 2), FiniteSet(x)
assert upretty(Union(a, Complement(b, c))) == ucode_str1
ucode_str2 = '{x} ∩ {y} ∩ ({z} \\ [1, 2])'
d, e, f, g = FiniteSet(x), FiniteSet(y), FiniteSet(z), Interval(1, 2)
assert upretty(Intersection(d, e, Complement(f, g))) == ucode_str2
def test_issue_13651():
expr1 = c + Mul(-1, a + b, evaluate=False)
assert pretty(expr1) == 'c - (a + b)'
expr2 = c + Mul(-1, a - b + d, evaluate=False)
assert pretty(expr2) == 'c - (a - b + d)'
def test_pretty_primenu():
from sympy.ntheory.factor_ import primenu
ascii_str1 = "nu(n)"
ucode_str1 = "ν(n)"
n = symbols('n', integer=True)
assert pretty(primenu(n)) == ascii_str1
assert upretty(primenu(n)) == ucode_str1
def test_pretty_primeomega():
from sympy.ntheory.factor_ import primeomega
ascii_str1 = "Omega(n)"
ucode_str1 = "Ω(n)"
n = symbols('n', integer=True)
assert pretty(primeomega(n)) == ascii_str1
assert upretty(primeomega(n)) == ucode_str1
def test_pretty_Mod():
from sympy.core import Mod
ascii_str1 = "x mod 7"
ucode_str1 = "x mod 7"
ascii_str2 = "(x + 1) mod 7"
ucode_str2 = "(x + 1) mod 7"
ascii_str3 = "2*x mod 7"
ucode_str3 = "2⋅x mod 7"
ascii_str4 = "(x mod 7) + 1"
ucode_str4 = "(x mod 7) + 1"
ascii_str5 = "2*(x mod 7)"
ucode_str5 = "2⋅(x mod 7)"
x = symbols('x', integer=True)
assert pretty(Mod(x, 7)) == ascii_str1
assert upretty(Mod(x, 7)) == ucode_str1
assert pretty(Mod(x + 1, 7)) == ascii_str2
assert upretty(Mod(x + 1, 7)) == ucode_str2
assert pretty(Mod(2 * x, 7)) == ascii_str3
assert upretty(Mod(2 * x, 7)) == ucode_str3
assert pretty(Mod(x, 7) + 1) == ascii_str4
assert upretty(Mod(x, 7) + 1) == ucode_str4
assert pretty(2 * Mod(x, 7)) == ascii_str5
assert upretty(2 * Mod(x, 7)) == ucode_str5
def test_issue_11801():
assert pretty(Symbol("")) == ""
assert upretty(Symbol("")) == ""
def test_pretty_UnevaluatedExpr():
x = symbols('x')
he = UnevaluatedExpr(1/x)
ucode_str = \
"""\
1\n\
─\n\
x\
"""
assert upretty(he) == ucode_str
ucode_str = \
"""\
2\n\
⎛1⎞ \n\
⎜─⎟ \n\
⎝x⎠ \
"""
assert upretty(he**2) == ucode_str
ucode_str = \
"""\
1\n\
1 + ─\n\
x\
"""
assert upretty(he + 1) == ucode_str
ucode_str = \
('''\
1\n\
x⋅─\n\
x\
''')
assert upretty(x*he) == ucode_str
def test_issue_10472():
M = (Matrix([[0, 0], [0, 0]]), Matrix([0, 0]))
ucode_str = \
"""\
⎛⎡0 0⎤ ⎡0⎤⎞
⎜⎢ ⎥, ⎢ ⎥⎟
⎝⎣0 0⎦ ⎣0⎦⎠\
"""
assert upretty(M) == ucode_str
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
ascii_str1 = "A_00"
ucode_str1 = "A₀₀"
assert pretty(A[0, 0]) == ascii_str1
assert upretty(A[0, 0]) == ucode_str1
ascii_str1 = "3*A_00"
ucode_str1 = "3⋅A₀₀"
assert pretty(3*A[0, 0]) == ascii_str1
assert upretty(3*A[0, 0]) == ucode_str1
ascii_str1 = "(-B + A)[0, 0]"
ucode_str1 = "(-B + A)[0, 0]"
F = C[0, 0].subs(C, A - B)
assert pretty(F) == ascii_str1
assert upretty(F) == ucode_str1
def test_issue_12675():
x, y, t, j = symbols('x y t j')
e = CoordSys3D('e')
ucode_str = \
"""\
⎛ t⎞ \n\
⎜⎛x⎞ ⎟ j_e\n\
⎜⎜─⎟ ⎟ \n\
⎝⎝y⎠ ⎠ \
"""
assert upretty((x/y)**t*e.j) == ucode_str
ucode_str = \
"""\
⎛1⎞ \n\
⎜─⎟ j_e\n\
⎝y⎠ \
"""
assert upretty((1/y)*e.j) == ucode_str
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert pretty(-A*B*C) == "-A*B*C"
assert pretty(A - B) == "-B + A"
assert pretty(A*B*C - A*B - B*C) == "-A*B -B*C + A*B*C"
# issue #14814
x = MatrixSymbol('x', n, n)
y = MatrixSymbol('y*', n, n)
assert pretty(x + y) == "x + y*"
ascii_str = \
"""\
2 \n\
-2*y* -a*x\
"""
assert pretty(-a*x + -2*y*y) == ascii_str
def test_degree_printing():
expr1 = 90*degree
assert pretty(expr1) == '90°'
expr2 = x*degree
assert pretty(expr2) == 'x°'
expr3 = cos(x*degree + 90*degree)
assert pretty(expr3) == 'cos(x° + 90°)'
def test_vector_expr_pretty_printing():
A = CoordSys3D('A')
assert upretty(Cross(A.i, A.x*A.i+3*A.y*A.j)) == "(i_A)×((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(x*Cross(A.i, A.j)) == 'x⋅(i_A)×(j_A)'
assert upretty(Curl(A.x*A.i + 3*A.y*A.j)) == "∇×((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(Divergence(A.x*A.i + 3*A.y*A.j)) == "∇⋅((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(Dot(A.i, A.x*A.i+3*A.y*A.j)) == "(i_A)⋅((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(Gradient(A.x+3*A.y)) == "∇(x_A + 3⋅y_A)"
assert upretty(Laplacian(A.x+3*A.y)) == "∆(x_A + 3⋅y_A)"
# TODO: add support for ASCII pretty.
def test_pretty_print_tensor_expr():
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
expr = -i
ascii_str = \
"""\
-i\
"""
ucode_str = \
"""\
-i\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)
ascii_str = \
"""\
i\n\
A \n\
\
"""
ucode_str = \
"""\
i\n\
A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i0)
ascii_str = \
"""\
i_0\n\
A \n\
\
"""
ucode_str = \
"""\
i₀\n\
A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(-i)
ascii_str = \
"""\
\n\
A \n\
i\
"""
ucode_str = \
"""\
\n\
A \n\
i\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -3*A(-i)
ascii_str = \
"""\
\n\
-3*A \n\
i\
"""
ucode_str = \
"""\
\n\
-3⋅A \n\
i\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)
ascii_str = \
"""\
i \n\
H \n\
j\
"""
ucode_str = \
"""\
i \n\
H \n\
j\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -i)
ascii_str = \
"""\
L_0 \n\
H \n\
L_0\
"""
ucode_str = \
"""\
L₀ \n\
H \n\
L₀\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)*A(j)*B(k)
ascii_str = \
"""\
i L_0 k\n\
H *A *B \n\
L_0 \
"""
ucode_str = \
"""\
i L₀ k\n\
H ⋅A ⋅B \n\
L₀ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1+x)*A(i)
ascii_str = \
"""\
i\n\
(x + 1)*A \n\
\
"""
ucode_str = \
"""\
i\n\
(x + 1)⋅A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i) + 3*B(i)
ascii_str = \
"""\
i i\n\
3*B + A \n\
\
"""
ucode_str = \
"""\
i i\n\
3⋅B + A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_print_tensor_partial_deriv():
from sympy.tensor.toperators import PartialDerivative
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
expr = PartialDerivative(A(i), A(j))
ascii_str = \
"""\
d / i\\\n\
---|A |\n\
j\\ /\n\
dA \n\
\
"""
ucode_str = \
"""\
∂ ⎛ i⎞\n\
───⎜A ⎟\n\
j⎝ ⎠\n\
∂A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k \\\n\
A *---|H |\n\
j\\ L_0/\n\
dA \n\
\
"""
ucode_str = \
"""\
L₀ ∂ ⎛ k ⎞\n\
A ⋅───⎜H ⎟\n\
j⎝ L₀⎠\n\
∂A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k k \\\n\
A *---|3*H + B *C |\n\
j\\ L_0 L_0/\n\
dA \n\
\
"""
ucode_str = \
"""\
L₀ ∂ ⎛ k k ⎞\n\
A ⋅───⎜3⋅H + B ⋅C ⎟\n\
j⎝ L₀ L₀⎠\n\
∂A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(j), D(j))
ascii_str = \
"""\
/ i i\\ d / L_0\\\n\
|A + B |*-----|C |\n\
\\ / L_0\\ /\n\
dD \n\
\
"""
ucode_str = \
"""\
⎛ i i⎞ ∂ ⎛ L₀⎞\n\
⎜A + B ⎟⋅────⎜C ⎟\n\
⎝ ⎠ L₀⎝ ⎠\n\
∂D \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-i), D(j))
ascii_str = \
"""\
/ L_0 L_0\\ d / \\\n\
|A + B |*---|C |\n\
\\ / j\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
"""\
⎛ L₀ L₀⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅───⎜C ⎟\n\
⎝ ⎠ j⎝ L₀⎠\n\
∂D \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n))
ucode_str = """\
2 \n\
∂ ⎛ ⎞\n\
───────⎜A + B ⎟\n\
⎝ i i⎠\n\
∂A ∂A \n\
n j \
"""
assert upretty(expr) == ucode_str
expr = PartialDerivative(3*A(-i), A(-j), A(-n))
ucode_str = """\
2 \n\
∂ ⎛ ⎞\n\
───────⎜3⋅A ⎟\n\
⎝ i⎠\n\
∂A ∂A \n\
n j \
"""
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1})
ascii_str = \
"""\
i=1,j\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i: 1, j: 1})
ascii_str = \
"""\
i=1,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {j: 1})
ascii_str = \
"""\
i,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
expr = TensorElement(H(-i, j), {-i: 1})
ascii_str = \
"""\
j\n\
H \n\
i=1 \
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_15560():
a = MatrixSymbol('a', 1, 1)
e = pretty(a*(KroneckerProduct(a, a)))
result = 'a*(a x a)'
assert e == result
def test_print_lerchphi():
# Part of issue 6013
a = Symbol('a')
pretty(lerchphi(a, 1, 2))
uresult = 'Φ(a, 1, 2)'
aresult = 'lerchphi(a, 1, 2)'
assert pretty(lerchphi(a, 1, 2)) == aresult
assert upretty(lerchphi(a, 1, 2)) == uresult
def test_issue_15583():
N = mechanics.ReferenceFrame('N')
result = '(n_x, n_y, n_z)'
e = pretty((N.x, N.y, N.z))
assert e == result
def test_matrixSymbolBold():
# Issue 15871
def boldpretty(expr):
return xpretty(expr, use_unicode=True, wrap_line=False, mat_symbol_style="bold")
from sympy.matrices.expressions.trace import trace
A = MatrixSymbol("A", 2, 2)
assert boldpretty(trace(A)) == 'tr(𝐀)'
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert boldpretty(-A) == '-𝐀'
assert boldpretty(A - A*B - B) == '-𝐁 -𝐀⋅𝐁 + 𝐀'
assert boldpretty(-A*B - A*B*C - B) == '-𝐁 -𝐀⋅𝐁 -𝐀⋅𝐁⋅𝐂'
A = MatrixSymbol("Addot", 3, 3)
assert boldpretty(A) == '𝐀̈'
omega = MatrixSymbol("omega", 3, 3)
assert boldpretty(omega) == 'ω'
omega = MatrixSymbol("omeganorm", 3, 3)
assert boldpretty(omega) == '‖ω‖'
a = Symbol('alpha')
b = Symbol('b')
c = MatrixSymbol("c", 3, 1)
d = MatrixSymbol("d", 3, 1)
assert boldpretty(a*B*c+b*d) == 'b⋅𝐝 + α⋅𝐁⋅𝐜'
d = MatrixSymbol("delta", 3, 1)
B = MatrixSymbol("Beta", 3, 3)
assert boldpretty(a*B*c+b*d) == 'b⋅δ + α⋅Β⋅𝐜'
A = MatrixSymbol("A_2", 3, 3)
assert boldpretty(A) == '𝐀₂'
def test_center_accent():
assert center_accent('a', '\N{COMBINING TILDE}') == 'ã'
assert center_accent('aa', '\N{COMBINING TILDE}') == 'aã'
assert center_accent('aaa', '\N{COMBINING TILDE}') == 'aãa'
assert center_accent('aaaa', '\N{COMBINING TILDE}') == 'aaãa'
assert center_accent('aaaaa', '\N{COMBINING TILDE}') == 'aaãaa'
assert center_accent('abcdefg', '\N{COMBINING FOUR DOTS ABOVE}') == 'abcd⃜efg'
def test_imaginary_unit():
from sympy.printing.pretty import pretty # b/c it was redefined above
assert pretty(1 + I, use_unicode=False) == '1 + I'
assert pretty(1 + I, use_unicode=True) == '1 + ⅈ'
assert pretty(1 + I, use_unicode=False, imaginary_unit='j') == '1 + I'
assert pretty(1 + I, use_unicode=True, imaginary_unit='j') == '1 + ⅉ'
raises(TypeError, lambda: pretty(I, imaginary_unit=I))
raises(ValueError, lambda: pretty(I, imaginary_unit="kkk"))
def test_str_special_matrices():
from sympy.matrices import Identity, ZeroMatrix, OneMatrix
assert pretty(Identity(4)) == 'I'
assert upretty(Identity(4)) == '𝕀'
assert pretty(ZeroMatrix(2, 2)) == '0'
assert upretty(ZeroMatrix(2, 2)) == '𝟘'
assert pretty(OneMatrix(2, 2)) == '1'
assert upretty(OneMatrix(2, 2)) == '𝟙'
def test_pretty_misc_functions():
assert pretty(LambertW(x)) == 'W(x)'
assert upretty(LambertW(x)) == 'W(x)'
assert pretty(LambertW(x, y)) == 'W(x, y)'
assert upretty(LambertW(x, y)) == 'W(x, y)'
assert pretty(airyai(x)) == 'Ai(x)'
assert upretty(airyai(x)) == 'Ai(x)'
assert pretty(airybi(x)) == 'Bi(x)'
assert upretty(airybi(x)) == 'Bi(x)'
assert pretty(airyaiprime(x)) == "Ai'(x)"
assert upretty(airyaiprime(x)) == "Ai'(x)"
assert pretty(airybiprime(x)) == "Bi'(x)"
assert upretty(airybiprime(x)) == "Bi'(x)"
assert pretty(fresnelc(x)) == 'C(x)'
assert upretty(fresnelc(x)) == 'C(x)'
assert pretty(fresnels(x)) == 'S(x)'
assert upretty(fresnels(x)) == 'S(x)'
assert pretty(Heaviside(x)) == 'Heaviside(x)'
assert upretty(Heaviside(x)) == 'θ(x)'
assert pretty(Heaviside(x, y)) == 'Heaviside(x, y)'
assert upretty(Heaviside(x, y)) == 'θ(x, y)'
assert pretty(dirichlet_eta(x)) == 'dirichlet_eta(x)'
assert upretty(dirichlet_eta(x)) == 'η(x)'
def test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
# Testing printer:
expr = hadamard_power(A, n)
ascii_str = \
"""\
.n\n\
A \
"""
ucode_str = \
"""\
∘n\n\
A \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hadamard_power(A, 1+n)
ascii_str = \
"""\
.(n + 1)\n\
A \
"""
ucode_str = \
"""\
∘(n + 1)\n\
A \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hadamard_power(A*B.T, 1+n)
ascii_str = \
"""\
.(n + 1)\n\
/ T\\ \n\
\\A*B / \
"""
ucode_str = \
"""\
∘(n + 1)\n\
⎛ T⎞ \n\
⎝A⋅B ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_17258():
n = Symbol('n', integer=True)
assert pretty(Sum(n, (n, -oo, 1))) == \
' 1 \n'\
' __ \n'\
' \\ ` \n'\
' ) n\n'\
' /_, \n'\
'n = -oo '
assert upretty(Sum(n, (n, -oo, 1))) == \
"""\
1 \n\
___ \n\
╲ \n\
╲ \n\
╱ n\n\
╱ \n\
‾‾‾ \n\
n = -∞ \
"""
def test_is_combining():
line = "v̇_m"
assert [is_combining(sym) for sym in line] == \
[False, True, False, False]
def test_issue_17616():
assert pretty(pi**(1/exp(1))) == \
' / -1\\\n'\
' \\e /\n'\
'pi '
assert upretty(pi**(1/exp(1))) == \
' ⎛ -1⎞\n'\
' ⎝ℯ ⎠\n'\
'π '
assert pretty(pi**(1/pi)) == \
' 1 \n'\
' --\n'\
' pi\n'\
'pi '
assert upretty(pi**(1/pi)) == \
' 1\n'\
' ─\n'\
' π\n'\
'π '
assert pretty(pi**(1/EulerGamma)) == \
' 1 \n'\
' ----------\n'\
' EulerGamma\n'\
'pi '
assert upretty(pi**(1/EulerGamma)) == \
' 1\n'\
' ─\n'\
' γ\n'\
'π '
z = Symbol("x_17")
assert upretty(7**(1/z)) == \
'x₁₇___\n'\
' ╲╱ 7 '
assert pretty(7**(1/z)) == \
'x_17___\n'\
' \\/ 7 '
def test_issue_17857():
assert pretty(Range(-oo, oo)) == '{..., -1, 0, 1, ...}'
assert pretty(Range(oo, -oo, -1)) == '{..., 1, 0, -1, ...}'
def test_issue_18272():
x = Symbol('x')
n = Symbol('n')
assert upretty(ConditionSet(x, Eq(-x + exp(x), 0), S.Complexes)) == \
'⎧ │ ⎛ x ⎞⎫\n'\
'⎨x │ x ∊ ℂ ∧ ⎝-x + ℯ = 0⎠⎬\n'\
'⎩ │ ⎭'
assert upretty(ConditionSet(x, Contains(n/2, Interval(0, oo)), FiniteSet(-n/2, n/2))) == \
'⎧ │ ⎧-n n⎫ ⎛n ⎞⎫\n'\
'⎨x │ x ∊ ⎨───, ─⎬ ∧ ⎜─ ∈ [0, ∞)⎟⎬\n'\
'⎩ │ ⎩ 2 2⎭ ⎝2 ⎠⎭'
assert upretty(ConditionSet(x, Eq(Piecewise((1, x >= 3), (x/2 - 1/2, x >= 2), (1/2, x >= 1),
(x/2, True)) - 1/2, 0), Interval(0, 3))) == \
'⎧ │ ⎛⎛⎧ 1 for x ≥ 3⎞ ⎞⎫\n'\
'⎪ │ ⎜⎜⎪ ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪x ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪─ - 0.5 for x ≥ 2⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪2 ⎟ ⎟⎪\n'\
'⎨x │ x ∊ [0, 3] ∧ ⎜⎜⎨ ⎟ - 0.5 = 0⎟⎬\n'\
'⎪ │ ⎜⎜⎪ 0.5 for x ≥ 1⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪ ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪ x ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪ ─ otherwise⎟ ⎟⎪\n'\
'⎩ │ ⎝⎝⎩ 2 ⎠ ⎠⎭'
def test_Str():
from sympy.core.symbol import Str
assert pretty(Str('x')) == 'x'
def test_symbolic_probability():
mu = symbols("mu")
sigma = symbols("sigma", positive=True)
X = Normal("X", mu, sigma)
assert pretty(Expectation(X)) == r'E[X]'
assert pretty(Variance(X)) == r'Var(X)'
assert pretty(Probability(X > 0)) == r'P(X > 0)'
Y = Normal("Y", mu, sigma)
assert pretty(Covariance(X, Y)) == 'Cov(X, Y)'
def test_issue_21758():
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.series.fourier import FourierSeries
x = Symbol('x')
k, n = symbols('k n')
fo = FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), SeqFormula(
Piecewise((-2*pi*cos(n*pi)/n + 2*sin(n*pi)/n**2, (n > -oo) & (n < oo) & Ne(n, 0)),
(0, True))*sin(n*x)/pi, (n, 1, oo))))
assert upretty(piecewise_fold(fo)) == \
'⎧ 2⋅sin(3⋅x) \n'\
'⎪2⋅sin(x) - sin(2⋅x) + ────────── + … for n > -∞ ∧ n < ∞ ∧ n ≠ 0\n'\
'⎨ 3 \n'\
'⎪ \n'\
'⎩ 0 otherwise '
assert pretty(FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)),
SeqFormula(0, (n, 1, oo))))) == '0'
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
x,y = symbols('x y', real=True)
m = Manifold('M', 2)
assert pretty(m) == 'M'
p = Patch('P', m)
assert pretty(p) == "P"
rect = CoordSystem('rect', p, [x, y])
assert pretty(rect) == "rect"
b = BaseScalarField(rect, 0)
assert pretty(b) == "x"
def test_deprecated_prettyForm():
with warns_deprecated_sympy():
from sympy.printing.pretty.pretty_symbology import xstr
assert xstr(1) == '1'
with warns_deprecated_sympy():
from sympy.printing.pretty.stringpict import prettyForm
p = prettyForm('s', unicode='s')
with warns_deprecated_sympy():
assert p.unicode == p.s == 's'
|
d95789d70fba878a93d7f94845d2714d6d1a7777fa434423712ce41800e941d2 | from sympy.concrete.summations import (Sum, summation)
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import (Derivative, Function, Lambda, diff)
from sympy.core import EulerGamma
from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.complexes import (Abs, im, polar_lift, re, sign)
from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, cosh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, sinc, tan)
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.error_functions import (Ci, Ei, Si, erf, erfc, erfi, fresnelc, li)
from sympy.functions.special.gamma_functions import (gamma, polygamma)
from sympy.functions.special.hyper import (hyper, meijerg)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import lerchphi
from sympy.integrals.integrals import integrate
from sympy.logic.boolalg import And
from sympy.matrices.dense import Matrix
from sympy.polys.polytools import (Poly, factor)
from sympy.printing.str import sstr
from sympy.series.order import O
from sympy.sets.sets import Interval
from sympy.simplify.gammasimp import gammasimp
from sympy.simplify.simplify import simplify
from sympy.simplify.trigsimp import trigsimp
from sympy.tensor.indexed import (Idx, IndexedBase)
from sympy.core.expr import unchanged
from sympy.functions.elementary.integers import floor
from sympy.integrals.integrals import Integral
from sympy.integrals.risch import NonElementaryIntegral
from sympy.physics import units
from sympy.testing.pytest import (raises, slow, skip, ON_TRAVIS,
warns_deprecated_sympy, warns)
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.random import verify_numerically
x, y, a, t, x_1, x_2, z, s, b = symbols('x y a t x_1 x_2 z s b')
n = Symbol('n', integer=True)
f = Function('f')
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_poly_deprecated():
p = Poly(2*x, x)
assert p.integrate(x) == Poly(x**2, x, domain='QQ')
# The stacklevel is based on Integral(Poly)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
integrate(p, x)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Integral(p, (x,))
@slow
def test_principal_value():
g = 1 / x
assert Integral(g, (x, -oo, oo)).principal_value() == 0
assert Integral(g, (y, -oo, oo)).principal_value() == oo * sign(1 / x)
raises(ValueError, lambda: Integral(g, (x)).principal_value())
raises(ValueError, lambda: Integral(g).principal_value())
l = 1 / ((x ** 3) - 1)
assert Integral(l, (x, -oo, oo)).principal_value().together() == -sqrt(3)*pi/3
raises(ValueError, lambda: Integral(l, (x, -oo, 1)).principal_value())
d = 1 / (x ** 2 - 1)
assert Integral(d, (x, -oo, oo)).principal_value() == 0
assert Integral(d, (x, -2, 2)).principal_value() == -log(3)
v = x / (x ** 2 - 1)
assert Integral(v, (x, -oo, oo)).principal_value() == 0
assert Integral(v, (x, -2, 2)).principal_value() == 0
s = x ** 2 / (x ** 2 - 1)
assert Integral(s, (x, -oo, oo)).principal_value() is oo
assert Integral(s, (x, -2, 2)).principal_value() == -log(3) + 4
f = 1 / ((x ** 2 - 1) * (1 + x ** 2))
assert Integral(f, (x, -oo, oo)).principal_value() == -pi / 2
assert Integral(f, (x, -2, 2)).principal_value() == -atan(2) - log(3) / 2
def diff_test(i):
"""Return the set of symbols, s, which were used in testing that
i.diff(s) agrees with i.doit().diff(s). If there is an error then
the assertion will fail, causing the test to fail."""
syms = i.free_symbols
for s in syms:
assert (i.diff(s).doit() - i.doit().diff(s)).expand() == 0
return syms
def test_improper_integral():
assert integrate(log(x), (x, 0, 1)) == -1
assert integrate(x**(-2), (x, 1, oo)) == 1
assert integrate(1/(1 + exp(x)), (x, 0, oo)) == log(2)
def test_constructor():
# this is shared by Sum, so testing Integral's constructor
# is equivalent to testing Sum's
s1 = Integral(n, n)
assert s1.limits == (Tuple(n),)
s2 = Integral(n, (n,))
assert s2.limits == (Tuple(n),)
s3 = Integral(Sum(x, (x, 1, y)))
assert s3.limits == (Tuple(y),)
s4 = Integral(n, Tuple(n,))
assert s4.limits == (Tuple(n),)
s5 = Integral(n, (n, Interval(1, 2)))
assert s5.limits == (Tuple(n, 1, 2),)
# Testing constructor with inequalities:
s6 = Integral(n, n > 10)
assert s6.limits == (Tuple(n, 10, oo),)
s7 = Integral(n, (n > 2) & (n < 5))
assert s7.limits == (Tuple(n, 2, 5),)
def test_basics():
assert Integral(0, x) != 0
assert Integral(x, (x, 1, 1)) != 0
assert Integral(oo, x) != oo
assert Integral(S.NaN, x) is S.NaN
assert diff(Integral(y, y), x) == 0
assert diff(Integral(x, (x, 0, 1)), x) == 0
assert diff(Integral(x, x), x) == x
assert diff(Integral(t, (t, 0, x)), x) == x
e = (t + 1)**2
assert diff(integrate(e, (t, 0, x)), x) == \
diff(Integral(e, (t, 0, x)), x).doit().expand() == \
((1 + x)**2).expand()
assert diff(integrate(e, (t, 0, x)), t) == \
diff(Integral(e, (t, 0, x)), t) == 0
assert diff(integrate(e, (t, 0, x)), a) == \
diff(Integral(e, (t, 0, x)), a) == 0
assert diff(integrate(e, t), a) == diff(Integral(e, t), a) == 0
assert integrate(e, (t, a, x)).diff(x) == \
Integral(e, (t, a, x)).diff(x).doit().expand()
assert Integral(e, (t, a, x)).diff(x).doit() == ((1 + x)**2)
assert integrate(e, (t, x, a)).diff(x).doit() == (-(1 + x)**2).expand()
assert integrate(t**2, (t, x, 2*x)).diff(x) == 7*x**2
assert Integral(x, x).atoms() == {x}
assert Integral(f(x), (x, 0, 1)).atoms() == {S.Zero, S.One, x}
assert diff_test(Integral(x, (x, 3*y))) == {y}
assert diff_test(Integral(x, (a, 3*y))) == {x, y}
assert integrate(x, (x, oo, oo)) == 0 #issue 8171
assert integrate(x, (x, -oo, -oo)) == 0
# sum integral of terms
assert integrate(y + x + exp(x), x) == x*y + x**2/2 + exp(x)
assert Integral(x).is_commutative
n = Symbol('n', commutative=False)
assert Integral(n + x, x).is_commutative is False
def test_diff_wrt():
class Test(Expr):
_diff_wrt = True
is_commutative = True
t = Test()
assert integrate(t + 1, t) == t**2/2 + t
assert integrate(t + 1, (t, 0, 1)) == Rational(3, 2)
raises(ValueError, lambda: integrate(x + 1, x + 1))
raises(ValueError, lambda: integrate(x + 1, (x + 1, 0, 1)))
def test_basics_multiple():
assert diff_test(Integral(x, (x, 3*x, 5*y), (y, x, 2*x))) == {x}
assert diff_test(Integral(x, (x, 5*y), (y, x, 2*x))) == {x}
assert diff_test(Integral(x, (x, 5*y), (y, y, 2*x))) == {x, y}
assert diff_test(Integral(y, y, x)) == {x, y}
assert diff_test(Integral(y*x, x, y)) == {x, y}
assert diff_test(Integral(x + y, y, (y, 1, x))) == {x}
assert diff_test(Integral(x + y, (x, x, y), (y, y, x))) == {x, y}
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
x = Symbol("x", complex=True)
p = Integral(A*B, (x,))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
x = Symbol("x", real=True)
p = Integral(A*B, (x,))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
def test_integration():
assert integrate(0, (t, 0, x)) == 0
assert integrate(3, (t, 0, x)) == 3*x
assert integrate(t, (t, 0, x)) == x**2/2
assert integrate(3*t, (t, 0, x)) == 3*x**2/2
assert integrate(3*t**2, (t, 0, x)) == x**3
assert integrate(1/t, (t, 1, x)) == log(x)
assert integrate(-1/t**2, (t, 1, x)) == 1/x - 1
assert integrate(t**2 + 5*t - 8, (t, 0, x)) == x**3/3 + 5*x**2/2 - 8*x
assert integrate(x**2, x) == x**3/3
assert integrate((3*t*x)**5, x) == (3*t)**5 * x**6 / 6
b = Symbol("b")
c = Symbol("c")
assert integrate(a*t, (t, 0, x)) == a*x**2/2
assert integrate(a*t**4, (t, 0, x)) == a*x**5/5
assert integrate(a*t**2 + b*t + c, (t, 0, x)) == a*x**3/3 + b*x**2/2 + c*x
def test_multiple_integration():
assert integrate((x**2)*(y**2), (x, 0, 1), (y, -1, 2)) == Rational(1)
assert integrate((y**2)*(x**2), x, y) == Rational(1, 9)*(x**3)*(y**3)
assert integrate(1/(x + 3)/(1 + x)**3, x) == \
log(3 + x)*Rational(-1, 8) + log(1 + x)*Rational(1, 8) + x/(4 + 8*x + 4*x**2)
assert integrate(sin(x*y)*y, (x, 0, 1), (y, 0, 1)) == -sin(1) + 1
def test_issue_3532():
assert integrate(exp(-x), (x, 0, oo)) == 1
def test_issue_3560():
assert integrate(sqrt(x)**3, x) == 2*sqrt(x)**5/5
assert integrate(sqrt(x), x) == 2*sqrt(x)**3/3
assert integrate(1/sqrt(x)**3, x) == -2/sqrt(x)
def test_issue_18038():
raises(AttributeError, lambda: integrate((x, x)))
def test_integrate_poly():
p = Poly(x + x**2*y + y**3, x, y)
# The stacklevel is based on Integral(Poly)
with warns_deprecated_sympy():
qx = Integral(p, x)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
qx = integrate(p, x)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
qy = integrate(p, y)
assert isinstance(qx, Poly) is True
assert isinstance(qy, Poly) is True
assert qx.gens == (x, y)
assert qy.gens == (x, y)
assert qx.as_expr() == x**2/2 + x**3*y/3 + x*y**3
assert qy.as_expr() == x*y + x**2*y**2/2 + y**4/4
def test_integrate_poly_definite():
p = Poly(x + x**2*y + y**3, x, y)
with warns_deprecated_sympy():
Qx = Integral(p, (x, 0, 1))
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Qx = integrate(p, (x, 0, 1))
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Qy = integrate(p, (y, 0, pi))
assert isinstance(Qx, Poly) is True
assert isinstance(Qy, Poly) is True
assert Qx.gens == (y,)
assert Qy.gens == (x,)
assert Qx.as_expr() == S.Half + y/3 + y**3
assert Qy.as_expr() == pi**4/4 + pi*x + pi**2*x**2/2
def test_integrate_omit_var():
y = Symbol('y')
assert integrate(x) == x**2/2
raises(ValueError, lambda: integrate(2))
raises(ValueError, lambda: integrate(x*y))
def test_integrate_poly_accurately():
y = Symbol('y')
assert integrate(x*sin(y), x) == x**2*sin(y)/2
# when passed to risch_norman, this will be a CPU hog, so this really
# checks, that integrated function is recognized as polynomial
assert integrate(x**1000*sin(y), x) == x**1001*sin(y)/1001
def test_issue_3635():
y = Symbol('y')
assert integrate(x**2, y) == x**2*y
assert integrate(x**2, (y, -1, 1)) == 2*x**2
# works in SymPy and py.test but hangs in `setup.py test`
def test_integrate_linearterm_pow():
# check integrate((a*x+b)^c, x) -- issue 3499
y = Symbol('y', positive=True)
# TODO: Remove conds='none' below, let the assumption take care of it.
assert integrate(x**y, x, conds='none') == x**(y + 1)/(y + 1)
assert integrate((exp(y)*x + 1/y)**(1 + sin(y)), x, conds='none') == \
exp(-y)*(exp(y)*x + 1/y)**(2 + sin(y)) / (2 + sin(y))
def test_issue_3618():
assert integrate(pi*sqrt(x), x) == 2*pi*sqrt(x)**3/3
assert integrate(pi*sqrt(x) + E*sqrt(x)**3, x) == \
2*pi*sqrt(x)**3/3 + 2*E *sqrt(x)**5/5
def test_issue_3623():
assert integrate(cos((n + 1)*x), x) == Piecewise(
(sin(x*(n + 1))/(n + 1), Ne(n + 1, 0)), (x, True))
assert integrate(cos((n - 1)*x), x) == Piecewise(
(sin(x*(n - 1))/(n - 1), Ne(n - 1, 0)), (x, True))
assert integrate(cos((n + 1)*x) + cos((n - 1)*x), x) == \
Piecewise((sin(x*(n - 1))/(n - 1), Ne(n - 1, 0)), (x, True)) + \
Piecewise((sin(x*(n + 1))/(n + 1), Ne(n + 1, 0)), (x, True))
def test_issue_3664():
n = Symbol('n', integer=True, nonzero=True)
assert integrate(-1./2 * x * sin(n * pi * x/2), [x, -2, 0]) == \
2.0*cos(pi*n)/(pi*n)
assert integrate(x * sin(n * pi * x/2) * Rational(-1, 2), [x, -2, 0]) == \
2*cos(pi*n)/(pi*n)
def test_issue_3679():
# definite integration of rational functions gives wrong answers
assert NS(Integral(1/(x**2 - 8*x + 17), (x, 2, 4))) == '1.10714871779409'
def test_issue_3686(): # remove this when fresnel itegrals are implemented
from sympy.core.function import expand_func
from sympy.functions.special.error_functions import fresnels
assert expand_func(integrate(sin(x**2), x)) == \
sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))/2
def test_integrate_units():
m = units.m
s = units.s
assert integrate(x * m/s, (x, 1*s, 5*s)) == 12*m*s
def test_transcendental_functions():
assert integrate(LambertW(2*x), x) == \
-x + x*LambertW(2*x) + x/LambertW(2*x)
def test_log_polylog():
assert integrate(log(1 - x)/x, (x, 0, 1)) == -pi**2/6
assert integrate(log(x)*(1 - x)**(-1), (x, 0, 1)) == -pi**2/6
def test_issue_3740():
f = 4*log(x) - 2*log(x)**2
fid = diff(integrate(f, x), x)
assert abs(f.subs(x, 42).evalf() - fid.subs(x, 42).evalf()) < 1e-10
def test_issue_3788():
assert integrate(1/(1 + x**2), x) == atan(x)
def test_issue_3952():
f = sin(x)
assert integrate(f, x) == -cos(x)
raises(ValueError, lambda: integrate(f, 2*x))
def test_issue_4516():
assert integrate(2**x - 2*x, x) == 2**x/log(2) - x**2
def test_issue_7450():
ans = integrate(exp(-(1 + I)*x), (x, 0, oo))
assert re(ans) == S.Half and im(ans) == Rational(-1, 2)
def test_issue_8623():
assert integrate((1 + cos(2*x)) / (3 - 2*cos(2*x)), (x, 0, pi)) == -pi/2 + sqrt(5)*pi/2
assert integrate((1 + cos(2*x))/(3 - 2*cos(2*x))) == -x/2 + sqrt(5)*(atan(sqrt(5)*tan(x)) + \
pi*floor((x - pi/2)/pi))/2
def test_issue_9569():
assert integrate(1 / (2 - cos(x)), (x, 0, pi)) == pi/sqrt(3)
assert integrate(1/(2 - cos(x))) == 2*sqrt(3)*(atan(sqrt(3)*tan(x/2)) + pi*floor((x/2 - pi/2)/pi))/3
def test_issue_13733():
s = Symbol('s', positive=True)
pz = exp(-(z - y)**2/(2*s*s))/sqrt(2*pi*s*s)
pzgx = integrate(pz, (z, x, oo))
assert integrate(pzgx, (x, 0, oo)) == sqrt(2)*s*exp(-y**2/(2*s**2))/(2*sqrt(pi)) + \
y*erf(sqrt(2)*y/(2*s))/2 + y/2
def test_issue_13749():
assert integrate(1 / (2 + cos(x)), (x, 0, pi)) == pi/sqrt(3)
assert integrate(1/(2 + cos(x))) == 2*sqrt(3)*(atan(sqrt(3)*tan(x/2)/3) + pi*floor((x/2 - pi/2)/pi))/3
def test_issue_18133():
assert integrate(exp(x)/(1 + x)**2, x) == NonElementaryIntegral(exp(x)/(x + 1)**2, x)
def test_issue_21741():
a = Float('3999999.9999999995', precision=53)
b = Float('2.5000000000000004e-7', precision=53)
r = Piecewise((b*I*exp(-a*I*pi*t*y)*exp(-a*I*pi*x*z)/(pi*x),
Ne(1.0*pi*x*exp(a*I*pi*t*y), 0)),
(z*exp(-a*I*pi*t*y), True))
fun = E**((-2*I*pi*(z*x+t*y))/(500*10**(-9)))
assert integrate(fun, z) == r
def test_matrices():
M = Matrix(2, 2, lambda i, j: (i + j + 1)*sin((i + j + 1)*x))
assert integrate(M, x) == Matrix([
[-cos(x), -cos(2*x)],
[-cos(2*x), -cos(3*x)],
])
def test_integrate_functions():
# issue 4111
assert integrate(f(x), x) == Integral(f(x), x)
assert integrate(f(x), (x, 0, 1)) == Integral(f(x), (x, 0, 1))
assert integrate(f(x)*diff(f(x), x), x) == f(x)**2/2
assert integrate(diff(f(x), x) / f(x), x) == log(f(x))
def test_integrate_derivatives():
assert integrate(Derivative(f(x), x), x) == f(x)
assert integrate(Derivative(f(y), y), x) == x*Derivative(f(y), y)
assert integrate(Derivative(f(x), x)**2, x) == \
Integral(Derivative(f(x), x)**2, x)
def test_transform():
a = Integral(x**2 + 1, (x, -1, 2))
fx = x
fy = 3*y + 1
assert a.doit() == a.transform(fx, fy).doit()
assert a.transform(fx, fy).transform(fy, fx) == a
fx = 3*x + 1
fy = y
assert a.transform(fx, fy).transform(fy, fx) == a
a = Integral(sin(1/x), (x, 0, 1))
assert a.transform(x, 1/y) == Integral(sin(y)/y**2, (y, 1, oo))
assert a.transform(x, 1/y).transform(y, 1/x) == a
a = Integral(exp(-x**2), (x, -oo, oo))
assert a.transform(x, 2*y) == Integral(2*exp(-4*y**2), (y, -oo, oo))
# < 3 arg limit handled properly
assert Integral(x, x).transform(x, a*y).doit() == \
Integral(y*a**2, y).doit()
_3 = S(3)
assert Integral(x, (x, 0, -_3)).transform(x, 1/y).doit() == \
Integral(-1/x**3, (x, -oo, -1/_3)).doit()
assert Integral(x, (x, 0, _3)).transform(x, 1/y) == \
Integral(y**(-3), (y, 1/_3, oo))
# issue 8400
i = Integral(x + y, (x, 1, 2), (y, 1, 2))
assert i.transform(x, (x + 2*y, x)).doit() == \
i.transform(x, (x + 2*z, x)).doit() == 3
i = Integral(x, (x, a, b))
assert i.transform(x, 2*s) == Integral(4*s, (s, a/2, b/2))
raises(ValueError, lambda: i.transform(x, 1))
raises(ValueError, lambda: i.transform(x, s*t))
raises(ValueError, lambda: i.transform(x, -s))
raises(ValueError, lambda: i.transform(x, (s, t)))
raises(ValueError, lambda: i.transform(2*x, 2*s))
i = Integral(x**2, (x, 1, 2))
raises(ValueError, lambda: i.transform(x**2, s))
am = Symbol('a', negative=True)
bp = Symbol('b', positive=True)
i = Integral(x, (x, bp, am))
i.transform(x, 2*s)
assert i.transform(x, 2*s) == Integral(-4*s, (s, am/2, bp/2))
i = Integral(x, (x, a))
assert i.transform(x, 2*s) == Integral(4*s, (s, a/2))
def test_issue_4052():
f = S.Half*asin(x) + x*sqrt(1 - x**2)/2
assert integrate(cos(asin(x)), x) == f
assert integrate(sin(acos(x)), x) == f
@slow
def test_evalf_integrals():
assert NS(Integral(x, (x, 2, 5)), 15) == '10.5000000000000'
gauss = Integral(exp(-x**2), (x, -oo, oo))
assert NS(gauss, 15) == '1.77245385090552'
assert NS(gauss**2 - pi + E*Rational(
1, 10**20), 15) in ('2.71828182845904e-20', '2.71828182845905e-20')
# A monster of an integral from http://mathworld.wolfram.com/DefiniteIntegral.html
t = Symbol('t')
a = 8*sqrt(3)/(1 + 3*t**2)
b = 16*sqrt(2)*(3*t + 1)*sqrt(4*t**2 + t + 1)**3
c = (3*t**2 + 1)*(11*t**2 + 2*t + 3)**2
d = sqrt(2)*(249*t**2 + 54*t + 65)/(11*t**2 + 2*t + 3)**2
f = a - b/c - d
assert NS(Integral(f, (t, 0, 1)), 50) == \
NS((3*sqrt(2) - 49*pi + 162*atan(sqrt(2)))/12, 50)
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(1/x))/(1 + x + x**2), (x, 0, 1)), 15) == \
NS('pi/sqrt(3) * log(2*pi**(5/6) / gamma(1/6))', 15)
# http://mathworld.wolfram.com/AhmedsIntegral.html
assert NS(Integral(atan(sqrt(x**2 + 2))/(sqrt(x**2 + 2)*(x**2 + 1)), (x,
0, 1)), 15) == NS(5*pi**2/96, 15)
# http://mathworld.wolfram.com/AbelsIntegral.html
assert NS(Integral(x/((exp(pi*x) - exp(
-pi*x))*(x**2 + 1)), (x, 0, oo)), 15) == NS('log(2)/2-1/4', 15)
# Complex part trimming
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(sin(x)/cos(x))), (x, pi/4, pi/2)), 15, chop=True) == \
NS('pi/4*log(4*pi**3/gamma(1/4)**4)', 15)
#
# Endpoints causing trouble (rounding error in integration points -> complex log)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 17, chop=True) == NS(2, 17)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 20, chop=True) == NS(2, 20)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 22, chop=True) == NS(2, 22)
# Needs zero handling
assert NS(pi - 4*Integral(
'sqrt(1-x**2)', (x, 0, 1)), 15, maxn=30, chop=True) in ('0.0', '0')
# Oscillatory quadrature
a = Integral(sin(x)/x**2, (x, 1, oo)).evalf(maxn=15)
assert 0.49 < a < 0.51
assert NS(
Integral(sin(x)/x**2, (x, 1, oo)), quad='osc') == '0.504067061906928'
assert NS(Integral(
cos(pi*x + 1)/x, (x, -oo, -1)), quad='osc') == '0.276374705640365'
# indefinite integrals aren't evaluated
assert NS(Integral(x, x)) == 'Integral(x, x)'
assert NS(Integral(x, (x, y))) == 'Integral(x, (x, y))'
def test_evalf_issue_939():
# https://github.com/sympy/sympy/issues/4038
# The output form of an integral may differ by a step function between
# revisions, making this test a bit useless. This can't be said about
# other two tests. For now, all values of this evaluation are used here,
# but in future this should be reconsidered.
assert NS(integrate(1/(x**5 + 1), x).subs(x, 4), chop=True) in \
['-0.000976138910649103', '0.965906660135753', '1.93278945918216']
assert NS(Integral(1/(x**5 + 1), (x, 2, 4))) == '0.0144361088886740'
assert NS(
integrate(1/(x**5 + 1), (x, 2, 4)), chop=True) == '0.0144361088886740'
def test_double_previously_failing_integrals():
# Double integrals not implemented <- Sure it is!
res = integrate(sqrt(x) + x*y, (x, 1, 2), (y, -1, 1))
# Old numerical test
assert NS(res, 15) == '2.43790283299492'
# Symbolic test
assert res == Rational(-4, 3) + 8*sqrt(2)/3
# double integral + zero detection
assert integrate(sin(x + x*y), (x, -1, 1), (y, -1, 1)) is S.Zero
def test_integrate_SingularityFunction():
in_1 = SingularityFunction(x, a, 3) + SingularityFunction(x, 5, -1)
out_1 = SingularityFunction(x, a, 4)/4 + SingularityFunction(x, 5, 0)
assert integrate(in_1, x) == out_1
in_2 = 10*SingularityFunction(x, 4, 0) - 5*SingularityFunction(x, -6, -2)
out_2 = 10*SingularityFunction(x, 4, 1) - 5*SingularityFunction(x, -6, -1)
assert integrate(in_2, x) == out_2
in_3 = 2*x**2*y -10*SingularityFunction(x, -4, 7) - 2*SingularityFunction(y, 10, -2)
out_3_1 = 2*x**3*y/3 - 2*x*SingularityFunction(y, 10, -2) - 5*SingularityFunction(x, -4, 8)/4
out_3_2 = x**2*y**2 - 10*y*SingularityFunction(x, -4, 7) - 2*SingularityFunction(y, 10, -1)
assert integrate(in_3, x) == out_3_1
assert integrate(in_3, y) == out_3_2
assert unchanged(Integral, in_3, (x,))
assert Integral(in_3, x) == Integral(in_3, (x,))
assert Integral(in_3, x).doit() == out_3_1
in_4 = 10*SingularityFunction(x, -4, 7) - 2*SingularityFunction(x, 10, -2)
out_4 = 5*SingularityFunction(x, -4, 8)/4 - 2*SingularityFunction(x, 10, -1)
assert integrate(in_4, (x, -oo, x)) == out_4
assert integrate(SingularityFunction(x, 5, -1), x) == SingularityFunction(x, 5, 0)
assert integrate(SingularityFunction(x, 0, -1), (x, -oo, oo)) == 1
assert integrate(5*SingularityFunction(x, 5, -1), (x, -oo, oo)) == 5
assert integrate(SingularityFunction(x, 5, -1) * f(x), (x, -oo, oo)) == f(5)
def test_integrate_DiracDelta():
# This is here to check that deltaintegrate is being called, but also
# to test definite integrals. More tests are in test_deltafunctions.py
assert integrate(DiracDelta(x) * f(x), (x, -oo, oo)) == f(0)
assert integrate(DiracDelta(x)**2, (x, -oo, oo)) == DiracDelta(0)
# issue 4522
assert integrate(integrate((4 - 4*x + x*y - 4*y) * \
DiracDelta(x)*DiracDelta(y - 1), (x, 0, 1)), (y, 0, 1)) == 0
# issue 5729
p = exp(-(x**2 + y**2))/pi
assert integrate(p*DiracDelta(x - 10*y), (x, -oo, oo), (y, -oo, oo)) == \
integrate(p*DiracDelta(x - 10*y), (y, -oo, oo), (x, -oo, oo)) == \
integrate(p*DiracDelta(10*x - y), (x, -oo, oo), (y, -oo, oo)) == \
integrate(p*DiracDelta(10*x - y), (y, -oo, oo), (x, -oo, oo)) == \
1/sqrt(101*pi)
def test_integrate_returns_piecewise():
assert integrate(x**y, x) == Piecewise(
(x**(y + 1)/(y + 1), Ne(y, -1)), (log(x), True))
assert integrate(x**y, y) == Piecewise(
(x**y/log(x), Ne(log(x), 0)), (y, True))
assert integrate(exp(n*x), x) == Piecewise(
(exp(n*x)/n, Ne(n, 0)), (x, True))
assert integrate(x*exp(n*x), x) == Piecewise(
((n*x - 1)*exp(n*x)/n**2, Ne(n**2, 0)), (x**2/2, True))
assert integrate(x**(n*y), x) == Piecewise(
(x**(n*y + 1)/(n*y + 1), Ne(n*y, -1)), (log(x), True))
assert integrate(x**(n*y), y) == Piecewise(
(x**(n*y)/(n*log(x)), Ne(n*log(x), 0)), (y, True))
assert integrate(cos(n*x), x) == Piecewise(
(sin(n*x)/n, Ne(n, 0)), (x, True))
assert integrate(cos(n*x)**2, x) == Piecewise(
((n*x/2 + sin(n*x)*cos(n*x)/2)/n, Ne(n, 0)), (x, True))
assert integrate(x*cos(n*x), x) == Piecewise(
(x*sin(n*x)/n + cos(n*x)/n**2, Ne(n, 0)), (x**2/2, True))
assert integrate(sin(n*x), x) == Piecewise(
(-cos(n*x)/n, Ne(n, 0)), (0, True))
assert integrate(sin(n*x)**2, x) == Piecewise(
((n*x/2 - sin(n*x)*cos(n*x)/2)/n, Ne(n, 0)), (0, True))
assert integrate(x*sin(n*x), x) == Piecewise(
(-x*cos(n*x)/n + sin(n*x)/n**2, Ne(n, 0)), (0, True))
assert integrate(exp(x*y), (x, 0, z)) == Piecewise(
(exp(y*z)/y - 1/y, (y > -oo) & (y < oo) & Ne(y, 0)), (z, True))
def test_integrate_max_min():
x = symbols('x', real=True)
assert integrate(Min(x, 2), (x, 0, 3)) == 4
assert integrate(Max(x**2, x**3), (x, 0, 2)) == Rational(49, 12)
assert integrate(Min(exp(x), exp(-x))**2, x) == Piecewise( \
(exp(2*x)/2, x <= 0), (1 - exp(-2*x)/2, True))
# issue 7907
c = symbols('c', extended_real=True)
int1 = integrate(Max(c, x)*exp(-x**2), (x, -oo, oo))
int2 = integrate(c*exp(-x**2), (x, -oo, c))
int3 = integrate(x*exp(-x**2), (x, c, oo))
assert int1 == int2 + int3 == sqrt(pi)*c*erf(c)/2 + \
sqrt(pi)*c/2 + exp(-c**2)/2
def test_integrate_Abs_sign():
assert integrate(Abs(x), (x, -2, 1)) == Rational(5, 2)
assert integrate(Abs(x), (x, 0, 1)) == S.Half
assert integrate(Abs(x + 1), (x, 0, 1)) == Rational(3, 2)
assert integrate(Abs(x**2 - 1), (x, -2, 2)) == 4
assert integrate(Abs(x**2 - 3*x), (x, -15, 15)) == 2259
assert integrate(sign(x), (x, -1, 2)) == 1
assert integrate(sign(x)*sin(x), (x, -pi, pi)) == 4
assert integrate(sign(x - 2) * x**2, (x, 0, 3)) == Rational(11, 3)
t, s = symbols('t s', real=True)
assert integrate(Abs(t), t) == Piecewise(
(-t**2/2, t <= 0), (t**2/2, True))
assert integrate(Abs(2*t - 6), t) == Piecewise(
(-t**2 + 6*t, t <= 3), (t**2 - 6*t + 18, True))
assert (integrate(abs(t - s**2), (t, 0, 2)) ==
2*s**2*Min(2, s**2) - 2*s**2 - Min(2, s**2)**2 + 2)
assert integrate(exp(-Abs(t)), t) == Piecewise(
(exp(t), t <= 0), (2 - exp(-t), True))
assert integrate(sign(2*t - 6), t) == Piecewise(
(-t, t < 3), (t - 6, True))
assert integrate(2*t*sign(t**2 - 1), t) == Piecewise(
(t**2, t < -1), (-t**2 + 2, t < 1), (t**2, True))
assert integrate(sign(t), (t, s + 1)) == Piecewise(
(s + 1, s + 1 > 0), (-s - 1, s + 1 < 0), (0, True))
def test_subs1():
e = Integral(exp(x - y), x)
assert e.subs(y, 3) == Integral(exp(x - 3), x)
e = Integral(exp(x - y), (x, 0, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo))
def test_subs2():
e = Integral(exp(x - y), x, t)
assert e.subs(y, 3) == Integral(exp(x - 3), x, t)
e = Integral(exp(x - y), (x, 0, 1), (t, 0, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 1), (t, 0, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo), (t, 0, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs3():
e = Integral(exp(x - y), (x, 0, y), (t, y, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 3), (t, 3, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo), (t, x, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs4():
e = Integral(exp(x), (x, 0, y), (t, y, 1))
assert e.subs(y, 3) == Integral(exp(x), (x, 0, 3), (t, 3, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(y)*f(y), (y, -oo, oo), (t, x, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs5():
e = Integral(exp(-x**2), (x, -oo, oo))
assert e.subs(x, 5) == e
e = Integral(exp(-x**2 + y), x)
assert e.subs(y, 5) == Integral(exp(-x**2 + 5), x)
e = Integral(exp(-x**2 + y), (x, x))
assert e.subs(x, 5) == Integral(exp(y - x**2), (x, 5))
assert e.subs(y, 5) == Integral(exp(-x**2 + 5), x)
e = Integral(exp(-x**2 + y), (y, -oo, oo), (x, -oo, oo))
assert e.subs(x, 5) == e
assert e.subs(y, 5) == e
# Test evaluation of antiderivatives
e = Integral(exp(-x**2), (x, x))
assert e.subs(x, 5) == Integral(exp(-x**2), (x, 5))
e = Integral(exp(x), x)
assert (e.subs(x,1) - e.subs(x,0) - Integral(exp(x), (x, 0, 1))
).doit().is_zero
def test_subs6():
a, b = symbols('a b')
e = Integral(x*y, (x, f(x), f(y)))
assert e.subs(x, 1) == Integral(x*y, (x, f(1), f(y)))
assert e.subs(y, 1) == Integral(x, (x, f(x), f(1)))
e = Integral(x*y, (x, f(x), f(y)), (y, f(x), f(y)))
assert e.subs(x, 1) == Integral(x*y, (x, f(1), f(y)), (y, f(1), f(y)))
assert e.subs(y, 1) == Integral(x*y, (x, f(x), f(y)), (y, f(x), f(1)))
e = Integral(x*y, (x, f(x), f(a)), (y, f(x), f(a)))
assert e.subs(a, 1) == Integral(x*y, (x, f(x), f(1)), (y, f(x), f(1)))
def test_subs7():
e = Integral(x, (x, 1, y), (y, 1, 2))
assert e.subs({x: 1, y: 2}) == e
e = Integral(sin(x) + sin(y), (x, sin(x), sin(y)),
(y, 1, 2))
assert e.subs(sin(y), 1) == e
assert e.subs(sin(x), 1) == Integral(sin(x) + sin(y), (x, 1, sin(y)),
(y, 1, 2))
def test_expand():
e = Integral(f(x)+f(x**2), (x, 1, y))
assert e.expand() == Integral(f(x), (x, 1, y)) + Integral(f(x**2), (x, 1, y))
def test_integration_variable():
raises(ValueError, lambda: Integral(exp(-x**2), 3))
raises(ValueError, lambda: Integral(exp(-x**2), (3, -oo, oo)))
def test_expand_integral():
assert Integral(cos(x**2)*(sin(x**2) + 1), (x, 0, 1)).expand() == \
Integral(cos(x**2)*sin(x**2), (x, 0, 1)) + \
Integral(cos(x**2), (x, 0, 1))
assert Integral(cos(x**2)*(sin(x**2) + 1), x).expand() == \
Integral(cos(x**2)*sin(x**2), x) + \
Integral(cos(x**2), x)
def test_as_sum_midpoint1():
e = Integral(sqrt(x**3 + 1), (x, 2, 10))
assert e.as_sum(1, method="midpoint") == 8*sqrt(217)
assert e.as_sum(2, method="midpoint") == 4*sqrt(65) + 12*sqrt(57)
assert e.as_sum(3, method="midpoint") == 8*sqrt(217)/3 + \
8*sqrt(3081)/27 + 8*sqrt(52809)/27
assert e.as_sum(4, method="midpoint") == 2*sqrt(730) + \
4*sqrt(7) + 4*sqrt(86) + 6*sqrt(14)
assert abs(e.as_sum(4, method="midpoint").n() - e.n()) < 0.5
e = Integral(sqrt(x**3 + y**3), (x, 2, 10), (y, 0, 10))
raises(NotImplementedError, lambda: e.as_sum(4))
def test_as_sum_midpoint2():
e = Integral((x + y)**2, (x, 0, 1))
n = Symbol('n', positive=True, integer=True)
assert e.as_sum(1, method="midpoint").expand() == Rational(1, 4) + y + y**2
assert e.as_sum(2, method="midpoint").expand() == Rational(5, 16) + y + y**2
assert e.as_sum(3, method="midpoint").expand() == Rational(35, 108) + y + y**2
assert e.as_sum(4, method="midpoint").expand() == Rational(21, 64) + y + y**2
assert e.as_sum(n, method="midpoint").expand() == \
y**2 + y + Rational(1, 3) - 1/(12*n**2)
def test_as_sum_left():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="left").expand() == y**2
assert e.as_sum(2, method="left").expand() == Rational(1, 8) + y/2 + y**2
assert e.as_sum(3, method="left").expand() == Rational(5, 27) + y*Rational(2, 3) + y**2
assert e.as_sum(4, method="left").expand() == Rational(7, 32) + y*Rational(3, 4) + y**2
assert e.as_sum(n, method="left").expand() == \
y**2 + y + Rational(1, 3) - y/n - 1/(2*n) + 1/(6*n**2)
assert e.as_sum(10, method="left", evaluate=False).has(Sum)
def test_as_sum_right():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="right").expand() == 1 + 2*y + y**2
assert e.as_sum(2, method="right").expand() == Rational(5, 8) + y*Rational(3, 2) + y**2
assert e.as_sum(3, method="right").expand() == Rational(14, 27) + y*Rational(4, 3) + y**2
assert e.as_sum(4, method="right").expand() == Rational(15, 32) + y*Rational(5, 4) + y**2
assert e.as_sum(n, method="right").expand() == \
y**2 + y + Rational(1, 3) + y/n + 1/(2*n) + 1/(6*n**2)
def test_as_sum_trapezoid():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="trapezoid").expand() == y**2 + y + S.Half
assert e.as_sum(2, method="trapezoid").expand() == y**2 + y + Rational(3, 8)
assert e.as_sum(3, method="trapezoid").expand() == y**2 + y + Rational(19, 54)
assert e.as_sum(4, method="trapezoid").expand() == y**2 + y + Rational(11, 32)
assert e.as_sum(n, method="trapezoid").expand() == \
y**2 + y + Rational(1, 3) + 1/(6*n**2)
assert Integral(sign(x), (x, 0, 1)).as_sum(1, 'trapezoid') == S.Half
def test_as_sum_raises():
e = Integral((x + y)**2, (x, 0, 1))
raises(ValueError, lambda: e.as_sum(-1))
raises(ValueError, lambda: e.as_sum(0))
raises(ValueError, lambda: Integral(x).as_sum(3))
raises(ValueError, lambda: e.as_sum(oo))
raises(ValueError, lambda: e.as_sum(3, method='xxxx2'))
def test_nested_doit():
e = Integral(Integral(x, x), x)
f = Integral(x, x, x)
assert e.doit() == f.doit()
def test_issue_4665():
# Allow only upper or lower limit evaluation
e = Integral(x**2, (x, None, 1))
f = Integral(x**2, (x, 1, None))
assert e.doit() == Rational(1, 3)
assert f.doit() == Rational(-1, 3)
assert Integral(x*y, (x, None, y)).subs(y, t) == Integral(x*t, (x, None, t))
assert Integral(x*y, (x, y, None)).subs(y, t) == Integral(x*t, (x, t, None))
assert integrate(x**2, (x, None, 1)) == Rational(1, 3)
assert integrate(x**2, (x, 1, None)) == Rational(-1, 3)
assert integrate("x**2", ("x", "1", None)) == Rational(-1, 3)
def test_integral_reconstruct():
e = Integral(x**2, (x, -1, 1))
assert e == Integral(*e.args)
def test_doit_integrals():
e = Integral(Integral(2*x), (x, 0, 1))
assert e.doit() == Rational(1, 3)
assert e.doit(deep=False) == Rational(1, 3)
f = Function('f')
# doesn't matter if the integral can't be performed
assert Integral(f(x), (x, 1, 1)).doit() == 0
# doesn't matter if the limits can't be evaluated
assert Integral(0, (x, 1, Integral(f(x), x))).doit() == 0
assert Integral(x, (a, 0)).doit() == 0
limits = ((a, 1, exp(x)), (x, 0))
assert Integral(a, *limits).doit() == Rational(1, 4)
assert Integral(a, *list(reversed(limits))).doit() == 0
def test_issue_4884():
assert integrate(sqrt(x)*(1 + x)) == \
Piecewise(
(2*sqrt(x)*(x + 1)**2/5 - 2*sqrt(x)*(x + 1)/15 - 4*sqrt(x)/15,
Abs(x + 1) > 1),
(2*I*sqrt(-x)*(x + 1)**2/5 - 2*I*sqrt(-x)*(x + 1)/15 -
4*I*sqrt(-x)/15, True))
assert integrate(x**x*(1 + log(x))) == x**x
def test_issue_18153():
assert integrate(x**n*log(x),x) == \
Piecewise(
(n*x*x**n*log(x)/(n**2 + 2*n + 1) +
x*x**n*log(x)/(n**2 + 2*n + 1) - x*x**n/(n**2 + 2*n + 1)
, Ne(n, -1)), (log(x)**2/2, True)
)
def test_is_number():
from sympy.abc import x, y, z
assert Integral(x).is_number is False
assert Integral(1, x).is_number is False
assert Integral(1, (x, 1)).is_number is True
assert Integral(1, (x, 1, 2)).is_number is True
assert Integral(1, (x, 1, y)).is_number is False
assert Integral(1, (x, y)).is_number is False
assert Integral(x, y).is_number is False
assert Integral(x, (y, 1, x)).is_number is False
assert Integral(x, (y, 1, 2)).is_number is False
assert Integral(x, (x, 1, 2)).is_number is True
# `foo.is_number` should always be equivalent to `not foo.free_symbols`
# in each of these cases, there are pseudo-free symbols
i = Integral(x, (y, 1, 1))
assert i.is_number is False and i.n() == 0
i = Integral(x, (y, z, z))
assert i.is_number is False and i.n() == 0
i = Integral(1, (y, z, z + 2))
assert i.is_number is False and i.n() == 2
assert Integral(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
assert Integral(x*y, (x, 1, 2), (y, 1, z)).is_number is False
assert Integral(x, (x, 1)).is_number is True
assert Integral(x, (x, 1, Integral(y, (y, 1, 2)))).is_number is True
assert Integral(Sum(z, (z, 1, 2)), (x, 1, 2)).is_number is True
# it is possible to get a false negative if the integrand is
# actually an unsimplified zero, but this is true of is_number in general.
assert Integral(sin(x)**2 + cos(x)**2 - 1, x).is_number is False
assert Integral(f(x), (x, 0, 1)).is_number is True
def test_free_symbols():
from sympy.abc import x, y, z
assert Integral(0, x).free_symbols == {x}
assert Integral(x).free_symbols == {x}
assert Integral(x, (x, None, y)).free_symbols == {y}
assert Integral(x, (x, y, None)).free_symbols == {y}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert Integral(x, (x, y, 1)).free_symbols == {y}
assert Integral(x, (x, x, y)).free_symbols == {x, y}
assert Integral(x, x, y).free_symbols == {x, y}
assert Integral(x, (x, 1, 2)).free_symbols == set()
assert Integral(x, (y, 1, 2)).free_symbols == {x}
# pseudo-free in this case
assert Integral(x, (y, z, z)).free_symbols == {x, z}
assert Integral(x, (y, 1, 2), (y, None, None)
).free_symbols == {x, y}
assert Integral(x, (y, 1, 2), (x, 1, y)
).free_symbols == {y}
assert Integral(2, (y, 1, 2), (y, 1, x), (x, 1, 2)
).free_symbols == set()
assert Integral(2, (y, x, 2), (y, 1, x), (x, 1, 2)
).free_symbols == set()
assert Integral(2, (x, 1, 2), (y, x, 2), (y, 1, 2)
).free_symbols == {x}
assert Integral(f(x), (f(x), 1, y)).free_symbols == {y}
assert Integral(f(x), (f(x), 1, x)).free_symbols == {x}
def test_is_zero():
from sympy.abc import x, m
assert Integral(0, (x, 1, x)).is_zero
assert Integral(1, (x, 1, 1)).is_zero
assert Integral(1, (x, 1, 2), (y, 2)).is_zero is False
assert Integral(x, (m, 0)).is_zero
assert Integral(x + m, (m, 0)).is_zero is None
i = Integral(m, (m, 1, exp(x)), (x, 0))
assert i.is_zero is None
assert Integral(m, (x, 0), (m, 1, exp(x))).is_zero is True
assert Integral(x, (x, oo, oo)).is_zero # issue 8171
assert Integral(x, (x, -oo, -oo)).is_zero
# this is zero but is beyond the scope of what is_zero
# should be doing
assert Integral(sin(x), (x, 0, 2*pi)).is_zero is None
def test_series():
from sympy.abc import x
i = Integral(cos(x), (x, x))
e = i.lseries(x)
assert i.nseries(x, n=8).removeO() == Add(*[next(e) for j in range(4)])
def test_trig_nonelementary_integrals():
x = Symbol('x')
assert integrate((1 + sin(x))/x, x) == log(x) + Si(x)
# next one comes out as log(x) + log(x**2)/2 + Ci(x)
# so not hardcoding this log ugliness
assert integrate((cos(x) + 2)/x, x).has(Ci)
def test_issue_4403():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', positive=True)
assert integrate(sqrt(x**2 + z**2), x) == \
z**2*asinh(x/z)/2 + x*sqrt(x**2 + z**2)/2
assert integrate(sqrt(x**2 - z**2), x) == \
-z**2*acosh(x/z)/2 + x*sqrt(x**2 - z**2)/2
x = Symbol('x', real=True)
y = Symbol('y', positive=True)
assert integrate(1/(x**2 + y**2)**S('3/2'), x) == \
x/(y**2*sqrt(x**2 + y**2))
# If y is real and nonzero, we get x*Abs(y)/(y**3*sqrt(x**2 + y**2)),
# which results from sqrt(1 + x**2/y**2) = sqrt(x**2 + y**2)/|y|.
def test_issue_4403_2():
assert integrate(sqrt(-x**2 - 4), x) == \
-2*atan(x/sqrt(-4 - x**2)) + x*sqrt(-4 - x**2)/2
def test_issue_4100():
R = Symbol('R', positive=True)
assert integrate(sqrt(R**2 - x**2), (x, 0, R)) == pi*R**2/4
def test_issue_5167():
from sympy.abc import w, x, y, z
f = Function('f')
assert Integral(Integral(f(x), x), x) == Integral(f(x), x, x)
assert Integral(f(x)).args == (f(x), Tuple(x))
assert Integral(Integral(f(x))).args == (f(x), Tuple(x), Tuple(x))
assert Integral(Integral(f(x)), y).args == (f(x), Tuple(x), Tuple(y))
assert Integral(Integral(f(x), z), y).args == (f(x), Tuple(z), Tuple(y))
assert Integral(Integral(Integral(f(x), x), y), z).args == \
(f(x), Tuple(x), Tuple(y), Tuple(z))
assert integrate(Integral(f(x), x), x) == Integral(f(x), x, x)
assert integrate(Integral(f(x), y), x) == y*Integral(f(x), x)
assert integrate(Integral(f(x), x), y) in [Integral(y*f(x), x), y*Integral(f(x), x)]
assert integrate(Integral(2, x), x) == x**2
assert integrate(Integral(2, x), y) == 2*x*y
# don't re-order given limits
assert Integral(1, x, y).args != Integral(1, y, x).args
# do as many as possible
assert Integral(f(x), y, x, y, x).doit() == y**2*Integral(f(x), x, x)/2
assert Integral(f(x), (x, 1, 2), (w, 1, x), (z, 1, y)).doit() == \
y*(x - 1)*Integral(f(x), (x, 1, 2)) - (x - 1)*Integral(f(x), (x, 1, 2))
def test_issue_4890():
z = Symbol('z', positive=True)
assert integrate(exp(-log(x)**2), x) == \
sqrt(pi)*exp(Rational(1, 4))*erf(log(x) - S.Half)/2
assert integrate(exp(log(x)**2), x) == \
sqrt(pi)*exp(Rational(-1, 4))*erfi(log(x)+S.Half)/2
assert integrate(exp(-z*log(x)**2), x) == \
sqrt(pi)*exp(1/(4*z))*erf(sqrt(z)*log(x) - 1/(2*sqrt(z)))/(2*sqrt(z))
def test_issue_4551():
assert not integrate(1/(x*sqrt(1 - x**2)), x).has(Integral)
def test_issue_4376():
n = Symbol('n', integer=True, positive=True)
assert simplify(integrate(n*(x**(1/n) - 1), (x, 0, S.Half)) -
(n**2 - 2**(1/n)*n**2 - n*2**(1/n))/(2**(1 + 1/n) + n*2**(1 + 1/n))) == 0
def test_issue_4517():
assert integrate((sqrt(x) - x**3)/x**Rational(1, 3), x) == \
6*x**Rational(7, 6)/7 - 3*x**Rational(11, 3)/11
def test_issue_4527():
k, m = symbols('k m', integer=True)
assert integrate(sin(k*x)*sin(m*x), (x, 0, pi)).simplify() == \
Piecewise((0, Eq(k, 0) | Eq(m, 0)),
(-pi/2, Eq(k, -m) | (Eq(k, 0) & Eq(m, 0))),
(pi/2, Eq(k, m) | (Eq(k, 0) & Eq(m, 0))),
(0, True))
# Should be possible to further simplify to:
# Piecewise(
# (0, Eq(k, 0) | Eq(m, 0)),
# (-pi/2, Eq(k, -m)),
# (pi/2, Eq(k, m)),
# (0, True))
assert integrate(sin(k*x)*sin(m*x), (x,)) == Piecewise(
(0, And(Eq(k, 0), Eq(m, 0))),
(-x*sin(m*x)**2/2 - x*cos(m*x)**2/2 + sin(m*x)*cos(m*x)/(2*m), Eq(k, -m)),
(x*sin(m*x)**2/2 + x*cos(m*x)**2/2 - sin(m*x)*cos(m*x)/(2*m), Eq(k, m)),
(m*sin(k*x)*cos(m*x)/(k**2 - m**2) -
k*sin(m*x)*cos(k*x)/(k**2 - m**2), True))
def test_issue_4199():
ypos = Symbol('y', positive=True)
# TODO: Remove conds='none' below, let the assumption take care of it.
assert integrate(exp(-I*2*pi*ypos*x)*x, (x, -oo, oo), conds='none') == \
Integral(exp(-I*2*pi*ypos*x)*x, (x, -oo, oo))
def test_issue_3940():
a, b, c, d = symbols('a:d', positive=True)
assert integrate(exp(-x**2 + I*c*x), x) == \
-sqrt(pi)*exp(-c**2/4)*erf(I*c/2 - x)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
sqrt(pi)*exp(c)*exp(-b**2/(4*a))*erfi(sqrt(a)*x + b/(2*sqrt(a)))/(2*sqrt(a))
from sympy.core.function import expand_mul
from sympy.abc import k
assert expand_mul(integrate(exp(-x**2)*exp(I*k*x), (x, -oo, oo))) == \
sqrt(pi)*exp(-k**2/4)
a, d = symbols('a d', positive=True)
assert expand_mul(integrate(exp(-a*x**2 + 2*d*x), (x, -oo, oo))) == \
sqrt(pi)*exp(d**2/a)/sqrt(a)
def test_issue_5413():
# Note that this is not the same as testing ratint() because integrate()
# pulls out the coefficient.
assert integrate(-a/(a**2 + x**2), x) == I*log(-I*a + x)/2 - I*log(I*a + x)/2
def test_issue_4892a():
A, z = symbols('A z')
c = Symbol('c', nonzero=True)
P1 = -A*exp(-z)
P2 = -A/(c*t)*(sin(x)**2 + cos(y)**2)
h1 = -sin(x)**2 - cos(y)**2
h2 = -sin(x)**2 + sin(y)**2 - 1
# there is still some non-deterministic behavior in integrate
# or trigsimp which permits one of the following
assert integrate(c*(P2 - P1), t) in [
c*(-A*(-h1)*log(c*t)/c + A*t*exp(-z)),
c*(-A*(-h2)*log(c*t)/c + A*t*exp(-z)),
c*( A* h1 *log(c*t)/c + A*t*exp(-z)),
c*( A* h2 *log(c*t)/c + A*t*exp(-z)),
(A*c*t - A*(-h1)*log(t)*exp(z))*exp(-z),
(A*c*t - A*(-h2)*log(t)*exp(z))*exp(-z),
]
def test_issue_4892b():
# Issues relating to issue 4596 are making the actual result of this hard
# to test. The answer should be something like
#
# (-sin(y) + sqrt(-72 + 48*cos(y) - 8*cos(y)**2)/2)*log(x + sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/(2*(3 - cos(y)))) + (-sin(y) - sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/2)*log(x - sqrt(-72 + 48*cos(y) -
# 8*cos(y)**2)/(2*(3 - cos(y)))) + x**2*sin(y)/2 + 2*x*cos(y)
expr = (sin(y)*x**3 + 2*cos(y)*x**2 + 12)/(x**2 + 2)
assert trigsimp(factor(integrate(expr, x).diff(x) - expr)) == 0
def test_issue_5178():
assert integrate(sin(x)*f(y, z), (x, 0, pi), (y, 0, pi), (z, 0, pi)) == \
2*Integral(f(y, z), (y, 0, pi), (z, 0, pi))
def test_integrate_series():
f = sin(x).series(x, 0, 10)
g = x**2/2 - x**4/24 + x**6/720 - x**8/40320 + x**10/3628800 + O(x**11)
assert integrate(f, x) == g
assert diff(integrate(f, x), x) == f
assert integrate(O(x**5), x) == O(x**6)
def test_atom_bug():
from sympy.integrals.heurisch import heurisch
assert heurisch(meijerg([], [], [1], [], x), x) is None
def test_limit_bug():
z = Symbol('z', zero=False)
assert integrate(sin(x*y*z), (x, 0, pi), (y, 0, pi)).together() == \
(log(z) - Ci(pi**2*z) + EulerGamma + 2*log(pi))/z
def test_issue_4703():
g = Function('g')
assert integrate(exp(x)*g(x), x).has(Integral)
def test_issue_1888():
f = Function('f')
assert integrate(f(x).diff(x)**2, x).has(Integral)
# The following tests work using meijerint.
def test_issue_3558():
assert integrate(cos(x*y), (x, -pi/2, pi/2), (y, 0, pi)) == 2*Si(pi**2/2)
def test_issue_4422():
assert integrate(1/sqrt(16 + 4*x**2), x) == asinh(x/2) / 2
def test_issue_4493():
assert simplify(integrate(x*sqrt(1 + 2*x), x)) == \
sqrt(2*x + 1)*(6*x**2 + x - 1)/15
def test_issue_4737():
assert integrate(sin(x)/x, (x, -oo, oo)) == pi
assert integrate(sin(x)/x, (x, 0, oo)) == pi/2
assert integrate(sin(x)/x, x) == Si(x)
def test_issue_4992():
# Note: psi in _check_antecedents becomes NaN.
from sympy.core.function import expand_func
a = Symbol('a', positive=True)
assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \
(a*polygamma(0, a) + 1)*gamma(a)
def test_issue_4487():
from sympy.functions.special.gamma_functions import lowergamma
assert simplify(integrate(exp(-x)*x**y, x)) == lowergamma(y + 1, x)
def test_issue_4215():
x = Symbol("x")
assert integrate(1/(x**2), (x, -1, 1)) is oo
def test_issue_4400():
n = Symbol('n', integer=True, positive=True)
assert integrate((x**n)*log(x), x) == \
n*x*x**n*log(x)/(n**2 + 2*n + 1) + x*x**n*log(x)/(n**2 + 2*n + 1) - \
x*x**n/(n**2 + 2*n + 1)
def test_issue_6253():
# Note: this used to raise NotImplementedError
# Note: psi in _check_antecedents becomes NaN.
assert integrate((sqrt(1 - x) + sqrt(1 + x))**2/x, x, meijerg=True) == \
Integral((sqrt(-x + 1) + sqrt(x + 1))**2/x, x)
def test_issue_4153():
assert integrate(1/(1 + x + y + z), (x, 0, 1), (y, 0, 1), (z, 0, 1)) in [
-12*log(3) - 3*log(6)/2 + 3*log(8)/2 + 5*log(2) + 7*log(4),
6*log(2) + 8*log(4) - 27*log(3)/2, 22*log(2) - 27*log(3)/2,
-12*log(3) - 3*log(6)/2 + 47*log(2)/2]
def test_issue_4326():
R, b, h = symbols('R b h')
# It doesn't matter if we can do the integral. Just make sure the result
# doesn't contain nan. This is really a test against _eval_interval.
e = integrate(((h*(x - R + b))/b)*sqrt(R**2 - x**2), (x, R - b, R))
assert not e.has(nan)
# See that it evaluates
assert not e.has(Integral)
def test_powers():
assert integrate(2**x + 3**x, x) == 2**x/log(2) + 3**x/log(3)
def test_manual_option():
raises(ValueError, lambda: integrate(1/x, x, manual=True, meijerg=True))
# an example of a function that manual integration cannot handle
assert integrate(log(1+x)/x, (x, 0, 1), manual=True).has(Integral)
def test_meijerg_option():
raises(ValueError, lambda: integrate(1/x, x, meijerg=True, risch=True))
# an example of a function that meijerg integration cannot handle
assert integrate(tan(x), x, meijerg=True) == Integral(tan(x), x)
def test_risch_option():
# risch=True only allowed on indefinite integrals
raises(ValueError, lambda: integrate(1/log(x), (x, 0, oo), risch=True))
assert integrate(exp(-x**2), x, risch=True) == NonElementaryIntegral(exp(-x**2), x)
assert integrate(log(1/x)*y, x, y, risch=True) == y**2*(x*log(1/x)/2 + x/2)
assert integrate(erf(x), x, risch=True) == Integral(erf(x), x)
# TODO: How to test risch=False?
@slow
def test_heurisch_option():
raises(ValueError, lambda: integrate(1/x, x, risch=True, heurisch=True))
# an integral that heurisch can handle
assert integrate(exp(x**2), x, heurisch=True) == sqrt(pi)*erfi(x)/2
# an integral that heurisch currently cannot handle
assert integrate(exp(x)/x, x, heurisch=True) == Integral(exp(x)/x, x)
# an integral where heurisch currently hangs, issue 15471
assert integrate(log(x)*cos(log(x))/x**Rational(3, 4), x, heurisch=False) == (
-128*x**Rational(1, 4)*sin(log(x))/289 + 240*x**Rational(1, 4)*cos(log(x))/289 +
(16*x**Rational(1, 4)*sin(log(x))/17 + 4*x**Rational(1, 4)*cos(log(x))/17)*log(x))
def test_issue_6828():
f = 1/(1.08*x**2 - 4.3)
g = integrate(f, x).diff(x)
assert verify_numerically(f, g, tol=1e-12)
def test_issue_4803():
x_max = Symbol("x_max")
assert integrate(y/pi*exp(-(x_max - x)/cos(a)), x) == \
y*exp((x - x_max)/cos(a))*cos(a)/pi
def test_issue_4234():
assert integrate(1/sqrt(1 + tan(x)**2)) == tan(x)/sqrt(1 + tan(x)**2)
def test_issue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)).factor(
deep=True) == Piecewise(
(I*(2*x**5 - 15*x**3 + 25*x - 25*sqrt(x**2 - 5)*acosh(sqrt(5)*x/5)) /
(8*sqrt(x**2 - 5)), (x > sqrt(5)) | (x < -sqrt(5))),
((2*x**5 - 15*x**3 + 25*x - 25*sqrt(5 - x**2)*asin(sqrt(5)*x/5)) /
(-8*sqrt(-x**2 + 5)), True))
def test_issue_2708():
# This test needs to use an integration function that can
# not be evaluated in closed form. Update as needed.
f = 1/(a + z + log(z))
integral_f = NonElementaryIntegral(f, (z, 2, 3))
assert Integral(f, (z, 2, 3)).doit() == integral_f
assert integrate(f + exp(z), (z, 2, 3)) == integral_f - exp(2) + exp(3)
assert integrate(2*f + exp(z), (z, 2, 3)) == \
2*integral_f - exp(2) + exp(3)
assert integrate(exp(1.2*n*s*z*(-t + z)/t), (z, 0, x)) == \
NonElementaryIntegral(exp(-1.2*n*s*z)*exp(1.2*n*s*z**2/t),
(z, 0, x))
def test_issue_2884():
f = (4.000002016020*x + 4.000002016020*y + 4.000006024032)*exp(10.0*x)
e = integrate(f, (x, 0.1, 0.2))
assert str(e) == '1.86831064982608*y + 2.16387491480008'
def test_issue_8368i():
from sympy.functions.elementary.complexes import arg, Abs
assert integrate(exp(-s*x)*cosh(x), (x, 0, oo)) == \
Piecewise(
( pi*Piecewise(
( -s/(pi*(-s**2 + 1)),
Abs(s**2) < 1),
( 1/(pi*s*(1 - 1/s**2)),
Abs(s**(-2)) < 1),
( meijerg(
((S.Half,), (0, 0)),
((0, S.Half), (0,)),
polar_lift(s)**2),
True)
),
s**2 > 1
),
(
Integral(exp(-s*x)*cosh(x), (x, 0, oo)),
True))
assert integrate(exp(-s*x)*sinh(x), (x, 0, oo)) == \
Piecewise(
( -1/(s + 1)/2 - 1/(-s + 1)/2,
And(
Abs(s) > 1,
Abs(arg(s)) < pi/2,
Abs(arg(s)) <= pi/2
)),
( Integral(exp(-s*x)*sinh(x), (x, 0, oo)),
True))
def test_issue_8901():
assert integrate(sinh(1.0*x)) == 1.0*cosh(1.0*x)
assert integrate(tanh(1.0*x)) == 1.0*x - 1.0*log(tanh(1.0*x) + 1)
assert integrate(tanh(x)) == x - log(tanh(x) + 1)
@slow
def test_issue_8945():
assert integrate(sin(x)**3/x, (x, 0, 1)) == -Si(3)/4 + 3*Si(1)/4
assert integrate(sin(x)**3/x, (x, 0, oo)) == pi/4
assert integrate(cos(x)**2/x**2, x) == -Si(2*x) - cos(2*x)/(2*x) - 1/(2*x)
@slow
def test_issue_7130():
if ON_TRAVIS:
skip("Too slow for travis.")
i, L, a, b = symbols('i L a b')
integrand = (cos(pi*i*x/L)**2 / (a + b*x)).rewrite(exp)
assert x not in integrate(integrand, (x, 0, L)).free_symbols
def test_issue_10567():
a, b, c, t = symbols('a b c t')
vt = Matrix([a*t, b, c])
assert integrate(vt, t) == Integral(vt, t).doit()
assert integrate(vt, t) == Matrix([[a*t**2/2], [b*t], [c*t]])
def test_issue_11856():
t = symbols('t')
assert integrate(sinc(pi*t), t) == Si(pi*t)/pi
@slow
def test_issue_11876():
assert integrate(sqrt(log(1/x)), (x, 0, 1)) == sqrt(pi)/2
def test_issue_4950():
assert integrate((-60*exp(x) - 19.2*exp(4*x))*exp(4*x), x) ==\
-2.4*exp(8*x) - 12.0*exp(5*x)
def test_issue_4968():
assert integrate(sin(log(x**2))) == x*sin(log(x**2))/5 - 2*x*cos(log(x**2))/5
def test_singularities():
assert integrate(1/x**2, (x, -oo, oo)) is oo
assert integrate(1/x**2, (x, -1, 1)) is oo
assert integrate(1/(x - 1)**2, (x, -2, 2)) is oo
assert integrate(1/x**2, (x, 1, -1)) is -oo
assert integrate(1/(x - 1)**2, (x, 2, -2)) is -oo
def test_issue_12645():
x, y = symbols('x y', real=True)
assert (integrate(sin(x*x*x + y*y),
(x, -sqrt(pi - y*y), sqrt(pi - y*y)),
(y, -sqrt(pi), sqrt(pi)))
== Integral(sin(x**3 + y**2),
(x, -sqrt(-y**2 + pi), sqrt(-y**2 + pi)),
(y, -sqrt(pi), sqrt(pi))))
def test_issue_12677():
assert integrate(sin(x) / (cos(x)**3), (x, 0, pi/6)) == Rational(1, 6)
def test_issue_14078():
assert integrate((cos(3*x)-cos(x))/x, (x, 0, oo)) == -log(3)
def test_issue_14064():
assert integrate(1/cosh(x), (x, 0, oo)) == pi/2
def test_issue_14027():
assert integrate(1/(1 + exp(x - S.Half)/(1 + exp(x))), x) == \
x - exp(S.Half)*log(exp(x) + exp(S.Half)/(1 + exp(S.Half)))/(exp(S.Half) + E)
def test_issue_8170():
assert integrate(tan(x), (x, 0, pi/2)) is S.Infinity
def test_issue_8440_14040():
assert integrate(1/x, (x, -1, 1)) is S.NaN
assert integrate(1/(x + 1), (x, -2, 3)) is S.NaN
def test_issue_14096():
assert integrate(1/(x + y)**2, (x, 0, 1)) == -1/(y + 1) + 1/y
assert integrate(1/(1 + x + y + z)**2, (x, 0, 1), (y, 0, 1), (z, 0, 1)) == \
-4*log(4) - 6*log(2) + 9*log(3)
def test_issue_14144():
assert Abs(integrate(1/sqrt(1 - x**3), (x, 0, 1)).n() - 1.402182) < 1e-6
assert Abs(integrate(sqrt(1 - x**3), (x, 0, 1)).n() - 0.841309) < 1e-6
def test_issue_14375():
# This raised a TypeError. The antiderivative has exp_polar, which
# may be possible to unpolarify, so the exact output is not asserted here.
assert integrate(exp(I*x)*log(x), x).has(Ei)
def test_issue_14437():
f = Function('f')(x, y, z)
assert integrate(f, (x, 0, 1), (y, 0, 2), (z, 0, 3)) == \
Integral(f, (x, 0, 1), (y, 0, 2), (z, 0, 3))
def test_issue_14470():
assert integrate(1/sqrt(exp(x) + 1), x) == \
log(-1 + 1/sqrt(exp(x) + 1)) - log(1 + 1/sqrt(exp(x) + 1))
def test_issue_14877():
f = exp(1 - exp(x**2)*x + 2*x**2)*(2*x**3 + x)/(1 - exp(x**2)*x)**2
assert integrate(f, x) == \
-exp(2*x**2 - x*exp(x**2) + 1)/(x*exp(3*x**2) - exp(2*x**2))
def test_issue_14782():
f = sqrt(-x**2 + 1)*(-x**2 + x)
assert integrate(f, [x, -1, 1]) == - pi / 8
@slow
def test_issue_14782_slow():
f = sqrt(-x**2 + 1)*(-x**2 + x)
assert integrate(f, [x, 0, 1]) == S.One / 3 - pi / 16
def test_issue_12081():
f = x**(Rational(-3, 2))*exp(-x)
assert integrate(f, [x, 0, oo]) is oo
def test_issue_15285():
y = 1/x - 1
f = 4*y*exp(-2*y)/x**2
assert integrate(f, [x, 0, 1]) == 1
def test_issue_15432():
assert integrate(x**n * exp(-x) * log(x), (x, 0, oo)).gammasimp() == Piecewise(
(gamma(n + 1)*polygamma(0, n) + gamma(n + 1)/n, re(n) + 1 > 0),
(Integral(x**n*exp(-x)*log(x), (x, 0, oo)), True))
def test_issue_15124():
omega = IndexedBase('omega')
m, p = symbols('m p', cls=Idx)
assert integrate(exp(x*I*(omega[m] + omega[p])), x, conds='none') == \
-I*exp(I*x*omega[m])*exp(I*x*omega[p])/(omega[m] + omega[p])
def test_issue_15218():
with warns_deprecated_sympy():
Integral(Eq(x, y))
with warns_deprecated_sympy():
assert Integral(Eq(x, y), x) == Eq(Integral(x, x), Integral(y, x))
with warns_deprecated_sympy():
assert Integral(Eq(x, y), x).doit() == Eq(x**2/2, x*y)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
# The warning is made in the ExprWithLimits superclass. The stacklevel
# is correct for integrate(Eq) but not Eq.integrate
assert Eq(x, y).integrate(x) == Eq(x**2/2, x*y)
# These are not deprecated because they are definite integrals
assert integrate(Eq(x, y), (x, 0, 1)) == Eq(S.Half, y)
assert Eq(x, y).integrate((x, 0, 1)) == Eq(S.Half, y)
def test_issue_15292():
res = integrate(exp(-x**2*cos(2*t)) * cos(x**2*sin(2*t)), (x, 0, oo))
assert isinstance(res, Piecewise)
assert gammasimp((res - sqrt(pi)/2 * cos(t)).subs(t, pi/6)) == 0
def test_issue_4514():
assert integrate(sin(2*x)/sin(x), x) == 2*sin(x)
def test_issue_15457():
x, a, b = symbols('x a b', real=True)
definite = integrate(exp(Abs(x-2)), (x, a, b))
indefinite = integrate(exp(Abs(x-2)), x)
assert definite.subs({a: 1, b: 3}) == -2 + 2*E
assert indefinite.subs(x, 3) - indefinite.subs(x, 1) == -2 + 2*E
assert definite.subs({a: -3, b: -1}) == -exp(3) + exp(5)
assert indefinite.subs(x, -1) - indefinite.subs(x, -3) == -exp(3) + exp(5)
def test_issue_15431():
assert integrate(x*exp(x)*log(x), x) == \
(x*exp(x) - exp(x))*log(x) - exp(x) + Ei(x)
def test_issue_15640_log_substitutions():
f = x/log(x)
F = Ei(2*log(x))
assert integrate(f, x) == F and F.diff(x) == f
f = x**3/log(x)**2
F = -x**4/log(x) + 4*Ei(4*log(x))
assert integrate(f, x) == F and F.diff(x) == f
f = sqrt(log(x))/x**2
F = -sqrt(pi)*erfc(sqrt(log(x)))/2 - sqrt(log(x))/x
assert integrate(f, x) == F and F.diff(x) == f
def test_issue_15509():
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
x = N.x
assert integrate(cos(a*x + b), (x, x_1, x_2), heurisch=True) == Piecewise(
(-sin(a*x_1 + b)/a + sin(a*x_2 + b)/a, (a > -oo) & (a < oo) & Ne(a, 0)), \
(-x_1*cos(b) + x_2*cos(b), True))
def test_issue_4311_fast():
x = symbols('x', real=True)
assert integrate(x*abs(9-x**2), x) == Piecewise(
(x**4/4 - 9*x**2/2, x <= -3),
(-x**4/4 + 9*x**2/2 - Rational(81, 2), x <= 3),
(x**4/4 - 9*x**2/2, True))
def test_integrate_with_complex_constants():
K = Symbol('K', positive=True)
x = Symbol('x', real=True)
m = Symbol('m', real=True)
t = Symbol('t', real=True)
assert integrate(exp(-I*K*x**2+m*x), x) == sqrt(I)*sqrt(pi)*exp(-I*m**2
/(4*K))*erfi((-2*I*K*x + m)/(2*sqrt(K)*sqrt(-I)))/(2*sqrt(K))
assert integrate(1/(1 + I*x**2), x) == (-I*(sqrt(-I)*log(x - I*sqrt(-I))/2
- sqrt(-I)*log(x + I*sqrt(-I))/2))
assert integrate(exp(-I*x**2), x) == sqrt(pi)*erf(sqrt(I)*x)/(2*sqrt(I))
assert integrate((1/(exp(I*t)-2)), t) == -t/2 - I*log(exp(I*t) - 2)/2
assert integrate((1/(exp(I*t)-2)), (t, 0, 2*pi)) == -pi
def test_issue_14241():
x = Symbol('x')
n = Symbol('n', positive=True, integer=True)
assert integrate(n * x ** (n - 1) / (x + 1), x) == \
n**2*x**n*lerchphi(x*exp_polar(I*pi), 1, n)*gamma(n)/gamma(n + 1)
def test_issue_13112():
assert integrate(sin(t)**2 / (5 - 4*cos(t)), [t, 0, 2*pi]) == pi / 4
@slow
def test_issue_14709b():
h = Symbol('h', positive=True)
i = integrate(x*acos(1 - 2*x/h), (x, 0, h))
assert i == 5*h**2*pi/16
def test_issue_8614():
x = Symbol('x')
t = Symbol('t')
assert integrate(exp(t)/t, (t, -oo, x)) == Ei(x)
assert integrate((exp(-x) - exp(-2*x))/x, (x, 0, oo)) == log(2)
@slow
def test_issue_15494():
s = symbols('s', positive=True)
integrand = (exp(s/2) - 2*exp(1.6*s) + exp(s))*exp(s)
solution = integrate(integrand, s)
assert solution != S.NaN
# Not sure how to test this properly as it is a symbolic expression with floats
# assert str(solution) == '0.666666666666667*exp(1.5*s) + 0.5*exp(2.0*s) - 0.769230769230769*exp(2.6*s)'
# Maybe
assert abs(solution.subs(s, 1) - (-3.67440080236188)) <= 1e-8
integrand = (exp(s/2) - 2*exp(S(8)/5*s) + exp(s))*exp(s)
assert integrate(integrand, s) == -10*exp(13*s/5)/13 + 2*exp(3*s/2)/3 + exp(2*s)/2
def test_li_integral():
y = Symbol('y')
assert Integral(li(y*x**2), x).doit() == Piecewise((x*li(x**2*y) - \
x*Ei(3*log(x**2*y)/2)/sqrt(x**2*y),
Ne(y, 0)), (0, True))
def test_issue_17473():
x = Symbol('x')
n = Symbol('n')
assert integrate(sin(x**n), x) == \
x*x**n*gamma(S(1)/2 + 1/(2*n))*hyper((S(1)/2 + 1/(2*n),),
(S(3)/2, S(3)/2 + 1/(2*n)),
-x**(2*n)/4)/(2*n*gamma(S(3)/2 + 1/(2*n)))
def test_issue_17671():
assert integrate(log(log(x)) / x**2, [x, 1, oo]) == -EulerGamma
assert integrate(log(log(x)) / x**3, [x, 1, oo]) == -log(2)/2 - EulerGamma/2
assert integrate(log(log(x)) / x**10, [x, 1, oo]) == -2*log(3)/9 - EulerGamma/9
def test_issue_2975():
w = Symbol('w')
C = Symbol('C')
y = Symbol('y')
assert integrate(1/(y**2+C)**(S(3)/2), (y, -w/2, w/2)) == w/(C**(S(3)/2)*sqrt(1 + w**2/(4*C)))
def test_issue_7827():
x, n, M = symbols('x n M')
N = Symbol('N', integer=True)
assert integrate(summation(x*n, (n, 1, N)), x) == x**2*(N**2/4 + N/4)
assert integrate(summation(x*sin(n), (n,1,N)), x) == \
Sum(x**2*sin(n)/2, (n, 1, N))
assert integrate(summation(sin(n*x), (n,1,N)), x) == \
Sum(Piecewise((-cos(n*x)/n, Ne(n, 0)), (0, True)), (n, 1, N))
assert integrate(integrate(summation(sin(n*x), (n,1,N)), x), x) == \
Piecewise((Sum(Piecewise((-sin(n*x)/n**2, Ne(n, 0)), (-x/n, True)),
(n, 1, N)), (n > -oo) & (n < oo) & Ne(n, 0)), (0, True))
assert integrate(Sum(x, (n, 1, M)), x) == M*x**2/2
raises(ValueError, lambda: integrate(Sum(x, (x, y, n)), y))
raises(ValueError, lambda: integrate(Sum(x, (x, 1, n)), n))
raises(ValueError, lambda: integrate(Sum(x, (x, 1, y)), x))
def test_issue_4231():
f = (1 + 2*x + sqrt(x + log(x))*(1 + 3*x) + x**2)/(x*(x + sqrt(x + log(x)))*sqrt(x + log(x)))
assert integrate(f, x) == 2*sqrt(x + log(x)) + 2*log(x + sqrt(x + log(x)))
def test_issue_17841():
f = diff(1/(x**2+x+I), x)
assert integrate(f, x) == 1/(x**2 + x + I)
def test_issue_21034():
x = Symbol('x', real=True, nonzero=True)
f1 = x*(-x**4/asin(5)**4 - x*sinh(x + log(asin(5))) + 5)
f2 = (x + cosh(cos(4)))/(x*(x + 1/(12*x)))
assert integrate(f1, x) == \
-x**6/(6*asin(5)**4) - x**2*cosh(x + log(asin(5))) + 5*x**2/2 + 2*x*sinh(x + log(asin(5))) - 2*cosh(x + log(asin(5)))
assert integrate(f2, x) == \
log(x**2 + S(1)/12)/2 + 2*sqrt(3)*cosh(cos(4))*atan(2*sqrt(3)*x)
def test_issue_4187():
assert integrate(log(x)*exp(-x), x) == Ei(-x) - exp(-x)*log(x)
assert integrate(log(x)*exp(-x), (x, 0, oo)) == -EulerGamma
def test_issue_5547():
L = Symbol('L')
z = Symbol('z')
r0 = Symbol('r0')
R0 = Symbol('R0')
assert integrate(r0**2*cos(z)**2, (z, -L/2, L/2)) == -r0**2*(-L/4 -
sin(L/2)*cos(L/2)/2) + r0**2*(L/4 + sin(L/2)*cos(L/2)/2)
assert integrate(r0**2*cos(R0*z)**2, (z, -L/2, L/2)) == Piecewise(
(-r0**2*(-L*R0/4 - sin(L*R0/2)*cos(L*R0/2)/2)/R0 +
r0**2*(L*R0/4 + sin(L*R0/2)*cos(L*R0/2)/2)/R0, (R0 > -oo) & (R0 < oo) & Ne(R0, 0)),
(L*r0**2, True))
w = 2*pi*z/L
sol = sqrt(2)*sqrt(L)*r0**2*fresnelc(sqrt(2)*sqrt(L))*gamma(S.One/4)/(16*gamma(S(5)/4)) + L*r0**2/2
assert integrate(r0**2*cos(w*z)**2, (z, -L/2, L/2)) == sol
def test_issue_15810():
assert integrate(1/(2**(2*x/3) + 1), (x, 0, oo)) == Rational(3, 2)
def test_issue_21024():
x = Symbol('x', real=True, nonzero=True)
f = log(x)*log(4*x) + log(3*x + exp(2))
F = x*log(x)**2 + x*(1 - 2*log(2)) + (-2*x + 2*x*log(2))*log(x) + \
(x + exp(2)/6)*log(3*x + exp(2)) + exp(2)*log(3*x + exp(2))/6
assert F == integrate(f, x)
f = (x + exp(3))/x**2
F = log(x) - exp(3)/x
assert F == integrate(f, x)
f = (x**2 + exp(5))/x
F = x**2/2 + exp(5)*log(x)
assert F == integrate(f, x)
f = x/(2*x + tanh(1))
F = x/2 - log(2*x + tanh(1))*tanh(1)/4
assert F == integrate(f, x)
f = x - sinh(4)/x
F = x**2/2 - log(x)*sinh(4)
assert F == integrate(f, x)
f = log(x + exp(5)/x)
F = x*log(x + exp(5)/x) - x + 2*exp(Rational(5, 2))*atan(x*exp(Rational(-5, 2)))
assert F == integrate(f, x)
f = x**5/(x + E)
F = x**5/5 - E*x**4/4 + x**3*exp(2)/3 - x**2*exp(3)/2 + x*exp(4) - exp(5)*log(x + E)
assert F == integrate(f, x)
f = 4*x/(x + sinh(5))
F = 4*x - 4*log(x + sinh(5))*sinh(5)
assert F == integrate(f, x)
f = x**2/(2*x + sinh(2))
F = x**2/4 - x*sinh(2)/4 + log(2*x + sinh(2))*sinh(2)**2/8
assert F == integrate(f, x)
f = -x**2/(x + E)
F = -x**2/2 + E*x - exp(2)*log(x + E)
assert F == integrate(f, x)
f = (2*x + 3)*exp(5)/x
F = 2*x*exp(5) + 3*exp(5)*log(x)
assert F == integrate(f, x)
f = x + 2 + cosh(3)/x
F = x**2/2 + 2*x + log(x)*cosh(3)
assert F == integrate(f, x)
f = x - tanh(1)/x**3
F = x**2/2 + tanh(1)/(2*x**2)
assert F == integrate(f, x)
f = (3*x - exp(6))/x
F = 3*x - exp(6)*log(x)
assert F == integrate(f, x)
f = x**4/(x + exp(5))**2 + x
F = x**3/3 + x**2*(Rational(1, 2) - exp(5)) + 3*x*exp(10) - 4*exp(15)*log(x + exp(5)) - exp(20)/(x + exp(5))
assert F == integrate(f, x)
f = x*(x + exp(10)/x**2) + x
F = x**3/3 + x**2/2 + exp(10)*log(x)
assert F == integrate(f, x)
f = x + x/(5*x + sinh(3))
F = x**2/2 + x/5 - log(5*x + sinh(3))*sinh(3)/25
assert F == integrate(f, x)
f = (x + exp(3))/(2*x**2 + 2*x)
F = exp(3)*log(x)/2 - exp(3)*log(x + 1)/2 + log(x + 1)/2
assert F == integrate(f, x).expand()
f = log(x + 4*sinh(4))
F = x*log(x + 4*sinh(4)) - x + 4*log(x + 4*sinh(4))*sinh(4)
assert F == integrate(f, x)
f = -x + 20*(exp(-5) - atan(4)/x)**3*sin(4)/x
F = (-x**2*exp(15)/2 + 20*log(x)*sin(4) - (-180*x**2*exp(5)*sin(4)*atan(4) + 90*x*exp(10)*sin(4)*atan(4)**2 - \
20*exp(15)*sin(4)*atan(4)**3)/(3*x**3))*exp(-15)
assert F == integrate(f, x)
f = 2*x**2*exp(-4) + 6/x
F_true = (2*x**3/3 + 6*exp(4)*log(x))*exp(-4)
assert F_true == integrate(f, x)
def test_issue_21831():
theta = symbols('theta')
assert integrate(cos(3*theta)/(5-4*cos(theta)), (theta, 0, 2*pi)) == pi/12
integrand = cos(2*theta)/(5 - 4*cos(theta))
assert integrate(integrand, (theta, 0, 2*pi)) == pi/6
@slow
def test_issue_22033_integral():
assert integrate((x**2 - Rational(1, 4))**2 * sqrt(1 - x**2), (x, -1, 1)) == pi/32
@slow
def test_issue_21671():
assert integrate(1,(z,x**2+y**2,2-x**2-y**2),(y,-sqrt(1-x**2),sqrt(1-x**2)),(x,-1,1)) == pi
assert integrate(-4*(1 - x**2)**(S(3)/2)/3 + 2*sqrt(1 - x**2)*(2 - 2*x**2), (x, -1, 1)) == pi
def test_issue_18527():
# The manual integrator can not currently solve this. Assert that it does
# not give an incorrect result involving Abs when x has real assumptions.
xr = symbols('xr', real=True)
expr = (cos(x)/(4+(sin(x))**2))
res_real = integrate(expr.subs(x, xr), xr, manual=True).subs(xr, x)
assert integrate(expr, x, manual=True) == res_real == Integral(expr, x)
|
72682f316f06f8b676d195f2fb0839573e68808fe8ab6a766c94b0332f48305d | from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import symbols
from sympy.core.singleton import S
from sympy.core.function import expand, Function
from sympy.core.numbers import I
from sympy.integrals.integrals import Integral
from sympy.polys.polytools import factor
from sympy.core.traversal import preorder_traversal, use, postorder_traversal, iterargs, iterfreeargs
from sympy.functions.elementary.piecewise import ExprCondPair, Piecewise
from sympy.testing.pytest import warns_deprecated_sympy
from sympy.utilities.iterables import capture
b1 = Basic()
b2 = Basic(b1)
b3 = Basic(b2)
b21 = Basic(b2, b1)
def test_preorder_traversal():
expr = Basic(b21, b3)
assert list(
preorder_traversal(expr)) == [expr, b21, b2, b1, b1, b3, b2, b1]
assert list(preorder_traversal(('abc', ('d', 'ef')))) == [
('abc', ('d', 'ef')), 'abc', ('d', 'ef'), 'd', 'ef']
result = []
pt = preorder_traversal(expr)
for i in pt:
result.append(i)
if i == b2:
pt.skip()
assert result == [expr, b21, b2, b1, b3, b2]
w, x, y, z = symbols('w:z')
expr = z + w*(x + y)
assert list(preorder_traversal([expr], keys=default_sort_key)) == \
[[w*(x + y) + z], w*(x + y) + z, z, w*(x + y), w, x + y, x, y]
assert list(preorder_traversal((x + y)*z, keys=True)) == \
[z*(x + y), z, x + y, x, y]
def test_use():
x, y = symbols('x y')
assert use(0, expand) == 0
f = (x + y)**2*x + 1
assert use(f, expand, level=0) == x**3 + 2*x**2*y + x*y**2 + + 1
assert use(f, expand, level=1) == x**3 + 2*x**2*y + x*y**2 + + 1
assert use(f, expand, level=2) == 1 + x*(2*x*y + x**2 + y**2)
assert use(f, expand, level=3) == (x + y)**2*x + 1
f = (x**2 + 1)**2 - 1
kwargs = {'gaussian': True}
assert use(f, factor, level=0, kwargs=kwargs) == x**2*(x**2 + 2)
assert use(f, factor, level=1, kwargs=kwargs) == (x + I)**2*(x - I)**2 - 1
assert use(f, factor, level=2, kwargs=kwargs) == (x + I)**2*(x - I)**2 - 1
assert use(f, factor, level=3, kwargs=kwargs) == (x**2 + 1)**2 - 1
def test_postorder_traversal():
x, y, z, w = symbols('x y z w')
expr = z + w*(x + y)
expected = [z, w, x, y, x + y, w*(x + y), w*(x + y) + z]
assert list(postorder_traversal(expr, keys=default_sort_key)) == expected
assert list(postorder_traversal(expr, keys=True)) == expected
expr = Piecewise((x, x < 1), (x**2, True))
expected = [
x, 1, x, x < 1, ExprCondPair(x, x < 1),
2, x, x**2, S.true,
ExprCondPair(x**2, True), Piecewise((x, x < 1), (x**2, True))
]
assert list(postorder_traversal(expr, keys=default_sort_key)) == expected
assert list(postorder_traversal(
[expr], keys=default_sort_key)) == expected + [[expr]]
assert list(postorder_traversal(Integral(x**2, (x, 0, 1)),
keys=default_sort_key)) == [
2, x, x**2, 0, 1, x, Tuple(x, 0, 1),
Integral(x**2, Tuple(x, 0, 1))
]
assert list(postorder_traversal(('abc', ('d', 'ef')))) == [
'abc', 'd', 'ef', ('d', 'ef'), ('abc', ('d', 'ef'))]
def test_iterargs():
f = Function('f')
x = symbols('x')
assert list(iterfreeargs(Integral(f(x), (f(x), 1)))) == [
Integral(f(x), (f(x), 1)), 1]
assert list(iterargs(Integral(f(x), (f(x), 1)))) == [
Integral(f(x), (f(x), 1)), f(x), (f(x), 1), x, f(x), 1, x]
def test_deprecated_imports():
x = symbols('x')
with warns_deprecated_sympy():
from sympy.core.basic import preorder_traversal
preorder_traversal(x)
with warns_deprecated_sympy():
from sympy.simplify.simplify import bottom_up
bottom_up(x, lambda x: x)
with warns_deprecated_sympy():
from sympy.simplify.simplify import walk
walk(x, lambda x: x)
with warns_deprecated_sympy():
from sympy.simplify.traversaltools import use
use(x, lambda x: x)
with warns_deprecated_sympy():
from sympy.utilities.iterables import postorder_traversal
postorder_traversal(x)
with warns_deprecated_sympy():
from sympy.utilities.iterables import interactive_traversal
capture(lambda: interactive_traversal(x))
|
9e04b0f59d7c9d5cfeef46d11186cae99c9e02018a627cf1ac1457c551f8a53c | from sympy.assumptions.refine import refine
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import (ExprBuilder, unchanged, Expr,
UnevaluatedExpr)
from sympy.core.function import (Function, expand, WildFunction,
AppliedUndef, Derivative, diff, Subs)
from sympy.core.mul import Mul
from sympy.core.numbers import (NumberSymbol, E, zoo, oo, Float, I,
Rational, nan, Integer, Number, pi)
from sympy.core.power import Pow
from sympy.core.relational import Ge, Lt, Gt, Le
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Symbol, symbols, Dummy, Wild
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.exponential import exp_polar, exp, log
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import tan, sin, cos
from sympy.functions.special.delta_functions import (Heaviside,
DiracDelta)
from sympy.functions.special.error_functions import Si
from sympy.functions.special.gamma_functions import gamma
from sympy.integrals.integrals import integrate, Integral
from sympy.physics.secondquant import FockState
from sympy.polys.partfrac import apart
from sympy.polys.polytools import factor, cancel, Poly
from sympy.polys.rationaltools import together
from sympy.series.order import O
from sympy.simplify.combsimp import combsimp
from sympy.simplify.gammasimp import gammasimp
from sympy.simplify.powsimp import powsimp
from sympy.simplify.radsimp import collect, radsimp
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import simplify, nsimplify
from sympy.simplify.trigsimp import trigsimp
from sympy.tensor.indexed import Indexed
from sympy.physics.units import meter
from sympy.testing.pytest import raises, XFAIL
from sympy.abc import a, b, c, n, t, u, x, y, z
f, g, h = symbols('f,g,h', cls=Function)
class DummyNumber:
"""
Minimal implementation of a number that works with SymPy.
If one has a Number class (e.g. Sage Integer, or some other custom class)
that one wants to work well with SymPy, one has to implement at least the
methods of this class DummyNumber, resp. its subclasses I5 and F1_1.
Basically, one just needs to implement either __int__() or __float__() and
then one needs to make sure that the class works with Python integers and
with itself.
"""
def __radd__(self, a):
if isinstance(a, (int, float)):
return a + self.number
return NotImplemented
def __add__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number + a
return NotImplemented
def __rsub__(self, a):
if isinstance(a, (int, float)):
return a - self.number
return NotImplemented
def __sub__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number - a
return NotImplemented
def __rmul__(self, a):
if isinstance(a, (int, float)):
return a * self.number
return NotImplemented
def __mul__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number * a
return NotImplemented
def __rtruediv__(self, a):
if isinstance(a, (int, float)):
return a / self.number
return NotImplemented
def __truediv__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number / a
return NotImplemented
def __rpow__(self, a):
if isinstance(a, (int, float)):
return a ** self.number
return NotImplemented
def __pow__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number ** a
return NotImplemented
def __pos__(self):
return self.number
def __neg__(self):
return - self.number
class I5(DummyNumber):
number = 5
def __int__(self):
return self.number
class F1_1(DummyNumber):
number = 1.1
def __float__(self):
return self.number
i5 = I5()
f1_1 = F1_1()
# basic SymPy objects
basic_objs = [
Rational(2),
Float("1.3"),
x,
y,
pow(x, y)*y,
]
# all supported objects
all_objs = basic_objs + [
5,
5.5,
i5,
f1_1
]
def dotest(s):
for xo in all_objs:
for yo in all_objs:
s(xo, yo)
return True
def test_basic():
def j(a, b):
x = a
x = +a
x = -a
x = a + b
x = a - b
x = a*b
x = a/b
x = a**b
del x
assert dotest(j)
def test_ibasic():
def s(a, b):
x = a
x += b
x = a
x -= b
x = a
x *= b
x = a
x /= b
assert dotest(s)
class NonBasic:
'''This class represents an object that knows how to implement binary
operations like +, -, etc with Expr but is not a subclass of Basic itself.
The NonExpr subclass below does subclass Basic but not Expr.
For both NonBasic and NonExpr it should be possible for them to override
Expr.__add__ etc because Expr.__add__ should be returning NotImplemented
for non Expr classes. Otherwise Expr.__add__ would create meaningless
objects like Add(Integer(1), FiniteSet(2)) and it wouldn't be possible for
other classes to override these operations when interacting with Expr.
'''
def __add__(self, other):
return SpecialOp('+', self, other)
def __radd__(self, other):
return SpecialOp('+', other, self)
def __sub__(self, other):
return SpecialOp('-', self, other)
def __rsub__(self, other):
return SpecialOp('-', other, self)
def __mul__(self, other):
return SpecialOp('*', self, other)
def __rmul__(self, other):
return SpecialOp('*', other, self)
def __truediv__(self, other):
return SpecialOp('/', self, other)
def __rtruediv__(self, other):
return SpecialOp('/', other, self)
def __floordiv__(self, other):
return SpecialOp('//', self, other)
def __rfloordiv__(self, other):
return SpecialOp('//', other, self)
def __mod__(self, other):
return SpecialOp('%', self, other)
def __rmod__(self, other):
return SpecialOp('%', other, self)
def __divmod__(self, other):
return SpecialOp('divmod', self, other)
def __rdivmod__(self, other):
return SpecialOp('divmod', other, self)
def __pow__(self, other):
return SpecialOp('**', self, other)
def __rpow__(self, other):
return SpecialOp('**', other, self)
def __lt__(self, other):
return SpecialOp('<', self, other)
def __gt__(self, other):
return SpecialOp('>', self, other)
def __le__(self, other):
return SpecialOp('<=', self, other)
def __ge__(self, other):
return SpecialOp('>=', self, other)
class NonExpr(Basic, NonBasic):
'''Like NonBasic above except this is a subclass of Basic but not Expr'''
pass
class SpecialOp():
'''Represents the results of operations with NonBasic and NonExpr'''
def __new__(cls, op, arg1, arg2):
obj = object.__new__(cls)
obj.args = (op, arg1, arg2)
return obj
class NonArithmetic(Basic):
'''Represents a Basic subclass that does not support arithmetic operations'''
pass
def test_cooperative_operations():
'''Tests that Expr uses binary operations cooperatively.
In particular it should be possible for non-Expr classes to override
binary operators like +, - etc when used with Expr instances. This should
work for non-Expr classes whether they are Basic subclasses or not. Also
non-Expr classes that do not define binary operators with Expr should give
TypeError.
'''
# A bunch of instances of Expr subclasses
exprs = [
Expr(),
S.Zero,
S.One,
S.Infinity,
S.NegativeInfinity,
S.ComplexInfinity,
S.Half,
Float(0.5),
Integer(2),
Symbol('x'),
Mul(2, Symbol('x')),
Add(2, Symbol('x')),
Pow(2, Symbol('x')),
]
for e in exprs:
# Test that these classes can override arithmetic operations in
# combination with various Expr types.
for ne in [NonBasic(), NonExpr()]:
results = [
(ne + e, ('+', ne, e)),
(e + ne, ('+', e, ne)),
(ne - e, ('-', ne, e)),
(e - ne, ('-', e, ne)),
(ne * e, ('*', ne, e)),
(e * ne, ('*', e, ne)),
(ne / e, ('/', ne, e)),
(e / ne, ('/', e, ne)),
(ne // e, ('//', ne, e)),
(e // ne, ('//', e, ne)),
(ne % e, ('%', ne, e)),
(e % ne, ('%', e, ne)),
(divmod(ne, e), ('divmod', ne, e)),
(divmod(e, ne), ('divmod', e, ne)),
(ne ** e, ('**', ne, e)),
(e ** ne, ('**', e, ne)),
(e < ne, ('>', ne, e)),
(ne < e, ('<', ne, e)),
(e > ne, ('<', ne, e)),
(ne > e, ('>', ne, e)),
(e <= ne, ('>=', ne, e)),
(ne <= e, ('<=', ne, e)),
(e >= ne, ('<=', ne, e)),
(ne >= e, ('>=', ne, e)),
]
for res, args in results:
assert type(res) is SpecialOp and res.args == args
# These classes do not support binary operators with Expr. Every
# operation should raise in combination with any of the Expr types.
for na in [NonArithmetic(), object()]:
raises(TypeError, lambda : e + na)
raises(TypeError, lambda : na + e)
raises(TypeError, lambda : e - na)
raises(TypeError, lambda : na - e)
raises(TypeError, lambda : e * na)
raises(TypeError, lambda : na * e)
raises(TypeError, lambda : e / na)
raises(TypeError, lambda : na / e)
raises(TypeError, lambda : e // na)
raises(TypeError, lambda : na // e)
raises(TypeError, lambda : e % na)
raises(TypeError, lambda : na % e)
raises(TypeError, lambda : divmod(e, na))
raises(TypeError, lambda : divmod(na, e))
raises(TypeError, lambda : e ** na)
raises(TypeError, lambda : na ** e)
raises(TypeError, lambda : e > na)
raises(TypeError, lambda : na > e)
raises(TypeError, lambda : e < na)
raises(TypeError, lambda : na < e)
raises(TypeError, lambda : e >= na)
raises(TypeError, lambda : na >= e)
raises(TypeError, lambda : e <= na)
raises(TypeError, lambda : na <= e)
def test_relational():
from sympy.core.relational import Lt
assert (pi < 3) is S.false
assert (pi <= 3) is S.false
assert (pi > 3) is S.true
assert (pi >= 3) is S.true
assert (-pi < 3) is S.true
assert (-pi <= 3) is S.true
assert (-pi > 3) is S.false
assert (-pi >= 3) is S.false
r = Symbol('r', real=True)
assert (r - 2 < r - 3) is S.false
assert Lt(x + I, x + I + 2).func == Lt # issue 8288
def test_relational_assumptions():
m1 = Symbol("m1", nonnegative=False)
m2 = Symbol("m2", positive=False)
m3 = Symbol("m3", nonpositive=False)
m4 = Symbol("m4", negative=False)
assert (m1 < 0) == Lt(m1, 0)
assert (m2 <= 0) == Le(m2, 0)
assert (m3 > 0) == Gt(m3, 0)
assert (m4 >= 0) == Ge(m4, 0)
m1 = Symbol("m1", nonnegative=False, real=True)
m2 = Symbol("m2", positive=False, real=True)
m3 = Symbol("m3", nonpositive=False, real=True)
m4 = Symbol("m4", negative=False, real=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=True)
m2 = Symbol("m2", nonpositive=True)
m3 = Symbol("m3", positive=True)
m4 = Symbol("m4", nonnegative=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=False, real=True)
m2 = Symbol("m2", nonpositive=False, real=True)
m3 = Symbol("m3", positive=False, real=True)
m4 = Symbol("m4", nonnegative=False, real=True)
assert (m1 < 0) is S.false
assert (m2 <= 0) is S.false
assert (m3 > 0) is S.false
assert (m4 >= 0) is S.false
# See https://github.com/sympy/sympy/issues/17708
#def test_relational_noncommutative():
# from sympy import Lt, Gt, Le, Ge
# A, B = symbols('A,B', commutative=False)
# assert (A < B) == Lt(A, B)
# assert (A <= B) == Le(A, B)
# assert (A > B) == Gt(A, B)
# assert (A >= B) == Ge(A, B)
def test_basic_nostr():
for obj in basic_objs:
raises(TypeError, lambda: obj + '1')
raises(TypeError, lambda: obj - '1')
if obj == 2:
assert obj * '1' == '11'
else:
raises(TypeError, lambda: obj * '1')
raises(TypeError, lambda: obj / '1')
raises(TypeError, lambda: obj ** '1')
def test_series_expansion_for_uniform_order():
assert (1/x + y + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + x).series(x, 0, 1) == 1/x + y + O(x)
assert (1/x + 1 + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + 1 + x).series(x, 0, 1) == 1/x + 1 + O(x)
assert (1/x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 1) == 1/x + y + O(x)
def test_leadterm():
assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0)
assert (1/x**2 + 1 + x + x**2).leadterm(x)[1] == -2
assert (1/x + 1 + x + x**2).leadterm(x)[1] == -1
assert (x**2 + 1/x).leadterm(x)[1] == -1
assert (1 + x**2).leadterm(x)[1] == 0
assert (x + 1).leadterm(x)[1] == 0
assert (x + x**2).leadterm(x)[1] == 1
assert (x**2).leadterm(x)[1] == 2
def test_as_leading_term():
assert (3 + 2*x**(log(3)/log(2) - 1)).as_leading_term(x) == 3
assert (1/x**2 + 1 + x + x**2).as_leading_term(x) == 1/x**2
assert (1/x + 1 + x + x**2).as_leading_term(x) == 1/x
assert (x**2 + 1/x).as_leading_term(x) == 1/x
assert (1 + x**2).as_leading_term(x) == 1
assert (x + 1).as_leading_term(x) == 1
assert (x + x**2).as_leading_term(x) == x
assert (x**2).as_leading_term(x) == x**2
assert (x + oo).as_leading_term(x) is oo
raises(ValueError, lambda: (x + 1).as_leading_term(1))
# https://github.com/sympy/sympy/issues/21177
e = -3*x + (x + Rational(3, 2) - sqrt(3)*S.ImaginaryUnit/2)**2\
- Rational(3, 2) + 3*sqrt(3)*S.ImaginaryUnit/2
assert e.as_leading_term(x) == \
(12*sqrt(3)*x - 12*S.ImaginaryUnit*x)/(4*sqrt(3) + 12*S.ImaginaryUnit)
# https://github.com/sympy/sympy/issues/21245
e = 1 - x - x**2
d = (1 + sqrt(5))/2
assert e.subs(x, y + 1/d).as_leading_term(y) == \
(-576*sqrt(5)*y - 1280*y)/(256*sqrt(5) + 576)
def test_leadterm2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).leadterm(x) == \
(sin(1 + sin(1)), 0)
def test_leadterm3():
assert (y + z + x).leadterm(x) == (y + z, 0)
def test_as_leading_term2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).as_leading_term(x) == \
sin(1 + sin(1))
def test_as_leading_term3():
assert (2 + pi + x).as_leading_term(x) == 2 + pi
assert (2*x + pi*x + x**2).as_leading_term(x) == 2*x + pi*x
def test_as_leading_term4():
# see issue 6843
n = Symbol('n', integer=True, positive=True)
r = -n**3/(2*n**2 + 4*n + 2) - n**2/(n**2 + 2*n + 1) + \
n**2/(n + 1) - n/(2*n**2 + 4*n + 2) + n/(n*x + x) + 2*n/(n + 1) - \
1 + 1/(n*x + x) + 1/(n + 1) - 1/x
assert r.as_leading_term(x).cancel() == n/2
def test_as_leading_term_stub():
class foo(Function):
pass
assert foo(1/x).as_leading_term(x) == foo(1/x)
assert foo(1).as_leading_term(x) == foo(1)
raises(NotImplementedError, lambda: foo(x).as_leading_term(x))
def test_as_leading_term_deriv_integral():
# related to issue 11313
assert Derivative(x ** 3, x).as_leading_term(x) == 3*x**2
assert Derivative(x ** 3, y).as_leading_term(x) == 0
assert Integral(x ** 3, x).as_leading_term(x) == x**4/4
assert Integral(x ** 3, y).as_leading_term(x) == y*x**3
assert Derivative(exp(x), x).as_leading_term(x) == 1
assert Derivative(log(x), x).as_leading_term(x) == (1/x).as_leading_term(x)
def test_atoms():
assert x.atoms() == {x}
assert (1 + x).atoms() == {x, S.One}
assert (1 + 2*cos(x)).atoms(Symbol) == {x}
assert (1 + 2*cos(x)).atoms(Symbol, Number) == {S.One, S(2), x}
assert (2*(x**(y**x))).atoms() == {S(2), x, y}
assert S.Half.atoms() == {S.Half}
assert S.Half.atoms(Symbol) == set()
assert sin(oo).atoms(oo) == set()
assert Poly(0, x).atoms() == {S.Zero, x}
assert Poly(1, x).atoms() == {S.One, x}
assert Poly(x, x).atoms() == {x}
assert Poly(x, x, y).atoms() == {x, y}
assert Poly(x + y, x, y).atoms() == {x, y}
assert Poly(x + y, x, y, z).atoms() == {x, y, z}
assert Poly(x + y*t, x, y, z).atoms() == {t, x, y, z}
assert (I*pi).atoms(NumberSymbol) == {pi}
assert (I*pi).atoms(NumberSymbol, I) == \
(I*pi).atoms(I, NumberSymbol) == {pi, I}
assert exp(exp(x)).atoms(exp) == {exp(exp(x)), exp(x)}
assert (1 + x*(2 + y) + exp(3 + z)).atoms(Add) == \
{1 + x*(2 + y) + exp(3 + z), 2 + y, 3 + z}
# issue 6132
e = (f(x) + sin(x) + 2)
assert e.atoms(AppliedUndef) == \
{f(x)}
assert e.atoms(AppliedUndef, Function) == \
{f(x), sin(x)}
assert e.atoms(Function) == \
{f(x), sin(x)}
assert e.atoms(AppliedUndef, Number) == \
{f(x), S(2)}
assert e.atoms(Function, Number) == \
{S(2), sin(x), f(x)}
def test_is_polynomial():
k = Symbol('k', nonnegative=True, integer=True)
assert Rational(2).is_polynomial(x, y, z) is True
assert (S.Pi).is_polynomial(x, y, z) is True
assert x.is_polynomial(x) is True
assert x.is_polynomial(y) is True
assert (x**2).is_polynomial(x) is True
assert (x**2).is_polynomial(y) is True
assert (x**(-2)).is_polynomial(x) is False
assert (x**(-2)).is_polynomial(y) is True
assert (2**x).is_polynomial(x) is False
assert (2**x).is_polynomial(y) is True
assert (x**k).is_polynomial(x) is False
assert (x**k).is_polynomial(k) is False
assert (x**x).is_polynomial(x) is False
assert (k**k).is_polynomial(k) is False
assert (k**x).is_polynomial(k) is False
assert (x**(-k)).is_polynomial(x) is False
assert ((2*x)**k).is_polynomial(x) is False
assert (x**2 + 3*x - 8).is_polynomial(x) is True
assert (x**2 + 3*x - 8).is_polynomial(y) is True
assert (x**2 + 3*x - 8).is_polynomial() is True
assert sqrt(x).is_polynomial(x) is False
assert (sqrt(x)**3).is_polynomial(x) is False
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(x) is True
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(y) is False
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial() is True
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial() is False
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial(x, y) is True
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial(x, y) is False
assert (1/f(x) + 1).is_polynomial(f(x)) is False
def test_is_rational_function():
assert Integer(1).is_rational_function() is True
assert Integer(1).is_rational_function(x) is True
assert Rational(17, 54).is_rational_function() is True
assert Rational(17, 54).is_rational_function(x) is True
assert (12/x).is_rational_function() is True
assert (12/x).is_rational_function(x) is True
assert (x/y).is_rational_function() is True
assert (x/y).is_rational_function(x) is True
assert (x/y).is_rational_function(x, y) is True
assert (x**2 + 1/x/y).is_rational_function() is True
assert (x**2 + 1/x/y).is_rational_function(x) is True
assert (x**2 + 1/x/y).is_rational_function(x, y) is True
assert (sin(y)/x).is_rational_function() is False
assert (sin(y)/x).is_rational_function(y) is False
assert (sin(y)/x).is_rational_function(x) is True
assert (sin(y)/x).is_rational_function(x, y) is False
assert (S.NaN).is_rational_function() is False
assert (S.Infinity).is_rational_function() is False
assert (S.NegativeInfinity).is_rational_function() is False
assert (S.ComplexInfinity).is_rational_function() is False
def test_is_meromorphic():
f = a/x**2 + b + x + c*x**2
assert f.is_meromorphic(x, 0) is True
assert f.is_meromorphic(x, 1) is True
assert f.is_meromorphic(x, zoo) is True
g = 3 + 2*x**(log(3)/log(2) - 1)
assert g.is_meromorphic(x, 0) is False
assert g.is_meromorphic(x, 1) is True
assert g.is_meromorphic(x, zoo) is False
n = Symbol('n', integer=True)
e = sin(1/x)**n*x
assert e.is_meromorphic(x, 0) is False
assert e.is_meromorphic(x, 1) is True
assert e.is_meromorphic(x, zoo) is False
e = log(x)**pi
assert e.is_meromorphic(x, 0) is False
assert e.is_meromorphic(x, 1) is False
assert e.is_meromorphic(x, 2) is True
assert e.is_meromorphic(x, zoo) is False
assert (log(x)**a).is_meromorphic(x, 0) is False
assert (log(x)**a).is_meromorphic(x, 1) is False
assert (a**log(x)).is_meromorphic(x, 0) is None
assert (3**log(x)).is_meromorphic(x, 0) is False
assert (3**log(x)).is_meromorphic(x, 1) is True
def test_is_algebraic_expr():
assert sqrt(3).is_algebraic_expr(x) is True
assert sqrt(3).is_algebraic_expr() is True
eq = ((1 + x**2)/(1 - y**2))**(S.One/3)
assert eq.is_algebraic_expr(x) is True
assert eq.is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(x) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr() is True
assert (cos(y)/sqrt(x)).is_algebraic_expr() is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x) is True
assert (cos(y)/sqrt(x)).is_algebraic_expr(y) is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x, y) is False
def test_SAGE1():
#see https://github.com/sympy/sympy/issues/3346
class MyInt:
def _sympy_(self):
return Integer(5)
m = MyInt()
e = Rational(2)*m
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE2():
class MyInt:
def __int__(self):
return 5
assert sympify(MyInt()) == 5
e = Rational(2)*MyInt()
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE3():
class MySymbol:
def __rmul__(self, other):
return ('mys', other, self)
o = MySymbol()
e = x*o
assert e == ('mys', x, o)
def test_len():
e = x*y
assert len(e.args) == 2
e = x + y + z
assert len(e.args) == 3
def test_doit():
a = Integral(x**2, x)
assert isinstance(a.doit(), Integral) is False
assert isinstance(a.doit(integrals=True), Integral) is False
assert isinstance(a.doit(integrals=False), Integral) is True
assert (2*Integral(x, x)).doit() == x**2
def test_attribute_error():
raises(AttributeError, lambda: x.cos())
raises(AttributeError, lambda: x.sin())
raises(AttributeError, lambda: x.exp())
def test_args():
assert (x*y).args in ((x, y), (y, x))
assert (x + y).args in ((x, y), (y, x))
assert (x*y + 1).args in ((x*y, 1), (1, x*y))
assert sin(x*y).args == (x*y,)
assert sin(x*y).args[0] == x*y
assert (x**y).args == (x, y)
assert (x**y).args[0] == x
assert (x**y).args[1] == y
def test_noncommutative_expand_issue_3757():
A, B, C = symbols('A,B,C', commutative=False)
assert A*B - B*A != 0
assert (A*(A + B)*B).expand() == A**2*B + A*B**2
assert (A*(A + B + C)*B).expand() == A**2*B + A*B**2 + A*C*B
def test_as_numer_denom():
a, b, c = symbols('a, b, c')
assert nan.as_numer_denom() == (nan, 1)
assert oo.as_numer_denom() == (oo, 1)
assert (-oo).as_numer_denom() == (-oo, 1)
assert zoo.as_numer_denom() == (zoo, 1)
assert (-zoo).as_numer_denom() == (zoo, 1)
assert x.as_numer_denom() == (x, 1)
assert (1/x).as_numer_denom() == (1, x)
assert (x/y).as_numer_denom() == (x, y)
assert (x/2).as_numer_denom() == (x, 2)
assert (x*y/z).as_numer_denom() == (x*y, z)
assert (x/(y*z)).as_numer_denom() == (x, y*z)
assert S.Half.as_numer_denom() == (1, 2)
assert (1/y**2).as_numer_denom() == (1, y**2)
assert (x/y**2).as_numer_denom() == (x, y**2)
assert ((x**2 + 1)/y).as_numer_denom() == (x**2 + 1, y)
assert (x*(y + 1)/y**7).as_numer_denom() == (x*(y + 1), y**7)
assert (x**-2).as_numer_denom() == (1, x**2)
assert (a/x + b/2/x + c/3/x).as_numer_denom() == \
(6*a + 3*b + 2*c, 6*x)
assert (a/x + b/2/x + c/3/y).as_numer_denom() == \
(2*c*x + y*(6*a + 3*b), 6*x*y)
assert (a/x + b/2/x + c/.5/x).as_numer_denom() == \
(2*a + b + 4.0*c, 2*x)
# this should take no more than a few seconds
assert int(log(Add(*[Dummy()/i/x for i in range(1, 705)]
).as_numer_denom()[1]/x).n(4)) == 705
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).as_numer_denom() == \
(x + i, 3)
assert (S.Infinity + x/3 + y/4).as_numer_denom() == \
(4*x + 3*y + S.Infinity, 12)
assert (oo*x + zoo*y).as_numer_denom() == \
(zoo*y + oo*x, 1)
A, B, C = symbols('A,B,C', commutative=False)
assert (A*B*C**-1).as_numer_denom() == (A*B*C**-1, 1)
assert (A*B*C**-1/x).as_numer_denom() == (A*B*C**-1, x)
assert (C**-1*A*B).as_numer_denom() == (C**-1*A*B, 1)
assert (C**-1*A*B/x).as_numer_denom() == (C**-1*A*B, x)
assert ((A*B*C)**-1).as_numer_denom() == ((A*B*C)**-1, 1)
assert ((A*B*C)**-1/x).as_numer_denom() == ((A*B*C)**-1, x)
# the following morphs from Add to Mul during processing
assert Add(0, (x + y)/z/-2, evaluate=False).as_numer_denom(
) == (-x - y, 2*z)
def test_trunc():
import math
x, y = symbols('x y')
assert math.trunc(2) == 2
assert math.trunc(4.57) == 4
assert math.trunc(-5.79) == -5
assert math.trunc(pi) == 3
assert math.trunc(log(7)) == 1
assert math.trunc(exp(5)) == 148
assert math.trunc(cos(pi)) == -1
assert math.trunc(sin(5)) == 0
raises(TypeError, lambda: math.trunc(x))
raises(TypeError, lambda: math.trunc(x + y**2))
raises(TypeError, lambda: math.trunc(oo))
def test_as_independent():
assert S.Zero.as_independent(x, as_Add=True) == (0, 0)
assert S.Zero.as_independent(x, as_Add=False) == (0, 0)
assert (2*x*sin(x) + y + x).as_independent(x) == (y, x + 2*x*sin(x))
assert (2*x*sin(x) + y + x).as_independent(y) == (x + 2*x*sin(x), y)
assert (2*x*sin(x) + y + x).as_independent(x, y) == (0, y + x + 2*x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(x) == (cos(y), x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(y) == (x*sin(x), cos(y))
assert (x*sin(x)*cos(y)).as_independent(x, y) == (1, x*sin(x)*cos(y))
assert (sin(x)).as_independent(x) == (1, sin(x))
assert (sin(x)).as_independent(y) == (sin(x), 1)
assert (2*sin(x)).as_independent(x) == (2, sin(x))
assert (2*sin(x)).as_independent(y) == (2*sin(x), 1)
# issue 4903 = 1766b
n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
assert (n1 + n1*n2).as_independent(n2) == (n1, n1*n2)
assert (n2*n1 + n1*n2).as_independent(n2) == (0, n1*n2 + n2*n1)
assert (n1*n2*n1).as_independent(n2) == (n1, n2*n1)
assert (n1*n2*n1).as_independent(n1) == (1, n1*n2*n1)
assert (3*x).as_independent(x, as_Add=True) == (0, 3*x)
assert (3*x).as_independent(x, as_Add=False) == (3, x)
assert (3 + x).as_independent(x, as_Add=True) == (3, x)
assert (3 + x).as_independent(x, as_Add=False) == (1, 3 + x)
# issue 5479
assert (3*x).as_independent(Symbol) == (3, x)
# issue 5648
assert (n1*x*y).as_independent(x) == (n1*y, x)
assert ((x + n1)*(x - y)).as_independent(x) == (1, (x + n1)*(x - y))
assert ((x + n1)*(x - y)).as_independent(y) == (x + n1, x - y)
assert (DiracDelta(x - n1)*DiracDelta(x - y)).as_independent(x) \
== (1, DiracDelta(x - n1)*DiracDelta(x - y))
assert (x*y*n1*n2*n3).as_independent(n2) == (x*y*n1, n2*n3)
assert (x*y*n1*n2*n3).as_independent(n1) == (x*y, n1*n2*n3)
assert (x*y*n1*n2*n3).as_independent(n3) == (x*y*n1*n2, n3)
assert (DiracDelta(x - n1)*DiracDelta(y - n1)*DiracDelta(x - n2)).as_independent(y) == \
(DiracDelta(x - n1)*DiracDelta(x - n2), DiracDelta(y - n1))
# issue 5784
assert (x + Integral(x, (x, 1, 2))).as_independent(x, strict=True) == \
(Integral(x, (x, 1, 2)), x)
eq = Add(x, -x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-1, Add(x, -x, evaluate=False))
eq = Mul(x, 1/x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-6, Mul(x, 1/x, evaluate=False))
assert (x*y).as_independent(z, as_Add=True) == (x*y, 0)
@XFAIL
def test_call_2():
# TODO UndefinedFunction does not subclass Expr
assert (2*f)(x) == 2*f(x)
def test_replace():
e = log(sin(x)) + tan(sin(x**2))
assert e.replace(sin, cos) == log(cos(x)) + tan(cos(x**2))
assert e.replace(
sin, lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
a = Wild('a')
b = Wild('b')
assert e.replace(sin(a), cos(a)) == log(cos(x)) + tan(cos(x**2))
assert e.replace(
sin(a), lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
# test exact
assert (2*x).replace(a*x + b, b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, b - a) == 2*x
assert (2*x).replace(a*x + b, b - a, exact=False) == 2/x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=False) == 2/x
g = 2*sin(x**3)
assert g.replace(
lambda expr: expr.is_Number, lambda expr: expr**2) == 4*sin(x**9)
assert cos(x).replace(cos, sin, map=True) == (sin(x), {cos(x): sin(x)})
assert sin(x).replace(cos, sin) == sin(x)
cond, func = lambda x: x.is_Mul, lambda x: 2*x
assert (x*y).replace(cond, func, map=True) == (2*x*y, {x*y: 2*x*y})
assert (x*(1 + x*y)).replace(cond, func, map=True) == \
(2*x*(2*x*y + 1), {x*(2*x*y + 1): 2*x*(2*x*y + 1), x*y: 2*x*y})
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y, map=True) == \
(sin(x), {sin(x): sin(x)/y})
# if not simultaneous then y*sin(x) -> y*sin(x)/y = sin(x) -> sin(x)/y
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y,
simultaneous=False) == sin(x)/y
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e
) == x**2/2 + O(x**3)
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e,
simultaneous=False) == x**2/2 + O(x**3)
assert (x*(x*y + 3)).replace(lambda x: x.is_Mul, lambda x: 2 + x) == \
x*(x*y + 5) + 2
e = (x*y + 1)*(2*x*y + 1) + 1
assert e.replace(cond, func, map=True) == (
2*((2*x*y + 1)*(4*x*y + 1)) + 1,
{2*x*y: 4*x*y, x*y: 2*x*y, (2*x*y + 1)*(4*x*y + 1):
2*((2*x*y + 1)*(4*x*y + 1))})
assert x.replace(x, y) == y
assert (x + 1).replace(1, 2) == x + 2
# https://groups.google.com/forum/#!topic/sympy/8wCgeC95tz0
n1, n2, n3 = symbols('n1:4', commutative=False)
assert (n1*f(n2)).replace(f, lambda x: x) == n1*n2
assert (n3*f(n2)).replace(f, lambda x: x) == n3*n2
# issue 16725
assert S.Zero.replace(Wild('x'), 1) == 1
# let the user override the default decision of False
assert S.Zero.replace(Wild('x'), 1, exact=True) == 0
def test_find():
expr = (x + y + 2 + sin(3*x))
assert expr.find(lambda u: u.is_Integer) == {S(2), S(3)}
assert expr.find(lambda u: u.is_Symbol) == {x, y}
assert expr.find(lambda u: u.is_Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(lambda u: u.is_Symbol, group=True) == {x: 2, y: 1}
assert expr.find(Integer) == {S(2), S(3)}
assert expr.find(Symbol) == {x, y}
assert expr.find(Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(Symbol, group=True) == {x: 2, y: 1}
a = Wild('a')
expr = sin(sin(x)) + sin(x) + cos(x) + x
assert expr.find(lambda u: type(u) is sin) == {sin(x), sin(sin(x))}
assert expr.find(
lambda u: type(u) is sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin(a)) == {sin(x), sin(sin(x))}
assert expr.find(sin(a), group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin) == {sin(x), sin(sin(x))}
assert expr.find(sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
def test_count():
expr = (x + y + 2 + sin(3*x))
assert expr.count(lambda u: u.is_Integer) == 2
assert expr.count(lambda u: u.is_Symbol) == 3
assert expr.count(Integer) == 2
assert expr.count(Symbol) == 3
assert expr.count(2) == 1
a = Wild('a')
assert expr.count(sin) == 1
assert expr.count(sin(a)) == 1
assert expr.count(lambda u: type(u) is sin) == 1
assert f(x).count(f(x)) == 1
assert f(x).diff(x).count(f(x)) == 1
assert f(x).diff(x).count(x) == 2
def test_has_basics():
p = Wild('p')
assert sin(x).has(x)
assert sin(x).has(sin)
assert not sin(x).has(y)
assert not sin(x).has(cos)
assert f(x).has(x)
assert f(x).has(f)
assert not f(x).has(y)
assert not f(x).has(g)
assert f(x).diff(x).has(x)
assert f(x).diff(x).has(f)
assert f(x).diff(x).has(Derivative)
assert not f(x).diff(x).has(y)
assert not f(x).diff(x).has(g)
assert not f(x).diff(x).has(sin)
assert (x**2).has(Symbol)
assert not (x**2).has(Wild)
assert (2*p).has(Wild)
assert not x.has()
def test_has_multiple():
f = x**2*y + sin(2**t + log(z))
assert f.has(x)
assert f.has(y)
assert f.has(z)
assert f.has(t)
assert not f.has(u)
assert f.has(x, y, z, t)
assert f.has(x, y, z, t, u)
i = Integer(4400)
assert not i.has(x)
assert (i*x**i).has(x)
assert not (i*y**i).has(x)
assert (i*y**i).has(x, y)
assert not (i*y**i).has(x, z)
def test_has_piecewise():
f = (x*y + 3/y)**(3 + 2)
p = Piecewise((g(x), x < -1), (1, x <= 1), (f, True))
assert p.has(x)
assert p.has(y)
assert not p.has(z)
assert p.has(1)
assert p.has(3)
assert not p.has(4)
assert p.has(f)
assert p.has(g)
assert not p.has(h)
def test_has_iterative():
A, B, C = symbols('A,B,C', commutative=False)
f = x*gamma(x)*sin(x)*exp(x*y)*A*B*C*cos(x*A*B)
assert f.has(x)
assert f.has(x*y)
assert f.has(x*sin(x))
assert not f.has(x*sin(y))
assert f.has(x*A)
assert f.has(x*A*B)
assert not f.has(x*A*C)
assert f.has(x*A*B*C)
assert not f.has(x*A*C*B)
assert f.has(x*sin(x)*A*B*C)
assert not f.has(x*sin(x)*A*C*B)
assert not f.has(x*sin(y)*A*B*C)
assert f.has(x*gamma(x))
assert not f.has(x + sin(x))
assert (x & y & z).has(x & z)
def test_has_integrals():
f = Integral(x**2 + sin(x*y*z), (x, 0, x + y + z))
assert f.has(x + y)
assert f.has(x + z)
assert f.has(y + z)
assert f.has(x*y)
assert f.has(x*z)
assert f.has(y*z)
assert not f.has(2*x + y)
assert not f.has(2*x*y)
def test_has_tuple():
assert Tuple(x, y).has(x)
assert not Tuple(x, y).has(z)
assert Tuple(f(x), g(x)).has(x)
assert not Tuple(f(x), g(x)).has(y)
assert Tuple(f(x), g(x)).has(f)
assert Tuple(f(x), g(x)).has(f(x))
# XXX to be deprecated
#assert not Tuple(f, g).has(x)
#assert Tuple(f, g).has(f)
#assert not Tuple(f, g).has(h)
assert Tuple(True).has(True)
assert Tuple(True).has(S.true)
assert not Tuple(True).has(1)
def test_has_units():
from sympy.physics.units import m, s
assert (x*m/s).has(x)
assert (x*m/s).has(y, z) is False
def test_has_polys():
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x)
assert poly.has(x, y, z)
assert poly.has(x, y, z, t)
def test_has_physics():
assert FockState((x, y)).has(x)
def test_as_poly_as_expr():
f = x**2 + 2*x*y
assert f.as_poly().as_expr() == f
assert f.as_poly(x, y).as_expr() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
# https://github.com/sympy/sympy/issues/20610
assert S(2).as_poly() is None
assert sqrt(2).as_poly(extension=True) is None
raises(AttributeError, lambda: Tuple(x, x).as_poly(x))
raises(AttributeError, lambda: Tuple(x ** 2, x, y).as_poly(x))
def test_nonzero():
assert bool(S.Zero) is False
assert bool(S.One) is True
assert bool(x) is True
assert bool(x + y) is True
assert bool(x - x) is False
assert bool(x*y) is True
assert bool(x*1) is True
assert bool(x*0) is False
def test_is_number():
assert Float(3.14).is_number is True
assert Integer(737).is_number is True
assert Rational(3, 2).is_number is True
assert Rational(8).is_number is True
assert x.is_number is False
assert (2*x).is_number is False
assert (x + y).is_number is False
assert log(2).is_number is True
assert log(x).is_number is False
assert (2 + log(2)).is_number is True
assert (8 + log(2)).is_number is True
assert (2 + log(x)).is_number is False
assert (8 + log(2) + x).is_number is False
assert (1 + x**2/x - x).is_number is True
assert Tuple(Integer(1)).is_number is False
assert Add(2, x).is_number is False
assert Mul(3, 4).is_number is True
assert Pow(log(2), 2).is_number is True
assert oo.is_number is True
g = WildFunction('g')
assert g.is_number is False
assert (2*g).is_number is False
assert (x**2).subs(x, 3).is_number is True
# test extensibility of .is_number
# on subinstances of Basic
class A(Basic):
pass
a = A()
assert a.is_number is False
def test_as_coeff_add():
assert S(2).as_coeff_add() == (2, ())
assert S(3.0).as_coeff_add() == (0, (S(3.0),))
assert S(-3.0).as_coeff_add() == (0, (S(-3.0),))
assert x.as_coeff_add() == (0, (x,))
assert (x - 1).as_coeff_add() == (-1, (x,))
assert (x + 1).as_coeff_add() == (1, (x,))
assert (x + 2).as_coeff_add() == (2, (x,))
assert (x + y).as_coeff_add(y) == (x, (y,))
assert (3*x).as_coeff_add(y) == (3*x, ())
# don't do expansion
e = (x + y)**2
assert e.as_coeff_add(y) == (0, (e,))
def test_as_coeff_mul():
assert S(2).as_coeff_mul() == (2, ())
assert S(3.0).as_coeff_mul() == (1, (S(3.0),))
assert S(-3.0).as_coeff_mul() == (-1, (S(3.0),))
assert S(-3.0).as_coeff_mul(rational=False) == (-S(3.0), ())
assert x.as_coeff_mul() == (1, (x,))
assert (-x).as_coeff_mul() == (-1, (x,))
assert (2*x).as_coeff_mul() == (2, (x,))
assert (x*y).as_coeff_mul(y) == (x, (y,))
assert (3 + x).as_coeff_mul() == (1, (3 + x,))
assert (3 + x).as_coeff_mul(y) == (3 + x, ())
# don't do expansion
e = exp(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
e = 2**(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
assert (1.1*x).as_coeff_mul(rational=False) == (1.1, (x,))
assert (1.1*x).as_coeff_mul() == (1, (1.1, x))
assert (-oo*x).as_coeff_mul(rational=True) == (-1, (oo, x))
def test_as_coeff_exponent():
assert (3*x**4).as_coeff_exponent(x) == (3, 4)
assert (2*x**3).as_coeff_exponent(x) == (2, 3)
assert (4*x**2).as_coeff_exponent(x) == (4, 2)
assert (6*x**1).as_coeff_exponent(x) == (6, 1)
assert (3*x**0).as_coeff_exponent(x) == (3, 0)
assert (2*x**0).as_coeff_exponent(x) == (2, 0)
assert (1*x**0).as_coeff_exponent(x) == (1, 0)
assert (0*x**0).as_coeff_exponent(x) == (0, 0)
assert (-1*x**0).as_coeff_exponent(x) == (-1, 0)
assert (-2*x**0).as_coeff_exponent(x) == (-2, 0)
assert (2*x**3 + pi*x**3).as_coeff_exponent(x) == (2 + pi, 3)
assert (x*log(2)/(2*x + pi*x)).as_coeff_exponent(x) == \
(log(2)/(2 + pi), 0)
# issue 4784
D = Derivative
fx = D(f(x), x)
assert fx.as_coeff_exponent(f(x)) == (fx, 0)
def test_extractions():
for base in (2, S.Exp1):
assert Pow(base**x, 3, evaluate=False
).extract_multiplicatively(base**x) == base**(2*x)
assert (base**(5*x)).extract_multiplicatively(
base**(3*x)) == base**(2*x)
assert ((x*y)**3).extract_multiplicatively(x**2 * y) == x*y**2
assert ((x*y)**3).extract_multiplicatively(x**4 * y) is None
assert (2*x).extract_multiplicatively(2) == x
assert (2*x).extract_multiplicatively(3) is None
assert (2*x).extract_multiplicatively(-1) is None
assert (S.Half*x).extract_multiplicatively(3) == x/6
assert (sqrt(x)).extract_multiplicatively(x) is None
assert (sqrt(x)).extract_multiplicatively(1/x) is None
assert x.extract_multiplicatively(-x) is None
assert (-2 - 4*I).extract_multiplicatively(-2) == 1 + 2*I
assert (-2 - 4*I).extract_multiplicatively(3) is None
assert (-2*x - 4*y - 8).extract_multiplicatively(-2) == x + 2*y + 4
assert (-2*x*y - 4*x**2*y).extract_multiplicatively(-2*y) == 2*x**2 + x
assert (2*x*y + 4*x**2*y).extract_multiplicatively(2*y) == 2*x**2 + x
assert (-4*y**2*x).extract_multiplicatively(-3*y) is None
assert (2*x).extract_multiplicatively(1) == 2*x
assert (-oo).extract_multiplicatively(5) is -oo
assert (oo).extract_multiplicatively(5) is oo
assert ((x*y)**3).extract_additively(1) is None
assert (x + 1).extract_additively(x) == 1
assert (x + 1).extract_additively(2*x) is None
assert (x + 1).extract_additively(-x) is None
assert (-x + 1).extract_additively(2*x) is None
assert (2*x + 3).extract_additively(x) == x + 3
assert (2*x + 3).extract_additively(2) == 2*x + 1
assert (2*x + 3).extract_additively(3) == 2*x
assert (2*x + 3).extract_additively(-2) is None
assert (2*x + 3).extract_additively(3*x) is None
assert (2*x + 3).extract_additively(2*x) == 3
assert x.extract_additively(0) == x
assert S(2).extract_additively(x) is None
assert S(2.).extract_additively(2) is S.Zero
assert S(2*x + 3).extract_additively(x + 1) == x + 2
assert S(2*x + 3).extract_additively(y + 1) is None
assert S(2*x - 3).extract_additively(x + 1) is None
assert S(2*x - 3).extract_additively(y + z) is None
assert ((a + 1)*x*4 + y).extract_additively(x).expand() == \
4*a*x + 3*x + y
assert ((a + 1)*x*4 + 3*y).extract_additively(x + 2*y).expand() == \
4*a*x + 3*x + y
assert (y*(x + 1)).extract_additively(x + 1) is None
assert ((y + 1)*(x + 1) + 3).extract_additively(x + 1) == \
y*(x + 1) + 3
assert ((x + y)*(x + 1) + x + y + 3).extract_additively(x + y) == \
x*(x + y) + 3
assert (x + y + 2*((x + y)*(x + 1)) + 3).extract_additively((x + y)*(x + 1)) == \
x + y + (x + 1)*(x + y) + 3
assert ((y + 1)*(x + 2*y + 1) + 3).extract_additively(y + 1) == \
(x + 2*y)*(y + 1) + 3
assert (-x - x*I).extract_additively(-x) == -I*x
# extraction does not leave artificats, now
assert (4*x*(y + 1) + y).extract_additively(x) == x*(4*y + 3) + y
n = Symbol("n", integer=True)
assert (Integer(-3)).could_extract_minus_sign() is True
assert (-n*x + x).could_extract_minus_sign() != \
(n*x - x).could_extract_minus_sign()
assert (x - y).could_extract_minus_sign() != \
(-x + y).could_extract_minus_sign()
assert (1 - x - y).could_extract_minus_sign() is True
assert (1 - x + y).could_extract_minus_sign() is False
assert ((-x - x*y)/y).could_extract_minus_sign() is False
assert ((x + x*y)/(-y)).could_extract_minus_sign() is True
assert ((x + x*y)/y).could_extract_minus_sign() is False
assert ((-x - y)/(x + y)).could_extract_minus_sign() is False
class sign_invariant(Function, Expr):
nargs = 1
def __neg__(self):
return self
foo = sign_invariant(x)
assert foo == -foo
assert foo.could_extract_minus_sign() is False
assert (x - y).could_extract_minus_sign() is False
assert (-x + y).could_extract_minus_sign() is True
assert (x - 1).could_extract_minus_sign() is False
assert (1 - x).could_extract_minus_sign() is True
assert (sqrt(2) - 1).could_extract_minus_sign() is True
assert (1 - sqrt(2)).could_extract_minus_sign() is False
# check that result is canonical
eq = (3*x + 15*y).extract_multiplicatively(3)
assert eq.args == eq.func(*eq.args).args
def test_nan_extractions():
for r in (1, 0, I, nan):
assert nan.extract_additively(r) is None
assert nan.extract_multiplicatively(r) is None
def test_coeff():
assert (x + 1).coeff(x + 1) == 1
assert (3*x).coeff(0) == 0
assert (z*(1 + x)*x**2).coeff(1 + x) == z*x**2
assert (1 + 2*x*x**(1 + x)).coeff(x*x**(1 + x)) == 2
assert (1 + 2*x**(y + z)).coeff(x**(y + z)) == 2
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (3 + 2*x + 4*x**2).coeff(-1) == 0
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (-x/8 + x*y).coeff(x) == Rational(-1, 8) + y
assert (-x/8 + x*y).coeff(-x) == S.One/8
assert (4*x).coeff(2*x) == 0
assert (2*x).coeff(2*x) == 1
assert (-oo*x).coeff(x*oo) == -1
assert (10*x).coeff(x, 0) == 0
assert (10*x).coeff(10*x, 0) == 0
n1, n2 = symbols('n1 n2', commutative=False)
assert (n1*n2).coeff(n1) == 1
assert (n1*n2).coeff(n2) == n1
assert (n1*n2 + x*n1).coeff(n1) == 1 # 1*n1*(n2+x)
assert (n2*n1 + x*n1).coeff(n1) == n2 + x
assert (n2*n1 + x*n1**2).coeff(n1) == n2
assert (n1**x).coeff(n1) == 0
assert (n1*n2 + n2*n1).coeff(n1) == 0
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=1) == n2
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=0) == 2
assert (2*f(x) + 3*f(x).diff(x)).coeff(f(x)) == 2
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr.coeff(x + y) == 0
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
assert (x + y + 3*z).coeff(1) == x + y
assert (-x + 2*y).coeff(-1) == x
assert (x - 2*y).coeff(-1) == 2*y
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (-x - 2*y).coeff(2) == -y
assert (x + sqrt(2)*x).coeff(sqrt(2)) == x
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (z*(x + y)**2).coeff((x + y)**2) == z
assert (z*(x + y)**2).coeff(x + y) == 0
assert (2 + 2*x + (x + 1)*y).coeff(x + 1) == y
assert (x + 2*y + 3).coeff(1) == x
assert (x + 2*y + 3).coeff(x, 0) == 2*y + 3
assert (x**2 + 2*y + 3*x).coeff(x**2, 0) == 2*y + 3*x
assert x.coeff(0, 0) == 0
assert x.coeff(x, 0) == 0
n, m, o, l = symbols('n m o l', commutative=False)
assert n.coeff(n) == 1
assert y.coeff(n) == 0
assert (3*n).coeff(n) == 3
assert (2 + n).coeff(x*m) == 0
assert (2*x*n*m).coeff(x) == 2*n*m
assert (2 + n).coeff(x*m*n + y) == 0
assert (2*x*n*m).coeff(3*n) == 0
assert (n*m + m*n*m).coeff(n) == 1 + m
assert (n*m + m*n*m).coeff(n, right=True) == m # = (1 + m)*n*m
assert (n*m + m*n).coeff(n) == 0
assert (n*m + o*m*n).coeff(m*n) == o
assert (n*m + o*m*n).coeff(m*n, right=True) == 1
assert (n*m + n*m*n).coeff(n*m, right=True) == 1 + n # = n*m*(n + 1)
assert (x*y).coeff(z, 0) == x*y
assert (x*n + y*n + z*m).coeff(n) == x + y
assert (n*m + n*o + o*l).coeff(n, right=True) == m + o
assert (x*n*m*n + y*n*m*o + z*l).coeff(m, right=True) == x*n + y*o
assert (x*n*m*n + x*n*m*o + z*l).coeff(m, right=True) == n + o
assert (x*n*m*n + x*n*m*o + z*l).coeff(m) == x*n
def test_coeff2():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r)) == 2/r
def test_coeff2_0():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r, 2)) == 1
def test_coeff_expand():
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
def test_integrate():
assert x.integrate(x) == x**2/2
assert x.integrate((x, 0, 1)) == S.Half
def test_as_base_exp():
assert x.as_base_exp() == (x, S.One)
assert (x*y*z).as_base_exp() == (x*y*z, S.One)
assert (x + y + z).as_base_exp() == (x + y + z, S.One)
assert ((x + y)**z).as_base_exp() == (x + y, z)
def test_issue_4963():
assert hasattr(Mul(x, y), "is_commutative")
assert hasattr(Mul(x, y, evaluate=False), "is_commutative")
assert hasattr(Pow(x, y), "is_commutative")
assert hasattr(Pow(x, y, evaluate=False), "is_commutative")
expr = Mul(Pow(2, 2, evaluate=False), 3, evaluate=False) + 1
assert hasattr(expr, "is_commutative")
def test_action_verbs():
assert nsimplify(1/(exp(3*pi*x/5) + 1)) == \
(1/(exp(3*pi*x/5) + 1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep=True)
assert radsimp(1/(2 + sqrt(2))) == (1/(2 + sqrt(2))).radsimp()
assert radsimp(1/(a + b*sqrt(c)), symbolic=False) == \
(1/(a + b*sqrt(c))).radsimp(symbolic=False)
assert powsimp(x**y*x**z*y**z, combine='all') == \
(x**y*x**z*y**z).powsimp(combine='all')
assert (x**t*y**t).powsimp(force=True) == (x*y)**t
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == \
(a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y + 2)/(y + 1), y) == (y/(y + 2)/(y + 1)).apart(y)
assert combsimp(y/(x + 2)/(x + 1)) == (y/(x + 2)/(x + 1)).combsimp()
assert gammasimp(gamma(x)/gamma(x-5)) == (gamma(x)/gamma(x-5)).gammasimp()
assert factor(x**2 + 5*x + 6) == (x**2 + 5*x + 6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2 + 5*x + 6)/(x + 2)) == ((x**2 + 5*x + 6)/(x + 2)).cancel()
def test_as_powers_dict():
assert x.as_powers_dict() == {x: 1}
assert (x**y*z).as_powers_dict() == {x: y, z: 1}
assert Mul(2, 2, evaluate=False).as_powers_dict() == {S(2): S(2)}
assert (x*y).as_powers_dict()[z] == 0
assert (x + y).as_powers_dict()[z] == 0
def test_as_coefficients_dict():
check = [S.One, x, y, x*y, 1]
assert [Add(3*x, 2*x, y, 3).as_coefficients_dict()[i] for i in check] == \
[3, 5, 1, 0, 3]
assert [Add(3*x, 2*x, y, 3, evaluate=False).as_coefficients_dict()[i]
for i in check] == [3, 5, 1, 0, 3]
assert [(3*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3, 0]
assert [(3.0*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3.0, 0]
assert (3.0*x*y).as_coefficients_dict()[3.0*x*y] == 0
def test_args_cnc():
A = symbols('A', commutative=False)
assert (x + A).args_cnc() == \
[[], [x + A]]
assert (x + a).args_cnc() == \
[[a + x], []]
assert (x*a).args_cnc() == \
[[a, x], []]
assert (x*y*A*(A + 1)).args_cnc(cset=True) == \
[{x, y}, [A, 1 + A]]
assert Mul(x, x, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x}, []]
assert Mul(x, x**2, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x, x**2}, []]
raises(ValueError, lambda: Mul(x, x, evaluate=False).args_cnc(cset=True))
assert Mul(x, y, x, evaluate=False).args_cnc() == \
[[x, y, x], []]
# always split -1 from leading number
assert (-1.*x).args_cnc() == [[-1, 1.0, x], []]
def test_new_rawargs():
n = Symbol('n', commutative=False)
a = x + n
assert a.is_commutative is False
assert a._new_rawargs(x).is_commutative
assert a._new_rawargs(x, y).is_commutative
assert a._new_rawargs(x, n).is_commutative is False
assert a._new_rawargs(x, y, n).is_commutative is False
m = x*n
assert m.is_commutative is False
assert m._new_rawargs(x).is_commutative
assert m._new_rawargs(n).is_commutative is False
assert m._new_rawargs(x, y).is_commutative
assert m._new_rawargs(x, n).is_commutative is False
assert m._new_rawargs(x, y, n).is_commutative is False
assert m._new_rawargs(x, n, reeval=False).is_commutative is False
assert m._new_rawargs(S.One) is S.One
def test_issue_5226():
assert Add(evaluate=False) == 0
assert Mul(evaluate=False) == 1
assert Mul(x + y, evaluate=False).is_Add
def test_free_symbols():
# free_symbols should return the free symbols of an object
assert S.One.free_symbols == set()
assert x.free_symbols == {x}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert (-Integral(x, (x, 1, y))).free_symbols == {y}
assert meter.free_symbols == set()
assert (meter**x).free_symbols == {x}
def test_has_free():
assert x.has_free(x)
assert not x.has_free(y)
assert (x + y).has_free(x)
assert (x + y).has_free(*(x, z))
assert f(x).has_free(x)
assert f(x).has_free(f(x))
assert Integral(f(x), (f(x), 1, y)).has_free(y)
assert not Integral(f(x), (f(x), 1, y)).has_free(x)
assert not Integral(f(x), (f(x), 1, y)).has_free(f(x))
def test_issue_5300():
x = Symbol('x', commutative=False)
assert x*sqrt(2)/sqrt(6) == x*sqrt(3)/3
def test_floordiv():
from sympy.functions.elementary.integers import floor
assert x // y == floor(x / y)
def test_as_coeff_Mul():
assert Integer(3).as_coeff_Mul() == (Integer(3), Integer(1))
assert Rational(3, 4).as_coeff_Mul() == (Rational(3, 4), Integer(1))
assert Float(5.0).as_coeff_Mul() == (Float(5.0), Integer(1))
assert (Integer(3)*x).as_coeff_Mul() == (Integer(3), x)
assert (Rational(3, 4)*x).as_coeff_Mul() == (Rational(3, 4), x)
assert (Float(5.0)*x).as_coeff_Mul() == (Float(5.0), x)
assert (Integer(3)*x*y).as_coeff_Mul() == (Integer(3), x*y)
assert (Rational(3, 4)*x*y).as_coeff_Mul() == (Rational(3, 4), x*y)
assert (Float(5.0)*x*y).as_coeff_Mul() == (Float(5.0), x*y)
assert (x).as_coeff_Mul() == (S.One, x)
assert (x*y).as_coeff_Mul() == (S.One, x*y)
assert (-oo*x).as_coeff_Mul(rational=True) == (-1, oo*x)
def test_as_coeff_Add():
assert Integer(3).as_coeff_Add() == (Integer(3), Integer(0))
assert Rational(3, 4).as_coeff_Add() == (Rational(3, 4), Integer(0))
assert Float(5.0).as_coeff_Add() == (Float(5.0), Integer(0))
assert (Integer(3) + x).as_coeff_Add() == (Integer(3), x)
assert (Rational(3, 4) + x).as_coeff_Add() == (Rational(3, 4), x)
assert (Float(5.0) + x).as_coeff_Add() == (Float(5.0), x)
assert (Float(5.0) + x).as_coeff_Add(rational=True) == (0, Float(5.0) + x)
assert (Integer(3) + x + y).as_coeff_Add() == (Integer(3), x + y)
assert (Rational(3, 4) + x + y).as_coeff_Add() == (Rational(3, 4), x + y)
assert (Float(5.0) + x + y).as_coeff_Add() == (Float(5.0), x + y)
assert (x).as_coeff_Add() == (S.Zero, x)
assert (x*y).as_coeff_Add() == (S.Zero, x*y)
def test_expr_sorting():
exprs = [1/x**2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)**3, x**2]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x, 2*x, 2*x**2, 2*x**3, x**n, 2*x**n, sin(x), sin(x)**n,
sin(x**2), cos(x), cos(x**2), tan(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x + 1, x**2 + x + 1, x**3 + x**2 + x + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [S(4), x - 3*I/2, x + 3*I/2, x - 4*I + 1, x + 4*I + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[3], [1, 2]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [1, 2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{x: -y}, {x: y}]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{1}, {1, 2}]
assert sorted(exprs, key=default_sort_key) == exprs
a, b = exprs = [Dummy('x'), Dummy('x')]
assert sorted([b, a], key=default_sort_key) == exprs
def test_as_ordered_factors():
assert x.as_ordered_factors() == [x]
assert (2*x*x**n*sin(x)*cos(x)).as_ordered_factors() \
== [Integer(2), x, x**n, sin(x), cos(x)]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Mul(*args)
assert expr.as_ordered_factors() == args
A, B = symbols('A,B', commutative=False)
assert (A*B).as_ordered_factors() == [A, B]
assert (B*A).as_ordered_factors() == [B, A]
def test_as_ordered_terms():
assert x.as_ordered_terms() == [x]
assert (sin(x)**2*cos(x) + sin(x)*cos(x)**2 + 1).as_ordered_terms() \
== [sin(x)**2*cos(x), sin(x)*cos(x)**2, 1]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Add(*args)
assert expr.as_ordered_terms() == args
assert (1 + 4*sqrt(3)*pi*x).as_ordered_terms() == [4*pi*x*sqrt(3), 1]
assert ( 2 + 3*I).as_ordered_terms() == [2, 3*I]
assert (-2 + 3*I).as_ordered_terms() == [-2, 3*I]
assert ( 2 - 3*I).as_ordered_terms() == [2, -3*I]
assert (-2 - 3*I).as_ordered_terms() == [-2, -3*I]
assert ( 4 + 3*I).as_ordered_terms() == [4, 3*I]
assert (-4 + 3*I).as_ordered_terms() == [-4, 3*I]
assert ( 4 - 3*I).as_ordered_terms() == [4, -3*I]
assert (-4 - 3*I).as_ordered_terms() == [-4, -3*I]
e = x**2*y**2 + x*y**4 + y + 2
assert e.as_ordered_terms(order="lex") == [x**2*y**2, x*y**4, y, 2]
assert e.as_ordered_terms(order="grlex") == [x*y**4, x**2*y**2, y, 2]
assert e.as_ordered_terms(order="rev-lex") == [2, y, x*y**4, x**2*y**2]
assert e.as_ordered_terms(order="rev-grlex") == [2, y, x**2*y**2, x*y**4]
k = symbols('k')
assert k.as_ordered_terms(data=True) == ([(k, ((1.0, 0.0), (1,), ()))], [k])
def test_sort_key_atomic_expr():
from sympy.physics.units import m, s
assert sorted([-m, s], key=lambda arg: arg.sort_key()) == [-m, s]
def test_eval_interval():
assert exp(x)._eval_interval(*Tuple(x, 0, 1)) == exp(1) - exp(0)
# issue 4199
a = x/y
raises(NotImplementedError, lambda: a._eval_interval(x, S.Zero, oo)._eval_interval(y, oo, S.Zero))
raises(NotImplementedError, lambda: a._eval_interval(x, S.Zero, oo)._eval_interval(y, S.Zero, oo))
a = x - y
raises(NotImplementedError, lambda: a._eval_interval(x, S.One, oo)._eval_interval(y, oo, S.One))
raises(ValueError, lambda: x._eval_interval(x, None, None))
a = -y*Heaviside(x - y)
assert a._eval_interval(x, -oo, oo) == -y
assert a._eval_interval(x, oo, -oo) == y
def test_eval_interval_zoo():
# Test that limit is used when zoo is returned
assert Si(1/x)._eval_interval(x, S.Zero, S.One) == -pi/2 + Si(1)
def test_primitive():
assert (3*(x + 1)**2).primitive() == (3, (x + 1)**2)
assert (6*x + 2).primitive() == (2, 3*x + 1)
assert (x/2 + 3).primitive() == (S.Half, x + 6)
eq = (6*x + 2)*(x/2 + 3)
assert eq.primitive()[0] == 1
eq = (2 + 2*x)**2
assert eq.primitive()[0] == 1
assert (4.0*x).primitive() == (1, 4.0*x)
assert (4.0*x + y/2).primitive() == (S.Half, 8.0*x + y)
assert (-2*x).primitive() == (2, -x)
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).primitive() == \
(S.One/14, 7.0*x + 21*y + 10*z)
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).primitive() == \
(S.One/3, i + x)
assert (S.Infinity + 2*x/3 + 4*y/7).primitive() == \
(S.One/21, 14*x + 12*y + oo)
assert S.Zero.primitive() == (S.One, S.Zero)
def test_issue_5843():
a = 1 + x
assert (2*a).extract_multiplicatively(a) == 2
assert (4*a).extract_multiplicatively(2*a) == 2
assert ((3*a)*(2*a)).extract_multiplicatively(a) == 6*a
def test_is_constant():
from sympy.solvers.solvers import checksol
assert Sum(x, (x, 1, 10)).is_constant() is True
assert Sum(x, (x, 1, n)).is_constant() is False
assert Sum(x, (x, 1, n)).is_constant(y) is True
assert Sum(x, (x, 1, n)).is_constant(n) is False
assert Sum(x, (x, 1, n)).is_constant(x) is True
eq = a*cos(x)**2 + a*sin(x)**2 - a
assert eq.is_constant() is True
assert eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
assert x.is_constant() is False
assert x.is_constant(y) is True
assert log(x/y).is_constant() is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert f(1).is_constant
assert checksol(x, x, f(x)) is False
assert Pow(x, S.Zero, evaluate=False).is_constant() is True # == 1
assert Pow(S.Zero, x, evaluate=False).is_constant() is False # == 0 or 1
assert (2**x).is_constant() is False
assert Pow(S(2), S(3), evaluate=False).is_constant() is True
z1, z2 = symbols('z1 z2', zero=True)
assert (z1 + 2*z2).is_constant() is True
assert meter.is_constant() is True
assert (3*meter).is_constant() is True
assert (x*meter).is_constant() is False
def test_equals():
assert (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2).equals(0)
assert (x**2 - 1).equals((x + 1)*(x - 1))
assert (cos(x)**2 + sin(x)**2).equals(1)
assert (a*cos(x)**2 + a*sin(x)**2).equals(a)
r = sqrt(2)
assert (-1/(r + r*x) + 1/r/(1 + x)).equals(0)
assert factorial(x + 1).equals((x + 1)*factorial(x))
assert sqrt(3).equals(2*sqrt(3)) is False
assert (sqrt(5)*sqrt(3)).equals(sqrt(3)) is False
assert (sqrt(5) + sqrt(3)).equals(0) is False
assert (sqrt(5) + pi).equals(0) is False
assert meter.equals(0) is False
assert (3*meter**2).equals(0) is False
eq = -(-1)**(S(3)/4)*6**(S.One/4) + (-6)**(S.One/4)*I
if eq != 0: # if canonicalization makes this zero, skip the test
assert eq.equals(0)
assert sqrt(x).equals(0) is False
# from integrate(x*sqrt(1 + 2*x), x);
# diff is zero only when assumptions allow
i = 2*sqrt(2)*x**(S(5)/2)*(1 + 1/(2*x))**(S(5)/2)/5 + \
2*sqrt(2)*x**(S(3)/2)*(1 + 1/(2*x))**(S(5)/2)/(-6 - 3/x)
ans = sqrt(2*x + 1)*(6*x**2 + x - 1)/15
diff = i - ans
assert diff.equals(0) is None # should be False, but previously this was False due to wrong intermediate result
assert diff.subs(x, Rational(-1, 2)/2) == 7*sqrt(2)/120
# there are regions for x for which the expression is True, for
# example, when x < -1/2 or x > 0 the expression is zero
p = Symbol('p', positive=True)
assert diff.subs(x, p).equals(0) is True
assert diff.subs(x, -1).equals(0) is True
# prove via minimal_polynomial or self-consistency
eq = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert eq.equals(0)
q = 3**Rational(1, 3) + 3
p = expand(q**3)**Rational(1, 3)
assert (p - q).equals(0)
# issue 6829
# eq = q*x + q/4 + x**4 + x**3 + 2*x**2 - S.One/3
# z = eq.subs(x, solve(eq, x)[0])
q = symbols('q')
z = (q*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12)/2 - sqrt((2*q - S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/6)/2 - S.One/4) + q/4 + (-sqrt(-2*(-(q
- S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q
- S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**4 + (-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**3 + 2*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**2 - Rational(1, 3))
assert z.equals(0)
def test_random():
from sympy.functions.combinatorial.numbers import lucas
from sympy.simplify.simplify import posify
assert posify(x)[0]._random() is not None
assert lucas(n)._random(2, -2, 0, -1, 1) is None
# issue 8662
assert Piecewise((Max(x, y), z))._random() is None
def test_round():
assert str(Float('0.1249999').round(2)) == '0.12'
d20 = 12345678901234567890
ans = S(d20).round(2)
assert ans.is_Integer and ans == d20
ans = S(d20).round(-2)
assert ans.is_Integer and ans == 12345678901234567900
assert str(S('1/7').round(4)) == '0.1429'
assert str(S('.[12345]').round(4)) == '0.1235'
assert str(S('.1349').round(2)) == '0.13'
n = S(12345)
ans = n.round()
assert ans.is_Integer
assert ans == n
ans = n.round(1)
assert ans.is_Integer
assert ans == n
ans = n.round(4)
assert ans.is_Integer
assert ans == n
assert n.round(-1) == 12340
r = Float(str(n)).round(-4)
assert r == 10000
assert n.round(-5) == 0
assert str((pi + sqrt(2)).round(2)) == '4.56'
assert (10*(pi + sqrt(2))).round(-1) == 50
raises(TypeError, lambda: round(x + 2, 2))
assert str(S(2.3).round(1)) == '2.3'
# rounding in SymPy (as in Decimal) should be
# exact for the given precision; we check here
# that when a 5 follows the last digit that
# the rounded digit will be even.
for i in range(-99, 100):
# construct a decimal that ends in 5, e.g. 123 -> 0.1235
s = str(abs(i))
p = len(s) # we are going to round to the last digit of i
n = '0.%s5' % s # put a 5 after i's digits
j = p + 2 # 2 for '0.'
if i < 0: # 1 for '-'
j += 1
n = '-' + n
v = str(Float(n).round(p))[:j] # pertinent digits
if v.endswith('.'):
continue # it ends with 0 which is even
L = int(v[-1]) # last digit
assert L % 2 == 0, (n, '->', v)
assert (Float(.3, 3) + 2*pi).round() == 7
assert (Float(.3, 3) + 2*pi*100).round() == 629
assert (pi + 2*E*I).round() == 3 + 5*I
# don't let request for extra precision give more than
# what is known (in this case, only 3 digits)
assert str((Float(.03, 3) + 2*pi/100).round(5)) == '0.0928'
assert str((Float(.03, 3) + 2*pi/100).round(4)) == '0.0928'
assert S.Zero.round() == 0
a = (Add(1, Float('1.' + '9'*27, ''), evaluate=0))
assert a.round(10) == Float('3.0000000000', '')
assert a.round(25) == Float('3.0000000000000000000000000', '')
assert a.round(26) == Float('3.00000000000000000000000000', '')
assert a.round(27) == Float('2.999999999999999999999999999', '')
assert a.round(30) == Float('2.999999999999999999999999999', '')
raises(TypeError, lambda: x.round())
raises(TypeError, lambda: f(1).round())
# exact magnitude of 10
assert str(S.One.round()) == '1'
assert str(S(100).round()) == '100'
# applied to real and imaginary portions
assert (2*pi + E*I).round() == 6 + 3*I
assert (2*pi + I/10).round() == 6
assert (pi/10 + 2*I).round() == 2*I
# the lhs re and im parts are Float with dps of 2
# and those on the right have dps of 15 so they won't compare
# equal unless we use string or compare components (which will
# then coerce the floats to the same precision) or re-create
# the floats
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
assert str((pi/10 + E*I).round(2).as_real_imag()) == '(0.31, 2.72)'
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
# issue 6914
assert (I**(I + 3)).round(3) == Float('-0.208', '')*I
# issue 8720
assert S(-123.6).round() == -124
assert S(-1.5).round() == -2
assert S(-100.5).round() == -100
assert S(-1.5 - 10.5*I).round() == -2 - 10*I
# issue 7961
assert str(S(0.006).round(2)) == '0.01'
assert str(S(0.00106).round(4)) == '0.0011'
# issue 8147
assert S.NaN.round() is S.NaN
assert S.Infinity.round() is S.Infinity
assert S.NegativeInfinity.round() is S.NegativeInfinity
assert S.ComplexInfinity.round() is S.ComplexInfinity
# check that types match
for i in range(2):
fi = float(i)
# 2 args
assert all(type(round(i, p)) is int for p in (-1, 0, 1))
assert all(S(i).round(p).is_Integer for p in (-1, 0, 1))
assert all(type(round(fi, p)) is float for p in (-1, 0, 1))
assert all(S(fi).round(p).is_Float for p in (-1, 0, 1))
# 1 arg (p is None)
assert type(round(i)) is int
assert S(i).round().is_Integer
assert type(round(fi)) is int
assert S(fi).round().is_Integer
def test_held_expression_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
e1 = x*he
assert isinstance(e1, Mul)
assert e1.args == (x, he)
assert e1.doit() == 1
assert UnevaluatedExpr(Derivative(x, x)).doit(deep=False
) == Derivative(x, x)
assert UnevaluatedExpr(Derivative(x, x)).doit() == 1
xx = Mul(x, x, evaluate=False)
assert xx != x**2
ue2 = UnevaluatedExpr(xx)
assert isinstance(ue2, UnevaluatedExpr)
assert ue2.args == (xx,)
assert ue2.doit() == x**2
assert ue2.doit(deep=False) == xx
x2 = UnevaluatedExpr(2)*2
assert type(x2) is Mul
assert x2.args == (2, UnevaluatedExpr(2))
def test_round_exception_nostr():
# Don't use the string form of the expression in the round exception, as
# it's too slow
s = Symbol('bad')
try:
s.round()
except TypeError as e:
assert 'bad' not in str(e)
else:
# Did not raise
raise AssertionError("Did not raise")
def test_extract_branch_factor():
assert exp_polar(2.0*I*pi).extract_branch_factor() == (1, 1)
def test_identity_removal():
assert Add.make_args(x + 0) == (x,)
assert Mul.make_args(x*1) == (x,)
def test_float_0():
assert Float(0.0) + 1 == Float(1.0)
@XFAIL
def test_float_0_fail():
assert Float(0.0)*x == Float(0.0)
assert (x + Float(0.0)).is_Add
def test_issue_6325():
ans = (b**2 + z**2 - (b*(a + b*t) + z*(c + t*z))**2/(
(a + b*t)**2 + (c + t*z)**2))/sqrt((a + b*t)**2 + (c + t*z)**2)
e = sqrt((a + b*t)**2 + (c + z*t)**2)
assert diff(e, t, 2) == ans
assert e.diff(t, 2) == ans
assert diff(e, t, 2, simplify=False) != ans
def test_issue_7426():
f1 = a % c
f2 = x % z
assert f1.equals(f2) is None
def test_issue_11122():
x = Symbol('x', extended_positive=False)
assert unchanged(Gt, x, 0) # (x > 0)
# (x > 0) should remain unevaluated after PR #16956
x = Symbol('x', positive=False, real=True)
assert (x > 0) is S.false
def test_issue_10651():
x = Symbol('x', real=True)
e1 = (-1 + x)/(1 - x)
e3 = (4*x**2 - 4)/((1 - x)*(1 + x))
e4 = 1/(cos(x)**2) - (tan(x))**2
x = Symbol('x', positive=True)
e5 = (1 + x)/x
assert e1.is_constant() is None
assert e3.is_constant() is None
assert e4.is_constant() is None
assert e5.is_constant() is False
def test_issue_10161():
x = symbols('x', real=True)
assert x*abs(x)*abs(x) == x**3
def test_issue_10755():
x = symbols('x')
raises(TypeError, lambda: int(log(x)))
raises(TypeError, lambda: log(x).round(2))
def test_issue_11877():
x = symbols('x')
assert integrate(log(S.Half - x), (x, 0, S.Half)) == Rational(-1, 2) -log(2)/2
def test_normal():
x = symbols('x')
e = Mul(S.Half, 1 + x, evaluate=False)
assert e.normal() == e
def test_expr():
x = symbols('x')
raises(TypeError, lambda: tan(x).series(x, 2, oo, "+"))
def test_ExprBuilder():
eb = ExprBuilder(Mul)
eb.args.extend([x, x])
assert eb.build() == x**2
def test_issue_22020():
from sympy.parsing.sympy_parser import parse_expr
x = parse_expr("log((2*V/3-V)/C)/-(R+r)*C")
y = parse_expr("log((2*V/3-V)/C)/-(R+r)*2")
assert x.equals(y) is False
def test_non_string_equality():
# Expressions should not compare equal to strings
x = symbols('x')
one = sympify(1)
assert (x == 'x') is False
assert (x != 'x') is True
assert (one == '1') is False
assert (one != '1') is True
assert (x + 1 == 'x + 1') is False
assert (x + 1 != 'x + 1') is True
# Make sure == doesn't try to convert the resulting expression to a string
# (e.g., by calling sympify() instead of _sympify())
class BadRepr:
def __repr__(self):
raise RuntimeError
assert (x == BadRepr()) is False
assert (x != BadRepr()) is True
def test_21494():
from sympy.testing.pytest import warns_deprecated_sympy
with warns_deprecated_sympy():
assert x.expr_free_symbols == {x}
with warns_deprecated_sympy():
assert Basic().expr_free_symbols == set()
with warns_deprecated_sympy():
assert S(2).expr_free_symbols == {S(2)}
with warns_deprecated_sympy():
assert Indexed("A", x).expr_free_symbols == {Indexed("A", x)}
with warns_deprecated_sympy():
assert Subs(x, x, 0).expr_free_symbols == set()
def test_Expr__eq__iterable_handling():
assert x != range(3)
|
ee9bcd504188711adf5a5ddda2a1e59b701d3641a570284eebd828b37164d3d0 | from sympy.testing.pytest import warns_deprecated_sympy
def test_compatibility_submodule():
# Test the sympy.core.compatibility deprecation warning
with warns_deprecated_sympy():
import sympy.core.compatibility # noqa:F401
|
6b3d875df5a07683268ca94189de44db0261155e4d7ef4dcba240b9021b43c45 | """Test whether all elements of cls.args are instances of Basic. """
# NOTE: keep tests sorted by (module, class name) key. If a class can't
# be instantiated, add it here anyway with @SKIP("abstract class) (see
# e.g. Function).
import os
import re
from sympy.assumptions.ask import Q
from sympy.core.basic import Basic
from sympy.core.function import (Function, Lambda)
from sympy.core.numbers import (Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import sin
from sympy.testing.pytest import SKIP
a, b, c, x, y, z = symbols('a,b,c,x,y,z')
whitelist = [
"sympy.assumptions.predicates", # tested by test_predicates()
"sympy.assumptions.relation.equality", # tested by test_predicates()
]
def test_all_classes_are_tested():
this = os.path.split(__file__)[0]
path = os.path.join(this, os.pardir, os.pardir)
sympy_path = os.path.abspath(path)
prefix = os.path.split(sympy_path)[0] + os.sep
re_cls = re.compile(r"^class ([A-Za-z][A-Za-z0-9_]*)\s*\(", re.MULTILINE)
modules = {}
for root, dirs, files in os.walk(sympy_path):
module = root.replace(prefix, "").replace(os.sep, ".")
for file in files:
if file.startswith(("_", "test_", "bench_")):
continue
if not file.endswith(".py"):
continue
with open(os.path.join(root, file), encoding='utf-8') as f:
text = f.read()
submodule = module + '.' + file[:-3]
if any(submodule.startswith(wpath) for wpath in whitelist):
continue
names = re_cls.findall(text)
if not names:
continue
try:
mod = __import__(submodule, fromlist=names)
except ImportError:
continue
def is_Basic(name):
cls = getattr(mod, name)
if hasattr(cls, '_sympy_deprecated_func'):
cls = cls._sympy_deprecated_func
if not isinstance(cls, type):
# check instance of singleton class with same name
cls = type(cls)
return issubclass(cls, Basic)
names = list(filter(is_Basic, names))
if names:
modules[submodule] = names
ns = globals()
failed = []
for module, names in modules.items():
mod = module.replace('.', '__')
for name in names:
test = 'test_' + mod + '__' + name
if test not in ns:
failed.append(module + '.' + name)
assert not failed, "Missing classes: %s. Please add tests for these to sympy/core/tests/test_args.py." % ", ".join(failed)
def _test_args(obj):
all_basic = all(isinstance(arg, Basic) for arg in obj.args)
# Ideally obj.func(*obj.args) would always recreate the object, but for
# now, we only require it for objects with non-empty .args
recreatable = not obj.args or obj.func(*obj.args) == obj
return all_basic and recreatable
def test_sympy__algebras__quaternion__Quaternion():
from sympy.algebras.quaternion import Quaternion
assert _test_args(Quaternion(x, 1, 2, 3))
def test_sympy__assumptions__assume__AppliedPredicate():
from sympy.assumptions.assume import AppliedPredicate, Predicate
assert _test_args(AppliedPredicate(Predicate("test"), 2))
assert _test_args(Q.is_true(True))
@SKIP("abstract class")
def test_sympy__assumptions__assume__Predicate():
pass
def test_predicates():
predicates = [
getattr(Q, attr)
for attr in Q.__class__.__dict__
if not attr.startswith('__')]
for p in predicates:
assert _test_args(p)
def test_sympy__assumptions__assume__UndefinedPredicate():
from sympy.assumptions.assume import Predicate
assert _test_args(Predicate("test"))
@SKIP('abstract class')
def test_sympy__assumptions__relation__binrel__BinaryRelation():
pass
def test_sympy__assumptions__relation__binrel__AppliedBinaryRelation():
assert _test_args(Q.eq(1, 2))
def test_sympy__assumptions__wrapper__AssumptionsWrapper():
from sympy.assumptions.wrapper import AssumptionsWrapper
assert _test_args(AssumptionsWrapper(x, Q.positive(x)))
@SKIP("abstract Class")
def test_sympy__codegen__ast__CodegenAST():
from sympy.codegen.ast import CodegenAST
assert _test_args(CodegenAST())
@SKIP("abstract Class")
def test_sympy__codegen__ast__AssignmentBase():
from sympy.codegen.ast import AssignmentBase
assert _test_args(AssignmentBase(x, 1))
@SKIP("abstract Class")
def test_sympy__codegen__ast__AugmentedAssignment():
from sympy.codegen.ast import AugmentedAssignment
assert _test_args(AugmentedAssignment(x, 1))
def test_sympy__codegen__ast__AddAugmentedAssignment():
from sympy.codegen.ast import AddAugmentedAssignment
assert _test_args(AddAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__SubAugmentedAssignment():
from sympy.codegen.ast import SubAugmentedAssignment
assert _test_args(SubAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__MulAugmentedAssignment():
from sympy.codegen.ast import MulAugmentedAssignment
assert _test_args(MulAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__DivAugmentedAssignment():
from sympy.codegen.ast import DivAugmentedAssignment
assert _test_args(DivAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__ModAugmentedAssignment():
from sympy.codegen.ast import ModAugmentedAssignment
assert _test_args(ModAugmentedAssignment(x, 1))
def test_sympy__codegen__ast__CodeBlock():
from sympy.codegen.ast import CodeBlock, Assignment
assert _test_args(CodeBlock(Assignment(x, 1), Assignment(y, 2)))
def test_sympy__codegen__ast__For():
from sympy.codegen.ast import For, CodeBlock, AddAugmentedAssignment
from sympy.sets import Range
assert _test_args(For(x, Range(10), CodeBlock(AddAugmentedAssignment(y, 1))))
def test_sympy__codegen__ast__Token():
from sympy.codegen.ast import Token
assert _test_args(Token())
def test_sympy__codegen__ast__ContinueToken():
from sympy.codegen.ast import ContinueToken
assert _test_args(ContinueToken())
def test_sympy__codegen__ast__BreakToken():
from sympy.codegen.ast import BreakToken
assert _test_args(BreakToken())
def test_sympy__codegen__ast__NoneToken():
from sympy.codegen.ast import NoneToken
assert _test_args(NoneToken())
def test_sympy__codegen__ast__String():
from sympy.codegen.ast import String
assert _test_args(String('foobar'))
def test_sympy__codegen__ast__QuotedString():
from sympy.codegen.ast import QuotedString
assert _test_args(QuotedString('foobar'))
def test_sympy__codegen__ast__Comment():
from sympy.codegen.ast import Comment
assert _test_args(Comment('this is a comment'))
def test_sympy__codegen__ast__Node():
from sympy.codegen.ast import Node
assert _test_args(Node())
assert _test_args(Node(attrs={1, 2, 3}))
def test_sympy__codegen__ast__Type():
from sympy.codegen.ast import Type
assert _test_args(Type('float128'))
def test_sympy__codegen__ast__IntBaseType():
from sympy.codegen.ast import IntBaseType
assert _test_args(IntBaseType('bigint'))
def test_sympy__codegen__ast___SizedIntType():
from sympy.codegen.ast import _SizedIntType
assert _test_args(_SizedIntType('int128', 128))
def test_sympy__codegen__ast__SignedIntType():
from sympy.codegen.ast import SignedIntType
assert _test_args(SignedIntType('int128_with_sign', 128))
def test_sympy__codegen__ast__UnsignedIntType():
from sympy.codegen.ast import UnsignedIntType
assert _test_args(UnsignedIntType('unt128', 128))
def test_sympy__codegen__ast__FloatBaseType():
from sympy.codegen.ast import FloatBaseType
assert _test_args(FloatBaseType('positive_real'))
def test_sympy__codegen__ast__FloatType():
from sympy.codegen.ast import FloatType
assert _test_args(FloatType('float242', 242, nmant=142, nexp=99))
def test_sympy__codegen__ast__ComplexBaseType():
from sympy.codegen.ast import ComplexBaseType
assert _test_args(ComplexBaseType('positive_cmplx'))
def test_sympy__codegen__ast__ComplexType():
from sympy.codegen.ast import ComplexType
assert _test_args(ComplexType('complex42', 42, nmant=15, nexp=5))
def test_sympy__codegen__ast__Attribute():
from sympy.codegen.ast import Attribute
assert _test_args(Attribute('noexcept'))
def test_sympy__codegen__ast__Variable():
from sympy.codegen.ast import Variable, Type, value_const
assert _test_args(Variable(x))
assert _test_args(Variable(y, Type('float32'), {value_const}))
assert _test_args(Variable(z, type=Type('float64')))
def test_sympy__codegen__ast__Pointer():
from sympy.codegen.ast import Pointer, Type, pointer_const
assert _test_args(Pointer(x))
assert _test_args(Pointer(y, type=Type('float32')))
assert _test_args(Pointer(z, Type('float64'), {pointer_const}))
def test_sympy__codegen__ast__Declaration():
from sympy.codegen.ast import Declaration, Variable, Type
vx = Variable(x, type=Type('float'))
assert _test_args(Declaration(vx))
def test_sympy__codegen__ast__While():
from sympy.codegen.ast import While, AddAugmentedAssignment
assert _test_args(While(abs(x) < 1, [AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Scope():
from sympy.codegen.ast import Scope, AddAugmentedAssignment
assert _test_args(Scope([AddAugmentedAssignment(x, -1)]))
def test_sympy__codegen__ast__Stream():
from sympy.codegen.ast import Stream
assert _test_args(Stream('stdin'))
def test_sympy__codegen__ast__Print():
from sympy.codegen.ast import Print
assert _test_args(Print([x, y]))
assert _test_args(Print([x, y], "%d %d"))
def test_sympy__codegen__ast__FunctionPrototype():
from sympy.codegen.ast import FunctionPrototype, real, Declaration, Variable
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionPrototype(real, 'pwer', [inp_x]))
def test_sympy__codegen__ast__FunctionDefinition():
from sympy.codegen.ast import FunctionDefinition, real, Declaration, Variable, Assignment
inp_x = Declaration(Variable(x, type=real))
assert _test_args(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)]))
def test_sympy__codegen__ast__Return():
from sympy.codegen.ast import Return
assert _test_args(Return(x))
def test_sympy__codegen__ast__FunctionCall():
from sympy.codegen.ast import FunctionCall
assert _test_args(FunctionCall('pwer', [x]))
def test_sympy__codegen__ast__Element():
from sympy.codegen.ast import Element
assert _test_args(Element('x', range(3)))
def test_sympy__codegen__cnodes__CommaOperator():
from sympy.codegen.cnodes import CommaOperator
assert _test_args(CommaOperator(1, 2))
def test_sympy__codegen__cnodes__goto():
from sympy.codegen.cnodes import goto
assert _test_args(goto('early_exit'))
def test_sympy__codegen__cnodes__Label():
from sympy.codegen.cnodes import Label
assert _test_args(Label('early_exit'))
def test_sympy__codegen__cnodes__PreDecrement():
from sympy.codegen.cnodes import PreDecrement
assert _test_args(PreDecrement(x))
def test_sympy__codegen__cnodes__PostDecrement():
from sympy.codegen.cnodes import PostDecrement
assert _test_args(PostDecrement(x))
def test_sympy__codegen__cnodes__PreIncrement():
from sympy.codegen.cnodes import PreIncrement
assert _test_args(PreIncrement(x))
def test_sympy__codegen__cnodes__PostIncrement():
from sympy.codegen.cnodes import PostIncrement
assert _test_args(PostIncrement(x))
def test_sympy__codegen__cnodes__struct():
from sympy.codegen.ast import real, Variable
from sympy.codegen.cnodes import struct
assert _test_args(struct(declarations=[
Variable(x, type=real),
Variable(y, type=real)
]))
def test_sympy__codegen__cnodes__union():
from sympy.codegen.ast import float32, int32, Variable
from sympy.codegen.cnodes import union
assert _test_args(union(declarations=[
Variable(x, type=float32),
Variable(y, type=int32)
]))
def test_sympy__codegen__cxxnodes__using():
from sympy.codegen.cxxnodes import using
assert _test_args(using('std::vector'))
assert _test_args(using('std::vector', 'vec'))
def test_sympy__codegen__fnodes__Program():
from sympy.codegen.fnodes import Program
assert _test_args(Program('foobar', []))
def test_sympy__codegen__fnodes__Module():
from sympy.codegen.fnodes import Module
assert _test_args(Module('foobar', [], []))
def test_sympy__codegen__fnodes__Subroutine():
from sympy.codegen.fnodes import Subroutine
x = symbols('x', real=True)
assert _test_args(Subroutine('foo', [x], []))
def test_sympy__codegen__fnodes__GoTo():
from sympy.codegen.fnodes import GoTo
assert _test_args(GoTo([10]))
assert _test_args(GoTo([10, 20], x > 1))
def test_sympy__codegen__fnodes__FortranReturn():
from sympy.codegen.fnodes import FortranReturn
assert _test_args(FortranReturn(10))
def test_sympy__codegen__fnodes__Extent():
from sympy.codegen.fnodes import Extent
assert _test_args(Extent())
assert _test_args(Extent(None))
assert _test_args(Extent(':'))
assert _test_args(Extent(-3, 4))
assert _test_args(Extent(x, y))
def test_sympy__codegen__fnodes__use_rename():
from sympy.codegen.fnodes import use_rename
assert _test_args(use_rename('loc', 'glob'))
def test_sympy__codegen__fnodes__use():
from sympy.codegen.fnodes import use
assert _test_args(use('modfoo', only='bar'))
def test_sympy__codegen__fnodes__SubroutineCall():
from sympy.codegen.fnodes import SubroutineCall
assert _test_args(SubroutineCall('foo', ['bar', 'baz']))
def test_sympy__codegen__fnodes__Do():
from sympy.codegen.fnodes import Do
assert _test_args(Do([], 'i', 1, 42))
def test_sympy__codegen__fnodes__ImpliedDoLoop():
from sympy.codegen.fnodes import ImpliedDoLoop
assert _test_args(ImpliedDoLoop('i', 'i', 1, 42))
def test_sympy__codegen__fnodes__ArrayConstructor():
from sympy.codegen.fnodes import ArrayConstructor
assert _test_args(ArrayConstructor([1, 2, 3]))
from sympy.codegen.fnodes import ImpliedDoLoop
idl = ImpliedDoLoop('i', 'i', 1, 42)
assert _test_args(ArrayConstructor([1, idl, 3]))
def test_sympy__codegen__fnodes__sum_():
from sympy.codegen.fnodes import sum_
assert _test_args(sum_('arr'))
def test_sympy__codegen__fnodes__product_():
from sympy.codegen.fnodes import product_
assert _test_args(product_('arr'))
def test_sympy__codegen__numpy_nodes__logaddexp():
from sympy.codegen.numpy_nodes import logaddexp
assert _test_args(logaddexp(x, y))
def test_sympy__codegen__numpy_nodes__logaddexp2():
from sympy.codegen.numpy_nodes import logaddexp2
assert _test_args(logaddexp2(x, y))
def test_sympy__codegen__pynodes__List():
from sympy.codegen.pynodes import List
assert _test_args(List(1, 2, 3))
def test_sympy__codegen__pynodes__NumExprEvaluate():
from sympy.codegen.pynodes import NumExprEvaluate
assert _test_args(NumExprEvaluate(x))
def test_sympy__codegen__scipy_nodes__cosm1():
from sympy.codegen.scipy_nodes import cosm1
assert _test_args(cosm1(x))
def test_sympy__codegen__abstract_nodes__List():
from sympy.codegen.abstract_nodes import List
assert _test_args(List(1, 2, 3))
def test_sympy__combinatorics__graycode__GrayCode():
from sympy.combinatorics.graycode import GrayCode
# an integer is given and returned from GrayCode as the arg
assert _test_args(GrayCode(3, start='100'))
assert _test_args(GrayCode(3, rank=1))
def test_sympy__combinatorics__permutations__Permutation():
from sympy.combinatorics.permutations import Permutation
assert _test_args(Permutation([0, 1, 2, 3]))
def test_sympy__combinatorics__permutations__AppliedPermutation():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.permutations import AppliedPermutation
p = Permutation([0, 1, 2, 3])
assert _test_args(AppliedPermutation(p, x))
def test_sympy__combinatorics__perm_groups__PermutationGroup():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.perm_groups import PermutationGroup
assert _test_args(PermutationGroup([Permutation([0, 1])]))
def test_sympy__combinatorics__polyhedron__Polyhedron():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.polyhedron import Polyhedron
from sympy.abc import w, x, y, z
pgroup = [Permutation([[0, 1, 2], [3]]),
Permutation([[0, 1, 3], [2]]),
Permutation([[0, 2, 3], [1]]),
Permutation([[1, 2, 3], [0]]),
Permutation([[0, 1], [2, 3]]),
Permutation([[0, 2], [1, 3]]),
Permutation([[0, 3], [1, 2]]),
Permutation([[0, 1, 2, 3]])]
corners = [w, x, y, z]
faces = [(w, x, y), (w, y, z), (w, z, x), (x, y, z)]
assert _test_args(Polyhedron(corners, faces, pgroup))
def test_sympy__combinatorics__prufer__Prufer():
from sympy.combinatorics.prufer import Prufer
assert _test_args(Prufer([[0, 1], [0, 2], [0, 3]], 4))
def test_sympy__combinatorics__partitions__Partition():
from sympy.combinatorics.partitions import Partition
assert _test_args(Partition([1]))
def test_sympy__combinatorics__partitions__IntegerPartition():
from sympy.combinatorics.partitions import IntegerPartition
assert _test_args(IntegerPartition([1]))
def test_sympy__concrete__products__Product():
from sympy.concrete.products import Product
assert _test_args(Product(x, (x, 0, 10)))
assert _test_args(Product(x, (x, 0, y), (y, 0, 10)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__ExprWithLimits():
from sympy.concrete.expr_with_limits import ExprWithLimits
assert _test_args(ExprWithLimits(x, (x, 0, 10)))
assert _test_args(ExprWithLimits(x*y, (x, 0, 10.),(y,1.,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_limits__AddWithLimits():
from sympy.concrete.expr_with_limits import AddWithLimits
assert _test_args(AddWithLimits(x, (x, 0, 10)))
assert _test_args(AddWithLimits(x*y, (x, 0, 10),(y,1,3)))
@SKIP("abstract Class")
def test_sympy__concrete__expr_with_intlimits__ExprWithIntLimits():
from sympy.concrete.expr_with_intlimits import ExprWithIntLimits
assert _test_args(ExprWithIntLimits(x, (x, 0, 10)))
assert _test_args(ExprWithIntLimits(x*y, (x, 0, 10),(y,1,3)))
def test_sympy__concrete__summations__Sum():
from sympy.concrete.summations import Sum
assert _test_args(Sum(x, (x, 0, 10)))
assert _test_args(Sum(x, (x, 0, y), (y, 0, 10)))
def test_sympy__core__add__Add():
from sympy.core.add import Add
assert _test_args(Add(x, y, z, 2))
def test_sympy__core__basic__Atom():
from sympy.core.basic import Atom
assert _test_args(Atom())
def test_sympy__core__basic__Basic():
from sympy.core.basic import Basic
assert _test_args(Basic())
def test_sympy__core__containers__Dict():
from sympy.core.containers import Dict
assert _test_args(Dict({x: y, y: z}))
def test_sympy__core__containers__Tuple():
from sympy.core.containers import Tuple
assert _test_args(Tuple(x, y, z, 2))
def test_sympy__core__expr__AtomicExpr():
from sympy.core.expr import AtomicExpr
assert _test_args(AtomicExpr())
def test_sympy__core__expr__Expr():
from sympy.core.expr import Expr
assert _test_args(Expr())
def test_sympy__core__expr__UnevaluatedExpr():
from sympy.core.expr import UnevaluatedExpr
from sympy.abc import x
assert _test_args(UnevaluatedExpr(x))
def test_sympy__core__function__Application():
from sympy.core.function import Application
assert _test_args(Application(1, 2, 3))
def test_sympy__core__function__AppliedUndef():
from sympy.core.function import AppliedUndef
assert _test_args(AppliedUndef(1, 2, 3))
def test_sympy__core__function__Derivative():
from sympy.core.function import Derivative
assert _test_args(Derivative(2, x, y, 3))
@SKIP("abstract class")
def test_sympy__core__function__Function():
pass
def test_sympy__core__function__Lambda():
assert _test_args(Lambda((x, y), x + y + z))
def test_sympy__core__function__Subs():
from sympy.core.function import Subs
assert _test_args(Subs(x + y, x, 2))
def test_sympy__core__function__WildFunction():
from sympy.core.function import WildFunction
assert _test_args(WildFunction('f'))
def test_sympy__core__mod__Mod():
from sympy.core.mod import Mod
assert _test_args(Mod(x, 2))
def test_sympy__core__mul__Mul():
from sympy.core.mul import Mul
assert _test_args(Mul(2, x, y, z))
def test_sympy__core__numbers__Catalan():
from sympy.core.numbers import Catalan
assert _test_args(Catalan())
def test_sympy__core__numbers__ComplexInfinity():
from sympy.core.numbers import ComplexInfinity
assert _test_args(ComplexInfinity())
def test_sympy__core__numbers__EulerGamma():
from sympy.core.numbers import EulerGamma
assert _test_args(EulerGamma())
def test_sympy__core__numbers__Exp1():
from sympy.core.numbers import Exp1
assert _test_args(Exp1())
def test_sympy__core__numbers__Float():
from sympy.core.numbers import Float
assert _test_args(Float(1.23))
def test_sympy__core__numbers__GoldenRatio():
from sympy.core.numbers import GoldenRatio
assert _test_args(GoldenRatio())
def test_sympy__core__numbers__TribonacciConstant():
from sympy.core.numbers import TribonacciConstant
assert _test_args(TribonacciConstant())
def test_sympy__core__numbers__Half():
from sympy.core.numbers import Half
assert _test_args(Half())
def test_sympy__core__numbers__ImaginaryUnit():
from sympy.core.numbers import ImaginaryUnit
assert _test_args(ImaginaryUnit())
def test_sympy__core__numbers__Infinity():
from sympy.core.numbers import Infinity
assert _test_args(Infinity())
def test_sympy__core__numbers__Integer():
from sympy.core.numbers import Integer
assert _test_args(Integer(7))
@SKIP("abstract class")
def test_sympy__core__numbers__IntegerConstant():
pass
def test_sympy__core__numbers__NaN():
from sympy.core.numbers import NaN
assert _test_args(NaN())
def test_sympy__core__numbers__NegativeInfinity():
from sympy.core.numbers import NegativeInfinity
assert _test_args(NegativeInfinity())
def test_sympy__core__numbers__NegativeOne():
from sympy.core.numbers import NegativeOne
assert _test_args(NegativeOne())
def test_sympy__core__numbers__Number():
from sympy.core.numbers import Number
assert _test_args(Number(1, 7))
def test_sympy__core__numbers__NumberSymbol():
from sympy.core.numbers import NumberSymbol
assert _test_args(NumberSymbol())
def test_sympy__core__numbers__One():
from sympy.core.numbers import One
assert _test_args(One())
def test_sympy__core__numbers__Pi():
from sympy.core.numbers import Pi
assert _test_args(Pi())
def test_sympy__core__numbers__Rational():
from sympy.core.numbers import Rational
assert _test_args(Rational(1, 7))
@SKIP("abstract class")
def test_sympy__core__numbers__RationalConstant():
pass
def test_sympy__core__numbers__Zero():
from sympy.core.numbers import Zero
assert _test_args(Zero())
@SKIP("abstract class")
def test_sympy__core__operations__AssocOp():
pass
@SKIP("abstract class")
def test_sympy__core__operations__LatticeOp():
pass
def test_sympy__core__power__Pow():
from sympy.core.power import Pow
assert _test_args(Pow(x, 2))
def test_sympy__core__relational__Equality():
from sympy.core.relational import Equality
assert _test_args(Equality(x, 2))
def test_sympy__core__relational__GreaterThan():
from sympy.core.relational import GreaterThan
assert _test_args(GreaterThan(x, 2))
def test_sympy__core__relational__LessThan():
from sympy.core.relational import LessThan
assert _test_args(LessThan(x, 2))
@SKIP("abstract class")
def test_sympy__core__relational__Relational():
pass
def test_sympy__core__relational__StrictGreaterThan():
from sympy.core.relational import StrictGreaterThan
assert _test_args(StrictGreaterThan(x, 2))
def test_sympy__core__relational__StrictLessThan():
from sympy.core.relational import StrictLessThan
assert _test_args(StrictLessThan(x, 2))
def test_sympy__core__relational__Unequality():
from sympy.core.relational import Unequality
assert _test_args(Unequality(x, 2))
def test_sympy__sandbox__indexed_integrals__IndexedIntegral():
from sympy.tensor import IndexedBase, Idx
from sympy.sandbox.indexed_integrals import IndexedIntegral
A = IndexedBase('A')
i, j = symbols('i j', integer=True)
a1, a2 = symbols('a1:3', cls=Idx)
assert _test_args(IndexedIntegral(A[a1], A[a2]))
assert _test_args(IndexedIntegral(A[i], A[j]))
def test_sympy__calculus__accumulationbounds__AccumulationBounds():
from sympy.calculus.accumulationbounds import AccumulationBounds
assert _test_args(AccumulationBounds(0, 1))
def test_sympy__sets__ordinals__OmegaPower():
from sympy.sets.ordinals import OmegaPower
assert _test_args(OmegaPower(1, 1))
def test_sympy__sets__ordinals__Ordinal():
from sympy.sets.ordinals import Ordinal, OmegaPower
assert _test_args(Ordinal(OmegaPower(2, 1)))
def test_sympy__sets__ordinals__OrdinalOmega():
from sympy.sets.ordinals import OrdinalOmega
assert _test_args(OrdinalOmega())
def test_sympy__sets__ordinals__OrdinalZero():
from sympy.sets.ordinals import OrdinalZero
assert _test_args(OrdinalZero())
def test_sympy__sets__powerset__PowerSet():
from sympy.sets.powerset import PowerSet
from sympy.core.singleton import S
assert _test_args(PowerSet(S.EmptySet))
def test_sympy__sets__sets__EmptySet():
from sympy.sets.sets import EmptySet
assert _test_args(EmptySet())
def test_sympy__sets__sets__UniversalSet():
from sympy.sets.sets import UniversalSet
assert _test_args(UniversalSet())
def test_sympy__sets__sets__FiniteSet():
from sympy.sets.sets import FiniteSet
assert _test_args(FiniteSet(x, y, z))
def test_sympy__sets__sets__Interval():
from sympy.sets.sets import Interval
assert _test_args(Interval(0, 1))
def test_sympy__sets__sets__ProductSet():
from sympy.sets.sets import ProductSet, Interval
assert _test_args(ProductSet(Interval(0, 1), Interval(0, 1)))
@SKIP("does it make sense to test this?")
def test_sympy__sets__sets__Set():
from sympy.sets.sets import Set
assert _test_args(Set())
def test_sympy__sets__sets__Intersection():
from sympy.sets.sets import Intersection, Interval
from sympy.core.symbol import Symbol
x = Symbol('x')
y = Symbol('y')
S = Intersection(Interval(0, x), Interval(y, 1))
assert isinstance(S, Intersection)
assert _test_args(S)
def test_sympy__sets__sets__Union():
from sympy.sets.sets import Union, Interval
assert _test_args(Union(Interval(0, 1), Interval(2, 3)))
def test_sympy__sets__sets__Complement():
from sympy.sets.sets import Complement, Interval
assert _test_args(Complement(Interval(0, 2), Interval(0, 1)))
def test_sympy__sets__sets__SymmetricDifference():
from sympy.sets.sets import FiniteSet, SymmetricDifference
assert _test_args(SymmetricDifference(FiniteSet(1, 2, 3), \
FiniteSet(2, 3, 4)))
def test_sympy__sets__sets__DisjointUnion():
from sympy.sets.sets import FiniteSet, DisjointUnion
assert _test_args(DisjointUnion(FiniteSet(1, 2, 3), \
FiniteSet(2, 3, 4)))
def test_sympy__physics__quantum__trace__Tr():
from sympy.physics.quantum.trace import Tr
a, b = symbols('a b', commutative=False)
assert _test_args(Tr(a + b))
def test_sympy__sets__setexpr__SetExpr():
from sympy.sets.setexpr import SetExpr
from sympy.sets.sets import Interval
assert _test_args(SetExpr(Interval(0, 1)))
def test_sympy__sets__fancysets__Rationals():
from sympy.sets.fancysets import Rationals
assert _test_args(Rationals())
def test_sympy__sets__fancysets__Naturals():
from sympy.sets.fancysets import Naturals
assert _test_args(Naturals())
def test_sympy__sets__fancysets__Naturals0():
from sympy.sets.fancysets import Naturals0
assert _test_args(Naturals0())
def test_sympy__sets__fancysets__Integers():
from sympy.sets.fancysets import Integers
assert _test_args(Integers())
def test_sympy__sets__fancysets__Reals():
from sympy.sets.fancysets import Reals
assert _test_args(Reals())
def test_sympy__sets__fancysets__Complexes():
from sympy.sets.fancysets import Complexes
assert _test_args(Complexes())
def test_sympy__sets__fancysets__ComplexRegion():
from sympy.sets.fancysets import ComplexRegion
from sympy.core.singleton import S
from sympy.sets import Interval
a = Interval(0, 1)
b = Interval(2, 3)
theta = Interval(0, 2*S.Pi)
assert _test_args(ComplexRegion(a*b))
assert _test_args(ComplexRegion(a*theta, polar=True))
def test_sympy__sets__fancysets__CartesianComplexRegion():
from sympy.sets.fancysets import CartesianComplexRegion
from sympy.sets import Interval
a = Interval(0, 1)
b = Interval(2, 3)
assert _test_args(CartesianComplexRegion(a*b))
def test_sympy__sets__fancysets__PolarComplexRegion():
from sympy.sets.fancysets import PolarComplexRegion
from sympy.core.singleton import S
from sympy.sets import Interval
a = Interval(0, 1)
theta = Interval(0, 2*S.Pi)
assert _test_args(PolarComplexRegion(a*theta))
def test_sympy__sets__fancysets__ImageSet():
from sympy.sets.fancysets import ImageSet
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
x = Symbol('x')
assert _test_args(ImageSet(Lambda(x, x**2), S.Naturals))
def test_sympy__sets__fancysets__Range():
from sympy.sets.fancysets import Range
assert _test_args(Range(1, 5, 1))
def test_sympy__sets__conditionset__ConditionSet():
from sympy.sets.conditionset import ConditionSet
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
x = Symbol('x')
assert _test_args(ConditionSet(x, Eq(x**2, 1), S.Reals))
def test_sympy__sets__contains__Contains():
from sympy.sets.fancysets import Range
from sympy.sets.contains import Contains
assert _test_args(Contains(x, Range(0, 10, 2)))
# STATS
from sympy.stats.crv_types import NormalDistribution
nd = NormalDistribution(0, 1)
from sympy.stats.frv_types import DieDistribution
die = DieDistribution(6)
def test_sympy__stats__crv__ContinuousDomain():
from sympy.sets.sets import Interval
from sympy.stats.crv import ContinuousDomain
assert _test_args(ContinuousDomain({x}, Interval(-oo, oo)))
def test_sympy__stats__crv__SingleContinuousDomain():
from sympy.sets.sets import Interval
from sympy.stats.crv import SingleContinuousDomain
assert _test_args(SingleContinuousDomain(x, Interval(-oo, oo)))
def test_sympy__stats__crv__ProductContinuousDomain():
from sympy.sets.sets import Interval
from sympy.stats.crv import SingleContinuousDomain, ProductContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
E = SingleContinuousDomain(y, Interval(0, oo))
assert _test_args(ProductContinuousDomain(D, E))
def test_sympy__stats__crv__ConditionalContinuousDomain():
from sympy.sets.sets import Interval
from sympy.stats.crv import (SingleContinuousDomain,
ConditionalContinuousDomain)
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ConditionalContinuousDomain(D, x > 0))
def test_sympy__stats__crv__ContinuousPSpace():
from sympy.sets.sets import Interval
from sympy.stats.crv import ContinuousPSpace, SingleContinuousDomain
D = SingleContinuousDomain(x, Interval(-oo, oo))
assert _test_args(ContinuousPSpace(D, nd))
def test_sympy__stats__crv__SingleContinuousPSpace():
from sympy.stats.crv import SingleContinuousPSpace
assert _test_args(SingleContinuousPSpace(x, nd))
@SKIP("abstract class")
def test_sympy__stats__rv__Distribution():
pass
@SKIP("abstract class")
def test_sympy__stats__crv__SingleContinuousDistribution():
pass
def test_sympy__stats__drv__SingleDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain
assert _test_args(SingleDiscreteDomain(x, S.Naturals))
def test_sympy__stats__drv__ProductDiscreteDomain():
from sympy.stats.drv import SingleDiscreteDomain, ProductDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals)
Y = SingleDiscreteDomain(y, S.Integers)
assert _test_args(ProductDiscreteDomain(X, Y))
def test_sympy__stats__drv__SingleDiscretePSpace():
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(SingleDiscretePSpace(x, PoissonDistribution(1)))
def test_sympy__stats__drv__DiscretePSpace():
from sympy.stats.drv import DiscretePSpace, SingleDiscreteDomain
density = Lambda(x, 2**(-x))
domain = SingleDiscreteDomain(x, S.Naturals)
assert _test_args(DiscretePSpace(domain, density))
def test_sympy__stats__drv__ConditionalDiscreteDomain():
from sympy.stats.drv import ConditionalDiscreteDomain, SingleDiscreteDomain
X = SingleDiscreteDomain(x, S.Naturals0)
assert _test_args(ConditionalDiscreteDomain(X, x > 2))
def test_sympy__stats__joint_rv__JointPSpace():
from sympy.stats.joint_rv import JointPSpace, JointDistribution
assert _test_args(JointPSpace('X', JointDistribution(1)))
def test_sympy__stats__joint_rv__JointRandomSymbol():
from sympy.stats.joint_rv import JointRandomSymbol
assert _test_args(JointRandomSymbol(x))
def test_sympy__stats__joint_rv_types__JointDistributionHandmade():
from sympy.tensor.indexed import Indexed
from sympy.stats.joint_rv_types import JointDistributionHandmade
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert _test_args(JointDistributionHandmade(x1 + x2, S.Reals**2))
def test_sympy__stats__joint_rv__MarginalDistribution():
from sympy.stats.rv import RandomSymbol
from sympy.stats.joint_rv import MarginalDistribution
r = RandomSymbol(S('r'))
assert _test_args(MarginalDistribution(r, (r,)))
def test_sympy__stats__compound_rv__CompoundDistribution():
from sympy.stats.compound_rv import CompoundDistribution
from sympy.stats.drv_types import PoissonDistribution, Poisson
r = Poisson('r', 10)
assert _test_args(CompoundDistribution(PoissonDistribution(r)))
def test_sympy__stats__compound_rv__CompoundPSpace():
from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution
from sympy.stats.drv_types import PoissonDistribution, Poisson
r = Poisson('r', 5)
C = CompoundDistribution(PoissonDistribution(r))
assert _test_args(CompoundPSpace('C', C))
@SKIP("abstract class")
def test_sympy__stats__drv__SingleDiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__drv__DiscreteDomain():
pass
def test_sympy__stats__rv__RandomDomain():
from sympy.stats.rv import RandomDomain
from sympy.sets.sets import FiniteSet
assert _test_args(RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__SingleDomain():
from sympy.stats.rv import SingleDomain
from sympy.sets.sets import FiniteSet
assert _test_args(SingleDomain(x, FiniteSet(1, 2, 3)))
def test_sympy__stats__rv__ConditionalDomain():
from sympy.stats.rv import ConditionalDomain, RandomDomain
from sympy.sets.sets import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2))
assert _test_args(ConditionalDomain(D, x > 1))
def test_sympy__stats__rv__MatrixDomain():
from sympy.stats.rv import MatrixDomain
from sympy.matrices import MatrixSet
from sympy.core.singleton import S
assert _test_args(MatrixDomain(x, MatrixSet(2, 2, S.Reals)))
def test_sympy__stats__rv__PSpace():
from sympy.stats.rv import PSpace, RandomDomain
from sympy.sets.sets import FiniteSet
D = RandomDomain(FiniteSet(x), FiniteSet(1, 2, 3, 4, 5, 6))
assert _test_args(PSpace(D, die))
@SKIP("abstract Class")
def test_sympy__stats__rv__SinglePSpace():
pass
def test_sympy__stats__rv__RandomSymbol():
from sympy.stats.rv import RandomSymbol
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
assert _test_args(RandomSymbol(x, A))
@SKIP("abstract Class")
def test_sympy__stats__rv__ProductPSpace():
pass
def test_sympy__stats__rv__IndependentProductPSpace():
from sympy.stats.rv import IndependentProductPSpace
from sympy.stats.crv import SingleContinuousPSpace
A = SingleContinuousPSpace(x, nd)
B = SingleContinuousPSpace(y, nd)
assert _test_args(IndependentProductPSpace(A, B))
def test_sympy__stats__rv__ProductDomain():
from sympy.sets.sets import Interval
from sympy.stats.rv import ProductDomain, SingleDomain
D = SingleDomain(x, Interval(-oo, oo))
E = SingleDomain(y, Interval(0, oo))
assert _test_args(ProductDomain(D, E))
def test_sympy__stats__symbolic_probability__Probability():
from sympy.stats.symbolic_probability import Probability
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Probability(X > 0))
def test_sympy__stats__symbolic_probability__Expectation():
from sympy.stats.symbolic_probability import Expectation
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Expectation(X > 0))
def test_sympy__stats__symbolic_probability__Covariance():
from sympy.stats.symbolic_probability import Covariance
from sympy.stats import Normal
X = Normal('X', 0, 1)
Y = Normal('Y', 0, 3)
assert _test_args(Covariance(X, Y))
def test_sympy__stats__symbolic_probability__Variance():
from sympy.stats.symbolic_probability import Variance
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Variance(X))
def test_sympy__stats__symbolic_probability__Moment():
from sympy.stats.symbolic_probability import Moment
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(Moment(X, 3, 2, X > 3))
def test_sympy__stats__symbolic_probability__CentralMoment():
from sympy.stats.symbolic_probability import CentralMoment
from sympy.stats import Normal
X = Normal('X', 0, 1)
assert _test_args(CentralMoment(X, 2, X > 1))
def test_sympy__stats__frv_types__DiscreteUniformDistribution():
from sympy.stats.frv_types import DiscreteUniformDistribution
from sympy.core.containers import Tuple
assert _test_args(DiscreteUniformDistribution(Tuple(*list(range(6)))))
def test_sympy__stats__frv_types__DieDistribution():
assert _test_args(die)
def test_sympy__stats__frv_types__BernoulliDistribution():
from sympy.stats.frv_types import BernoulliDistribution
assert _test_args(BernoulliDistribution(S.Half, 0, 1))
def test_sympy__stats__frv_types__BinomialDistribution():
from sympy.stats.frv_types import BinomialDistribution
assert _test_args(BinomialDistribution(5, S.Half, 1, 0))
def test_sympy__stats__frv_types__BetaBinomialDistribution():
from sympy.stats.frv_types import BetaBinomialDistribution
assert _test_args(BetaBinomialDistribution(5, 1, 1))
def test_sympy__stats__frv_types__HypergeometricDistribution():
from sympy.stats.frv_types import HypergeometricDistribution
assert _test_args(HypergeometricDistribution(10, 5, 3))
def test_sympy__stats__frv_types__RademacherDistribution():
from sympy.stats.frv_types import RademacherDistribution
assert _test_args(RademacherDistribution())
def test_sympy__stats__frv_types__IdealSolitonDistribution():
from sympy.stats.frv_types import IdealSolitonDistribution
assert _test_args(IdealSolitonDistribution(10))
def test_sympy__stats__frv_types__RobustSolitonDistribution():
from sympy.stats.frv_types import RobustSolitonDistribution
assert _test_args(RobustSolitonDistribution(1000, 0.5, 0.1))
def test_sympy__stats__frv__FiniteDomain():
from sympy.stats.frv import FiniteDomain
assert _test_args(FiniteDomain({(x, 1), (x, 2)})) # x can be 1 or 2
def test_sympy__stats__frv__SingleFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain
assert _test_args(SingleFiniteDomain(x, {1, 2})) # x can be 1 or 2
def test_sympy__stats__frv__ProductFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ProductFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
yd = SingleFiniteDomain(y, {1, 2})
assert _test_args(ProductFiniteDomain(xd, yd))
def test_sympy__stats__frv__ConditionalFiniteDomain():
from sympy.stats.frv import SingleFiniteDomain, ConditionalFiniteDomain
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(ConditionalFiniteDomain(xd, x > 1))
def test_sympy__stats__frv__FinitePSpace():
from sympy.stats.frv import FinitePSpace, SingleFiniteDomain
xd = SingleFiniteDomain(x, {1, 2, 3, 4, 5, 6})
assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half}))
xd = SingleFiniteDomain(x, {1, 2})
assert _test_args(FinitePSpace(xd, {(x, 1): S.Half, (x, 2): S.Half}))
def test_sympy__stats__frv__SingleFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace
from sympy.core.symbol import Symbol
assert _test_args(SingleFinitePSpace(Symbol('x'), die))
def test_sympy__stats__frv__ProductFinitePSpace():
from sympy.stats.frv import SingleFinitePSpace, ProductFinitePSpace
from sympy.core.symbol import Symbol
xp = SingleFinitePSpace(Symbol('x'), die)
yp = SingleFinitePSpace(Symbol('y'), die)
assert _test_args(ProductFinitePSpace(xp, yp))
@SKIP("abstract class")
def test_sympy__stats__frv__SingleFiniteDistribution():
pass
@SKIP("abstract class")
def test_sympy__stats__crv__ContinuousDistribution():
pass
def test_sympy__stats__frv_types__FiniteDistributionHandmade():
from sympy.stats.frv_types import FiniteDistributionHandmade
from sympy.core.containers import Dict
assert _test_args(FiniteDistributionHandmade(Dict({1: 1})))
def test_sympy__stats__crv_types__ContinuousDistributionHandmade():
from sympy.stats.crv_types import ContinuousDistributionHandmade
from sympy.core.function import Lambda
from sympy.sets.sets import Interval
from sympy.abc import x
assert _test_args(ContinuousDistributionHandmade(Lambda(x, 2*x),
Interval(0, 1)))
def test_sympy__stats__drv_types__DiscreteDistributionHandmade():
from sympy.stats.drv_types import DiscreteDistributionHandmade
from sympy.core.function import Lambda
from sympy.sets.sets import FiniteSet
from sympy.abc import x
assert _test_args(DiscreteDistributionHandmade(Lambda(x, Rational(1, 10)),
FiniteSet(*range(10))))
def test_sympy__stats__rv__Density():
from sympy.stats.rv import Density
from sympy.stats.crv_types import Normal
assert _test_args(Density(Normal('x', 0, 1)))
def test_sympy__stats__crv_types__ArcsinDistribution():
from sympy.stats.crv_types import ArcsinDistribution
assert _test_args(ArcsinDistribution(0, 1))
def test_sympy__stats__crv_types__BeniniDistribution():
from sympy.stats.crv_types import BeniniDistribution
assert _test_args(BeniniDistribution(1, 1, 1))
def test_sympy__stats__crv_types__BetaDistribution():
from sympy.stats.crv_types import BetaDistribution
assert _test_args(BetaDistribution(1, 1))
def test_sympy__stats__crv_types__BetaNoncentralDistribution():
from sympy.stats.crv_types import BetaNoncentralDistribution
assert _test_args(BetaNoncentralDistribution(1, 1, 1))
def test_sympy__stats__crv_types__BetaPrimeDistribution():
from sympy.stats.crv_types import BetaPrimeDistribution
assert _test_args(BetaPrimeDistribution(1, 1))
def test_sympy__stats__crv_types__BoundedParetoDistribution():
from sympy.stats.crv_types import BoundedParetoDistribution
assert _test_args(BoundedParetoDistribution(1, 1, 2))
def test_sympy__stats__crv_types__CauchyDistribution():
from sympy.stats.crv_types import CauchyDistribution
assert _test_args(CauchyDistribution(0, 1))
def test_sympy__stats__crv_types__ChiDistribution():
from sympy.stats.crv_types import ChiDistribution
assert _test_args(ChiDistribution(1))
def test_sympy__stats__crv_types__ChiNoncentralDistribution():
from sympy.stats.crv_types import ChiNoncentralDistribution
assert _test_args(ChiNoncentralDistribution(1,1))
def test_sympy__stats__crv_types__ChiSquaredDistribution():
from sympy.stats.crv_types import ChiSquaredDistribution
assert _test_args(ChiSquaredDistribution(1))
def test_sympy__stats__crv_types__DagumDistribution():
from sympy.stats.crv_types import DagumDistribution
assert _test_args(DagumDistribution(1, 1, 1))
def test_sympy__stats__crv_types__ExGaussianDistribution():
from sympy.stats.crv_types import ExGaussianDistribution
assert _test_args(ExGaussianDistribution(1, 1, 1))
def test_sympy__stats__crv_types__ExponentialDistribution():
from sympy.stats.crv_types import ExponentialDistribution
assert _test_args(ExponentialDistribution(1))
def test_sympy__stats__crv_types__ExponentialPowerDistribution():
from sympy.stats.crv_types import ExponentialPowerDistribution
assert _test_args(ExponentialPowerDistribution(0, 1, 1))
def test_sympy__stats__crv_types__FDistributionDistribution():
from sympy.stats.crv_types import FDistributionDistribution
assert _test_args(FDistributionDistribution(1, 1))
def test_sympy__stats__crv_types__FisherZDistribution():
from sympy.stats.crv_types import FisherZDistribution
assert _test_args(FisherZDistribution(1, 1))
def test_sympy__stats__crv_types__FrechetDistribution():
from sympy.stats.crv_types import FrechetDistribution
assert _test_args(FrechetDistribution(1, 1, 1))
def test_sympy__stats__crv_types__GammaInverseDistribution():
from sympy.stats.crv_types import GammaInverseDistribution
assert _test_args(GammaInverseDistribution(1, 1))
def test_sympy__stats__crv_types__GammaDistribution():
from sympy.stats.crv_types import GammaDistribution
assert _test_args(GammaDistribution(1, 1))
def test_sympy__stats__crv_types__GumbelDistribution():
from sympy.stats.crv_types import GumbelDistribution
assert _test_args(GumbelDistribution(1, 1, False))
def test_sympy__stats__crv_types__GompertzDistribution():
from sympy.stats.crv_types import GompertzDistribution
assert _test_args(GompertzDistribution(1, 1))
def test_sympy__stats__crv_types__KumaraswamyDistribution():
from sympy.stats.crv_types import KumaraswamyDistribution
assert _test_args(KumaraswamyDistribution(1, 1))
def test_sympy__stats__crv_types__LaplaceDistribution():
from sympy.stats.crv_types import LaplaceDistribution
assert _test_args(LaplaceDistribution(0, 1))
def test_sympy__stats__crv_types__LevyDistribution():
from sympy.stats.crv_types import LevyDistribution
assert _test_args(LevyDistribution(0, 1))
def test_sympy__stats__crv_types__LogCauchyDistribution():
from sympy.stats.crv_types import LogCauchyDistribution
assert _test_args(LogCauchyDistribution(0, 1))
def test_sympy__stats__crv_types__LogisticDistribution():
from sympy.stats.crv_types import LogisticDistribution
assert _test_args(LogisticDistribution(0, 1))
def test_sympy__stats__crv_types__LogLogisticDistribution():
from sympy.stats.crv_types import LogLogisticDistribution
assert _test_args(LogLogisticDistribution(1, 1))
def test_sympy__stats__crv_types__LogitNormalDistribution():
from sympy.stats.crv_types import LogitNormalDistribution
assert _test_args(LogitNormalDistribution(0, 1))
def test_sympy__stats__crv_types__LogNormalDistribution():
from sympy.stats.crv_types import LogNormalDistribution
assert _test_args(LogNormalDistribution(0, 1))
def test_sympy__stats__crv_types__LomaxDistribution():
from sympy.stats.crv_types import LomaxDistribution
assert _test_args(LomaxDistribution(1, 2))
def test_sympy__stats__crv_types__MaxwellDistribution():
from sympy.stats.crv_types import MaxwellDistribution
assert _test_args(MaxwellDistribution(1))
def test_sympy__stats__crv_types__MoyalDistribution():
from sympy.stats.crv_types import MoyalDistribution
assert _test_args(MoyalDistribution(1,2))
def test_sympy__stats__crv_types__NakagamiDistribution():
from sympy.stats.crv_types import NakagamiDistribution
assert _test_args(NakagamiDistribution(1, 1))
def test_sympy__stats__crv_types__NormalDistribution():
from sympy.stats.crv_types import NormalDistribution
assert _test_args(NormalDistribution(0, 1))
def test_sympy__stats__crv_types__GaussianInverseDistribution():
from sympy.stats.crv_types import GaussianInverseDistribution
assert _test_args(GaussianInverseDistribution(1, 1))
def test_sympy__stats__crv_types__ParetoDistribution():
from sympy.stats.crv_types import ParetoDistribution
assert _test_args(ParetoDistribution(1, 1))
def test_sympy__stats__crv_types__PowerFunctionDistribution():
from sympy.stats.crv_types import PowerFunctionDistribution
assert _test_args(PowerFunctionDistribution(2,0,1))
def test_sympy__stats__crv_types__QuadraticUDistribution():
from sympy.stats.crv_types import QuadraticUDistribution
assert _test_args(QuadraticUDistribution(1, 2))
def test_sympy__stats__crv_types__RaisedCosineDistribution():
from sympy.stats.crv_types import RaisedCosineDistribution
assert _test_args(RaisedCosineDistribution(1, 1))
def test_sympy__stats__crv_types__RayleighDistribution():
from sympy.stats.crv_types import RayleighDistribution
assert _test_args(RayleighDistribution(1))
def test_sympy__stats__crv_types__ReciprocalDistribution():
from sympy.stats.crv_types import ReciprocalDistribution
assert _test_args(ReciprocalDistribution(5, 30))
def test_sympy__stats__crv_types__ShiftedGompertzDistribution():
from sympy.stats.crv_types import ShiftedGompertzDistribution
assert _test_args(ShiftedGompertzDistribution(1, 1))
def test_sympy__stats__crv_types__StudentTDistribution():
from sympy.stats.crv_types import StudentTDistribution
assert _test_args(StudentTDistribution(1))
def test_sympy__stats__crv_types__TrapezoidalDistribution():
from sympy.stats.crv_types import TrapezoidalDistribution
assert _test_args(TrapezoidalDistribution(1, 2, 3, 4))
def test_sympy__stats__crv_types__TriangularDistribution():
from sympy.stats.crv_types import TriangularDistribution
assert _test_args(TriangularDistribution(-1, 0, 1))
def test_sympy__stats__crv_types__UniformDistribution():
from sympy.stats.crv_types import UniformDistribution
assert _test_args(UniformDistribution(0, 1))
def test_sympy__stats__crv_types__UniformSumDistribution():
from sympy.stats.crv_types import UniformSumDistribution
assert _test_args(UniformSumDistribution(1))
def test_sympy__stats__crv_types__VonMisesDistribution():
from sympy.stats.crv_types import VonMisesDistribution
assert _test_args(VonMisesDistribution(1, 1))
def test_sympy__stats__crv_types__WeibullDistribution():
from sympy.stats.crv_types import WeibullDistribution
assert _test_args(WeibullDistribution(1, 1))
def test_sympy__stats__crv_types__WignerSemicircleDistribution():
from sympy.stats.crv_types import WignerSemicircleDistribution
assert _test_args(WignerSemicircleDistribution(1))
def test_sympy__stats__drv_types__GeometricDistribution():
from sympy.stats.drv_types import GeometricDistribution
assert _test_args(GeometricDistribution(.5))
def test_sympy__stats__drv_types__HermiteDistribution():
from sympy.stats.drv_types import HermiteDistribution
assert _test_args(HermiteDistribution(1, 2))
def test_sympy__stats__drv_types__LogarithmicDistribution():
from sympy.stats.drv_types import LogarithmicDistribution
assert _test_args(LogarithmicDistribution(.5))
def test_sympy__stats__drv_types__NegativeBinomialDistribution():
from sympy.stats.drv_types import NegativeBinomialDistribution
assert _test_args(NegativeBinomialDistribution(.5, .5))
def test_sympy__stats__drv_types__FlorySchulzDistribution():
from sympy.stats.drv_types import FlorySchulzDistribution
assert _test_args(FlorySchulzDistribution(.5))
def test_sympy__stats__drv_types__PoissonDistribution():
from sympy.stats.drv_types import PoissonDistribution
assert _test_args(PoissonDistribution(1))
def test_sympy__stats__drv_types__SkellamDistribution():
from sympy.stats.drv_types import SkellamDistribution
assert _test_args(SkellamDistribution(1, 1))
def test_sympy__stats__drv_types__YuleSimonDistribution():
from sympy.stats.drv_types import YuleSimonDistribution
assert _test_args(YuleSimonDistribution(.5))
def test_sympy__stats__drv_types__ZetaDistribution():
from sympy.stats.drv_types import ZetaDistribution
assert _test_args(ZetaDistribution(1.5))
def test_sympy__stats__joint_rv__JointDistribution():
from sympy.stats.joint_rv import JointDistribution
assert _test_args(JointDistribution(1, 2, 3, 4))
def test_sympy__stats__joint_rv_types__MultivariateNormalDistribution():
from sympy.stats.joint_rv_types import MultivariateNormalDistribution
assert _test_args(
MultivariateNormalDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateLaplaceDistribution():
from sympy.stats.joint_rv_types import MultivariateLaplaceDistribution
assert _test_args(MultivariateLaplaceDistribution([0, 1], [[1, 0],[0, 1]]))
def test_sympy__stats__joint_rv_types__MultivariateTDistribution():
from sympy.stats.joint_rv_types import MultivariateTDistribution
assert _test_args(MultivariateTDistribution([0, 1], [[1, 0],[0, 1]], 1))
def test_sympy__stats__joint_rv_types__NormalGammaDistribution():
from sympy.stats.joint_rv_types import NormalGammaDistribution
assert _test_args(NormalGammaDistribution(1, 2, 3, 4))
def test_sympy__stats__joint_rv_types__GeneralizedMultivariateLogGammaDistribution():
from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaDistribution
v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4])
assert _test_args(GeneralizedMultivariateLogGammaDistribution(S.Half, v, l, mu))
def test_sympy__stats__joint_rv_types__MultivariateBetaDistribution():
from sympy.stats.joint_rv_types import MultivariateBetaDistribution
assert _test_args(MultivariateBetaDistribution([1, 2, 3]))
def test_sympy__stats__joint_rv_types__MultivariateEwensDistribution():
from sympy.stats.joint_rv_types import MultivariateEwensDistribution
assert _test_args(MultivariateEwensDistribution(5, 1))
def test_sympy__stats__joint_rv_types__MultinomialDistribution():
from sympy.stats.joint_rv_types import MultinomialDistribution
assert _test_args(MultinomialDistribution(5, [0.5, 0.1, 0.3]))
def test_sympy__stats__joint_rv_types__NegativeMultinomialDistribution():
from sympy.stats.joint_rv_types import NegativeMultinomialDistribution
assert _test_args(NegativeMultinomialDistribution(5, [0.5, 0.1, 0.3]))
def test_sympy__stats__rv__RandomIndexedSymbol():
from sympy.stats.rv import RandomIndexedSymbol, pspace
from sympy.stats.stochastic_process_types import DiscreteMarkovChain
X = DiscreteMarkovChain("X")
assert _test_args(RandomIndexedSymbol(X[0].symbol, pspace(X[0])))
def test_sympy__stats__rv__RandomMatrixSymbol():
from sympy.stats.rv import RandomMatrixSymbol
from sympy.stats.random_matrix import RandomMatrixPSpace
pspace = RandomMatrixPSpace('P')
assert _test_args(RandomMatrixSymbol('M', 3, 3, pspace))
def test_sympy__stats__stochastic_process__StochasticPSpace():
from sympy.stats.stochastic_process import StochasticPSpace
from sympy.stats.stochastic_process_types import StochasticProcess
from sympy.stats.frv_types import BernoulliDistribution
assert _test_args(StochasticPSpace("Y", StochasticProcess("Y", [1, 2, 3]), BernoulliDistribution(S.Half, 1, 0)))
def test_sympy__stats__stochastic_process_types__StochasticProcess():
from sympy.stats.stochastic_process_types import StochasticProcess
assert _test_args(StochasticProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__MarkovProcess():
from sympy.stats.stochastic_process_types import MarkovProcess
assert _test_args(MarkovProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__DiscreteTimeStochasticProcess():
from sympy.stats.stochastic_process_types import DiscreteTimeStochasticProcess
assert _test_args(DiscreteTimeStochasticProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__ContinuousTimeStochasticProcess():
from sympy.stats.stochastic_process_types import ContinuousTimeStochasticProcess
assert _test_args(ContinuousTimeStochasticProcess("Y", [1, 2, 3]))
def test_sympy__stats__stochastic_process_types__TransitionMatrixOf():
from sympy.stats.stochastic_process_types import TransitionMatrixOf, DiscreteMarkovChain
from sympy.matrices.expressions.matexpr import MatrixSymbol
DMC = DiscreteMarkovChain("Y")
assert _test_args(TransitionMatrixOf(DMC, MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__GeneratorMatrixOf():
from sympy.stats.stochastic_process_types import GeneratorMatrixOf, ContinuousMarkovChain
from sympy.matrices.expressions.matexpr import MatrixSymbol
DMC = ContinuousMarkovChain("Y")
assert _test_args(GeneratorMatrixOf(DMC, MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__StochasticStateSpaceOf():
from sympy.stats.stochastic_process_types import StochasticStateSpaceOf, DiscreteMarkovChain
DMC = DiscreteMarkovChain("Y")
assert _test_args(StochasticStateSpaceOf(DMC, [0, 1, 2]))
def test_sympy__stats__stochastic_process_types__DiscreteMarkovChain():
from sympy.stats.stochastic_process_types import DiscreteMarkovChain
from sympy.matrices.expressions.matexpr import MatrixSymbol
assert _test_args(DiscreteMarkovChain("Y", [0, 1, 2], MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__ContinuousMarkovChain():
from sympy.stats.stochastic_process_types import ContinuousMarkovChain
from sympy.matrices.expressions.matexpr import MatrixSymbol
assert _test_args(ContinuousMarkovChain("Y", [0, 1, 2], MatrixSymbol('T', 3, 3)))
def test_sympy__stats__stochastic_process_types__BernoulliProcess():
from sympy.stats.stochastic_process_types import BernoulliProcess
assert _test_args(BernoulliProcess("B", 0.5, 1, 0))
def test_sympy__stats__stochastic_process_types__CountingProcess():
from sympy.stats.stochastic_process_types import CountingProcess
assert _test_args(CountingProcess("C"))
def test_sympy__stats__stochastic_process_types__PoissonProcess():
from sympy.stats.stochastic_process_types import PoissonProcess
assert _test_args(PoissonProcess("X", 2))
def test_sympy__stats__stochastic_process_types__WienerProcess():
from sympy.stats.stochastic_process_types import WienerProcess
assert _test_args(WienerProcess("X"))
def test_sympy__stats__stochastic_process_types__GammaProcess():
from sympy.stats.stochastic_process_types import GammaProcess
assert _test_args(GammaProcess("X", 1, 2))
def test_sympy__stats__random_matrix__RandomMatrixPSpace():
from sympy.stats.random_matrix import RandomMatrixPSpace
from sympy.stats.random_matrix_models import RandomMatrixEnsembleModel
model = RandomMatrixEnsembleModel('R', 3)
assert _test_args(RandomMatrixPSpace('P', model=model))
def test_sympy__stats__random_matrix_models__RandomMatrixEnsembleModel():
from sympy.stats.random_matrix_models import RandomMatrixEnsembleModel
assert _test_args(RandomMatrixEnsembleModel('R', 3))
def test_sympy__stats__random_matrix_models__GaussianEnsembleModel():
from sympy.stats.random_matrix_models import GaussianEnsembleModel
assert _test_args(GaussianEnsembleModel('G', 3))
def test_sympy__stats__random_matrix_models__GaussianUnitaryEnsembleModel():
from sympy.stats.random_matrix_models import GaussianUnitaryEnsembleModel
assert _test_args(GaussianUnitaryEnsembleModel('U', 3))
def test_sympy__stats__random_matrix_models__GaussianOrthogonalEnsembleModel():
from sympy.stats.random_matrix_models import GaussianOrthogonalEnsembleModel
assert _test_args(GaussianOrthogonalEnsembleModel('U', 3))
def test_sympy__stats__random_matrix_models__GaussianSymplecticEnsembleModel():
from sympy.stats.random_matrix_models import GaussianSymplecticEnsembleModel
assert _test_args(GaussianSymplecticEnsembleModel('U', 3))
def test_sympy__stats__random_matrix_models__CircularEnsembleModel():
from sympy.stats.random_matrix_models import CircularEnsembleModel
assert _test_args(CircularEnsembleModel('C', 3))
def test_sympy__stats__random_matrix_models__CircularUnitaryEnsembleModel():
from sympy.stats.random_matrix_models import CircularUnitaryEnsembleModel
assert _test_args(CircularUnitaryEnsembleModel('U', 3))
def test_sympy__stats__random_matrix_models__CircularOrthogonalEnsembleModel():
from sympy.stats.random_matrix_models import CircularOrthogonalEnsembleModel
assert _test_args(CircularOrthogonalEnsembleModel('O', 3))
def test_sympy__stats__random_matrix_models__CircularSymplecticEnsembleModel():
from sympy.stats.random_matrix_models import CircularSymplecticEnsembleModel
assert _test_args(CircularSymplecticEnsembleModel('S', 3))
def test_sympy__stats__symbolic_multivariate_probability__ExpectationMatrix():
from sympy.stats import ExpectationMatrix
from sympy.stats.rv import RandomMatrixSymbol
assert _test_args(ExpectationMatrix(RandomMatrixSymbol('R', 2, 1)))
def test_sympy__stats__symbolic_multivariate_probability__VarianceMatrix():
from sympy.stats import VarianceMatrix
from sympy.stats.rv import RandomMatrixSymbol
assert _test_args(VarianceMatrix(RandomMatrixSymbol('R', 3, 1)))
def test_sympy__stats__symbolic_multivariate_probability__CrossCovarianceMatrix():
from sympy.stats import CrossCovarianceMatrix
from sympy.stats.rv import RandomMatrixSymbol
assert _test_args(CrossCovarianceMatrix(RandomMatrixSymbol('R', 3, 1),
RandomMatrixSymbol('X', 3, 1)))
def test_sympy__stats__matrix_distributions__MatrixPSpace():
from sympy.stats.matrix_distributions import MatrixDistribution, MatrixPSpace
from sympy.matrices.dense import Matrix
M = MatrixDistribution(1, Matrix([[1, 0], [0, 1]]))
assert _test_args(MatrixPSpace('M', M, 2, 2))
def test_sympy__stats__matrix_distributions__MatrixDistribution():
from sympy.stats.matrix_distributions import MatrixDistribution
from sympy.matrices.dense import Matrix
assert _test_args(MatrixDistribution(1, Matrix([[1, 0], [0, 1]])))
def test_sympy__stats__matrix_distributions__MatrixGammaDistribution():
from sympy.stats.matrix_distributions import MatrixGammaDistribution
from sympy.matrices.dense import Matrix
assert _test_args(MatrixGammaDistribution(3, 4, Matrix([[1, 0], [0, 1]])))
def test_sympy__stats__matrix_distributions__WishartDistribution():
from sympy.stats.matrix_distributions import WishartDistribution
from sympy.matrices.dense import Matrix
assert _test_args(WishartDistribution(3, Matrix([[1, 0], [0, 1]])))
def test_sympy__stats__matrix_distributions__MatrixNormalDistribution():
from sympy.stats.matrix_distributions import MatrixNormalDistribution
from sympy.matrices.expressions.matexpr import MatrixSymbol
L = MatrixSymbol('L', 1, 2)
S1 = MatrixSymbol('S1', 1, 1)
S2 = MatrixSymbol('S2', 2, 2)
assert _test_args(MatrixNormalDistribution(L, S1, S2))
def test_sympy__stats__matrix_distributions__MatrixStudentTDistribution():
from sympy.stats.matrix_distributions import MatrixStudentTDistribution
from sympy.matrices.expressions.matexpr import MatrixSymbol
v = symbols('v', positive=True)
Omega = MatrixSymbol('Omega', 3, 3)
Sigma = MatrixSymbol('Sigma', 1, 1)
Location = MatrixSymbol('Location', 1, 3)
assert _test_args(MatrixStudentTDistribution(v, Location, Omega, Sigma))
def test_sympy__utilities__matchpy_connector__WildDot():
from sympy.utilities.matchpy_connector import WildDot
assert _test_args(WildDot("w_"))
def test_sympy__utilities__matchpy_connector__WildPlus():
from sympy.utilities.matchpy_connector import WildPlus
assert _test_args(WildPlus("w__"))
def test_sympy__utilities__matchpy_connector__WildStar():
from sympy.utilities.matchpy_connector import WildStar
assert _test_args(WildStar("w___"))
def test_sympy__core__symbol__Str():
from sympy.core.symbol import Str
assert _test_args(Str('t'))
def test_sympy__core__symbol__Dummy():
from sympy.core.symbol import Dummy
assert _test_args(Dummy('t'))
def test_sympy__core__symbol__Symbol():
from sympy.core.symbol import Symbol
assert _test_args(Symbol('t'))
def test_sympy__core__symbol__Wild():
from sympy.core.symbol import Wild
assert _test_args(Wild('x', exclude=[x]))
@SKIP("abstract class")
def test_sympy__functions__combinatorial__factorials__CombinatorialFunction():
pass
def test_sympy__functions__combinatorial__factorials__FallingFactorial():
from sympy.functions.combinatorial.factorials import FallingFactorial
assert _test_args(FallingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__MultiFactorial():
from sympy.functions.combinatorial.factorials import MultiFactorial
assert _test_args(MultiFactorial(x))
def test_sympy__functions__combinatorial__factorials__RisingFactorial():
from sympy.functions.combinatorial.factorials import RisingFactorial
assert _test_args(RisingFactorial(2, x))
def test_sympy__functions__combinatorial__factorials__binomial():
from sympy.functions.combinatorial.factorials import binomial
assert _test_args(binomial(2, x))
def test_sympy__functions__combinatorial__factorials__subfactorial():
from sympy.functions.combinatorial.factorials import subfactorial
assert _test_args(subfactorial(x))
def test_sympy__functions__combinatorial__factorials__factorial():
from sympy.functions.combinatorial.factorials import factorial
assert _test_args(factorial(x))
def test_sympy__functions__combinatorial__factorials__factorial2():
from sympy.functions.combinatorial.factorials import factorial2
assert _test_args(factorial2(x))
def test_sympy__functions__combinatorial__numbers__bell():
from sympy.functions.combinatorial.numbers import bell
assert _test_args(bell(x, y))
def test_sympy__functions__combinatorial__numbers__bernoulli():
from sympy.functions.combinatorial.numbers import bernoulli
assert _test_args(bernoulli(x))
def test_sympy__functions__combinatorial__numbers__catalan():
from sympy.functions.combinatorial.numbers import catalan
assert _test_args(catalan(x))
def test_sympy__functions__combinatorial__numbers__genocchi():
from sympy.functions.combinatorial.numbers import genocchi
assert _test_args(genocchi(x))
def test_sympy__functions__combinatorial__numbers__euler():
from sympy.functions.combinatorial.numbers import euler
assert _test_args(euler(x))
def test_sympy__functions__combinatorial__numbers__carmichael():
from sympy.functions.combinatorial.numbers import carmichael
assert _test_args(carmichael(x))
def test_sympy__functions__combinatorial__numbers__motzkin():
from sympy.functions.combinatorial.numbers import motzkin
assert _test_args(motzkin(5))
def test_sympy__functions__combinatorial__numbers__fibonacci():
from sympy.functions.combinatorial.numbers import fibonacci
assert _test_args(fibonacci(x))
def test_sympy__functions__combinatorial__numbers__tribonacci():
from sympy.functions.combinatorial.numbers import tribonacci
assert _test_args(tribonacci(x))
def test_sympy__functions__combinatorial__numbers__harmonic():
from sympy.functions.combinatorial.numbers import harmonic
assert _test_args(harmonic(x, 2))
def test_sympy__functions__combinatorial__numbers__lucas():
from sympy.functions.combinatorial.numbers import lucas
assert _test_args(lucas(x))
def test_sympy__functions__combinatorial__numbers__partition():
from sympy.core.symbol import Symbol
from sympy.functions.combinatorial.numbers import partition
assert _test_args(partition(Symbol('a', integer=True)))
def test_sympy__functions__elementary__complexes__Abs():
from sympy.functions.elementary.complexes import Abs
assert _test_args(Abs(x))
def test_sympy__functions__elementary__complexes__adjoint():
from sympy.functions.elementary.complexes import adjoint
assert _test_args(adjoint(x))
def test_sympy__functions__elementary__complexes__arg():
from sympy.functions.elementary.complexes import arg
assert _test_args(arg(x))
def test_sympy__functions__elementary__complexes__conjugate():
from sympy.functions.elementary.complexes import conjugate
assert _test_args(conjugate(x))
def test_sympy__functions__elementary__complexes__im():
from sympy.functions.elementary.complexes import im
assert _test_args(im(x))
def test_sympy__functions__elementary__complexes__re():
from sympy.functions.elementary.complexes import re
assert _test_args(re(x))
def test_sympy__functions__elementary__complexes__sign():
from sympy.functions.elementary.complexes import sign
assert _test_args(sign(x))
def test_sympy__functions__elementary__complexes__polar_lift():
from sympy.functions.elementary.complexes import polar_lift
assert _test_args(polar_lift(x))
def test_sympy__functions__elementary__complexes__periodic_argument():
from sympy.functions.elementary.complexes import periodic_argument
assert _test_args(periodic_argument(x, y))
def test_sympy__functions__elementary__complexes__principal_branch():
from sympy.functions.elementary.complexes import principal_branch
assert _test_args(principal_branch(x, y))
def test_sympy__functions__elementary__complexes__transpose():
from sympy.functions.elementary.complexes import transpose
assert _test_args(transpose(x))
def test_sympy__functions__elementary__exponential__LambertW():
from sympy.functions.elementary.exponential import LambertW
assert _test_args(LambertW(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__exponential__ExpBase():
pass
def test_sympy__functions__elementary__exponential__exp():
from sympy.functions.elementary.exponential import exp
assert _test_args(exp(2))
def test_sympy__functions__elementary__exponential__exp_polar():
from sympy.functions.elementary.exponential import exp_polar
assert _test_args(exp_polar(2))
def test_sympy__functions__elementary__exponential__log():
from sympy.functions.elementary.exponential import log
assert _test_args(log(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__HyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__ReciprocalHyperbolicFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__hyperbolic__InverseHyperbolicFunction():
pass
def test_sympy__functions__elementary__hyperbolic__acosh():
from sympy.functions.elementary.hyperbolic import acosh
assert _test_args(acosh(2))
def test_sympy__functions__elementary__hyperbolic__acoth():
from sympy.functions.elementary.hyperbolic import acoth
assert _test_args(acoth(2))
def test_sympy__functions__elementary__hyperbolic__asinh():
from sympy.functions.elementary.hyperbolic import asinh
assert _test_args(asinh(2))
def test_sympy__functions__elementary__hyperbolic__atanh():
from sympy.functions.elementary.hyperbolic import atanh
assert _test_args(atanh(2))
def test_sympy__functions__elementary__hyperbolic__asech():
from sympy.functions.elementary.hyperbolic import asech
assert _test_args(asech(x))
def test_sympy__functions__elementary__hyperbolic__acsch():
from sympy.functions.elementary.hyperbolic import acsch
assert _test_args(acsch(x))
def test_sympy__functions__elementary__hyperbolic__cosh():
from sympy.functions.elementary.hyperbolic import cosh
assert _test_args(cosh(2))
def test_sympy__functions__elementary__hyperbolic__coth():
from sympy.functions.elementary.hyperbolic import coth
assert _test_args(coth(2))
def test_sympy__functions__elementary__hyperbolic__csch():
from sympy.functions.elementary.hyperbolic import csch
assert _test_args(csch(2))
def test_sympy__functions__elementary__hyperbolic__sech():
from sympy.functions.elementary.hyperbolic import sech
assert _test_args(sech(2))
def test_sympy__functions__elementary__hyperbolic__sinh():
from sympy.functions.elementary.hyperbolic import sinh
assert _test_args(sinh(2))
def test_sympy__functions__elementary__hyperbolic__tanh():
from sympy.functions.elementary.hyperbolic import tanh
assert _test_args(tanh(2))
@SKIP("abstract class")
def test_sympy__functions__elementary__integers__RoundFunction():
pass
def test_sympy__functions__elementary__integers__ceiling():
from sympy.functions.elementary.integers import ceiling
assert _test_args(ceiling(x))
def test_sympy__functions__elementary__integers__floor():
from sympy.functions.elementary.integers import floor
assert _test_args(floor(x))
def test_sympy__functions__elementary__integers__frac():
from sympy.functions.elementary.integers import frac
assert _test_args(frac(x))
def test_sympy__functions__elementary__miscellaneous__IdentityFunction():
from sympy.functions.elementary.miscellaneous import IdentityFunction
assert _test_args(IdentityFunction())
def test_sympy__functions__elementary__miscellaneous__Max():
from sympy.functions.elementary.miscellaneous import Max
assert _test_args(Max(x, 2))
def test_sympy__functions__elementary__miscellaneous__Min():
from sympy.functions.elementary.miscellaneous import Min
assert _test_args(Min(x, 2))
@SKIP("abstract class")
def test_sympy__functions__elementary__miscellaneous__MinMaxBase():
pass
def test_sympy__functions__elementary__miscellaneous__Rem():
from sympy.functions.elementary.miscellaneous import Rem
assert _test_args(Rem(x, 2))
def test_sympy__functions__elementary__piecewise__ExprCondPair():
from sympy.functions.elementary.piecewise import ExprCondPair
assert _test_args(ExprCondPair(1, True))
def test_sympy__functions__elementary__piecewise__Piecewise():
from sympy.functions.elementary.piecewise import Piecewise
assert _test_args(Piecewise((1, x >= 0), (0, True)))
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__TrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__ReciprocalTrigonometricFunction():
pass
@SKIP("abstract class")
def test_sympy__functions__elementary__trigonometric__InverseTrigonometricFunction():
pass
def test_sympy__functions__elementary__trigonometric__acos():
from sympy.functions.elementary.trigonometric import acos
assert _test_args(acos(2))
def test_sympy__functions__elementary__trigonometric__acot():
from sympy.functions.elementary.trigonometric import acot
assert _test_args(acot(2))
def test_sympy__functions__elementary__trigonometric__asin():
from sympy.functions.elementary.trigonometric import asin
assert _test_args(asin(2))
def test_sympy__functions__elementary__trigonometric__asec():
from sympy.functions.elementary.trigonometric import asec
assert _test_args(asec(x))
def test_sympy__functions__elementary__trigonometric__acsc():
from sympy.functions.elementary.trigonometric import acsc
assert _test_args(acsc(x))
def test_sympy__functions__elementary__trigonometric__atan():
from sympy.functions.elementary.trigonometric import atan
assert _test_args(atan(2))
def test_sympy__functions__elementary__trigonometric__atan2():
from sympy.functions.elementary.trigonometric import atan2
assert _test_args(atan2(2, 3))
def test_sympy__functions__elementary__trigonometric__cos():
from sympy.functions.elementary.trigonometric import cos
assert _test_args(cos(2))
def test_sympy__functions__elementary__trigonometric__csc():
from sympy.functions.elementary.trigonometric import csc
assert _test_args(csc(2))
def test_sympy__functions__elementary__trigonometric__cot():
from sympy.functions.elementary.trigonometric import cot
assert _test_args(cot(2))
def test_sympy__functions__elementary__trigonometric__sin():
assert _test_args(sin(2))
def test_sympy__functions__elementary__trigonometric__sinc():
from sympy.functions.elementary.trigonometric import sinc
assert _test_args(sinc(2))
def test_sympy__functions__elementary__trigonometric__sec():
from sympy.functions.elementary.trigonometric import sec
assert _test_args(sec(2))
def test_sympy__functions__elementary__trigonometric__tan():
from sympy.functions.elementary.trigonometric import tan
assert _test_args(tan(2))
@SKIP("abstract class")
def test_sympy__functions__special__bessel__BesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalBesselBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__bessel__SphericalHankelBase():
pass
def test_sympy__functions__special__bessel__besseli():
from sympy.functions.special.bessel import besseli
assert _test_args(besseli(x, 1))
def test_sympy__functions__special__bessel__besselj():
from sympy.functions.special.bessel import besselj
assert _test_args(besselj(x, 1))
def test_sympy__functions__special__bessel__besselk():
from sympy.functions.special.bessel import besselk
assert _test_args(besselk(x, 1))
def test_sympy__functions__special__bessel__bessely():
from sympy.functions.special.bessel import bessely
assert _test_args(bessely(x, 1))
def test_sympy__functions__special__bessel__hankel1():
from sympy.functions.special.bessel import hankel1
assert _test_args(hankel1(x, 1))
def test_sympy__functions__special__bessel__hankel2():
from sympy.functions.special.bessel import hankel2
assert _test_args(hankel2(x, 1))
def test_sympy__functions__special__bessel__jn():
from sympy.functions.special.bessel import jn
assert _test_args(jn(0, x))
def test_sympy__functions__special__bessel__yn():
from sympy.functions.special.bessel import yn
assert _test_args(yn(0, x))
def test_sympy__functions__special__bessel__hn1():
from sympy.functions.special.bessel import hn1
assert _test_args(hn1(0, x))
def test_sympy__functions__special__bessel__hn2():
from sympy.functions.special.bessel import hn2
assert _test_args(hn2(0, x))
def test_sympy__functions__special__bessel__AiryBase():
pass
def test_sympy__functions__special__bessel__airyai():
from sympy.functions.special.bessel import airyai
assert _test_args(airyai(2))
def test_sympy__functions__special__bessel__airybi():
from sympy.functions.special.bessel import airybi
assert _test_args(airybi(2))
def test_sympy__functions__special__bessel__airyaiprime():
from sympy.functions.special.bessel import airyaiprime
assert _test_args(airyaiprime(2))
def test_sympy__functions__special__bessel__airybiprime():
from sympy.functions.special.bessel import airybiprime
assert _test_args(airybiprime(2))
def test_sympy__functions__special__bessel__marcumq():
from sympy.functions.special.bessel import marcumq
assert _test_args(marcumq(x, y, z))
def test_sympy__functions__special__elliptic_integrals__elliptic_k():
from sympy.functions.special.elliptic_integrals import elliptic_k as K
assert _test_args(K(x))
def test_sympy__functions__special__elliptic_integrals__elliptic_f():
from sympy.functions.special.elliptic_integrals import elliptic_f as F
assert _test_args(F(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_e():
from sympy.functions.special.elliptic_integrals import elliptic_e as E
assert _test_args(E(x))
assert _test_args(E(x, y))
def test_sympy__functions__special__elliptic_integrals__elliptic_pi():
from sympy.functions.special.elliptic_integrals import elliptic_pi as P
assert _test_args(P(x, y))
assert _test_args(P(x, y, z))
def test_sympy__functions__special__delta_functions__DiracDelta():
from sympy.functions.special.delta_functions import DiracDelta
assert _test_args(DiracDelta(x, 1))
def test_sympy__functions__special__singularity_functions__SingularityFunction():
from sympy.functions.special.singularity_functions import SingularityFunction
assert _test_args(SingularityFunction(x, y, z))
def test_sympy__functions__special__delta_functions__Heaviside():
from sympy.functions.special.delta_functions import Heaviside
assert _test_args(Heaviside(x))
def test_sympy__functions__special__error_functions__erf():
from sympy.functions.special.error_functions import erf
assert _test_args(erf(2))
def test_sympy__functions__special__error_functions__erfc():
from sympy.functions.special.error_functions import erfc
assert _test_args(erfc(2))
def test_sympy__functions__special__error_functions__erfi():
from sympy.functions.special.error_functions import erfi
assert _test_args(erfi(2))
def test_sympy__functions__special__error_functions__erf2():
from sympy.functions.special.error_functions import erf2
assert _test_args(erf2(2, 3))
def test_sympy__functions__special__error_functions__erfinv():
from sympy.functions.special.error_functions import erfinv
assert _test_args(erfinv(2))
def test_sympy__functions__special__error_functions__erfcinv():
from sympy.functions.special.error_functions import erfcinv
assert _test_args(erfcinv(2))
def test_sympy__functions__special__error_functions__erf2inv():
from sympy.functions.special.error_functions import erf2inv
assert _test_args(erf2inv(2, 3))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__FresnelIntegral():
pass
def test_sympy__functions__special__error_functions__fresnels():
from sympy.functions.special.error_functions import fresnels
assert _test_args(fresnels(2))
def test_sympy__functions__special__error_functions__fresnelc():
from sympy.functions.special.error_functions import fresnelc
assert _test_args(fresnelc(2))
def test_sympy__functions__special__error_functions__erfs():
from sympy.functions.special.error_functions import _erfs
assert _test_args(_erfs(2))
def test_sympy__functions__special__error_functions__Ei():
from sympy.functions.special.error_functions import Ei
assert _test_args(Ei(2))
def test_sympy__functions__special__error_functions__li():
from sympy.functions.special.error_functions import li
assert _test_args(li(2))
def test_sympy__functions__special__error_functions__Li():
from sympy.functions.special.error_functions import Li
assert _test_args(Li(5))
@SKIP("abstract class")
def test_sympy__functions__special__error_functions__TrigonometricIntegral():
pass
def test_sympy__functions__special__error_functions__Si():
from sympy.functions.special.error_functions import Si
assert _test_args(Si(2))
def test_sympy__functions__special__error_functions__Ci():
from sympy.functions.special.error_functions import Ci
assert _test_args(Ci(2))
def test_sympy__functions__special__error_functions__Shi():
from sympy.functions.special.error_functions import Shi
assert _test_args(Shi(2))
def test_sympy__functions__special__error_functions__Chi():
from sympy.functions.special.error_functions import Chi
assert _test_args(Chi(2))
def test_sympy__functions__special__error_functions__expint():
from sympy.functions.special.error_functions import expint
assert _test_args(expint(y, x))
def test_sympy__functions__special__gamma_functions__gamma():
from sympy.functions.special.gamma_functions import gamma
assert _test_args(gamma(x))
def test_sympy__functions__special__gamma_functions__loggamma():
from sympy.functions.special.gamma_functions import loggamma
assert _test_args(loggamma(x))
def test_sympy__functions__special__gamma_functions__lowergamma():
from sympy.functions.special.gamma_functions import lowergamma
assert _test_args(lowergamma(x, 2))
def test_sympy__functions__special__gamma_functions__polygamma():
from sympy.functions.special.gamma_functions import polygamma
assert _test_args(polygamma(x, 2))
def test_sympy__functions__special__gamma_functions__digamma():
from sympy.functions.special.gamma_functions import digamma
assert _test_args(digamma(x))
def test_sympy__functions__special__gamma_functions__trigamma():
from sympy.functions.special.gamma_functions import trigamma
assert _test_args(trigamma(x))
def test_sympy__functions__special__gamma_functions__uppergamma():
from sympy.functions.special.gamma_functions import uppergamma
assert _test_args(uppergamma(x, 2))
def test_sympy__functions__special__gamma_functions__multigamma():
from sympy.functions.special.gamma_functions import multigamma
assert _test_args(multigamma(x, 1))
def test_sympy__functions__special__beta_functions__beta():
from sympy.functions.special.beta_functions import beta
assert _test_args(beta(x))
assert _test_args(beta(x, x))
def test_sympy__functions__special__beta_functions__betainc():
from sympy.functions.special.beta_functions import betainc
assert _test_args(betainc(a, b, x, y))
def test_sympy__functions__special__beta_functions__betainc_regularized():
from sympy.functions.special.beta_functions import betainc_regularized
assert _test_args(betainc_regularized(a, b, x, y))
def test_sympy__functions__special__mathieu_functions__MathieuBase():
pass
def test_sympy__functions__special__mathieu_functions__mathieus():
from sympy.functions.special.mathieu_functions import mathieus
assert _test_args(mathieus(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieuc():
from sympy.functions.special.mathieu_functions import mathieuc
assert _test_args(mathieuc(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieusprime():
from sympy.functions.special.mathieu_functions import mathieusprime
assert _test_args(mathieusprime(1, 1, 1))
def test_sympy__functions__special__mathieu_functions__mathieucprime():
from sympy.functions.special.mathieu_functions import mathieucprime
assert _test_args(mathieucprime(1, 1, 1))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleParametersBase():
pass
@SKIP("abstract class")
def test_sympy__functions__special__hyper__TupleArg():
pass
def test_sympy__functions__special__hyper__hyper():
from sympy.functions.special.hyper import hyper
assert _test_args(hyper([1, 2, 3], [4, 5], x))
def test_sympy__functions__special__hyper__meijerg():
from sympy.functions.special.hyper import meijerg
assert _test_args(meijerg([1, 2, 3], [4, 5], [6], [], x))
@SKIP("abstract class")
def test_sympy__functions__special__hyper__HyperRep():
pass
def test_sympy__functions__special__hyper__HyperRep_power1():
from sympy.functions.special.hyper import HyperRep_power1
assert _test_args(HyperRep_power1(x, y))
def test_sympy__functions__special__hyper__HyperRep_power2():
from sympy.functions.special.hyper import HyperRep_power2
assert _test_args(HyperRep_power2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log1():
from sympy.functions.special.hyper import HyperRep_log1
assert _test_args(HyperRep_log1(x))
def test_sympy__functions__special__hyper__HyperRep_atanh():
from sympy.functions.special.hyper import HyperRep_atanh
assert _test_args(HyperRep_atanh(x))
def test_sympy__functions__special__hyper__HyperRep_asin1():
from sympy.functions.special.hyper import HyperRep_asin1
assert _test_args(HyperRep_asin1(x))
def test_sympy__functions__special__hyper__HyperRep_asin2():
from sympy.functions.special.hyper import HyperRep_asin2
assert _test_args(HyperRep_asin2(x))
def test_sympy__functions__special__hyper__HyperRep_sqrts1():
from sympy.functions.special.hyper import HyperRep_sqrts1
assert _test_args(HyperRep_sqrts1(x, y))
def test_sympy__functions__special__hyper__HyperRep_sqrts2():
from sympy.functions.special.hyper import HyperRep_sqrts2
assert _test_args(HyperRep_sqrts2(x, y))
def test_sympy__functions__special__hyper__HyperRep_log2():
from sympy.functions.special.hyper import HyperRep_log2
assert _test_args(HyperRep_log2(x))
def test_sympy__functions__special__hyper__HyperRep_cosasin():
from sympy.functions.special.hyper import HyperRep_cosasin
assert _test_args(HyperRep_cosasin(x, y))
def test_sympy__functions__special__hyper__HyperRep_sinasin():
from sympy.functions.special.hyper import HyperRep_sinasin
assert _test_args(HyperRep_sinasin(x, y))
def test_sympy__functions__special__hyper__appellf1():
from sympy.functions.special.hyper import appellf1
a, b1, b2, c, x, y = symbols('a b1 b2 c x y')
assert _test_args(appellf1(a, b1, b2, c, x, y))
@SKIP("abstract class")
def test_sympy__functions__special__polynomials__OrthogonalPolynomial():
pass
def test_sympy__functions__special__polynomials__jacobi():
from sympy.functions.special.polynomials import jacobi
assert _test_args(jacobi(x, y, 2, 2))
def test_sympy__functions__special__polynomials__gegenbauer():
from sympy.functions.special.polynomials import gegenbauer
assert _test_args(gegenbauer(x, 2, 2))
def test_sympy__functions__special__polynomials__chebyshevt():
from sympy.functions.special.polynomials import chebyshevt
assert _test_args(chebyshevt(x, 2))
def test_sympy__functions__special__polynomials__chebyshevt_root():
from sympy.functions.special.polynomials import chebyshevt_root
assert _test_args(chebyshevt_root(3, 2))
def test_sympy__functions__special__polynomials__chebyshevu():
from sympy.functions.special.polynomials import chebyshevu
assert _test_args(chebyshevu(x, 2))
def test_sympy__functions__special__polynomials__chebyshevu_root():
from sympy.functions.special.polynomials import chebyshevu_root
assert _test_args(chebyshevu_root(3, 2))
def test_sympy__functions__special__polynomials__hermite():
from sympy.functions.special.polynomials import hermite
assert _test_args(hermite(x, 2))
def test_sympy__functions__special__polynomials__legendre():
from sympy.functions.special.polynomials import legendre
assert _test_args(legendre(x, 2))
def test_sympy__functions__special__polynomials__assoc_legendre():
from sympy.functions.special.polynomials import assoc_legendre
assert _test_args(assoc_legendre(x, 0, y))
def test_sympy__functions__special__polynomials__laguerre():
from sympy.functions.special.polynomials import laguerre
assert _test_args(laguerre(x, 2))
def test_sympy__functions__special__polynomials__assoc_laguerre():
from sympy.functions.special.polynomials import assoc_laguerre
assert _test_args(assoc_laguerre(x, 0, y))
def test_sympy__functions__special__spherical_harmonics__Ynm():
from sympy.functions.special.spherical_harmonics import Ynm
assert _test_args(Ynm(1, 1, x, y))
def test_sympy__functions__special__spherical_harmonics__Znm():
from sympy.functions.special.spherical_harmonics import Znm
assert _test_args(Znm(x, y, 1, 1))
def test_sympy__functions__special__tensor_functions__LeviCivita():
from sympy.functions.special.tensor_functions import LeviCivita
assert _test_args(LeviCivita(x, y, 2))
def test_sympy__functions__special__tensor_functions__KroneckerDelta():
from sympy.functions.special.tensor_functions import KroneckerDelta
assert _test_args(KroneckerDelta(x, y))
def test_sympy__functions__special__zeta_functions__dirichlet_eta():
from sympy.functions.special.zeta_functions import dirichlet_eta
assert _test_args(dirichlet_eta(x))
def test_sympy__functions__special__zeta_functions__riemann_xi():
from sympy.functions.special.zeta_functions import riemann_xi
assert _test_args(riemann_xi(x))
def test_sympy__functions__special__zeta_functions__zeta():
from sympy.functions.special.zeta_functions import zeta
assert _test_args(zeta(101))
def test_sympy__functions__special__zeta_functions__lerchphi():
from sympy.functions.special.zeta_functions import lerchphi
assert _test_args(lerchphi(x, y, z))
def test_sympy__functions__special__zeta_functions__polylog():
from sympy.functions.special.zeta_functions import polylog
assert _test_args(polylog(x, y))
def test_sympy__functions__special__zeta_functions__stieltjes():
from sympy.functions.special.zeta_functions import stieltjes
assert _test_args(stieltjes(x, y))
def test_sympy__integrals__integrals__Integral():
from sympy.integrals.integrals import Integral
assert _test_args(Integral(2, (x, 0, 1)))
def test_sympy__integrals__risch__NonElementaryIntegral():
from sympy.integrals.risch import NonElementaryIntegral
assert _test_args(NonElementaryIntegral(exp(-x**2), x))
@SKIP("abstract class")
def test_sympy__integrals__transforms__IntegralTransform():
pass
def test_sympy__integrals__transforms__MellinTransform():
from sympy.integrals.transforms import MellinTransform
assert _test_args(MellinTransform(2, x, y))
def test_sympy__integrals__transforms__InverseMellinTransform():
from sympy.integrals.transforms import InverseMellinTransform
assert _test_args(InverseMellinTransform(2, x, y, 0, 1))
def test_sympy__integrals__transforms__LaplaceTransform():
from sympy.integrals.transforms import LaplaceTransform
assert _test_args(LaplaceTransform(2, x, y))
def test_sympy__integrals__transforms__InverseLaplaceTransform():
from sympy.integrals.transforms import InverseLaplaceTransform
assert _test_args(InverseLaplaceTransform(2, x, y, 0))
@SKIP("abstract class")
def test_sympy__integrals__transforms__FourierTypeTransform():
pass
def test_sympy__integrals__transforms__InverseFourierTransform():
from sympy.integrals.transforms import InverseFourierTransform
assert _test_args(InverseFourierTransform(2, x, y))
def test_sympy__integrals__transforms__FourierTransform():
from sympy.integrals.transforms import FourierTransform
assert _test_args(FourierTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__SineCosineTypeTransform():
pass
def test_sympy__integrals__transforms__InverseSineTransform():
from sympy.integrals.transforms import InverseSineTransform
assert _test_args(InverseSineTransform(2, x, y))
def test_sympy__integrals__transforms__SineTransform():
from sympy.integrals.transforms import SineTransform
assert _test_args(SineTransform(2, x, y))
def test_sympy__integrals__transforms__InverseCosineTransform():
from sympy.integrals.transforms import InverseCosineTransform
assert _test_args(InverseCosineTransform(2, x, y))
def test_sympy__integrals__transforms__CosineTransform():
from sympy.integrals.transforms import CosineTransform
assert _test_args(CosineTransform(2, x, y))
@SKIP("abstract class")
def test_sympy__integrals__transforms__HankelTypeTransform():
pass
def test_sympy__integrals__transforms__InverseHankelTransform():
from sympy.integrals.transforms import InverseHankelTransform
assert _test_args(InverseHankelTransform(2, x, y, 0))
def test_sympy__integrals__transforms__HankelTransform():
from sympy.integrals.transforms import HankelTransform
assert _test_args(HankelTransform(2, x, y, 0))
def test_sympy__liealgebras__cartan_type__Standard_Cartan():
from sympy.liealgebras.cartan_type import Standard_Cartan
assert _test_args(Standard_Cartan("A", 2))
def test_sympy__liealgebras__weyl_group__WeylGroup():
from sympy.liealgebras.weyl_group import WeylGroup
assert _test_args(WeylGroup("B4"))
def test_sympy__liealgebras__root_system__RootSystem():
from sympy.liealgebras.root_system import RootSystem
assert _test_args(RootSystem("A2"))
def test_sympy__liealgebras__type_a__TypeA():
from sympy.liealgebras.type_a import TypeA
assert _test_args(TypeA(2))
def test_sympy__liealgebras__type_b__TypeB():
from sympy.liealgebras.type_b import TypeB
assert _test_args(TypeB(4))
def test_sympy__liealgebras__type_c__TypeC():
from sympy.liealgebras.type_c import TypeC
assert _test_args(TypeC(4))
def test_sympy__liealgebras__type_d__TypeD():
from sympy.liealgebras.type_d import TypeD
assert _test_args(TypeD(4))
def test_sympy__liealgebras__type_e__TypeE():
from sympy.liealgebras.type_e import TypeE
assert _test_args(TypeE(6))
def test_sympy__liealgebras__type_f__TypeF():
from sympy.liealgebras.type_f import TypeF
assert _test_args(TypeF(4))
def test_sympy__liealgebras__type_g__TypeG():
from sympy.liealgebras.type_g import TypeG
assert _test_args(TypeG(2))
def test_sympy__logic__boolalg__And():
from sympy.logic.boolalg import And
assert _test_args(And(x, y, 1))
@SKIP("abstract class")
def test_sympy__logic__boolalg__Boolean():
pass
def test_sympy__logic__boolalg__BooleanFunction():
from sympy.logic.boolalg import BooleanFunction
assert _test_args(BooleanFunction(1, 2, 3))
@SKIP("abstract class")
def test_sympy__logic__boolalg__BooleanAtom():
pass
def test_sympy__logic__boolalg__BooleanTrue():
from sympy.logic.boolalg import true
assert _test_args(true)
def test_sympy__logic__boolalg__BooleanFalse():
from sympy.logic.boolalg import false
assert _test_args(false)
def test_sympy__logic__boolalg__Equivalent():
from sympy.logic.boolalg import Equivalent
assert _test_args(Equivalent(x, 2))
def test_sympy__logic__boolalg__ITE():
from sympy.logic.boolalg import ITE
assert _test_args(ITE(x, y, 1))
def test_sympy__logic__boolalg__Implies():
from sympy.logic.boolalg import Implies
assert _test_args(Implies(x, y))
def test_sympy__logic__boolalg__Nand():
from sympy.logic.boolalg import Nand
assert _test_args(Nand(x, y, 1))
def test_sympy__logic__boolalg__Nor():
from sympy.logic.boolalg import Nor
assert _test_args(Nor(x, y))
def test_sympy__logic__boolalg__Not():
from sympy.logic.boolalg import Not
assert _test_args(Not(x))
def test_sympy__logic__boolalg__Or():
from sympy.logic.boolalg import Or
assert _test_args(Or(x, y))
def test_sympy__logic__boolalg__Xor():
from sympy.logic.boolalg import Xor
assert _test_args(Xor(x, y, 2))
def test_sympy__logic__boolalg__Xnor():
from sympy.logic.boolalg import Xnor
assert _test_args(Xnor(x, y, 2))
def test_sympy__logic__boolalg__Exclusive():
from sympy.logic.boolalg import Exclusive
assert _test_args(Exclusive(x, y, z))
def test_sympy__matrices__matrices__DeferredVector():
from sympy.matrices.matrices import DeferredVector
assert _test_args(DeferredVector("X"))
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixBase():
pass
@SKIP("abstract class")
def test_sympy__matrices__immutable__ImmutableRepMatrix():
pass
def test_sympy__matrices__immutable__ImmutableDenseMatrix():
from sympy.matrices.immutable import ImmutableDenseMatrix
m = ImmutableDenseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableDenseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__immutable__ImmutableSparseMatrix():
from sympy.matrices.immutable import ImmutableSparseMatrix
m = ImmutableSparseMatrix([[1, 2], [3, 4]])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, {(0, 0): 1})
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(1, 1, [1])
assert _test_args(m)
assert _test_args(Basic(*list(m)))
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1)
assert m[0, 0] is S.One
m = ImmutableSparseMatrix(2, 2, lambda i, j: 1/(1 + i) + 1/(1 + j))
assert m[1, 1] is S.One # true div. will give 1.0 if i,j not sympified
assert _test_args(m)
assert _test_args(Basic(*list(m)))
def test_sympy__matrices__expressions__slice__MatrixSlice():
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 4, 4)
assert _test_args(MatrixSlice(X, (0, 2), (0, 2)))
def test_sympy__matrices__expressions__applyfunc__ElementwiseApplyFunction():
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol("X", x, x)
func = Lambda(x, x**2)
assert _test_args(ElementwiseApplyFunction(func, X))
def test_sympy__matrices__expressions__blockmatrix__BlockDiagMatrix():
from sympy.matrices.expressions.blockmatrix import BlockDiagMatrix
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
assert _test_args(BlockDiagMatrix(X, Y))
def test_sympy__matrices__expressions__blockmatrix__BlockMatrix():
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
Z = MatrixSymbol('Z', x, y)
O = ZeroMatrix(y, x)
assert _test_args(BlockMatrix([[X, Z], [O, Y]]))
def test_sympy__matrices__expressions__inverse__Inverse():
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Inverse(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__matadd__MatAdd():
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(MatAdd(X, Y))
@SKIP("abstract class")
def test_sympy__matrices__expressions__matexpr__MatrixExpr():
pass
def test_sympy__matrices__expressions__matexpr__MatrixElement():
from sympy.matrices.expressions.matexpr import MatrixSymbol, MatrixElement
from sympy.core.singleton import S
assert _test_args(MatrixElement(MatrixSymbol('A', 3, 5), S(2), S(3)))
def test_sympy__matrices__expressions__matexpr__MatrixSymbol():
from sympy.matrices.expressions.matexpr import MatrixSymbol
assert _test_args(MatrixSymbol('A', 3, 5))
def test_sympy__matrices__expressions__special__OneMatrix():
from sympy.matrices.expressions.special import OneMatrix
assert _test_args(OneMatrix(3, 5))
def test_sympy__matrices__expressions__special__ZeroMatrix():
from sympy.matrices.expressions.special import ZeroMatrix
assert _test_args(ZeroMatrix(3, 5))
def test_sympy__matrices__expressions__special__GenericZeroMatrix():
from sympy.matrices.expressions.special import GenericZeroMatrix
assert _test_args(GenericZeroMatrix())
def test_sympy__matrices__expressions__special__Identity():
from sympy.matrices.expressions.special import Identity
assert _test_args(Identity(3))
def test_sympy__matrices__expressions__special__GenericIdentity():
from sympy.matrices.expressions.special import GenericIdentity
assert _test_args(GenericIdentity())
def test_sympy__matrices__expressions__sets__MatrixSet():
from sympy.matrices.expressions.sets import MatrixSet
from sympy.core.singleton import S
assert _test_args(MatrixSet(2, 2, S.Reals))
def test_sympy__matrices__expressions__matmul__MatMul():
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', y, x)
assert _test_args(MatMul(X, Y))
def test_sympy__matrices__expressions__dotproduct__DotProduct():
from sympy.matrices.expressions.dotproduct import DotProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, 1)
Y = MatrixSymbol('Y', x, 1)
assert _test_args(DotProduct(X, Y))
def test_sympy__matrices__expressions__diagonal__DiagonalMatrix():
from sympy.matrices.expressions.diagonal import DiagonalMatrix
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagonalMatrix(x))
def test_sympy__matrices__expressions__diagonal__DiagonalOf():
from sympy.matrices.expressions.diagonal import DiagonalOf
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('x', 10, 10)
assert _test_args(DiagonalOf(X))
def test_sympy__matrices__expressions__diagonal__DiagMatrix():
from sympy.matrices.expressions.diagonal import DiagMatrix
from sympy.matrices.expressions import MatrixSymbol
x = MatrixSymbol('x', 10, 1)
assert _test_args(DiagMatrix(x))
def test_sympy__matrices__expressions__hadamard__HadamardProduct():
from sympy.matrices.expressions.hadamard import HadamardProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(HadamardProduct(X, Y))
def test_sympy__matrices__expressions__hadamard__HadamardPower():
from sympy.matrices.expressions.hadamard import HadamardPower
from sympy.matrices.expressions import MatrixSymbol
from sympy.core.symbol import Symbol
X = MatrixSymbol('X', x, y)
n = Symbol("n")
assert _test_args(HadamardPower(X, n))
def test_sympy__matrices__expressions__kronecker__KroneckerProduct():
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, y)
Y = MatrixSymbol('Y', x, y)
assert _test_args(KroneckerProduct(X, Y))
def test_sympy__matrices__expressions__matpow__MatPow():
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', x, x)
assert _test_args(MatPow(X, 2))
def test_sympy__matrices__expressions__transpose__Transpose():
from sympy.matrices.expressions.transpose import Transpose
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Transpose(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__adjoint__Adjoint():
from sympy.matrices.expressions.adjoint import Adjoint
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Adjoint(MatrixSymbol('A', 3, 5)))
def test_sympy__matrices__expressions__trace__Trace():
from sympy.matrices.expressions.trace import Trace
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Trace(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__determinant__Determinant():
from sympy.matrices.expressions.determinant import Determinant
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Determinant(MatrixSymbol('A', 3, 3)))
def test_sympy__matrices__expressions__determinant__Permanent():
from sympy.matrices.expressions.determinant import Permanent
from sympy.matrices.expressions import MatrixSymbol
assert _test_args(Permanent(MatrixSymbol('A', 3, 4)))
def test_sympy__matrices__expressions__funcmatrix__FunctionMatrix():
from sympy.matrices.expressions.funcmatrix import FunctionMatrix
from sympy.core.symbol import symbols
i, j = symbols('i,j')
assert _test_args(FunctionMatrix(3, 3, Lambda((i, j), i - j) ))
def test_sympy__matrices__expressions__fourier__DFT():
from sympy.matrices.expressions.fourier import DFT
from sympy.core.singleton import S
assert _test_args(DFT(S(2)))
def test_sympy__matrices__expressions__fourier__IDFT():
from sympy.matrices.expressions.fourier import IDFT
from sympy.core.singleton import S
assert _test_args(IDFT(S(2)))
from sympy.matrices.expressions import MatrixSymbol
X = MatrixSymbol('X', 10, 10)
def test_sympy__matrices__expressions__factorizations__LofLU():
from sympy.matrices.expressions.factorizations import LofLU
assert _test_args(LofLU(X))
def test_sympy__matrices__expressions__factorizations__UofLU():
from sympy.matrices.expressions.factorizations import UofLU
assert _test_args(UofLU(X))
def test_sympy__matrices__expressions__factorizations__QofQR():
from sympy.matrices.expressions.factorizations import QofQR
assert _test_args(QofQR(X))
def test_sympy__matrices__expressions__factorizations__RofQR():
from sympy.matrices.expressions.factorizations import RofQR
assert _test_args(RofQR(X))
def test_sympy__matrices__expressions__factorizations__LofCholesky():
from sympy.matrices.expressions.factorizations import LofCholesky
assert _test_args(LofCholesky(X))
def test_sympy__matrices__expressions__factorizations__UofCholesky():
from sympy.matrices.expressions.factorizations import UofCholesky
assert _test_args(UofCholesky(X))
def test_sympy__matrices__expressions__factorizations__EigenVectors():
from sympy.matrices.expressions.factorizations import EigenVectors
assert _test_args(EigenVectors(X))
def test_sympy__matrices__expressions__factorizations__EigenValues():
from sympy.matrices.expressions.factorizations import EigenValues
assert _test_args(EigenValues(X))
def test_sympy__matrices__expressions__factorizations__UofSVD():
from sympy.matrices.expressions.factorizations import UofSVD
assert _test_args(UofSVD(X))
def test_sympy__matrices__expressions__factorizations__VofSVD():
from sympy.matrices.expressions.factorizations import VofSVD
assert _test_args(VofSVD(X))
def test_sympy__matrices__expressions__factorizations__SofSVD():
from sympy.matrices.expressions.factorizations import SofSVD
assert _test_args(SofSVD(X))
@SKIP("abstract class")
def test_sympy__matrices__expressions__factorizations__Factorization():
pass
def test_sympy__matrices__expressions__permutation__PermutationMatrix():
from sympy.combinatorics import Permutation
from sympy.matrices.expressions.permutation import PermutationMatrix
assert _test_args(PermutationMatrix(Permutation([2, 0, 1])))
def test_sympy__matrices__expressions__permutation__MatrixPermute():
from sympy.combinatorics import Permutation
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.permutation import MatrixPermute
A = MatrixSymbol('A', 3, 3)
assert _test_args(MatrixPermute(A, Permutation([2, 0, 1])))
def test_sympy__matrices__expressions__companion__CompanionMatrix():
from sympy.core.symbol import Symbol
from sympy.matrices.expressions.companion import CompanionMatrix
from sympy.polys.polytools import Poly
x = Symbol('x')
p = Poly([1, 2, 3], x)
assert _test_args(CompanionMatrix(p))
def test_sympy__physics__vector__frame__CoordinateSym():
from sympy.physics.vector import CoordinateSym
from sympy.physics.vector import ReferenceFrame
assert _test_args(CoordinateSym('R_x', ReferenceFrame('R'), 0))
def test_sympy__physics__paulialgebra__Pauli():
from sympy.physics.paulialgebra import Pauli
assert _test_args(Pauli(1))
def test_sympy__physics__quantum__anticommutator__AntiCommutator():
from sympy.physics.quantum.anticommutator import AntiCommutator
assert _test_args(AntiCommutator(x, y))
def test_sympy__physics__quantum__cartesian__PositionBra3D():
from sympy.physics.quantum.cartesian import PositionBra3D
assert _test_args(PositionBra3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionKet3D():
from sympy.physics.quantum.cartesian import PositionKet3D
assert _test_args(PositionKet3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PositionState3D():
from sympy.physics.quantum.cartesian import PositionState3D
assert _test_args(PositionState3D(x, y, z))
def test_sympy__physics__quantum__cartesian__PxBra():
from sympy.physics.quantum.cartesian import PxBra
assert _test_args(PxBra(x, y, z))
def test_sympy__physics__quantum__cartesian__PxKet():
from sympy.physics.quantum.cartesian import PxKet
assert _test_args(PxKet(x, y, z))
def test_sympy__physics__quantum__cartesian__PxOp():
from sympy.physics.quantum.cartesian import PxOp
assert _test_args(PxOp(x, y, z))
def test_sympy__physics__quantum__cartesian__XBra():
from sympy.physics.quantum.cartesian import XBra
assert _test_args(XBra(x))
def test_sympy__physics__quantum__cartesian__XKet():
from sympy.physics.quantum.cartesian import XKet
assert _test_args(XKet(x))
def test_sympy__physics__quantum__cartesian__XOp():
from sympy.physics.quantum.cartesian import XOp
assert _test_args(XOp(x))
def test_sympy__physics__quantum__cartesian__YOp():
from sympy.physics.quantum.cartesian import YOp
assert _test_args(YOp(x))
def test_sympy__physics__quantum__cartesian__ZOp():
from sympy.physics.quantum.cartesian import ZOp
assert _test_args(ZOp(x))
def test_sympy__physics__quantum__cg__CG():
from sympy.physics.quantum.cg import CG
from sympy.core.singleton import S
assert _test_args(CG(Rational(3, 2), Rational(3, 2), S.Half, Rational(-1, 2), 1, 1))
def test_sympy__physics__quantum__cg__Wigner3j():
from sympy.physics.quantum.cg import Wigner3j
assert _test_args(Wigner3j(6, 0, 4, 0, 2, 0))
def test_sympy__physics__quantum__cg__Wigner6j():
from sympy.physics.quantum.cg import Wigner6j
assert _test_args(Wigner6j(1, 2, 3, 2, 1, 2))
def test_sympy__physics__quantum__cg__Wigner9j():
from sympy.physics.quantum.cg import Wigner9j
assert _test_args(Wigner9j(2, 1, 1, Rational(3, 2), S.Half, 1, S.Half, S.Half, 0))
def test_sympy__physics__quantum__circuitplot__Mz():
from sympy.physics.quantum.circuitplot import Mz
assert _test_args(Mz(0))
def test_sympy__physics__quantum__circuitplot__Mx():
from sympy.physics.quantum.circuitplot import Mx
assert _test_args(Mx(0))
def test_sympy__physics__quantum__commutator__Commutator():
from sympy.physics.quantum.commutator import Commutator
A, B = symbols('A,B', commutative=False)
assert _test_args(Commutator(A, B))
def test_sympy__physics__quantum__constants__HBar():
from sympy.physics.quantum.constants import HBar
assert _test_args(HBar())
def test_sympy__physics__quantum__dagger__Dagger():
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.state import Ket
assert _test_args(Dagger(Dagger(Ket('psi'))))
def test_sympy__physics__quantum__gate__CGate():
from sympy.physics.quantum.gate import CGate, Gate
assert _test_args(CGate((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CGateS():
from sympy.physics.quantum.gate import CGateS, Gate
assert _test_args(CGateS((0, 1), Gate(2)))
def test_sympy__physics__quantum__gate__CNotGate():
from sympy.physics.quantum.gate import CNotGate
assert _test_args(CNotGate(0, 1))
def test_sympy__physics__quantum__gate__Gate():
from sympy.physics.quantum.gate import Gate
assert _test_args(Gate(0))
def test_sympy__physics__quantum__gate__HadamardGate():
from sympy.physics.quantum.gate import HadamardGate
assert _test_args(HadamardGate(0))
def test_sympy__physics__quantum__gate__IdentityGate():
from sympy.physics.quantum.gate import IdentityGate
assert _test_args(IdentityGate(0))
def test_sympy__physics__quantum__gate__OneQubitGate():
from sympy.physics.quantum.gate import OneQubitGate
assert _test_args(OneQubitGate(0))
def test_sympy__physics__quantum__gate__PhaseGate():
from sympy.physics.quantum.gate import PhaseGate
assert _test_args(PhaseGate(0))
def test_sympy__physics__quantum__gate__SwapGate():
from sympy.physics.quantum.gate import SwapGate
assert _test_args(SwapGate(0, 1))
def test_sympy__physics__quantum__gate__TGate():
from sympy.physics.quantum.gate import TGate
assert _test_args(TGate(0))
def test_sympy__physics__quantum__gate__TwoQubitGate():
from sympy.physics.quantum.gate import TwoQubitGate
assert _test_args(TwoQubitGate(0))
def test_sympy__physics__quantum__gate__UGate():
from sympy.physics.quantum.gate import UGate
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.core.containers import Tuple
from sympy.core.numbers import Integer
assert _test_args(
UGate(Tuple(Integer(1)), ImmutableDenseMatrix([[1, 0], [0, 2]])))
def test_sympy__physics__quantum__gate__XGate():
from sympy.physics.quantum.gate import XGate
assert _test_args(XGate(0))
def test_sympy__physics__quantum__gate__YGate():
from sympy.physics.quantum.gate import YGate
assert _test_args(YGate(0))
def test_sympy__physics__quantum__gate__ZGate():
from sympy.physics.quantum.gate import ZGate
assert _test_args(ZGate(0))
def test_sympy__physics__quantum__grover__OracleGateFunction():
from sympy.physics.quantum.grover import OracleGateFunction
@OracleGateFunction
def f(qubit):
return
assert _test_args(f)
def test_sympy__physics__quantum__grover__OracleGate():
from sympy.physics.quantum.grover import OracleGate
def f(qubit):
return
assert _test_args(OracleGate(1,f))
def test_sympy__physics__quantum__grover__WGate():
from sympy.physics.quantum.grover import WGate
assert _test_args(WGate(1))
def test_sympy__physics__quantum__hilbert__ComplexSpace():
from sympy.physics.quantum.hilbert import ComplexSpace
assert _test_args(ComplexSpace(x))
def test_sympy__physics__quantum__hilbert__DirectSumHilbertSpace():
from sympy.physics.quantum.hilbert import DirectSumHilbertSpace, ComplexSpace, FockSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(DirectSumHilbertSpace(c, f))
def test_sympy__physics__quantum__hilbert__FockSpace():
from sympy.physics.quantum.hilbert import FockSpace
assert _test_args(FockSpace())
def test_sympy__physics__quantum__hilbert__HilbertSpace():
from sympy.physics.quantum.hilbert import HilbertSpace
assert _test_args(HilbertSpace())
def test_sympy__physics__quantum__hilbert__L2():
from sympy.physics.quantum.hilbert import L2
from sympy.core.numbers import oo
from sympy.sets.sets import Interval
assert _test_args(L2(Interval(0, oo)))
def test_sympy__physics__quantum__hilbert__TensorPowerHilbertSpace():
from sympy.physics.quantum.hilbert import TensorPowerHilbertSpace, FockSpace
f = FockSpace()
assert _test_args(TensorPowerHilbertSpace(f, 2))
def test_sympy__physics__quantum__hilbert__TensorProductHilbertSpace():
from sympy.physics.quantum.hilbert import TensorProductHilbertSpace, FockSpace, ComplexSpace
c = ComplexSpace(2)
f = FockSpace()
assert _test_args(TensorProductHilbertSpace(f, c))
def test_sympy__physics__quantum__innerproduct__InnerProduct():
from sympy.physics.quantum import Bra, Ket, InnerProduct
b = Bra('b')
k = Ket('k')
assert _test_args(InnerProduct(b, k))
def test_sympy__physics__quantum__operator__DifferentialOperator():
from sympy.physics.quantum.operator import DifferentialOperator
from sympy.core.function import (Derivative, Function)
f = Function('f')
assert _test_args(DifferentialOperator(1/x*Derivative(f(x), x), f(x)))
def test_sympy__physics__quantum__operator__HermitianOperator():
from sympy.physics.quantum.operator import HermitianOperator
assert _test_args(HermitianOperator('H'))
def test_sympy__physics__quantum__operator__IdentityOperator():
from sympy.physics.quantum.operator import IdentityOperator
assert _test_args(IdentityOperator(5))
def test_sympy__physics__quantum__operator__Operator():
from sympy.physics.quantum.operator import Operator
assert _test_args(Operator('A'))
def test_sympy__physics__quantum__operator__OuterProduct():
from sympy.physics.quantum.operator import OuterProduct
from sympy.physics.quantum import Ket, Bra
b = Bra('b')
k = Ket('k')
assert _test_args(OuterProduct(k, b))
def test_sympy__physics__quantum__operator__UnitaryOperator():
from sympy.physics.quantum.operator import UnitaryOperator
assert _test_args(UnitaryOperator('U'))
def test_sympy__physics__quantum__piab__PIABBra():
from sympy.physics.quantum.piab import PIABBra
assert _test_args(PIABBra('B'))
def test_sympy__physics__quantum__boson__BosonOp():
from sympy.physics.quantum.boson import BosonOp
assert _test_args(BosonOp('a'))
assert _test_args(BosonOp('a', False))
def test_sympy__physics__quantum__boson__BosonFockKet():
from sympy.physics.quantum.boson import BosonFockKet
assert _test_args(BosonFockKet(1))
def test_sympy__physics__quantum__boson__BosonFockBra():
from sympy.physics.quantum.boson import BosonFockBra
assert _test_args(BosonFockBra(1))
def test_sympy__physics__quantum__boson__BosonCoherentKet():
from sympy.physics.quantum.boson import BosonCoherentKet
assert _test_args(BosonCoherentKet(1))
def test_sympy__physics__quantum__boson__BosonCoherentBra():
from sympy.physics.quantum.boson import BosonCoherentBra
assert _test_args(BosonCoherentBra(1))
def test_sympy__physics__quantum__fermion__FermionOp():
from sympy.physics.quantum.fermion import FermionOp
assert _test_args(FermionOp('c'))
assert _test_args(FermionOp('c', False))
def test_sympy__physics__quantum__fermion__FermionFockKet():
from sympy.physics.quantum.fermion import FermionFockKet
assert _test_args(FermionFockKet(1))
def test_sympy__physics__quantum__fermion__FermionFockBra():
from sympy.physics.quantum.fermion import FermionFockBra
assert _test_args(FermionFockBra(1))
def test_sympy__physics__quantum__pauli__SigmaOpBase():
from sympy.physics.quantum.pauli import SigmaOpBase
assert _test_args(SigmaOpBase())
def test_sympy__physics__quantum__pauli__SigmaX():
from sympy.physics.quantum.pauli import SigmaX
assert _test_args(SigmaX())
def test_sympy__physics__quantum__pauli__SigmaY():
from sympy.physics.quantum.pauli import SigmaY
assert _test_args(SigmaY())
def test_sympy__physics__quantum__pauli__SigmaZ():
from sympy.physics.quantum.pauli import SigmaZ
assert _test_args(SigmaZ())
def test_sympy__physics__quantum__pauli__SigmaMinus():
from sympy.physics.quantum.pauli import SigmaMinus
assert _test_args(SigmaMinus())
def test_sympy__physics__quantum__pauli__SigmaPlus():
from sympy.physics.quantum.pauli import SigmaPlus
assert _test_args(SigmaPlus())
def test_sympy__physics__quantum__pauli__SigmaZKet():
from sympy.physics.quantum.pauli import SigmaZKet
assert _test_args(SigmaZKet(0))
def test_sympy__physics__quantum__pauli__SigmaZBra():
from sympy.physics.quantum.pauli import SigmaZBra
assert _test_args(SigmaZBra(0))
def test_sympy__physics__quantum__piab__PIABHamiltonian():
from sympy.physics.quantum.piab import PIABHamiltonian
assert _test_args(PIABHamiltonian('P'))
def test_sympy__physics__quantum__piab__PIABKet():
from sympy.physics.quantum.piab import PIABKet
assert _test_args(PIABKet('K'))
def test_sympy__physics__quantum__qexpr__QExpr():
from sympy.physics.quantum.qexpr import QExpr
assert _test_args(QExpr(0))
def test_sympy__physics__quantum__qft__Fourier():
from sympy.physics.quantum.qft import Fourier
assert _test_args(Fourier(0, 1))
def test_sympy__physics__quantum__qft__IQFT():
from sympy.physics.quantum.qft import IQFT
assert _test_args(IQFT(0, 1))
def test_sympy__physics__quantum__qft__QFT():
from sympy.physics.quantum.qft import QFT
assert _test_args(QFT(0, 1))
def test_sympy__physics__quantum__qft__RkGate():
from sympy.physics.quantum.qft import RkGate
assert _test_args(RkGate(0, 1))
def test_sympy__physics__quantum__qubit__IntQubit():
from sympy.physics.quantum.qubit import IntQubit
assert _test_args(IntQubit(0))
def test_sympy__physics__quantum__qubit__IntQubitBra():
from sympy.physics.quantum.qubit import IntQubitBra
assert _test_args(IntQubitBra(0))
def test_sympy__physics__quantum__qubit__IntQubitState():
from sympy.physics.quantum.qubit import IntQubitState, QubitState
assert _test_args(IntQubitState(QubitState(0, 1)))
def test_sympy__physics__quantum__qubit__Qubit():
from sympy.physics.quantum.qubit import Qubit
assert _test_args(Qubit(0, 0, 0))
def test_sympy__physics__quantum__qubit__QubitBra():
from sympy.physics.quantum.qubit import QubitBra
assert _test_args(QubitBra('1', 0))
def test_sympy__physics__quantum__qubit__QubitState():
from sympy.physics.quantum.qubit import QubitState
assert _test_args(QubitState(0, 1))
def test_sympy__physics__quantum__density__Density():
from sympy.physics.quantum.density import Density
from sympy.physics.quantum.state import Ket
assert _test_args(Density([Ket(0), 0.5], [Ket(1), 0.5]))
@SKIP("TODO: sympy.physics.quantum.shor: Cmod Not Implemented")
def test_sympy__physics__quantum__shor__CMod():
from sympy.physics.quantum.shor import CMod
assert _test_args(CMod())
def test_sympy__physics__quantum__spin__CoupledSpinState():
from sympy.physics.quantum.spin import CoupledSpinState
assert _test_args(CoupledSpinState(1, 0, (1, 1)))
assert _test_args(CoupledSpinState(1, 0, (1, S.Half, S.Half)))
assert _test_args(CoupledSpinState(
1, 0, (1, S.Half, S.Half), ((2, 3, S.Half), (1, 2, 1)) ))
j, m, j1, j2, j3, j12, x = symbols('j m j1:4 j12 x')
assert CoupledSpinState(
j, m, (j1, j2, j3)).subs(j2, x) == CoupledSpinState(j, m, (j1, x, j3))
assert CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, j12), (1, 2, j)) ).subs(j12, x) == \
CoupledSpinState(j, m, (j1, j2, j3), ((1, 3, x), (1, 2, j)) )
def test_sympy__physics__quantum__spin__J2Op():
from sympy.physics.quantum.spin import J2Op
assert _test_args(J2Op('J'))
def test_sympy__physics__quantum__spin__JminusOp():
from sympy.physics.quantum.spin import JminusOp
assert _test_args(JminusOp('J'))
def test_sympy__physics__quantum__spin__JplusOp():
from sympy.physics.quantum.spin import JplusOp
assert _test_args(JplusOp('J'))
def test_sympy__physics__quantum__spin__JxBra():
from sympy.physics.quantum.spin import JxBra
assert _test_args(JxBra(1, 0))
def test_sympy__physics__quantum__spin__JxBraCoupled():
from sympy.physics.quantum.spin import JxBraCoupled
assert _test_args(JxBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxKet():
from sympy.physics.quantum.spin import JxKet
assert _test_args(JxKet(1, 0))
def test_sympy__physics__quantum__spin__JxKetCoupled():
from sympy.physics.quantum.spin import JxKetCoupled
assert _test_args(JxKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JxOp():
from sympy.physics.quantum.spin import JxOp
assert _test_args(JxOp('J'))
def test_sympy__physics__quantum__spin__JyBra():
from sympy.physics.quantum.spin import JyBra
assert _test_args(JyBra(1, 0))
def test_sympy__physics__quantum__spin__JyBraCoupled():
from sympy.physics.quantum.spin import JyBraCoupled
assert _test_args(JyBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyKet():
from sympy.physics.quantum.spin import JyKet
assert _test_args(JyKet(1, 0))
def test_sympy__physics__quantum__spin__JyKetCoupled():
from sympy.physics.quantum.spin import JyKetCoupled
assert _test_args(JyKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JyOp():
from sympy.physics.quantum.spin import JyOp
assert _test_args(JyOp('J'))
def test_sympy__physics__quantum__spin__JzBra():
from sympy.physics.quantum.spin import JzBra
assert _test_args(JzBra(1, 0))
def test_sympy__physics__quantum__spin__JzBraCoupled():
from sympy.physics.quantum.spin import JzBraCoupled
assert _test_args(JzBraCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzKet():
from sympy.physics.quantum.spin import JzKet
assert _test_args(JzKet(1, 0))
def test_sympy__physics__quantum__spin__JzKetCoupled():
from sympy.physics.quantum.spin import JzKetCoupled
assert _test_args(JzKetCoupled(1, 0, (1, 1)))
def test_sympy__physics__quantum__spin__JzOp():
from sympy.physics.quantum.spin import JzOp
assert _test_args(JzOp('J'))
def test_sympy__physics__quantum__spin__Rotation():
from sympy.physics.quantum.spin import Rotation
assert _test_args(Rotation(pi, 0, pi/2))
def test_sympy__physics__quantum__spin__SpinState():
from sympy.physics.quantum.spin import SpinState
assert _test_args(SpinState(1, 0))
def test_sympy__physics__quantum__spin__WignerD():
from sympy.physics.quantum.spin import WignerD
assert _test_args(WignerD(0, 1, 2, 3, 4, 5))
def test_sympy__physics__quantum__state__Bra():
from sympy.physics.quantum.state import Bra
assert _test_args(Bra(0))
def test_sympy__physics__quantum__state__BraBase():
from sympy.physics.quantum.state import BraBase
assert _test_args(BraBase(0))
def test_sympy__physics__quantum__state__Ket():
from sympy.physics.quantum.state import Ket
assert _test_args(Ket(0))
def test_sympy__physics__quantum__state__KetBase():
from sympy.physics.quantum.state import KetBase
assert _test_args(KetBase(0))
def test_sympy__physics__quantum__state__State():
from sympy.physics.quantum.state import State
assert _test_args(State(0))
def test_sympy__physics__quantum__state__StateBase():
from sympy.physics.quantum.state import StateBase
assert _test_args(StateBase(0))
def test_sympy__physics__quantum__state__OrthogonalBra():
from sympy.physics.quantum.state import OrthogonalBra
assert _test_args(OrthogonalBra(0))
def test_sympy__physics__quantum__state__OrthogonalKet():
from sympy.physics.quantum.state import OrthogonalKet
assert _test_args(OrthogonalKet(0))
def test_sympy__physics__quantum__state__OrthogonalState():
from sympy.physics.quantum.state import OrthogonalState
assert _test_args(OrthogonalState(0))
def test_sympy__physics__quantum__state__TimeDepBra():
from sympy.physics.quantum.state import TimeDepBra
assert _test_args(TimeDepBra('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepKet():
from sympy.physics.quantum.state import TimeDepKet
assert _test_args(TimeDepKet('psi', 't'))
def test_sympy__physics__quantum__state__TimeDepState():
from sympy.physics.quantum.state import TimeDepState
assert _test_args(TimeDepState('psi', 't'))
def test_sympy__physics__quantum__state__Wavefunction():
from sympy.physics.quantum.state import Wavefunction
from sympy.functions import sin
from sympy.functions.elementary.piecewise import Piecewise
n = 1
L = 1
g = Piecewise((0, x < 0), (0, x > L), (sqrt(2//L)*sin(n*pi*x/L), True))
assert _test_args(Wavefunction(g, x))
def test_sympy__physics__quantum__tensorproduct__TensorProduct():
from sympy.physics.quantum.tensorproduct import TensorProduct
x, y = symbols("x y", commutative=False)
assert _test_args(TensorProduct(x, y))
def test_sympy__physics__quantum__identitysearch__GateIdentity():
from sympy.physics.quantum.gate import X
from sympy.physics.quantum.identitysearch import GateIdentity
assert _test_args(GateIdentity(X(0), X(0)))
def test_sympy__physics__quantum__sho1d__SHOOp():
from sympy.physics.quantum.sho1d import SHOOp
assert _test_args(SHOOp('a'))
def test_sympy__physics__quantum__sho1d__RaisingOp():
from sympy.physics.quantum.sho1d import RaisingOp
assert _test_args(RaisingOp('a'))
def test_sympy__physics__quantum__sho1d__LoweringOp():
from sympy.physics.quantum.sho1d import LoweringOp
assert _test_args(LoweringOp('a'))
def test_sympy__physics__quantum__sho1d__NumberOp():
from sympy.physics.quantum.sho1d import NumberOp
assert _test_args(NumberOp('N'))
def test_sympy__physics__quantum__sho1d__Hamiltonian():
from sympy.physics.quantum.sho1d import Hamiltonian
assert _test_args(Hamiltonian('H'))
def test_sympy__physics__quantum__sho1d__SHOState():
from sympy.physics.quantum.sho1d import SHOState
assert _test_args(SHOState(0))
def test_sympy__physics__quantum__sho1d__SHOKet():
from sympy.physics.quantum.sho1d import SHOKet
assert _test_args(SHOKet(0))
def test_sympy__physics__quantum__sho1d__SHOBra():
from sympy.physics.quantum.sho1d import SHOBra
assert _test_args(SHOBra(0))
def test_sympy__physics__secondquant__AnnihilateBoson():
from sympy.physics.secondquant import AnnihilateBoson
assert _test_args(AnnihilateBoson(0))
def test_sympy__physics__secondquant__AnnihilateFermion():
from sympy.physics.secondquant import AnnihilateFermion
assert _test_args(AnnihilateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Annihilator():
pass
def test_sympy__physics__secondquant__AntiSymmetricTensor():
from sympy.physics.secondquant import AntiSymmetricTensor
i, j = symbols('i j', below_fermi=True)
a, b = symbols('a b', above_fermi=True)
assert _test_args(AntiSymmetricTensor('v', (a, i), (b, j)))
def test_sympy__physics__secondquant__BosonState():
from sympy.physics.secondquant import BosonState
assert _test_args(BosonState((0, 1)))
@SKIP("abstract class")
def test_sympy__physics__secondquant__BosonicOperator():
pass
def test_sympy__physics__secondquant__Commutator():
from sympy.physics.secondquant import Commutator
x, y = symbols('x y', commutative=False)
assert _test_args(Commutator(x, y))
def test_sympy__physics__secondquant__CreateBoson():
from sympy.physics.secondquant import CreateBoson
assert _test_args(CreateBoson(0))
def test_sympy__physics__secondquant__CreateFermion():
from sympy.physics.secondquant import CreateFermion
assert _test_args(CreateFermion(0))
@SKIP("abstract class")
def test_sympy__physics__secondquant__Creator():
pass
def test_sympy__physics__secondquant__Dagger():
from sympy.physics.secondquant import Dagger
assert _test_args(Dagger(x))
def test_sympy__physics__secondquant__FermionState():
from sympy.physics.secondquant import FermionState
assert _test_args(FermionState((0, 1)))
def test_sympy__physics__secondquant__FermionicOperator():
from sympy.physics.secondquant import FermionicOperator
assert _test_args(FermionicOperator(0))
def test_sympy__physics__secondquant__FockState():
from sympy.physics.secondquant import FockState
assert _test_args(FockState((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonBra():
from sympy.physics.secondquant import FockStateBosonBra
assert _test_args(FockStateBosonBra((0, 1)))
def test_sympy__physics__secondquant__FockStateBosonKet():
from sympy.physics.secondquant import FockStateBosonKet
assert _test_args(FockStateBosonKet((0, 1)))
def test_sympy__physics__secondquant__FockStateBra():
from sympy.physics.secondquant import FockStateBra
assert _test_args(FockStateBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionBra():
from sympy.physics.secondquant import FockStateFermionBra
assert _test_args(FockStateFermionBra((0, 1)))
def test_sympy__physics__secondquant__FockStateFermionKet():
from sympy.physics.secondquant import FockStateFermionKet
assert _test_args(FockStateFermionKet((0, 1)))
def test_sympy__physics__secondquant__FockStateKet():
from sympy.physics.secondquant import FockStateKet
assert _test_args(FockStateKet((0, 1)))
def test_sympy__physics__secondquant__InnerProduct():
from sympy.physics.secondquant import InnerProduct
from sympy.physics.secondquant import FockStateKet, FockStateBra
assert _test_args(InnerProduct(FockStateBra((0, 1)), FockStateKet((0, 1))))
def test_sympy__physics__secondquant__NO():
from sympy.physics.secondquant import NO, F, Fd
assert _test_args(NO(Fd(x)*F(y)))
def test_sympy__physics__secondquant__PermutationOperator():
from sympy.physics.secondquant import PermutationOperator
assert _test_args(PermutationOperator(0, 1))
def test_sympy__physics__secondquant__SqOperator():
from sympy.physics.secondquant import SqOperator
assert _test_args(SqOperator(0))
def test_sympy__physics__secondquant__TensorSymbol():
from sympy.physics.secondquant import TensorSymbol
assert _test_args(TensorSymbol(x))
def test_sympy__physics__control__lti__LinearTimeInvariant():
# Direct instances of LinearTimeInvariant class are not allowed.
# func(*args) tests for its derived classes (TransferFunction,
# Series, Parallel and TransferFunctionMatrix) should pass.
pass
def test_sympy__physics__control__lti__SISOLinearTimeInvariant():
# Direct instances of SISOLinearTimeInvariant class are not allowed.
pass
def test_sympy__physics__control__lti__MIMOLinearTimeInvariant():
# Direct instances of MIMOLinearTimeInvariant class are not allowed.
pass
def test_sympy__physics__control__lti__TransferFunction():
from sympy.physics.control.lti import TransferFunction
assert _test_args(TransferFunction(2, 3, x))
def test_sympy__physics__control__lti__Series():
from sympy.physics.control import Series, TransferFunction
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
assert _test_args(Series(tf1, tf2))
def test_sympy__physics__control__lti__MIMOSeries():
from sympy.physics.control import MIMOSeries, TransferFunction, TransferFunctionMatrix
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
tfm_1 = TransferFunctionMatrix([[tf2, tf1]])
tfm_2 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
tfm_3 = TransferFunctionMatrix([[tf1], [tf2]])
assert _test_args(MIMOSeries(tfm_3, tfm_2, tfm_1))
def test_sympy__physics__control__lti__Parallel():
from sympy.physics.control import Parallel, TransferFunction
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
assert _test_args(Parallel(tf1, tf2))
def test_sympy__physics__control__lti__MIMOParallel():
from sympy.physics.control import MIMOParallel, TransferFunction, TransferFunctionMatrix
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
assert _test_args(MIMOParallel(tfm_1, tfm_2))
def test_sympy__physics__control__lti__Feedback():
from sympy.physics.control import TransferFunction, Feedback
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
assert _test_args(Feedback(tf1, tf2))
assert _test_args(Feedback(tf1, tf2, 1))
def test_sympy__physics__control__lti__MIMOFeedback():
from sympy.physics.control import TransferFunction, MIMOFeedback, TransferFunctionMatrix
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
tfm_1 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
tfm_2 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
assert _test_args(MIMOFeedback(tfm_1, tfm_2))
assert _test_args(MIMOFeedback(tfm_1, tfm_2, 1))
def test_sympy__physics__control__lti__TransferFunctionMatrix():
from sympy.physics.control import TransferFunction, TransferFunctionMatrix
tf1 = TransferFunction(x**2 - y**3, y - z, x)
tf2 = TransferFunction(y - x, z + y, x)
assert _test_args(TransferFunctionMatrix([[tf1, tf2]]))
def test_sympy__physics__units__dimensions__Dimension():
from sympy.physics.units.dimensions import Dimension
assert _test_args(Dimension("length", "L"))
def test_sympy__physics__units__dimensions__DimensionSystem():
from sympy.physics.units.dimensions import DimensionSystem
from sympy.physics.units.definitions.dimension_definitions import length, time, velocity
assert _test_args(DimensionSystem((length, time), (velocity,)))
def test_sympy__physics__units__quantities__Quantity():
from sympy.physics.units.quantities import Quantity
assert _test_args(Quantity("dam"))
def test_sympy__physics__units__prefixes__Prefix():
from sympy.physics.units.prefixes import Prefix
assert _test_args(Prefix('kilo', 'k', 3))
def test_sympy__core__numbers__AlgebraicNumber():
from sympy.core.numbers import AlgebraicNumber
assert _test_args(AlgebraicNumber(sqrt(2), [1, 2, 3]))
def test_sympy__polys__polytools__GroebnerBasis():
from sympy.polys.polytools import GroebnerBasis
assert _test_args(GroebnerBasis([x, y, z], x, y, z))
def test_sympy__polys__polytools__Poly():
from sympy.polys.polytools import Poly
assert _test_args(Poly(2, x, y))
def test_sympy__polys__polytools__PurePoly():
from sympy.polys.polytools import PurePoly
assert _test_args(PurePoly(2, x, y))
@SKIP('abstract class')
def test_sympy__polys__rootoftools__RootOf():
pass
def test_sympy__polys__rootoftools__ComplexRootOf():
from sympy.polys.rootoftools import ComplexRootOf
assert _test_args(ComplexRootOf(x**3 + x + 1, 0))
def test_sympy__polys__rootoftools__RootSum():
from sympy.polys.rootoftools import RootSum
assert _test_args(RootSum(x**3 + x + 1, sin))
def test_sympy__series__limits__Limit():
from sympy.series.limits import Limit
assert _test_args(Limit(x, x, 0, dir='-'))
def test_sympy__series__order__Order():
from sympy.series.order import Order
assert _test_args(Order(1, x, y))
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqBase():
pass
def test_sympy__series__sequences__EmptySequence():
# Need to imort the instance from series not the class from
# series.sequence
from sympy.series import EmptySequence
assert _test_args(EmptySequence)
@SKIP('Abstract Class')
def test_sympy__series__sequences__SeqExpr():
pass
def test_sympy__series__sequences__SeqPer():
from sympy.series.sequences import SeqPer
assert _test_args(SeqPer((1, 2, 3), (0, 10)))
def test_sympy__series__sequences__SeqFormula():
from sympy.series.sequences import SeqFormula
assert _test_args(SeqFormula(x**2, (0, 10)))
def test_sympy__series__sequences__RecursiveSeq():
from sympy.series.sequences import RecursiveSeq
y = Function("y")
n = symbols("n")
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y(n), n, (0, 1)))
assert _test_args(RecursiveSeq(y(n - 1) + y(n - 2), y(n), n))
def test_sympy__series__sequences__SeqExprOp():
from sympy.series.sequences import SeqExprOp, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqExprOp(s1, s2))
def test_sympy__series__sequences__SeqAdd():
from sympy.series.sequences import SeqAdd, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqAdd(s1, s2))
def test_sympy__series__sequences__SeqMul():
from sympy.series.sequences import SeqMul, sequence
s1 = sequence((1, 2, 3))
s2 = sequence(x**2)
assert _test_args(SeqMul(s1, s2))
@SKIP('Abstract Class')
def test_sympy__series__series_class__SeriesBase():
pass
def test_sympy__series__fourier__FourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(x, (x, -pi, pi)))
def test_sympy__series__fourier__FiniteFourierSeries():
from sympy.series.fourier import fourier_series
assert _test_args(fourier_series(sin(pi*x), (x, -1, 1)))
def test_sympy__series__formal__FormalPowerSeries():
from sympy.series.formal import fps
assert _test_args(fps(log(1 + x), x))
def test_sympy__series__formal__Coeff():
from sympy.series.formal import fps
assert _test_args(fps(x**2 + x + 1, x))
@SKIP('Abstract Class')
def test_sympy__series__formal__FiniteFormalPowerSeries():
pass
def test_sympy__series__formal__FormalPowerSeriesProduct():
from sympy.series.formal import fps
f1, f2 = fps(sin(x)), fps(exp(x))
assert _test_args(f1.product(f2, x))
def test_sympy__series__formal__FormalPowerSeriesCompose():
from sympy.series.formal import fps
f1, f2 = fps(exp(x)), fps(sin(x))
assert _test_args(f1.compose(f2, x))
def test_sympy__series__formal__FormalPowerSeriesInverse():
from sympy.series.formal import fps
f1 = fps(exp(x))
assert _test_args(f1.inverse(x))
def test_sympy__simplify__hyperexpand__Hyper_Function():
from sympy.simplify.hyperexpand import Hyper_Function
assert _test_args(Hyper_Function([2], [1]))
def test_sympy__simplify__hyperexpand__G_Function():
from sympy.simplify.hyperexpand import G_Function
assert _test_args(G_Function([2], [1], [], []))
@SKIP("abstract class")
def test_sympy__tensor__array__ndim_array__ImmutableNDimArray():
pass
def test_sympy__tensor__array__dense_ndim_array__ImmutableDenseNDimArray():
from sympy.tensor.array.dense_ndim_array import ImmutableDenseNDimArray
densarr = ImmutableDenseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(densarr)
def test_sympy__tensor__array__sparse_ndim_array__ImmutableSparseNDimArray():
from sympy.tensor.array.sparse_ndim_array import ImmutableSparseNDimArray
sparr = ImmutableSparseNDimArray(range(10, 34), (2, 3, 4))
assert _test_args(sparr)
def test_sympy__tensor__array__array_comprehension__ArrayComprehension():
from sympy.tensor.array.array_comprehension import ArrayComprehension
arrcom = ArrayComprehension(x, (x, 1, 5))
assert _test_args(arrcom)
def test_sympy__tensor__array__array_comprehension__ArrayComprehensionMap():
from sympy.tensor.array.array_comprehension import ArrayComprehensionMap
arrcomma = ArrayComprehensionMap(lambda: 0, (x, 1, 5))
assert _test_args(arrcomma)
def test_sympy__tensor__array__array_derivatives__ArrayDerivative():
from sympy.tensor.array.array_derivatives import ArrayDerivative
A = MatrixSymbol("A", 2, 2)
arrder = ArrayDerivative(A, A, evaluate=False)
assert _test_args(arrder)
def test_sympy__tensor__array__expressions__array_expressions__ArraySymbol():
from sympy.tensor.array.expressions.array_expressions import ArraySymbol
m, n, k = symbols("m n k")
array = ArraySymbol("A", (m, n, k, 2))
assert _test_args(array)
def test_sympy__tensor__array__expressions__array_expressions__ArrayElement():
from sympy.tensor.array.expressions.array_expressions import ArrayElement
m, n, k = symbols("m n k")
ae = ArrayElement("A", (m, n, k, 2))
assert _test_args(ae)
def test_sympy__tensor__array__expressions__array_expressions__ZeroArray():
from sympy.tensor.array.expressions.array_expressions import ZeroArray
m, n, k = symbols("m n k")
za = ZeroArray(m, n, k, 2)
assert _test_args(za)
def test_sympy__tensor__array__expressions__array_expressions__OneArray():
from sympy.tensor.array.expressions.array_expressions import OneArray
m, n, k = symbols("m n k")
za = OneArray(m, n, k, 2)
assert _test_args(za)
def test_sympy__tensor__functions__TensorProduct():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 3, 3)
tp = TensorProduct(A, B)
assert _test_args(tp)
def test_sympy__tensor__indexed__Idx():
from sympy.tensor.indexed import Idx
assert _test_args(Idx('test'))
assert _test_args(Idx('test', (0, 10)))
assert _test_args(Idx('test', 2))
assert _test_args(Idx('test', x))
def test_sympy__tensor__indexed__Indexed():
from sympy.tensor.indexed import Indexed, Idx
assert _test_args(Indexed('A', Idx('i'), Idx('j')))
def test_sympy__tensor__indexed__IndexedBase():
from sympy.tensor.indexed import IndexedBase
assert _test_args(IndexedBase('A', shape=(x, y)))
assert _test_args(IndexedBase('A', 1))
assert _test_args(IndexedBase('A')[0, 1])
def test_sympy__tensor__tensor__TensorIndexType():
from sympy.tensor.tensor import TensorIndexType
assert _test_args(TensorIndexType('Lorentz'))
@SKIP("deprecated class")
def test_sympy__tensor__tensor__TensorType():
pass
def test_sympy__tensor__tensor__TensorSymmetry():
from sympy.tensor.tensor import TensorSymmetry, get_symmetric_group_sgs
assert _test_args(TensorSymmetry(get_symmetric_group_sgs(2)))
def test_sympy__tensor__tensor__TensorHead():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, TensorHead
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
sym = TensorSymmetry(get_symmetric_group_sgs(1))
assert _test_args(TensorHead('p', [Lorentz], sym, 0))
def test_sympy__tensor__tensor__TensorIndex():
from sympy.tensor.tensor import TensorIndexType, TensorIndex
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
assert _test_args(TensorIndex('i', Lorentz))
@SKIP("abstract class")
def test_sympy__tensor__tensor__TensExpr():
pass
def test_sympy__tensor__tensor__TensAdd():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, TensAdd, tensor_heads
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
p, q = tensor_heads('p,q', [Lorentz], sym)
t1 = p(a)
t2 = q(a)
assert _test_args(TensAdd(t1, t2))
def test_sympy__tensor__tensor__Tensor():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, TensorHead
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
p = TensorHead('p', [Lorentz], sym)
assert _test_args(p(a))
def test_sympy__tensor__tensor__TensMul():
from sympy.tensor.tensor import TensorIndexType, TensorSymmetry, get_symmetric_group_sgs, tensor_indices, tensor_heads
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
sym = TensorSymmetry(get_symmetric_group_sgs(1))
p, q = tensor_heads('p, q', [Lorentz], sym)
assert _test_args(3*p(a)*q(b))
def test_sympy__tensor__tensor__TensorElement():
from sympy.tensor.tensor import TensorIndexType, TensorHead, TensorElement
L = TensorIndexType("L")
A = TensorHead("A", [L, L])
telem = TensorElement(A(x, y), {x: 1})
assert _test_args(telem)
def test_sympy__tensor__toperators__PartialDerivative():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead
from sympy.tensor.toperators import PartialDerivative
Lorentz = TensorIndexType('Lorentz', dummy_name='L')
a, b = tensor_indices('a,b', Lorentz)
A = TensorHead("A", [Lorentz])
assert _test_args(PartialDerivative(A(a), A(b)))
def test_as_coeff_add():
assert (7, (3*x, 4*x**2)) == (7 + 3*x + 4*x**2).as_coeff_add()
def test_sympy__geometry__curve__Curve():
from sympy.geometry.curve import Curve
assert _test_args(Curve((x, 1), (x, 0, 1)))
def test_sympy__geometry__point__Point():
from sympy.geometry.point import Point
assert _test_args(Point(0, 1))
def test_sympy__geometry__point__Point2D():
from sympy.geometry.point import Point2D
assert _test_args(Point2D(0, 1))
def test_sympy__geometry__point__Point3D():
from sympy.geometry.point import Point3D
assert _test_args(Point3D(0, 1, 2))
def test_sympy__geometry__ellipse__Ellipse():
from sympy.geometry.ellipse import Ellipse
assert _test_args(Ellipse((0, 1), 2, 3))
def test_sympy__geometry__ellipse__Circle():
from sympy.geometry.ellipse import Circle
assert _test_args(Circle((0, 1), 2))
def test_sympy__geometry__parabola__Parabola():
from sympy.geometry.parabola import Parabola
from sympy.geometry.line import Line
assert _test_args(Parabola((0, 0), Line((2, 3), (4, 3))))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity():
pass
def test_sympy__geometry__line__Line():
from sympy.geometry.line import Line
assert _test_args(Line((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray():
from sympy.geometry.line import Ray
assert _test_args(Ray((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment():
from sympy.geometry.line import Segment
assert _test_args(Segment((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity2D():
pass
def test_sympy__geometry__line__Line2D():
from sympy.geometry.line import Line2D
assert _test_args(Line2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Ray2D():
from sympy.geometry.line import Ray2D
assert _test_args(Ray2D((0, 1), (2, 3)))
def test_sympy__geometry__line__Segment2D():
from sympy.geometry.line import Segment2D
assert _test_args(Segment2D((0, 1), (2, 3)))
@SKIP("abstract class")
def test_sympy__geometry__line__LinearEntity3D():
pass
def test_sympy__geometry__line__Line3D():
from sympy.geometry.line import Line3D
assert _test_args(Line3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Segment3D():
from sympy.geometry.line import Segment3D
assert _test_args(Segment3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__line__Ray3D():
from sympy.geometry.line import Ray3D
assert _test_args(Ray3D((0, 1, 1), (2, 3, 4)))
def test_sympy__geometry__plane__Plane():
from sympy.geometry.plane import Plane
assert _test_args(Plane((1, 1, 1), (-3, 4, -2), (1, 2, 3)))
def test_sympy__geometry__polygon__Polygon():
from sympy.geometry.polygon import Polygon
assert _test_args(Polygon((0, 1), (2, 3), (4, 5), (6, 7)))
def test_sympy__geometry__polygon__RegularPolygon():
from sympy.geometry.polygon import RegularPolygon
assert _test_args(RegularPolygon((0, 1), 2, 3, 4))
def test_sympy__geometry__polygon__Triangle():
from sympy.geometry.polygon import Triangle
assert _test_args(Triangle((0, 1), (2, 3), (4, 5)))
def test_sympy__geometry__entity__GeometryEntity():
from sympy.geometry.entity import GeometryEntity
from sympy.geometry.point import Point
assert _test_args(GeometryEntity(Point(1, 0), 1, [1, 2]))
@SKIP("abstract class")
def test_sympy__geometry__entity__GeometrySet():
pass
def test_sympy__diffgeom__diffgeom__Manifold():
from sympy.diffgeom import Manifold
assert _test_args(Manifold('name', 3))
def test_sympy__diffgeom__diffgeom__Patch():
from sympy.diffgeom import Manifold, Patch
assert _test_args(Patch('name', Manifold('name', 3)))
def test_sympy__diffgeom__diffgeom__CoordSystem():
from sympy.diffgeom import Manifold, Patch, CoordSystem
assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3))))
assert _test_args(CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]))
def test_sympy__diffgeom__diffgeom__CoordinateSymbol():
from sympy.diffgeom import Manifold, Patch, CoordSystem, CoordinateSymbol
assert _test_args(CoordinateSymbol(CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]), 0))
def test_sympy__diffgeom__diffgeom__Point():
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
assert _test_args(Point(
CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c]), [x, y]))
def test_sympy__diffgeom__diffgeom__BaseScalarField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
assert _test_args(BaseScalarField(cs, 0))
def test_sympy__diffgeom__diffgeom__BaseVectorField():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
assert _test_args(BaseVectorField(cs, 0))
def test_sympy__diffgeom__diffgeom__Differential():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
assert _test_args(Differential(BaseScalarField(cs, 0)))
def test_sympy__diffgeom__diffgeom__Commutator():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, Commutator
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
cs1 = CoordSystem('name1', Patch('name', Manifold('name', 3)), [a, b, c])
v = BaseVectorField(cs, 0)
v1 = BaseVectorField(cs1, 0)
assert _test_args(Commutator(v, v1))
def test_sympy__diffgeom__diffgeom__TensorProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, TensorProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
d = Differential(BaseScalarField(cs, 0))
assert _test_args(TensorProduct(d, d))
def test_sympy__diffgeom__diffgeom__WedgeProduct():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, WedgeProduct
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
d = Differential(BaseScalarField(cs, 0))
d1 = Differential(BaseScalarField(cs, 1))
assert _test_args(WedgeProduct(d, d1))
def test_sympy__diffgeom__diffgeom__LieDerivative():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential, BaseVectorField, LieDerivative
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
d = Differential(BaseScalarField(cs, 0))
v = BaseVectorField(cs, 0)
assert _test_args(LieDerivative(v, d))
def test_sympy__diffgeom__diffgeom__BaseCovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseCovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
assert _test_args(BaseCovarDerivativeOp(cs, 0, [[[0, ]*3, ]*3, ]*3))
def test_sympy__diffgeom__diffgeom__CovarDerivativeOp():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseVectorField, CovarDerivativeOp
cs = CoordSystem('name', Patch('name', Manifold('name', 3)), [a, b, c])
v = BaseVectorField(cs, 0)
_test_args(CovarDerivativeOp(v, [[[0, ]*3, ]*3, ]*3))
def test_sympy__categories__baseclasses__Class():
from sympy.categories.baseclasses import Class
assert _test_args(Class())
def test_sympy__categories__baseclasses__Object():
from sympy.categories import Object
assert _test_args(Object("A"))
@SKIP("abstract class")
def test_sympy__categories__baseclasses__Morphism():
pass
def test_sympy__categories__baseclasses__IdentityMorphism():
from sympy.categories import Object, IdentityMorphism
assert _test_args(IdentityMorphism(Object("A")))
def test_sympy__categories__baseclasses__NamedMorphism():
from sympy.categories import Object, NamedMorphism
assert _test_args(NamedMorphism(Object("A"), Object("B"), "f"))
def test_sympy__categories__baseclasses__CompositeMorphism():
from sympy.categories import Object, NamedMorphism, CompositeMorphism
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
assert _test_args(CompositeMorphism(f, g))
def test_sympy__categories__baseclasses__Diagram():
from sympy.categories import Object, NamedMorphism, Diagram
A = Object("A")
B = Object("B")
f = NamedMorphism(A, B, "f")
d = Diagram([f])
assert _test_args(d)
def test_sympy__categories__baseclasses__Category():
from sympy.categories import Object, NamedMorphism, Diagram, Category
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d1 = Diagram([f, g])
d2 = Diagram([f])
K = Category("K", commutative_diagrams=[d1, d2])
assert _test_args(K)
def test_sympy__ntheory__factor___totient():
from sympy.ntheory.factor_ import totient
k = symbols('k', integer=True)
t = totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___reduced_totient():
from sympy.ntheory.factor_ import reduced_totient
k = symbols('k', integer=True)
t = reduced_totient(k)
assert _test_args(t)
def test_sympy__ntheory__factor___divisor_sigma():
from sympy.ntheory.factor_ import divisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = divisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___udivisor_sigma():
from sympy.ntheory.factor_ import udivisor_sigma
k = symbols('k', integer=True)
n = symbols('n', integer=True)
t = udivisor_sigma(n, k)
assert _test_args(t)
def test_sympy__ntheory__factor___primenu():
from sympy.ntheory.factor_ import primenu
n = symbols('n', integer=True)
t = primenu(n)
assert _test_args(t)
def test_sympy__ntheory__factor___primeomega():
from sympy.ntheory.factor_ import primeomega
n = symbols('n', integer=True)
t = primeomega(n)
assert _test_args(t)
def test_sympy__ntheory__residue_ntheory__mobius():
from sympy.ntheory import mobius
assert _test_args(mobius(2))
def test_sympy__ntheory__generate__primepi():
from sympy.ntheory import primepi
n = symbols('n')
t = primepi(n)
assert _test_args(t)
def test_sympy__physics__optics__waves__TWave():
from sympy.physics.optics import TWave
A, f, phi = symbols('A, f, phi')
assert _test_args(TWave(A, f, phi))
def test_sympy__physics__optics__gaussopt__BeamParameter():
from sympy.physics.optics import BeamParameter
assert _test_args(BeamParameter(530e-9, 1, w=1e-3, n=1))
def test_sympy__physics__optics__medium__Medium():
from sympy.physics.optics import Medium
assert _test_args(Medium('m'))
def test_sympy__physics__optics__medium__MediumN():
from sympy.physics.optics.medium import Medium
assert _test_args(Medium('m', n=2))
def test_sympy__physics__optics__medium__MediumPP():
from sympy.physics.optics.medium import Medium
assert _test_args(Medium('m', permittivity=2, permeability=2))
def test_sympy__tensor__array__expressions__array_expressions__ArrayContraction():
from sympy.tensor.array.expressions.array_expressions import ArrayContraction
from sympy.tensor.indexed import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(ArrayContraction(A, (0, 1)))
def test_sympy__tensor__array__expressions__array_expressions__ArrayDiagonal():
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy.tensor.indexed import IndexedBase
A = symbols("A", cls=IndexedBase)
assert _test_args(ArrayDiagonal(A, (0, 1)))
def test_sympy__tensor__array__expressions__array_expressions__ArrayTensorProduct():
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.tensor.indexed import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(ArrayTensorProduct(A, B))
def test_sympy__tensor__array__expressions__array_expressions__ArrayAdd():
from sympy.tensor.array.expressions.array_expressions import ArrayAdd
from sympy.tensor.indexed import IndexedBase
A, B = symbols("A B", cls=IndexedBase)
assert _test_args(ArrayAdd(A, B))
def test_sympy__tensor__array__expressions__array_expressions__PermuteDims():
from sympy.tensor.array.expressions.array_expressions import PermuteDims
A = MatrixSymbol("A", 4, 4)
assert _test_args(PermuteDims(A, (1, 0)))
def test_sympy__tensor__array__expressions__array_expressions__ArrayElementwiseApplyFunc():
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElementwiseApplyFunc
A = ArraySymbol("A", (4,))
assert _test_args(ArrayElementwiseApplyFunc(exp, A))
def test_sympy__tensor__array__expressions__array_expressions__Reshape():
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, Reshape
A = ArraySymbol("A", (4,))
assert _test_args(Reshape(A, (2, 2)))
def test_sympy__codegen__ast__Assignment():
from sympy.codegen.ast import Assignment
assert _test_args(Assignment(x, y))
def test_sympy__codegen__cfunctions__expm1():
from sympy.codegen.cfunctions import expm1
assert _test_args(expm1(x))
def test_sympy__codegen__cfunctions__log1p():
from sympy.codegen.cfunctions import log1p
assert _test_args(log1p(x))
def test_sympy__codegen__cfunctions__exp2():
from sympy.codegen.cfunctions import exp2
assert _test_args(exp2(x))
def test_sympy__codegen__cfunctions__log2():
from sympy.codegen.cfunctions import log2
assert _test_args(log2(x))
def test_sympy__codegen__cfunctions__fma():
from sympy.codegen.cfunctions import fma
assert _test_args(fma(x, y, z))
def test_sympy__codegen__cfunctions__log10():
from sympy.codegen.cfunctions import log10
assert _test_args(log10(x))
def test_sympy__codegen__cfunctions__Sqrt():
from sympy.codegen.cfunctions import Sqrt
assert _test_args(Sqrt(x))
def test_sympy__codegen__cfunctions__Cbrt():
from sympy.codegen.cfunctions import Cbrt
assert _test_args(Cbrt(x))
def test_sympy__codegen__cfunctions__hypot():
from sympy.codegen.cfunctions import hypot
assert _test_args(hypot(x, y))
def test_sympy__codegen__fnodes__FFunction():
from sympy.codegen.fnodes import FFunction
assert _test_args(FFunction('f'))
def test_sympy__codegen__fnodes__F95Function():
from sympy.codegen.fnodes import F95Function
assert _test_args(F95Function('f'))
def test_sympy__codegen__fnodes__isign():
from sympy.codegen.fnodes import isign
assert _test_args(isign(1, x))
def test_sympy__codegen__fnodes__dsign():
from sympy.codegen.fnodes import dsign
assert _test_args(dsign(1, x))
def test_sympy__codegen__fnodes__cmplx():
from sympy.codegen.fnodes import cmplx
assert _test_args(cmplx(x, y))
def test_sympy__codegen__fnodes__kind():
from sympy.codegen.fnodes import kind
assert _test_args(kind(x))
def test_sympy__codegen__fnodes__merge():
from sympy.codegen.fnodes import merge
assert _test_args(merge(1, 2, Eq(x, 0)))
def test_sympy__codegen__fnodes___literal():
from sympy.codegen.fnodes import _literal
assert _test_args(_literal(1))
def test_sympy__codegen__fnodes__literal_sp():
from sympy.codegen.fnodes import literal_sp
assert _test_args(literal_sp(1))
def test_sympy__codegen__fnodes__literal_dp():
from sympy.codegen.fnodes import literal_dp
assert _test_args(literal_dp(1))
def test_sympy__codegen__matrix_nodes__MatrixSolve():
from sympy.matrices import MatrixSymbol
from sympy.codegen.matrix_nodes import MatrixSolve
A = MatrixSymbol('A', 3, 3)
v = MatrixSymbol('x', 3, 1)
assert _test_args(MatrixSolve(A, v))
def test_sympy__vector__coordsysrect__CoordSys3D():
from sympy.vector.coordsysrect import CoordSys3D
assert _test_args(CoordSys3D('C'))
def test_sympy__vector__point__Point():
from sympy.vector.point import Point
assert _test_args(Point('P'))
def test_sympy__vector__basisdependent__BasisDependent():
#from sympy.vector.basisdependent import BasisDependent
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__basisdependent__BasisDependentMul():
#from sympy.vector.basisdependent import BasisDependentMul
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__basisdependent__BasisDependentAdd():
#from sympy.vector.basisdependent import BasisDependentAdd
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__basisdependent__BasisDependentZero():
#from sympy.vector.basisdependent import BasisDependentZero
#These classes have been created to maintain an OOP hierarchy
#for Vectors and Dyadics. Are NOT meant to be initialized
pass
def test_sympy__vector__vector__BaseVector():
from sympy.vector.vector import BaseVector
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseVector(0, C, ' ', ' '))
def test_sympy__vector__vector__VectorAdd():
from sympy.vector.vector import VectorAdd, VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a, b, c, x, y, z
v1 = a*C.i + b*C.j + c*C.k
v2 = x*C.i + y*C.j + z*C.k
assert _test_args(VectorAdd(v1, v2))
assert _test_args(VectorMul(x, v1))
def test_sympy__vector__vector__VectorMul():
from sympy.vector.vector import VectorMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
from sympy.abc import a
assert _test_args(VectorMul(a, C.i))
def test_sympy__vector__vector__VectorZero():
from sympy.vector.vector import VectorZero
assert _test_args(VectorZero())
def test_sympy__vector__vector__Vector():
#from sympy.vector.vector import Vector
#Vector is never to be initialized using args
pass
def test_sympy__vector__vector__Cross():
from sympy.vector.vector import Cross
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Cross(C.i, C.j))
def test_sympy__vector__vector__Dot():
from sympy.vector.vector import Dot
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
_test_args(Dot(C.i, C.j))
def test_sympy__vector__dyadic__Dyadic():
#from sympy.vector.dyadic import Dyadic
#Dyadic is never to be initialized using args
pass
def test_sympy__vector__dyadic__BaseDyadic():
from sympy.vector.dyadic import BaseDyadic
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseDyadic(C.i, C.j))
def test_sympy__vector__dyadic__DyadicMul():
from sympy.vector.dyadic import BaseDyadic, DyadicMul
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(DyadicMul(3, BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicAdd():
from sympy.vector.dyadic import BaseDyadic, DyadicAdd
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(2 * DyadicAdd(BaseDyadic(C.i, C.i),
BaseDyadic(C.i, C.j)))
def test_sympy__vector__dyadic__DyadicZero():
from sympy.vector.dyadic import DyadicZero
assert _test_args(DyadicZero())
def test_sympy__vector__deloperator__Del():
from sympy.vector.deloperator import Del
assert _test_args(Del())
def test_sympy__vector__implicitregion__ImplicitRegion():
from sympy.vector.implicitregion import ImplicitRegion
from sympy.abc import x, y
assert _test_args(ImplicitRegion((x, y), y**3 - 4*x))
def test_sympy__vector__integrals__ParametricIntegral():
from sympy.vector.integrals import ParametricIntegral
from sympy.vector.parametricregion import ParametricRegion
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(ParametricIntegral(C.y*C.i - 10*C.j,\
ParametricRegion((x, y), (x, 1, 3), (y, -2, 2))))
def test_sympy__vector__operators__Curl():
from sympy.vector.operators import Curl
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Curl(C.i))
def test_sympy__vector__operators__Laplacian():
from sympy.vector.operators import Laplacian
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Laplacian(C.i))
def test_sympy__vector__operators__Divergence():
from sympy.vector.operators import Divergence
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Divergence(C.i))
def test_sympy__vector__operators__Gradient():
from sympy.vector.operators import Gradient
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(Gradient(C.x))
def test_sympy__vector__orienters__Orienter():
#from sympy.vector.orienters import Orienter
#Not to be initialized
pass
def test_sympy__vector__orienters__ThreeAngleOrienter():
#from sympy.vector.orienters import ThreeAngleOrienter
#Not to be initialized
pass
def test_sympy__vector__orienters__AxisOrienter():
from sympy.vector.orienters import AxisOrienter
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(AxisOrienter(x, C.i))
def test_sympy__vector__orienters__BodyOrienter():
from sympy.vector.orienters import BodyOrienter
assert _test_args(BodyOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__SpaceOrienter():
from sympy.vector.orienters import SpaceOrienter
assert _test_args(SpaceOrienter(x, y, z, '123'))
def test_sympy__vector__orienters__QuaternionOrienter():
from sympy.vector.orienters import QuaternionOrienter
a, b, c, d = symbols('a b c d')
assert _test_args(QuaternionOrienter(a, b, c, d))
def test_sympy__vector__parametricregion__ParametricRegion():
from sympy.abc import t
from sympy.vector.parametricregion import ParametricRegion
assert _test_args(ParametricRegion((t, t**3), (t, 0, 2)))
def test_sympy__vector__scalar__BaseScalar():
from sympy.vector.scalar import BaseScalar
from sympy.vector.coordsysrect import CoordSys3D
C = CoordSys3D('C')
assert _test_args(BaseScalar(0, C, ' ', ' '))
def test_sympy__physics__wigner__Wigner3j():
from sympy.physics.wigner import Wigner3j
assert _test_args(Wigner3j(0, 0, 0, 0, 0, 0))
def test_sympy__integrals__rubi__symbol__matchpyWC():
from sympy.integrals.rubi.symbol import matchpyWC
assert _test_args(matchpyWC(1, True, 'a'))
def test_sympy__integrals__rubi__utility_function__rubi_unevaluated_expr():
from sympy.integrals.rubi.utility_function import rubi_unevaluated_expr
a = symbols('a')
assert _test_args(rubi_unevaluated_expr(a))
def test_sympy__integrals__rubi__utility_function__rubi_exp():
from sympy.integrals.rubi.utility_function import rubi_exp
assert _test_args(rubi_exp(5))
def test_sympy__integrals__rubi__utility_function__rubi_log():
from sympy.integrals.rubi.utility_function import rubi_log
assert _test_args(rubi_log(5))
def test_sympy__integrals__rubi__utility_function__Int():
from sympy.integrals.rubi.utility_function import Int
assert _test_args(Int(5, x))
def test_sympy__integrals__rubi__utility_function__Util_Coefficient():
from sympy.integrals.rubi.utility_function import Util_Coefficient
a, x = symbols('a x')
assert _test_args(Util_Coefficient(a, x))
def test_sympy__integrals__rubi__utility_function__Gamma():
from sympy.integrals.rubi.utility_function import Gamma
assert _test_args(Gamma(x))
def test_sympy__integrals__rubi__utility_function__Util_Part():
from sympy.integrals.rubi.utility_function import Util_Part
a, b = symbols('a b')
assert _test_args(Util_Part(a + b, 0))
def test_sympy__integrals__rubi__utility_function__PolyGamma():
from sympy.integrals.rubi.utility_function import PolyGamma
assert _test_args(PolyGamma(1, x))
def test_sympy__integrals__rubi__utility_function__ProductLog():
from sympy.integrals.rubi.utility_function import ProductLog
assert _test_args(ProductLog(1))
def test_sympy__combinatorics__schur_number__SchurNumber():
from sympy.combinatorics.schur_number import SchurNumber
assert _test_args(SchurNumber(x))
def test_sympy__combinatorics__perm_groups__SymmetricPermutationGroup():
from sympy.combinatorics.perm_groups import SymmetricPermutationGroup
assert _test_args(SymmetricPermutationGroup(5))
def test_sympy__combinatorics__perm_groups__Coset():
from sympy.combinatorics.permutations import Permutation
from sympy.combinatorics.perm_groups import PermutationGroup, Coset
a = Permutation(1, 2)
b = Permutation(0, 1)
G = PermutationGroup([a, b])
assert _test_args(Coset(a, G))
|
9d117e679ead093f1a9e7ffa11a9a9d3bd5bd03a42df0c522ec82731145c5861 | from sympy.core.numbers import (I, Rational, pi)
from sympy.core.relational import (GreaterThan, LessThan, StrictGreaterThan, StrictLessThan)
from sympy.core.symbol import (Dummy, Symbol, Wild, symbols)
from sympy.core.sympify import sympify # can't import as S yet
from sympy.core.symbol import uniquely_named_symbol, _symbol, Str
from sympy.testing.pytest import raises
from sympy.core.symbol import disambiguate
def test_Str():
a1 = Str('a')
a2 = Str('a')
b = Str('b')
assert a1 == a2 != b
raises(TypeError, lambda: Str())
def test_Symbol():
a = Symbol("a")
x1 = Symbol("x")
x2 = Symbol("x")
xdummy1 = Dummy("x")
xdummy2 = Dummy("x")
assert a != x1
assert a != x2
assert x1 == x2
assert x1 != xdummy1
assert xdummy1 != xdummy2
assert Symbol("x") == Symbol("x")
assert Dummy("x") != Dummy("x")
d = symbols('d', cls=Dummy)
assert isinstance(d, Dummy)
c, d = symbols('c,d', cls=Dummy)
assert isinstance(c, Dummy)
assert isinstance(d, Dummy)
raises(TypeError, lambda: Symbol())
def test_Dummy():
assert Dummy() != Dummy()
def test_Dummy_force_dummy_index():
raises(AssertionError, lambda: Dummy(dummy_index=1))
assert Dummy('d', dummy_index=2) == Dummy('d', dummy_index=2)
assert Dummy('d1', dummy_index=2) != Dummy('d2', dummy_index=2)
d1 = Dummy('d', dummy_index=3)
d2 = Dummy('d')
# might fail if d1 were created with dummy_index >= 10**6
assert d1 != d2
d3 = Dummy('d', dummy_index=3)
assert d1 == d3
assert Dummy()._count == Dummy('d', dummy_index=3)._count
def test_lt_gt():
S = sympify
x, y = Symbol('x'), Symbol('y')
assert (x >= y) == GreaterThan(x, y)
assert (x >= 0) == GreaterThan(x, 0)
assert (x <= y) == LessThan(x, y)
assert (x <= 0) == LessThan(x, 0)
assert (0 <= x) == GreaterThan(x, 0)
assert (0 >= x) == LessThan(x, 0)
assert (S(0) >= x) == GreaterThan(0, x)
assert (S(0) <= x) == LessThan(0, x)
assert (x > y) == StrictGreaterThan(x, y)
assert (x > 0) == StrictGreaterThan(x, 0)
assert (x < y) == StrictLessThan(x, y)
assert (x < 0) == StrictLessThan(x, 0)
assert (0 < x) == StrictGreaterThan(x, 0)
assert (0 > x) == StrictLessThan(x, 0)
assert (S(0) > x) == StrictGreaterThan(0, x)
assert (S(0) < x) == StrictLessThan(0, x)
e = x**2 + 4*x + 1
assert (e >= 0) == GreaterThan(e, 0)
assert (0 <= e) == GreaterThan(e, 0)
assert (e > 0) == StrictGreaterThan(e, 0)
assert (0 < e) == StrictGreaterThan(e, 0)
assert (e <= 0) == LessThan(e, 0)
assert (0 >= e) == LessThan(e, 0)
assert (e < 0) == StrictLessThan(e, 0)
assert (0 > e) == StrictLessThan(e, 0)
assert (S(0) >= e) == GreaterThan(0, e)
assert (S(0) <= e) == LessThan(0, e)
assert (S(0) < e) == StrictLessThan(0, e)
assert (S(0) > e) == StrictGreaterThan(0, e)
def test_no_len():
# there should be no len for numbers
x = Symbol('x')
raises(TypeError, lambda: len(x))
def test_ineq_unequal():
S = sympify
x, y, z = symbols('x,y,z')
e = (
S(-1) >= x, S(-1) >= y, S(-1) >= z,
S(-1) > x, S(-1) > y, S(-1) > z,
S(-1) <= x, S(-1) <= y, S(-1) <= z,
S(-1) < x, S(-1) < y, S(-1) < z,
S(0) >= x, S(0) >= y, S(0) >= z,
S(0) > x, S(0) > y, S(0) > z,
S(0) <= x, S(0) <= y, S(0) <= z,
S(0) < x, S(0) < y, S(0) < z,
S('3/7') >= x, S('3/7') >= y, S('3/7') >= z,
S('3/7') > x, S('3/7') > y, S('3/7') > z,
S('3/7') <= x, S('3/7') <= y, S('3/7') <= z,
S('3/7') < x, S('3/7') < y, S('3/7') < z,
S(1.5) >= x, S(1.5) >= y, S(1.5) >= z,
S(1.5) > x, S(1.5) > y, S(1.5) > z,
S(1.5) <= x, S(1.5) <= y, S(1.5) <= z,
S(1.5) < x, S(1.5) < y, S(1.5) < z,
S(2) >= x, S(2) >= y, S(2) >= z,
S(2) > x, S(2) > y, S(2) > z,
S(2) <= x, S(2) <= y, S(2) <= z,
S(2) < x, S(2) < y, S(2) < z,
x >= -1, y >= -1, z >= -1,
x > -1, y > -1, z > -1,
x <= -1, y <= -1, z <= -1,
x < -1, y < -1, z < -1,
x >= 0, y >= 0, z >= 0,
x > 0, y > 0, z > 0,
x <= 0, y <= 0, z <= 0,
x < 0, y < 0, z < 0,
x >= 1.5, y >= 1.5, z >= 1.5,
x > 1.5, y > 1.5, z > 1.5,
x <= 1.5, y <= 1.5, z <= 1.5,
x < 1.5, y < 1.5, z < 1.5,
x >= 2, y >= 2, z >= 2,
x > 2, y > 2, z > 2,
x <= 2, y <= 2, z <= 2,
x < 2, y < 2, z < 2,
x >= y, x >= z, y >= x, y >= z, z >= x, z >= y,
x > y, x > z, y > x, y > z, z > x, z > y,
x <= y, x <= z, y <= x, y <= z, z <= x, z <= y,
x < y, x < z, y < x, y < z, z < x, z < y,
x - pi >= y + z, y - pi >= x + z, z - pi >= x + y,
x - pi > y + z, y - pi > x + z, z - pi > x + y,
x - pi <= y + z, y - pi <= x + z, z - pi <= x + y,
x - pi < y + z, y - pi < x + z, z - pi < x + y,
True, False
)
left_e = e[:-1]
for i, e1 in enumerate( left_e ):
for e2 in e[i + 1:]:
assert e1 != e2
def test_Wild_properties():
S = sympify
# these tests only include Atoms
x = Symbol("x")
y = Symbol("y")
p = Symbol("p", positive=True)
k = Symbol("k", integer=True)
n = Symbol("n", integer=True, positive=True)
given_patterns = [ x, y, p, k, -k, n, -n, S(-3), S(3),
pi, Rational(3, 2), I ]
integerp = lambda k: k.is_integer
positivep = lambda k: k.is_positive
symbolp = lambda k: k.is_Symbol
realp = lambda k: k.is_extended_real
S = Wild("S", properties=[symbolp])
R = Wild("R", properties=[realp])
Y = Wild("Y", exclude=[x, p, k, n])
P = Wild("P", properties=[positivep])
K = Wild("K", properties=[integerp])
N = Wild("N", properties=[positivep, integerp])
given_wildcards = [ S, R, Y, P, K, N ]
goodmatch = {
S: (x, y, p, k, n),
R: (p, k, -k, n, -n, -3, 3, pi, Rational(3, 2)),
Y: (y, -3, 3, pi, Rational(3, 2), I ),
P: (p, n, 3, pi, Rational(3, 2)),
K: (k, -k, n, -n, -3, 3),
N: (n, 3)}
for A in given_wildcards:
for pat in given_patterns:
d = pat.match(A)
if pat in goodmatch[A]:
assert d[A] in goodmatch[A]
else:
assert d is None
def test_symbols():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
assert symbols('x') == x
assert symbols('x ') == x
assert symbols(' x ') == x
assert symbols('x,') == (x,)
assert symbols('x, ') == (x,)
assert symbols('x ,') == (x,)
assert symbols('x , y') == (x, y)
assert symbols('x,y,z') == (x, y, z)
assert symbols('x y z') == (x, y, z)
assert symbols('x,y,z,') == (x, y, z)
assert symbols('x y z ') == (x, y, z)
xyz = Symbol('xyz')
abc = Symbol('abc')
assert symbols('xyz') == xyz
assert symbols('xyz,') == (xyz,)
assert symbols('xyz,abc') == (xyz, abc)
assert symbols(('xyz',)) == (xyz,)
assert symbols(('xyz,',)) == ((xyz,),)
assert symbols(('x,y,z,',)) == ((x, y, z),)
assert symbols(('xyz', 'abc')) == (xyz, abc)
assert symbols(('xyz,abc',)) == ((xyz, abc),)
assert symbols(('xyz,abc', 'x,y,z')) == ((xyz, abc), (x, y, z))
assert symbols(('x', 'y', 'z')) == (x, y, z)
assert symbols(['x', 'y', 'z']) == [x, y, z]
assert symbols({'x', 'y', 'z'}) == {x, y, z}
raises(ValueError, lambda: symbols(''))
raises(ValueError, lambda: symbols(','))
raises(ValueError, lambda: symbols('x,,y,,z'))
raises(ValueError, lambda: symbols(('x', '', 'y', '', 'z')))
a, b = symbols('x,y', real=True)
assert a.is_real and b.is_real
x0 = Symbol('x0')
x1 = Symbol('x1')
x2 = Symbol('x2')
y0 = Symbol('y0')
y1 = Symbol('y1')
assert symbols('x0:0') == ()
assert symbols('x0:1') == (x0,)
assert symbols('x0:2') == (x0, x1)
assert symbols('x0:3') == (x0, x1, x2)
assert symbols('x:0') == ()
assert symbols('x:1') == (x0,)
assert symbols('x:2') == (x0, x1)
assert symbols('x:3') == (x0, x1, x2)
assert symbols('x1:1') == ()
assert symbols('x1:2') == (x1,)
assert symbols('x1:3') == (x1, x2)
assert symbols('x1:3,x,y,z') == (x1, x2, x, y, z)
assert symbols('x:3,y:2') == (x0, x1, x2, y0, y1)
assert symbols(('x:3', 'y:2')) == ((x0, x1, x2), (y0, y1))
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')
d = Symbol('d')
assert symbols('x:z') == (x, y, z)
assert symbols('a:d,x:z') == (a, b, c, d, x, y, z)
assert symbols(('a:d', 'x:z')) == ((a, b, c, d), (x, y, z))
aa = Symbol('aa')
ab = Symbol('ab')
ac = Symbol('ac')
ad = Symbol('ad')
assert symbols('aa:d') == (aa, ab, ac, ad)
assert symbols('aa:d,x:z') == (aa, ab, ac, ad, x, y, z)
assert symbols(('aa:d','x:z')) == ((aa, ab, ac, ad), (x, y, z))
# issue 6675
def sym(s):
return str(symbols(s))
assert sym('a0:4') == '(a0, a1, a2, a3)'
assert sym('a2:4,b1:3') == '(a2, a3, b1, b2)'
assert sym('a1(2:4)') == '(a12, a13)'
assert sym('a0:2.0:2') == '(a0.0, a0.1, a1.0, a1.1)'
assert sym('aa:cz') == '(aaz, abz, acz)'
assert sym('aa:c0:2') == '(aa0, aa1, ab0, ab1, ac0, ac1)'
assert sym('aa:ba:b') == '(aaa, aab, aba, abb)'
assert sym('a:3b') == '(a0b, a1b, a2b)'
assert sym('a-1:3b') == '(a-1b, a-2b)'
assert sym(r'a:2\,:2' + chr(0)) == '(a0,0%s, a0,1%s, a1,0%s, a1,1%s)' % (
(chr(0),)*4)
assert sym('x(:a:3)') == '(x(a0), x(a1), x(a2))'
assert sym('x(:c):1') == '(xa0, xb0, xc0)'
assert sym('x((:a)):3') == '(x(a)0, x(a)1, x(a)2)'
assert sym('x(:a:3') == '(x(a0, x(a1, x(a2)'
assert sym(':2') == '(0, 1)'
assert sym(':b') == '(a, b)'
assert sym(':b:2') == '(a0, a1, b0, b1)'
assert sym(':2:2') == '(00, 01, 10, 11)'
assert sym(':b:b') == '(aa, ab, ba, bb)'
raises(ValueError, lambda: symbols(':'))
raises(ValueError, lambda: symbols('a:'))
raises(ValueError, lambda: symbols('::'))
raises(ValueError, lambda: symbols('a::'))
raises(ValueError, lambda: symbols(':a:'))
raises(ValueError, lambda: symbols('::a'))
def test_symbols_become_functions_issue_3539():
from sympy.abc import alpha, phi, beta, t
raises(TypeError, lambda: beta(2))
raises(TypeError, lambda: beta(2.5))
raises(TypeError, lambda: phi(2.5))
raises(TypeError, lambda: alpha(2.5))
raises(TypeError, lambda: phi(t))
def test_unicode():
xu = Symbol('x')
x = Symbol('x')
assert x == xu
raises(TypeError, lambda: Symbol(1))
def test_uniquely_named_symbol_and_Symbol():
F = uniquely_named_symbol
x = Symbol('x')
assert F(x) == x
assert F('x') == x
assert str(F('x', x)) == 'x0'
assert str(F('x', (x + 1, 1/x))) == 'x0'
_x = Symbol('x', real=True)
assert F(('x', _x)) == _x
assert F((x, _x)) == _x
assert F('x', real=True).is_real
y = Symbol('y')
assert F(('x', y), real=True).is_real
r = Symbol('x', real=True)
assert F(('x', r)).is_real
assert F(('x', r), real=False).is_real
assert F('x1', Symbol('x1'),
compare=lambda i: str(i).rstrip('1')).name == 'x0'
assert F('x1', Symbol('x1'),
modify=lambda i: i + '_').name == 'x1_'
assert _symbol(x, _x) == x
def test_disambiguate():
x, y, y_1, _x, x_1, x_2 = symbols('x y y_1 _x x_1 x_2')
t1 = Dummy('y'), _x, Dummy('x'), Dummy('x')
t2 = Dummy('x'), Dummy('x')
t3 = Dummy('x'), Dummy('y')
t4 = x, Dummy('x')
t5 = Symbol('x', integer=True), x, Symbol('x_1')
assert disambiguate(*t1) == (y, x_2, x, x_1)
assert disambiguate(*t2) == (x, x_1)
assert disambiguate(*t3) == (x, y)
assert disambiguate(*t4) == (x_1, x)
assert disambiguate(*t5) == (t5[0], x_2, x_1)
assert disambiguate(*t5)[0] != x # assumptions are retained
t6 = _x, Dummy('x')/y
t7 = y*Dummy('y'), y
assert disambiguate(*t6) == (x_1, x/y)
assert disambiguate(*t7) == (y*y_1, y_1)
assert disambiguate(Dummy('x_1'), Dummy('x_1')
) == (x_1, Symbol('x_1_1'))
|
2cdf38bb676faf5788be522609cdc8bc1c2ed13f4b7dc55118223263e150c785 | from sympy.concrete.summations import Sum
from sympy.core.basic import Basic, _aresame
from sympy.core.cache import clear_cache
from sympy.core.containers import Dict, Tuple
from sympy.core.expr import Expr, unchanged
from sympy.core.function import (Subs, Function, diff, Lambda, expand,
nfloat, Derivative)
from sympy.core.numbers import E, Float, zoo, Rational, pi, I, oo, nan
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import symbols, Dummy, Symbol
from sympy.functions.elementary.complexes import im, re
from sympy.functions.elementary.exponential import log, exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import sin, cos, acos
from sympy.functions.special.error_functions import expint
from sympy.functions.special.gamma_functions import loggamma, polygamma
from sympy.matrices.dense import Matrix
from sympy.printing.str import sstr
from sympy.series.order import O
from sympy.tensor.indexed import Indexed
from sympy.core.function import (PoleError, _mexpand, arity,
BadSignatureError, BadArgumentsError)
from sympy.core.parameters import _exp_is_pow
from sympy.core.sympify import sympify
from sympy.matrices import MutableMatrix, ImmutableMatrix
from sympy.sets.sets import FiniteSet
from sympy.solvers.solveset import solveset
from sympy.tensor.array import NDimArray
from sympy.utilities.iterables import subsets, variations
from sympy.testing.pytest import XFAIL, raises, warns_deprecated_sympy, _both_exp_pow
from sympy.abc import t, w, x, y, z
f, g, h = symbols('f g h', cls=Function)
_xi_1, _xi_2, _xi_3 = [Dummy() for i in range(3)]
def test_f_expand_complex():
x = Symbol('x', real=True)
assert f(x).expand(complex=True) == I*im(f(x)) + re(f(x))
assert exp(x).expand(complex=True) == exp(x)
assert exp(I*x).expand(complex=True) == cos(x) + I*sin(x)
assert exp(z).expand(complex=True) == cos(im(z))*exp(re(z)) + \
I*sin(im(z))*exp(re(z))
def test_bug1():
e = sqrt(-log(w))
assert e.subs(log(w), -x) == sqrt(x)
e = sqrt(-5*log(w))
assert e.subs(log(w), -x) == sqrt(5*x)
def test_general_function():
nu = Function('nu')
e = nu(x)
edx = e.diff(x)
edy = e.diff(y)
edxdx = e.diff(x).diff(x)
edxdy = e.diff(x).diff(y)
assert e == nu(x)
assert edx != nu(x)
assert edx == diff(nu(x), x)
assert edy == 0
assert edxdx == diff(diff(nu(x), x), x)
assert edxdy == 0
def test_general_function_nullary():
nu = Function('nu')
e = nu()
edx = e.diff(x)
edxdx = e.diff(x).diff(x)
assert e == nu()
assert edx != nu()
assert edx == 0
assert edxdx == 0
def test_derivative_subs_bug():
e = diff(g(x), x)
assert e.subs(g(x), f(x)) != e
assert e.subs(g(x), f(x)) == Derivative(f(x), x)
assert e.subs(g(x), -f(x)) == Derivative(-f(x), x)
assert e.subs(x, y) == Derivative(g(y), y)
def test_derivative_subs_self_bug():
d = diff(f(x), x)
assert d.subs(d, y) == y
def test_derivative_linearity():
assert diff(-f(x), x) == -diff(f(x), x)
assert diff(8*f(x), x) == 8*diff(f(x), x)
assert diff(8*f(x), x) != 7*diff(f(x), x)
assert diff(8*f(x)*x, x) == 8*f(x) + 8*x*diff(f(x), x)
assert diff(8*f(x)*y*x, x).expand() == 8*y*f(x) + 8*y*x*diff(f(x), x)
def test_derivative_evaluate():
assert Derivative(sin(x), x) != diff(sin(x), x)
assert Derivative(sin(x), x).doit() == diff(sin(x), x)
assert Derivative(Derivative(f(x), x), x) == diff(f(x), x, x)
assert Derivative(sin(x), x, 0) == sin(x)
assert Derivative(sin(x), (x, y), (x, -y)) == sin(x)
def test_diff_symbols():
assert diff(f(x, y, z), x, y, z) == Derivative(f(x, y, z), x, y, z)
assert diff(f(x, y, z), x, x, x) == Derivative(f(x, y, z), x, x, x) == Derivative(f(x, y, z), (x, 3))
assert diff(f(x, y, z), x, 3) == Derivative(f(x, y, z), x, 3)
# issue 5028
assert [diff(-z + x/y, sym) for sym in (z, x, y)] == [-1, 1/y, -x/y**2]
assert diff(f(x, y, z), x, y, z, 2) == Derivative(f(x, y, z), x, y, z, z)
assert diff(f(x, y, z), x, y, z, 2, evaluate=False) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(f(x, y, z), x, y, z)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z, z)
assert Derivative(Derivative(f(x, y, z), x), y)._eval_derivative(z) == \
Derivative(f(x, y, z), x, y, z)
raises(TypeError, lambda: cos(x).diff((x, y)).variables)
assert cos(x).diff((x, y))._wrt_variables == [x]
# issue 23222
assert sympify("a*x+b").diff("x") == sympify("a")
def test_Function():
class myfunc(Function):
@classmethod
def eval(cls): # zero args
return
assert myfunc.nargs == FiniteSet(0)
assert myfunc().nargs == FiniteSet(0)
raises(TypeError, lambda: myfunc(x).nargs)
class myfunc(Function):
@classmethod
def eval(cls, x): # one arg
return
assert myfunc.nargs == FiniteSet(1)
assert myfunc(x).nargs == FiniteSet(1)
raises(TypeError, lambda: myfunc(x, y).nargs)
class myfunc(Function):
@classmethod
def eval(cls, *x): # star args
return
assert myfunc.nargs == S.Naturals0
assert myfunc(x).nargs == S.Naturals0
def test_nargs():
f = Function('f')
assert f.nargs == S.Naturals0
assert f(1).nargs == S.Naturals0
assert Function('f', nargs=2)(1, 2).nargs == FiniteSet(2)
assert sin.nargs == FiniteSet(1)
assert sin(2).nargs == FiniteSet(1)
assert log.nargs == FiniteSet(1, 2)
assert log(2).nargs == FiniteSet(1, 2)
assert Function('f', nargs=2).nargs == FiniteSet(2)
assert Function('f', nargs=0).nargs == FiniteSet(0)
assert Function('f', nargs=(0, 1)).nargs == FiniteSet(0, 1)
assert Function('f', nargs=None).nargs == S.Naturals0
raises(ValueError, lambda: Function('f', nargs=()))
def test_nargs_inheritance():
class f1(Function):
nargs = 2
class f2(f1):
pass
class f3(f2):
pass
class f4(f3):
nargs = 1,2
class f5(f4):
pass
class f6(f5):
pass
class f7(f6):
nargs=None
class f8(f7):
pass
class f9(f8):
pass
class f10(f9):
nargs = 1
class f11(f10):
pass
assert f1.nargs == FiniteSet(2)
assert f2.nargs == FiniteSet(2)
assert f3.nargs == FiniteSet(2)
assert f4.nargs == FiniteSet(1, 2)
assert f5.nargs == FiniteSet(1, 2)
assert f6.nargs == FiniteSet(1, 2)
assert f7.nargs == S.Naturals0
assert f8.nargs == S.Naturals0
assert f9.nargs == S.Naturals0
assert f10.nargs == FiniteSet(1)
assert f11.nargs == FiniteSet(1)
def test_arity():
f = lambda x, y: 1
assert arity(f) == 2
def f(x, y, z=None):
pass
assert arity(f) == (2, 3)
assert arity(lambda *x: x) is None
assert arity(log) == (1, 2)
def test_Lambda():
e = Lambda(x, x**2)
assert e(4) == 16
assert e(x) == x**2
assert e(y) == y**2
assert Lambda((), 42)() == 42
assert unchanged(Lambda, (), 42)
assert Lambda((), 42) != Lambda((), 43)
assert Lambda((), f(x))() == f(x)
assert Lambda((), 42).nargs == FiniteSet(0)
assert unchanged(Lambda, (x,), x**2)
assert Lambda(x, x**2) == Lambda((x,), x**2)
assert Lambda(x, x**2) != Lambda(x, x**2 + 1)
assert Lambda((x, y), x**y) != Lambda((y, x), y**x)
assert Lambda((x, y), x**y) != Lambda((x, y), y**x)
assert Lambda((x, y), x**y)(x, y) == x**y
assert Lambda((x, y), x**y)(3, 3) == 3**3
assert Lambda((x, y), x**y)(x, 3) == x**3
assert Lambda((x, y), x**y)(3, y) == 3**y
assert Lambda(x, f(x))(x) == f(x)
assert Lambda(x, x**2)(e(x)) == x**4
assert e(e(x)) == x**4
x1, x2 = (Indexed('x', i) for i in (1, 2))
assert Lambda((x1, x2), x1 + x2)(x, y) == x + y
assert Lambda((x, y), x + y).nargs == FiniteSet(2)
p = x, y, z, t
assert Lambda(p, t*(x + y + z))(*p) == t * (x + y + z)
eq = Lambda(x, 2*x) + Lambda(y, 2*y)
assert eq != 2*Lambda(x, 2*x)
assert eq.as_dummy() == 2*Lambda(x, 2*x).as_dummy()
assert Lambda(x, 2*x) not in [ Lambda(x, x) ]
raises(BadSignatureError, lambda: Lambda(1, x))
assert Lambda(x, 1)(1) is S.One
raises(BadSignatureError, lambda: Lambda((x, x), x + 2))
raises(BadSignatureError, lambda: Lambda(((x, x), y), x))
raises(BadSignatureError, lambda: Lambda(((y, x), x), x))
raises(BadSignatureError, lambda: Lambda(((y, 1), 2), x))
with warns_deprecated_sympy():
assert Lambda([x, y], x+y) == Lambda((x, y), x+y)
flam = Lambda(((x, y),), x + y)
assert flam((2, 3)) == 5
flam = Lambda(((x, y), z), x + y + z)
assert flam((2, 3), 1) == 6
flam = Lambda((((x, y), z),), x + y + z)
assert flam(((2, 3), 1)) == 6
raises(BadArgumentsError, lambda: flam(1, 2, 3))
flam = Lambda( (x,), (x, x))
assert flam(1,) == (1, 1)
assert flam((1,)) == ((1,), (1,))
flam = Lambda( ((x,),), (x, x))
raises(BadArgumentsError, lambda: flam(1))
assert flam((1,)) == (1, 1)
# Previously TypeError was raised so this is potentially needed for
# backwards compatibility.
assert issubclass(BadSignatureError, TypeError)
assert issubclass(BadArgumentsError, TypeError)
# These are tested to see they don't raise:
hash(Lambda(x, 2*x))
hash(Lambda(x, x)) # IdentityFunction subclass
def test_IdentityFunction():
assert Lambda(x, x) is Lambda(y, y) is S.IdentityFunction
assert Lambda(x, 2*x) is not S.IdentityFunction
assert Lambda((x, y), x) is not S.IdentityFunction
def test_Lambda_symbols():
assert Lambda(x, 2*x).free_symbols == set()
assert Lambda(x, x*y).free_symbols == {y}
assert Lambda((), 42).free_symbols == set()
assert Lambda((), x*y).free_symbols == {x,y}
def test_functionclas_symbols():
assert f.free_symbols == set()
def test_Lambda_arguments():
raises(TypeError, lambda: Lambda(x, 2*x)(x, y))
raises(TypeError, lambda: Lambda((x, y), x + y)(x))
raises(TypeError, lambda: Lambda((), 42)(x))
def test_Lambda_equality():
assert Lambda((x, y), 2*x) == Lambda((x, y), 2*x)
# these, of course, should never be equal
assert Lambda(x, 2*x) != Lambda((x, y), 2*x)
assert Lambda(x, 2*x) != 2*x
# But it is tempting to want expressions that differ only
# in bound symbols to compare the same. But this is not what
# Python's `==` is intended to do; two objects that compare
# as equal means that they are indistibguishable and cache to the
# same value. We wouldn't want to expression that are
# mathematically the same but written in different variables to be
# interchanged else what is the point of allowing for different
# variable names?
assert Lambda(x, 2*x) != Lambda(y, 2*y)
def test_Subs():
assert Subs(1, (), ()) is S.One
# check null subs influence on hashing
assert Subs(x, y, z) != Subs(x, y, 1)
# neutral subs works
assert Subs(x, x, 1).subs(x, y).has(y)
# self mapping var/point
assert Subs(Derivative(f(x), (x, 2)), x, x).doit() == f(x).diff(x, x)
assert Subs(x, x, 0).has(x) # it's a structural answer
assert not Subs(x, x, 0).free_symbols
assert Subs(Subs(x + y, x, 2), y, 1) == Subs(x + y, (x, y), (2, 1))
assert Subs(x, (x,), (0,)) == Subs(x, x, 0)
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(x, y, 2).subs(x, y).doit() == 2
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x)*y, (x, y), (0, 1)) == Subs(f(y)*x, (y, x), (0, 1))
assert Subs(f(x)*y, (x, y), (1, 1)) == Subs(f(y)*x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y*f(x), x, y).subs(y, 2) == Subs(2*f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2*y
assert Subs(f(x), x, 0).free_symbols == set()
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2*f(x), x, 0).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), x, y)
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2*Subs(f(x), x, 0)
e1 = Subs(z*f(x), x, 1)
e2 = Subs(z*f(y), y, 1)
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), x, 1) not in [ e1, e2 ]
assert Derivative(f(x), x).subs(x, g(x)) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs(x, x + y) == Subs(Derivative(f(x), x),
x, x + y)
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x*f(x).diff(x).subs(x, 0)).subs(x, y) == y*f(x).diff(x).subs(x, 0)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit() == 2*exp(x)
assert Subs(Derivative(g(x)**2, g(x), x), g(x), exp(x)
).doit(deep=False) == 2*Derivative(exp(x), x)
assert Derivative(f(x, g(x)), x).doit() == Derivative(
f(x, g(x)), g(x))*Derivative(g(x), x) + Subs(Derivative(
f(y, g(x)), y), y, x)
def test_doitdoit():
done = Derivative(f(x, g(x)), x, g(x)).doit()
assert done == done.doit()
@XFAIL
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), x**2, x).doit() == f(sqrt(x))
def test_expand_function():
assert expand(x + y) == x + y
assert expand(x + y, complex=True) == I*im(x) + I*im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
def test_function_comparable():
assert sin(x).is_comparable is False
assert cos(x).is_comparable is False
assert sin(Float('0.1')).is_comparable is True
assert cos(Float('0.1')).is_comparable is True
assert sin(E).is_comparable is True
assert cos(E).is_comparable is True
assert sin(Rational(1, 3)).is_comparable is True
assert cos(Rational(1, 3)).is_comparable is True
def test_function_comparable_infinities():
assert sin(oo).is_comparable is False
assert sin(-oo).is_comparable is False
assert sin(zoo).is_comparable is False
assert sin(nan).is_comparable is False
def test_deriv1():
# These all require derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(2*x).diff(x) == 2*Subs(Derivative(f(x), x), x, 2*x)
assert (f(x)**3).diff(x) == 3*f(x)**2*f(x).diff(x)
assert (f(2*x)**3).diff(x) == 6*f(2*x)**2*Subs(
Derivative(f(x), x), x, 2*x)
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), x, x + 2)
assert f(2 + 3*x).diff(x) == 3*Subs(
Derivative(f(x), x), x, 3*x + 2)
assert f(3*sin(x)).diff(x) == 3*cos(x)*Subs(
Derivative(f(x), x), x, 3*sin(x))
# See issue 8510
assert f(x, x + z).diff(x) == (
Subs(Derivative(f(y, x + z), y), y, x) +
Subs(Derivative(f(x, y), y), y, x + z))
assert f(x, x**2).diff(x) == (
2*x*Subs(Derivative(f(x, y), y), y, x**2) +
Subs(Derivative(f(y, x**2), y), y, x))
# but Subs is not always necessary
assert f(x, g(y)).diff(g(y)) == Derivative(f(x, g(y)), g(y))
def test_deriv2():
assert (x**3).diff(x) == 3*x**2
assert (x**3).diff(x, evaluate=False) != 3*x**2
assert (x**3).diff(x, evaluate=False) == Derivative(x**3, x)
assert diff(x**3, x) == 3*x**2
assert diff(x**3, x, evaluate=False) != 3*x**2
assert diff(x**3, x, evaluate=False) == Derivative(x**3, x)
def test_func_deriv():
assert f(x).diff(x) == Derivative(f(x), x)
# issue 4534
assert f(x, y).diff(x, y) - f(x, y).diff(y, x) == 0
assert Derivative(f(x, y), x, y).args[1:] == ((x, 1), (y, 1))
assert Derivative(f(x, y), y, x).args[1:] == ((y, 1), (x, 1))
assert (Derivative(f(x, y), x, y) - Derivative(f(x, y), y, x)).doit() == 0
def test_suppressed_evaluation():
a = sin(0, evaluate=False)
assert a != 0
assert a.func is sin
assert a.args == (0,)
def test_function_evalf():
def eq(a, b, eps):
return abs(a - b) < eps
assert eq(sin(1).evalf(15), Float("0.841470984807897"), 1e-13)
assert eq(
sin(2).evalf(25), Float("0.9092974268256816953960199", 25), 1e-23)
assert eq(sin(1 + I).evalf(
15), Float("1.29845758141598") + Float("0.634963914784736")*I, 1e-13)
assert eq(exp(1 + I).evalf(15), Float(
"1.46869393991588") + Float("2.28735528717884239")*I, 1e-13)
assert eq(exp(-0.5 + 1.5*I).evalf(15), Float(
"0.0429042815937374") + Float("0.605011292285002")*I, 1e-13)
assert eq(log(pi + sqrt(2)*I).evalf(
15), Float("1.23699044022052") + Float("0.422985442737893")*I, 1e-13)
assert eq(cos(100).evalf(15), Float("0.86231887228768"), 1e-13)
def test_extensibility_eval():
class MyFunc(Function):
@classmethod
def eval(cls, *args):
return (0, 0, 0)
assert MyFunc(0) == (0, 0, 0)
@_both_exp_pow
def test_function_non_commutative():
x = Symbol('x', commutative=False)
assert f(x).is_commutative is False
assert sin(x).is_commutative is False
assert exp(x).is_commutative is False
assert log(x).is_commutative is False
assert f(x).is_complex is False
assert sin(x).is_complex is False
assert exp(x).is_complex is False
assert log(x).is_complex is False
def test_function_complex():
x = Symbol('x', complex=True)
xzf = Symbol('x', complex=True, zero=False)
assert f(x).is_commutative is True
assert sin(x).is_commutative is True
assert exp(x).is_commutative is True
assert log(x).is_commutative is True
assert f(x).is_complex is None
assert sin(x).is_complex is True
assert exp(x).is_complex is True
assert log(x).is_complex is None
assert log(xzf).is_complex is True
def test_function__eval_nseries():
n = Symbol('n')
assert sin(x)._eval_nseries(x, 2, None) == x + O(x**2)
assert sin(x + 1)._eval_nseries(x, 2, None) == x*cos(1) + sin(1) + O(x**2)
assert sin(pi*(1 - x))._eval_nseries(x, 2, None) == pi*x + O(x**2)
assert acos(1 - x**2)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x**2) + O(x**2)
assert polygamma(n, x + 1)._eval_nseries(x, 2, None) == \
polygamma(n, 1) + polygamma(n + 1, 1)*x + O(x**2)
raises(PoleError, lambda: sin(1/x)._eval_nseries(x, 2, None))
assert acos(1 - x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(x) + sqrt(2)*x**(S(3)/2)/12 + O(x**2)
assert acos(1 + x)._eval_nseries(x, 2, None) == sqrt(2)*sqrt(-x) + sqrt(2)*(-x)**(S(3)/2)/12 + O(x**2)
assert loggamma(1/x)._eval_nseries(x, 0, None) == \
log(x)/2 - log(x)/x - 1/x + O(1, x)
assert loggamma(log(1/x)).nseries(x, n=1, logx=y) == loggamma(-y)
# issue 6725:
assert expint(Rational(3, 2), -x)._eval_nseries(x, 5, None) == \
2 - 2*sqrt(pi)*sqrt(-x) - 2*x + x**2 + x**3/3 + x**4/12 + 4*I*x**(S(3)/2)*sqrt(-x)/3 + \
2*I*x**(S(5)/2)*sqrt(-x)/5 + 2*I*x**(S(7)/2)*sqrt(-x)/21 + O(x**5)
assert sin(sqrt(x))._eval_nseries(x, 3, None) == \
sqrt(x) - x**Rational(3, 2)/6 + x**Rational(5, 2)/120 + O(x**3)
# issue 19065:
s1 = f(x,y).series(y, n=2)
assert {i.name for i in s1.atoms(Symbol)} == {'x', 'xi', 'y'}
xi = Symbol('xi')
s2 = f(xi, y).series(y, n=2)
assert {i.name for i in s2.atoms(Symbol)} == {'xi', 'xi0', 'y'}
def test_doit():
n = Symbol('n', integer=True)
f = Sum(2 * n * x, (n, 1, 3))
d = Derivative(f, x)
assert d.doit() == 12
assert d.doit(deep=False) == Sum(2*n, (n, 1, 3))
def test_evalf_default():
from sympy.functions.special.gamma_functions import polygamma
assert type(sin(4.0)) == Float
assert type(re(sin(I + 1.0))) == Float
assert type(im(sin(I + 1.0))) == Float
assert type(sin(4)) == sin
assert type(polygamma(2.0, 4.0)) == Float
assert type(sin(Rational(1, 4))) == sin
def test_issue_5399():
args = [x, y, S(2), S.Half]
def ok(a):
"""Return True if the input args for diff are ok"""
if not a:
return False
if a[0].is_Symbol is False:
return False
s_at = [i for i in range(len(a)) if a[i].is_Symbol]
n_at = [i for i in range(len(a)) if not a[i].is_Symbol]
# every symbol is followed by symbol or int
# every number is followed by a symbol
return (all(a[i + 1].is_Symbol or a[i + 1].is_Integer
for i in s_at if i + 1 < len(a)) and
all(a[i + 1].is_Symbol
for i in n_at if i + 1 < len(a)))
eq = x**10*y**8
for a in subsets(args):
for v in variations(a, len(a)):
if ok(v):
eq.diff(*v) # does not raise
else:
raises(ValueError, lambda: eq.diff(*v))
def test_derivative_numerically():
z0 = x._random()
assert abs(Derivative(sin(x), x).doit_numerically(z0) - cos(z0)) < 1e-15
def test_fdiff_argument_index_error():
from sympy.core.function import ArgumentIndexError
class myfunc(Function):
nargs = 1 # define since there is no eval routine
def fdiff(self, idx):
raise ArgumentIndexError
mf = myfunc(x)
assert mf.diff(x) == Derivative(mf, x)
raises(TypeError, lambda: myfunc(x, x))
def test_deriv_wrt_function():
x = f(t)
xd = diff(x, t)
xdd = diff(xd, t)
y = g(t)
yd = diff(y, t)
assert diff(x, t) == xd
assert diff(2 * x + 4, t) == 2 * xd
assert diff(2 * x + 4 + y, t) == 2 * xd + yd
assert diff(2 * x + 4 + y * x, t) == 2 * xd + x * yd + xd * y
assert diff(2 * x + 4 + y * x, x) == 2 + y
assert (diff(4 * x**2 + 3 * x + x * y, t) == 3 * xd + x * yd + xd * y +
8 * x * xd)
assert (diff(4 * x**2 + 3 * xd + x * y, t) == 3 * xdd + x * yd + xd * y +
8 * x * xd)
assert diff(4 * x**2 + 3 * xd + x * y, xd) == 3
assert diff(4 * x**2 + 3 * xd + x * y, xdd) == 0
assert diff(sin(x), t) == xd * cos(x)
assert diff(exp(x), t) == xd * exp(x)
assert diff(sqrt(x), t) == xd / (2 * sqrt(x))
def test_diff_wrt_value():
assert Expr()._diff_wrt is False
assert x._diff_wrt is True
assert f(x)._diff_wrt is True
assert Derivative(f(x), x)._diff_wrt is True
assert Derivative(x**2, x)._diff_wrt is False
def test_diff_wrt():
fx = f(x)
dfx = diff(f(x), x)
ddfx = diff(f(x), x, x)
assert diff(sin(fx) + fx**2, fx) == cos(fx) + 2*fx
assert diff(sin(dfx) + dfx**2, dfx) == cos(dfx) + 2*dfx
assert diff(sin(ddfx) + ddfx**2, ddfx) == cos(ddfx) + 2*ddfx
assert diff(fx**2, dfx) == 0
assert diff(fx**2, ddfx) == 0
assert diff(dfx**2, fx) == 0
assert diff(dfx**2, ddfx) == 0
assert diff(ddfx**2, dfx) == 0
assert diff(fx*dfx*ddfx, fx) == dfx*ddfx
assert diff(fx*dfx*ddfx, dfx) == fx*ddfx
assert diff(fx*dfx*ddfx, ddfx) == fx*dfx
assert diff(f(x), x).diff(f(x)) == 0
assert (sin(f(x)) - cos(diff(f(x), x))).diff(f(x)) == cos(f(x))
assert diff(sin(fx), fx, x) == diff(sin(fx), x, fx)
# Chain rule cases
assert f(g(x)).diff(x) == \
Derivative(g(x), x)*Derivative(f(g(x)), g(x))
assert diff(f(g(x), h(y)), x) == \
Derivative(g(x), x)*Derivative(f(g(x), h(y)), g(x))
assert diff(f(g(x), h(x)), x) == (
Derivative(f(g(x), h(x)), g(x))*Derivative(g(x), x) +
Derivative(f(g(x), h(x)), h(x))*Derivative(h(x), x))
assert f(
sin(x)).diff(x) == cos(x)*Subs(Derivative(f(x), x), x, sin(x))
assert diff(f(g(x)), g(x)) == Derivative(f(g(x)), g(x))
def test_diff_wrt_func_subs():
assert f(g(x)).diff(x).subs(g, Lambda(x, 2*x)).doit() == f(2*x).diff(x)
def test_subs_in_derivative():
expr = sin(x*exp(y))
u = Function('u')
v = Function('v')
assert Derivative(expr, y).subs(expr, y) == Derivative(y, y)
assert Derivative(expr, y).subs(y, x).doit() == \
Derivative(expr, y).doit().subs(y, x)
assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y, x)
assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x, y)
assert Derivative(f(x, y), y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y, g(x, y)).doit()
assert Derivative(f(x, y), y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x, g(x, y))
assert Derivative(f(x, y), g(y)).subs(x, g(x, y)) == Derivative(f(g(x, y), y), g(y))
assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y)).doit()
assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
Derivative(f(g(x), u(y)), u(y))
assert Derivative(f(x, f(x, x)), f(x, x)).subs(
f, Lambda((x, y), x + y)) == Subs(
Derivative(z + x, z), z, 2*x)
assert Subs(Derivative(f(f(x)), x), f, cos).doit() == sin(x)*sin(cos(x))
assert Subs(Derivative(f(f(x)), f(x)), f, cos).doit() == -sin(cos(x))
# Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
# This is also related to issues 13791 and 13795; issue 15190
F = Lambda((x, y), exp(2*x + 3*y))
abstract = f(x, f(x, x)).diff(x, 2)
concrete = F(x, F(x, x)).diff(x, 2)
assert (abstract.subs(f, F).doit() - concrete).simplify() == 0
# don't introduce a new symbol if not necessary
assert x in f(x).diff(x).subs(x, 0).atoms()
# case (4)
assert Derivative(f(x,f(x,y)), x, y).subs(x, g(y)
) == Subs(Derivative(f(x, f(x, y)), x, y), x, g(y))
assert Derivative(f(x, x), x).subs(x, 0
) == Subs(Derivative(f(x, x), x), x, 0)
# issue 15194
assert Derivative(f(y, g(x)), (x, z)).subs(z, x
) == Derivative(f(y, g(x)), (x, x))
df = f(x).diff(x)
assert df.subs(df, 1) is S.One
assert df.diff(df) is S.One
dxy = Derivative(f(x, y), x, y)
dyx = Derivative(f(x, y), y, x)
assert dxy.subs(Derivative(f(x, y), y, x), 1) is S.One
assert dxy.diff(dyx) is S.One
assert Derivative(f(x, y), x, 2, y, 3).subs(
dyx, g(x, y)) == Derivative(g(x, y), x, 1, y, 2)
assert Derivative(f(x, x - y), y).subs(x, x + y) == Subs(
Derivative(f(x, x - y), y), x, x + y)
def test_diff_wrt_not_allowed():
# issue 7027 included
for wrt in (
cos(x), re(x), x**2, x*y, 1 + x,
Derivative(cos(x), x), Derivative(f(f(x)), x)):
raises(ValueError, lambda: diff(f(x), wrt))
# if we don't differentiate wrt then don't raise error
assert diff(exp(x*y), x*y, 0) == exp(x*y)
def test_diff_wrt_intlike():
class Two:
def __int__(self):
return 2
assert cos(x).diff(x, Two()) == -cos(x)
def test_klein_gordon_lagrangian():
m = Symbol('m')
phi = f(x, t)
L = -(diff(phi, t)**2 - diff(phi, x)**2 - m**2*phi**2)/2
eqna = Eq(
diff(L, phi) - diff(L, diff(phi, x), x) - diff(L, diff(phi, t), t), 0)
eqnb = Eq(diff(phi, t, t) - diff(phi, x, x) + m**2*phi, 0)
assert eqna == eqnb
def test_sho_lagrangian():
m = Symbol('m')
k = Symbol('k')
x = f(t)
L = m*diff(x, t)**2/2 - k*x**2/2
eqna = Eq(diff(L, x), diff(L, diff(x, t), t))
eqnb = Eq(-k*x, m*diff(x, t, t))
assert eqna == eqnb
assert diff(L, x, t) == diff(L, t, x)
assert diff(L, diff(x, t), t) == m*diff(x, t, 2)
assert diff(L, t, diff(x, t)) == -k*x + m*diff(x, t, 2)
def test_straight_line():
F = f(x)
Fd = F.diff(x)
L = sqrt(1 + Fd**2)
assert diff(L, F) == 0
assert diff(L, Fd) == Fd/sqrt(1 + Fd**2)
def test_sort_variable():
vsort = Derivative._sort_variable_count
def vsort0(*v, reverse=False):
return [i[0] for i in vsort([(i, 0) for i in (
reversed(v) if reverse else v)])]
for R in range(2):
assert vsort0(y, x, reverse=R) == [x, y]
assert vsort0(f(x), x, reverse=R) == [x, f(x)]
assert vsort0(f(y), f(x), reverse=R) == [f(x), f(y)]
assert vsort0(g(x), f(y), reverse=R) == [f(y), g(x)]
assert vsort0(f(x, y), f(x), reverse=R) == [f(x), f(x, y)]
fx = f(x).diff(x)
assert vsort0(fx, y, reverse=R) == [y, fx]
fy = f(y).diff(y)
assert vsort0(fy, fx, reverse=R) == [fx, fy]
fxx = fx.diff(x)
assert vsort0(fxx, fx, reverse=R) == [fx, fxx]
assert vsort0(Basic(x), f(x), reverse=R) == [f(x), Basic(x)]
assert vsort0(Basic(y), Basic(x), reverse=R) == [Basic(x), Basic(y)]
assert vsort0(Basic(y, z), Basic(x), reverse=R) == [
Basic(x), Basic(y, z)]
assert vsort0(fx, x, reverse=R) == [
x, fx] if R else [fx, x]
assert vsort0(Basic(x), x, reverse=R) == [
x, Basic(x)] if R else [Basic(x), x]
assert vsort0(Basic(f(x)), f(x), reverse=R) == [
f(x), Basic(f(x))] if R else [Basic(f(x)), f(x)]
assert vsort0(Basic(x, z), Basic(x), reverse=R) == [
Basic(x), Basic(x, z)] if R else [Basic(x, z), Basic(x)]
assert vsort([]) == []
assert _aresame(vsort([(x, 1)]), [Tuple(x, 1)])
assert vsort([(x, y), (x, z)]) == [(x, y + z)]
assert vsort([(y, 1), (x, 1 + y)]) == [(x, 1 + y), (y, 1)]
# coverage complete; legacy tests below
assert vsort([(x, 3), (y, 2), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(x), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 2), (x, 3), (h(x), 1), (g(x), 1),
(f(x), 1)]) == [(x, 3), (y, 2), (z, 1), (f(x), 1), (g(x), 1),
(h(x), 1)]
assert vsort([(x, 1), (f(x), 1), (y, 1), (f(y), 1)]) == [(x, 1),
(y, 1), (f(x), 1), (f(y), 1)]
assert vsort([(y, 1), (x, 2), (g(x), 1), (f(x), 1), (z, 1),
(h(x), 1), (y, 2), (x, 1)]) == [(x, 3), (y, 3), (z, 1),
(f(x), 1), (g(x), 1), (h(x), 1)]
assert vsort([(z, 1), (y, 1), (f(x), 1), (x, 1), (f(x), 1),
(g(x), 1)]) == [(x, 1), (y, 1), (z, 1), (f(x), 2), (g(x), 1)]
assert vsort([(z, 1), (y, 2), (f(x), 1), (x, 2), (f(x), 2),
(g(x), 1), (z, 2), (z, 1), (y, 1), (x, 1)]) == [(x, 3), (y, 3),
(z, 4), (f(x), 3), (g(x), 1)]
assert vsort(((y, 2), (x, 1), (y, 1), (x, 1))) == [(x, 2), (y, 3)]
assert isinstance(vsort([(x, 3), (y, 2), (z, 1)])[0], Tuple)
assert vsort([(x, 1), (f(x), 1), (x, 1)]) == [(x, 2), (f(x), 1)]
assert vsort([(y, 2), (x, 3), (z, 1)]) == [(x, 3), (y, 2), (z, 1)]
assert vsort([(h(y), 1), (g(x), 1), (f(x), 1)]) == [
(f(x), 1), (g(x), 1), (h(y), 1)]
assert vsort([(x, 1), (y, 1), (x, 1)]) == [(x, 2), (y, 1)]
assert vsort([(f(x), 1), (f(y), 1), (f(x), 1)]) == [
(f(x), 2), (f(y), 1)]
dfx = f(x).diff(x)
self = [(dfx, 1), (x, 1)]
assert vsort(self) == self
assert vsort([
(dfx, 1), (y, 1), (f(x), 1), (x, 1), (f(y), 1), (x, 1)]) == [
(y, 1), (f(x), 1), (f(y), 1), (dfx, 1), (x, 2)]
dfy = f(y).diff(y)
assert vsort([(dfy, 1), (dfx, 1)]) == [(dfx, 1), (dfy, 1)]
d2fx = dfx.diff(x)
assert vsort([(d2fx, 1), (dfx, 1)]) == [(dfx, 1), (d2fx, 1)]
def test_multiple_derivative():
# Issue #15007
assert f(x, y).diff(y, y, x, y, x
) == Derivative(f(x, y), (x, 2), (y, 3))
def test_unhandled():
class MyExpr(Expr):
def _eval_derivative(self, s):
if not s.name.startswith('xi'):
return self
else:
return None
eq = MyExpr(f(x), y, z)
assert diff(eq, x, y, f(x), z) == Derivative(eq, f(x))
assert diff(eq, f(x), x) == Derivative(eq, f(x))
assert f(x, y).diff(x,(y, z)) == Derivative(f(x, y), x, (y, z))
assert f(x, y).diff(x,(y, 0)) == Derivative(f(x, y), x)
def test_nfloat():
from sympy.core.basic import _aresame
from sympy.polys.rootoftools import rootof
x = Symbol("x")
eq = x**Rational(4, 3) + 4*x**(S.One/3)/3
assert _aresame(nfloat(eq), x**Rational(4, 3) + (4.0/3)*x**(S.One/3))
assert _aresame(nfloat(eq, exponent=True), x**(4.0/3) + (4.0/3)*x**(1.0/3))
eq = x**Rational(4, 3) + 4*x**(x/3)/3
assert _aresame(nfloat(eq), x**Rational(4, 3) + (4.0/3)*x**(x/3))
big = 12345678901234567890
# specify precision to match value used in nfloat
Float_big = Float(big, 15)
assert _aresame(nfloat(big), Float_big)
assert _aresame(nfloat(big*x), Float_big*x)
assert _aresame(nfloat(x**big, exponent=True), x**Float_big)
assert nfloat(cos(x + sqrt(2))) == cos(x + nfloat(sqrt(2)))
# issue 6342
f = S('x*lamda + lamda**3*(x/2 + 1/2) + lamda**2 + 1/4')
assert not any(a.free_symbols for a in solveset(f.subs(x, -0.139)))
# issue 6632
assert nfloat(-100000*sqrt(2500000001) + 5000000001) == \
9.99999999800000e-11
# issue 7122
eq = cos(3*x**4 + y)*rootof(x**5 + 3*x**3 + 1, 0)
assert str(nfloat(eq, exponent=False, n=1)) == '-0.7*cos(3.0*x**4 + y)'
# issue 10933
for ti in (dict, Dict):
d = ti({S.Half: S.Half})
n = nfloat(d)
assert isinstance(n, ti)
assert _aresame(list(n.items()).pop(), (S.Half, Float(.5)))
for ti in (dict, Dict):
d = ti({S.Half: S.Half})
n = nfloat(d, dkeys=True)
assert isinstance(n, ti)
assert _aresame(list(n.items()).pop(), (Float(.5), Float(.5)))
d = [S.Half]
n = nfloat(d)
assert type(n) is list
assert _aresame(n[0], Float(.5))
assert _aresame(nfloat(Eq(x, S.Half)).rhs, Float(.5))
assert _aresame(nfloat(S(True)), S(True))
assert _aresame(nfloat(Tuple(S.Half))[0], Float(.5))
assert nfloat(Eq((3 - I)**2/2 + I, 0)) == S.false
# pass along kwargs
assert nfloat([{S.Half: x}], dkeys=True) == [{Float(0.5): x}]
# Issue 17706
A = MutableMatrix([[1, 2], [3, 4]])
B = MutableMatrix(
[[Float('1.0', precision=53), Float('2.0', precision=53)],
[Float('3.0', precision=53), Float('4.0', precision=53)]])
assert _aresame(nfloat(A), B)
A = ImmutableMatrix([[1, 2], [3, 4]])
B = ImmutableMatrix(
[[Float('1.0', precision=53), Float('2.0', precision=53)],
[Float('3.0', precision=53), Float('4.0', precision=53)]])
assert _aresame(nfloat(A), B)
# issue 22524
f = Function('f')
assert not nfloat(f(2)).atoms(Float)
def test_issue_7068():
from sympy.abc import a, b
f = Function('f')
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), a, y1)
z2 = Subs(f(a), a, y2)
assert z1 != z2
def test_issue_7231():
from sympy.abc import a
ans1 = f(x).series(x, a)
res = (f(a) + (-a + x)*Subs(Derivative(f(y), y), y, a) +
(-a + x)**2*Subs(Derivative(f(y), y, y), y, a)/2 +
(-a + x)**3*Subs(Derivative(f(y), y, y, y),
y, a)/6 +
(-a + x)**4*Subs(Derivative(f(y), y, y, y, y),
y, a)/24 +
(-a + x)**5*Subs(Derivative(f(y), y, y, y, y, y),
y, a)/120 + O((-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def test_issue_7687():
from sympy.core.function import Function
from sympy.abc import x
f = Function('f')(x)
ff = Function('f')(x)
match_with_cache = ff.matches(f)
assert isinstance(f, type(ff))
clear_cache()
ff = Function('f')(x)
assert isinstance(f, type(ff))
assert match_with_cache == ff.matches(f)
def test_issue_7688():
from sympy.core.function import Function, UndefinedFunction
f = Function('f') # actually an UndefinedFunction
clear_cache()
class A(UndefinedFunction):
pass
a = A('f')
assert isinstance(a, type(f))
def test_mexpand():
from sympy.abc import x
assert _mexpand(None) is None
assert _mexpand(1) is S.One
assert _mexpand(x*(x + 1)**2) == (x*(x + 1)**2).expand()
def test_issue_8469():
# This should not take forever to run
N = 40
def g(w, theta):
return 1/(1+exp(w-theta))
ws = symbols(['w%i'%i for i in range(N)])
import functools
expr = functools.reduce(g, ws)
assert isinstance(expr, Pow)
def test_issue_12996():
# foo=True imitates the sort of arguments that Derivative can get
# from Integral when it passes doit to the expression
assert Derivative(im(x), x).doit(foo=True) == Derivative(im(x), x)
def test_should_evalf():
# This should not take forever to run (see #8506)
assert isinstance(sin((1.0 + 1.0*I)**10000 + 1), sin)
def test_Derivative_as_finite_difference():
# Central 1st derivative at gridpoint
x, h = symbols('x h', real=True)
dfdx = f(x).diff(x)
assert (dfdx.as_finite_difference([x-2, x-1, x, x+1, x+2]) -
(S.One/12*(f(x-2)-f(x+2)) + Rational(2, 3)*(f(x+1)-f(x-1)))).simplify() == 0
# Central 1st derivative "half-way"
assert (dfdx.as_finite_difference() -
(f(x + S.Half)-f(x - S.Half))).simplify() == 0
assert (dfdx.as_finite_difference(h) -
(f(x + h/S(2))-f(x - h/S(2)))/h).simplify() == 0
assert (dfdx.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(S(9)/(8*2*h)*(f(x+h) - f(x-h)) +
S.One/(24*2*h)*(f(x - 3*h) - f(x + 3*h)))).simplify() == 0
# One sided 1st derivative at gridpoint
assert (dfdx.as_finite_difference([0, 1, 2], 0) -
(Rational(-3, 2)*f(0) + 2*f(1) - f(2)/2)).simplify() == 0
assert (dfdx.as_finite_difference([x, x+h], x) -
(f(x+h) - f(x))/h).simplify() == 0
assert (dfdx.as_finite_difference([x-h, x, x+h], x-h) -
(-S(3)/(2*h)*f(x-h) + 2/h*f(x) -
S.One/(2*h)*f(x+h))).simplify() == 0
# One sided 1st derivative "half-way"
assert (dfdx.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h, x + 7*h])
- 1/(2*h)*(-S(11)/(12)*f(x-h) + S(17)/(24)*f(x+h)
+ Rational(3, 8)*f(x + 3*h) - Rational(5, 24)*f(x + 5*h)
+ S.One/24*f(x + 7*h))).simplify() == 0
d2fdx2 = f(x).diff(x, 2)
# Central 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x-h, x, x+h]) -
h**-2 * (f(x-h) + f(x+h) - 2*f(x))).simplify() == 0
assert (d2fdx2.as_finite_difference([x - 2*h, x-h, x, x+h, x + 2*h]) -
h**-2 * (Rational(-1, 12)*(f(x - 2*h) + f(x + 2*h)) +
Rational(4, 3)*(f(x+h) + f(x-h)) - Rational(5, 2)*f(x))).simplify() == 0
# Central 2nd derivative "half-way"
assert (d2fdx2.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(2*h)**-2 * (S.Half*(f(x - 3*h) + f(x + 3*h)) -
S.Half*(f(x+h) + f(x-h)))).simplify() == 0
# One sided 2nd derivative at gridpoint
assert (d2fdx2.as_finite_difference([x, x+h, x + 2*h, x + 3*h]) -
h**-2 * (2*f(x) - 5*f(x+h) +
4*f(x+2*h) - f(x+3*h))).simplify() == 0
# One sided 2nd derivative at "half-way"
assert (d2fdx2.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h]) -
(2*h)**-2 * (Rational(3, 2)*f(x-h) - Rational(7, 2)*f(x+h) + Rational(5, 2)*f(x + 3*h) -
S.Half*f(x + 5*h))).simplify() == 0
d3fdx3 = f(x).diff(x, 3)
# Central 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference() -
(-f(x - Rational(3, 2)) + 3*f(x - S.Half) -
3*f(x + S.Half) + f(x + Rational(3, 2)))).simplify() == 0
assert (d3fdx3.as_finite_difference(
[x - 3*h, x - 2*h, x-h, x, x+h, x + 2*h, x + 3*h]) -
h**-3 * (S.One/8*(f(x - 3*h) - f(x + 3*h)) - f(x - 2*h) +
f(x + 2*h) + Rational(13, 8)*(f(x-h) - f(x+h)))).simplify() == 0
# Central 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x - 3*h, x-h, x+h, x + 3*h]) -
(2*h)**-3 * (f(x + 3*h)-f(x - 3*h) +
3*(f(x-h)-f(x+h)))).simplify() == 0
# One sided 3rd derivative at gridpoint
assert (d3fdx3.as_finite_difference([x, x+h, x + 2*h, x + 3*h]) -
h**-3 * (f(x + 3*h)-f(x) + 3*(f(x+h)-f(x + 2*h)))).simplify() == 0
# One sided 3rd derivative at "half-way"
assert (d3fdx3.as_finite_difference([x-h, x+h, x + 3*h, x + 5*h]) -
(2*h)**-3 * (f(x + 5*h)-f(x-h) +
3*(f(x+h)-f(x + 3*h)))).simplify() == 0
# issue 11007
y = Symbol('y', real=True)
d2fdxdy = f(x, y).diff(x, y)
ref0 = Derivative(f(x + S.Half, y), y) - Derivative(f(x - S.Half, y), y)
assert (d2fdxdy.as_finite_difference(wrt=x) - ref0).simplify() == 0
half = S.Half
xm, xp, ym, yp = x-half, x+half, y-half, y+half
ref2 = f(xm, ym) + f(xp, yp) - f(xp, ym) - f(xm, yp)
assert (d2fdxdy.as_finite_difference() - ref2).simplify() == 0
def test_issue_11159():
# Tests Application._eval_subs
with _exp_is_pow(False):
expr1 = E
expr0 = expr1 * expr1
expr1 = expr0.subs(expr1,expr0)
assert expr0 == expr1
with _exp_is_pow(True):
expr1 = E
expr0 = expr1 * expr1
expr2 = expr0.subs(expr1, expr0)
assert expr2 == E ** 4
def test_issue_12005():
e1 = Subs(Derivative(f(x), x), x, x)
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), x, x**2 + 1)
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), x, x**2 + 1)
e3 = Subs(Derivative(f(x) + y**2 - y, y), y, y**2)
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), y, (x**2))
assert e4.diff(y) is S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Derivative(f(g(x)), g(x), g(x))
def test_issue_13843():
x = symbols('x')
f = Function('f')
m, n = symbols('m n', integer=True)
assert Derivative(Derivative(f(x), (x, m)), (x, n)) == Derivative(f(x), (x, m + n))
assert Derivative(Derivative(f(x), (x, m+5)), (x, n+3)) == Derivative(f(x), (x, m + n + 8))
assert Derivative(f(x), (x, n)).doit() == Derivative(f(x), (x, n))
def test_order_could_be_zero():
x, y = symbols('x, y')
n = symbols('n', integer=True, nonnegative=True)
m = symbols('m', integer=True, positive=True)
assert diff(y, (x, n)) == Piecewise((y, Eq(n, 0)), (0, True))
assert diff(y, (x, n + 1)) is S.Zero
assert diff(y, (x, m)) is S.Zero
def test_undefined_function_eq():
f = Function('f')
f2 = Function('f')
g = Function('g')
f_real = Function('f', is_real=True)
# This test may only be meaningful if the cache is turned off
assert f == f2
assert hash(f) == hash(f2)
assert f == f
assert f != g
assert f != f_real
def test_function_assumptions():
x = Symbol('x')
f = Function('f')
f_real = Function('f', real=True)
f_real1 = Function('f', real=1)
f_real_inherit = Function(Symbol('f', real=True))
assert f_real == f_real1 # assumptions are sanitized
assert f != f_real
assert f(x) != f_real(x)
assert f(x).is_real is None
assert f_real(x).is_real is True
assert f_real_inherit(x).is_real is True and f_real_inherit.name == 'f'
# Can also do it this way, but it won't be equal to f_real because of the
# way UndefinedFunction.__new__ works. Any non-recognized assumptions
# are just added literally as something which is used in the hash
f_real2 = Function('f', is_real=True)
assert f_real2(x).is_real is True
def test_undef_fcn_float_issue_6938():
f = Function('ceil')
assert not f(0.3).is_number
f = Function('sin')
assert not f(0.3).is_number
assert not f(pi).evalf().is_number
x = Symbol('x')
assert not f(x).evalf(subs={x:1.2}).is_number
def test_undefined_function_eval():
# Issue 15170. Make sure UndefinedFunction with eval defined works
# properly. The issue there was that the hash was determined before _nargs
# was set, which is included in the hash, hence changing the hash. The
# class is added to sympy.core.core.all_classes before the hash is
# changed, meaning "temp in all_classes" would fail, causing sympify(temp(t))
# to give a new class. We will eventually remove all_classes, but make
# sure this continues to work.
fdiff = lambda self, argindex=1: cos(self.args[argindex - 1])
eval = classmethod(lambda cls, t: None)
_imp_ = classmethod(lambda cls, t: sin(t))
temp = Function('temp', fdiff=fdiff, eval=eval, _imp_=_imp_)
expr = temp(t)
assert sympify(expr) == expr
assert type(sympify(expr)).fdiff.__name__ == "<lambda>"
assert expr.diff(t) == cos(t)
def test_issue_15241():
F = f(x)
Fx = F.diff(x)
assert (F + x*Fx).diff(x, Fx) == 2
assert (F + x*Fx).diff(Fx, x) == 1
assert (x*F + x*Fx*F).diff(F, x) == x*Fx.diff(x) + Fx + 1
assert (x*F + x*Fx*F).diff(x, F) == x*Fx.diff(x) + Fx + 1
y = f(x)
G = f(y)
Gy = G.diff(y)
assert (G + y*Gy).diff(y, Gy) == 2
assert (G + y*Gy).diff(Gy, y) == 1
assert (y*G + y*Gy*G).diff(G, y) == y*Gy.diff(y) + Gy + 1
assert (y*G + y*Gy*G).diff(y, G) == y*Gy.diff(y) + Gy + 1
def test_issue_15226():
assert Subs(Derivative(f(y), x, y), y, g(x)).doit() != 0
def test_issue_7027():
for wrt in (cos(x), re(x), Derivative(cos(x), x)):
raises(ValueError, lambda: diff(f(x), wrt))
def test_derivative_quick_exit():
assert f(x).diff(y) == 0
assert f(x).diff(y, f(x)) == 0
assert f(x).diff(x, f(y)) == 0
assert f(f(x)).diff(x, f(x), f(y)) == 0
assert f(f(x)).diff(x, f(x), y) == 0
assert f(x).diff(g(x)) == 0
assert f(x).diff(x, f(x).diff(x)) == 1
df = f(x).diff(x)
assert f(x).diff(df) == 0
dg = g(x).diff(x)
assert dg.diff(df).doit() == 0
def test_issue_15084_13166():
eq = f(x, g(x))
assert eq.diff((g(x), y)) == Derivative(f(x, g(x)), (g(x), y))
# issue 13166
assert eq.diff(x, 2).doit() == (
(Derivative(f(x, g(x)), (g(x), 2))*Derivative(g(x), x) +
Subs(Derivative(f(x, _xi_2), _xi_2, x), _xi_2, g(x)))*Derivative(g(x),
x) + Derivative(f(x, g(x)), g(x))*Derivative(g(x), (x, 2)) +
Derivative(g(x), x)*Subs(Derivative(f(_xi_1, g(x)), _xi_1, g(x)),
_xi_1, x) + Subs(Derivative(f(_xi_1, g(x)), (_xi_1, 2)), _xi_1, x))
# issue 6681
assert diff(f(x, t, g(x, t)), x).doit() == (
Derivative(f(x, t, g(x, t)), g(x, t))*Derivative(g(x, t), x) +
Subs(Derivative(f(_xi_1, t, g(x, t)), _xi_1), _xi_1, x))
# make sure the order doesn't matter when using diff
assert eq.diff(x, g(x)) == eq.diff(g(x), x)
def test_negative_counts():
# issue 13873
raises(ValueError, lambda: sin(x).diff(x, -1))
def test_Derivative__new__():
raises(TypeError, lambda: f(x).diff((x, 2), 0))
assert f(x, y).diff([(x, y), 0]) == f(x, y)
assert f(x, y).diff([(x, y), 1]) == NDimArray([
Derivative(f(x, y), x), Derivative(f(x, y), y)])
assert f(x,y).diff(y, (x, z), y, x) == Derivative(
f(x, y), (x, z + 1), (y, 2))
assert Matrix([x]).diff(x, 2) == Matrix([0]) # is_zero exit
def test_issue_14719_10150():
class V(Expr):
_diff_wrt = True
is_scalar = False
assert V().diff(V()) == Derivative(V(), V())
assert (2*V()).diff(V()) == 2*Derivative(V(), V())
class X(Expr):
_diff_wrt = True
assert X().diff(X()) == 1
assert (2*X()).diff(X()) == 2
def test_noncommutative_issue_15131():
x = Symbol('x', commutative=False)
t = Symbol('t', commutative=False)
fx = Function('Fx', commutative=False)(x)
ft = Function('Ft', commutative=False)(t)
A = Symbol('A', commutative=False)
eq = fx * A * ft
eqdt = eq.diff(t)
assert eqdt.args[-1] == ft.diff(t)
def test_Subs_Derivative():
a = Derivative(f(g(x), h(x)), g(x), h(x),x)
b = Derivative(Derivative(f(g(x), h(x)), g(x), h(x)),x)
c = f(g(x), h(x)).diff(g(x), h(x), x)
d = f(g(x), h(x)).diff(g(x), h(x)).diff(x)
e = Derivative(f(g(x), h(x)), x)
eqs = (a, b, c, d, e)
subs = lambda arg: arg.subs(f, Lambda((x, y), exp(x + y))
).subs(g(x), 1/x).subs(h(x), x**3)
ans = 3*x**2*exp(1/x)*exp(x**3) - exp(1/x)*exp(x**3)/x**2
assert all(subs(i).doit().expand() == ans for i in eqs)
assert all(subs(i.doit()).doit().expand() == ans for i in eqs)
def test_issue_15360():
f = Function('f')
assert f.name == 'f'
def test_issue_15947():
assert f._diff_wrt is False
raises(TypeError, lambda: f(f))
raises(TypeError, lambda: f(x).diff(f))
def test_Derivative_free_symbols():
f = Function('f')
n = Symbol('n', integer=True, positive=True)
assert diff(f(x), (x, n)).free_symbols == {n, x}
def test_issue_20683():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
y = Derivative(z, x).subs(x,0)
assert y.doit() == 0
y = Derivative(8, x).subs(x,0)
assert y.doit() == 0
def test_issue_10503():
f = exp(x**3)*cos(x**6)
assert f.series(x, 0, 14) == 1 + x**3 + x**6/2 + x**9/6 - 11*x**12/24 + O(x**14)
def test_issue_17382():
# copied from sympy/core/tests/test_evalf.py
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
x = Symbol('x')
expr = solveset(2 * cos(x) * cos(2 * x) - 1, x, S.Reals)
expected = "Union(" \
"ImageSet(Lambda(_n, 6.28318530717959*_n + 5.79812359592087), Integers), " \
"ImageSet(Lambda(_n, 6.28318530717959*_n + 0.485061711258717), Integers))"
assert NS(expr) == expected
|
14bb633466d47ad237063d1e90740f2896be12ea890c8ddb8bcb06b1e7e98019 | from sympy.core import (
Basic, Rational, Symbol, S, Float, Integer, Mul, Number, Pow,
Expr, I, nan, pi, symbols, oo, zoo, N)
from sympy.core.parameters import global_parameters
from sympy.core.tests.test_evalf import NS
from sympy.core.function import expand_multinomial
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.special.error_functions import erf
from sympy.functions.elementary.trigonometric import (
sin, cos, tan, sec, csc, atan)
from sympy.functions.elementary.hyperbolic import cosh, sinh, tanh
from sympy.polys import Poly
from sympy.series.order import O
from sympy.sets import FiniteSet
from sympy.core.power import power
from sympy.testing.pytest import warns, _both_exp_pow
from sympy.utilities.exceptions import SymPyDeprecationWarning
def test_rational():
a = Rational(1, 5)
r = sqrt(5)/5
assert sqrt(a) == r
assert 2*sqrt(a) == 2*r
r = a*a**S.Half
assert a**Rational(3, 2) == r
assert 2*a**Rational(3, 2) == 2*r
r = a**5*a**Rational(2, 3)
assert a**Rational(17, 3) == r
assert 2 * a**Rational(17, 3) == 2*r
def test_large_rational():
e = (Rational(123712**12 - 1, 7) + Rational(1, 7))**Rational(1, 3)
assert e == 234232585392159195136 * (Rational(1, 7)**Rational(1, 3))
def test_negative_real():
def feq(a, b):
return abs(a - b) < 1E-10
assert feq(S.One / Float(-0.5), -Integer(2))
def test_expand():
x = Symbol('x')
assert (2**(-1 - x)).expand() == S.Half*2**(-x)
def test_issue_3449():
#test if powers are simplified correctly
#see also issue 3995
x = Symbol('x')
assert ((x**Rational(1, 3))**Rational(2)) == x**Rational(2, 3)
assert (
(x**Rational(3))**Rational(2, 5)) == (x**Rational(3))**Rational(2, 5)
a = Symbol('a', real=True)
b = Symbol('b', real=True)
assert (a**2)**b == (abs(a)**b)**2
assert sqrt(1/a) != 1/sqrt(a) # e.g. for a = -1
assert (a**3)**Rational(1, 3) != a
assert (x**a)**b != x**(a*b) # e.g. x = -1, a=2, b=1/2
assert (x**.5)**b == x**(.5*b)
assert (x**.5)**.5 == x**.25
assert (x**2.5)**.5 != x**1.25 # e.g. for x = 5*I
k = Symbol('k', integer=True)
m = Symbol('m', integer=True)
assert (x**k)**m == x**(k*m)
assert Number(5)**Rational(2, 3) == Number(25)**Rational(1, 3)
assert (x**.5)**2 == x**1.0
assert (x**2)**k == (x**k)**2 == x**(2*k)
a = Symbol('a', positive=True)
assert (a**3)**Rational(2, 5) == a**Rational(6, 5)
assert (a**2)**b == (a**b)**2
assert (a**Rational(2, 3))**x == a**(x*Rational(2, 3)) != (a**x)**Rational(2, 3)
def test_issue_3866():
assert --sqrt(sqrt(5) - 1) == sqrt(sqrt(5) - 1)
def test_negative_one():
x = Symbol('x', complex=True)
y = Symbol('y', complex=True)
assert 1/x**y == x**(-y)
def test_issue_4362():
neg = Symbol('neg', negative=True)
nonneg = Symbol('nonneg', nonnegative=True)
any = Symbol('any')
num, den = sqrt(1/neg).as_numer_denom()
assert num == sqrt(-1)
assert den == sqrt(-neg)
num, den = sqrt(1/nonneg).as_numer_denom()
assert num == 1
assert den == sqrt(nonneg)
num, den = sqrt(1/any).as_numer_denom()
assert num == sqrt(1/any)
assert den == 1
def eqn(num, den, pow):
return (num/den)**pow
npos = 1
nneg = -1
dpos = 2 - sqrt(3)
dneg = 1 - sqrt(3)
assert dpos > 0 and dneg < 0 and npos > 0 and nneg < 0
# pos or neg integer
eq = eqn(npos, dpos, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dpos**2)
eq = eqn(npos, dneg, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dneg**2)
eq = eqn(nneg, dpos, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dpos**2)
eq = eqn(nneg, dneg, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dneg**2)
eq = eqn(npos, dpos, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**2, 1)
eq = eqn(npos, dneg, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dneg**2, 1)
eq = eqn(nneg, dpos, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**2, 1)
eq = eqn(nneg, dneg, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dneg**2, 1)
# pos or neg rational
pow = S.Half
eq = eqn(npos, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (npos**pow, dpos**pow)
eq = eqn(npos, dneg, pow)
assert eq.is_Pow is False and eq.as_numer_denom() == ((-npos)**pow, (-dneg)**pow)
eq = eqn(nneg, dpos, pow)
assert not eq.is_Pow or eq.as_numer_denom() == (nneg**pow, dpos**pow)
eq = eqn(nneg, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-nneg)**pow, (-dneg)**pow)
eq = eqn(npos, dpos, -pow)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**pow, npos**pow)
eq = eqn(npos, dneg, -pow)
assert eq.is_Pow is False and eq.as_numer_denom() == (-(-npos)**pow*(-dneg)**pow, npos)
eq = eqn(nneg, dpos, -pow)
assert not eq.is_Pow or eq.as_numer_denom() == (dpos**pow, nneg**pow)
eq = eqn(nneg, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-nneg)**pow)
# unknown exponent
pow = 2*any
eq = eqn(npos, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (npos**pow, dpos**pow)
eq = eqn(npos, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-npos)**pow, (-dneg)**pow)
eq = eqn(nneg, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (nneg**pow, dpos**pow)
eq = eqn(nneg, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-nneg)**pow, (-dneg)**pow)
eq = eqn(npos, dpos, -pow)
assert eq.as_numer_denom() == (dpos**pow, npos**pow)
eq = eqn(npos, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-npos)**pow)
eq = eqn(nneg, dpos, -pow)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**pow, nneg**pow)
eq = eqn(nneg, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-nneg)**pow)
x = Symbol('x')
y = Symbol('y')
assert ((1/(1 + x/3))**(-S.One)).as_numer_denom() == (3 + x, 3)
notp = Symbol('notp', positive=False) # not positive does not imply real
b = ((1 + x/notp)**-2)
assert (b**(-y)).as_numer_denom() == (1, b**y)
assert (b**(-S.One)).as_numer_denom() == ((notp + x)**2, notp**2)
nonp = Symbol('nonp', nonpositive=True)
assert (((1 + x/nonp)**-2)**(-S.One)).as_numer_denom() == ((-nonp -
x)**2, nonp**2)
n = Symbol('n', negative=True)
assert (x**n).as_numer_denom() == (1, x**-n)
assert sqrt(1/n).as_numer_denom() == (S.ImaginaryUnit, sqrt(-n))
n = Symbol('0 or neg', nonpositive=True)
# if x and n are split up without negating each term and n is negative
# then the answer might be wrong; if n is 0 it won't matter since
# 1/oo and 1/zoo are both zero as is sqrt(0)/sqrt(-x) unless x is also
# zero (in which case the negative sign doesn't matter):
# 1/sqrt(1/-1) = -I but sqrt(-1)/sqrt(1) = I
assert (1/sqrt(x/n)).as_numer_denom() == (sqrt(-n), sqrt(-x))
c = Symbol('c', complex=True)
e = sqrt(1/c)
assert e.as_numer_denom() == (e, 1)
i = Symbol('i', integer=True)
assert ((1 + x/y)**i).as_numer_denom() == ((x + y)**i, y**i)
def test_Pow_Expr_args():
x = Symbol('x')
bases = [Basic(), Poly(x, x), FiniteSet(x)]
for base in bases:
# The cache can mess with the stacklevel test
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Pow(base, S.One)
def test_Pow_signs():
"""Cf. issues 4595 and 5250"""
x = Symbol('x')
y = Symbol('y')
n = Symbol('n', even=True)
assert (3 - y)**2 != (y - 3)**2
assert (3 - y)**n != (y - 3)**n
assert (-3 + y - x)**2 != (3 - y + x)**2
assert (y - 3)**3 != -(3 - y)**3
def test_power_with_noncommutative_mul_as_base():
x = Symbol('x', commutative=False)
y = Symbol('y', commutative=False)
assert not (x*y)**3 == x**3*y**3
assert (2*x*y)**3 == 8*(x*y)**3
@_both_exp_pow
def test_power_rewrite_exp():
assert (I**I).rewrite(exp) == exp(-pi/2)
expr = (2 + 3*I)**(4 + 5*I)
assert expr.rewrite(exp) == exp((4 + 5*I)*(log(sqrt(13)) + I*atan(Rational(3, 2))))
assert expr.rewrite(exp).expand() == \
169*exp(5*I*log(13)/2)*exp(4*I*atan(Rational(3, 2)))*exp(-5*atan(Rational(3, 2)))
assert ((6 + 7*I)**5).rewrite(exp) == 7225*sqrt(85)*exp(5*I*atan(Rational(7, 6)))
expr = 5**(6 + 7*I)
assert expr.rewrite(exp) == exp((6 + 7*I)*log(5))
assert expr.rewrite(exp).expand() == 15625*exp(7*I*log(5))
assert Pow(123, 789, evaluate=False).rewrite(exp) == 123**789
assert (1**I).rewrite(exp) == 1**I
assert (0**I).rewrite(exp) == 0**I
expr = (-2)**(2 + 5*I)
assert expr.rewrite(exp) == exp((2 + 5*I)*(log(2) + I*pi))
assert expr.rewrite(exp).expand() == 4*exp(-5*pi)*exp(5*I*log(2))
assert ((-2)**S(-5)).rewrite(exp) == (-2)**S(-5)
x, y = symbols('x y')
assert (x**y).rewrite(exp) == exp(y*log(x))
if global_parameters.exp_is_pow:
assert (7**x).rewrite(exp) == Pow(S.Exp1, x*log(7), evaluate=False)
else:
assert (7**x).rewrite(exp) == exp(x*log(7), evaluate=False)
assert ((2 + 3*I)**x).rewrite(exp) == exp(x*(log(sqrt(13)) + I*atan(Rational(3, 2))))
assert (y**(5 + 6*I)).rewrite(exp) == exp(log(y)*(5 + 6*I))
assert all((1/func(x)).rewrite(exp) == 1/(func(x).rewrite(exp)) for func in
(sin, cos, tan, sec, csc, sinh, cosh, tanh))
def test_zero():
x = Symbol('x')
y = Symbol('y')
assert 0**x != 0
assert 0**(2*x) == 0**x
assert 0**(1.0*x) == 0**x
assert 0**(2.0*x) == 0**x
assert (0**(2 - x)).as_base_exp() == (0, 2 - x)
assert 0**(x - 2) != S.Infinity**(2 - x)
assert 0**(2*x*y) == 0**(x*y)
assert 0**(-2*x*y) == S.ComplexInfinity**(x*y)
assert Float(0)**2 is not S.Zero
assert Float(0)**2 == 0.0
assert Float(0)**-2 is zoo
assert Float(0)**oo is S.Zero
#Test issue 19572
assert 0 ** -oo is zoo
assert power(0, -oo) is zoo
assert Float(0)**-oo is zoo
def test_pow_as_base_exp():
x = Symbol('x')
assert (S.Infinity**(2 - x)).as_base_exp() == (S.Infinity, 2 - x)
assert (S.Infinity**(x - 2)).as_base_exp() == (S.Infinity, x - 2)
p = S.Half**x
assert p.base, p.exp == p.as_base_exp() == (S(2), -x)
# issue 8344:
assert Pow(1, 2, evaluate=False).as_base_exp() == (S.One, S(2))
def test_nseries():
x = Symbol('x')
assert sqrt(I*x - 1)._eval_nseries(x, 4, None, 1) == I + x/2 + I*x**2/8 - x**3/16 + O(x**4)
assert sqrt(I*x - 1)._eval_nseries(x, 4, None, -1) == -I - x/2 - I*x**2/8 + x**3/16 + O(x**4)
assert cbrt(I*x - 1)._eval_nseries(x, 4, None, 1) == (-1)**(S(1)/3) - (-1)**(S(5)/6)*x/3 + \
(-1)**(S(1)/3)*x**2/9 + 5*(-1)**(S(5)/6)*x**3/81 + O(x**4)
assert cbrt(I*x - 1)._eval_nseries(x, 4, None, -1) == (-1)**(S(1)/3)*exp(-2*I*pi/3) - \
(-1)**(S(5)/6)*x*exp(-2*I*pi/3)/3 + (-1)**(S(1)/3)*x**2*exp(-2*I*pi/3)/9 + \
5*(-1)**(S(5)/6)*x**3*exp(-2*I*pi/3)/81 + O(x**4)
assert (1 / (exp(-1/x) + 1/x))._eval_nseries(x, 2, None) == x + O(x**2)
def test_issue_6100_12942_4473():
x = Symbol('x')
y = Symbol('y')
assert x**1.0 != x
assert x != x**1.0
assert True != x**1.0
assert x**1.0 is not True
assert x is not True
assert x*y != (x*y)**1.0
# Pow != Symbol
assert (x**1.0)**1.0 != x
assert (x**1.0)**2.0 != x**2
b = Expr()
assert Pow(b, 1.0, evaluate=False) != b
# if the following gets distributed as a Mul (x**1.0*y**1.0 then
# __eq__ methods could be added to Symbol and Pow to detect the
# power-of-1.0 case.
assert ((x*y)**1.0).func is Pow
def test_issue_6208():
from sympy.functions.elementary.miscellaneous import root
assert sqrt(33**(I*9/10)) == -33**(I*9/20)
assert root((6*I)**(2*I), 3).as_base_exp()[1] == Rational(1, 3) # != 2*I/3
assert root((6*I)**(I/3), 3).as_base_exp()[1] == I/9
assert sqrt(exp(3*I)) == exp(3*I/2)
assert sqrt(-sqrt(3)*(1 + 2*I)) == sqrt(sqrt(3))*sqrt(-1 - 2*I)
assert sqrt(exp(5*I)) == -exp(5*I/2)
assert root(exp(5*I), 3).exp == Rational(1, 3)
def test_issue_6990():
x = Symbol('x')
a = Symbol('a')
b = Symbol('b')
assert (sqrt(a + b*x + x**2)).series(x, 0, 3).removeO() == \
sqrt(a)*x**2*(1/(2*a) - b**2/(8*a**2)) + sqrt(a) + b*x/(2*sqrt(a))
def test_issue_6068():
x = Symbol('x')
assert sqrt(sin(x)).series(x, 0, 7) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 + O(x**7)
assert sqrt(sin(x)).series(x, 0, 9) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 - 67*x**Rational(17, 2)/29030400 + O(x**9)
assert sqrt(sin(x**3)).series(x, 0, 19) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 + O(x**19)
assert sqrt(sin(x**3)).series(x, 0, 20) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 - \
x**Rational(39, 2)/24192 + O(x**20)
def test_issue_6782():
x = Symbol('x')
assert sqrt(sin(x**3)).series(x, 0, 7) == x**Rational(3, 2) + O(x**7)
assert sqrt(sin(x**4)).series(x, 0, 3) == x**2 + O(x**3)
def test_issue_6653():
x = Symbol('x')
assert (1 / sqrt(1 + sin(x**2))).series(x, 0, 3) == 1 - x**2/2 + O(x**3)
def test_issue_6429():
x = Symbol('x')
c = Symbol('c')
f = (c**2 + x)**(0.5)
assert f.series(x, x0=0, n=1) == (c**2)**0.5 + O(x)
assert f.taylor_term(0, x) == (c**2)**0.5
assert f.taylor_term(1, x) == 0.5*x*(c**2)**(-0.5)
assert f.taylor_term(2, x) == -0.125*x**2*(c**2)**(-1.5)
def test_issue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**Rational(1, 3)
assert (((1 + I)**(I*(1 + 7*f)))**Rational(1, 3)).exp == Rational(1, 3)
r = symbols('r', real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**Rational(2, 3)
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt(1 - I)/sqrt(2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp in [S.Half,
Rational(3, 2) + I/2]
assert sqrt(r**Rational(4, 3)) != r**Rational(2, 3)
assert sqrt((p + I)**Rational(4, 3)) == (p + I)**Rational(2, 3)
assert sqrt((p - p**2*I)**2) == p - p**2*I
assert sqrt((p**2*I - p)**2) == p**2*I - p # XXX ok?
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_issue_8582():
assert 1**oo is nan
assert 1**(-oo) is nan
assert 1**zoo is nan
assert 1**(oo + I) is nan
assert 1**(1 + I*oo) is nan
assert 1**(oo + I*oo) is nan
def test_issue_8650():
n = Symbol('n', integer=True, nonnegative=True)
assert (n**n).is_positive is True
x = 5*n + 5
assert (x**(5*(n + 1))).is_positive is True
def test_issue_13914():
b = Symbol('b')
assert (-1)**zoo is nan
assert 2**zoo is nan
assert (S.Half)**(1 + zoo) is nan
assert I**(zoo + I) is nan
assert b**(I + zoo) is nan
def test_better_sqrt():
n = Symbol('n', integer=True, nonnegative=True)
assert sqrt(3 + 4*I) == 2 + I
assert sqrt(3 - 4*I) == 2 - I
assert sqrt(-3 - 4*I) == 1 - 2*I
assert sqrt(-3 + 4*I) == 1 + 2*I
assert sqrt(32 + 24*I) == 6 + 2*I
assert sqrt(32 - 24*I) == 6 - 2*I
assert sqrt(-32 - 24*I) == 2 - 6*I
assert sqrt(-32 + 24*I) == 2 + 6*I
# triple (3, 4, 5):
# parity of 3 matches parity of 5 and
# den, 4, is a square
assert sqrt((3 + 4*I)/4) == 1 + I/2
# triple (8, 15, 17)
# parity of 8 doesn't match parity of 17 but
# den/2, 8/2, is a square
assert sqrt((8 + 15*I)/8) == (5 + 3*I)/4
# handle the denominator
assert sqrt((3 - 4*I)/25) == (2 - I)/5
assert sqrt((3 - 4*I)/26) == (2 - I)/sqrt(26)
# mul
# issue #12739
assert sqrt((3 + 4*I)/(3 - 4*I)) == (3 + 4*I)/5
assert sqrt(2/(3 + 4*I)) == sqrt(2)/5*(2 - I)
assert sqrt(n/(3 + 4*I)).subs(n, 2) == sqrt(2)/5*(2 - I)
assert sqrt(-2/(3 + 4*I)) == sqrt(2)/5*(1 + 2*I)
assert sqrt(-n/(3 + 4*I)).subs(n, 2) == sqrt(2)/5*(1 + 2*I)
# power
assert sqrt(1/(3 + I*4)) == (2 - I)/5
assert sqrt(1/(3 - I)) == sqrt(10)*sqrt(3 + I)/10
# symbolic
i = symbols('i', imaginary=True)
assert sqrt(3/i) == Mul(sqrt(3), 1/sqrt(i), evaluate=False)
# multiples of 1/2; don't make this too automatic
assert sqrt(3 + 4*I)**3 == (2 + I)**3
assert Pow(3 + 4*I, Rational(3, 2)) == 2 + 11*I
assert Pow(6 + 8*I, Rational(3, 2)) == 2*sqrt(2)*(2 + 11*I)
n, d = (3 + 4*I), (3 - 4*I)**3
a = n/d
assert a.args == (1/d, n)
eq = sqrt(a)
assert eq.args == (a, S.Half)
assert expand_multinomial(eq) == sqrt((-117 + 44*I)*(3 + 4*I))/125
assert eq.expand() == (7 - 24*I)/125
# issue 12775
# pos im part
assert sqrt(2*I) == (1 + I)
assert sqrt(2*9*I) == Mul(3, 1 + I, evaluate=False)
assert Pow(2*I, 3*S.Half) == (1 + I)**3
# neg im part
assert sqrt(-I/2) == Mul(S.Half, 1 - I, evaluate=False)
# fractional im part
assert Pow(Rational(-9, 2)*I, Rational(3, 2)) == 27*(1 - I)**3/8
def test_issue_2993():
x = Symbol('x')
assert str((2.3*x - 4)**0.3) == '1.5157165665104*(0.575*x - 1)**0.3'
assert str((2.3*x + 4)**0.3) == '1.5157165665104*(0.575*x + 1)**0.3'
assert str((-2.3*x + 4)**0.3) == '1.5157165665104*(1 - 0.575*x)**0.3'
assert str((-2.3*x - 4)**0.3) == '1.5157165665104*(-0.575*x - 1)**0.3'
assert str((2.3*x - 2)**0.3) == '1.28386201800527*(x - 0.869565217391304)**0.3'
assert str((-2.3*x - 2)**0.3) == '1.28386201800527*(-x - 0.869565217391304)**0.3'
assert str((-2.3*x + 2)**0.3) == '1.28386201800527*(0.869565217391304 - x)**0.3'
assert str((2.3*x + 2)**0.3) == '1.28386201800527*(x + 0.869565217391304)**0.3'
assert str((2.3*x - 4)**Rational(1, 3)) == '2**(2/3)*(0.575*x - 1)**(1/3)'
eq = (2.3*x + 4)
assert eq**2 == 16*(0.575*x + 1)**2
assert (1/eq).args == (eq, -1) # don't change trivial power
# issue 17735
q=.5*exp(x) - .5*exp(-x) + 0.1
assert int((q**2).subs(x, 1)) == 1
# issue 17756
y = Symbol('y')
assert len(sqrt(x/(x + y)**2 + Float('0.008', 30)).subs(y, pi.n(25)).atoms(Float)) == 2
# issue 17756
a, b, c, d, e, f, g = symbols('a:g')
expr = sqrt(1 + a*(c**4 + g*d - 2*g*e - f*(-g + d))**2/
(c**3*b**2*(d - 3*e + 2*f)**2))/2
r = [
(a, N('0.0170992456333788667034850458615', 30)),
(b, N('0.0966594956075474769169134801223', 30)),
(c, N('0.390911862903463913632151616184', 30)),
(d, N('0.152812084558656566271750185933', 30)),
(e, N('0.137562344465103337106561623432', 30)),
(f, N('0.174259178881496659302933610355', 30)),
(g, N('0.220745448491223779615401870086', 30))]
tru = expr.n(30, subs=dict(r))
seq = expr.subs(r)
# although `tru` is the right way to evaluate
# expr with numerical values, `seq` will have
# significant loss of precision if extraction of
# the largest coefficient of a power's base's terms
# is done improperly
assert seq == tru
def test_issue_17450():
assert (erf(cosh(1)**7)**I).is_real is None
assert (erf(cosh(1)**7)**I).is_imaginary is False
assert (Pow(exp(1+sqrt(2)), ((1-sqrt(2))*I*pi), evaluate=False)).is_real is None
assert ((-10)**(10*I*pi/3)).is_real is False
assert ((-5)**(4*I*pi)).is_real is False
def test_issue_18190():
assert sqrt(1 / tan(1 + I)) == 1 / sqrt(tan(1 + I))
def test_issue_14815():
x = Symbol('x', real=True)
assert sqrt(x).is_extended_negative is False
x = Symbol('x', real=False)
assert sqrt(x).is_extended_negative is None
x = Symbol('x', complex=True)
assert sqrt(x).is_extended_negative is False
x = Symbol('x', extended_real=True)
assert sqrt(x).is_extended_negative is False
assert sqrt(zoo, evaluate=False).is_extended_negative is None
assert sqrt(nan, evaluate=False).is_extended_negative is None
def test_issue_18509():
x = Symbol('x', prime=True)
assert x**oo is oo
assert (1/x)**oo is S.Zero
assert (-1/x)**oo is S.Zero
assert (-x)**oo is zoo
assert (-oo)**(-1 + I) is S.Zero
assert (-oo)**(1 + I) is zoo
assert (oo)**(-1 + I) is S.Zero
assert (oo)**(1 + I) is zoo
def test_issue_18762():
e, p = symbols('e p')
g0 = sqrt(1 + e**2 - 2*e*cos(p))
assert len(g0.series(e, 1, 3).args) == 4
def test_issue_21860():
x = Symbol('x')
e = 3*2**Rational(66666666667,200000000000)*3**Rational(16666666667,50000000000)*x**Rational(66666666667, 200000000000)
ans = Mul(Rational(3, 2),
Pow(Integer(2), Rational(33333333333, 100000000000)),
Pow(Integer(3), Rational(26666666667, 40000000000)))
assert e.xreplace({x: Rational(3,8)}) == ans
def test_issue_21647():
x = Symbol('x')
e = log((Integer(567)/500)**(811*(Integer(567)/500)**x/100))
ans = log(Mul(Rational(64701150190720499096094005280169087619821081527,
76293945312500000000000000000000000000000000000),
Pow(Integer(2), Rational(396204892125479941, 781250000000000000)),
Pow(Integer(3), Rational(385045107874520059, 390625000000000000)),
Pow(Integer(5), Rational(407364676376439823, 1562500000000000000)),
Pow(Integer(7), Rational(385045107874520059, 1562500000000000000))))
assert e.xreplace({x: 6}) == ans
def test_issue_21762():
x = Symbol('x')
e = (x**2 + 6)**(Integer(33333333333333333)/50000000000000000)
ans = Mul(Rational(5, 4),
Pow(Integer(2), Rational(16666666666666667, 25000000000000000)),
Pow(Integer(5), Rational(8333333333333333, 25000000000000000)))
assert e.xreplace({x: S.Half}) == ans
def test_rational_powers_larger_than_one():
assert Rational(2, 3)**Rational(3, 2) == 2*sqrt(6)/9
assert Rational(1, 6)**Rational(9, 4) == 6**Rational(3, 4)/216
assert Rational(3, 7)**Rational(7, 3) == 9*3**Rational(1, 3)*7**Rational(2, 3)/343
def test_power_dispatcher():
class NewBase(Expr):
pass
class NewPow(NewBase, Pow):
pass
a, b = Symbol('a'), NewBase()
@power.register(Expr, NewBase)
@power.register(NewBase, Expr)
@power.register(NewBase, NewBase)
def _(a, b):
return NewPow(a, b)
# Pow called as fallback
assert power(2, 3) == 8*S.One
assert power(a, 2) == Pow(a, 2)
assert power(a, a) == Pow(a, a)
# NewPow called by dispatch
assert power(a, b) == NewPow(a, b)
assert power(b, a) == NewPow(b, a)
assert power(b, b) == NewPow(b, b)
def test_powers_of_I():
assert [sqrt(I)**i for i in range(13)] == [
1, sqrt(I), I, sqrt(I)**3, -1, -sqrt(I), -I, -sqrt(I)**3,
1, sqrt(I), I, sqrt(I)**3, -1]
assert sqrt(I)**(S(9)/2) == -I**(S(1)/4)
|
df837fb9cae41f754849aafb7ace4a1f4fbe807c93fbdae92d50112b3510c17b | from collections import defaultdict
from sympy.core.basic import Basic
from sympy.core.containers import (Dict, Tuple)
from sympy.core.numbers import Integer
from sympy.core.kind import NumberKind
from sympy.matrices.common import MatrixKind
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.core.sympify import sympify
from sympy.matrices.dense import Matrix
from sympy.sets.sets import FiniteSet
from sympy.core.containers import tuple_wrapper, TupleKind
from sympy.core.expr import unchanged
from sympy.core.function import Function, Lambda
from sympy.core.relational import Eq
from sympy.testing.pytest import raises
from sympy.utilities.iterables import is_sequence, iterable
from sympy.abc import x, y
def test_Tuple():
t = (1, 2, 3, 4)
st = Tuple(*t)
assert set(sympify(t)) == set(st)
assert len(t) == len(st)
assert set(sympify(t[:2])) == set(st[:2])
assert isinstance(st[:], Tuple)
assert st == Tuple(1, 2, 3, 4)
assert st.func(*st.args) == st
p, q, r, s = symbols('p q r s')
t2 = (p, q, r, s)
st2 = Tuple(*t2)
assert st2.atoms() == set(t2)
assert st == st2.subs({p: 1, q: 2, r: 3, s: 4})
# issue 5505
assert all(isinstance(arg, Basic) for arg in st.args)
assert Tuple(p, 1).subs(p, 0) == Tuple(0, 1)
assert Tuple(p, Tuple(p, 1)).subs(p, 0) == Tuple(0, Tuple(0, 1))
assert Tuple(t2) == Tuple(Tuple(*t2))
assert Tuple.fromiter(t2) == Tuple(*t2)
assert Tuple.fromiter(x for x in range(4)) == Tuple(0, 1, 2, 3)
assert st2.fromiter(st2.args) == st2
def test_Tuple_contains():
t1, t2 = Tuple(1), Tuple(2)
assert t1 in Tuple(1, 2, 3, t1, Tuple(t2))
assert t2 not in Tuple(1, 2, 3, t1, Tuple(t2))
def test_Tuple_concatenation():
assert Tuple(1, 2) + Tuple(3, 4) == Tuple(1, 2, 3, 4)
assert (1, 2) + Tuple(3, 4) == Tuple(1, 2, 3, 4)
assert Tuple(1, 2) + (3, 4) == Tuple(1, 2, 3, 4)
raises(TypeError, lambda: Tuple(1, 2) + 3)
raises(TypeError, lambda: 1 + Tuple(2, 3))
#the Tuple case in __radd__ is only reached when a subclass is involved
class Tuple2(Tuple):
def __radd__(self, other):
return Tuple.__radd__(self, other + other)
assert Tuple(1, 2) + Tuple2(3, 4) == Tuple(1, 2, 1, 2, 3, 4)
assert Tuple2(1, 2) + Tuple(3, 4) == Tuple(1, 2, 3, 4)
def test_Tuple_equality():
assert not isinstance(Tuple(1, 2), tuple)
assert (Tuple(1, 2) == (1, 2)) is True
assert (Tuple(1, 2) != (1, 2)) is False
assert (Tuple(1, 2) == (1, 3)) is False
assert (Tuple(1, 2) != (1, 3)) is True
assert (Tuple(1, 2) == Tuple(1, 2)) is True
assert (Tuple(1, 2) != Tuple(1, 2)) is False
assert (Tuple(1, 2) == Tuple(1, 3)) is False
assert (Tuple(1, 2) != Tuple(1, 3)) is True
def test_Tuple_Eq():
assert Eq(Tuple(), Tuple()) is S.true
assert Eq(Tuple(1), 1) is S.false
assert Eq(Tuple(1, 2), Tuple(1)) is S.false
assert Eq(Tuple(1), Tuple(1)) is S.true
assert Eq(Tuple(1, 2), Tuple(1, 3)) is S.false
assert Eq(Tuple(1, 2), Tuple(1, 2)) is S.true
assert unchanged(Eq, Tuple(1, x), Tuple(1, 2))
assert Eq(Tuple(1, x), Tuple(1, 2)).subs(x, 2) is S.true
assert unchanged(Eq, Tuple(1, 2), x)
f = Function('f')
assert unchanged(Eq, Tuple(1), f(x))
assert Eq(Tuple(1), f(x)).subs(x, 1).subs(f, Lambda(y, (y,))) is S.true
def test_Tuple_comparision():
assert (Tuple(1, 3) >= Tuple(-10, 30)) is S.true
assert (Tuple(1, 3) <= Tuple(-10, 30)) is S.false
assert (Tuple(1, 3) >= Tuple(1, 3)) is S.true
assert (Tuple(1, 3) <= Tuple(1, 3)) is S.true
def test_Tuple_tuple_count():
assert Tuple(0, 1, 2, 3).tuple_count(4) == 0
assert Tuple(0, 4, 1, 2, 3).tuple_count(4) == 1
assert Tuple(0, 4, 1, 4, 2, 3).tuple_count(4) == 2
assert Tuple(0, 4, 1, 4, 2, 4, 3).tuple_count(4) == 3
def test_Tuple_index():
assert Tuple(4, 0, 1, 2, 3).index(4) == 0
assert Tuple(0, 4, 1, 2, 3).index(4) == 1
assert Tuple(0, 1, 4, 2, 3).index(4) == 2
assert Tuple(0, 1, 2, 4, 3).index(4) == 3
assert Tuple(0, 1, 2, 3, 4).index(4) == 4
raises(ValueError, lambda: Tuple(0, 1, 2, 3).index(4))
raises(ValueError, lambda: Tuple(4, 0, 1, 2, 3).index(4, 1))
raises(ValueError, lambda: Tuple(0, 1, 2, 3, 4).index(4, 1, 4))
def test_Tuple_mul():
assert Tuple(1, 2, 3)*2 == Tuple(1, 2, 3, 1, 2, 3)
assert 2*Tuple(1, 2, 3) == Tuple(1, 2, 3, 1, 2, 3)
assert Tuple(1, 2, 3)*Integer(2) == Tuple(1, 2, 3, 1, 2, 3)
assert Integer(2)*Tuple(1, 2, 3) == Tuple(1, 2, 3, 1, 2, 3)
raises(TypeError, lambda: Tuple(1, 2, 3)*S.Half)
raises(TypeError, lambda: S.Half*Tuple(1, 2, 3))
def test_tuple_wrapper():
@tuple_wrapper
def wrap_tuples_and_return(*t):
return t
p = symbols('p')
assert wrap_tuples_and_return(p, 1) == (p, 1)
assert wrap_tuples_and_return((p, 1)) == (Tuple(p, 1),)
assert wrap_tuples_and_return(1, (p, 2), 3) == (1, Tuple(p, 2), 3)
def test_iterable_is_sequence():
ordered = [list(), tuple(), Tuple(), Matrix([[]])]
unordered = [set()]
not_sympy_iterable = [{}, '', '']
assert all(is_sequence(i) for i in ordered)
assert all(not is_sequence(i) for i in unordered)
assert all(iterable(i) for i in ordered + unordered)
assert all(not iterable(i) for i in not_sympy_iterable)
assert all(iterable(i, exclude=None) for i in not_sympy_iterable)
def test_TupleKind():
kind = TupleKind(NumberKind, MatrixKind(NumberKind))
assert Tuple(1, Matrix([1, 2])).kind is kind
assert Tuple(1, 2).kind is TupleKind(NumberKind, NumberKind)
assert Tuple(1, 2).kind.element_kind == (NumberKind, NumberKind)
def test_Dict():
x, y, z = symbols('x y z')
d = Dict({x: 1, y: 2, z: 3})
assert d[x] == 1
assert d[y] == 2
raises(KeyError, lambda: d[2])
raises(KeyError, lambda: d['2'])
assert len(d) == 3
assert set(d.keys()) == {x, y, z}
assert set(d.values()) == {S.One, S(2), S(3)}
assert d.get(5, 'default') == 'default'
assert d.get('5', 'default') == 'default'
assert x in d and z in d and 5 not in d and '5' not in d
assert d.has(x) and d.has(1) # SymPy Basic .has method
# Test input types
# input - a Python dict
# input - items as args - SymPy style
assert (Dict({x: 1, y: 2, z: 3}) ==
Dict((x, 1), (y, 2), (z, 3)))
raises(TypeError, lambda: Dict(((x, 1), (y, 2), (z, 3))))
with raises(NotImplementedError):
d[5] = 6 # assert immutability
assert set(
d.items()) == {Tuple(x, S.One), Tuple(y, S(2)), Tuple(z, S(3))}
assert set(d) == {x, y, z}
assert str(d) == '{x: 1, y: 2, z: 3}'
assert d.__repr__() == '{x: 1, y: 2, z: 3}'
# Test creating a Dict from a Dict.
d = Dict({x: 1, y: 2, z: 3})
assert d == Dict(d)
# Test for supporting defaultdict
d = defaultdict(int)
assert d[x] == 0
assert d[y] == 0
assert d[z] == 0
assert Dict(d)
d = Dict(d)
assert len(d) == 3
assert set(d.keys()) == {x, y, z}
assert set(d.values()) == {S.Zero, S.Zero, S.Zero}
def test_issue_5788():
args = [(1, 2), (2, 1)]
for o in [Dict, Tuple, FiniteSet]:
# __eq__ and arg handling
if o != Tuple:
assert o(*args) == o(*reversed(args))
pair = [o(*args), o(*reversed(args))]
assert sorted(pair) == sorted(reversed(pair))
assert set(o(*args)) # doesn't fail
|
6e8e814d7b197b755b83b7b36d5d8257f5fb0b0d752272f91c8c2f6a2c9976d4 | import math
from sympy.concrete.products import (Product, product)
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.evalf import N
from sympy.core.function import (Function, nfloat)
from sympy.core.mul import Mul
from sympy.core import (GoldenRatio)
from sympy.core.numbers import (AlgebraicNumber, E, Float, I, Rational,
oo, zoo, nan, pi)
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.elementary.complexes import (Abs, re, im)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (acosh, cosh)
from sympy.functions.elementary.integers import (ceiling, floor)
from sympy.functions.elementary.miscellaneous import (Max, sqrt)
from sympy.functions.elementary.trigonometric import (acos, atan, cos, sin, tan)
from sympy.integrals.integrals import (Integral, integrate)
from sympy.polys.polytools import factor
from sympy.polys.rootoftools import CRootOf
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.printing import srepr
from sympy.printing.str import sstr
from sympy.simplify.simplify import simplify
from sympy.core.numbers import comp
from sympy.core.evalf import (complex_accuracy, PrecisionExhausted,
scaled_zero, get_integer_part, as_mpmath, evalf, _evalf_with_bounded_error)
from mpmath import inf, ninf, make_mpc
from mpmath.libmp.libmpf import from_float, fzero
from sympy.core.expr import unchanged
from sympy.testing.pytest import raises, XFAIL
from sympy.abc import n, x, y
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_evalf_helpers():
from mpmath.libmp import finf
assert complex_accuracy((from_float(2.0), None, 35, None)) == 35
assert complex_accuracy((from_float(2.0), from_float(10.0), 35, 100)) == 37
assert complex_accuracy(
(from_float(2.0), from_float(1000.0), 35, 100)) == 43
assert complex_accuracy((from_float(2.0), from_float(10.0), 100, 35)) == 35
assert complex_accuracy(
(from_float(2.0), from_float(1000.0), 100, 35)) == 35
assert complex_accuracy(finf) == math.inf
assert complex_accuracy(zoo) == math.inf
raises(ValueError, lambda: get_integer_part(zoo, 1, {}))
def test_evalf_basic():
assert NS('pi', 15) == '3.14159265358979'
assert NS('2/3', 10) == '0.6666666667'
assert NS('355/113-pi', 6) == '2.66764e-7'
assert NS('16*atan(1/5)-4*atan(1/239)', 15) == '3.14159265358979'
def test_cancellation():
assert NS(Add(pi, Rational(1, 10**1000), -pi, evaluate=False), 15,
maxn=1200) == '1.00000000000000e-1000'
def test_evalf_powers():
assert NS('pi**(10**20)', 10) == '1.339148777e+49714987269413385435'
assert NS(pi**(10**100), 10) == ('4.946362032e+4971498726941338543512682882'
'9089887365167832438044244613405349992494711208'
'95526746555473864642912223')
assert NS('2**(1/10**50)', 15) == '1.00000000000000'
assert NS('2**(1/10**50)-1', 15) == '6.93147180559945e-51'
# Evaluation of Rump's ill-conditioned polynomial
def test_evalf_rump():
a = 1335*y**6/4 + x**2*(11*x**2*y**2 - y**6 - 121*y**4 - 2) + 11*y**8/2 + x/(2*y)
assert NS(a, 15, subs={x: 77617, y: 33096}) == '-0.827396059946821'
def test_evalf_complex():
assert NS('2*sqrt(pi)*I', 10) == '3.544907702*I'
assert NS('3+3*I', 15) == '3.00000000000000 + 3.00000000000000*I'
assert NS('E+pi*I', 15) == '2.71828182845905 + 3.14159265358979*I'
assert NS('pi * (3+4*I)', 15) == '9.42477796076938 + 12.5663706143592*I'
assert NS('I*(2+I)', 15) == '-1.00000000000000 + 2.00000000000000*I'
@XFAIL
def test_evalf_complex_bug():
assert NS('(pi+E*I)*(E+pi*I)', 15) in ('0.e-15 + 17.25866050002*I',
'0.e-17 + 17.25866050002*I', '-0.e-17 + 17.25866050002*I')
def test_evalf_complex_powers():
assert NS('(E+pi*I)**100000000000000000') == \
'-3.58896782867793e+61850354284995199 + 4.58581754997159e+61850354284995199*I'
# XXX: rewrite if a+a*I simplification introduced in SymPy
#assert NS('(pi + pi*I)**2') in ('0.e-15 + 19.7392088021787*I', '0.e-16 + 19.7392088021787*I')
assert NS('(pi + pi*I)**2', chop=True) == '19.7392088021787*I'
assert NS(
'(pi + 1/10**8 + pi*I)**2') == '6.2831853e-8 + 19.7392088650106*I'
assert NS('(pi + 1/10**12 + pi*I)**2') == '6.283e-12 + 19.7392088021850*I'
assert NS('(pi + pi*I)**4', chop=True) == '-389.636364136010'
assert NS(
'(pi + 1/10**8 + pi*I)**4') == '-389.636366616512 + 2.4805021e-6*I'
assert NS('(pi + 1/10**12 + pi*I)**4') == '-389.636364136258 + 2.481e-10*I'
assert NS(
'(10000*pi + 10000*pi*I)**4', chop=True) == '-3.89636364136010e+18'
@XFAIL
def test_evalf_complex_powers_bug():
assert NS('(pi + pi*I)**4') == '-389.63636413601 + 0.e-14*I'
def test_evalf_exponentiation():
assert NS(sqrt(-pi)) == '1.77245385090552*I'
assert NS(Pow(pi*I, Rational(
1, 2), evaluate=False)) == '1.25331413731550 + 1.25331413731550*I'
assert NS(pi**I) == '0.413292116101594 + 0.910598499212615*I'
assert NS(pi**(E + I/3)) == '20.8438653991931 + 8.36343473930031*I'
assert NS((pi + I/3)**(E + I/3)) == '17.2442906093590 + 13.6839376767037*I'
assert NS(exp(pi)) == '23.1406926327793'
assert NS(exp(pi + E*I)) == '-21.0981542849657 + 9.50576358282422*I'
assert NS(pi**pi) == '36.4621596072079'
assert NS((-pi)**pi) == '-32.9138577418939 - 15.6897116534332*I'
assert NS((-pi)**(-pi)) == '-0.0247567717232697 + 0.0118013091280262*I'
# An example from Smith, "Multiple Precision Complex Arithmetic and Functions"
def test_evalf_complex_cancellation():
A = Rational('63287/100000')
B = Rational('52498/100000')
C = Rational('69301/100000')
D = Rational('83542/100000')
F = Rational('2231321613/2500000000')
# XXX: the number of returned mantissa digits in the real part could
# change with the implementation. What matters is that the returned digits are
# correct; those that are showing now are correct.
# >>> ((A+B*I)*(C+D*I)).expand()
# 64471/10000000000 + 2231321613*I/2500000000
# >>> 2231321613*4
# 8925286452L
assert NS((A + B*I)*(C + D*I), 6) == '6.44710e-6 + 0.892529*I'
assert NS((A + B*I)*(C + D*I), 10) == '6.447100000e-6 + 0.8925286452*I'
assert NS((A + B*I)*(
C + D*I) - F*I, 5) in ('6.4471e-6 + 0.e-14*I', '6.4471e-6 - 0.e-14*I')
def test_evalf_logs():
assert NS("log(3+pi*I)", 15) == '1.46877619736226 + 0.808448792630022*I'
assert NS("log(pi*I)", 15) == '1.14472988584940 + 1.57079632679490*I'
assert NS('log(-1 + 0.00001)', 2) == '-1.0e-5 + 3.1*I'
assert NS('log(100, 10, evaluate=False)', 15) == '2.00000000000000'
assert NS('-2*I*log(-(-1)**(S(1)/9))', 15) == '-5.58505360638185'
def test_evalf_trig():
assert NS('sin(1)', 15) == '0.841470984807897'
assert NS('cos(1)', 15) == '0.540302305868140'
assert NS('sin(10**-6)', 15) == '9.99999999999833e-7'
assert NS('cos(10**-6)', 15) == '0.999999999999500'
assert NS('sin(E*10**100)', 15) == '0.409160531722613'
# Some input near roots
assert NS(sin(exp(pi*sqrt(163))*pi), 15) == '-2.35596641936785e-12'
assert NS(sin(pi*10**100 + Rational(7, 10**5), evaluate=False), 15, maxn=120) == \
'6.99999999428333e-5'
assert NS(sin(Rational(7, 10**5), evaluate=False), 15) == \
'6.99999999428333e-5'
# Check detection of various false identities
def test_evalf_near_integers():
# Binet's formula
f = lambda n: ((1 + sqrt(5))**n)/(2**n * sqrt(5))
assert NS(f(5000) - fibonacci(5000), 10, maxn=1500) == '5.156009964e-1046'
# Some near-integer identities from
# http://mathworld.wolfram.com/AlmostInteger.html
assert NS('sin(2017*2**(1/5))', 15) == '-1.00000000000000'
assert NS('sin(2017*2**(1/5))', 20) == '-0.99999999999999997857'
assert NS('1+sin(2017*2**(1/5))', 15) == '2.14322287389390e-17'
assert NS('45 - 613*E/37 + 35/991', 15) == '6.03764498766326e-11'
def test_evalf_ramanujan():
assert NS(exp(pi*sqrt(163)) - 640320**3 - 744, 10) == '-7.499274028e-13'
# A related identity
A = 262537412640768744*exp(-pi*sqrt(163))
B = 196884*exp(-2*pi*sqrt(163))
C = 103378831900730205293632*exp(-3*pi*sqrt(163))
assert NS(1 - A - B + C, 10) == '1.613679005e-59'
# Input that for various reasons have failed at some point
def test_evalf_bugs():
assert NS(sin(1) + exp(-10**10), 10) == NS(sin(1), 10)
assert NS(exp(10**10) + sin(1), 10) == NS(exp(10**10), 10)
assert NS('expand_log(log(1+1/10**50))', 20) == '1.0000000000000000000e-50'
assert NS('log(10**100,10)', 10) == '100.0000000'
assert NS('log(2)', 10) == '0.6931471806'
assert NS(
'(sin(x)-x)/x**3', 15, subs={x: '1/10**50'}) == '-0.166666666666667'
assert NS(sin(1) + Rational(
1, 10**100)*I, 15) == '0.841470984807897 + 1.00000000000000e-100*I'
assert x.evalf() == x
assert NS((1 + I)**2*I, 6) == '-2.00000'
d = {n: (
-1)**Rational(6, 7), y: (-1)**Rational(4, 7), x: (-1)**Rational(2, 7)}
assert NS((x*(1 + y*(1 + n))).subs(d).evalf(), 6) == '0.346011 + 0.433884*I'
assert NS(((-I - sqrt(2)*I)**2).evalf()) == '-5.82842712474619'
assert NS((1 + I)**2*I, 15) == '-2.00000000000000'
# issue 4758 (1/2):
assert NS(pi.evalf(69) - pi) == '-4.43863937855894e-71'
# issue 4758 (2/2): With the bug present, this still only fails if the
# terms are in the order given here. This is not generally the case,
# because the order depends on the hashes of the terms.
assert NS(20 - 5008329267844*n**25 - 477638700*n**37 - 19*n,
subs={n: .01}) == '19.8100000000000'
assert NS(((x - 1)*(1 - x)**1000).n()
) == '(1.00000000000000 - x)**1000*(x - 1.00000000000000)'
assert NS((-x).n()) == '-x'
assert NS((-2*x).n()) == '-2.00000000000000*x'
assert NS((-2*x*y).n()) == '-2.00000000000000*x*y'
assert cos(x).n(subs={x: 1+I}) == cos(x).subs(x, 1+I).n()
# issue 6660. Also NaN != mpmath.nan
# In this order:
# 0*nan, 0/nan, 0*inf, 0/inf
# 0+nan, 0-nan, 0+inf, 0-inf
# >>> n = Some Number
# n*nan, n/nan, n*inf, n/inf
# n+nan, n-nan, n+inf, n-inf
assert (0*E**(oo)).n() is S.NaN
assert (0/E**(oo)).n() is S.Zero
assert (0+E**(oo)).n() is S.Infinity
assert (0-E**(oo)).n() is S.NegativeInfinity
assert (5*E**(oo)).n() is S.Infinity
assert (5/E**(oo)).n() is S.Zero
assert (5+E**(oo)).n() is S.Infinity
assert (5-E**(oo)).n() is S.NegativeInfinity
#issue 7416
assert as_mpmath(0.0, 10, {'chop': True}) == 0
#issue 5412
assert ((oo*I).n() == S.Infinity*I)
assert ((oo+oo*I).n() == S.Infinity + S.Infinity*I)
#issue 11518
assert NS(2*x**2.5, 5) == '2.0000*x**2.5000'
#issue 13076
assert NS(Mul(Max(0, y), x, evaluate=False).evalf()) == 'x*Max(0, y)'
#issue 18516
assert NS(log(S(3273390607896141870013189696827599152216642046043064789483291368096133796404674554883270092325904157150886684127560071009217256545885393053328527589376)/36360291795869936842385267079543319118023385026001623040346035832580600191583895484198508262979388783308179702534403855752855931517013066142992430916562025780021771247847643450125342836565813209972590371590152578728008385990139795377610001).evalf(15, chop=True)) == '-oo'
def test_evalf_integer_parts():
a = floor(log(8)/log(2) - exp(-1000), evaluate=False)
b = floor(log(8)/log(2), evaluate=False)
assert a.evalf() == 3
assert b.evalf() == 3
# equals, as a fallback, can still fail but it might succeed as here
assert ceiling(10*(sin(1)**2 + cos(1)**2)) == 10
assert int(floor(factorial(50)/E, evaluate=False).evalf(70)) == \
int(11188719610782480504630258070757734324011354208865721592720336800)
assert int(ceiling(factorial(50)/E, evaluate=False).evalf(70)) == \
int(11188719610782480504630258070757734324011354208865721592720336801)
assert int(floor(GoldenRatio**999 / sqrt(5) + S.Half)
.evalf(1000)) == fibonacci(999)
assert int(floor(GoldenRatio**1000 / sqrt(5) + S.Half)
.evalf(1000)) == fibonacci(1000)
assert ceiling(x).evalf(subs={x: 3}) == 3
assert ceiling(x).evalf(subs={x: 3*I}) == 3.0*I
assert ceiling(x).evalf(subs={x: 2 + 3*I}) == 2.0 + 3.0*I
assert ceiling(x).evalf(subs={x: 3.}) == 3
assert ceiling(x).evalf(subs={x: 3.*I}) == 3.0*I
assert ceiling(x).evalf(subs={x: 2. + 3*I}) == 2.0 + 3.0*I
assert float((floor(1.5, evaluate=False)+1/9).evalf()) == 1 + 1/9
assert float((floor(0.5, evaluate=False)+20).evalf()) == 20
# issue 19991
n = 1169809367327212570704813632106852886389036911
r = 744723773141314414542111064094745678855643068
assert floor(n / (pi / 2)) == r
assert floor(80782 * sqrt(2)) == 114242
# issue 20076
assert 260515 - floor(260515/pi + 1/2) * pi == atan(tan(260515))
def test_evalf_trig_zero_detection():
a = sin(160*pi, evaluate=False)
t = a.evalf(maxn=100)
assert abs(t) < 1e-100
assert t._prec < 2
assert a.evalf(chop=True) == 0
raises(PrecisionExhausted, lambda: a.evalf(strict=True))
def test_evalf_sum():
assert Sum(n,(n,1,2)).evalf() == 3.
assert Sum(n,(n,1,2)).doit().evalf() == 3.
# the next test should return instantly
assert Sum(1/n,(n,1,2)).evalf() == 1.5
# issue 8219
assert Sum(E/factorial(n), (n, 0, oo)).evalf() == (E*E).evalf()
# issue 8254
assert Sum(2**n*n/factorial(n), (n, 0, oo)).evalf() == (2*E*E).evalf()
# issue 8411
s = Sum(1/x**2, (x, 100, oo))
assert s.n() == s.doit().n()
def test_evalf_divergent_series():
raises(ValueError, lambda: Sum(1/n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum(n/(n**2 + 1), (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((-1)**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((-1)**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum(n**2, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum(2**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((-2)**n, (n, 1, oo)).evalf())
raises(ValueError, lambda: Sum((2*n + 3)/(3*n**2 + 4), (n, 0, oo)).evalf())
raises(ValueError, lambda: Sum((0.5*n**3)/(n**4 + 1), (n, 0, oo)).evalf())
def test_evalf_product():
assert Product(n, (n, 1, 10)).evalf() == 3628800.
assert comp(Product(1 - S.Half**2/n**2, (n, 1, oo)).n(5), 0.63662)
assert Product(n, (n, -1, 3)).evalf() == 0
def test_evalf_py_methods():
assert abs(float(pi + 1) - 4.1415926535897932) < 1e-10
assert abs(complex(pi + 1) - 4.1415926535897932) < 1e-10
assert abs(
complex(pi + E*I) - (3.1415926535897931 + 2.7182818284590451j)) < 1e-10
raises(TypeError, lambda: float(pi + x))
def test_evalf_power_subs_bugs():
assert (x**2).evalf(subs={x: 0}) == 0
assert sqrt(x).evalf(subs={x: 0}) == 0
assert (x**Rational(2, 3)).evalf(subs={x: 0}) == 0
assert (x**x).evalf(subs={x: 0}) == 1
assert (3**x).evalf(subs={x: 0}) == 1
assert exp(x).evalf(subs={x: 0}) == 1
assert ((2 + I)**x).evalf(subs={x: 0}) == 1
assert (0**x).evalf(subs={x: 0}) == 1
def test_evalf_arguments():
raises(TypeError, lambda: pi.evalf(method="garbage"))
def test_implemented_function_evalf():
from sympy.utilities.lambdify import implemented_function
f = Function('f')
f = implemented_function(f, lambda x: x + 1)
assert str(f(x)) == "f(x)"
assert str(f(2)) == "f(2)"
assert f(2).evalf() == 3
assert f(x).evalf() == f(x)
f = implemented_function(Function('sin'), lambda x: x + 1)
assert f(2).evalf() != sin(2)
del f._imp_ # XXX: due to caching _imp_ would influence all other tests
def test_evaluate_false():
for no in [0, False]:
assert Add(3, 2, evaluate=no).is_Add
assert Mul(3, 2, evaluate=no).is_Mul
assert Pow(3, 2, evaluate=no).is_Pow
assert Pow(y, 2, evaluate=True) - Pow(y, 2, evaluate=True) == 0
def test_evalf_relational():
assert Eq(x/5, y/10).evalf() == Eq(0.2*x, 0.1*y)
# if this first assertion fails it should be replaced with
# one that doesn't
assert unchanged(Eq, (3 - I)**2/2 + I, 0)
assert Eq((3 - I)**2/2 + I, 0).n() is S.false
assert nfloat(Eq((3 - I)**2 + I, 0)) == S.false
def test_issue_5486():
assert not cos(sqrt(0.5 + I)).n().is_Function
def test_issue_5486_bug():
from sympy.core.expr import Expr
from sympy.core.numbers import I
assert abs(Expr._from_mpmath(I._to_mpmath(15), 15) - I) < 1.0e-15
def test_bugs():
from sympy.functions.elementary.complexes import (polar_lift, re)
assert abs(re((1 + I)**2)) < 1e-15
# anything that evalf's to 0 will do in place of polar_lift
assert abs(polar_lift(0)).n() == 0
def test_subs():
assert NS('besseli(-x, y) - besseli(x, y)', subs={x: 3.5, y: 20.0}) == \
'-4.92535585957223e-10'
assert NS('Piecewise((x, x>0)) + Piecewise((1-x, x>0))', subs={x: 0.1}) == \
'1.00000000000000'
raises(TypeError, lambda: x.evalf(subs=(x, 1)))
def test_issue_4956_5204():
# issue 4956
v = S('''(-27*12**(1/3)*sqrt(31)*I +
27*2**(2/3)*3**(1/3)*sqrt(31)*I)/(-2511*2**(2/3)*3**(1/3) +
(29*18**(1/3) + 9*2**(1/3)*3**(2/3)*sqrt(31)*I +
87*2**(1/3)*3**(1/6)*I)**2)''')
assert NS(v, 1) == '0.e-118 - 0.e-118*I'
# issue 5204
v = S('''-(357587765856 + 18873261792*249**(1/2) + 56619785376*I*83**(1/2) +
108755765856*I*3**(1/2) + 41281887168*6**(1/3)*(1422 +
54*249**(1/2))**(1/3) - 1239810624*6**(1/3)*249**(1/2)*(1422 +
54*249**(1/2))**(1/3) - 3110400000*I*6**(1/3)*83**(1/2)*(1422 +
54*249**(1/2))**(1/3) + 13478400000*I*3**(1/2)*6**(1/3)*(1422 +
54*249**(1/2))**(1/3) + 1274950152*6**(2/3)*(1422 +
54*249**(1/2))**(2/3) + 32347944*6**(2/3)*249**(1/2)*(1422 +
54*249**(1/2))**(2/3) - 1758790152*I*3**(1/2)*6**(2/3)*(1422 +
54*249**(1/2))**(2/3) - 304403832*I*6**(2/3)*83**(1/2)*(1422 +
4*249**(1/2))**(2/3))/(175732658352 + (1106028 + 25596*249**(1/2) +
76788*I*83**(1/2))**2)''')
assert NS(v, 5) == '0.077284 + 1.1104*I'
assert NS(v, 1) == '0.08 + 1.*I'
def test_old_docstring():
a = (E + pi*I)*(E - pi*I)
assert NS(a) == '17.2586605000200'
assert a.n() == 17.25866050002001
def test_issue_4806():
assert integrate(atan(x)**2, (x, -1, 1)).evalf().round(1) == 0.5
assert atan(0, evaluate=False).n() == 0
def test_evalf_mul():
# SymPy should not try to expand this; it should be handled term-wise
# in evalf through mpmath
assert NS(product(1 + sqrt(n)*I, (n, 1, 500)), 1) == '5.e+567 + 2.e+568*I'
def test_scaled_zero():
a, b = (([0], 1, 100, 1), -1)
assert scaled_zero(100) == (a, b)
assert scaled_zero(a) == (0, 1, 100, 1)
a, b = (([1], 1, 100, 1), -1)
assert scaled_zero(100, -1) == (a, b)
assert scaled_zero(a) == (1, 1, 100, 1)
raises(ValueError, lambda: scaled_zero(scaled_zero(100)))
raises(ValueError, lambda: scaled_zero(100, 2))
raises(ValueError, lambda: scaled_zero(100, 0))
raises(ValueError, lambda: scaled_zero((1, 5, 1, 3)))
def test_chop_value():
for i in range(-27, 28):
assert (Pow(10, i)*2).n(chop=10**i) and not (Pow(10, i)).n(chop=10**i)
def test_infinities():
assert oo.evalf(chop=True) == inf
assert (-oo).evalf(chop=True) == ninf
def test_to_mpmath():
assert sqrt(3)._to_mpmath(20)._mpf_ == (0, int(908093), -19, 20)
assert S(3.2)._to_mpmath(20)._mpf_ == (0, int(838861), -18, 20)
def test_issue_6632_evalf():
add = (-100000*sqrt(2500000001) + 5000000001)
assert add.n() == 9.999999998e-11
assert (add*add).n() == 9.999999996e-21
def test_issue_4945():
from sympy.abc import H
assert (H/0).evalf(subs={H:1}) == zoo
def test_evalf_integral():
# test that workprec has to increase in order to get a result other than 0
eps = Rational(1, 1000000)
assert Integral(sin(x), (x, -pi, pi + eps)).n(2)._prec == 10
def test_issue_8821_highprec_from_str():
s = str(pi.evalf(128))
p = N(s)
assert Abs(sin(p)) < 1e-15
p = N(s, 64)
assert Abs(sin(p)) < 1e-64
def test_issue_8853():
p = Symbol('x', even=True, positive=True)
assert floor(-p - S.Half).is_even == False
assert floor(-p + S.Half).is_even == True
assert ceiling(p - S.Half).is_even == True
assert ceiling(p + S.Half).is_even == False
assert get_integer_part(S.Half, -1, {}, True) == (0, 0)
assert get_integer_part(S.Half, 1, {}, True) == (1, 0)
assert get_integer_part(Rational(-1, 2), -1, {}, True) == (-1, 0)
assert get_integer_part(Rational(-1, 2), 1, {}, True) == (0, 0)
def test_issue_17681():
class identity_func(Function):
def _eval_evalf(self, *args, **kwargs):
return self.args[0].evalf(*args, **kwargs)
assert floor(identity_func(S(0))) == 0
assert get_integer_part(S(0), 1, {}, True) == (0, 0)
def test_issue_9326():
from sympy.core.symbol import Dummy
d1 = Dummy('d')
d2 = Dummy('d')
e = d1 + d2
assert e.evalf(subs = {d1: 1, d2: 2}) == 3
def test_issue_10323():
assert ceiling(sqrt(2**30 + 1)) == 2**15 + 1
def test_AssocOp_Function():
# the first arg of Min is not comparable in the imaginary part
raises(ValueError, lambda: S('''
Min(-sqrt(3)*cos(pi/18)/6 + re(1/((-1/2 - sqrt(3)*I/2)*(1/6 +
sqrt(3)*I/18)**(1/3)))/3 + sin(pi/18)/2 + 2 + I*(-cos(pi/18)/2 -
sqrt(3)*sin(pi/18)/6 + im(1/((-1/2 - sqrt(3)*I/2)*(1/6 +
sqrt(3)*I/18)**(1/3)))/3), re(1/((-1/2 + sqrt(3)*I/2)*(1/6 +
sqrt(3)*I/18)**(1/3)))/3 - sqrt(3)*cos(pi/18)/6 - sin(pi/18)/2 + 2 +
I*(im(1/((-1/2 + sqrt(3)*I/2)*(1/6 + sqrt(3)*I/18)**(1/3)))/3 -
sqrt(3)*sin(pi/18)/6 + cos(pi/18)/2))'''))
# if that is changed so a non-comparable number remains as
# an arg, then the Min/Max instantiation needs to be changed
# to watch out for non-comparable args when making simplifications
# and the following test should be added instead (with e being
# the sympified expression above):
# raises(ValueError, lambda: e._eval_evalf(2))
def test_issue_10395():
eq = x*Max(0, y)
assert nfloat(eq) == eq
eq = x*Max(y, -1.1)
assert nfloat(eq) == eq
assert Max(y, 4).n() == Max(4.0, y)
def test_issue_13098():
assert floor(log(S('9.'+'9'*20), 10)) == 0
assert ceiling(log(S('9.'+'9'*20), 10)) == 1
assert floor(log(20 - S('9.'+'9'*20), 10)) == 1
assert ceiling(log(20 - S('9.'+'9'*20), 10)) == 2
def test_issue_14601():
e = 5*x*y/2 - y*(35*(x**3)/2 - 15*x/2)
subst = {x:0.0, y:0.0}
e2 = e.evalf(subs=subst)
assert float(e2) == 0.0
assert float((x + x*(x**2 + x)).evalf(subs={x: 0.0})) == 0.0
def test_issue_11151():
z = S.Zero
e = Sum(z, (x, 1, 2))
assert e != z # it shouldn't evaluate
# when it does evaluate, this is what it should give
assert evalf(e, 15, {}) == \
evalf(z, 15, {}) == (None, None, 15, None)
# so this shouldn't fail
assert (e/2).n() == 0
# this was where the issue appeared
expr0 = Sum(x**2 + x, (x, 1, 2))
expr1 = Sum(0, (x, 1, 2))
expr2 = expr1/expr0
assert simplify(factor(expr2) - expr2) == 0
def test_issue_13425():
assert N('2**.5', 30) == N('sqrt(2)', 30)
assert N('x - x', 30) == 0
assert abs((N('pi*.1', 22)*10 - pi).n()) < 1e-22
def test_issue_17421():
assert N(acos(-I + acosh(cosh(cosh(1) + I)))) == 1.0*I
def test_issue_20291():
from sympy.sets import EmptySet, Reals
from sympy.sets.sets import (Complement, FiniteSet, Intersection)
a = Symbol('a')
b = Symbol('b')
A = FiniteSet(a, b)
assert A.evalf(subs={a: 1, b: 2}) == FiniteSet(1.0, 2.0)
B = FiniteSet(a-b, 1)
assert B.evalf(subs={a: 1, b: 2}) == FiniteSet(-1.0, 1.0)
sol = Complement(Intersection(FiniteSet(-b/2 - sqrt(b**2-4*pi)/2), Reals), FiniteSet(0))
assert sol.evalf(subs={b: 1}) == EmptySet
def test_evalf_with_zoo():
assert (1/x).evalf(subs={x: 0}) == zoo # issue 8242
assert (-1/x).evalf(subs={x: 0}) == zoo # PR 16150
assert (0 ** x).evalf(subs={x: -1}) == zoo # PR 16150
assert (0 ** x).evalf(subs={x: -1 + I}) == nan
assert Mul(2, Pow(0, -1, evaluate=False), evaluate=False).evalf() == zoo # issue 21147
assert Mul(x, 1/x, evaluate=False).evalf(subs={x: 0}) == Mul(x, 1/x, evaluate=False).subs(x, 0) == nan
assert Mul(1/x, 1/x, evaluate=False).evalf(subs={x: 0}) == zoo
assert Mul(1/x, Abs(1/x), evaluate=False).evalf(subs={x: 0}) == zoo
assert Abs(zoo, evaluate=False).evalf() == oo
assert re(zoo, evaluate=False).evalf() == nan
assert im(zoo, evaluate=False).evalf() == nan
assert Add(zoo, zoo, evaluate=False).evalf() == nan
assert Add(oo, zoo, evaluate=False).evalf() == nan
assert Pow(zoo, -1, evaluate=False).evalf() == 0
assert Pow(zoo, Rational(-1, 3), evaluate=False).evalf() == 0
assert Pow(zoo, Rational(1, 3), evaluate=False).evalf() == zoo
assert Pow(zoo, S.Half, evaluate=False).evalf() == zoo
assert Pow(zoo, 2, evaluate=False).evalf() == zoo
assert Pow(0, zoo, evaluate=False).evalf() == nan
assert log(zoo, evaluate=False).evalf() == zoo
assert zoo.evalf(chop=True) == zoo
assert x.evalf(subs={x: zoo}) == zoo
def test_evalf_with_bounded_error():
cases = [
# zero
(Rational(0), None, 1),
# zero im part
(pi, None, 10),
# zero real part
(pi*I, None, 10),
# re and im nonzero
(2-3*I, None, 5),
# similar tests again, but using eps instead of m
(Rational(0), Rational(1, 2), None),
(pi, Rational(1, 1000), None),
(pi * I, Rational(1, 1000), None),
(2 - 3 * I, Rational(1, 1000), None),
# very large eps
(2 - 3 * I, Rational(1000), None),
# case where x already small, hence some cancelation in p = m + n - 1
(Rational(1234, 10**8), Rational(1, 10**12), None),
]
for x0, eps, m in cases:
a, b, _, _ = evalf(x0, 53, {})
c, d, _, _ = _evalf_with_bounded_error(x0, eps, m)
if eps is None:
eps = 2**(-m)
z = make_mpc((a or fzero, b or fzero))
w = make_mpc((c or fzero, d or fzero))
assert abs(w - z) < eps
# eps must be positive
raises(ValueError, lambda: _evalf_with_bounded_error(pi, Rational(0)))
raises(ValueError, lambda: _evalf_with_bounded_error(pi, -pi))
raises(ValueError, lambda: _evalf_with_bounded_error(pi, I))
def test_issue_22849():
a = -8 + 3 * sqrt(3)
x = AlgebraicNumber(a)
assert evalf(a, 1, {}) == evalf(x, 1, {})
def test_evalf_real_alg_num():
# This test demonstrates why the entry for `AlgebraicNumber` in
# `sympy.core.evalf._create_evalf_table()` has to use `x.to_root()`,
# instead of `x.as_expr()`. If the latter is used, then `z` will be
# a complex number with `0.e-20` for imaginary part, even though `a5`
# is a real number.
zeta = Symbol('zeta')
a5 = AlgebraicNumber(CRootOf(cyclotomic_poly(5), -1), [-1, -1, 0, 0], alias=zeta)
z = a5.evalf()
assert isinstance(z, Float)
assert not hasattr(z, '_mpc_')
assert hasattr(z, '_mpf_')
def test_issue_20733():
expr = 1/((x - 9)*(x - 8)*(x - 7)*(x - 4)**2*(x - 3)**3*(x - 2))
assert str(expr.evalf(1, subs={x:1})) == '-4.e-5'
assert str(expr.evalf(2, subs={x:1})) == '-4.1e-5'
assert str(expr.evalf(11, subs={x:1})) == '-4.1335978836e-5'
assert str(expr.evalf(20, subs={x:1})) == '-0.000041335978835978835979'
expr = Mul(*((x - i) for i in range(2, 1000)))
assert srepr(expr.evalf(2, subs={x: 1})) == "Float('4.0271e+2561', precision=10)"
assert srepr(expr.evalf(10, subs={x: 1})) == "Float('4.02790050126e+2561', precision=37)"
assert srepr(expr.evalf(53, subs={x: 1})) == "Float('4.0279005012722099453824067459760158730668154575647110393e+2561', precision=179)"
|
4e57c347f1cf49fd58307ea8a6df665100696a9c50c4258f7dc883d9ce11a727 | from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core.numbers import (Float, I, Integer, Rational, comp, nan,
oo, pi, zoo)
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.complexes import (im, re, sign)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import (Max, sqrt)
from sympy.functions.elementary.trigonometric import (atan, cos, sin)
from sympy.polys.polytools import Poly
from sympy.sets.sets import FiniteSet
from sympy.core.parameters import distribute
from sympy.core.expr import unchanged
from sympy.utilities.iterables import permutations
from sympy.testing.pytest import XFAIL, raises, warns
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.random import verify_numerically
from sympy.functions.elementary.trigonometric import asin
from itertools import product
a, c, x, y, z = symbols('a,c,x,y,z')
b = Symbol("b", positive=True)
def same_and_same_prec(a, b):
# stricter matching for Floats
return a == b and a._prec == b._prec
def test_bug1():
assert re(x) != x
x.series(x, 0, 1)
assert re(x) != x
def test_Symbol():
e = a*b
assert e == a*b
assert a*b*b == a*b**2
assert a*b*b + c == c + a*b**2
assert a*b*b - c == -c + a*b**2
x = Symbol('x', complex=True, real=False)
assert x.is_imaginary is None # could be I or 1 + I
x = Symbol('x', complex=True, imaginary=False)
assert x.is_real is None # could be 1 or 1 + I
x = Symbol('x', real=True)
assert x.is_complex
x = Symbol('x', imaginary=True)
assert x.is_complex
x = Symbol('x', real=False, imaginary=False)
assert x.is_complex is None # might be a non-number
def test_arit0():
p = Rational(5)
e = a*b
assert e == a*b
e = a*b + b*a
assert e == 2*a*b
e = a*b + b*a + a*b + p*b*a
assert e == 8*a*b
e = a*b + b*a + a*b + p*b*a + a
assert e == a + 8*a*b
e = a + a
assert e == 2*a
e = a + b + a
assert e == b + 2*a
e = a + b*b + a + b*b
assert e == 2*a + 2*b**2
e = a + Rational(2) + b*b + a + b*b + p
assert e == 7 + 2*a + 2*b**2
e = (a + b*b + a + b*b)*p
assert e == 5*(2*a + 2*b**2)
e = (a*b*c + c*b*a + b*a*c)*p
assert e == 15*a*b*c
e = (a*b*c + c*b*a + b*a*c)*p - Rational(15)*a*b*c
assert e == Rational(0)
e = Rational(50)*(a - a)
assert e == Rational(0)
e = b*a - b - a*b + b
assert e == Rational(0)
e = a*b + c**p
assert e == a*b + c**5
e = a/b
assert e == a*b**(-1)
e = a*2*2
assert e == 4*a
e = 2 + a*2/2
assert e == 2 + a
e = 2 - a - 2
assert e == -a
e = 2*a*2
assert e == 4*a
e = 2/a/2
assert e == a**(-1)
e = 2**a**2
assert e == 2**(a**2)
e = -(1 + a)
assert e == -1 - a
e = S.Half*(1 + a)
assert e == S.Half + a/2
def test_div():
e = a/b
assert e == a*b**(-1)
e = a/b + c/2
assert e == a*b**(-1) + Rational(1)/2*c
e = (1 - b)/(b - 1)
assert e == (1 + -b)*((-1) + b)**(-1)
def test_pow():
n1 = Rational(1)
n2 = Rational(2)
n5 = Rational(5)
e = a*a
assert e == a**2
e = a*a*a
assert e == a**3
e = a*a*a*a**Rational(6)
assert e == a**9
e = a*a*a*a**Rational(6) - a**Rational(9)
assert e == Rational(0)
e = a**(b - b)
assert e == Rational(1)
e = (a + Rational(1) - a)**b
assert e == Rational(1)
e = (a + b + c)**n2
assert e == (a + b + c)**2
assert e.expand() == 2*b*c + 2*a*c + 2*a*b + a**2 + c**2 + b**2
e = (a + b)**n2
assert e == (a + b)**2
assert e.expand() == 2*a*b + a**2 + b**2
e = (a + b)**(n1/n2)
assert e == sqrt(a + b)
assert e.expand() == sqrt(a + b)
n = n5**(n1/n2)
assert n == sqrt(5)
e = n*a*b - n*b*a
assert e == Rational(0)
e = n*a*b + n*b*a
assert e == 2*a*b*sqrt(5)
assert e.diff(a) == 2*b*sqrt(5)
assert e.diff(a) == 2*b*sqrt(5)
e = a/b**2
assert e == a*b**(-2)
assert sqrt(2*(1 + sqrt(2))) == (2*(1 + 2**S.Half))**S.Half
x = Symbol('x')
y = Symbol('y')
assert ((x*y)**3).expand() == y**3 * x**3
assert ((x*y)**-3).expand() == y**-3 * x**-3
assert (x**5*(3*x)**(3)).expand() == 27 * x**8
assert (x**5*(-3*x)**(3)).expand() == -27 * x**8
assert (x**5*(3*x)**(-3)).expand() == x**2 * Rational(1, 27)
assert (x**5*(-3*x)**(-3)).expand() == x**2 * Rational(-1, 27)
# expand_power_exp
assert (x**(y**(x + exp(x + y)) + z)).expand(deep=False) == \
x**z*x**(y**(x + exp(x + y)))
assert (x**(y**(x + exp(x + y)) + z)).expand() == \
x**z*x**(y**x*y**(exp(x)*exp(y)))
n = Symbol('n', even=False)
k = Symbol('k', even=True)
o = Symbol('o', odd=True)
assert unchanged(Pow, -1, x)
assert unchanged(Pow, -1, n)
assert (-2)**k == 2**k
assert (-1)**k == 1
assert (-1)**o == -1
def test_pow2():
# x**(2*y) is always (x**y)**2 but is only (x**2)**y if
# x.is_positive or y.is_integer
# let x = 1 to see why the following are not true.
assert (-x)**Rational(2, 3) != x**Rational(2, 3)
assert (-x)**Rational(5, 7) != -x**Rational(5, 7)
assert ((-x)**2)**Rational(1, 3) != ((-x)**Rational(1, 3))**2
assert sqrt(x**2) != x
def test_pow3():
assert sqrt(2)**3 == 2 * sqrt(2)
assert sqrt(2)**3 == sqrt(8)
def test_mod_pow():
for s, t, u, v in [(4, 13, 497, 445), (4, -3, 497, 365),
(3.2, 2.1, 1.9, 0.1031015682350942), (S(3)/2, 5, S(5)/6, S(3)/32)]:
assert pow(S(s), t, u) == v
assert pow(S(s), S(t), u) == v
assert pow(S(s), t, S(u)) == v
assert pow(S(s), S(t), S(u)) == v
assert pow(S(2), S(10000000000), S(3)) == 1
assert pow(x, y, z) == x**y%z
raises(TypeError, lambda: pow(S(4), "13", 497))
raises(TypeError, lambda: pow(S(4), 13, "497"))
def test_pow_E():
assert 2**(y/log(2)) == S.Exp1**y
assert 2**(y/log(2)/3) == S.Exp1**(y/3)
assert 3**(1/log(-3)) != S.Exp1
assert (3 + 2*I)**(1/(log(-3 - 2*I) + I*pi)) == S.Exp1
assert (4 + 2*I)**(1/(log(-4 - 2*I) + I*pi)) == S.Exp1
assert (3 + 2*I)**(1/(log(-3 - 2*I, 3)/2 + I*pi/log(3)/2)) == 9
assert (3 + 2*I)**(1/(log(3 + 2*I, 3)/2)) == 9
# every time tests are run they will affirm with a different random
# value that this identity holds
while 1:
b = x._random()
r, i = b.as_real_imag()
if i:
break
assert verify_numerically(b**(1/(log(-b) + sign(i)*I*pi).n()), S.Exp1)
def test_pow_issue_3516():
assert 4**Rational(1, 4) == sqrt(2)
def test_pow_im():
for m in (-2, -1, 2):
for d in (3, 4, 5):
b = m*I
for i in range(1, 4*d + 1):
e = Rational(i, d)
assert (b**e - b.n()**e.n()).n(2, chop=1e-10) == 0
e = Rational(7, 3)
assert (2*x*I)**e == 4*2**Rational(1, 3)*(I*x)**e # same as Wolfram Alpha
im = symbols('im', imaginary=True)
assert (2*im*I)**e == 4*2**Rational(1, 3)*(I*im)**e
args = [I, I, I, I, 2]
e = Rational(1, 3)
ans = 2**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args = [I, I, I, 2]
e = Rational(1, 3)
ans = 2**e*(-I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-3)
ans = (6*I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-1)
ans = (-6*I)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args = [I, I, 2]
e = Rational(1, 3)
ans = (-2)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-3)
ans = (6)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
args.append(-1)
ans = (-6)**e
assert Mul(*args, evaluate=False)**e == ans
assert Mul(*args)**e == ans
assert Mul(Pow(-1, Rational(3, 2), evaluate=False), I, I) == I
assert Mul(I*Pow(I, S.Half, evaluate=False)) == sqrt(I)*I
def test_real_mul():
assert Float(0) * pi * x == 0
assert set((Float(1) * pi * x).args) == {Float(1), pi, x}
def test_ncmul():
A = Symbol("A", commutative=False)
B = Symbol("B", commutative=False)
C = Symbol("C", commutative=False)
assert A*B != B*A
assert A*B*C != C*B*A
assert A*b*B*3*C == 3*b*A*B*C
assert A*b*B*3*C != 3*b*B*A*C
assert A*b*B*3*C == 3*A*B*C*b
assert A + B == B + A
assert (A + B)*C != C*(A + B)
assert C*(A + B)*C != C*C*(A + B)
assert A*A == A**2
assert (A + B)*(A + B) == (A + B)**2
assert A**-1 * A == 1
assert A/A == 1
assert A/(A**2) == 1/A
assert A/(1 + A) == A/(1 + A)
assert set((A + B + 2*(A + B)).args) == \
{A, B, 2*(A + B)}
def test_mul_add_identity():
m = Mul(1, 2)
assert isinstance(m, Rational) and m.p == 2 and m.q == 1
m = Mul(1, 2, evaluate=False)
assert isinstance(m, Mul) and m.args == (1, 2)
m = Mul(0, 1)
assert m is S.Zero
m = Mul(0, 1, evaluate=False)
assert isinstance(m, Mul) and m.args == (0, 1)
m = Add(0, 1)
assert m is S.One
m = Add(0, 1, evaluate=False)
assert isinstance(m, Add) and m.args == (0, 1)
def test_ncpow():
x = Symbol('x', commutative=False)
y = Symbol('y', commutative=False)
z = Symbol('z', commutative=False)
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')
assert (x**2)*(y**2) != (y**2)*(x**2)
assert (x**-2)*y != y*(x**2)
assert 2**x*2**y != 2**(x + y)
assert 2**x*2**y*2**z != 2**(x + y + z)
assert 2**x*2**(2*x) == 2**(3*x)
assert 2**x*2**(2*x)*2**x == 2**(4*x)
assert exp(x)*exp(y) != exp(y)*exp(x)
assert exp(x)*exp(y)*exp(z) != exp(y)*exp(x)*exp(z)
assert exp(x)*exp(y)*exp(z) != exp(x + y + z)
assert x**a*x**b != x**(a + b)
assert x**a*x**b*x**c != x**(a + b + c)
assert x**3*x**4 == x**7
assert x**3*x**4*x**2 == x**9
assert x**a*x**(4*a) == x**(5*a)
assert x**a*x**(4*a)*x**a == x**(6*a)
def test_powerbug():
x = Symbol("x")
assert x**1 != (-x)**1
assert x**2 == (-x)**2
assert x**3 != (-x)**3
assert x**4 == (-x)**4
assert x**5 != (-x)**5
assert x**6 == (-x)**6
assert x**128 == (-x)**128
assert x**129 != (-x)**129
assert (2*x)**2 == (-2*x)**2
def test_Mul_doesnt_expand_exp():
x = Symbol('x')
y = Symbol('y')
assert unchanged(Mul, exp(x), exp(y))
assert unchanged(Mul, 2**x, 2**y)
assert x**2*x**3 == x**5
assert 2**x*3**x == 6**x
assert x**(y)*x**(2*y) == x**(3*y)
assert sqrt(2)*sqrt(2) == 2
assert 2**x*2**(2*x) == 2**(3*x)
assert sqrt(2)*2**Rational(1, 4)*5**Rational(3, 4) == 10**Rational(3, 4)
assert (x**(-log(5)/log(3))*x)/(x*x**( - log(5)/log(3))) == sympify(1)
def test_Mul_is_integer():
k = Symbol('k', integer=True)
n = Symbol('n', integer=True)
nr = Symbol('nr', rational=False)
nz = Symbol('nz', integer=True, zero=False)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
i2 = Symbol('2', prime=True, even=True)
assert (k/3).is_integer is None
assert (nz/3).is_integer is None
assert (nr/3).is_integer is False
assert (x*k*n).is_integer is None
assert (e/2).is_integer is True
assert (e**2/2).is_integer is True
assert (2/k).is_integer is None
assert (2/k**2).is_integer is None
assert ((-1)**k*n).is_integer is True
assert (3*k*e/2).is_integer is True
assert (2*k*e/3).is_integer is None
assert (e/o).is_integer is None
assert (o/e).is_integer is False
assert (o/i2).is_integer is False
assert Mul(k, 1/k, evaluate=False).is_integer is None
assert Mul(2., S.Half, evaluate=False).is_integer is None
assert (2*sqrt(k)).is_integer is None
assert (2*k**n).is_integer is None
s = 2**2**2**Pow(2, 1000, evaluate=False)
m = Mul(s, s, evaluate=False)
assert m.is_integer
# broken in 1.6 and before, see #20161
xq = Symbol('xq', rational=True)
yq = Symbol('yq', rational=True)
assert (xq*yq).is_integer is None
e_20161 = Mul(-1,Mul(1,Pow(2,-1,evaluate=False),evaluate=False),evaluate=False)
assert e_20161.is_integer is not True # expand(e_20161) -> -1/2, but no need to see that in the assumption without evaluation
def test_Add_Mul_is_integer():
x = Symbol('x')
k = Symbol('k', integer=True)
n = Symbol('n', integer=True)
nk = Symbol('nk', integer=False)
nr = Symbol('nr', rational=False)
nz = Symbol('nz', integer=True, zero=False)
assert (-nk).is_integer is None
assert (-nr).is_integer is False
assert (2*k).is_integer is True
assert (-k).is_integer is True
assert (k + nk).is_integer is False
assert (k + n).is_integer is True
assert (k + x).is_integer is None
assert (k + n*x).is_integer is None
assert (k + n/3).is_integer is None
assert (k + nz/3).is_integer is None
assert (k + nr/3).is_integer is False
assert ((1 + sqrt(3))*(-sqrt(3) + 1)).is_integer is not False
assert (1 + (1 + sqrt(3))*(-sqrt(3) + 1)).is_integer is not False
def test_Add_Mul_is_finite():
x = Symbol('x', extended_real=True, finite=False)
assert sin(x).is_finite is True
assert (x*sin(x)).is_finite is None
assert (x*atan(x)).is_finite is False
assert (1024*sin(x)).is_finite is True
assert (sin(x)*exp(x)).is_finite is None
assert (sin(x)*cos(x)).is_finite is True
assert (x*sin(x)*exp(x)).is_finite is None
assert (sin(x) - 67).is_finite is True
assert (sin(x) + exp(x)).is_finite is not True
assert (1 + x).is_finite is False
assert (1 + x**2 + (1 + x)*(1 - x)).is_finite is None
assert (sqrt(2)*(1 + x)).is_finite is False
assert (sqrt(2)*(1 + x)*(1 - x)).is_finite is False
def test_Mul_is_even_odd():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
n = Symbol('n', odd=True)
m = Symbol('m', even=True)
assert (2*x).is_even is True
assert (2*x).is_odd is False
assert (3*x).is_even is None
assert (3*x).is_odd is None
assert (k/3).is_integer is None
assert (k/3).is_even is None
assert (k/3).is_odd is None
assert (2*n).is_even is True
assert (2*n).is_odd is False
assert (2*m).is_even is True
assert (2*m).is_odd is False
assert (-n).is_even is False
assert (-n).is_odd is True
assert (k*n).is_even is False
assert (k*n).is_odd is True
assert (k*m).is_even is True
assert (k*m).is_odd is False
assert (k*n*m).is_even is True
assert (k*n*m).is_odd is False
assert (k*m*x).is_even is True
assert (k*m*x).is_odd is False
# issue 6791:
assert (x/2).is_integer is None
assert (k/2).is_integer is False
assert (m/2).is_integer is True
assert (x*y).is_even is None
assert (x*x).is_even is None
assert (x*(x + k)).is_even is True
assert (x*(x + m)).is_even is None
assert (x*y).is_odd is None
assert (x*x).is_odd is None
assert (x*(x + k)).is_odd is False
assert (x*(x + m)).is_odd is None
# issue 8648
assert (m**2/2).is_even
assert (m**2/3).is_even is False
assert (2/m**2).is_odd is False
assert (2/m).is_odd is None
@XFAIL
def test_evenness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
assert (x*y*(y + k)).is_even is True
assert (y*x*(x + k)).is_even is True
def test_evenness_in_ternary_integer_product_with_even():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
m = Symbol('m', even=True)
assert (x*y*(y + m)).is_even is None
@XFAIL
def test_oddness_in_ternary_integer_product_with_odd():
# Tests that oddness inference is independent of term ordering.
# Term ordering at the point of testing depends on SymPy's symbol order, so
# we try to force a different order by modifying symbol names.
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
k = Symbol('k', odd=True)
assert (x*y*(y + k)).is_odd is False
assert (y*x*(x + k)).is_odd is False
def test_oddness_in_ternary_integer_product_with_even():
x = Symbol('x', integer=True)
y = Symbol('y', integer=True)
m = Symbol('m', even=True)
assert (x*y*(y + m)).is_odd is None
def test_Mul_is_rational():
x = Symbol('x')
n = Symbol('n', integer=True)
m = Symbol('m', integer=True, nonzero=True)
assert (n/m).is_rational is True
assert (x/pi).is_rational is None
assert (x/n).is_rational is None
assert (m/pi).is_rational is False
r = Symbol('r', rational=True)
assert (pi*r).is_rational is None
# issue 8008
z = Symbol('z', zero=True)
i = Symbol('i', imaginary=True)
assert (z*i).is_rational is True
bi = Symbol('i', imaginary=True, finite=True)
assert (z*bi).is_zero is True
def test_Add_is_rational():
x = Symbol('x')
n = Symbol('n', rational=True)
m = Symbol('m', rational=True)
assert (n + m).is_rational is True
assert (x + pi).is_rational is None
assert (x + n).is_rational is None
assert (n + pi).is_rational is False
def test_Add_is_even_odd():
x = Symbol('x', integer=True)
k = Symbol('k', odd=True)
n = Symbol('n', odd=True)
m = Symbol('m', even=True)
assert (k + 7).is_even is True
assert (k + 7).is_odd is False
assert (-k + 7).is_even is True
assert (-k + 7).is_odd is False
assert (k - 12).is_even is False
assert (k - 12).is_odd is True
assert (-k - 12).is_even is False
assert (-k - 12).is_odd is True
assert (k + n).is_even is True
assert (k + n).is_odd is False
assert (k + m).is_even is False
assert (k + m).is_odd is True
assert (k + n + m).is_even is True
assert (k + n + m).is_odd is False
assert (k + n + x + m).is_even is None
assert (k + n + x + m).is_odd is None
def test_Mul_is_negative_positive():
x = Symbol('x', real=True)
y = Symbol('y', extended_real=False, complex=True)
z = Symbol('z', zero=True)
e = 2*z
assert e.is_Mul and e.is_positive is False and e.is_negative is False
neg = Symbol('neg', negative=True)
pos = Symbol('pos', positive=True)
nneg = Symbol('nneg', nonnegative=True)
npos = Symbol('npos', nonpositive=True)
assert neg.is_negative is True
assert (-neg).is_negative is False
assert (2*neg).is_negative is True
assert (2*pos)._eval_is_extended_negative() is False
assert (2*pos).is_negative is False
assert pos.is_negative is False
assert (-pos).is_negative is True
assert (2*pos).is_negative is False
assert (pos*neg).is_negative is True
assert (2*pos*neg).is_negative is True
assert (-pos*neg).is_negative is False
assert (pos*neg*y).is_negative is False # y.is_real=F; !real -> !neg
assert nneg.is_negative is False
assert (-nneg).is_negative is None
assert (2*nneg).is_negative is False
assert npos.is_negative is None
assert (-npos).is_negative is False
assert (2*npos).is_negative is None
assert (nneg*npos).is_negative is None
assert (neg*nneg).is_negative is None
assert (neg*npos).is_negative is False
assert (pos*nneg).is_negative is False
assert (pos*npos).is_negative is None
assert (npos*neg*nneg).is_negative is False
assert (npos*pos*nneg).is_negative is None
assert (-npos*neg*nneg).is_negative is None
assert (-npos*pos*nneg).is_negative is False
assert (17*npos*neg*nneg).is_negative is False
assert (17*npos*pos*nneg).is_negative is None
assert (neg*npos*pos*nneg).is_negative is False
assert (x*neg).is_negative is None
assert (nneg*npos*pos*x*neg).is_negative is None
assert neg.is_positive is False
assert (-neg).is_positive is True
assert (2*neg).is_positive is False
assert pos.is_positive is True
assert (-pos).is_positive is False
assert (2*pos).is_positive is True
assert (pos*neg).is_positive is False
assert (2*pos*neg).is_positive is False
assert (-pos*neg).is_positive is True
assert (-pos*neg*y).is_positive is False # y.is_real=F; !real -> !neg
assert nneg.is_positive is None
assert (-nneg).is_positive is False
assert (2*nneg).is_positive is None
assert npos.is_positive is False
assert (-npos).is_positive is None
assert (2*npos).is_positive is False
assert (nneg*npos).is_positive is False
assert (neg*nneg).is_positive is False
assert (neg*npos).is_positive is None
assert (pos*nneg).is_positive is None
assert (pos*npos).is_positive is False
assert (npos*neg*nneg).is_positive is None
assert (npos*pos*nneg).is_positive is False
assert (-npos*neg*nneg).is_positive is False
assert (-npos*pos*nneg).is_positive is None
assert (17*npos*neg*nneg).is_positive is None
assert (17*npos*pos*nneg).is_positive is False
assert (neg*npos*pos*nneg).is_positive is None
assert (x*neg).is_positive is None
assert (nneg*npos*pos*x*neg).is_positive is None
def test_Mul_is_negative_positive_2():
a = Symbol('a', nonnegative=True)
b = Symbol('b', nonnegative=True)
c = Symbol('c', nonpositive=True)
d = Symbol('d', nonpositive=True)
assert (a*b).is_nonnegative is True
assert (a*b).is_negative is False
assert (a*b).is_zero is None
assert (a*b).is_positive is None
assert (c*d).is_nonnegative is True
assert (c*d).is_negative is False
assert (c*d).is_zero is None
assert (c*d).is_positive is None
assert (a*c).is_nonpositive is True
assert (a*c).is_positive is False
assert (a*c).is_zero is None
assert (a*c).is_negative is None
def test_Mul_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert k.is_nonpositive is True
assert (-k).is_nonpositive is False
assert (2*k).is_nonpositive is True
assert n.is_nonpositive is False
assert (-n).is_nonpositive is True
assert (2*n).is_nonpositive is False
assert (n*k).is_nonpositive is True
assert (2*n*k).is_nonpositive is True
assert (-n*k).is_nonpositive is False
assert u.is_nonpositive is None
assert (-u).is_nonpositive is True
assert (2*u).is_nonpositive is None
assert v.is_nonpositive is True
assert (-v).is_nonpositive is None
assert (2*v).is_nonpositive is True
assert (u*v).is_nonpositive is True
assert (k*u).is_nonpositive is True
assert (k*v).is_nonpositive is None
assert (n*u).is_nonpositive is None
assert (n*v).is_nonpositive is True
assert (v*k*u).is_nonpositive is None
assert (v*n*u).is_nonpositive is True
assert (-v*k*u).is_nonpositive is True
assert (-v*n*u).is_nonpositive is None
assert (17*v*k*u).is_nonpositive is None
assert (17*v*n*u).is_nonpositive is True
assert (k*v*n*u).is_nonpositive is None
assert (x*k).is_nonpositive is None
assert (u*v*n*x*k).is_nonpositive is None
assert k.is_nonnegative is False
assert (-k).is_nonnegative is True
assert (2*k).is_nonnegative is False
assert n.is_nonnegative is True
assert (-n).is_nonnegative is False
assert (2*n).is_nonnegative is True
assert (n*k).is_nonnegative is False
assert (2*n*k).is_nonnegative is False
assert (-n*k).is_nonnegative is True
assert u.is_nonnegative is True
assert (-u).is_nonnegative is None
assert (2*u).is_nonnegative is True
assert v.is_nonnegative is None
assert (-v).is_nonnegative is True
assert (2*v).is_nonnegative is None
assert (u*v).is_nonnegative is None
assert (k*u).is_nonnegative is None
assert (k*v).is_nonnegative is True
assert (n*u).is_nonnegative is True
assert (n*v).is_nonnegative is None
assert (v*k*u).is_nonnegative is True
assert (v*n*u).is_nonnegative is None
assert (-v*k*u).is_nonnegative is None
assert (-v*n*u).is_nonnegative is True
assert (17*v*k*u).is_nonnegative is True
assert (17*v*n*u).is_nonnegative is None
assert (k*v*n*u).is_nonnegative is True
assert (x*k).is_nonnegative is None
assert (u*v*n*x*k).is_nonnegative is None
def test_Add_is_negative_positive():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert (k - 2).is_negative is True
assert (k + 17).is_negative is None
assert (-k - 5).is_negative is None
assert (-k + 123).is_negative is False
assert (k - n).is_negative is True
assert (k + n).is_negative is None
assert (-k - n).is_negative is None
assert (-k + n).is_negative is False
assert (k - n - 2).is_negative is True
assert (k + n + 17).is_negative is None
assert (-k - n - 5).is_negative is None
assert (-k + n + 123).is_negative is False
assert (-2*k + 123*n + 17).is_negative is False
assert (k + u).is_negative is None
assert (k + v).is_negative is True
assert (n + u).is_negative is False
assert (n + v).is_negative is None
assert (u - v).is_negative is False
assert (u + v).is_negative is None
assert (-u - v).is_negative is None
assert (-u + v).is_negative is None
assert (u - v + n + 2).is_negative is False
assert (u + v + n + 2).is_negative is None
assert (-u - v + n + 2).is_negative is None
assert (-u + v + n + 2).is_negative is None
assert (k + x).is_negative is None
assert (k + x - n).is_negative is None
assert (k - 2).is_positive is False
assert (k + 17).is_positive is None
assert (-k - 5).is_positive is None
assert (-k + 123).is_positive is True
assert (k - n).is_positive is False
assert (k + n).is_positive is None
assert (-k - n).is_positive is None
assert (-k + n).is_positive is True
assert (k - n - 2).is_positive is False
assert (k + n + 17).is_positive is None
assert (-k - n - 5).is_positive is None
assert (-k + n + 123).is_positive is True
assert (-2*k + 123*n + 17).is_positive is True
assert (k + u).is_positive is None
assert (k + v).is_positive is False
assert (n + u).is_positive is True
assert (n + v).is_positive is None
assert (u - v).is_positive is None
assert (u + v).is_positive is None
assert (-u - v).is_positive is None
assert (-u + v).is_positive is False
assert (u - v - n - 2).is_positive is None
assert (u + v - n - 2).is_positive is None
assert (-u - v - n - 2).is_positive is None
assert (-u + v - n - 2).is_positive is False
assert (n + x).is_positive is None
assert (n + x - k).is_positive is None
z = (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2)
assert z.is_zero
z = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert z.is_zero
def test_Add_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', negative=True)
n = Symbol('n', positive=True)
u = Symbol('u', nonnegative=True)
v = Symbol('v', nonpositive=True)
assert (u - 2).is_nonpositive is None
assert (u + 17).is_nonpositive is False
assert (-u - 5).is_nonpositive is True
assert (-u + 123).is_nonpositive is None
assert (u - v).is_nonpositive is None
assert (u + v).is_nonpositive is None
assert (-u - v).is_nonpositive is None
assert (-u + v).is_nonpositive is True
assert (u - v - 2).is_nonpositive is None
assert (u + v + 17).is_nonpositive is None
assert (-u - v - 5).is_nonpositive is None
assert (-u + v - 123).is_nonpositive is True
assert (-2*u + 123*v - 17).is_nonpositive is True
assert (k + u).is_nonpositive is None
assert (k + v).is_nonpositive is True
assert (n + u).is_nonpositive is False
assert (n + v).is_nonpositive is None
assert (k - n).is_nonpositive is True
assert (k + n).is_nonpositive is None
assert (-k - n).is_nonpositive is None
assert (-k + n).is_nonpositive is False
assert (k - n + u + 2).is_nonpositive is None
assert (k + n + u + 2).is_nonpositive is None
assert (-k - n + u + 2).is_nonpositive is None
assert (-k + n + u + 2).is_nonpositive is False
assert (u + x).is_nonpositive is None
assert (v - x - n).is_nonpositive is None
assert (u - 2).is_nonnegative is None
assert (u + 17).is_nonnegative is True
assert (-u - 5).is_nonnegative is False
assert (-u + 123).is_nonnegative is None
assert (u - v).is_nonnegative is True
assert (u + v).is_nonnegative is None
assert (-u - v).is_nonnegative is None
assert (-u + v).is_nonnegative is None
assert (u - v + 2).is_nonnegative is True
assert (u + v + 17).is_nonnegative is None
assert (-u - v - 5).is_nonnegative is None
assert (-u + v - 123).is_nonnegative is False
assert (2*u - 123*v + 17).is_nonnegative is True
assert (k + u).is_nonnegative is None
assert (k + v).is_nonnegative is False
assert (n + u).is_nonnegative is True
assert (n + v).is_nonnegative is None
assert (k - n).is_nonnegative is False
assert (k + n).is_nonnegative is None
assert (-k - n).is_nonnegative is None
assert (-k + n).is_nonnegative is True
assert (k - n - u - 2).is_nonnegative is False
assert (k + n - u - 2).is_nonnegative is None
assert (-k - n - u - 2).is_nonnegative is None
assert (-k + n - u - 2).is_nonnegative is None
assert (u - x).is_nonnegative is None
assert (v + x + n).is_nonnegative is None
def test_Pow_is_integer():
x = Symbol('x')
k = Symbol('k', integer=True)
n = Symbol('n', integer=True, nonnegative=True)
m = Symbol('m', integer=True, positive=True)
assert (k**2).is_integer is True
assert (k**(-2)).is_integer is None
assert ((m + 1)**(-2)).is_integer is False
assert (m**(-1)).is_integer is None # issue 8580
assert (2**k).is_integer is None
assert (2**(-k)).is_integer is None
assert (2**n).is_integer is True
assert (2**(-n)).is_integer is None
assert (2**m).is_integer is True
assert (2**(-m)).is_integer is False
assert (x**2).is_integer is None
assert (2**x).is_integer is None
assert (k**n).is_integer is True
assert (k**(-n)).is_integer is None
assert (k**x).is_integer is None
assert (x**k).is_integer is None
assert (k**(n*m)).is_integer is True
assert (k**(-n*m)).is_integer is None
assert sqrt(3).is_integer is False
assert sqrt(.3).is_integer is False
assert Pow(3, 2, evaluate=False).is_integer is True
assert Pow(3, 0, evaluate=False).is_integer is True
assert Pow(3, -2, evaluate=False).is_integer is False
assert Pow(S.Half, 3, evaluate=False).is_integer is False
# decided by re-evaluating
assert Pow(3, S.Half, evaluate=False).is_integer is False
assert Pow(3, S.Half, evaluate=False).is_integer is False
assert Pow(4, S.Half, evaluate=False).is_integer is True
assert Pow(S.Half, -2, evaluate=False).is_integer is True
assert ((-1)**k).is_integer
# issue 8641
x = Symbol('x', real=True, integer=False)
assert (x**2).is_integer is None
# issue 10458
x = Symbol('x', positive=True)
assert (1/(x + 1)).is_integer is False
assert (1/(-x - 1)).is_integer is False
assert (-1/(x + 1)).is_integer is False
# issue 8648-like
k = Symbol('k', even=True)
assert (k**3/2).is_integer
assert (k**3/8).is_integer
assert (k**3/16).is_integer is None
assert (2/k).is_integer is None
assert (2/k**2).is_integer is False
o = Symbol('o', odd=True)
assert (k/o).is_integer is None
o = Symbol('o', odd=True, prime=True)
assert (k/o).is_integer is False
def test_Pow_is_real():
x = Symbol('x', real=True)
y = Symbol('y', positive=True)
assert (x**2).is_real is True
assert (x**3).is_real is True
assert (x**x).is_real is None
assert (y**x).is_real is True
assert (x**Rational(1, 3)).is_real is None
assert (y**Rational(1, 3)).is_real is True
assert sqrt(-1 - sqrt(2)).is_real is False
i = Symbol('i', imaginary=True)
assert (i**i).is_real is None
assert (I**i).is_extended_real is True
assert ((-I)**i).is_extended_real is True
assert (2**i).is_real is None # (2**(pi/log(2) * I)) is real, 2**I is not
assert (2**I).is_real is False
assert (2**-I).is_real is False
assert (i**2).is_extended_real is True
assert (i**3).is_extended_real is False
assert (i**x).is_real is None # could be (-I)**(2/3)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
k = Symbol('k', integer=True)
assert (i**e).is_extended_real is True
assert (i**o).is_extended_real is False
assert (i**k).is_real is None
assert (i**(4*k)).is_extended_real is True
x = Symbol("x", nonnegative=True)
y = Symbol("y", nonnegative=True)
assert im(x**y).expand(complex=True) is S.Zero
assert (x**y).is_real is True
i = Symbol('i', imaginary=True)
assert (exp(i)**I).is_extended_real is True
assert log(exp(i)).is_imaginary is None # i could be 2*pi*I
c = Symbol('c', complex=True)
assert log(c).is_real is None # c could be 0 or 2, too
assert log(exp(c)).is_real is None # log(0), log(E), ...
n = Symbol('n', negative=False)
assert log(n).is_real is None
n = Symbol('n', nonnegative=True)
assert log(n).is_real is None
assert sqrt(-I).is_real is False # issue 7843
i = Symbol('i', integer=True)
assert (1/(i-1)).is_real is None
assert (1/(i-1)).is_extended_real is None
# test issue 20715
from sympy.core.parameters import evaluate
x = S(-1)
with evaluate(False):
assert x.is_negative is True
f = Pow(x, -1)
with evaluate(False):
assert f.is_imaginary is False
def test_real_Pow():
k = Symbol('k', integer=True, nonzero=True)
assert (k**(I*pi/log(k))).is_real
def test_Pow_is_finite():
xe = Symbol('xe', extended_real=True)
xr = Symbol('xr', real=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
i = Symbol('i', integer=True)
assert (xe**2).is_finite is None # xe could be oo
assert (xr**2).is_finite is True
assert (xe**xe).is_finite is None
assert (xr**xe).is_finite is None
assert (xe**xr).is_finite is None
# FIXME: The line below should be True rather than None
# assert (xr**xr).is_finite is True
assert (xr**xr).is_finite is None
assert (p**xe).is_finite is None
assert (p**xr).is_finite is True
assert (n**xe).is_finite is None
assert (n**xr).is_finite is True
assert (sin(xe)**2).is_finite is True
assert (sin(xr)**2).is_finite is True
assert (sin(xe)**xe).is_finite is None # xe, xr could be -pi
assert (sin(xr)**xr).is_finite is None
# FIXME: Should the line below be True rather than None?
assert (sin(xe)**exp(xe)).is_finite is None
assert (sin(xr)**exp(xr)).is_finite is True
assert (1/sin(xe)).is_finite is None # if zero, no, otherwise yes
assert (1/sin(xr)).is_finite is None
assert (1/exp(xe)).is_finite is None # xe could be -oo
assert (1/exp(xr)).is_finite is True
assert (1/S.Pi).is_finite is True
assert (1/(i-1)).is_finite is None
def test_Pow_is_even_odd():
x = Symbol('x')
k = Symbol('k', even=True)
n = Symbol('n', odd=True)
m = Symbol('m', integer=True, nonnegative=True)
p = Symbol('p', integer=True, positive=True)
assert ((-1)**n).is_odd
assert ((-1)**k).is_odd
assert ((-1)**(m - p)).is_odd
assert (k**2).is_even is True
assert (n**2).is_even is False
assert (2**k).is_even is None
assert (x**2).is_even is None
assert (k**m).is_even is None
assert (n**m).is_even is False
assert (k**p).is_even is True
assert (n**p).is_even is False
assert (m**k).is_even is None
assert (p**k).is_even is None
assert (m**n).is_even is None
assert (p**n).is_even is None
assert (k**x).is_even is None
assert (n**x).is_even is None
assert (k**2).is_odd is False
assert (n**2).is_odd is True
assert (3**k).is_odd is None
assert (k**m).is_odd is None
assert (n**m).is_odd is True
assert (k**p).is_odd is False
assert (n**p).is_odd is True
assert (m**k).is_odd is None
assert (p**k).is_odd is None
assert (m**n).is_odd is None
assert (p**n).is_odd is None
assert (k**x).is_odd is None
assert (n**x).is_odd is None
def test_Pow_is_negative_positive():
r = Symbol('r', real=True)
k = Symbol('k', integer=True, positive=True)
n = Symbol('n', even=True)
m = Symbol('m', odd=True)
x = Symbol('x')
assert (2**r).is_positive is True
assert ((-2)**r).is_positive is None
assert ((-2)**n).is_positive is True
assert ((-2)**m).is_positive is False
assert (k**2).is_positive is True
assert (k**(-2)).is_positive is True
assert (k**r).is_positive is True
assert ((-k)**r).is_positive is None
assert ((-k)**n).is_positive is True
assert ((-k)**m).is_positive is False
assert (2**r).is_negative is False
assert ((-2)**r).is_negative is None
assert ((-2)**n).is_negative is False
assert ((-2)**m).is_negative is True
assert (k**2).is_negative is False
assert (k**(-2)).is_negative is False
assert (k**r).is_negative is False
assert ((-k)**r).is_negative is None
assert ((-k)**n).is_negative is False
assert ((-k)**m).is_negative is True
assert (2**x).is_positive is None
assert (2**x).is_negative is None
def test_Pow_is_zero():
z = Symbol('z', zero=True)
e = z**2
assert e.is_zero
assert e.is_positive is False
assert e.is_negative is False
assert Pow(0, 0, evaluate=False).is_zero is False
assert Pow(0, 3, evaluate=False).is_zero
assert Pow(0, oo, evaluate=False).is_zero
assert Pow(0, -3, evaluate=False).is_zero is False
assert Pow(0, -oo, evaluate=False).is_zero is False
assert Pow(2, 2, evaluate=False).is_zero is False
a = Symbol('a', zero=False)
assert Pow(a, 3).is_zero is False # issue 7965
assert Pow(2, oo, evaluate=False).is_zero is False
assert Pow(2, -oo, evaluate=False).is_zero
assert Pow(S.Half, oo, evaluate=False).is_zero
assert Pow(S.Half, -oo, evaluate=False).is_zero is False
# All combinations of real/complex base/exponent
h = S.Half
T = True
F = False
N = None
pow_iszero = [
['**', 0, h, 1, 2, -h, -1,-2,-2*I,-I/2,I/2,1+I,oo,-oo,zoo],
[ 0, F, T, T, T, F, F, F, F, F, F, N, T, F, N],
[ h, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ 1, F, F, F, F, F, F, F, F, F, F, F, F, F, N],
[ 2, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[ -h, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ -1, F, F, F, F, F, F, F, F, F, F, F, F, F, N],
[ -2, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[-2*I, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[-I/2, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ I/2, F, F, F, F, F, F, F, F, F, F, F, T, F, N],
[ 1+I, F, F, F, F, F, F, F, F, F, F, F, F, T, N],
[ oo, F, F, F, F, T, T, T, F, F, F, F, F, T, N],
[ -oo, F, F, F, F, T, T, T, F, F, F, F, F, T, N],
[ zoo, F, F, F, F, T, T, T, N, N, N, N, F, T, N]
]
def test_table(table):
n = len(table[0])
for row in range(1, n):
base = table[row][0]
for col in range(1, n):
exp = table[0][col]
is_zero = table[row][col]
# The actual test here:
assert Pow(base, exp, evaluate=False).is_zero is is_zero
test_table(pow_iszero)
# A zero symbol...
zo, zo2 = symbols('zo, zo2', zero=True)
# All combinations of finite symbols
zf, zf2 = symbols('zf, zf2', finite=True)
wf, wf2 = symbols('wf, wf2', nonzero=True)
xf, xf2 = symbols('xf, xf2', real=True)
yf, yf2 = symbols('yf, yf2', nonzero=True)
af, af2 = symbols('af, af2', positive=True)
bf, bf2 = symbols('bf, bf2', nonnegative=True)
cf, cf2 = symbols('cf, cf2', negative=True)
df, df2 = symbols('df, df2', nonpositive=True)
# Without finiteness:
zi, zi2 = symbols('zi, zi2')
wi, wi2 = symbols('wi, wi2', zero=False)
xi, xi2 = symbols('xi, xi2', extended_real=True)
yi, yi2 = symbols('yi, yi2', zero=False, extended_real=True)
ai, ai2 = symbols('ai, ai2', extended_positive=True)
bi, bi2 = symbols('bi, bi2', extended_nonnegative=True)
ci, ci2 = symbols('ci, ci2', extended_negative=True)
di, di2 = symbols('di, di2', extended_nonpositive=True)
pow_iszero_sym = [
['**',zo,wf,yf,af,cf,zf,xf,bf,df,zi,wi,xi,yi,ai,bi,ci,di],
[ zo2, F, N, N, T, F, N, N, N, F, N, N, N, N, T, N, F, F],
[ wf2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ yf2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ af2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ cf2, F, F, F, F, F, F, F, F, F, N, N, N, N, N, N, N, N],
[ zf2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ xf2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ bf2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ df2, N, N, N, N, F, N, N, N, N, N, N, N, N, N, N, N, N],
[ zi2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N],
[ wi2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ xi2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N],
[ yi2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ ai2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ bi2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N],
[ ci2, F, N, N, F, N, N, N, F, N, N, N, N, N, N, N, N, N],
[ di2, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N]
]
test_table(pow_iszero_sym)
# In some cases (x**x).is_zero is different from (x**y).is_zero even if y
# has the same assumptions as x.
assert (zo ** zo).is_zero is False
assert (wf ** wf).is_zero is False
assert (yf ** yf).is_zero is False
assert (af ** af).is_zero is False
assert (cf ** cf).is_zero is False
assert (zf ** zf).is_zero is None
assert (xf ** xf).is_zero is None
assert (bf ** bf).is_zero is False # None in table
assert (df ** df).is_zero is None
assert (zi ** zi).is_zero is None
assert (wi ** wi).is_zero is None
assert (xi ** xi).is_zero is None
assert (yi ** yi).is_zero is None
assert (ai ** ai).is_zero is False # None in table
assert (bi ** bi).is_zero is False # None in table
assert (ci ** ci).is_zero is None
assert (di ** di).is_zero is None
def test_Pow_is_nonpositive_nonnegative():
x = Symbol('x', real=True)
k = Symbol('k', integer=True, nonnegative=True)
l = Symbol('l', integer=True, positive=True)
n = Symbol('n', even=True)
m = Symbol('m', odd=True)
assert (x**(4*k)).is_nonnegative is True
assert (2**x).is_nonnegative is True
assert ((-2)**x).is_nonnegative is None
assert ((-2)**n).is_nonnegative is True
assert ((-2)**m).is_nonnegative is False
assert (k**2).is_nonnegative is True
assert (k**(-2)).is_nonnegative is None
assert (k**k).is_nonnegative is True
assert (k**x).is_nonnegative is None # NOTE (0**x).is_real = U
assert (l**x).is_nonnegative is True
assert (l**x).is_positive is True
assert ((-k)**x).is_nonnegative is None
assert ((-k)**m).is_nonnegative is None
assert (2**x).is_nonpositive is False
assert ((-2)**x).is_nonpositive is None
assert ((-2)**n).is_nonpositive is False
assert ((-2)**m).is_nonpositive is True
assert (k**2).is_nonpositive is None
assert (k**(-2)).is_nonpositive is None
assert (k**x).is_nonpositive is None
assert ((-k)**x).is_nonpositive is None
assert ((-k)**n).is_nonpositive is None
assert (x**2).is_nonnegative is True
i = symbols('i', imaginary=True)
assert (i**2).is_nonpositive is True
assert (i**4).is_nonpositive is False
assert (i**3).is_nonpositive is False
assert (I**i).is_nonnegative is True
assert (exp(I)**i).is_nonnegative is True
assert ((-l)**n).is_nonnegative is True
assert ((-l)**m).is_nonpositive is True
assert ((-k)**n).is_nonnegative is None
assert ((-k)**m).is_nonpositive is None
def test_Mul_is_imaginary_real():
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
i1 = Symbol('i1', imaginary=True)
i2 = Symbol('i2', imaginary=True)
x = Symbol('x')
assert I.is_imaginary is True
assert I.is_real is False
assert (-I).is_imaginary is True
assert (-I).is_real is False
assert (3*I).is_imaginary is True
assert (3*I).is_real is False
assert (I*I).is_imaginary is False
assert (I*I).is_real is True
e = (p + p*I)
j = Symbol('j', integer=True, zero=False)
assert (e**j).is_real is None
assert (e**(2*j)).is_real is None
assert (e**j).is_imaginary is None
assert (e**(2*j)).is_imaginary is None
assert (e**-1).is_imaginary is False
assert (e**2).is_imaginary
assert (e**3).is_imaginary is False
assert (e**4).is_imaginary is False
assert (e**5).is_imaginary is False
assert (e**-1).is_real is False
assert (e**2).is_real is False
assert (e**3).is_real is False
assert (e**4).is_real is True
assert (e**5).is_real is False
assert (e**3).is_complex
assert (r*i1).is_imaginary is None
assert (r*i1).is_real is None
assert (x*i1).is_imaginary is None
assert (x*i1).is_real is None
assert (i1*i2).is_imaginary is False
assert (i1*i2).is_real is True
assert (r*i1*i2).is_imaginary is False
assert (r*i1*i2).is_real is True
# Github's issue 5874:
nr = Symbol('nr', real=False, complex=True) # e.g. I or 1 + I
a = Symbol('a', real=True, nonzero=True)
b = Symbol('b', real=True)
assert (i1*nr).is_real is None
assert (a*nr).is_real is False
assert (b*nr).is_real is None
ni = Symbol('ni', imaginary=False, complex=True) # e.g. 2 or 1 + I
a = Symbol('a', real=True, nonzero=True)
b = Symbol('b', real=True)
assert (i1*ni).is_real is False
assert (a*ni).is_real is None
assert (b*ni).is_real is None
def test_Mul_hermitian_antihermitian():
a = Symbol('a', hermitian=True, zero=False)
b = Symbol('b', hermitian=True)
c = Symbol('c', hermitian=False)
d = Symbol('d', antihermitian=True)
e1 = Mul(a, b, c, evaluate=False)
e2 = Mul(b, a, c, evaluate=False)
e3 = Mul(a, b, c, d, evaluate=False)
e4 = Mul(b, a, c, d, evaluate=False)
e5 = Mul(a, c, evaluate=False)
e6 = Mul(a, c, d, evaluate=False)
assert e1.is_hermitian is None
assert e2.is_hermitian is None
assert e1.is_antihermitian is None
assert e2.is_antihermitian is None
assert e3.is_antihermitian is None
assert e4.is_antihermitian is None
assert e5.is_antihermitian is None
assert e6.is_antihermitian is None
def test_Add_is_comparable():
assert (x + y).is_comparable is False
assert (x + 1).is_comparable is False
assert (Rational(1, 3) - sqrt(8)).is_comparable is True
def test_Mul_is_comparable():
assert (x*y).is_comparable is False
assert (x*2).is_comparable is False
assert (sqrt(2)*Rational(1, 3)).is_comparable is True
def test_Pow_is_comparable():
assert (x**y).is_comparable is False
assert (x**2).is_comparable is False
assert (sqrt(Rational(1, 3))).is_comparable is True
def test_Add_is_positive_2():
e = Rational(1, 3) - sqrt(8)
assert e.is_positive is False
assert e.is_negative is True
e = pi - 1
assert e.is_positive is True
assert e.is_negative is False
def test_Add_is_irrational():
i = Symbol('i', irrational=True)
assert i.is_irrational is True
assert i.is_rational is False
assert (i + 1).is_irrational is True
assert (i + 1).is_rational is False
def test_Mul_is_irrational():
expr = Mul(1, 2, 3, evaluate=False)
assert expr.is_irrational is False
expr = Mul(1, I, I, evaluate=False)
assert expr.is_rational is None # I * I = -1 but *no evaluation allowed*
# sqrt(2) * I * I = -sqrt(2) is irrational but
# this can't be determined without evaluating the
# expression and the eval_is routines shouldn't do that
expr = Mul(sqrt(2), I, I, evaluate=False)
assert expr.is_irrational is None
def test_issue_3531():
# https://github.com/sympy/sympy/issues/3531
# https://github.com/sympy/sympy/pull/18116
class MightyNumeric(tuple):
def __rtruediv__(self, other):
return "something"
assert sympify(1)/MightyNumeric((1, 2)) == "something"
def test_issue_3531b():
class Foo:
def __init__(self):
self.field = 1.0
def __mul__(self, other):
self.field = self.field * other
def __rmul__(self, other):
self.field = other * self.field
f = Foo()
x = Symbol("x")
assert f*x == x*f
def test_bug3():
a = Symbol("a")
b = Symbol("b", positive=True)
e = 2*a + b
f = b + 2*a
assert e == f
def test_suppressed_evaluation():
a = Add(0, 3, 2, evaluate=False)
b = Mul(1, 3, 2, evaluate=False)
c = Pow(3, 2, evaluate=False)
assert a != 6
assert a.func is Add
assert a.args == (0, 3, 2)
assert b != 6
assert b.func is Mul
assert b.args == (1, 3, 2)
assert c != 9
assert c.func is Pow
assert c.args == (3, 2)
def test_AssocOp_doit():
a = Add(x,x, evaluate=False)
b = Mul(y,y, evaluate=False)
c = Add(b,b, evaluate=False)
d = Mul(a,a, evaluate=False)
assert c.doit(deep=False).func == Mul
assert c.doit(deep=False).args == (2,y,y)
assert c.doit().func == Mul
assert c.doit().args == (2, Pow(y,2))
assert d.doit(deep=False).func == Pow
assert d.doit(deep=False).args == (a, 2*S.One)
assert d.doit().func == Mul
assert d.doit().args == (4*S.One, Pow(x,2))
def test_Add_Mul_Expr_args():
nonexpr = [Basic(), Poly(x, x), FiniteSet(x)]
for typ in [Add, Mul]:
for obj in nonexpr:
# The cache can mess with the stacklevel check
with warns(SymPyDeprecationWarning, test_stacklevel=False):
typ(obj, 1)
def test_Add_as_coeff_mul():
# issue 5524. These should all be (1, self)
assert (x + 1).as_coeff_mul() == (1, (x + 1,))
assert (x + 2).as_coeff_mul() == (1, (x + 2,))
assert (x + 3).as_coeff_mul() == (1, (x + 3,))
assert (x - 1).as_coeff_mul() == (1, (x - 1,))
assert (x - 2).as_coeff_mul() == (1, (x - 2,))
assert (x - 3).as_coeff_mul() == (1, (x - 3,))
n = Symbol('n', integer=True)
assert (n + 1).as_coeff_mul() == (1, (n + 1,))
assert (n + 2).as_coeff_mul() == (1, (n + 2,))
assert (n + 3).as_coeff_mul() == (1, (n + 3,))
assert (n - 1).as_coeff_mul() == (1, (n - 1,))
assert (n - 2).as_coeff_mul() == (1, (n - 2,))
assert (n - 3).as_coeff_mul() == (1, (n - 3,))
def test_Pow_as_coeff_mul_doesnt_expand():
assert exp(x + y).as_coeff_mul() == (1, (exp(x + y),))
assert exp(x + exp(x + y)) != exp(x + exp(x)*exp(y))
def test_issue_3514_18626():
assert sqrt(S.Half) * sqrt(6) == 2 * sqrt(3)/2
assert S.Half*sqrt(6)*sqrt(2) == sqrt(3)
assert sqrt(6)/2*sqrt(2) == sqrt(3)
assert sqrt(6)*sqrt(2)/2 == sqrt(3)
assert sqrt(8)**Rational(2, 3) == 2
def test_make_args():
assert Add.make_args(x) == (x,)
assert Mul.make_args(x) == (x,)
assert Add.make_args(x*y*z) == (x*y*z,)
assert Mul.make_args(x*y*z) == (x*y*z).args
assert Add.make_args(x + y + z) == (x + y + z).args
assert Mul.make_args(x + y + z) == (x + y + z,)
assert Add.make_args((x + y)**z) == ((x + y)**z,)
assert Mul.make_args((x + y)**z) == ((x + y)**z,)
def test_issue_5126():
assert (-2)**x*(-3)**x != 6**x
i = Symbol('i', integer=1)
assert (-2)**i*(-3)**i == 6**i
def test_Rational_as_content_primitive():
c, p = S.One, S.Zero
assert (c*p).as_content_primitive() == (c, p)
c, p = S.Half, S.One
assert (c*p).as_content_primitive() == (c, p)
def test_Add_as_content_primitive():
assert (x + 2).as_content_primitive() == (1, x + 2)
assert (3*x + 2).as_content_primitive() == (1, 3*x + 2)
assert (3*x + 3).as_content_primitive() == (3, x + 1)
assert (3*x + 6).as_content_primitive() == (3, x + 2)
assert (3*x + 2*y).as_content_primitive() == (1, 3*x + 2*y)
assert (3*x + 3*y).as_content_primitive() == (3, x + y)
assert (3*x + 6*y).as_content_primitive() == (3, x + 2*y)
assert (3/x + 2*x*y*z**2).as_content_primitive() == (1, 3/x + 2*x*y*z**2)
assert (3/x + 3*x*y*z**2).as_content_primitive() == (3, 1/x + x*y*z**2)
assert (3/x + 6*x*y*z**2).as_content_primitive() == (3, 1/x + 2*x*y*z**2)
assert (2*x/3 + 4*y/9).as_content_primitive() == \
(Rational(2, 9), 3*x + 2*y)
assert (2*x/3 + 2.5*y).as_content_primitive() == \
(Rational(1, 3), 2*x + 7.5*y)
# the coefficient may sort to a position other than 0
p = 3 + x + y
assert (2*p).expand().as_content_primitive() == (2, p)
assert (2.0*p).expand().as_content_primitive() == (1, 2.*p)
p *= -1
assert (2*p).expand().as_content_primitive() == (2, p)
def test_Mul_as_content_primitive():
assert (2*x).as_content_primitive() == (2, x)
assert (x*(2 + 2*x)).as_content_primitive() == (2, x*(1 + x))
assert (x*(2 + 2*y)*(3*x + 3)**2).as_content_primitive() == \
(18, x*(1 + y)*(x + 1)**2)
assert ((2 + 2*x)**2*(3 + 6*x) + S.Half).as_content_primitive() == \
(S.Half, 24*(x + 1)**2*(2*x + 1) + 1)
def test_Pow_as_content_primitive():
assert (x**y).as_content_primitive() == (1, x**y)
assert ((2*x + 2)**y).as_content_primitive() == \
(1, (Mul(2, (x + 1), evaluate=False))**y)
assert ((2*x + 2)**3).as_content_primitive() == (8, (x + 1)**3)
def test_issue_5460():
u = Mul(2, (1 + x), evaluate=False)
assert (2 + u).args == (2, u)
def test_product_irrational():
assert (I*pi).is_irrational is False
# The following used to be deduced from the above bug:
assert (I*pi).is_positive is False
def test_issue_5919():
assert (x/(y*(1 + y))).expand() == x/(y**2 + y)
def test_Mod():
assert Mod(x, 1).func is Mod
assert pi % pi is S.Zero
assert Mod(5, 3) == 2
assert Mod(-5, 3) == 1
assert Mod(5, -3) == -1
assert Mod(-5, -3) == -2
assert type(Mod(3.2, 2, evaluate=False)) == Mod
assert 5 % x == Mod(5, x)
assert x % 5 == Mod(x, 5)
assert x % y == Mod(x, y)
assert (x % y).subs({x: 5, y: 3}) == 2
assert Mod(nan, 1) is nan
assert Mod(1, nan) is nan
assert Mod(nan, nan) is nan
assert Mod(0, x) == 0
with raises(ZeroDivisionError):
Mod(x, 0)
k = Symbol('k', integer=True)
m = Symbol('m', integer=True, positive=True)
assert (x**m % x).func is Mod
assert (k**(-m) % k).func is Mod
assert k**m % k == 0
assert (-2*k)**m % k == 0
# Float handling
point3 = Float(3.3) % 1
assert (x - 3.3) % 1 == Mod(1.*x + 1 - point3, 1)
assert Mod(-3.3, 1) == 1 - point3
assert Mod(0.7, 1) == Float(0.7)
e = Mod(1.3, 1)
assert comp(e, .3) and e.is_Float
e = Mod(1.3, .7)
assert comp(e, .6) and e.is_Float
e = Mod(1.3, Rational(7, 10))
assert comp(e, .6) and e.is_Float
e = Mod(Rational(13, 10), 0.7)
assert comp(e, .6) and e.is_Float
e = Mod(Rational(13, 10), Rational(7, 10))
assert comp(e, .6) and e.is_Rational
# check that sign is right
r2 = sqrt(2)
r3 = sqrt(3)
for i in [-r3, -r2, r2, r3]:
for j in [-r3, -r2, r2, r3]:
assert verify_numerically(i % j, i.n() % j.n())
for _x in range(4):
for _y in range(9):
reps = [(x, _x), (y, _y)]
assert Mod(3*x + y, 9).subs(reps) == (3*_x + _y) % 9
# denesting
t = Symbol('t', real=True)
assert Mod(Mod(x, t), t) == Mod(x, t)
assert Mod(-Mod(x, t), t) == Mod(-x, t)
assert Mod(Mod(x, 2*t), t) == Mod(x, t)
assert Mod(-Mod(x, 2*t), t) == Mod(-x, t)
assert Mod(Mod(x, t), 2*t) == Mod(x, t)
assert Mod(-Mod(x, t), -2*t) == -Mod(x, t)
for i in [-4, -2, 2, 4]:
for j in [-4, -2, 2, 4]:
for k in range(4):
assert Mod(Mod(x, i), j).subs({x: k}) == (k % i) % j
assert Mod(-Mod(x, i), j).subs({x: k}) == -(k % i) % j
# known difference
assert Mod(5*sqrt(2), sqrt(5)) == 5*sqrt(2) - 3*sqrt(5)
p = symbols('p', positive=True)
assert Mod(2, p + 3) == 2
assert Mod(-2, p + 3) == p + 1
assert Mod(2, -p - 3) == -p - 1
assert Mod(-2, -p - 3) == -2
assert Mod(p + 5, p + 3) == 2
assert Mod(-p - 5, p + 3) == p + 1
assert Mod(p + 5, -p - 3) == -p - 1
assert Mod(-p - 5, -p - 3) == -2
assert Mod(p + 1, p - 1).func is Mod
# handling sums
assert (x + 3) % 1 == Mod(x, 1)
assert (x + 3.0) % 1 == Mod(1.*x, 1)
assert (x - S(33)/10) % 1 == Mod(x + S(7)/10, 1)
a = Mod(.6*x + y, .3*y)
b = Mod(0.1*y + 0.6*x, 0.3*y)
# Test that a, b are equal, with 1e-14 accuracy in coefficients
eps = 1e-14
assert abs((a.args[0] - b.args[0]).subs({x: 1, y: 1})) < eps
assert abs((a.args[1] - b.args[1]).subs({x: 1, y: 1})) < eps
assert (x + 1) % x == 1 % x
assert (x + y) % x == y % x
assert (x + y + 2) % x == (y + 2) % x
assert (a + 3*x + 1) % (2*x) == Mod(a + x + 1, 2*x)
assert (12*x + 18*y) % (3*x) == 3*Mod(6*y, x)
# gcd extraction
assert (-3*x) % (-2*y) == -Mod(3*x, 2*y)
assert (.6*pi) % (.3*x*pi) == 0.3*pi*Mod(2, x)
assert (.6*pi) % (.31*x*pi) == pi*Mod(0.6, 0.31*x)
assert (6*pi) % (.3*x*pi) == 0.3*pi*Mod(20, x)
assert (6*pi) % (.31*x*pi) == pi*Mod(6, 0.31*x)
assert (6*pi) % (.42*x*pi) == pi*Mod(6, 0.42*x)
assert (12*x) % (2*y) == 2*Mod(6*x, y)
assert (12*x) % (3*5*y) == 3*Mod(4*x, 5*y)
assert (12*x) % (15*x*y) == 3*x*Mod(4, 5*y)
assert (-2*pi) % (3*pi) == pi
assert (2*x + 2) % (x + 1) == 0
assert (x*(x + 1)) % (x + 1) == (x + 1)*Mod(x, 1)
assert Mod(5.0*x, 0.1*y) == 0.1*Mod(50*x, y)
i = Symbol('i', integer=True)
assert (3*i*x) % (2*i*y) == i*Mod(3*x, 2*y)
assert Mod(4*i, 4) == 0
# issue 8677
n = Symbol('n', integer=True, positive=True)
assert factorial(n) % n == 0
assert factorial(n + 2) % n == 0
assert (factorial(n + 4) % (n + 5)).func is Mod
# Wilson's theorem
assert factorial(18042, evaluate=False) % 18043 == 18042
p = Symbol('n', prime=True)
assert factorial(p - 1) % p == p - 1
assert factorial(p - 1) % -p == -1
assert (factorial(3, evaluate=False) % 4).doit() == 2
n = Symbol('n', composite=True, odd=True)
assert factorial(n - 1) % n == 0
# symbolic with known parity
n = Symbol('n', even=True)
assert Mod(n, 2) == 0
n = Symbol('n', odd=True)
assert Mod(n, 2) == 1
# issue 10963
assert (x**6000%400).args[1] == 400
#issue 13543
assert Mod(Mod(x + 1, 2) + 1, 2) == Mod(x, 2)
assert Mod(Mod(x + 2, 4)*(x + 4), 4) == Mod(x*(x + 2), 4)
assert Mod(Mod(x + 2, 4)*4, 4) == 0
# issue 15493
i, j = symbols('i j', integer=True, positive=True)
assert Mod(3*i, 2) == Mod(i, 2)
assert Mod(8*i/j, 4) == 4*Mod(2*i/j, 1)
assert Mod(8*i, 4) == 0
# rewrite
assert Mod(x, y).rewrite(floor) == x - y*floor(x/y)
assert ((x - Mod(x, y))/y).rewrite(floor) == floor(x/y)
# issue 21373
from sympy.functions.elementary.hyperbolic import sinh
from sympy.functions.elementary.piecewise import Piecewise
x_r, y_r = symbols('x_r y_r', real=True)
assert (Piecewise((x_r, y_r > x_r), (y_r, True)) / z) % 1
expr = exp(sinh(Piecewise((x_r, y_r > x_r), (y_r, True)) / z))
expr.subs({1: 1.0})
sinh(Piecewise((x_r, y_r > x_r), (y_r, True)) * z ** -1.0).is_zero
def test_Mod_Pow():
# modular exponentiation
assert isinstance(Mod(Pow(2, 2, evaluate=False), 3), Integer)
assert Mod(Pow(4, 13, evaluate=False), 497) == Mod(Pow(4, 13), 497)
assert Mod(Pow(2, 10000000000, evaluate=False), 3) == 1
assert Mod(Pow(32131231232, 9**10**6, evaluate=False),10**12) == \
pow(32131231232,9**10**6,10**12)
assert Mod(Pow(33284959323, 123**999, evaluate=False),11**13) == \
pow(33284959323,123**999,11**13)
assert Mod(Pow(78789849597, 333**555, evaluate=False),12**9) == \
pow(78789849597,333**555,12**9)
# modular nested exponentiation
expr = Pow(2, 2, evaluate=False)
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 16
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 6487
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 32191
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 18016
expr = Pow(2, expr, evaluate=False)
assert Mod(expr, 3**10) == 5137
expr = Pow(2, 2, evaluate=False)
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 16
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 256
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 6487
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 38281
expr = Pow(expr, 2, evaluate=False)
assert Mod(expr, 3**10) == 15928
expr = Pow(2, 2, evaluate=False)
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 256
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 9229
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 25708
expr = Pow(expr, expr, evaluate=False)
assert Mod(expr, 3**10) == 26608
expr = Pow(expr, expr, evaluate=False)
# XXX This used to fail in a nondeterministic way because of overflow
# error.
assert Mod(expr, 3**10) == 1966
def test_Mod_is_integer():
p = Symbol('p', integer=True)
q1 = Symbol('q1', integer=True)
q2 = Symbol('q2', integer=True, nonzero=True)
assert Mod(x, y).is_integer is None
assert Mod(p, q1).is_integer is None
assert Mod(x, q2).is_integer is None
assert Mod(p, q2).is_integer
def test_Mod_is_nonposneg():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, positive=True)
assert (n%3).is_nonnegative
assert Mod(n, -3).is_nonpositive
assert Mod(n, k).is_nonnegative
assert Mod(n, -k).is_nonpositive
assert Mod(k, n).is_nonnegative is None
def test_issue_6001():
A = Symbol("A", commutative=False)
eq = A + A**2
# it doesn't matter whether it's True or False; they should
# just all be the same
assert (
eq.is_commutative ==
(eq + 1).is_commutative ==
(A + 1).is_commutative)
B = Symbol("B", commutative=False)
# Although commutative terms could cancel we return True
# meaning "there are non-commutative symbols; aftersubstitution
# that definition can change, e.g. (A*B).subs(B,A**-1) -> 1
assert (sqrt(2)*A).is_commutative is False
assert (sqrt(2)*A*B).is_commutative is False
def test_polar():
from sympy.functions.elementary.complexes import polar_lift
p = Symbol('p', polar=True)
x = Symbol('x')
assert p.is_polar
assert x.is_polar is None
assert S.One.is_polar is None
assert (p**x).is_polar is True
assert (x**p).is_polar is None
assert ((2*p)**x).is_polar is True
assert (2*p).is_polar is True
assert (-2*p).is_polar is not True
assert (polar_lift(-2)*p).is_polar is True
q = Symbol('q', polar=True)
assert (p*q)**2 == p**2 * q**2
assert (2*q)**2 == 4 * q**2
assert ((p*q)**x).expand() == p**x * q**x
def test_issue_6040():
a, b = Pow(1, 2, evaluate=False), S.One
assert a != b
assert b != a
assert not (a == b)
assert not (b == a)
def test_issue_6082():
# Comparison is symmetric
assert Basic.compare(Max(x, 1), Max(x, 2)) == \
- Basic.compare(Max(x, 2), Max(x, 1))
# Equal expressions compare equal
assert Basic.compare(Max(x, 1), Max(x, 1)) == 0
# Basic subtypes (such as Max) compare different than standard types
assert Basic.compare(Max(1, x), frozenset((1, x))) != 0
def test_issue_6077():
assert x**2.0/x == x**1.0
assert x/x**2.0 == x**-1.0
assert x*x**2.0 == x**3.0
assert x**1.5*x**2.5 == x**4.0
assert 2**(2.0*x)/2**x == 2**(1.0*x)
assert 2**x/2**(2.0*x) == 2**(-1.0*x)
assert 2**x*2**(2.0*x) == 2**(3.0*x)
assert 2**(1.5*x)*2**(2.5*x) == 2**(4.0*x)
def test_mul_flatten_oo():
p = symbols('p', positive=True)
n, m = symbols('n,m', negative=True)
x_im = symbols('x_im', imaginary=True)
assert n*oo is -oo
assert n*m*oo is oo
assert p*oo is oo
assert x_im*oo != I*oo # i could be +/- 3*I -> +/-oo
def test_add_flatten():
# see https://github.com/sympy/sympy/issues/2633#issuecomment-29545524
a = oo + I*oo
b = oo - I*oo
assert a + b is nan
assert a - b is nan
# FIXME: This evaluates as:
# >>> 1/a
# 0*(oo + oo*I)
# which should not simplify to 0. Should be fixed in Pow.eval
#assert (1/a).simplify() == (1/b).simplify() == 0
a = Pow(2, 3, evaluate=False)
assert a + a == 16
def test_issue_5160_6087_6089_6090():
# issue 6087
assert ((-2*x*y**y)**3.2).n(2) == (2**3.2*(-x*y**y)**3.2).n(2)
# issue 6089
A, B, C = symbols('A,B,C', commutative=False)
assert (2.*B*C)**3 == 8.0*(B*C)**3
assert (-2.*B*C)**3 == -8.0*(B*C)**3
assert (-2*B*C)**2 == 4*(B*C)**2
# issue 5160
assert sqrt(-1.0*x) == 1.0*sqrt(-x)
assert sqrt(1.0*x) == 1.0*sqrt(x)
# issue 6090
assert (-2*x*y*A*B)**2 == 4*x**2*y**2*(A*B)**2
def test_float_int_round():
assert int(float(sqrt(10))) == int(sqrt(10))
assert int(pi**1000) % 10 == 2
assert int(Float('1.123456789012345678901234567890e20', '')) == \
int(112345678901234567890)
assert int(Float('1.123456789012345678901234567890e25', '')) == \
int(11234567890123456789012345)
# decimal forces float so it's not an exact integer ending in 000000
assert int(Float('1.123456789012345678901234567890e35', '')) == \
112345678901234567890123456789000192
assert int(Float('123456789012345678901234567890e5', '')) == \
12345678901234567890123456789000000
assert Integer(Float('1.123456789012345678901234567890e20', '')) == \
112345678901234567890
assert Integer(Float('1.123456789012345678901234567890e25', '')) == \
11234567890123456789012345
# decimal forces float so it's not an exact integer ending in 000000
assert Integer(Float('1.123456789012345678901234567890e35', '')) == \
112345678901234567890123456789000192
assert Integer(Float('123456789012345678901234567890e5', '')) == \
12345678901234567890123456789000000
assert same_and_same_prec(Float('123000e-2',''), Float('1230.00', ''))
assert same_and_same_prec(Float('123000e2',''), Float('12300000', ''))
assert int(1 + Rational('.9999999999999999999999999')) == 1
assert int(pi/1e20) == 0
assert int(1 + pi/1e20) == 1
assert int(Add(1.2, -2, evaluate=False)) == int(1.2 - 2)
assert int(Add(1.2, +2, evaluate=False)) == int(1.2 + 2)
assert int(Add(1 + Float('.99999999999999999', ''), evaluate=False)) == 1
raises(TypeError, lambda: float(x))
raises(TypeError, lambda: float(sqrt(-1)))
assert int(12345678901234567890 + cos(1)**2 + sin(1)**2) == \
12345678901234567891
def test_issue_6611a():
assert Mul.flatten([3**Rational(1, 3),
Pow(-Rational(1, 9), Rational(2, 3), evaluate=False)]) == \
([Rational(1, 3), (-1)**Rational(2, 3)], [], None)
def test_denest_add_mul():
# when working with evaluated expressions make sure they denest
eq = x + 1
eq = Add(eq, 2, evaluate=False)
eq = Add(eq, 2, evaluate=False)
assert Add(*eq.args) == x + 5
eq = x*2
eq = Mul(eq, 2, evaluate=False)
eq = Mul(eq, 2, evaluate=False)
assert Mul(*eq.args) == 8*x
# but don't let them denest unecessarily
eq = Mul(-2, x - 2, evaluate=False)
assert 2*eq == Mul(-4, x - 2, evaluate=False)
assert -eq == Mul(2, x - 2, evaluate=False)
def test_mul_coeff():
# It is important that all Numbers be removed from the seq;
# This can be tricky when powers combine to produce those numbers
p = exp(I*pi/3)
assert p**2*x*p*y*p*x*p**2 == x**2*y
def test_mul_zero_detection():
nz = Dummy(real=True, zero=False)
r = Dummy(extended_real=True)
c = Dummy(real=False, complex=True)
c2 = Dummy(real=False, complex=True)
i = Dummy(imaginary=True)
e = nz*r*c
assert e.is_imaginary is None
assert e.is_extended_real is None
e = nz*c
assert e.is_imaginary is None
assert e.is_extended_real is False
e = nz*i*c
assert e.is_imaginary is False
assert e.is_extended_real is None
# check for more than one complex; it is important to use
# uniquely named Symbols to ensure that two factors appear
# e.g. if the symbols have the same name they just become
# a single factor, a power.
e = nz*i*c*c2
assert e.is_imaginary is None
assert e.is_extended_real is None
# _eval_is_extended_real and _eval_is_zero both employ trapping of the
# zero value so args should be tested in both directions and
# TO AVOID GETTING THE CACHED RESULT, Dummy MUST BE USED
# real is unknown
def test(z, b, e):
if z.is_zero and b.is_finite:
assert e.is_extended_real and e.is_zero
else:
assert e.is_extended_real is None
if b.is_finite:
if z.is_zero:
assert e.is_zero
else:
assert e.is_zero is None
elif b.is_finite is False:
if z.is_zero is None:
assert e.is_zero is None
else:
assert e.is_zero is False
for iz, ib in product(*[[True, False, None]]*2):
z = Dummy('z', nonzero=iz)
b = Dummy('f', finite=ib)
e = Mul(z, b, evaluate=False)
test(z, b, e)
z = Dummy('nz', nonzero=iz)
b = Dummy('f', finite=ib)
e = Mul(b, z, evaluate=False)
test(z, b, e)
# real is True
def test(z, b, e):
if z.is_zero and not b.is_finite:
assert e.is_extended_real is None
else:
assert e.is_extended_real is True
for iz, ib in product(*[[True, False, None]]*2):
z = Dummy('z', nonzero=iz, extended_real=True)
b = Dummy('b', finite=ib, extended_real=True)
e = Mul(z, b, evaluate=False)
test(z, b, e)
z = Dummy('z', nonzero=iz, extended_real=True)
b = Dummy('b', finite=ib, extended_real=True)
e = Mul(b, z, evaluate=False)
test(z, b, e)
def test_Mul_with_zero_infinite():
zer = Dummy(zero=True)
inf = Dummy(finite=False)
e = Mul(zer, inf, evaluate=False)
assert e.is_extended_positive is None
assert e.is_hermitian is None
e = Mul(inf, zer, evaluate=False)
assert e.is_extended_positive is None
assert e.is_hermitian is None
def test_Mul_does_not_cancel_infinities():
a, b = symbols('a b')
assert ((zoo + 3*a)/(3*a + zoo)) is nan
assert ((b - oo)/(b - oo)) is nan
# issue 13904
expr = (1/(a+b) + 1/(a-b))/(1/(a+b) - 1/(a-b))
assert expr.subs(b, a) is nan
def test_Mul_does_not_distribute_infinity():
a, b = symbols('a b')
assert ((1 + I)*oo).is_Mul
assert ((a + b)*(-oo)).is_Mul
assert ((a + 1)*zoo).is_Mul
assert ((1 + I)*oo).is_finite is False
z = (1 + I)*oo
assert ((1 - I)*z).expand() is oo
def test_issue_8247_8354():
from sympy.functions.elementary.trigonometric import tan
z = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert z.is_positive is False # it's 0
z = S('''-2**(1/3)*(3*sqrt(93) + 29)**2 - 4*(3*sqrt(93) + 29)**(4/3) +
12*sqrt(93)*(3*sqrt(93) + 29)**(1/3) + 116*(3*sqrt(93) + 29)**(1/3) +
174*2**(1/3)*sqrt(93) + 1678*2**(1/3)''')
assert z.is_positive is False # it's 0
z = 2*(-3*tan(19*pi/90) + sqrt(3))*cos(11*pi/90)*cos(19*pi/90) - \
sqrt(3)*(-3 + 4*cos(19*pi/90)**2)
assert z.is_positive is not True # it's zero and it shouldn't hang
z = S('''9*(3*sqrt(93) + 29)**(2/3)*((3*sqrt(93) +
29)**(1/3)*(-2**(2/3)*(3*sqrt(93) + 29)**(1/3) - 2) - 2*2**(1/3))**3 +
72*(3*sqrt(93) + 29)**(2/3)*(81*sqrt(93) + 783) + (162*sqrt(93) +
1566)*((3*sqrt(93) + 29)**(1/3)*(-2**(2/3)*(3*sqrt(93) + 29)**(1/3) -
2) - 2*2**(1/3))**2''')
assert z.is_positive is False # it's 0 (and a single _mexpand isn't enough)
def test_Add_is_zero():
x, y = symbols('x y', zero=True)
assert (x + y).is_zero
# Issue 15873
e = -2*I + (1 + I)**2
assert e.is_zero is None
def test_issue_14392():
assert (sin(zoo)**2).as_real_imag() == (nan, nan)
def test_divmod():
assert divmod(x, y) == (x//y, x % y)
assert divmod(x, 3) == (x//3, x % 3)
assert divmod(3, x) == (3//x, 3 % x)
def test__neg__():
assert -(x*y) == -x*y
assert -(-x*y) == x*y
assert -(1.*x) == -1.*x
assert -(-1.*x) == 1.*x
assert -(2.*x) == -2.*x
assert -(-2.*x) == 2.*x
with distribute(False):
eq = -(x + y)
assert eq.is_Mul and eq.args == (-1, x + y)
def test_issue_18507():
assert Mul(zoo, zoo, 0) is nan
def test_issue_17130():
e = Add(b, -b, I, -I, evaluate=False)
assert e.is_zero is None # ideally this would be True
def test_issue_21034():
e = -I*log((re(asin(5)) + I*im(asin(5)))/sqrt(re(asin(5))**2 + im(asin(5))**2))/pi
assert e.round(2)
def test_issue_22021():
from sympy.calculus.accumulationbounds import AccumBounds
# these objects are special cases in Mul
from sympy.tensor.tensor import TensorIndexType, tensor_indices, tensor_heads
L = TensorIndexType("L")
i = tensor_indices("i", L)
A, B = tensor_heads("A B", [L])
e = A(i) + B(i)
assert -e == -1*e
e = zoo + x
assert -e == -1*e
a = AccumBounds(1, 2)
e = a + x
assert -e == -1*e
for args in permutations((zoo, a, x)):
e = Add(*args, evaluate=False)
assert -e == -1*e
assert 2*Add(1, x, x, evaluate=False) == 4*x + 2
def test_issue_22244():
assert -(zoo*x) == zoo*x
def test_issue_22453():
from sympy.utilities.iterables import cartes
e = Symbol('e', extended_positive=True)
for a, b in cartes(*[[oo, -oo, 3]]*2):
if a == b == 3:
continue
i = a + I*b
assert i**(1 + e) is S.ComplexInfinity
assert i**-e is S.Zero
assert unchanged(Pow, i, e)
assert 1/(oo + I*oo) is S.Zero
r, i = [Dummy(infinite=True, extended_real=True) for _ in range(2)]
assert 1/(r + I*i) is S.Zero
assert 1/(3 + I*i) is S.Zero
assert 1/(r + I*3) is S.Zero
def test_issue_22613():
assert (0**(x - 2)).as_content_primitive() == (1, 0**(x - 2))
assert (0**(x + 2)).as_content_primitive() == (1, 0**(x + 2))
|
22b1be333bff565080984fa1989547fddefb4ebb1fc9a1867c956653c145be56 | """Trait for implementing domain elements. """
from sympy.utilities import public
@public
class DomainElement:
"""
Represents an element of a domain.
Mix in this trait into a class whose instances should be recognized as
elements of a domain. Method ``parent()`` gives that domain.
"""
__slots__ = ()
def parent(self):
"""Get the domain associated with ``self``
Examples
========
>>> from sympy import ZZ, symbols
>>> x, y = symbols('x, y')
>>> K = ZZ[x,y]
>>> p = K(x)**2 + K(y)**2
>>> p
x**2 + y**2
>>> p.parent()
ZZ[x,y]
Notes
=====
This is used by :py:meth:`~.Domain.convert` to identify the domain
associated with a domain element.
"""
raise NotImplementedError("abstract method")
|
69e2d16da0076c09022960c543db4abea8a849bba6dede9d161d5bac7225f781 | """Implementation of :class:`Domain` class. """
from typing import Any, Optional, Type
from sympy.core.numbers import AlgebraicNumber
from sympy.core import Basic, sympify
from sympy.core.sorting import default_sort_key, ordered
from sympy.external.gmpy import HAS_GMPY
from sympy.polys.domains.domainelement import DomainElement
from sympy.polys.orderings import lex
from sympy.polys.polyerrors import UnificationFailed, CoercionFailed, DomainError
from sympy.polys.polyutils import _unify_gens, _not_a_coeff
from sympy.utilities import public
from sympy.utilities.iterables import is_sequence
@public
class Domain:
"""Superclass for all domains in the polys domains system.
See :ref:`polys-domainsintro` for an introductory explanation of the
domains system.
The :py:class:`~.Domain` class is an abstract base class for all of the
concrete domain types. There are many different :py:class:`~.Domain`
subclasses each of which has an associated ``dtype`` which is a class
representing the elements of the domain. The coefficients of a
:py:class:`~.Poly` are elements of a domain which must be a subclass of
:py:class:`~.Domain`.
Examples
========
The most common example domains are the integers :ref:`ZZ` and the
rationals :ref:`QQ`.
>>> from sympy import Poly, symbols, Domain
>>> x, y = symbols('x, y')
>>> p = Poly(x**2 + y)
>>> p
Poly(x**2 + y, x, y, domain='ZZ')
>>> p.domain
ZZ
>>> isinstance(p.domain, Domain)
True
>>> Poly(x**2 + y/2)
Poly(x**2 + 1/2*y, x, y, domain='QQ')
The domains can be used directly in which case the domain object e.g.
(:ref:`ZZ` or :ref:`QQ`) can be used as a constructor for elements of
``dtype``.
>>> from sympy import ZZ, QQ
>>> ZZ(2)
2
>>> ZZ.dtype # doctest: +SKIP
<class 'int'>
>>> type(ZZ(2)) # doctest: +SKIP
<class 'int'>
>>> QQ(1, 2)
1/2
>>> type(QQ(1, 2)) # doctest: +SKIP
<class 'sympy.polys.domains.pythonrational.PythonRational'>
The corresponding domain elements can be used with the arithmetic
operations ``+,-,*,**`` and depending on the domain some combination of
``/,//,%`` might be usable. For example in :ref:`ZZ` both ``//`` (floor
division) and ``%`` (modulo division) can be used but ``/`` (true
division) cannot. Since :ref:`QQ` is a :py:class:`~.Field` its elements
can be used with ``/`` but ``//`` and ``%`` should not be used. Some
domains have a :py:meth:`~.Domain.gcd` method.
>>> ZZ(2) + ZZ(3)
5
>>> ZZ(5) // ZZ(2)
2
>>> ZZ(5) % ZZ(2)
1
>>> QQ(1, 2) / QQ(2, 3)
3/4
>>> ZZ.gcd(ZZ(4), ZZ(2))
2
>>> QQ.gcd(QQ(2,7), QQ(5,3))
1/21
>>> ZZ.is_Field
False
>>> QQ.is_Field
True
There are also many other domains including:
1. :ref:`GF(p)` for finite fields of prime order.
2. :ref:`RR` for real (floating point) numbers.
3. :ref:`CC` for complex (floating point) numbers.
4. :ref:`QQ(a)` for algebraic number fields.
5. :ref:`K[x]` for polynomial rings.
6. :ref:`K(x)` for rational function fields.
7. :ref:`EX` for arbitrary expressions.
Each domain is represented by a domain object and also an implementation
class (``dtype``) for the elements of the domain. For example the
:ref:`K[x]` domains are represented by a domain object which is an
instance of :py:class:`~.PolynomialRing` and the elements are always
instances of :py:class:`~.PolyElement`. The implementation class
represents particular types of mathematical expressions in a way that is
more efficient than a normal SymPy expression which is of type
:py:class:`~.Expr`. The domain methods :py:meth:`~.Domain.from_sympy` and
:py:meth:`~.Domain.to_sympy` are used to convert from :py:class:`~.Expr`
to a domain element and vice versa.
>>> from sympy import Symbol, ZZ, Expr
>>> x = Symbol('x')
>>> K = ZZ[x] # polynomial ring domain
>>> K
ZZ[x]
>>> type(K) # class of the domain
<class 'sympy.polys.domains.polynomialring.PolynomialRing'>
>>> K.dtype # class of the elements
<class 'sympy.polys.rings.PolyElement'>
>>> p_expr = x**2 + 1 # Expr
>>> p_expr
x**2 + 1
>>> type(p_expr)
<class 'sympy.core.add.Add'>
>>> isinstance(p_expr, Expr)
True
>>> p_domain = K.from_sympy(p_expr)
>>> p_domain # domain element
x**2 + 1
>>> type(p_domain)
<class 'sympy.polys.rings.PolyElement'>
>>> K.to_sympy(p_domain) == p_expr
True
The :py:meth:`~.Domain.convert_from` method is used to convert domain
elements from one domain to another.
>>> from sympy import ZZ, QQ
>>> ez = ZZ(2)
>>> eq = QQ.convert_from(ez, ZZ)
>>> type(ez) # doctest: +SKIP
<class 'int'>
>>> type(eq) # doctest: +SKIP
<class 'sympy.polys.domains.pythonrational.PythonRational'>
Elements from different domains should not be mixed in arithmetic or other
operations: they should be converted to a common domain first. The domain
method :py:meth:`~.Domain.unify` is used to find a domain that can
represent all the elements of two given domains.
>>> from sympy import ZZ, QQ, symbols
>>> x, y = symbols('x, y')
>>> ZZ.unify(QQ)
QQ
>>> ZZ[x].unify(QQ)
QQ[x]
>>> ZZ[x].unify(QQ[y])
QQ[x,y]
If a domain is a :py:class:`~.Ring` then is might have an associated
:py:class:`~.Field` and vice versa. The :py:meth:`~.Domain.get_field` and
:py:meth:`~.Domain.get_ring` methods will find or create the associated
domain.
>>> from sympy import ZZ, QQ, Symbol
>>> x = Symbol('x')
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
>>> QQ.has_assoc_Ring
True
>>> QQ.get_ring()
ZZ
>>> K = QQ[x]
>>> K
QQ[x]
>>> K.get_field()
QQ(x)
See also
========
DomainElement: abstract base class for domain elements
construct_domain: construct a minimal domain for some expressions
"""
dtype = None # type: Optional[Type]
"""The type (class) of the elements of this :py:class:`~.Domain`:
>>> from sympy import ZZ, QQ, Symbol
>>> ZZ.dtype
<class 'int'>
>>> z = ZZ(2)
>>> z
2
>>> type(z)
<class 'int'>
>>> type(z) == ZZ.dtype
True
Every domain has an associated **dtype** ("datatype") which is the
class of the associated domain elements.
See also
========
of_type
"""
zero = None # type: Optional[Any]
"""The zero element of the :py:class:`~.Domain`:
>>> from sympy import QQ
>>> QQ.zero
0
>>> QQ.of_type(QQ.zero)
True
See also
========
of_type
one
"""
one = None # type: Optional[Any]
"""The one element of the :py:class:`~.Domain`:
>>> from sympy import QQ
>>> QQ.one
1
>>> QQ.of_type(QQ.one)
True
See also
========
of_type
zero
"""
is_Ring = False
"""Boolean flag indicating if the domain is a :py:class:`~.Ring`.
>>> from sympy import ZZ
>>> ZZ.is_Ring
True
Basically every :py:class:`~.Domain` represents a ring so this flag is
not that useful.
See also
========
is_PID
is_Field
get_ring
has_assoc_Ring
"""
is_Field = False
"""Boolean flag indicating if the domain is a :py:class:`~.Field`.
>>> from sympy import ZZ, QQ
>>> ZZ.is_Field
False
>>> QQ.is_Field
True
See also
========
is_PID
is_Ring
get_field
has_assoc_Field
"""
has_assoc_Ring = False
"""Boolean flag indicating if the domain has an associated
:py:class:`~.Ring`.
>>> from sympy import QQ
>>> QQ.has_assoc_Ring
True
>>> QQ.get_ring()
ZZ
See also
========
is_Field
get_ring
"""
has_assoc_Field = False
"""Boolean flag indicating if the domain has an associated
:py:class:`~.Field`.
>>> from sympy import ZZ
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
See also
========
is_Field
get_field
"""
is_FiniteField = is_FF = False
is_IntegerRing = is_ZZ = False
is_RationalField = is_QQ = False
is_GaussianRing = is_ZZ_I = False
is_GaussianField = is_QQ_I = False
is_RealField = is_RR = False
is_ComplexField = is_CC = False
is_AlgebraicField = is_Algebraic = False
is_PolynomialRing = is_Poly = False
is_FractionField = is_Frac = False
is_SymbolicDomain = is_EX = False
is_SymbolicRawDomain = is_EXRAW = False
is_FiniteExtension = False
is_Exact = True
is_Numerical = False
is_Simple = False
is_Composite = False
is_PID = False
"""Boolean flag indicating if the domain is a `principal ideal domain`_.
>>> from sympy import ZZ
>>> ZZ.has_assoc_Field
True
>>> ZZ.get_field()
QQ
.. _principal ideal domain: https://en.wikipedia.org/wiki/Principal_ideal_domain
See also
========
is_Field
get_field
"""
has_CharacteristicZero = False
rep = None # type: Optional[str]
alias = None # type: Optional[str]
def __init__(self):
raise NotImplementedError
def __str__(self):
return self.rep
def __repr__(self):
return str(self)
def __hash__(self):
return hash((self.__class__.__name__, self.dtype))
def new(self, *args):
return self.dtype(*args)
@property
def tp(self):
"""Alias for :py:attr:`~.Domain.dtype`"""
return self.dtype
def __call__(self, *args):
"""Construct an element of ``self`` domain from ``args``. """
return self.new(*args)
def normal(self, *args):
return self.dtype(*args)
def convert_from(self, element, base):
"""Convert ``element`` to ``self.dtype`` given the base domain. """
if base.alias is not None:
method = "from_" + base.alias
else:
method = "from_" + base.__class__.__name__
_convert = getattr(self, method)
if _convert is not None:
result = _convert(element, base)
if result is not None:
return result
raise CoercionFailed("Cannot convert %s of type %s from %s to %s" % (element, type(element), base, self))
def convert(self, element, base=None):
"""Convert ``element`` to ``self.dtype``. """
if base is not None:
if _not_a_coeff(element):
raise CoercionFailed('%s is not in any domain' % element)
return self.convert_from(element, base)
if self.of_type(element):
return element
if _not_a_coeff(element):
raise CoercionFailed('%s is not in any domain' % element)
from sympy.polys.domains import ZZ, QQ, RealField, ComplexField
if ZZ.of_type(element):
return self.convert_from(element, ZZ)
if isinstance(element, int):
return self.convert_from(ZZ(element), ZZ)
if HAS_GMPY:
integers = ZZ
if isinstance(element, integers.tp):
return self.convert_from(element, integers)
rationals = QQ
if isinstance(element, rationals.tp):
return self.convert_from(element, rationals)
if isinstance(element, float):
parent = RealField(tol=False)
return self.convert_from(parent(element), parent)
if isinstance(element, complex):
parent = ComplexField(tol=False)
return self.convert_from(parent(element), parent)
if isinstance(element, DomainElement):
return self.convert_from(element, element.parent())
# TODO: implement this in from_ methods
if self.is_Numerical and getattr(element, 'is_ground', False):
return self.convert(element.LC())
if isinstance(element, Basic):
try:
return self.from_sympy(element)
except (TypeError, ValueError):
pass
else: # TODO: remove this branch
if not is_sequence(element):
try:
element = sympify(element, strict=True)
if isinstance(element, Basic):
return self.from_sympy(element)
except (TypeError, ValueError):
pass
raise CoercionFailed("Cannot convert %s of type %s to %s" % (element, type(element), self))
def of_type(self, element):
"""Check if ``a`` is of type ``dtype``. """
return isinstance(element, self.tp) # XXX: this isn't correct, e.g. PolyElement
def __contains__(self, a):
"""Check if ``a`` belongs to this domain. """
try:
if _not_a_coeff(a):
raise CoercionFailed
self.convert(a) # this might raise, too
except CoercionFailed:
return False
return True
def to_sympy(self, a):
"""Convert domain element *a* to a SymPy expression (Expr).
Explanation
===========
Convert a :py:class:`~.Domain` element *a* to :py:class:`~.Expr`. Most
public SymPy functions work with objects of type :py:class:`~.Expr`.
The elements of a :py:class:`~.Domain` have a different internal
representation. It is not possible to mix domain elements with
:py:class:`~.Expr` so each domain has :py:meth:`~.Domain.to_sympy` and
:py:meth:`~.Domain.from_sympy` methods to convert its domain elements
to and from :py:class:`~.Expr`.
Parameters
==========
a: domain element
An element of this :py:class:`~.Domain`.
Returns
=======
expr: Expr
A normal SymPy expression of type :py:class:`~.Expr`.
Examples
========
Construct an element of the :ref:`QQ` domain and then convert it to
:py:class:`~.Expr`.
>>> from sympy import QQ, Expr
>>> q_domain = QQ(2)
>>> q_domain
2
>>> q_expr = QQ.to_sympy(q_domain)
>>> q_expr
2
Although the printed forms look similar these objects are not of the
same type.
>>> isinstance(q_domain, Expr)
False
>>> isinstance(q_expr, Expr)
True
Construct an element of :ref:`K[x]` and convert to
:py:class:`~.Expr`.
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> K = QQ[x]
>>> x_domain = K.gens[0] # generator x as a domain element
>>> p_domain = x_domain**2/3 + 1
>>> p_domain
1/3*x**2 + 1
>>> p_expr = K.to_sympy(p_domain)
>>> p_expr
x**2/3 + 1
The :py:meth:`~.Domain.from_sympy` method is used for the opposite
conversion from a normal SymPy expression to a domain element.
>>> p_domain == p_expr
False
>>> K.from_sympy(p_expr) == p_domain
True
>>> K.to_sympy(p_domain) == p_expr
True
>>> K.from_sympy(K.to_sympy(p_domain)) == p_domain
True
>>> K.to_sympy(K.from_sympy(p_expr)) == p_expr
True
The :py:meth:`~.Domain.from_sympy` method makes it easier to construct
domain elements interactively.
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> K = QQ[x]
>>> K.from_sympy(x**2/3 + 1)
1/3*x**2 + 1
See also
========
from_sympy
convert_from
"""
raise NotImplementedError
def from_sympy(self, a):
"""Convert a SymPy expression to an element of this domain.
Explanation
===========
See :py:meth:`~.Domain.to_sympy` for explanation and examples.
Parameters
==========
expr: Expr
A normal SymPy expression of type :py:class:`~.Expr`.
Returns
=======
a: domain element
An element of this :py:class:`~.Domain`.
See also
========
to_sympy
convert_from
"""
raise NotImplementedError
def sum(self, args):
return sum(args)
def from_FF(K1, a, K0):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return None
def from_FF_python(K1, a, K0):
"""Convert ``ModularInteger(int)`` to ``dtype``. """
return None
def from_ZZ_python(K1, a, K0):
"""Convert a Python ``int`` object to ``dtype``. """
return None
def from_QQ_python(K1, a, K0):
"""Convert a Python ``Fraction`` object to ``dtype``. """
return None
def from_FF_gmpy(K1, a, K0):
"""Convert ``ModularInteger(mpz)`` to ``dtype``. """
return None
def from_ZZ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpz`` object to ``dtype``. """
return None
def from_QQ_gmpy(K1, a, K0):
"""Convert a GMPY ``mpq`` object to ``dtype``. """
return None
def from_RealField(K1, a, K0):
"""Convert a real element object to ``dtype``. """
return None
def from_ComplexField(K1, a, K0):
"""Convert a complex element to ``dtype``. """
return None
def from_AlgebraicField(K1, a, K0):
"""Convert an algebraic number to ``dtype``. """
return None
def from_PolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
if a.is_ground:
return K1.convert(a.LC, K0.dom)
def from_FractionField(K1, a, K0):
"""Convert a rational function to ``dtype``. """
return None
def from_MonogenicFiniteExtension(K1, a, K0):
"""Convert an ``ExtensionElement`` to ``dtype``. """
return K1.convert_from(a.rep, K0.ring)
def from_ExpressionDomain(K1, a, K0):
"""Convert a ``EX`` object to ``dtype``. """
return K1.from_sympy(a.ex)
def from_ExpressionRawDomain(K1, a, K0):
"""Convert a ``EX`` object to ``dtype``. """
return K1.from_sympy(a)
def from_GlobalPolynomialRing(K1, a, K0):
"""Convert a polynomial to ``dtype``. """
if a.degree() <= 0:
return K1.convert(a.LC(), K0.dom)
def from_GeneralizedPolynomialRing(K1, a, K0):
return K1.from_FractionField(a, K0)
def unify_with_symbols(K0, K1, symbols):
if (K0.is_Composite and (set(K0.symbols) & set(symbols))) or (K1.is_Composite and (set(K1.symbols) & set(symbols))):
raise UnificationFailed("Cannot unify %s with %s, given %s generators" % (K0, K1, tuple(symbols)))
return K0.unify(K1)
def unify(K0, K1, symbols=None):
"""
Construct a minimal domain that contains elements of ``K0`` and ``K1``.
Known domains (from smallest to largest):
- ``GF(p)``
- ``ZZ``
- ``QQ``
- ``RR(prec, tol)``
- ``CC(prec, tol)``
- ``ALG(a, b, c)``
- ``K[x, y, z]``
- ``K(x, y, z)``
- ``EX``
"""
if symbols is not None:
return K0.unify_with_symbols(K1, symbols)
if K0 == K1:
return K0
if K0.is_EXRAW:
return K0
if K1.is_EXRAW:
return K1
if K0.is_EX:
return K0
if K1.is_EX:
return K1
if K0.is_FiniteExtension or K1.is_FiniteExtension:
if K1.is_FiniteExtension:
K0, K1 = K1, K0
if K1.is_FiniteExtension:
# Unifying two extensions.
# Try to ensure that K0.unify(K1) == K1.unify(K0)
if list(ordered([K0.modulus, K1.modulus]))[1] == K0.modulus:
K0, K1 = K1, K0
return K1.set_domain(K0)
else:
# Drop the generator from other and unify with the base domain
K1 = K1.drop(K0.symbol)
K1 = K0.domain.unify(K1)
return K0.set_domain(K1)
if K0.is_Composite or K1.is_Composite:
K0_ground = K0.dom if K0.is_Composite else K0
K1_ground = K1.dom if K1.is_Composite else K1
K0_symbols = K0.symbols if K0.is_Composite else ()
K1_symbols = K1.symbols if K1.is_Composite else ()
domain = K0_ground.unify(K1_ground)
symbols = _unify_gens(K0_symbols, K1_symbols)
order = K0.order if K0.is_Composite else K1.order
if ((K0.is_FractionField and K1.is_PolynomialRing or
K1.is_FractionField and K0.is_PolynomialRing) and
(not K0_ground.is_Field or not K1_ground.is_Field) and domain.is_Field
and domain.has_assoc_Ring):
domain = domain.get_ring()
if K0.is_Composite and (not K1.is_Composite or K0.is_FractionField or K1.is_PolynomialRing):
cls = K0.__class__
else:
cls = K1.__class__
from sympy.polys.domains.old_polynomialring import GlobalPolynomialRing
if cls == GlobalPolynomialRing:
return cls(domain, symbols)
return cls(domain, symbols, order)
def mkinexact(cls, K0, K1):
prec = max(K0.precision, K1.precision)
tol = max(K0.tolerance, K1.tolerance)
return cls(prec=prec, tol=tol)
if K1.is_ComplexField:
K0, K1 = K1, K0
if K0.is_ComplexField:
if K1.is_ComplexField or K1.is_RealField:
return mkinexact(K0.__class__, K0, K1)
else:
return K0
if K1.is_RealField:
K0, K1 = K1, K0
if K0.is_RealField:
if K1.is_RealField:
return mkinexact(K0.__class__, K0, K1)
elif K1.is_GaussianRing or K1.is_GaussianField:
from sympy.polys.domains.complexfield import ComplexField
return ComplexField(prec=K0.precision, tol=K0.tolerance)
else:
return K0
if K1.is_AlgebraicField:
K0, K1 = K1, K0
if K0.is_AlgebraicField:
if K1.is_GaussianRing:
K1 = K1.get_field()
if K1.is_GaussianField:
K1 = K1.as_AlgebraicField()
if K1.is_AlgebraicField:
return K0.__class__(K0.dom.unify(K1.dom), *_unify_gens(K0.orig_ext, K1.orig_ext))
else:
return K0
if K0.is_GaussianField:
return K0
if K1.is_GaussianField:
return K1
if K0.is_GaussianRing:
if K1.is_RationalField:
K0 = K0.get_field()
return K0
if K1.is_GaussianRing:
if K0.is_RationalField:
K1 = K1.get_field()
return K1
if K0.is_RationalField:
return K0
if K1.is_RationalField:
return K1
if K0.is_IntegerRing:
return K0
if K1.is_IntegerRing:
return K1
if K0.is_FiniteField and K1.is_FiniteField:
return K0.__class__(max(K0.mod, K1.mod, key=default_sort_key))
from sympy.polys.domains import EX
return EX
def __eq__(self, other):
"""Returns ``True`` if two domains are equivalent. """
return isinstance(other, Domain) and self.dtype == other.dtype
def __ne__(self, other):
"""Returns ``False`` if two domains are equivalent. """
return not self == other
def map(self, seq):
"""Rersively apply ``self`` to all elements of ``seq``. """
result = []
for elt in seq:
if isinstance(elt, list):
result.append(self.map(elt))
else:
result.append(self(elt))
return result
def get_ring(self):
"""Returns a ring associated with ``self``. """
raise DomainError('there is no ring associated with %s' % self)
def get_field(self):
"""Returns a field associated with ``self``. """
raise DomainError('there is no field associated with %s' % self)
def get_exact(self):
"""Returns an exact domain associated with ``self``. """
return self
def __getitem__(self, symbols):
"""The mathematical way to make a polynomial ring. """
if hasattr(symbols, '__iter__'):
return self.poly_ring(*symbols)
else:
return self.poly_ring(symbols)
def poly_ring(self, *symbols, order=lex):
"""Returns a polynomial ring, i.e. `K[X]`. """
from sympy.polys.domains.polynomialring import PolynomialRing
return PolynomialRing(self, symbols, order)
def frac_field(self, *symbols, order=lex):
"""Returns a fraction field, i.e. `K(X)`. """
from sympy.polys.domains.fractionfield import FractionField
return FractionField(self, symbols, order)
def old_poly_ring(self, *symbols, **kwargs):
"""Returns a polynomial ring, i.e. `K[X]`. """
from sympy.polys.domains.old_polynomialring import PolynomialRing
return PolynomialRing(self, *symbols, **kwargs)
def old_frac_field(self, *symbols, **kwargs):
"""Returns a fraction field, i.e. `K(X)`. """
from sympy.polys.domains.old_fractionfield import FractionField
return FractionField(self, *symbols, **kwargs)
def algebraic_field(self, *extension):
r"""Returns an algebraic field, i.e. `K(\alpha, \ldots)`. """
raise DomainError("Cannot create algebraic field over %s" % self)
def alg_field_from_poly(self, poly, alias=None, root_index=-1):
r"""
Convenience method to construct an algebraic extension on a root of a
polynomial, chosen by root index.
Parameters
==========
poly : :py:class:`~.Poly`
The polynomial whose root generates the extension.
alias : str, optional (default=None)
Symbol name for the generator of the extension.
E.g. "alpha" or "theta".
root_index : int, optional (default=-1)
Specifies which root of the polynomial is desired. The ordering is
as defined by the :py:class:`~.ComplexRootOf` class. The default of
``-1`` selects the most natural choice in the common cases of
quadratic and cyclotomic fields (the square root on the positive
real or imaginary axis, resp. $\mathrm{e}^{2\pi i/n}$).
Examples
========
>>> from sympy import QQ, Poly
>>> from sympy.abc import x
>>> f = Poly(x**2 - 2)
>>> K = QQ.alg_field_from_poly(f)
>>> K.ext.minpoly == f
True
>>> g = Poly(8*x**3 - 6*x - 1)
>>> L = QQ.alg_field_from_poly(g, "alpha")
>>> L.ext.minpoly == g
True
>>> L.to_sympy(L([1, 1, 1]))
alpha**2 + alpha + 1
"""
from sympy.polys.rootoftools import CRootOf
root = CRootOf(poly, root_index)
alpha = AlgebraicNumber(root, alias=alias)
return self.algebraic_field(alpha)
def cyclotomic_field(self, n, ss=False, alias="zeta", gen=None, root_index=-1):
r"""
Convenience method to construct a cyclotomic field.
Parameters
==========
n : int
Construct the nth cyclotomic field.
ss : boolean, optional (default=False)
If True, append *n* as a subscript on the alias string.
alias : str, optional (default="zeta")
Symbol name for the generator.
gen : :py:class:`~.Symbol`, optional (default=None)
Desired variable for the cyclotomic polynomial that defines the
field. If ``None``, a dummy variable will be used.
root_index : int, optional (default=-1)
Specifies which root of the polynomial is desired. The ordering is
as defined by the :py:class:`~.ComplexRootOf` class. The default of
``-1`` selects the root $\mathrm{e}^{2\pi i/n}$.
Examples
========
>>> from sympy import QQ, latex
>>> K = QQ.cyclotomic_field(5)
>>> K.to_sympy(K([-1, 1]))
1 - zeta
>>> L = QQ.cyclotomic_field(7, True)
>>> a = L.to_sympy(L([-1, 1]))
>>> print(a)
1 - zeta7
>>> print(latex(a))
1 - \zeta_{7}
"""
from sympy.polys.specialpolys import cyclotomic_poly
if ss:
alias += str(n)
return self.alg_field_from_poly(cyclotomic_poly(n, gen), alias=alias,
root_index=root_index)
def inject(self, *symbols):
"""Inject generators into this domain. """
raise NotImplementedError
def drop(self, *symbols):
"""Drop generators from this domain. """
if self.is_Simple:
return self
raise NotImplementedError # pragma: no cover
def is_zero(self, a):
"""Returns True if ``a`` is zero. """
return not a
def is_one(self, a):
"""Returns True if ``a`` is one. """
return a == self.one
def is_positive(self, a):
"""Returns True if ``a`` is positive. """
return a > 0
def is_negative(self, a):
"""Returns True if ``a`` is negative. """
return a < 0
def is_nonpositive(self, a):
"""Returns True if ``a`` is non-positive. """
return a <= 0
def is_nonnegative(self, a):
"""Returns True if ``a`` is non-negative. """
return a >= 0
def canonical_unit(self, a):
if self.is_negative(a):
return -self.one
else:
return self.one
def abs(self, a):
"""Absolute value of ``a``, implies ``__abs__``. """
return abs(a)
def neg(self, a):
"""Returns ``a`` negated, implies ``__neg__``. """
return -a
def pos(self, a):
"""Returns ``a`` positive, implies ``__pos__``. """
return +a
def add(self, a, b):
"""Sum of ``a`` and ``b``, implies ``__add__``. """
return a + b
def sub(self, a, b):
"""Difference of ``a`` and ``b``, implies ``__sub__``. """
return a - b
def mul(self, a, b):
"""Product of ``a`` and ``b``, implies ``__mul__``. """
return a * b
def pow(self, a, b):
"""Raise ``a`` to power ``b``, implies ``__pow__``. """
return a ** b
def exquo(self, a, b):
"""Exact quotient of *a* and *b*. Analogue of ``a / b``.
Explanation
===========
This is essentially the same as ``a / b`` except that an error will be
raised if the division is inexact (if there is any remainder) and the
result will always be a domain element. When working in a
:py:class:`~.Domain` that is not a :py:class:`~.Field` (e.g. :ref:`ZZ`
or :ref:`K[x]`) ``exquo`` should be used instead of ``/``.
The key invariant is that if ``q = K.exquo(a, b)`` (and ``exquo`` does
not raise an exception) then ``a == b*q``.
Examples
========
We can use ``K.exquo`` instead of ``/`` for exact division.
>>> from sympy import ZZ
>>> ZZ.exquo(ZZ(4), ZZ(2))
2
>>> ZZ.exquo(ZZ(5), ZZ(2))
Traceback (most recent call last):
...
ExactQuotientFailed: 2 does not divide 5 in ZZ
Over a :py:class:`~.Field` such as :ref:`QQ`, division (with nonzero
divisor) is always exact so in that case ``/`` can be used instead of
:py:meth:`~.Domain.exquo`.
>>> from sympy import QQ
>>> QQ.exquo(QQ(5), QQ(2))
5/2
>>> QQ(5) / QQ(2)
5/2
Parameters
==========
a: domain element
The dividend
b: domain element
The divisor
Returns
=======
q: domain element
The exact quotient
Raises
======
ExactQuotientFailed: if exact division is not possible.
ZeroDivisionError: when the divisor is zero.
See also
========
quo: Analogue of ``a // b``
rem: Analogue of ``a % b``
div: Analogue of ``divmod(a, b)``
Notes
=====
Since the default :py:attr:`~.Domain.dtype` for :ref:`ZZ` is ``int``
(or ``mpz``) division as ``a / b`` should not be used as it would give
a ``float``.
>>> ZZ(4) / ZZ(2)
2.0
>>> ZZ(5) / ZZ(2)
2.5
Using ``/`` with :ref:`ZZ` will lead to incorrect results so
:py:meth:`~.Domain.exquo` should be used instead.
"""
raise NotImplementedError
def quo(self, a, b):
"""Quotient of *a* and *b*. Analogue of ``a // b``.
``K.quo(a, b)`` is equivalent to ``K.div(a, b)[0]``. See
:py:meth:`~.Domain.div` for more explanation.
See also
========
rem: Analogue of ``a % b``
div: Analogue of ``divmod(a, b)``
exquo: Analogue of ``a / b``
"""
raise NotImplementedError
def rem(self, a, b):
"""Modulo division of *a* and *b*. Analogue of ``a % b``.
``K.rem(a, b)`` is equivalent to ``K.div(a, b)[1]``. See
:py:meth:`~.Domain.div` for more explanation.
See also
========
quo: Analogue of ``a // b``
div: Analogue of ``divmod(a, b)``
exquo: Analogue of ``a / b``
"""
raise NotImplementedError
def div(self, a, b):
"""Quotient and remainder for *a* and *b*. Analogue of ``divmod(a, b)``
Explanation
===========
This is essentially the same as ``divmod(a, b)`` except that is more
consistent when working over some :py:class:`~.Field` domains such as
:ref:`QQ`. When working over an arbitrary :py:class:`~.Domain` the
:py:meth:`~.Domain.div` method should be used instead of ``divmod``.
The key invariant is that if ``q, r = K.div(a, b)`` then
``a == b*q + r``.
The result of ``K.div(a, b)`` is the same as the tuple
``(K.quo(a, b), K.rem(a, b))`` except that if both quotient and
remainder are needed then it is more efficient to use
:py:meth:`~.Domain.div`.
Examples
========
We can use ``K.div`` instead of ``divmod`` for floor division and
remainder.
>>> from sympy import ZZ, QQ
>>> ZZ.div(ZZ(5), ZZ(2))
(2, 1)
If ``K`` is a :py:class:`~.Field` then the division is always exact
with a remainder of :py:attr:`~.Domain.zero`.
>>> QQ.div(QQ(5), QQ(2))
(5/2, 0)
Parameters
==========
a: domain element
The dividend
b: domain element
The divisor
Returns
=======
(q, r): tuple of domain elements
The quotient and remainder
Raises
======
ZeroDivisionError: when the divisor is zero.
See also
========
quo: Analogue of ``a // b``
rem: Analogue of ``a % b``
exquo: Analogue of ``a / b``
Notes
=====
If ``gmpy`` is installed then the ``gmpy.mpq`` type will be used as
the :py:attr:`~.Domain.dtype` for :ref:`QQ`. The ``gmpy.mpq`` type
defines ``divmod`` in a way that is undesirable so
:py:meth:`~.Domain.div` should be used instead of ``divmod``.
>>> a = QQ(1)
>>> b = QQ(3, 2)
>>> a # doctest: +SKIP
mpq(1,1)
>>> b # doctest: +SKIP
mpq(3,2)
>>> divmod(a, b) # doctest: +SKIP
(mpz(0), mpq(1,1))
>>> QQ.div(a, b) # doctest: +SKIP
(mpq(2,3), mpq(0,1))
Using ``//`` or ``%`` with :ref:`QQ` will lead to incorrect results so
:py:meth:`~.Domain.div` should be used instead.
"""
raise NotImplementedError
def invert(self, a, b):
"""Returns inversion of ``a mod b``, implies something. """
raise NotImplementedError
def revert(self, a):
"""Returns ``a**(-1)`` if possible. """
raise NotImplementedError
def numer(self, a):
"""Returns numerator of ``a``. """
raise NotImplementedError
def denom(self, a):
"""Returns denominator of ``a``. """
raise NotImplementedError
def half_gcdex(self, a, b):
"""Half extended GCD of ``a`` and ``b``. """
s, t, h = self.gcdex(a, b)
return s, h
def gcdex(self, a, b):
"""Extended GCD of ``a`` and ``b``. """
raise NotImplementedError
def cofactors(self, a, b):
"""Returns GCD and cofactors of ``a`` and ``b``. """
gcd = self.gcd(a, b)
cfa = self.quo(a, gcd)
cfb = self.quo(b, gcd)
return gcd, cfa, cfb
def gcd(self, a, b):
"""Returns GCD of ``a`` and ``b``. """
raise NotImplementedError
def lcm(self, a, b):
"""Returns LCM of ``a`` and ``b``. """
raise NotImplementedError
def log(self, a, b):
"""Returns b-base logarithm of ``a``. """
raise NotImplementedError
def sqrt(self, a):
"""Returns square root of ``a``. """
raise NotImplementedError
def evalf(self, a, prec=None, **options):
"""Returns numerical approximation of ``a``. """
return self.to_sympy(a).evalf(prec, **options)
n = evalf
def real(self, a):
return a
def imag(self, a):
return self.zero
def almosteq(self, a, b, tolerance=None):
"""Check if ``a`` and ``b`` are almost equal. """
return a == b
def characteristic(self):
"""Return the characteristic of this domain. """
raise NotImplementedError('characteristic()')
__all__ = ['Domain']
|
04360cb7f4687be0cd59d4d627d9eb08f47dee3692c96c0656a2adb1bc35ffe2 | from sympy.polys.domains import QQ, EX, RR
from sympy.polys.rings import ring
from sympy.polys.ring_series import (_invert_monoms, rs_integrate,
rs_trunc, rs_mul, rs_square, rs_pow, _has_constant_term, rs_hadamard_exp,
rs_series_from_list, rs_exp, rs_log, rs_newton, rs_series_inversion,
rs_compose_add, rs_asin, rs_atan, rs_atanh, rs_tan, rs_cot, rs_sin, rs_cos,
rs_cos_sin, rs_sinh, rs_cosh, rs_tanh, _tan1, rs_fun, rs_nth_root,
rs_LambertW, rs_series_reversion, rs_is_puiseux, rs_series)
from sympy.testing.pytest import raises, slow
from sympy.core.symbol import symbols
from sympy.functions import (sin, cos, exp, tan, cot, atan, atanh,
tanh, log, sqrt)
from sympy.core.numbers import Rational
from sympy.core import expand, S
def is_close(a, b):
tol = 10**(-10)
assert abs(a - b) < tol
def test_ring_series1():
R, x = ring('x', QQ)
p = x**4 + 2*x**3 + 3*x + 4
assert _invert_monoms(p) == 4*x**4 + 3*x**3 + 2*x + 1
assert rs_hadamard_exp(p) == x**4/24 + x**3/3 + 3*x + 4
R, x = ring('x', QQ)
p = x**4 + 2*x**3 + 3*x + 4
assert rs_integrate(p, x) == x**5/5 + x**4/2 + 3*x**2/2 + 4*x
R, x, y = ring('x, y', QQ)
p = x**2*y**2 + x + 1
assert rs_integrate(p, x) == x**3*y**2/3 + x**2/2 + x
assert rs_integrate(p, y) == x**2*y**3/3 + x*y + y
def test_trunc():
R, x, y, t = ring('x, y, t', QQ)
p = (y + t*x)**4
p1 = rs_trunc(p, x, 3)
assert p1 == y**4 + 4*y**3*t*x + 6*y**2*t**2*x**2
def test_mul_trunc():
R, x, y, t = ring('x, y, t', QQ)
p = 1 + t*x + t*y
for i in range(2):
p = rs_mul(p, p, t, 3)
assert p == 6*x**2*t**2 + 12*x*y*t**2 + 6*y**2*t**2 + 4*x*t + 4*y*t + 1
p = 1 + t*x + t*y + t**2*x*y
p1 = rs_mul(p, p, t, 2)
assert p1 == 1 + 2*t*x + 2*t*y
R1, z = ring('z', QQ)
raises(ValueError, lambda: rs_mul(p, z, x, 2))
p1 = 2 + 2*x + 3*x**2
p2 = 3 + x**2
assert rs_mul(p1, p2, x, 4) == 2*x**3 + 11*x**2 + 6*x + 6
def test_square_trunc():
R, x, y, t = ring('x, y, t', QQ)
p = (1 + t*x + t*y)*2
p1 = rs_mul(p, p, x, 3)
p2 = rs_square(p, x, 3)
assert p1 == p2
p = 1 + x + x**2 + x**3
assert rs_square(p, x, 4) == 4*x**3 + 3*x**2 + 2*x + 1
def test_pow_trunc():
R, x, y, z = ring('x, y, z', QQ)
p0 = y + x*z
p = p0**16
for xx in (x, y, z):
p1 = rs_trunc(p, xx, 8)
p2 = rs_pow(p0, 16, xx, 8)
assert p1 == p2
p = 1 + x
p1 = rs_pow(p, 3, x, 2)
assert p1 == 1 + 3*x
assert rs_pow(p, 0, x, 2) == 1
assert rs_pow(p, -2, x, 2) == 1 - 2*x
p = x + y
assert rs_pow(p, 3, y, 3) == x**3 + 3*x**2*y + 3*x*y**2
assert rs_pow(1 + x, Rational(2, 3), x, 4) == 4*x**3/81 - x**2/9 + x*Rational(2, 3) + 1
def test_has_constant_term():
R, x, y, z = ring('x, y, z', QQ)
p = y + x*z
assert _has_constant_term(p, x)
p = x + x**4
assert not _has_constant_term(p, x)
p = 1 + x + x**4
assert _has_constant_term(p, x)
p = x + y + x*z
def test_inversion():
R, x = ring('x', QQ)
p = 2 + x + 2*x**2
n = 5
p1 = rs_series_inversion(p, x, n)
assert rs_trunc(p*p1, x, n) == 1
R, x, y = ring('x, y', QQ)
p = 2 + x + 2*x**2 + y*x + x**2*y
p1 = rs_series_inversion(p, x, n)
assert rs_trunc(p*p1, x, n) == 1
R, x, y = ring('x, y', QQ)
p = 1 + x + y
raises(NotImplementedError, lambda: rs_series_inversion(p, x, 4))
p = R.zero
raises(ZeroDivisionError, lambda: rs_series_inversion(p, x, 3))
def test_series_reversion():
R, x, y = ring('x, y', QQ)
p = rs_tan(x, x, 10)
assert rs_series_reversion(p, x, 8, y) == rs_atan(y, y, 8)
p = rs_sin(x, x, 10)
assert rs_series_reversion(p, x, 8, y) == 5*y**7/112 + 3*y**5/40 + \
y**3/6 + y
def test_series_from_list():
R, x = ring('x', QQ)
p = 1 + 2*x + x**2 + 3*x**3
c = [1, 2, 0, 4, 4]
r = rs_series_from_list(p, c, x, 5)
pc = R.from_list(list(reversed(c)))
r1 = rs_trunc(pc.compose(x, p), x, 5)
assert r == r1
R, x, y = ring('x, y', QQ)
c = [1, 3, 5, 7]
p1 = rs_series_from_list(x + y, c, x, 3, concur=0)
p2 = rs_trunc((1 + 3*(x+y) + 5*(x+y)**2 + 7*(x+y)**3), x, 3)
assert p1 == p2
R, x = ring('x', QQ)
h = 25
p = rs_exp(x, x, h) - 1
p1 = rs_series_from_list(p, c, x, h)
p2 = 0
for i, cx in enumerate(c):
p2 += cx*rs_pow(p, i, x, h)
assert p1 == p2
def test_log():
R, x = ring('x', QQ)
p = 1 + x
p1 = rs_log(p, x, 4)/x**2
assert p1 == Rational(1, 3)*x - S.Half + x**(-1)
p = 1 + x +2*x**2/3
p1 = rs_log(p, x, 9)
assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \
7*x**4/36 - x**3/3 + x**2/6 + x
p2 = rs_series_inversion(p, x, 9)
p3 = rs_log(p2, x, 9)
assert p3 == -p1
R, x, y = ring('x, y', QQ)
p = 1 + x + 2*y*x**2
p1 = rs_log(p, x, 6)
assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y -
x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x)
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \
- EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a))
assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \
EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \
EX(1/a)*x + EX(log(a))
p = x + x**2 + 3
assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + Rational(19281291595, 9920232))
def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[exp(a), a])
assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
exp(a)*x**2/2 + exp(a)*x + exp(a)
assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)
R, x, y = ring('x, y', EX)
assert rs_exp(x + a, x, 5) == EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
EX(exp(a))*x + EX(exp(a))
def test_newton():
R, x = ring('x', QQ)
p = x**2 - 2
r = rs_newton(p, x, 4)
assert r == 8*x**4 + 4*x**2 + 2
def test_compose_add():
R, x = ring('x', QQ)
p1 = x**3 - 1
p2 = x**2 - 2
assert rs_compose_add(p1, p2) == x**6 - 6*x**4 - 2*x**3 + 12*x**2 - 12*x - 7
def test_fun():
R, x, y = ring('x, y', QQ)
p = x*y + x**2*y**3 + x**5*y
assert rs_fun(p, rs_tan, x, 10) == rs_tan(p, x, 10)
assert rs_fun(p, _tan1, x, 10) == _tan1(p, x, 10)
def test_nth_root():
R, x, y = ring('x, y', QQ)
assert rs_nth_root(1 + x**2*y, 4, x, 10) == -77*x**8*y**4/2048 + \
7*x**6*y**3/128 - 3*x**4*y**2/32 + x**2*y/4 + 1
assert rs_nth_root(1 + x*y + x**2*y**3, 3, x, 5) == -x**4*y**6/9 + \
5*x**4*y**5/27 - 10*x**4*y**4/243 - 2*x**3*y**4/9 + 5*x**3*y**3/81 + \
x**2*y**3/3 - x**2*y**2/9 + x*y/3 + 1
assert rs_nth_root(8*x, 3, x, 3) == 2*x**QQ(1, 3)
assert rs_nth_root(8*x + x**2 + x**3, 3, x, 3) == x**QQ(4,3)/12 + 2*x**QQ(1,3)
r = rs_nth_root(8*x + x**2*y + x**3, 3, x, 4)
assert r == -x**QQ(7,3)*y**2/288 + x**QQ(7,3)/12 + x**QQ(4,3)*y/12 + 2*x**QQ(1,3)
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_nth_root(x + a, 3, x, 4) == EX(5/(81*a**QQ(8, 3)))*x**3 - \
EX(1/(9*a**QQ(5, 3)))*x**2 + EX(1/(3*a**QQ(2, 3)))*x + EX(a**QQ(1, 3))
assert rs_nth_root(x**QQ(2, 3) + x**2*y + 5, 2, x, 3) == -EX(sqrt(5)/100)*\
x**QQ(8, 3)*y - EX(sqrt(5)/16000)*x**QQ(8, 3) + EX(sqrt(5)/10)*x**2*y + \
EX(sqrt(5)/2000)*x**2 - EX(sqrt(5)/200)*x**QQ(4, 3) + \
EX(sqrt(5)/10)*x**QQ(2, 3) + EX(sqrt(5))
def test_atan():
R, x, y = ring('x, y', QQ)
assert rs_atan(x, x, 9) == -x**7/7 + x**5/5 - x**3/3 + x
assert rs_atan(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 - x**8*y**9 + \
2*x**7*y**9 - x**7*y**7/7 - x**6*y**9/3 + x**6*y**7 - x**5*y**7 + \
x**5*y**5/5 - x**4*y**5 - x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_atan(x + a, x, 5) == -EX((a**3 - a)/(a**8 + 4*a**6 + 6*a**4 + \
4*a**2 + 1))*x**4 + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + \
9*a**2 + 3))*x**3 - EX(a/(a**4 + 2*a**2 + 1))*x**2 + \
EX(1/(a**2 + 1))*x + EX(atan(a))
assert rs_atan(x + x**2*y + a, x, 4) == -EX(2*a/(a**4 + 2*a**2 + 1)) \
*x**3*y + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + 9*a**2 + 3))*x**3 + \
EX(1/(a**2 + 1))*x**2*y - EX(a/(a**4 + 2*a**2 + 1))*x**2 + EX(1/(a**2 \
+ 1))*x + EX(atan(a))
def test_asin():
R, x, y = ring('x, y', QQ)
assert rs_asin(x + x*y, x, 5) == x**3*y**3/6 + x**3*y**2/2 + x**3*y/2 + \
x**3/6 + x*y + x
assert rs_asin(x*y + x**2*y**3, x, 6) == x**5*y**7/2 + 3*x**5*y**5/40 + \
x**4*y**5/2 + x**3*y**3/6 + x**2*y**3 + x*y
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9)/x**5 == \
Rational(17, 315)*x**2 + Rational(2, 15) + Rational(1, 3)*x**(-2) + x**(-4)
assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 +
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + Rational(1, 3))*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + Rational(4, 3)*tan(a)**2 + Rational(1, 3))*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
668083460499)
def test_cot():
R, x, y = ring('x, y', QQ)
assert rs_cot(x**6 + x**7, x, 8) == x**(-6) - x**(-5) + x**(-4) - \
x**(-3) + x**(-2) - x**(-1) + 1 - x + x**2 - x**3 + x**4 - x**5 + \
2*x**6/3 - 4*x**7/3
assert rs_cot(x + x**2*y, x, 5) == -x**4*y**5 - x**4*y/15 + x**3*y**4 - \
x**3/45 - x**2*y**3 - x**2*y/3 + x*y**2 - x/3 - y + x**(-1)
def test_sin():
R, x, y = ring('x, y', QQ)
assert rs_sin(x, x, 9)/x**5 == \
Rational(-1, 5040)*x**2 + Rational(1, 120) - Rational(1, 6)*x**(-2) + x**(-4)
assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \
x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \
x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \
x**3*y**3/6 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
sin(a)*x**2/2 + cos(a)*x + sin(a)
assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)
R, x, y = ring('x, y', EX)
assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
EX(cos(a))*x + EX(sin(a))
def test_cos():
R, x, y = ring('x, y', QQ)
assert rs_cos(x, x, 9)/x**5 == \
Rational(1, 40320)*x**3 - Rational(1, 720)*x + Rational(1, 24)*x**(-1) - S.Half*x**(-3) + x**(-5)
assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \
x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \
x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \
x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
cos(a)*x**2/2 - sin(a)*x + cos(a)
assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)
R, x, y = ring('x, y', EX)
assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
EX(sin(a))*x + EX(cos(a))
def test_cos_sin():
R, x, y = ring('x, y', QQ)
cos, sin = rs_cos_sin(x, x, 9)
assert cos == rs_cos(x, x, 9)
assert sin == rs_sin(x, x, 9)
cos, sin = rs_cos_sin(x + x*y, x, 5)
assert cos == rs_cos(x + x*y, x, 5)
assert sin == rs_sin(x + x*y, x, 5)
def test_atanh():
R, x, y = ring('x, y', QQ)
assert rs_atanh(x, x, 9)/x**5 == Rational(1, 7)*x**2 + Rational(1, 5) + Rational(1, 3)*x**(-2) + x**(-4)
assert rs_atanh(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 + x**8*y**9 + \
2*x**7*y**9 + x**7*y**7/7 + x**6*y**9/3 + x**6*y**7 + x**5*y**7 + \
x**5*y**5/5 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_atanh(x + a, x, 5) == EX((a**3 + a)/(a**8 - 4*a**6 + 6*a**4 - \
4*a**2 + 1))*x**4 - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + \
9*a**2 - 3))*x**3 + EX(a/(a**4 - 2*a**2 + 1))*x**2 - EX(1/(a**2 - \
1))*x + EX(atanh(a))
assert rs_atanh(x + x**2*y + a, x, 4) == EX(2*a/(a**4 - 2*a**2 + \
1))*x**3*y - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + 9*a**2 - 3))*x**3 - \
EX(1/(a**2 - 1))*x**2*y + EX(a/(a**4 - 2*a**2 + 1))*x**2 - \
EX(1/(a**2 - 1))*x + EX(atanh(a))
p = x + x**2 + 5
assert rs_atanh(p, x, 10).compose(x, 10) == EX(Rational(-733442653682135, 5079158784) \
+ atanh(5))
def test_sinh():
R, x, y = ring('x, y', QQ)
assert rs_sinh(x, x, 9)/x**5 == Rational(1, 5040)*x**2 + Rational(1, 120) + Rational(1, 6)*x**(-2) + x**(-4)
assert rs_sinh(x*y + x**2*y**3, x, 9) == x**8*y**11/12 + \
x**8*y**9/720 + x**7*y**9/12 + x**7*y**7/5040 + x**6*y**9/6 + \
x**6*y**7/24 + x**5*y**7/2 + x**5*y**5/120 + x**4*y**5/2 + \
x**3*y**3/6 + x**2*y**3 + x*y
def test_cosh():
R, x, y = ring('x, y', QQ)
assert rs_cosh(x, x, 9)/x**5 == Rational(1, 40320)*x**3 + Rational(1, 720)*x + Rational(1, 24)*x**(-1) + \
S.Half*x**(-3) + x**(-5)
assert rs_cosh(x*y + x**2*y**3, x, 9) == x**8*y**12/24 + \
x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 + \
x**7*y**8/120 + x**6*y**8/4 + x**6*y**6/720 + x**5*y**6/6 + \
x**4*y**6/2 + x**4*y**4/24 + x**3*y**4 + x**2*y**2/2 + 1
def test_tanh():
R, x, y = ring('x, y', QQ)
assert rs_tanh(x, x, 9)/x**5 == Rational(-17, 315)*x**2 + Rational(2, 15) - Rational(1, 3)*x**(-2) + x**(-4)
assert rs_tanh(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 - \
17*x**8*y**9/45 + 4*x**7*y**9/3 - 17*x**7*y**7/315 - x**6*y**9/3 + \
2*x**6*y**7/3 - x**5*y**7 + 2*x**5*y**5/15 - x**4*y**5 - \
x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_tanh(x + a, x, 5) == EX(tanh(a)**5 - 5*tanh(a)**3/3 +
2*tanh(a)/3)*x**4 + EX(-tanh(a)**4 + 4*tanh(a)**2/3 - QQ(1, 3))*x**3 + \
EX(tanh(a)**3 - tanh(a))*x**2 + EX(-tanh(a)**2 + 1)*x + EX(tanh(a))
p = rs_tanh(x + x**2*y + a, x, 4)
assert (p.compose(x, 10)).compose(y, 5) == EX(-1000*tanh(a)**4 + \
10100*tanh(a)**3 + 2470*tanh(a)**2/3 - 10099*tanh(a) + QQ(530, 3))
def test_RR():
rs_funcs = [rs_sin, rs_cos, rs_tan, rs_cot, rs_atan, rs_tanh]
sympy_funcs = [sin, cos, tan, cot, atan, tanh]
R, x, y = ring('x, y', RR)
a = symbols('a')
for rs_func, sympy_func in zip(rs_funcs, sympy_funcs):
p = rs_func(2 + x, x, 5).compose(x, 5)
q = sympy_func(2 + a).series(a, 0, 5).removeO()
is_close(p.as_expr(), q.subs(a, 5).n())
p = rs_nth_root(2 + x, 5, x, 5).compose(x, 5)
q = ((2 + a)**QQ(1, 5)).series(a, 0, 5).removeO()
is_close(p.as_expr(), q.subs(a, 5).n())
def test_is_regular():
R, x, y = ring('x, y', QQ)
p = 1 + 2*x + x**2 + 3*x**3
assert not rs_is_puiseux(p, x)
p = x + x**QQ(1,5)*y
assert rs_is_puiseux(p, x)
assert not rs_is_puiseux(p, y)
p = x + x**2*y**QQ(1,5)*y
assert not rs_is_puiseux(p, x)
def test_puiseux():
R, x, y = ring('x, y', QQ)
p = x**QQ(2,5) + x**QQ(2,3) + x
r = rs_series_inversion(p, x, 1)
r1 = -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + x**QQ(2,3) + \
2*x**QQ(7,15) - x**QQ(2,5) - x**QQ(1,5) + x**QQ(2,15) - x**QQ(-2,15) \
+ x**QQ(-2,5)
assert r == r1
r = rs_nth_root(1 + p, 3, x, 1)
assert r == -x**QQ(4,5)/9 + x**QQ(2,3)/3 + x**QQ(2,5)/3 + 1
r = rs_log(1 + p, x, 1)
assert r == -x**QQ(4,5)/2 + x**QQ(2,3) + x**QQ(2,5)
r = rs_LambertW(p, x, 1)
assert r == -x**QQ(4,5) + x**QQ(2,3) + x**QQ(2,5)
p1 = x + x**QQ(1,5)*y
r = rs_exp(p1, x, 1)
assert r == x**QQ(4,5)*y**4/24 + x**QQ(3,5)*y**3/6 + x**QQ(2,5)*y**2/2 + \
x**QQ(1,5)*y + 1
r = rs_atan(p, x, 2)
assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
x + x**QQ(2,3) + x**QQ(2,5)
r = rs_atan(p1, x, 2)
assert r == x**QQ(9,5)*y**9/9 + x**QQ(9,5)*y**4 - x**QQ(7,5)*y**7/7 - \
x**QQ(7,5)*y**2 + x*y**5/5 + x - x**QQ(3,5)*y**3/3 + x**QQ(1,5)*y
r = rs_asin(p, x, 2)
assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_cot(p, x, 1)
assert r == -x**QQ(14,15) + x**QQ(4,5) - 3*x**QQ(11,15) + \
2*x**QQ(2,3)/3 + 2*x**QQ(7,15) - 4*x**QQ(2,5)/3 - x**QQ(1,5) + \
x**QQ(2,15) - x**QQ(-2,15) + x**QQ(-2,5)
r = rs_cos_sin(p, x, 2)
assert r[0] == x**QQ(28,15)/6 - x**QQ(5,3) + x**QQ(8,5)/24 - x**QQ(7,5) - \
x**QQ(4,3)/2 - x**QQ(16,15) - x**QQ(4,5)/2 + 1
assert r[1] == -x**QQ(9,5)/2 - x**QQ(26,15)/2 - x**QQ(22,15)/2 - \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_atanh(p, x, 2)
assert r == x**QQ(9,5) + x**QQ(26,15) + x**QQ(22,15) + x**QQ(6,5)/3 + x + \
x**QQ(2,3) + x**QQ(2,5)
r = rs_sinh(p, x, 2)
assert r == x**QQ(9,5)/2 + x**QQ(26,15)/2 + x**QQ(22,15)/2 + \
x**QQ(6,5)/6 + x + x**QQ(2,3) + x**QQ(2,5)
r = rs_cosh(p, x, 2)
assert r == x**QQ(28,15)/6 + x**QQ(5,3) + x**QQ(8,5)/24 + x**QQ(7,5) + \
x**QQ(4,3)/2 + x**QQ(16,15) + x**QQ(4,5)/2 + 1
r = rs_tanh(p, x, 2)
assert r == -x**QQ(9,5) - x**QQ(26,15) - x**QQ(22,15) - x**QQ(6,5)/3 + \
x + x**QQ(2,3) + x**QQ(2,5)
def test1():
R, x = ring('x', QQ)
r = rs_sin(x, x, 15)*x**(-5)
assert r == x**8/6227020800 - x**6/39916800 + x**4/362880 - x**2/5040 + \
QQ(1,120) - x**-2/6 + x**-4
p = rs_sin(x, x, 10)
r = rs_nth_root(p, 2, x, 10)
assert r == -67*x**QQ(17,2)/29030400 - x**QQ(13,2)/24192 + \
x**QQ(9,2)/1440 - x**QQ(5,2)/12 + x**QQ(1,2)
p = rs_sin(x, x, 10)
r = rs_nth_root(p, 7, x, 10)
r = rs_pow(r, 5, x, 10)
assert r == -97*x**QQ(61,7)/124467840 - x**QQ(47,7)/16464 + \
11*x**QQ(33,7)/3528 - 5*x**QQ(19,7)/42 + x**QQ(5,7)
r = rs_exp(x**QQ(1,2), x, 10)
assert r == x**QQ(19,2)/121645100408832000 + x**9/6402373705728000 + \
x**QQ(17,2)/355687428096000 + x**8/20922789888000 + \
x**QQ(15,2)/1307674368000 + x**7/87178291200 + \
x**QQ(13,2)/6227020800 + x**6/479001600 + x**QQ(11,2)/39916800 + \
x**5/3628800 + x**QQ(9,2)/362880 + x**4/40320 + x**QQ(7,2)/5040 + \
x**3/720 + x**QQ(5,2)/120 + x**2/24 + x**QQ(3,2)/6 + x/2 + \
x**QQ(1,2) + 1
def test_puiseux2():
R, y = ring('y', QQ)
S, x = ring('x', R)
p = x + x**QQ(1,5)*y
r = rs_atan(p, x, 3)
assert r == (y**13/13 + y**8 + 2*y**3)*x**QQ(13,5) - (y**11/11 + y**6 +
y)*x**QQ(11,5) + (y**9/9 + y**4)*x**QQ(9,5) - (y**7/7 +
y**2)*x**QQ(7,5) + (y**5/5 + 1)*x - y**3*x**QQ(3,5)/3 + y*x**QQ(1,5)
@slow
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
cos(a)).series(a, 0, 5)).removeO()
assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
cos(a)).series(a, 0, 5)).removeO()
p = (sin(a) - a)*(cos(a**2) + a**4/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
6).removeO())
p = a**QQ(2,5) + a**QQ(2,3) + a
r = rs_series(tan(p), a, 2)
assert r.as_expr() == a**QQ(9,5) + a**QQ(26,15) + a**QQ(22,15) + a**QQ(6,5)/3 + \
a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(exp(p), a, 1)
assert r.as_expr() == a**QQ(4,5)/2 + a**QQ(2,3) + a**QQ(2,5) + 1
r = rs_series(sin(p), a, 2)
assert r.as_expr() == -a**QQ(9,5)/2 - a**QQ(26,15)/2 - a**QQ(22,15)/2 - \
a**QQ(6,5)/6 + a + a**QQ(2,3) + a**QQ(2,5)
r = rs_series(cos(p), a, 2)
assert r.as_expr() == a**QQ(28,15)/6 - a**QQ(5,3) + a**QQ(8,5)/24 - a**QQ(7,5) - \
a**QQ(4,3)/2 - a**QQ(16,15) - a**QQ(4,5)/2 + 1
assert rs_series(sin(a)/7, a, 5).as_expr() == (sin(a)/7).series(a, 0,
5).removeO()
assert rs_series(log(1 + x), x, 5).as_expr() == -x**4/4 + x**3/3 - \
x**2/2 + x
assert rs_series(log(1 + 4*x), x, 5).as_expr() == -64*x**4 + 64*x**3/3 - \
8*x**2 + 4*x
assert rs_series(log(1 + x + x**2), x, 10).as_expr() == -2*x**9/9 + \
x**8/8 + x**7/7 - x**6/3 + x**5/5 + x**4/4 - 2*x**3/3 + \
x**2/2 + x
assert rs_series(log(1 + x*a**2), x, 7).as_expr() == -x**6*a**12/6 + \
x**5*a**10/5 - x**4*a**8/4 + x**3*a**6/3 - \
x**2*a**4/2 + x*a**2
|
d580661094ed3d15a3e25dfd1cc9050fa7df914f21de922cee5cfe93ff571989 | """Tests for dense recursive polynomials' arithmetics. """
from sympy.polys.densebasic import (
dup_normal, dmp_normal,
)
from sympy.polys.densearith import (
dup_add_term, dmp_add_term,
dup_sub_term, dmp_sub_term,
dup_mul_term, dmp_mul_term,
dup_add_ground, dmp_add_ground,
dup_sub_ground, dmp_sub_ground,
dup_mul_ground, dmp_mul_ground,
dup_quo_ground, dmp_quo_ground,
dup_exquo_ground, dmp_exquo_ground,
dup_lshift, dup_rshift,
dup_abs, dmp_abs,
dup_neg, dmp_neg,
dup_add, dmp_add,
dup_sub, dmp_sub,
dup_mul, dmp_mul,
dup_sqr, dmp_sqr,
dup_pow, dmp_pow,
dup_add_mul, dmp_add_mul,
dup_sub_mul, dmp_sub_mul,
dup_pdiv, dup_prem, dup_pquo, dup_pexquo,
dmp_pdiv, dmp_prem, dmp_pquo, dmp_pexquo,
dup_rr_div, dmp_rr_div,
dup_ff_div, dmp_ff_div,
dup_div, dup_rem, dup_quo, dup_exquo,
dmp_div, dmp_rem, dmp_quo, dmp_exquo,
dup_max_norm, dmp_max_norm,
dup_l1_norm, dmp_l1_norm,
dup_l2_norm_squared, dmp_l2_norm_squared,
dup_expand, dmp_expand,
)
from sympy.polys.polyerrors import (
ExactQuotientFailed,
)
from sympy.polys.specialpolys import f_polys
from sympy.polys.domains import FF, ZZ, QQ
from sympy.testing.pytest import raises
f_0, f_1, f_2, f_3, f_4, f_5, f_6 = [ f.to_dense() for f in f_polys() ]
F_0 = dmp_mul_ground(dmp_normal(f_0, 2, QQ), QQ(1, 7), 2, QQ)
def test_dup_add_term():
f = dup_normal([], ZZ)
assert dup_add_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1], ZZ)
assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 0], ZZ)
assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 0, 0], ZZ)
f = dup_normal([1, 1, 1], ZZ)
assert dup_add_term(f, ZZ(1), 0, ZZ) == dup_normal([1, 1, 2], ZZ)
assert dup_add_term(f, ZZ(1), 1, ZZ) == dup_normal([1, 2, 1], ZZ)
assert dup_add_term(f, ZZ(1), 2, ZZ) == dup_normal([2, 1, 1], ZZ)
assert dup_add_term(f, ZZ(1), 3, ZZ) == dup_normal([1, 1, 1, 1], ZZ)
assert dup_add_term(f, ZZ(1), 4, ZZ) == dup_normal([1, 0, 1, 1, 1], ZZ)
assert dup_add_term(f, ZZ(1), 5, ZZ) == dup_normal([1, 0, 0, 1, 1, 1], ZZ)
assert dup_add_term(
f, ZZ(1), 6, ZZ) == dup_normal([1, 0, 0, 0, 1, 1, 1], ZZ)
assert dup_add_term(f, ZZ(-1), 2, ZZ) == dup_normal([1, 1], ZZ)
def test_dmp_add_term():
assert dmp_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \
dup_add_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ)
assert dmp_add_term(f_0, [[]], 3, 2, ZZ) == f_0
assert dmp_add_term(F_0, [[]], 3, 2, QQ) == F_0
def test_dup_sub_term():
f = dup_normal([], ZZ)
assert dup_sub_term(f, ZZ(0), 0, ZZ) == dup_normal([], ZZ)
assert dup_sub_term(f, ZZ(1), 0, ZZ) == dup_normal([-1], ZZ)
assert dup_sub_term(f, ZZ(1), 1, ZZ) == dup_normal([-1, 0], ZZ)
assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([-1, 0, 0], ZZ)
f = dup_normal([1, 1, 1], ZZ)
assert dup_sub_term(f, ZZ(2), 0, ZZ) == dup_normal([ 1, 1, -1], ZZ)
assert dup_sub_term(f, ZZ(2), 1, ZZ) == dup_normal([ 1, -1, 1], ZZ)
assert dup_sub_term(f, ZZ(2), 2, ZZ) == dup_normal([-1, 1, 1], ZZ)
assert dup_sub_term(f, ZZ(1), 3, ZZ) == dup_normal([-1, 1, 1, 1], ZZ)
assert dup_sub_term(f, ZZ(1), 4, ZZ) == dup_normal([-1, 0, 1, 1, 1], ZZ)
assert dup_sub_term(f, ZZ(1), 5, ZZ) == dup_normal([-1, 0, 0, 1, 1, 1], ZZ)
assert dup_sub_term(
f, ZZ(1), 6, ZZ) == dup_normal([-1, 0, 0, 0, 1, 1, 1], ZZ)
assert dup_sub_term(f, ZZ(1), 2, ZZ) == dup_normal([1, 1], ZZ)
def test_dmp_sub_term():
assert dmp_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, 0, ZZ) == \
dup_sub_term([ZZ(1), ZZ(1), ZZ(1)], ZZ(1), 2, ZZ)
assert dmp_sub_term(f_0, [[]], 3, 2, ZZ) == f_0
assert dmp_sub_term(F_0, [[]], 3, 2, QQ) == F_0
def test_dup_mul_term():
f = dup_normal([], ZZ)
assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([], ZZ)
f = dup_normal([1, 1], ZZ)
assert dup_mul_term(f, ZZ(0), 3, ZZ) == dup_normal([], ZZ)
f = dup_normal([1, 2, 3], ZZ)
assert dup_mul_term(f, ZZ(2), 0, ZZ) == dup_normal([2, 4, 6], ZZ)
assert dup_mul_term(f, ZZ(2), 1, ZZ) == dup_normal([2, 4, 6, 0], ZZ)
assert dup_mul_term(f, ZZ(2), 2, ZZ) == dup_normal([2, 4, 6, 0, 0], ZZ)
assert dup_mul_term(f, ZZ(2), 3, ZZ) == dup_normal([2, 4, 6, 0, 0, 0], ZZ)
def test_dmp_mul_term():
assert dmp_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, 0, ZZ) == \
dup_mul_term([ZZ(1), ZZ(2), ZZ(3)], ZZ(2), 1, ZZ)
assert dmp_mul_term([[]], [ZZ(2)], 3, 1, ZZ) == [[]]
assert dmp_mul_term([[ZZ(1)]], [], 3, 1, ZZ) == [[]]
assert dmp_mul_term([[ZZ(1), ZZ(2)], [ZZ(3)]], [ZZ(2)], 2, 1, ZZ) == \
[[ZZ(2), ZZ(4)], [ZZ(6)], [], []]
assert dmp_mul_term([[]], [QQ(2, 3)], 3, 1, QQ) == [[]]
assert dmp_mul_term([[QQ(1, 2)]], [], 3, 1, QQ) == [[]]
assert dmp_mul_term([[QQ(1, 5), QQ(2, 5)], [QQ(3, 5)]], [QQ(2, 3)], 2, 1, QQ) == \
[[QQ(2, 15), QQ(4, 15)], [QQ(6, 15)], [], []]
def test_dup_add_ground():
f = ZZ.map([1, 2, 3, 4])
g = ZZ.map([1, 2, 3, 8])
assert dup_add_ground(f, ZZ(4), ZZ) == g
def test_dmp_add_ground():
f = ZZ.map([[1], [2], [3], [4]])
g = ZZ.map([[1], [2], [3], [8]])
assert dmp_add_ground(f, ZZ(4), 1, ZZ) == g
def test_dup_sub_ground():
f = ZZ.map([1, 2, 3, 4])
g = ZZ.map([1, 2, 3, 0])
assert dup_sub_ground(f, ZZ(4), ZZ) == g
def test_dmp_sub_ground():
f = ZZ.map([[1], [2], [3], [4]])
g = ZZ.map([[1], [2], [3], []])
assert dmp_sub_ground(f, ZZ(4), 1, ZZ) == g
def test_dup_mul_ground():
f = dup_normal([], ZZ)
assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([], ZZ)
f = dup_normal([1, 2, 3], ZZ)
assert dup_mul_ground(f, ZZ(0), ZZ) == dup_normal([], ZZ)
assert dup_mul_ground(f, ZZ(2), ZZ) == dup_normal([2, 4, 6], ZZ)
def test_dmp_mul_ground():
assert dmp_mul_ground(f_0, ZZ(2), 2, ZZ) == [
[[ZZ(2), ZZ(4), ZZ(6)], [ZZ(4)]],
[[ZZ(6)]],
[[ZZ(8), ZZ(10), ZZ(12)], [ZZ(2), ZZ(4), ZZ(2)], [ZZ(2)]]
]
assert dmp_mul_ground(F_0, QQ(1, 2), 2, QQ) == [
[[QQ(1, 14), QQ(2, 14), QQ(3, 14)], [QQ(2, 14)]],
[[QQ(3, 14)]],
[[QQ(4, 14), QQ(5, 14), QQ(6, 14)], [QQ(1, 14), QQ(2, 14),
QQ(1, 14)], [QQ(1, 14)]]
]
def test_dup_quo_ground():
raises(ZeroDivisionError, lambda: dup_quo_ground(dup_normal([1, 2,
3], ZZ), ZZ(0), ZZ))
f = dup_normal([], ZZ)
assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ)
f = dup_normal([6, 2, 8], ZZ)
assert dup_quo_ground(f, ZZ(1), ZZ) == f
assert dup_quo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ)
assert dup_quo_ground(f, ZZ(3), ZZ) == dup_normal([2, 0, 2], ZZ)
f = dup_normal([6, 2, 8], QQ)
assert dup_quo_ground(f, QQ(1), QQ) == f
assert dup_quo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)]
assert dup_quo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)]
def test_dup_exquo_ground():
raises(ZeroDivisionError, lambda: dup_exquo_ground(dup_normal([1,
2, 3], ZZ), ZZ(0), ZZ))
raises(ExactQuotientFailed, lambda: dup_exquo_ground(dup_normal([1,
2, 3], ZZ), ZZ(3), ZZ))
f = dup_normal([], ZZ)
assert dup_exquo_ground(f, ZZ(3), ZZ) == dup_normal([], ZZ)
f = dup_normal([6, 2, 8], ZZ)
assert dup_exquo_ground(f, ZZ(1), ZZ) == f
assert dup_exquo_ground(f, ZZ(2), ZZ) == dup_normal([3, 1, 4], ZZ)
f = dup_normal([6, 2, 8], QQ)
assert dup_exquo_ground(f, QQ(1), QQ) == f
assert dup_exquo_ground(f, QQ(2), QQ) == [QQ(3), QQ(1), QQ(4)]
assert dup_exquo_ground(f, QQ(7), QQ) == [QQ(6, 7), QQ(2, 7), QQ(8, 7)]
def test_dmp_quo_ground():
f = dmp_normal([[6], [2], [8]], 1, ZZ)
assert dmp_quo_ground(f, ZZ(1), 1, ZZ) == f
assert dmp_quo_ground(
f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ)
assert dmp_normal(dmp_quo_ground(
f, ZZ(3), 1, ZZ), 1, ZZ) == dmp_normal([[2], [], [2]], 1, ZZ)
def test_dmp_exquo_ground():
f = dmp_normal([[6], [2], [8]], 1, ZZ)
assert dmp_exquo_ground(f, ZZ(1), 1, ZZ) == f
assert dmp_exquo_ground(
f, ZZ(2), 1, ZZ) == dmp_normal([[3], [1], [4]], 1, ZZ)
def test_dup_lshift():
assert dup_lshift([], 3, ZZ) == []
assert dup_lshift([1], 3, ZZ) == [1, 0, 0, 0]
def test_dup_rshift():
assert dup_rshift([], 3, ZZ) == []
assert dup_rshift([1, 0, 0, 0], 3, ZZ) == [1]
def test_dup_abs():
assert dup_abs([], ZZ) == []
assert dup_abs([ZZ( 1)], ZZ) == [ZZ(1)]
assert dup_abs([ZZ(-7)], ZZ) == [ZZ(7)]
assert dup_abs([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(2), ZZ(3)]
assert dup_abs([], QQ) == []
assert dup_abs([QQ( 1, 2)], QQ) == [QQ(1, 2)]
assert dup_abs([QQ(-7, 3)], QQ) == [QQ(7, 3)]
assert dup_abs(
[QQ(-1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(2, 7), QQ(3, 7)]
def test_dmp_abs():
assert dmp_abs([ZZ(-1)], 0, ZZ) == [ZZ(1)]
assert dmp_abs([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)]
assert dmp_abs([[[]]], 2, ZZ) == [[[]]]
assert dmp_abs([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_abs([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]
assert dmp_abs([[[]]], 2, QQ) == [[[]]]
assert dmp_abs([[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_abs([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]]
def test_dup_neg():
assert dup_neg([], ZZ) == []
assert dup_neg([ZZ(1)], ZZ) == [ZZ(-1)]
assert dup_neg([ZZ(-7)], ZZ) == [ZZ(7)]
assert dup_neg([ZZ(-1), ZZ(2), ZZ(3)], ZZ) == [ZZ(1), ZZ(-2), ZZ(-3)]
assert dup_neg([], QQ) == []
assert dup_neg([QQ(1, 2)], QQ) == [QQ(-1, 2)]
assert dup_neg([QQ(-7, 9)], QQ) == [QQ(7, 9)]
assert dup_neg([QQ(
-1, 7), QQ(2, 7), QQ(3, 7)], QQ) == [QQ(1, 7), QQ(-2, 7), QQ(-3, 7)]
def test_dmp_neg():
assert dmp_neg([ZZ(-1)], 0, ZZ) == [ZZ(1)]
assert dmp_neg([QQ(-1, 2)], 0, QQ) == [QQ(1, 2)]
assert dmp_neg([[[]]], 2, ZZ) == [[[]]]
assert dmp_neg([[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_neg([[[ZZ(-7)]]], 2, ZZ) == [[[ZZ(7)]]]
assert dmp_neg([[[]]], 2, QQ) == [[[]]]
assert dmp_neg([[[QQ(1, 9)]]], 2, QQ) == [[[QQ(-1, 9)]]]
assert dmp_neg([[[QQ(-7, 9)]]], 2, QQ) == [[[QQ(7, 9)]]]
def test_dup_add():
assert dup_add([], [], ZZ) == []
assert dup_add([ZZ(1)], [], ZZ) == [ZZ(1)]
assert dup_add([], [ZZ(1)], ZZ) == [ZZ(1)]
assert dup_add([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(2)]
assert dup_add([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(3)]
assert dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(3)]
assert dup_add([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(3)]
assert dup_add([ZZ(1), ZZ(
2), ZZ(3)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(9), ZZ(11), ZZ(13)]
assert dup_add([], [], QQ) == []
assert dup_add([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
assert dup_add([], [QQ(1, 2)], QQ) == [QQ(1, 2)]
assert dup_add([QQ(1, 4)], [QQ(1, 4)], QQ) == [QQ(1, 2)]
assert dup_add([QQ(1, 4)], [QQ(1, 2)], QQ) == [QQ(3, 4)]
assert dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ) == [QQ(1, 2), QQ(5, 3)]
assert dup_add([QQ(1)], [QQ(1, 2), QQ(2, 3)], QQ) == [QQ(1, 2), QQ(5, 3)]
assert dup_add([QQ(1, 7), QQ(2, 7), QQ(3, 7)], [QQ(
8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(9, 7), QQ(11, 7), QQ(13, 7)]
def test_dmp_add():
assert dmp_add([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
dup_add([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
assert dmp_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
dup_add([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
assert dmp_add([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_add([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_add([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_add([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(3)]]]
assert dmp_add([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(3)]]]
assert dmp_add([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_add([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_add([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_add([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
assert dmp_add([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(3, 7)]]]
def test_dup_sub():
assert dup_sub([], [], ZZ) == []
assert dup_sub([ZZ(1)], [], ZZ) == [ZZ(1)]
assert dup_sub([], [ZZ(1)], ZZ) == [ZZ(-1)]
assert dup_sub([ZZ(1)], [ZZ(1)], ZZ) == []
assert dup_sub([ZZ(1)], [ZZ(2)], ZZ) == [ZZ(-1)]
assert dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ) == [ZZ(1), ZZ(1)]
assert dup_sub([ZZ(1)], [ZZ(1), ZZ(2)], ZZ) == [ZZ(-1), ZZ(-1)]
assert dup_sub([ZZ(3), ZZ(
2), ZZ(1)], [ZZ(8), ZZ(9), ZZ(10)], ZZ) == [ZZ(-5), ZZ(-7), ZZ(-9)]
assert dup_sub([], [], QQ) == []
assert dup_sub([QQ(1, 2)], [], QQ) == [QQ(1, 2)]
assert dup_sub([], [QQ(1, 2)], QQ) == [QQ(-1, 2)]
assert dup_sub([QQ(1, 3)], [QQ(1, 3)], QQ) == []
assert dup_sub([QQ(1, 3)], [QQ(2, 3)], QQ) == [QQ(-1, 3)]
assert dup_sub([QQ(1, 7), QQ(2, 7)], [QQ(1)], QQ) == [QQ(1, 7), QQ(-5, 7)]
assert dup_sub([QQ(1)], [QQ(1, 7), QQ(2, 7)], QQ) == [QQ(-1, 7), QQ(5, 7)]
assert dup_sub([QQ(3, 7), QQ(2, 7), QQ(1, 7)], [QQ(
8, 7), QQ(9, 7), QQ(10, 7)], QQ) == [QQ(-5, 7), QQ(-7, 7), QQ(-9, 7)]
def test_dmp_sub():
assert dmp_sub([ZZ(1), ZZ(2)], [ZZ(1)], 0, ZZ) == \
dup_sub([ZZ(1), ZZ(2)], [ZZ(1)], ZZ)
assert dmp_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], 0, QQ) == \
dup_sub([QQ(1, 2), QQ(2, 3)], [QQ(1)], QQ)
assert dmp_sub([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_sub([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(1)]]]
assert dmp_sub([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(-1)]]]
assert dmp_sub([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_sub([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[QQ(1, 2)]]]
assert dmp_sub([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[QQ(-1, 2)]]]
assert dmp_sub([[[QQ(2, 7)]]], [[[QQ(1, 7)]]], 2, QQ) == [[[QQ(1, 7)]]]
assert dmp_sub([[[QQ(1, 7)]]], [[[QQ(2, 7)]]], 2, QQ) == [[[QQ(-1, 7)]]]
def test_dup_add_mul():
assert dup_add_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
[ZZ(1), ZZ(2)], ZZ) == [ZZ(3), ZZ(9), ZZ(7), ZZ(5)]
assert dmp_add_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
[[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(3)], [ZZ(3), ZZ(9)], [ZZ(4), ZZ(5)]]
def test_dup_sub_mul():
assert dup_sub_mul([ZZ(1), ZZ(2), ZZ(3)], [ZZ(3), ZZ(2), ZZ(1)],
[ZZ(1), ZZ(2)], ZZ) == [ZZ(-3), ZZ(-7), ZZ(-3), ZZ(1)]
assert dmp_sub_mul([[ZZ(1), ZZ(2)], [ZZ(3)]], [[ZZ(3)], [ZZ(2), ZZ(1)]],
[[ZZ(1)], [ZZ(2)]], 1, ZZ) == [[ZZ(-3)], [ZZ(-1), ZZ(-5)], [ZZ(-4), ZZ(1)]]
def test_dup_mul():
assert dup_mul([], [], ZZ) == []
assert dup_mul([], [ZZ(1)], ZZ) == []
assert dup_mul([ZZ(1)], [], ZZ) == []
assert dup_mul([ZZ(1)], [ZZ(1)], ZZ) == [ZZ(1)]
assert dup_mul([ZZ(5)], [ZZ(7)], ZZ) == [ZZ(35)]
assert dup_mul([], [], QQ) == []
assert dup_mul([], [QQ(1, 2)], QQ) == []
assert dup_mul([QQ(1, 2)], [], QQ) == []
assert dup_mul([QQ(1, 2)], [QQ(4, 7)], QQ) == [QQ(2, 7)]
assert dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ) == [QQ(15, 49)]
f = dup_normal([3, 0, 0, 6, 1, 2], ZZ)
g = dup_normal([4, 0, 1, 0], ZZ)
h = dup_normal([12, 0, 3, 24, 4, 14, 1, 2, 0], ZZ)
assert dup_mul(f, g, ZZ) == h
assert dup_mul(g, f, ZZ) == h
f = dup_normal([2, 0, 0, 1, 7], ZZ)
h = dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
assert dup_mul(f, f, ZZ) == h
K = FF(6)
assert dup_mul([K(2), K(1)], [K(3), K(4)], K) == [K(5), K(4)]
p1 = dup_normal([79, -1, 78, -94, -10, 11, 32, -19, 78, 2, -89, 30, 73, 42,
85, 77, 83, -30, -34, -2, 95, -81, 37, -49, -46, -58, -16, 37, 35, -11,
-57, -15, -31, 67, -20, 27, 76, 2, 70, 67, -65, 65, -26, -93, -44, -12,
-92, 57, -90, -57, -11, -67, -98, -69, 97, -41, 89, 33, 89, -50, 81,
-31, 60, -27, 43, 29, -77, 44, 21, -91, 32, -57, 33, 3, 53, -51, -38,
-99, -84, 23, -50, 66, -100, 1, -75, -25, 27, -60, 98, -51, -87, 6, 8,
78, -28, -95, -88, 12, -35, 26, -9, 16, -92, 55, -7, -86, 68, -39, -46,
84, 94, 45, 60, 92, 68, -75, -74, -19, 8, 75, 78, 91, 57, 34, 14, -3,
-49, 65, 78, -18, 6, -29, -80, -98, 17, 13, 58, 21, 20, 9, 37, 7, -30,
-53, -20, 34, 67, -42, 89, -22, 73, 43, -6, 5, 51, -8, -15, -52, -22,
-58, -72, -3, 43, -92, 82, 83, -2, -13, -23, -60, 16, -94, -8, -28,
-95, -72, 63, -90, 76, 6, -43, -100, -59, 76, 3, 3, 46, -85, 75, 62,
-71, -76, 88, 97, -72, -1, 30, -64, 72, -48, 14, -78, 58, 63, -91, 24,
-87, -27, -80, -100, -44, 98, 70, 100, -29, -38, 11, 77, 100, 52, 86,
65, -5, -42, -81, -38, -42, 43, -2, -70, -63, -52], ZZ)
p2 = dup_normal([65, -19, -47, 1, 90, 81, -15, -34, 25, -75, 9, -83, 50, -5,
-44, 31, 1, 70, -7, 78, 74, 80, 85, 65, 21, 41, 66, 19, -40, 63, -21,
-27, 32, 69, 83, 34, -35, 14, 81, 57, -75, 32, -67, -89, -100, -61, 46,
84, -78, -29, -50, -94, -24, -32, -68, -16, 100, -7, -72, -89, 35, 82,
58, 81, -92, 62, 5, -47, -39, -58, -72, -13, 84, 44, 55, -25, 48, -54,
-31, -56, -11, -50, -84, 10, 67, 17, 13, -14, 61, 76, -64, -44, -40,
-96, 11, -11, -94, 2, 6, 27, -6, 68, -54, 66, -74, -14, -1, -24, -73,
96, 89, -11, -89, 56, -53, 72, -43, 96, 25, 63, -31, 29, 68, 83, 91,
-93, -19, -38, -40, 40, -12, -19, -79, 44, 100, -66, -29, -77, 62, 39,
-8, 11, -97, 14, 87, 64, 21, -18, 13, 15, -59, -75, -99, -88, 57, 54,
56, -67, 6, -63, -59, -14, 28, 87, -20, -39, 84, -91, -2, 49, -75, 11,
-24, -95, 36, 66, 5, 25, -72, -40, 86, 90, 37, -33, 57, -35, 29, -18,
4, -79, 64, -17, -27, 21, 29, -5, -44, -87, -24, 52, 78, 11, -23, -53,
36, 42, 21, -68, 94, -91, -51, -21, 51, -76, 72, 31, 24, -48, -80, -9,
37, -47, -6, -8, -63, -91, 79, -79, -100, 38, -20, 38, 100, 83, -90,
87, 63, -36, 82, -19, 18, -98, -38, 26, 98, -70, 79, 92, 12, 12, 70,
74, 36, 48, -13, 31, 31, -47, -71, -12, -64, 36, -42, 32, -86, 60, 83,
70, 55, 0, 1, 29, -35, 8, -82, 8, -73, -46, -50, 43, 48, -5, -86, -72,
44, -90, 19, 19, 5, -20, 97, -13, -66, -5, 5, -69, 64, -30, 41, 51, 36,
13, -99, -61, 94, -12, 74, 98, 68, 24, 46, -97, -87, -6, -27, 82, 62,
-11, -77, 86, 66, -47, -49, -50, 13, 18, 89, -89, 46, -80, 13, 98, -35,
-36, -25, 12, 20, 26, -52, 79, 27, 79, 100, 8, 62, -58, -28, 37], ZZ)
res = dup_normal([5135, -1566, 1376, -7466, 4579, 11710, 8001, -7183,
-3737, -7439, 345, -10084, 24522, -1201, 1070, -10245, 9582, 9264,
1903, 23312, 18953, 10037, -15268, -5450, 6442, -6243, -3777, 5110,
10936, -16649, -6022, 16255, 31300, 24818, 31922, 32760, 7854, 27080,
15766, 29596, 7139, 31945, -19810, 465, -38026, -3971, 9641, 465,
-19375, 5524, -30112, -11960, -12813, 13535, 30670, 5925, -43725,
-14089, 11503, -22782, 6371, 43881, 37465, -33529, -33590, -39798,
-37854, -18466, -7908, -35825, -26020, -36923, -11332, -5699, 25166,
-3147, 19885, 12962, -20659, -1642, 27723, -56331, -24580, -11010,
-20206, 20087, -23772, -16038, 38580, 20901, -50731, 32037, -4299,
26508, 18038, -28357, 31846, -7405, -20172, -15894, 2096, 25110,
-45786, 45918, -55333, -31928, -49428, -29824, -58796, -24609, -15408,
69, -35415, -18439, 10123, -20360, -65949, 33356, -20333, 26476,
-32073, 33621, 930, 28803, -42791, 44716, 38164, 12302, -1739, 11421,
73385, -7613, 14297, 38155, -414, 77587, 24338, -21415, 29367, 42639,
13901, -288, 51027, -11827, 91260, 43407, 88521, -15186, 70572, -12049,
5090, -12208, -56374, 15520, -623, -7742, 50825, 11199, -14894, 40892,
59591, -31356, -28696, -57842, -87751, -33744, -28436, -28945, -40287,
37957, -35638, 33401, -61534, 14870, 40292, 70366, -10803, 102290,
-71719, -85251, 7902, -22409, 75009, 99927, 35298, -1175, -762, -34744,
-10587, -47574, -62629, -19581, -43659, -54369, -32250, -39545, 15225,
-24454, 11241, -67308, -30148, 39929, 37639, 14383, -73475, -77636,
-81048, -35992, 41601, -90143, 76937, -8112, 56588, 9124, -40094,
-32340, 13253, 10898, -51639, 36390, 12086, -1885, 100714, -28561,
-23784, -18735, 18916, 16286, 10742, -87360, -13697, 10689, -19477,
-29770, 5060, 20189, -8297, 112407, 47071, 47743, 45519, -4109, 17468,
-68831, 78325, -6481, -21641, -19459, 30919, 96115, 8607, 53341, 32105,
-16211, 23538, 57259, -76272, -40583, 62093, 38511, -34255, -40665,
-40604, -37606, -15274, 33156, -13885, 103636, 118678, -14101, -92682,
-100791, 2634, 63791, 98266, 19286, -34590, -21067, -71130, 25380,
-40839, -27614, -26060, 52358, -15537, 27138, -6749, 36269, -33306,
13207, -91084, -5540, -57116, 69548, 44169, -57742, -41234, -103327,
-62904, -8566, 41149, -12866, 71188, 23980, 1838, 58230, 73950, 5594,
43113, -8159, -15925, 6911, 85598, -75016, -16214, -62726, -39016,
8618, -63882, -4299, 23182, 49959, 49342, -3238, -24913, -37138, 78361,
32451, 6337, -11438, -36241, -37737, 8169, -3077, -24829, 57953, 53016,
-31511, -91168, 12599, -41849, 41576, 55275, -62539, 47814, -62319,
12300, -32076, -55137, -84881, -27546, 4312, -3433, -54382, 113288,
-30157, 74469, 18219, 79880, -2124, 98911, 17655, -33499, -32861,
47242, -37393, 99765, 14831, -44483, 10800, -31617, -52710, 37406,
22105, 29704, -20050, 13778, 43683, 36628, 8494, 60964, -22644, 31550,
-17693, 33805, -124879, -12302, 19343, 20400, -30937, -21574, -34037,
-33380, 56539, -24993, -75513, -1527, 53563, 65407, -101, 53577, 37991,
18717, -23795, -8090, -47987, -94717, 41967, 5170, -14815, -94311,
17896, -17734, -57718, -774, -38410, 24830, 29682, 76480, 58802,
-46416, -20348, -61353, -68225, -68306, 23822, -31598, 42972, 36327,
28968, -65638, -21638, 24354, -8356, 26777, 52982, -11783, -44051,
-26467, -44721, -28435, -53265, -25574, -2669, 44155, 22946, -18454,
-30718, -11252, 58420, 8711, 67447, 4425, 41749, 67543, 43162, 11793,
-41907, 20477, -13080, 6559, -6104, -13244, 42853, 42935, 29793, 36730,
-28087, 28657, 17946, 7503, 7204, 21491, -27450, -24241, -98156,
-18082, -42613, -24928, 10775, -14842, -44127, 55910, 14777, 31151, -2194,
39206, -2100, -4211, 11827, -8918, -19471, 72567, 36447, -65590, -34861,
-17147, -45303, 9025, -7333, -35473, 11101, 11638, 3441, 6626, -41800,
9416, 13679, 33508, 40502, -60542, 16358, 8392, -43242, -35864, -34127,
-48721, 35878, 30598, 28630, 20279, -19983, -14638, -24455, -1851, -11344,
45150, 42051, 26034, -28889, -32382, -3527, -14532, 22564, -22346, 477,
11706, 28338, -25972, -9185, -22867, -12522, 32120, -4424, 11339, -33913,
-7184, 5101, -23552, -17115, -31401, -6104, 21906, 25708, 8406, 6317,
-7525, 5014, 20750, 20179, 22724, 11692, 13297, 2493, -253, -16841, -17339,
-6753, -4808, 2976, -10881, -10228, -13816, -12686, 1385, 2316, 2190, -875,
-1924], ZZ)
assert dup_mul(p1, p2, ZZ) == res
p1 = dup_normal([83, -61, -86, -24, 12, 43, -88, -9, 42, 55, -66, 74, 95,
-25, -12, 68, -99, 4, 45, 6, -15, -19, 78, 65, -55, 47, -13, 17, 86,
81, -58, -27, 50, -40, -24, 39, -41, -92, 75, 90, -1, 40, -15, -27,
-35, 68, 70, -64, -40, 78, -88, -58, -39, 69, 46, 12, 28, -94, -37,
-50, -80, -96, -61, 25, 1, 71, 4, 12, 48, 4, 34, -47, -75, 5, 48, 82,
88, 23, 98, 35, 17, -10, 48, -61, -95, 47, 65, -19, -66, -57, -6, -51,
-42, -89, 66, -13, 18, 37, 90, -23, 72, 96, -53, 0, 40, -73, -52, -68,
32, -25, -53, 79, -52, 18, 44, 73, -81, 31, -90, 70, 3, 36, 48, 76,
-24, -44, 23, 98, -4, 73, 69, 88, -70, 14, -68, 94, -78, -15, -64, -97,
-70, -35, 65, 88, 49, -53, -7, 12, -45, -7, 59, -94, 99, -2, 67, -60,
-71, 29, -62, -77, 1, 51, 17, 80, -20, -47, -19, 24, -9, 39, -23, 21,
-84, 10, 84, 56, -17, -21, -66, 85, 70, 46, -51, -22, -95, 78, -60,
-96, -97, -45, 72, 35, 30, -61, -92, -93, -60, -61, 4, -4, -81, -73,
46, 53, -11, 26, 94, 45, 14, -78, 55, 84, -68, 98, 60, 23, 100, -63,
68, 96, -16, 3, 56, 21, -58, 62, -67, 66, 85, 41, -79, -22, 97, -67,
82, 82, -96, -20, -7, 48, -67, 48, -9, -39, 78], ZZ)
p2 = dup_normal([52, 88, 76, 66, 9, -64, 46, -20, -28, 69, 60, 96, -36,
-92, -30, -11, -35, 35, 55, 63, -92, -7, 25, -58, 74, 55, -6, 4, 47,
-92, -65, 67, -45, 74, -76, 59, -6, 69, 39, 24, -71, -7, 39, -45, 60,
-68, 98, 97, -79, 17, 4, 94, -64, 68, -100, -96, -2, 3, 22, 96, 54,
-77, -86, 67, 6, 57, 37, 40, 89, -78, 64, -94, -45, -92, 57, 87, -26,
36, 19, 97, 25, 77, -87, 24, 43, -5, 35, 57, 83, 71, 35, 63, 61, 96,
-22, 8, -1, 96, 43, 45, 94, -93, 36, 71, -41, -99, 85, -48, 59, 52,
-17, 5, 87, -16, -68, -54, 76, -18, 100, 91, -42, -70, -66, -88, -12,
1, 95, -82, 52, 43, -29, 3, 12, 72, -99, -43, -32, -93, -51, 16, -20,
-12, -11, 5, 33, -38, 93, -5, -74, 25, 74, -58, 93, 59, -63, -86, 63,
-20, -4, -74, -73, -95, 29, -28, 93, -91, -2, -38, -62, 77, -58, -85,
-28, 95, 38, 19, -69, 86, 94, 25, -2, -4, 47, 34, -59, 35, -48, 29,
-63, -53, 34, 29, 66, 73, 6, 92, -84, 89, 15, 81, 93, 97, 51, -72, -78,
25, 60, 90, -45, 39, 67, -84, -62, 57, 26, -32, -56, -14, -83, 76, 5,
-2, 99, -100, 28, 46, 94, -7, 53, -25, 16, -23, -36, 89, -78, -63, 31,
1, 84, -99, -52, 76, 48, 90, -76, 44, -19, 54, -36, -9, -73, -100, -69,
31, 42, 25, -39, 76, -26, -8, -14, 51, 3, 37, 45, 2, -54, 13, -34, -92,
17, -25, -65, 53, -63, 30, 4, -70, -67, 90, 52, 51, 18, -3, 31, -45,
-9, 59, 63, -87, 22, -32, 29, -38, 21, 36, -82, 27, -11], ZZ)
res = dup_normal([4316, 4132, -3532, -7974, -11303, -10069, 5484, -3330,
-5874, 7734, 4673, 11327, -9884, -8031, 17343, 21035, -10570, -9285,
15893, 3780, -14083, 8819, 17592, 10159, 7174, -11587, 8598, -16479,
3602, 25596, 9781, 12163, 150, 18749, -21782, -12307, 27578, -2757,
-12573, 12565, 6345, -18956, 19503, -15617, 1443, -16778, 36851, 23588,
-28474, 5749, 40695, -7521, -53669, -2497, -18530, 6770, 57038, 3926,
-6927, -15399, 1848, -64649, -27728, 3644, 49608, 15187, -8902, -9480,
-7398, -40425, 4824, 23767, -7594, -6905, 33089, 18786, 12192, 24670,
31114, 35334, -4501, -14676, 7107, -59018, -21352, 20777, 19661, 20653,
33754, -885, -43758, 6269, 51897, -28719, -97488, -9527, 13746, 11644,
17644, -21720, 23782, -10481, 47867, 20752, 33810, -1875, 39918, -7710,
-40840, 19808, -47075, 23066, 46616, 25201, 9287, 35436, -1602, 9645,
-11978, 13273, 15544, 33465, 20063, 44539, 11687, 27314, -6538, -37467,
14031, 32970, -27086, 41323, 29551, 65910, -39027, -37800, -22232,
8212, 46316, -28981, -55282, 50417, -44929, -44062, 73879, 37573,
-2596, -10877, -21893, -133218, -33707, -25753, -9531, 17530, 61126,
2748, -56235, 43874, -10872, -90459, -30387, 115267, -7264, -44452,
122626, 14839, -599, 10337, 57166, -67467, -54957, 63669, 1202, 18488,
52594, 7205, -97822, 612, 78069, -5403, -63562, 47236, 36873, -154827,
-26188, 82427, -39521, 5628, 7416, 5276, -53095, 47050, 26121, -42207,
79021, -13035, 2499, -66943, 29040, -72355, -23480, 23416, -12885,
-44225, -42688, -4224, 19858, 55299, 15735, 11465, 101876, -39169,
51786, 14723, 43280, -68697, 16410, 92295, 56767, 7183, 111850, 4550,
115451, -38443, -19642, -35058, 10230, 93829, 8925, 63047, 3146, 29250,
8530, 5255, -98117, -115517, -76817, -8724, 41044, 1312, -35974, 79333,
-28567, 7547, -10580, -24559, -16238, 10794, -3867, 24848, 57770,
-51536, -35040, 71033, 29853, 62029, -7125, -125585, -32169, -47907,
156811, -65176, -58006, -15757, -57861, 11963, 30225, -41901, -41681,
31310, 27982, 18613, 61760, 60746, -59096, 33499, 30097, -17997, 24032,
56442, -83042, 23747, -20931, -21978, -158752, -9883, -73598, -7987,
-7333, -125403, -116329, 30585, 53281, 51018, -29193, 88575, 8264,
-40147, -16289, 113088, 12810, -6508, 101552, -13037, 34440, -41840,
101643, 24263, 80532, 61748, 65574, 6423, -20672, 6591, -10834, -71716,
86919, -92626, 39161, 28490, 81319, 46676, 106720, 43530, 26998, 57456,
-8862, 60989, 13982, 3119, -2224, 14743, 55415, -49093, -29303, 28999,
1789, 55953, -84043, -7780, -65013, 57129, -47251, 61484, 61994,
-78361, -82778, 22487, -26894, 9756, -74637, -15519, -4360, 30115,
42433, 35475, 15286, 69768, 21509, -20214, 78675, -21163, 13596, 11443,
-10698, -53621, -53867, -24155, 64500, -42784, -33077, -16500, 873,
-52788, 14546, -38011, 36974, -39849, -34029, -94311, 83068, -50437,
-26169, -46746, 59185, 42259, -101379, -12943, 30089, -59086, 36271,
22723, -30253, -52472, -70826, -23289, 3331, -31687, 14183, -857,
-28627, 35246, -51284, 5636, -6933, 66539, 36654, 50927, 24783, 3457,
33276, 45281, 45650, -4938, -9968, -22590, 47995, 69229, 5214, -58365,
-17907, -14651, 18668, 18009, 12649, -11851, -13387, 20339, 52472,
-1087, -21458, -68647, 52295, 15849, 40608, 15323, 25164, -29368,
10352, -7055, 7159, 21695, -5373, -54849, 101103, -24963, -10511,
33227, 7659, 41042, -69588, 26718, -20515, 6441, 38135, -63, 24088,
-35364, -12785, -18709, 47843, 48533, -48575, 17251, -19394, 32878,
-9010, -9050, 504, -12407, 28076, -3429, 25324, -4210, -26119, 752,
-29203, 28251, -11324, -32140, -3366, -25135, 18702, -31588, -7047,
-24267, 49987, -14975, -33169, 37744, -7720, -9035, 16964, -2807, -421,
14114, -17097, -13662, 40628, -12139, -9427, 5369, 17551, -13232, -16211,
9804, -7422, 2677, 28635, -8280, -4906, 2908, -22558, 5604, 12459, 8756,
-3980, -4745, -18525, 7913, 5970, -16457, 20230, -6247, -13812, 2505,
11899, 1409, -15094, 22540, -18863, 137, 11123, -4516, 2290, -8594, 12150,
-10380, 3005, 5235, -7350, 2535, -858], ZZ)
assert dup_mul(p1, p2, ZZ) == res
def test_dmp_mul():
assert dmp_mul([ZZ(5)], [ZZ(7)], 0, ZZ) == \
dup_mul([ZZ(5)], [ZZ(7)], ZZ)
assert dmp_mul([QQ(5, 7)], [QQ(3, 7)], 0, QQ) == \
dup_mul([QQ(5, 7)], [QQ(3, 7)], QQ)
assert dmp_mul([[[]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_mul([[[ZZ(1)]]], [[[]]], 2, ZZ) == [[[]]]
assert dmp_mul([[[]]], [[[ZZ(1)]]], 2, ZZ) == [[[]]]
assert dmp_mul([[[ZZ(2)]]], [[[ZZ(1)]]], 2, ZZ) == [[[ZZ(2)]]]
assert dmp_mul([[[ZZ(1)]]], [[[ZZ(2)]]], 2, ZZ) == [[[ZZ(2)]]]
assert dmp_mul([[[]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_mul([[[QQ(1, 2)]]], [[[]]], 2, QQ) == [[[]]]
assert dmp_mul([[[]]], [[[QQ(1, 2)]]], 2, QQ) == [[[]]]
assert dmp_mul([[[QQ(2, 7)]]], [[[QQ(1, 3)]]], 2, QQ) == [[[QQ(2, 21)]]]
assert dmp_mul([[[QQ(1, 7)]]], [[[QQ(2, 3)]]], 2, QQ) == [[[QQ(2, 21)]]]
K = FF(6)
assert dmp_mul(
[[K(2)], [K(1)]], [[K(3)], [K(4)]], 1, K) == [[K(5)], [K(4)]]
def test_dup_sqr():
assert dup_sqr([], ZZ) == []
assert dup_sqr([ZZ(2)], ZZ) == [ZZ(4)]
assert dup_sqr([ZZ(1), ZZ(2)], ZZ) == [ZZ(1), ZZ(4), ZZ(4)]
assert dup_sqr([], QQ) == []
assert dup_sqr([QQ(2, 3)], QQ) == [QQ(4, 9)]
assert dup_sqr([QQ(1, 3), QQ(2, 3)], QQ) == [QQ(1, 9), QQ(4, 9), QQ(4, 9)]
f = dup_normal([2, 0, 0, 1, 7], ZZ)
assert dup_sqr(f, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
K = FF(9)
assert dup_sqr([K(3), K(4)], K) == [K(6), K(7)]
def test_dmp_sqr():
assert dmp_sqr([ZZ(1), ZZ(2)], 0, ZZ) == \
dup_sqr([ZZ(1), ZZ(2)], ZZ)
assert dmp_sqr([[[]]], 2, ZZ) == [[[]]]
assert dmp_sqr([[[ZZ(2)]]], 2, ZZ) == [[[ZZ(4)]]]
assert dmp_sqr([[[]]], 2, QQ) == [[[]]]
assert dmp_sqr([[[QQ(2, 3)]]], 2, QQ) == [[[QQ(4, 9)]]]
K = FF(9)
assert dmp_sqr([[K(3)], [K(4)]], 1, K) == [[K(6)], [K(7)]]
def test_dup_pow():
assert dup_pow([], 0, ZZ) == [ZZ(1)]
assert dup_pow([], 0, QQ) == [QQ(1)]
assert dup_pow([], 1, ZZ) == []
assert dup_pow([], 7, ZZ) == []
assert dup_pow([ZZ(1)], 0, ZZ) == [ZZ(1)]
assert dup_pow([ZZ(1)], 1, ZZ) == [ZZ(1)]
assert dup_pow([ZZ(1)], 7, ZZ) == [ZZ(1)]
assert dup_pow([ZZ(3)], 0, ZZ) == [ZZ(1)]
assert dup_pow([ZZ(3)], 1, ZZ) == [ZZ(3)]
assert dup_pow([ZZ(3)], 7, ZZ) == [ZZ(2187)]
assert dup_pow([QQ(1, 1)], 0, QQ) == [QQ(1, 1)]
assert dup_pow([QQ(1, 1)], 1, QQ) == [QQ(1, 1)]
assert dup_pow([QQ(1, 1)], 7, QQ) == [QQ(1, 1)]
assert dup_pow([QQ(3, 7)], 0, QQ) == [QQ(1, 1)]
assert dup_pow([QQ(3, 7)], 1, QQ) == [QQ(3, 7)]
assert dup_pow([QQ(3, 7)], 7, QQ) == [QQ(2187, 823543)]
f = dup_normal([2, 0, 0, 1, 7], ZZ)
assert dup_pow(f, 0, ZZ) == dup_normal([1], ZZ)
assert dup_pow(f, 1, ZZ) == dup_normal([2, 0, 0, 1, 7], ZZ)
assert dup_pow(f, 2, ZZ) == dup_normal([4, 0, 0, 4, 28, 0, 1, 14, 49], ZZ)
assert dup_pow(f, 3, ZZ) == dup_normal(
[8, 0, 0, 12, 84, 0, 6, 84, 294, 1, 21, 147, 343], ZZ)
def test_dmp_pow():
assert dmp_pow([[]], 0, 1, ZZ) == [[ZZ(1)]]
assert dmp_pow([[]], 0, 1, QQ) == [[QQ(1)]]
assert dmp_pow([[]], 1, 1, ZZ) == [[]]
assert dmp_pow([[]], 7, 1, ZZ) == [[]]
assert dmp_pow([[ZZ(1)]], 0, 1, ZZ) == [[ZZ(1)]]
assert dmp_pow([[ZZ(1)]], 1, 1, ZZ) == [[ZZ(1)]]
assert dmp_pow([[ZZ(1)]], 7, 1, ZZ) == [[ZZ(1)]]
assert dmp_pow([[QQ(3, 7)]], 0, 1, QQ) == [[QQ(1, 1)]]
assert dmp_pow([[QQ(3, 7)]], 1, 1, QQ) == [[QQ(3, 7)]]
assert dmp_pow([[QQ(3, 7)]], 7, 1, QQ) == [[QQ(2187, 823543)]]
f = dup_normal([2, 0, 0, 1, 7], ZZ)
assert dmp_pow(f, 2, 0, ZZ) == dup_pow(f, 2, ZZ)
def test_dup_pdiv():
f = dup_normal([3, 1, 1, 5], ZZ)
g = dup_normal([5, -3, 1], ZZ)
q = dup_normal([15, 14], ZZ)
r = dup_normal([52, 111], ZZ)
assert dup_pdiv(f, g, ZZ) == (q, r)
assert dup_pquo(f, g, ZZ) == q
assert dup_prem(f, g, ZZ) == r
raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, ZZ))
f = dup_normal([3, 1, 1, 5], QQ)
g = dup_normal([5, -3, 1], QQ)
q = dup_normal([15, 14], QQ)
r = dup_normal([52, 111], QQ)
assert dup_pdiv(f, g, QQ) == (q, r)
assert dup_pquo(f, g, QQ) == q
assert dup_prem(f, g, QQ) == r
raises(ExactQuotientFailed, lambda: dup_pexquo(f, g, QQ))
def test_dmp_pdiv():
f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
g = dmp_normal([[1], [-1, 0]], 1, ZZ)
q = dmp_normal([[1], [1, 0]], 1, ZZ)
r = dmp_normal([[2, 0, 0]], 1, ZZ)
assert dmp_pdiv(f, g, 1, ZZ) == (q, r)
assert dmp_pquo(f, g, 1, ZZ) == q
assert dmp_prem(f, g, 1, ZZ) == r
raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ))
f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
g = dmp_normal([[2], [-2, 0]], 1, ZZ)
q = dmp_normal([[2], [2, 0]], 1, ZZ)
r = dmp_normal([[8, 0, 0]], 1, ZZ)
assert dmp_pdiv(f, g, 1, ZZ) == (q, r)
assert dmp_pquo(f, g, 1, ZZ) == q
assert dmp_prem(f, g, 1, ZZ) == r
raises(ExactQuotientFailed, lambda: dmp_pexquo(f, g, 1, ZZ))
def test_dup_rr_div():
raises(ZeroDivisionError, lambda: dup_rr_div([1, 2, 3], [], ZZ))
f = dup_normal([3, 1, 1, 5], ZZ)
g = dup_normal([5, -3, 1], ZZ)
q, r = [], f
assert dup_rr_div(f, g, ZZ) == (q, r)
def test_dmp_rr_div():
raises(ZeroDivisionError, lambda: dmp_rr_div([[1, 2], [3]], [[]], 1, ZZ))
f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
g = dmp_normal([[1], [-1, 0]], 1, ZZ)
q = dmp_normal([[1], [1, 0]], 1, ZZ)
r = dmp_normal([[2, 0, 0]], 1, ZZ)
assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
g = dmp_normal([[-1], [1, 0]], 1, ZZ)
q = dmp_normal([[-1], [-1, 0]], 1, ZZ)
r = dmp_normal([[2, 0, 0]], 1, ZZ)
assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
f = dmp_normal([[1], [], [1, 0, 0]], 1, ZZ)
g = dmp_normal([[2], [-2, 0]], 1, ZZ)
q, r = [[]], f
assert dmp_rr_div(f, g, 1, ZZ) == (q, r)
def test_dup_ff_div():
raises(ZeroDivisionError, lambda: dup_ff_div([1, 2, 3], [], QQ))
f = dup_normal([3, 1, 1, 5], QQ)
g = dup_normal([5, -3, 1], QQ)
q = [QQ(3, 5), QQ(14, 25)]
r = [QQ(52, 25), QQ(111, 25)]
assert dup_ff_div(f, g, QQ) == (q, r)
def test_dup_ff_div_gmpy2():
try:
from gmpy2 import mpq
except ImportError:
return
from sympy.polys.domains import GMPYRationalField
K = GMPYRationalField()
f = [mpq(1,3), mpq(3,2)]
g = [mpq(2,1)]
assert dmp_ff_div(f, g, 0, K) == ([mpq(1,6), mpq(3,4)], [])
f = [mpq(1,2), mpq(1,3), mpq(1,4), mpq(1,5)]
g = [mpq(-1,1), mpq(1,1), mpq(-1,1)]
assert dmp_ff_div(f, g, 0, K) == ([mpq(-1,2), mpq(-5,6)], [mpq(7,12), mpq(-19,30)])
def test_dmp_ff_div():
raises(ZeroDivisionError, lambda: dmp_ff_div([[1, 2], [3]], [[]], 1, QQ))
f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
g = dmp_normal([[1], [-1, 0]], 1, QQ)
q = [[QQ(1, 1)], [QQ(1, 1), QQ(0, 1)]]
r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
assert dmp_ff_div(f, g, 1, QQ) == (q, r)
f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
g = dmp_normal([[-1], [1, 0]], 1, QQ)
q = [[QQ(-1, 1)], [QQ(-1, 1), QQ(0, 1)]]
r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
assert dmp_ff_div(f, g, 1, QQ) == (q, r)
f = dmp_normal([[1], [], [1, 0, 0]], 1, QQ)
g = dmp_normal([[2], [-2, 0]], 1, QQ)
q = [[QQ(1, 2)], [QQ(1, 2), QQ(0, 1)]]
r = [[QQ(2, 1), QQ(0, 1), QQ(0, 1)]]
assert dmp_ff_div(f, g, 1, QQ) == (q, r)
def test_dup_div():
f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1]
assert dup_div(f, g, ZZ) == (q, r)
assert dup_quo(f, g, ZZ) == q
assert dup_rem(f, g, ZZ) == r
raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
f, g, q, r = [5, 4, 3, 2, 1, 0], [1, 2, 0, 0, 9], [5, -6], [15, 2, -44, 54]
assert dup_div(f, g, ZZ) == (q, r)
assert dup_quo(f, g, ZZ) == q
assert dup_rem(f, g, ZZ) == r
raises(ExactQuotientFailed, lambda: dup_exquo(f, g, ZZ))
def test_dmp_div():
f, g, q, r = [5, 4, 3, 2, 1], [1, 2, 3], [5, -6, 0], [20, 1]
assert dmp_div(f, g, 0, ZZ) == (q, r)
assert dmp_quo(f, g, 0, ZZ) == q
assert dmp_rem(f, g, 0, ZZ) == r
raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 0, ZZ))
f, g, q, r = [[[1]]], [[[2]], [1]], [[[]]], [[[1]]]
assert dmp_div(f, g, 2, ZZ) == (q, r)
assert dmp_quo(f, g, 2, ZZ) == q
assert dmp_rem(f, g, 2, ZZ) == r
raises(ExactQuotientFailed, lambda: dmp_exquo(f, g, 2, ZZ))
def test_dup_max_norm():
assert dup_max_norm([], ZZ) == 0
assert dup_max_norm([1], ZZ) == 1
assert dup_max_norm([1, 4, 2, 3], ZZ) == 4
def test_dmp_max_norm():
assert dmp_max_norm([[[]]], 2, ZZ) == 0
assert dmp_max_norm([[[1]]], 2, ZZ) == 1
assert dmp_max_norm(f_0, 2, ZZ) == 6
def test_dup_l1_norm():
assert dup_l1_norm([], ZZ) == 0
assert dup_l1_norm([1], ZZ) == 1
assert dup_l1_norm([1, 4, 2, 3], ZZ) == 10
def test_dmp_l1_norm():
assert dmp_l1_norm([[[]]], 2, ZZ) == 0
assert dmp_l1_norm([[[1]]], 2, ZZ) == 1
assert dmp_l1_norm(f_0, 2, ZZ) == 31
def test_dup_l2_norm_squared():
assert dup_l2_norm_squared([], ZZ) == 0
assert dup_l2_norm_squared([1], ZZ) == 1
assert dup_l2_norm_squared([1, 4, 2, 3], ZZ) == 30
def test_dmp_l2_norm_squared():
assert dmp_l2_norm_squared([[[]]], 2, ZZ) == 0
assert dmp_l2_norm_squared([[[1]]], 2, ZZ) == 1
assert dmp_l2_norm_squared(f_0, 2, ZZ) == 111
def test_dup_expand():
assert dup_expand((), ZZ) == [1]
assert dup_expand(([1, 2, 3], [1, 2], [7, 5, 4, 3]), ZZ) == \
dup_mul([1, 2, 3], dup_mul([1, 2], [7, 5, 4, 3], ZZ), ZZ)
def test_dmp_expand():
assert dmp_expand((), 1, ZZ) == [[1]]
assert dmp_expand(([[1], [2], [3]], [[1], [2]], [[7], [5], [4], [3]]), 1, ZZ) == \
dmp_mul([[1], [2], [3]], dmp_mul([[1], [2]], [[7], [5], [
4], [3]], 1, ZZ), 1, ZZ)
|
c7fc4af2392f1a1f021a1c9248fd1ca1d09fedbd4ab78c28d25e28bf8464a416 | """Tests for user-friendly public interface to polynomial functions. """
import pickle
from sympy.polys.polytools import (
Poly, PurePoly, poly,
parallel_poly_from_expr,
degree, degree_list,
total_degree,
LC, LM, LT,
pdiv, prem, pquo, pexquo,
div, rem, quo, exquo,
half_gcdex, gcdex, invert,
subresultants,
resultant, discriminant,
terms_gcd, cofactors,
gcd, gcd_list,
lcm, lcm_list,
trunc,
monic, content, primitive,
compose, decompose,
sturm,
gff_list, gff,
sqf_norm, sqf_part, sqf_list, sqf,
factor_list, factor,
intervals, refine_root, count_roots,
real_roots, nroots, ground_roots,
nth_power_roots_poly,
cancel, reduced, groebner,
GroebnerBasis, is_zero_dimensional,
_torational_factor_list,
to_rational_coeffs)
from sympy.polys.polyerrors import (
MultivariatePolynomialError,
ExactQuotientFailed,
PolificationFailed,
ComputationFailed,
UnificationFailed,
RefinementFailed,
GeneratorsNeeded,
GeneratorsError,
PolynomialError,
CoercionFailed,
DomainError,
OptionError,
FlagError)
from sympy.polys.polyclasses import DMP
from sympy.polys.fields import field
from sympy.polys.domains import FF, ZZ, QQ, ZZ_I, QQ_I, RR, EX
from sympy.polys.domains.realfield import RealField
from sympy.polys.domains.complexfield import ComplexField
from sympy.polys.orderings import lex, grlex, grevlex
from sympy.core.add import Add
from sympy.core.basic import _aresame
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import (Derivative, diff, expand)
from sympy.core.mul import _keep_coeff, Mul
from sympy.core.numbers import (Float, I, Integer, Rational, oo, pi)
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.hyperbolic import tanh
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import sin
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.polys.rootoftools import rootof
from sympy.simplify.simplify import signsimp
from sympy.utilities.iterables import iterable
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.testing.pytest import raises, warns_deprecated_sympy, warns
from sympy.abc import a, b, c, d, p, q, t, w, x, y, z
def _epsilon_eq(a, b):
for u, v in zip(a, b):
if abs(u - v) > 1e-10:
return False
return True
def _strict_eq(a, b):
if type(a) == type(b):
if iterable(a):
if len(a) == len(b):
return all(_strict_eq(c, d) for c, d in zip(a, b))
else:
return False
else:
return isinstance(a, Poly) and a.eq(b, strict=True)
else:
return False
def test_Poly_mixed_operations():
p = Poly(x, x)
with warns_deprecated_sympy():
p * exp(x)
with warns_deprecated_sympy():
p + exp(x)
with warns_deprecated_sympy():
p - exp(x)
def test_Poly_from_dict():
K = FF(3)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict(
{0: 1, 1: 5}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict(
{(0,): 1, (1,): 5}, gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_dict({(0, 0): 1, (1, 1): 2}, gens=(
x, y), domain=K).rep == DMP([[K(2), K(0)], [K(1)]], K)
assert Poly.from_dict({0: 1, 1: 2}, gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{0: 1, 1: 2}, gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_dict(
{(0,): 1, (1,): 2}, gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_dict({(1,): sin(y)}, gens=x, composite=False) == \
Poly(sin(y)*x, x, domain='EX')
assert Poly.from_dict({(1,): y}, gens=x, composite=False) == \
Poly(y*x, x, domain='EX')
assert Poly.from_dict({(1, 1): 1}, gens=(x, y), composite=False) == \
Poly(x*y, x, y, domain='ZZ')
assert Poly.from_dict({(1, 0): y}, gens=(x, z), composite=False) == \
Poly(y*x, x, z, domain='EX')
def test_Poly_from_list():
K = FF(3)
assert Poly.from_list([2, 1], gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_list([5, 1], gens=x, domain=K).rep == DMP([K(2), K(1)], K)
assert Poly.from_list([2, 1], gens=x).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_list([2, 1], gens=x, field=True).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_list([2, 1], gens=x, domain=ZZ).rep == DMP([ZZ(2), ZZ(1)], ZZ)
assert Poly.from_list([2, 1], gens=x, domain=QQ).rep == DMP([QQ(2), QQ(1)], QQ)
assert Poly.from_list([0, 1.0], gens=x).rep == DMP([RR(1.0)], RR)
assert Poly.from_list([1.0, 0], gens=x).rep == DMP([RR(1.0), RR(0.0)], RR)
raises(MultivariatePolynomialError, lambda: Poly.from_list([[]], gens=(x, y)))
def test_Poly_from_poly():
f = Poly(x + 7, x, domain=ZZ)
g = Poly(x + 2, x, modulus=3)
h = Poly(x + y, x, y, domain=ZZ)
K = FF(3)
assert Poly.from_poly(f) == f
assert Poly.from_poly(f, domain=K).rep == DMP([K(1), K(1)], K)
assert Poly.from_poly(f, domain=ZZ).rep == DMP([1, 7], ZZ)
assert Poly.from_poly(f, domain=QQ).rep == DMP([1, 7], QQ)
assert Poly.from_poly(f, gens=x) == f
assert Poly.from_poly(f, gens=x, domain=K).rep == DMP([K(1), K(1)], K)
assert Poly.from_poly(f, gens=x, domain=ZZ).rep == DMP([1, 7], ZZ)
assert Poly.from_poly(f, gens=x, domain=QQ).rep == DMP([1, 7], QQ)
assert Poly.from_poly(f, gens=y) == Poly(x + 7, y, domain='ZZ[x]')
raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=K))
raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=ZZ))
raises(CoercionFailed, lambda: Poly.from_poly(f, gens=y, domain=QQ))
assert Poly.from_poly(f, gens=(x, y)) == Poly(x + 7, x, y, domain='ZZ')
assert Poly.from_poly(
f, gens=(x, y), domain=ZZ) == Poly(x + 7, x, y, domain='ZZ')
assert Poly.from_poly(
f, gens=(x, y), domain=QQ) == Poly(x + 7, x, y, domain='QQ')
assert Poly.from_poly(
f, gens=(x, y), modulus=3) == Poly(x + 7, x, y, domain='FF(3)')
K = FF(2)
assert Poly.from_poly(g) == g
assert Poly.from_poly(g, domain=ZZ).rep == DMP([1, -1], ZZ)
raises(CoercionFailed, lambda: Poly.from_poly(g, domain=QQ))
assert Poly.from_poly(g, domain=K).rep == DMP([K(1), K(0)], K)
assert Poly.from_poly(g, gens=x) == g
assert Poly.from_poly(g, gens=x, domain=ZZ).rep == DMP([1, -1], ZZ)
raises(CoercionFailed, lambda: Poly.from_poly(g, gens=x, domain=QQ))
assert Poly.from_poly(g, gens=x, domain=K).rep == DMP([K(1), K(0)], K)
K = FF(3)
assert Poly.from_poly(h) == h
assert Poly.from_poly(
h, domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(h, domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K)
assert Poly.from_poly(h, gens=x) == Poly(x + y, x, domain=ZZ[y])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, domain=ZZ))
assert Poly.from_poly(
h, gens=x, domain=ZZ[y]) == Poly(x + y, x, domain=ZZ[y])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, domain=QQ))
assert Poly.from_poly(
h, gens=x, domain=QQ[y]) == Poly(x + y, x, domain=QQ[y])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=x, modulus=3))
assert Poly.from_poly(h, gens=y) == Poly(x + y, y, domain=ZZ[x])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, domain=ZZ))
assert Poly.from_poly(
h, gens=y, domain=ZZ[x]) == Poly(x + y, y, domain=ZZ[x])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, domain=QQ))
assert Poly.from_poly(
h, gens=y, domain=QQ[x]) == Poly(x + y, y, domain=QQ[x])
raises(CoercionFailed, lambda: Poly.from_poly(h, gens=y, modulus=3))
assert Poly.from_poly(h, gens=(x, y)) == h
assert Poly.from_poly(
h, gens=(x, y), domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, gens=(x, y), domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(
h, gens=(x, y), domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K)
assert Poly.from_poly(
h, gens=(y, x)).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, gens=(y, x), domain=ZZ).rep == DMP([[ZZ(1)], [ZZ(1), ZZ(0)]], ZZ)
assert Poly.from_poly(
h, gens=(y, x), domain=QQ).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(
h, gens=(y, x), domain=K).rep == DMP([[K(1)], [K(1), K(0)]], K)
assert Poly.from_poly(
h, gens=(x, y), field=True).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
assert Poly.from_poly(
h, gens=(x, y), field=True).rep == DMP([[QQ(1)], [QQ(1), QQ(0)]], QQ)
def test_Poly_from_expr():
raises(GeneratorsNeeded, lambda: Poly.from_expr(S.Zero))
raises(GeneratorsNeeded, lambda: Poly.from_expr(S(7)))
F3 = FF(3)
assert Poly.from_expr(x + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(y + 5, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(x + 5, x, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(y + 5, y, domain=F3).rep == DMP([F3(1), F3(2)], F3)
assert Poly.from_expr(x + y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3)
assert Poly.from_expr(x + y, x, y, domain=F3).rep == DMP([[F3(1)], [F3(1), F3(0)]], F3)
assert Poly.from_expr(x + 5).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, y).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(y + 5, y, domain=ZZ).rep == DMP([1, 5], ZZ)
assert Poly.from_expr(x + 5, x, y, domain=ZZ).rep == DMP([[1], [5]], ZZ)
assert Poly.from_expr(y + 5, x, y, domain=ZZ).rep == DMP([[1, 5]], ZZ)
def test_poly_from_domain_element():
dom = ZZ[x]
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = dom.get_field()
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = QQ[x]
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = dom.get_field()
assert Poly(dom(x+1), y, domain=dom).rep == DMP([dom(x+1)], dom)
dom = ZZ.old_poly_ring(x)
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = dom.get_field()
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = QQ.old_poly_ring(x)
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = dom.get_field()
assert Poly(dom([1, 1]), y, domain=dom).rep == DMP([dom([1, 1])], dom)
dom = QQ.algebraic_field(I)
assert Poly(dom([1, 1]), x, domain=dom).rep == DMP([dom([1, 1])], dom)
def test_Poly__new__():
raises(GeneratorsError, lambda: Poly(x + 1, x, x))
raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[x]))
raises(GeneratorsError, lambda: Poly(x + y, x, y, domain=ZZ[y]))
raises(OptionError, lambda: Poly(x, x, symmetric=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, domain=QQ))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, gaussian=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, gaussian=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=[sqrt(3)]))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=[sqrt(3)]))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, extension=True))
raises(OptionError, lambda: Poly(x + 2, x, modulus=3, extension=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=True))
raises(OptionError, lambda: Poly(x + 2, x, domain=ZZ, greedy=False))
raises(OptionError, lambda: Poly(x + 2, x, domain=QQ, field=False))
raises(NotImplementedError, lambda: Poly(x + 1, x, modulus=3, order='grlex'))
raises(NotImplementedError, lambda: Poly(x + 1, x, order='grlex'))
raises(GeneratorsNeeded, lambda: Poly({1: 2, 0: 1}))
raises(GeneratorsNeeded, lambda: Poly([2, 1]))
raises(GeneratorsNeeded, lambda: Poly((2, 1)))
raises(GeneratorsNeeded, lambda: Poly(1))
f = a*x**2 + b*x + c
assert Poly({2: a, 1: b, 0: c}, x) == f
assert Poly(iter([a, b, c]), x) == f
assert Poly([a, b, c], x) == f
assert Poly((a, b, c), x) == f
f = Poly({}, x, y, z)
assert f.gens == (x, y, z) and f.as_expr() == 0
assert Poly(Poly(a*x + b*y, x, y), x) == Poly(a*x + b*y, x)
assert Poly(3*x**2 + 2*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1]
assert Poly(3*x**2 + 2*x + 1, domain='QQ').all_coeffs() == [3, 2, 1]
assert Poly(3*x**2 + 2*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0]
raises(CoercionFailed, lambda: Poly(3*x**2/5 + x*Rational(2, 5) + 1, domain='ZZ'))
assert Poly(
3*x**2/5 + x*Rational(2, 5) + 1, domain='QQ').all_coeffs() == [Rational(3, 5), Rational(2, 5), 1]
assert _epsilon_eq(
Poly(3*x**2/5 + x*Rational(2, 5) + 1, domain='RR').all_coeffs(), [0.6, 0.4, 1.0])
assert Poly(3.0*x**2 + 2.0*x + 1, domain='ZZ').all_coeffs() == [3, 2, 1]
assert Poly(3.0*x**2 + 2.0*x + 1, domain='QQ').all_coeffs() == [3, 2, 1]
assert Poly(
3.0*x**2 + 2.0*x + 1, domain='RR').all_coeffs() == [3.0, 2.0, 1.0]
raises(CoercionFailed, lambda: Poly(3.1*x**2 + 2.1*x + 1, domain='ZZ'))
assert Poly(3.1*x**2 + 2.1*x + 1, domain='QQ').all_coeffs() == [Rational(31, 10), Rational(21, 10), 1]
assert Poly(3.1*x**2 + 2.1*x + 1, domain='RR').all_coeffs() == [3.1, 2.1, 1.0]
assert Poly({(2, 1): 1, (1, 2): 2, (1, 1): 3}, x, y) == \
Poly(x**2*y + 2*x*y**2 + 3*x*y, x, y)
assert Poly(x**2 + 1, extension=I).get_domain() == QQ.algebraic_field(I)
f = 3*x**5 - x**4 + x**3 - x** 2 + 65538
assert Poly(f, x, modulus=65537, symmetric=True) == \
Poly(3*x**5 - x**4 + x**3 - x** 2 + 1, x, modulus=65537,
symmetric=True)
assert Poly(f, x, modulus=65537, symmetric=False) == \
Poly(3*x**5 + 65536*x**4 + x**3 + 65536*x** 2 + 1, x,
modulus=65537, symmetric=False)
assert isinstance(Poly(x**2 + x + 1.0).get_domain(), RealField)
assert isinstance(Poly(x**2 + x + I + 1.0).get_domain(), ComplexField)
def test_Poly__args():
assert Poly(x**2 + 1).args == (x**2 + 1, x)
def test_Poly__gens():
assert Poly((x - p)*(x - q), x).gens == (x,)
assert Poly((x - p)*(x - q), p).gens == (p,)
assert Poly((x - p)*(x - q), q).gens == (q,)
assert Poly((x - p)*(x - q), x, p).gens == (x, p)
assert Poly((x - p)*(x - q), x, q).gens == (x, q)
assert Poly((x - p)*(x - q), x, p, q).gens == (x, p, q)
assert Poly((x - p)*(x - q), p, x, q).gens == (p, x, q)
assert Poly((x - p)*(x - q), p, q, x).gens == (p, q, x)
assert Poly((x - p)*(x - q)).gens == (x, p, q)
assert Poly((x - p)*(x - q), sort='x > p > q').gens == (x, p, q)
assert Poly((x - p)*(x - q), sort='p > x > q').gens == (p, x, q)
assert Poly((x - p)*(x - q), sort='p > q > x').gens == (p, q, x)
assert Poly((x - p)*(x - q), x, p, q, sort='p > q > x').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='x').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='p').gens == (p, x, q)
assert Poly((x - p)*(x - q), wrt='q').gens == (q, x, p)
assert Poly((x - p)*(x - q), wrt=x).gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt=p).gens == (p, x, q)
assert Poly((x - p)*(x - q), wrt=q).gens == (q, x, p)
assert Poly((x - p)*(x - q), x, p, q, wrt='p').gens == (x, p, q)
assert Poly((x - p)*(x - q), wrt='p', sort='q > x').gens == (p, q, x)
assert Poly((x - p)*(x - q), wrt='q', sort='p > x').gens == (q, p, x)
def test_Poly_zero():
assert Poly(x).zero == Poly(0, x, domain=ZZ)
assert Poly(x/2).zero == Poly(0, x, domain=QQ)
def test_Poly_one():
assert Poly(x).one == Poly(1, x, domain=ZZ)
assert Poly(x/2).one == Poly(1, x, domain=QQ)
def test_Poly__unify():
raises(UnificationFailed, lambda: Poly(x)._unify(y))
F3 = FF(3)
F5 = FF(5)
assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=3))[2:] == (
DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3))
assert Poly(x, x, modulus=3)._unify(Poly(y, y, modulus=5))[2:] == (
DMP([[F5(1)], []], F5), DMP([[F5(1), F5(0)]], F5))
assert Poly(y, x, y)._unify(Poly(x, x, modulus=3))[2:] == (DMP([[F3(1), F3(0)]], F3), DMP([[F3(1)], []], F3))
assert Poly(x, x, modulus=3)._unify(Poly(y, x, y))[2:] == (DMP([[F3(1)], []], F3), DMP([[F3(1), F3(0)]], F3))
assert Poly(x + 1, x)._unify(Poly(x + 2, x))[2:] == (DMP([1, 1], ZZ), DMP([1, 2], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([1, 1], QQ), DMP([1, 2], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([1, 1], QQ), DMP([1, 2], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, y, x))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, x, domain='QQ')._unify(Poly(x + 2, y, x))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, x)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x))[2:] == (DMP([[1], [1]], ZZ), DMP([[1], [2]], ZZ))
assert Poly(x + 1, x, y, domain='QQ')._unify(Poly(x + 2, y, x))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, x, y)._unify(Poly(x + 2, y, x, domain='QQ'))[2:] == (DMP([[1], [1]], QQ), DMP([[1], [2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y))[2:] == (DMP([[1, 1]], ZZ), DMP([[1, 2]], ZZ))
assert Poly(x + 1, y, x, domain='QQ')._unify(Poly(x + 2, x, y))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x + 1, y, x)._unify(Poly(x + 2, x, y, domain='QQ'))[2:] == (DMP([[1, 1]], QQ), DMP([[1, 2]], QQ))
assert Poly(x**2 + I, x, domain=ZZ_I).unify(Poly(x**2 + sqrt(2), x, extension=True)) == \
(Poly(x**2 + I, x, domain='QQ<sqrt(2) + I>'), Poly(x**2 + sqrt(2), x, domain='QQ<sqrt(2) + I>'))
F, A, B = field("a,b", ZZ)
assert Poly(a*x, x, domain='ZZ[a]')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \
(DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain()))
assert Poly(a*x, x, domain='ZZ(a)')._unify(Poly(a*b*x, x, domain='ZZ(a,b)'))[2:] == \
(DMP([A, F(0)], F.to_domain()), DMP([A*B, F(0)], F.to_domain()))
raises(CoercionFailed, lambda: Poly(Poly(x**2 + x**2*z, y, field=True), domain='ZZ(x)'))
f = Poly(t**2 + t/3 + x, t, domain='QQ(x)')
g = Poly(t**2 + t/3 + x, t, domain='QQ[x]')
assert f._unify(g)[2:] == (f.rep, f.rep)
def test_Poly_free_symbols():
assert Poly(x**2 + 1).free_symbols == {x}
assert Poly(x**2 + y*z).free_symbols == {x, y, z}
assert Poly(x**2 + y*z, x).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z)).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z), x).free_symbols == {x, y, z}
assert Poly(x**2 + sin(y*z), x, domain=EX).free_symbols == {x, y, z}
assert Poly(1 + x + x**2, x, y, z).free_symbols == {x}
assert Poly(x + sin(y), z).free_symbols == {x, y}
def test_PurePoly_free_symbols():
assert PurePoly(x**2 + 1).free_symbols == set()
assert PurePoly(x**2 + y*z).free_symbols == set()
assert PurePoly(x**2 + y*z, x).free_symbols == {y, z}
assert PurePoly(x**2 + sin(y*z)).free_symbols == set()
assert PurePoly(x**2 + sin(y*z), x).free_symbols == {y, z}
assert PurePoly(x**2 + sin(y*z), x, domain=EX).free_symbols == {y, z}
def test_Poly__eq__():
assert (Poly(x, x) == Poly(x, x)) is True
assert (Poly(x, x, domain=QQ) == Poly(x, x)) is False
assert (Poly(x, x) == Poly(x, x, domain=QQ)) is False
assert (Poly(x, x, domain=ZZ[a]) == Poly(x, x)) is False
assert (Poly(x, x) == Poly(x, x, domain=ZZ[a])) is False
assert (Poly(x*y, x, y) == Poly(x, x)) is False
assert (Poly(x, x, y) == Poly(x, x)) is False
assert (Poly(x, x) == Poly(x, x, y)) is False
assert (Poly(x**2 + 1, x) == Poly(y**2 + 1, y)) is False
assert (Poly(y**2 + 1, y) == Poly(x**2 + 1, x)) is False
f = Poly(x, x, domain=ZZ)
g = Poly(x, x, domain=QQ)
assert f.eq(g) is False
assert f.ne(g) is True
assert f.eq(g, strict=True) is False
assert f.ne(g, strict=True) is True
t0 = Symbol('t0')
f = Poly((t0/2 + x**2)*t**2 - x**2*t, t, domain='QQ[x,t0]')
g = Poly((t0/2 + x**2)*t**2 - x**2*t, t, domain='ZZ(x,t0)')
assert (f == g) is False
def test_PurePoly__eq__():
assert (PurePoly(x, x) == PurePoly(x, x)) is True
assert (PurePoly(x, x, domain=QQ) == PurePoly(x, x)) is True
assert (PurePoly(x, x) == PurePoly(x, x, domain=QQ)) is True
assert (PurePoly(x, x, domain=ZZ[a]) == PurePoly(x, x)) is True
assert (PurePoly(x, x) == PurePoly(x, x, domain=ZZ[a])) is True
assert (PurePoly(x*y, x, y) == PurePoly(x, x)) is False
assert (PurePoly(x, x, y) == PurePoly(x, x)) is False
assert (PurePoly(x, x) == PurePoly(x, x, y)) is False
assert (PurePoly(x**2 + 1, x) == PurePoly(y**2 + 1, y)) is True
assert (PurePoly(y**2 + 1, y) == PurePoly(x**2 + 1, x)) is True
f = PurePoly(x, x, domain=ZZ)
g = PurePoly(x, x, domain=QQ)
assert f.eq(g) is True
assert f.ne(g) is False
assert f.eq(g, strict=True) is False
assert f.ne(g, strict=True) is True
f = PurePoly(x, x, domain=ZZ)
g = PurePoly(y, y, domain=QQ)
assert f.eq(g) is True
assert f.ne(g) is False
assert f.eq(g, strict=True) is False
assert f.ne(g, strict=True) is True
def test_PurePoly_Poly():
assert isinstance(PurePoly(Poly(x**2 + 1)), PurePoly) is True
assert isinstance(Poly(PurePoly(x**2 + 1)), Poly) is True
def test_Poly_get_domain():
assert Poly(2*x).get_domain() == ZZ
assert Poly(2*x, domain='ZZ').get_domain() == ZZ
assert Poly(2*x, domain='QQ').get_domain() == QQ
assert Poly(x/2).get_domain() == QQ
raises(CoercionFailed, lambda: Poly(x/2, domain='ZZ'))
assert Poly(x/2, domain='QQ').get_domain() == QQ
assert isinstance(Poly(0.2*x).get_domain(), RealField)
def test_Poly_set_domain():
assert Poly(2*x + 1).set_domain(ZZ) == Poly(2*x + 1)
assert Poly(2*x + 1).set_domain('ZZ') == Poly(2*x + 1)
assert Poly(2*x + 1).set_domain(QQ) == Poly(2*x + 1, domain='QQ')
assert Poly(2*x + 1).set_domain('QQ') == Poly(2*x + 1, domain='QQ')
assert Poly(Rational(2, 10)*x + Rational(1, 10)).set_domain('RR') == Poly(0.2*x + 0.1)
assert Poly(0.2*x + 0.1).set_domain('QQ') == Poly(Rational(2, 10)*x + Rational(1, 10))
raises(CoercionFailed, lambda: Poly(x/2 + 1).set_domain(ZZ))
raises(CoercionFailed, lambda: Poly(x + 1, modulus=2).set_domain(QQ))
raises(GeneratorsError, lambda: Poly(x*y, x, y).set_domain(ZZ[y]))
def test_Poly_get_modulus():
assert Poly(x**2 + 1, modulus=2).get_modulus() == 2
raises(PolynomialError, lambda: Poly(x**2 + 1).get_modulus())
def test_Poly_set_modulus():
assert Poly(
x**2 + 1, modulus=2).set_modulus(7) == Poly(x**2 + 1, modulus=7)
assert Poly(
x**2 + 5, modulus=7).set_modulus(2) == Poly(x**2 + 1, modulus=2)
assert Poly(x**2 + 1).set_modulus(2) == Poly(x**2 + 1, modulus=2)
raises(CoercionFailed, lambda: Poly(x/2 + 1).set_modulus(2))
def test_Poly_add_ground():
assert Poly(x + 1).add_ground(2) == Poly(x + 3)
def test_Poly_sub_ground():
assert Poly(x + 1).sub_ground(2) == Poly(x - 1)
def test_Poly_mul_ground():
assert Poly(x + 1).mul_ground(2) == Poly(2*x + 2)
def test_Poly_quo_ground():
assert Poly(2*x + 4).quo_ground(2) == Poly(x + 2)
assert Poly(2*x + 3).quo_ground(2) == Poly(x + 1)
def test_Poly_exquo_ground():
assert Poly(2*x + 4).exquo_ground(2) == Poly(x + 2)
raises(ExactQuotientFailed, lambda: Poly(2*x + 3).exquo_ground(2))
def test_Poly_abs():
assert Poly(-x + 1, x).abs() == abs(Poly(-x + 1, x)) == Poly(x + 1, x)
def test_Poly_neg():
assert Poly(-x + 1, x).neg() == -Poly(-x + 1, x) == Poly(x - 1, x)
def test_Poly_add():
assert Poly(0, x).add(Poly(0, x)) == Poly(0, x)
assert Poly(0, x) + Poly(0, x) == Poly(0, x)
assert Poly(1, x).add(Poly(0, x)) == Poly(1, x)
assert Poly(1, x, y) + Poly(0, x) == Poly(1, x, y)
assert Poly(0, x).add(Poly(1, x, y)) == Poly(1, x, y)
assert Poly(0, x, y) + Poly(1, x, y) == Poly(1, x, y)
assert Poly(1, x) + x == Poly(x + 1, x)
with warns_deprecated_sympy():
Poly(1, x) + sin(x)
assert Poly(x, x) + 1 == Poly(x + 1, x)
assert 1 + Poly(x, x) == Poly(x + 1, x)
def test_Poly_sub():
assert Poly(0, x).sub(Poly(0, x)) == Poly(0, x)
assert Poly(0, x) - Poly(0, x) == Poly(0, x)
assert Poly(1, x).sub(Poly(0, x)) == Poly(1, x)
assert Poly(1, x, y) - Poly(0, x) == Poly(1, x, y)
assert Poly(0, x).sub(Poly(1, x, y)) == Poly(-1, x, y)
assert Poly(0, x, y) - Poly(1, x, y) == Poly(-1, x, y)
assert Poly(1, x) - x == Poly(1 - x, x)
with warns_deprecated_sympy():
Poly(1, x) - sin(x)
assert Poly(x, x) - 1 == Poly(x - 1, x)
assert 1 - Poly(x, x) == Poly(1 - x, x)
def test_Poly_mul():
assert Poly(0, x).mul(Poly(0, x)) == Poly(0, x)
assert Poly(0, x) * Poly(0, x) == Poly(0, x)
assert Poly(2, x).mul(Poly(4, x)) == Poly(8, x)
assert Poly(2, x, y) * Poly(4, x) == Poly(8, x, y)
assert Poly(4, x).mul(Poly(2, x, y)) == Poly(8, x, y)
assert Poly(4, x, y) * Poly(2, x, y) == Poly(8, x, y)
assert Poly(1, x) * x == Poly(x, x)
with warns_deprecated_sympy():
Poly(1, x) * sin(x)
assert Poly(x, x) * 2 == Poly(2*x, x)
assert 2 * Poly(x, x) == Poly(2*x, x)
def test_issue_13079():
assert Poly(x)*x == Poly(x**2, x, domain='ZZ')
assert x*Poly(x) == Poly(x**2, x, domain='ZZ')
assert -2*Poly(x) == Poly(-2*x, x, domain='ZZ')
assert S(-2)*Poly(x) == Poly(-2*x, x, domain='ZZ')
assert Poly(x)*S(-2) == Poly(-2*x, x, domain='ZZ')
def test_Poly_sqr():
assert Poly(x*y, x, y).sqr() == Poly(x**2*y**2, x, y)
def test_Poly_pow():
assert Poly(x, x).pow(10) == Poly(x**10, x)
assert Poly(x, x).pow(Integer(10)) == Poly(x**10, x)
assert Poly(2*y, x, y).pow(4) == Poly(16*y**4, x, y)
assert Poly(2*y, x, y).pow(Integer(4)) == Poly(16*y**4, x, y)
assert Poly(7*x*y, x, y)**3 == Poly(343*x**3*y**3, x, y)
raises(TypeError, lambda: Poly(x*y + 1, x, y)**(-1))
raises(TypeError, lambda: Poly(x*y + 1, x, y)**x)
def test_Poly_divmod():
f, g = Poly(x**2), Poly(x)
q, r = g, Poly(0, x)
assert divmod(f, g) == (q, r)
assert f // g == q
assert f % g == r
assert divmod(f, x) == (q, r)
assert f // x == q
assert f % x == r
q, r = Poly(0, x), Poly(2, x)
assert divmod(2, g) == (q, r)
assert 2 // g == q
assert 2 % g == r
assert Poly(x)/Poly(x) == 1
assert Poly(x**2)/Poly(x) == x
assert Poly(x)/Poly(x**2) == 1/x
def test_Poly_eq_ne():
assert (Poly(x + y, x, y) == Poly(x + y, x, y)) is True
assert (Poly(x + y, x) == Poly(x + y, x, y)) is False
assert (Poly(x + y, x, y) == Poly(x + y, x)) is False
assert (Poly(x + y, x) == Poly(x + y, x)) is True
assert (Poly(x + y, y) == Poly(x + y, y)) is True
assert (Poly(x + y, x, y) == x + y) is True
assert (Poly(x + y, x) == x + y) is True
assert (Poly(x + y, x, y) == x + y) is True
assert (Poly(x + y, x) == x + y) is True
assert (Poly(x + y, y) == x + y) is True
assert (Poly(x + y, x, y) != Poly(x + y, x, y)) is False
assert (Poly(x + y, x) != Poly(x + y, x, y)) is True
assert (Poly(x + y, x, y) != Poly(x + y, x)) is True
assert (Poly(x + y, x) != Poly(x + y, x)) is False
assert (Poly(x + y, y) != Poly(x + y, y)) is False
assert (Poly(x + y, x, y) != x + y) is False
assert (Poly(x + y, x) != x + y) is False
assert (Poly(x + y, x, y) != x + y) is False
assert (Poly(x + y, x) != x + y) is False
assert (Poly(x + y, y) != x + y) is False
assert (Poly(x, x) == sin(x)) is False
assert (Poly(x, x) != sin(x)) is True
def test_Poly_nonzero():
assert not bool(Poly(0, x)) is True
assert not bool(Poly(1, x)) is False
def test_Poly_properties():
assert Poly(0, x).is_zero is True
assert Poly(1, x).is_zero is False
assert Poly(1, x).is_one is True
assert Poly(2, x).is_one is False
assert Poly(x - 1, x).is_sqf is True
assert Poly((x - 1)**2, x).is_sqf is False
assert Poly(x - 1, x).is_monic is True
assert Poly(2*x - 1, x).is_monic is False
assert Poly(3*x + 2, x).is_primitive is True
assert Poly(4*x + 2, x).is_primitive is False
assert Poly(1, x).is_ground is True
assert Poly(x, x).is_ground is False
assert Poly(x + y + z + 1).is_linear is True
assert Poly(x*y*z + 1).is_linear is False
assert Poly(x*y + z + 1).is_quadratic is True
assert Poly(x*y*z + 1).is_quadratic is False
assert Poly(x*y).is_monomial is True
assert Poly(x*y + 1).is_monomial is False
assert Poly(x**2 + x*y).is_homogeneous is True
assert Poly(x**3 + x*y).is_homogeneous is False
assert Poly(x).is_univariate is True
assert Poly(x*y).is_univariate is False
assert Poly(x*y).is_multivariate is True
assert Poly(x).is_multivariate is False
assert Poly(
x**16 + x**14 - x**10 + x**8 - x**6 + x**2 + 1).is_cyclotomic is False
assert Poly(
x**16 + x**14 - x**10 - x**8 - x**6 + x**2 + 1).is_cyclotomic is True
def test_Poly_is_irreducible():
assert Poly(x**2 + x + 1).is_irreducible is True
assert Poly(x**2 + 2*x + 1).is_irreducible is False
assert Poly(7*x + 3, modulus=11).is_irreducible is True
assert Poly(7*x**2 + 3*x + 1, modulus=11).is_irreducible is False
def test_Poly_subs():
assert Poly(x + 1).subs(x, 0) == 1
assert Poly(x + 1).subs(x, x) == Poly(x + 1)
assert Poly(x + 1).subs(x, y) == Poly(y + 1)
assert Poly(x*y, x).subs(y, x) == x**2
assert Poly(x*y, x).subs(x, y) == y**2
def test_Poly_replace():
assert Poly(x + 1).replace(x) == Poly(x + 1)
assert Poly(x + 1).replace(y) == Poly(y + 1)
raises(PolynomialError, lambda: Poly(x + y).replace(z))
assert Poly(x + 1).replace(x, x) == Poly(x + 1)
assert Poly(x + 1).replace(x, y) == Poly(y + 1)
assert Poly(x + y).replace(x, x) == Poly(x + y)
assert Poly(x + y).replace(x, z) == Poly(z + y, z, y)
assert Poly(x + y).replace(y, y) == Poly(x + y)
assert Poly(x + y).replace(y, z) == Poly(x + z, x, z)
assert Poly(x + y).replace(z, t) == Poly(x + y)
raises(PolynomialError, lambda: Poly(x + y).replace(x, y))
assert Poly(x + y, x).replace(x, z) == Poly(z + y, z)
assert Poly(x + y, y).replace(y, z) == Poly(x + z, z)
raises(PolynomialError, lambda: Poly(x + y, x).replace(x, y))
raises(PolynomialError, lambda: Poly(x + y, y).replace(y, x))
def test_Poly_reorder():
raises(PolynomialError, lambda: Poly(x + y).reorder(x, z))
assert Poly(x + y, x, y).reorder(x, y) == Poly(x + y, x, y)
assert Poly(x + y, x, y).reorder(y, x) == Poly(x + y, y, x)
assert Poly(x + y, y, x).reorder(x, y) == Poly(x + y, x, y)
assert Poly(x + y, y, x).reorder(y, x) == Poly(x + y, y, x)
assert Poly(x + y, x, y).reorder(wrt=x) == Poly(x + y, x, y)
assert Poly(x + y, x, y).reorder(wrt=y) == Poly(x + y, y, x)
def test_Poly_ltrim():
f = Poly(y**2 + y*z**2, x, y, z).ltrim(y)
assert f.as_expr() == y**2 + y*z**2 and f.gens == (y, z)
assert Poly(x*y - x, z, x, y).ltrim(1) == Poly(x*y - x, x, y)
raises(PolynomialError, lambda: Poly(x*y**2 + y**2, x, y).ltrim(y))
raises(PolynomialError, lambda: Poly(x*y - x, x, y).ltrim(-1))
def test_Poly_has_only_gens():
assert Poly(x*y + 1, x, y, z).has_only_gens(x, y) is True
assert Poly(x*y + z, x, y, z).has_only_gens(x, y) is False
raises(GeneratorsError, lambda: Poly(x*y**2 + y**2, x, y).has_only_gens(t))
def test_Poly_to_ring():
assert Poly(2*x + 1, domain='ZZ').to_ring() == Poly(2*x + 1, domain='ZZ')
assert Poly(2*x + 1, domain='QQ').to_ring() == Poly(2*x + 1, domain='ZZ')
raises(CoercionFailed, lambda: Poly(x/2 + 1).to_ring())
raises(DomainError, lambda: Poly(2*x + 1, modulus=3).to_ring())
def test_Poly_to_field():
assert Poly(2*x + 1, domain='ZZ').to_field() == Poly(2*x + 1, domain='QQ')
assert Poly(2*x + 1, domain='QQ').to_field() == Poly(2*x + 1, domain='QQ')
assert Poly(x/2 + 1, domain='QQ').to_field() == Poly(x/2 + 1, domain='QQ')
assert Poly(2*x + 1, modulus=3).to_field() == Poly(2*x + 1, modulus=3)
assert Poly(2.0*x + 1.0).to_field() == Poly(2.0*x + 1.0)
def test_Poly_to_exact():
assert Poly(2*x).to_exact() == Poly(2*x)
assert Poly(x/2).to_exact() == Poly(x/2)
assert Poly(0.1*x).to_exact() == Poly(x/10)
def test_Poly_retract():
f = Poly(x**2 + 1, x, domain=QQ[y])
assert f.retract() == Poly(x**2 + 1, x, domain='ZZ')
assert f.retract(field=True) == Poly(x**2 + 1, x, domain='QQ')
assert Poly(0, x, y).retract() == Poly(0, x, y)
def test_Poly_slice():
f = Poly(x**3 + 2*x**2 + 3*x + 4)
assert f.slice(0, 0) == Poly(0, x)
assert f.slice(0, 1) == Poly(4, x)
assert f.slice(0, 2) == Poly(3*x + 4, x)
assert f.slice(0, 3) == Poly(2*x**2 + 3*x + 4, x)
assert f.slice(0, 4) == Poly(x**3 + 2*x**2 + 3*x + 4, x)
assert f.slice(x, 0, 0) == Poly(0, x)
assert f.slice(x, 0, 1) == Poly(4, x)
assert f.slice(x, 0, 2) == Poly(3*x + 4, x)
assert f.slice(x, 0, 3) == Poly(2*x**2 + 3*x + 4, x)
assert f.slice(x, 0, 4) == Poly(x**3 + 2*x**2 + 3*x + 4, x)
def test_Poly_coeffs():
assert Poly(0, x).coeffs() == [0]
assert Poly(1, x).coeffs() == [1]
assert Poly(2*x + 1, x).coeffs() == [2, 1]
assert Poly(7*x**2 + 2*x + 1, x).coeffs() == [7, 2, 1]
assert Poly(7*x**4 + 2*x + 1, x).coeffs() == [7, 2, 1]
assert Poly(x*y**7 + 2*x**2*y**3).coeffs('lex') == [2, 1]
assert Poly(x*y**7 + 2*x**2*y**3).coeffs('grlex') == [1, 2]
def test_Poly_monoms():
assert Poly(0, x).monoms() == [(0,)]
assert Poly(1, x).monoms() == [(0,)]
assert Poly(2*x + 1, x).monoms() == [(1,), (0,)]
assert Poly(7*x**2 + 2*x + 1, x).monoms() == [(2,), (1,), (0,)]
assert Poly(7*x**4 + 2*x + 1, x).monoms() == [(4,), (1,), (0,)]
assert Poly(x*y**7 + 2*x**2*y**3).monoms('lex') == [(2, 3), (1, 7)]
assert Poly(x*y**7 + 2*x**2*y**3).monoms('grlex') == [(1, 7), (2, 3)]
def test_Poly_terms():
assert Poly(0, x).terms() == [((0,), 0)]
assert Poly(1, x).terms() == [((0,), 1)]
assert Poly(2*x + 1, x).terms() == [((1,), 2), ((0,), 1)]
assert Poly(7*x**2 + 2*x + 1, x).terms() == [((2,), 7), ((1,), 2), ((0,), 1)]
assert Poly(7*x**4 + 2*x + 1, x).terms() == [((4,), 7), ((1,), 2), ((0,), 1)]
assert Poly(
x*y**7 + 2*x**2*y**3).terms('lex') == [((2, 3), 2), ((1, 7), 1)]
assert Poly(
x*y**7 + 2*x**2*y**3).terms('grlex') == [((1, 7), 1), ((2, 3), 2)]
def test_Poly_all_coeffs():
assert Poly(0, x).all_coeffs() == [0]
assert Poly(1, x).all_coeffs() == [1]
assert Poly(2*x + 1, x).all_coeffs() == [2, 1]
assert Poly(7*x**2 + 2*x + 1, x).all_coeffs() == [7, 2, 1]
assert Poly(7*x**4 + 2*x + 1, x).all_coeffs() == [7, 0, 0, 2, 1]
def test_Poly_all_monoms():
assert Poly(0, x).all_monoms() == [(0,)]
assert Poly(1, x).all_monoms() == [(0,)]
assert Poly(2*x + 1, x).all_monoms() == [(1,), (0,)]
assert Poly(7*x**2 + 2*x + 1, x).all_monoms() == [(2,), (1,), (0,)]
assert Poly(7*x**4 + 2*x + 1, x).all_monoms() == [(4,), (3,), (2,), (1,), (0,)]
def test_Poly_all_terms():
assert Poly(0, x).all_terms() == [((0,), 0)]
assert Poly(1, x).all_terms() == [((0,), 1)]
assert Poly(2*x + 1, x).all_terms() == [((1,), 2), ((0,), 1)]
assert Poly(7*x**2 + 2*x + 1, x).all_terms() == \
[((2,), 7), ((1,), 2), ((0,), 1)]
assert Poly(7*x**4 + 2*x + 1, x).all_terms() == \
[((4,), 7), ((3,), 0), ((2,), 0), ((1,), 2), ((0,), 1)]
def test_Poly_termwise():
f = Poly(x**2 + 20*x + 400)
g = Poly(x**2 + 2*x + 4)
def func(monom, coeff):
(k,) = monom
return coeff//10**(2 - k)
assert f.termwise(func) == g
def func(monom, coeff):
(k,) = monom
return (k,), coeff//10**(2 - k)
assert f.termwise(func) == g
def test_Poly_length():
assert Poly(0, x).length() == 0
assert Poly(1, x).length() == 1
assert Poly(x, x).length() == 1
assert Poly(x + 1, x).length() == 2
assert Poly(x**2 + 1, x).length() == 2
assert Poly(x**2 + x + 1, x).length() == 3
def test_Poly_as_dict():
assert Poly(0, x).as_dict() == {}
assert Poly(0, x, y, z).as_dict() == {}
assert Poly(1, x).as_dict() == {(0,): 1}
assert Poly(1, x, y, z).as_dict() == {(0, 0, 0): 1}
assert Poly(x**2 + 3, x).as_dict() == {(2,): 1, (0,): 3}
assert Poly(x**2 + 3, x, y, z).as_dict() == {(2, 0, 0): 1, (0, 0, 0): 3}
assert Poly(3*x**2*y*z**3 + 4*x*y + 5*x*z).as_dict() == {(2, 1, 3): 3,
(1, 1, 0): 4, (1, 0, 1): 5}
def test_Poly_as_expr():
assert Poly(0, x).as_expr() == 0
assert Poly(0, x, y, z).as_expr() == 0
assert Poly(1, x).as_expr() == 1
assert Poly(1, x, y, z).as_expr() == 1
assert Poly(x**2 + 3, x).as_expr() == x**2 + 3
assert Poly(x**2 + 3, x, y, z).as_expr() == x**2 + 3
assert Poly(
3*x**2*y*z**3 + 4*x*y + 5*x*z).as_expr() == 3*x**2*y*z**3 + 4*x*y + 5*x*z
f = Poly(x**2 + 2*x*y**2 - y, x, y)
assert f.as_expr() == -y + x**2 + 2*x*y**2
assert f.as_expr({x: 5}) == 25 - y + 10*y**2
assert f.as_expr({y: 6}) == -6 + 72*x + x**2
assert f.as_expr({x: 5, y: 6}) == 379
assert f.as_expr(5, 6) == 379
raises(GeneratorsError, lambda: f.as_expr({z: 7}))
def test_Poly_lift():
assert Poly(x**4 - I*x + 17*I, x, gaussian=True).lift() == \
Poly(x**16 + 2*x**10 + 578*x**8 + x**4 - 578*x**2 + 83521,
x, domain='QQ')
def test_Poly_deflate():
assert Poly(0, x).deflate() == ((1,), Poly(0, x))
assert Poly(1, x).deflate() == ((1,), Poly(1, x))
assert Poly(x, x).deflate() == ((1,), Poly(x, x))
assert Poly(x**2, x).deflate() == ((2,), Poly(x, x))
assert Poly(x**17, x).deflate() == ((17,), Poly(x, x))
assert Poly(
x**2*y*z**11 + x**4*z**11).deflate() == ((2, 1, 11), Poly(x*y*z + x**2*z))
def test_Poly_inject():
f = Poly(x**2*y + x*y**3 + x*y + 1, x)
assert f.inject() == Poly(x**2*y + x*y**3 + x*y + 1, x, y)
assert f.inject(front=True) == Poly(y**3*x + y*x**2 + y*x + 1, y, x)
def test_Poly_eject():
f = Poly(x**2*y + x*y**3 + x*y + 1, x, y)
assert f.eject(x) == Poly(x*y**3 + (x**2 + x)*y + 1, y, domain='ZZ[x]')
assert f.eject(y) == Poly(y*x**2 + (y**3 + y)*x + 1, x, domain='ZZ[y]')
ex = x + y + z + t + w
g = Poly(ex, x, y, z, t, w)
assert g.eject(x) == Poly(ex, y, z, t, w, domain='ZZ[x]')
assert g.eject(x, y) == Poly(ex, z, t, w, domain='ZZ[x, y]')
assert g.eject(x, y, z) == Poly(ex, t, w, domain='ZZ[x, y, z]')
assert g.eject(w) == Poly(ex, x, y, z, t, domain='ZZ[w]')
assert g.eject(t, w) == Poly(ex, x, y, z, domain='ZZ[t, w]')
assert g.eject(z, t, w) == Poly(ex, x, y, domain='ZZ[z, t, w]')
raises(DomainError, lambda: Poly(x*y, x, y, domain=ZZ[z]).eject(y))
raises(NotImplementedError, lambda: Poly(x*y, x, y, z).eject(y))
def test_Poly_exclude():
assert Poly(x, x, y).exclude() == Poly(x, x)
assert Poly(x*y, x, y).exclude() == Poly(x*y, x, y)
assert Poly(1, x, y).exclude() == Poly(1, x, y)
def test_Poly__gen_to_level():
assert Poly(1, x, y)._gen_to_level(-2) == 0
assert Poly(1, x, y)._gen_to_level(-1) == 1
assert Poly(1, x, y)._gen_to_level( 0) == 0
assert Poly(1, x, y)._gen_to_level( 1) == 1
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level(-3))
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level( 2))
assert Poly(1, x, y)._gen_to_level(x) == 0
assert Poly(1, x, y)._gen_to_level(y) == 1
assert Poly(1, x, y)._gen_to_level('x') == 0
assert Poly(1, x, y)._gen_to_level('y') == 1
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level(z))
raises(PolynomialError, lambda: Poly(1, x, y)._gen_to_level('z'))
def test_Poly_degree():
assert Poly(0, x).degree() is -oo
assert Poly(1, x).degree() == 0
assert Poly(x, x).degree() == 1
assert Poly(0, x).degree(gen=0) is -oo
assert Poly(1, x).degree(gen=0) == 0
assert Poly(x, x).degree(gen=0) == 1
assert Poly(0, x).degree(gen=x) is -oo
assert Poly(1, x).degree(gen=x) == 0
assert Poly(x, x).degree(gen=x) == 1
assert Poly(0, x).degree(gen='x') is -oo
assert Poly(1, x).degree(gen='x') == 0
assert Poly(x, x).degree(gen='x') == 1
raises(PolynomialError, lambda: Poly(1, x).degree(gen=1))
raises(PolynomialError, lambda: Poly(1, x).degree(gen=y))
raises(PolynomialError, lambda: Poly(1, x).degree(gen='y'))
assert Poly(1, x, y).degree() == 0
assert Poly(2*y, x, y).degree() == 0
assert Poly(x*y, x, y).degree() == 1
assert Poly(1, x, y).degree(gen=x) == 0
assert Poly(2*y, x, y).degree(gen=x) == 0
assert Poly(x*y, x, y).degree(gen=x) == 1
assert Poly(1, x, y).degree(gen=y) == 0
assert Poly(2*y, x, y).degree(gen=y) == 1
assert Poly(x*y, x, y).degree(gen=y) == 1
assert degree(0, x) is -oo
assert degree(1, x) == 0
assert degree(x, x) == 1
assert degree(x*y**2, x) == 1
assert degree(x*y**2, y) == 2
assert degree(x*y**2, z) == 0
assert degree(pi) == 1
raises(TypeError, lambda: degree(y**2 + x**3))
raises(TypeError, lambda: degree(y**2 + x**3, 1))
raises(PolynomialError, lambda: degree(x, 1.1))
raises(PolynomialError, lambda: degree(x**2/(x**3 + 1), x))
assert degree(Poly(0,x),z) is -oo
assert degree(Poly(1,x),z) == 0
assert degree(Poly(x**2+y**3,y)) == 3
assert degree(Poly(y**2 + x**3, y, x), 1) == 3
assert degree(Poly(y**2 + x**3, x), z) == 0
assert degree(Poly(y**2 + x**3 + z**4, x), z) == 4
def test_Poly_degree_list():
assert Poly(0, x).degree_list() == (-oo,)
assert Poly(0, x, y).degree_list() == (-oo, -oo)
assert Poly(0, x, y, z).degree_list() == (-oo, -oo, -oo)
assert Poly(1, x).degree_list() == (0,)
assert Poly(1, x, y).degree_list() == (0, 0)
assert Poly(1, x, y, z).degree_list() == (0, 0, 0)
assert Poly(x**2*y + x**3*z**2 + 1).degree_list() == (3, 1, 2)
assert degree_list(1, x) == (0,)
assert degree_list(x, x) == (1,)
assert degree_list(x*y**2) == (1, 2)
raises(ComputationFailed, lambda: degree_list(1))
def test_Poly_total_degree():
assert Poly(x**2*y + x**3*z**2 + 1).total_degree() == 5
assert Poly(x**2 + z**3).total_degree() == 3
assert Poly(x*y*z + z**4).total_degree() == 4
assert Poly(x**3 + x + 1).total_degree() == 3
assert total_degree(x*y + z**3) == 3
assert total_degree(x*y + z**3, x, y) == 2
assert total_degree(1) == 0
assert total_degree(Poly(y**2 + x**3 + z**4)) == 4
assert total_degree(Poly(y**2 + x**3 + z**4, x)) == 3
assert total_degree(Poly(y**2 + x**3 + z**4, x), z) == 4
assert total_degree(Poly(x**9 + x*z*y + x**3*z**2 + z**7,x), z) == 7
def test_Poly_homogenize():
assert Poly(x**2+y).homogenize(z) == Poly(x**2+y*z)
assert Poly(x+y).homogenize(z) == Poly(x+y, x, y, z)
assert Poly(x+y**2).homogenize(y) == Poly(x*y+y**2)
def test_Poly_homogeneous_order():
assert Poly(0, x, y).homogeneous_order() is -oo
assert Poly(1, x, y).homogeneous_order() == 0
assert Poly(x, x, y).homogeneous_order() == 1
assert Poly(x*y, x, y).homogeneous_order() == 2
assert Poly(x + 1, x, y).homogeneous_order() is None
assert Poly(x*y + x, x, y).homogeneous_order() is None
assert Poly(x**5 + 2*x**3*y**2 + 9*x*y**4).homogeneous_order() == 5
assert Poly(x**5 + 2*x**3*y**3 + 9*x*y**4).homogeneous_order() is None
def test_Poly_LC():
assert Poly(0, x).LC() == 0
assert Poly(1, x).LC() == 1
assert Poly(2*x**2 + x, x).LC() == 2
assert Poly(x*y**7 + 2*x**2*y**3).LC('lex') == 2
assert Poly(x*y**7 + 2*x**2*y**3).LC('grlex') == 1
assert LC(x*y**7 + 2*x**2*y**3, order='lex') == 2
assert LC(x*y**7 + 2*x**2*y**3, order='grlex') == 1
def test_Poly_TC():
assert Poly(0, x).TC() == 0
assert Poly(1, x).TC() == 1
assert Poly(2*x**2 + x, x).TC() == 0
def test_Poly_EC():
assert Poly(0, x).EC() == 0
assert Poly(1, x).EC() == 1
assert Poly(2*x**2 + x, x).EC() == 1
assert Poly(x*y**7 + 2*x**2*y**3).EC('lex') == 1
assert Poly(x*y**7 + 2*x**2*y**3).EC('grlex') == 2
def test_Poly_coeff():
assert Poly(0, x).coeff_monomial(1) == 0
assert Poly(0, x).coeff_monomial(x) == 0
assert Poly(1, x).coeff_monomial(1) == 1
assert Poly(1, x).coeff_monomial(x) == 0
assert Poly(x**8, x).coeff_monomial(1) == 0
assert Poly(x**8, x).coeff_monomial(x**7) == 0
assert Poly(x**8, x).coeff_monomial(x**8) == 1
assert Poly(x**8, x).coeff_monomial(x**9) == 0
assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(1) == 1
assert Poly(3*x*y**2 + 1, x, y).coeff_monomial(x*y**2) == 3
p = Poly(24*x*y*exp(8) + 23*x, x, y)
assert p.coeff_monomial(x) == 23
assert p.coeff_monomial(y) == 0
assert p.coeff_monomial(x*y) == 24*exp(8)
assert p.as_expr().coeff(x) == 24*y*exp(8) + 23
raises(NotImplementedError, lambda: p.coeff(x))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(0))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x))
raises(ValueError, lambda: Poly(x + 1).coeff_monomial(3*x*y))
def test_Poly_nth():
assert Poly(0, x).nth(0) == 0
assert Poly(0, x).nth(1) == 0
assert Poly(1, x).nth(0) == 1
assert Poly(1, x).nth(1) == 0
assert Poly(x**8, x).nth(0) == 0
assert Poly(x**8, x).nth(7) == 0
assert Poly(x**8, x).nth(8) == 1
assert Poly(x**8, x).nth(9) == 0
assert Poly(3*x*y**2 + 1, x, y).nth(0, 0) == 1
assert Poly(3*x*y**2 + 1, x, y).nth(1, 2) == 3
raises(ValueError, lambda: Poly(x*y + 1, x, y).nth(1))
def test_Poly_LM():
assert Poly(0, x).LM() == (0,)
assert Poly(1, x).LM() == (0,)
assert Poly(2*x**2 + x, x).LM() == (2,)
assert Poly(x*y**7 + 2*x**2*y**3).LM('lex') == (2, 3)
assert Poly(x*y**7 + 2*x**2*y**3).LM('grlex') == (1, 7)
assert LM(x*y**7 + 2*x**2*y**3, order='lex') == x**2*y**3
assert LM(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7
def test_Poly_LM_custom_order():
f = Poly(x**2*y**3*z + x**2*y*z**3 + x*y*z + 1)
rev_lex = lambda monom: tuple(reversed(monom))
assert f.LM(order='lex') == (2, 3, 1)
assert f.LM(order=rev_lex) == (2, 1, 3)
def test_Poly_EM():
assert Poly(0, x).EM() == (0,)
assert Poly(1, x).EM() == (0,)
assert Poly(2*x**2 + x, x).EM() == (1,)
assert Poly(x*y**7 + 2*x**2*y**3).EM('lex') == (1, 7)
assert Poly(x*y**7 + 2*x**2*y**3).EM('grlex') == (2, 3)
def test_Poly_LT():
assert Poly(0, x).LT() == ((0,), 0)
assert Poly(1, x).LT() == ((0,), 1)
assert Poly(2*x**2 + x, x).LT() == ((2,), 2)
assert Poly(x*y**7 + 2*x**2*y**3).LT('lex') == ((2, 3), 2)
assert Poly(x*y**7 + 2*x**2*y**3).LT('grlex') == ((1, 7), 1)
assert LT(x*y**7 + 2*x**2*y**3, order='lex') == 2*x**2*y**3
assert LT(x*y**7 + 2*x**2*y**3, order='grlex') == x*y**7
def test_Poly_ET():
assert Poly(0, x).ET() == ((0,), 0)
assert Poly(1, x).ET() == ((0,), 1)
assert Poly(2*x**2 + x, x).ET() == ((1,), 1)
assert Poly(x*y**7 + 2*x**2*y**3).ET('lex') == ((1, 7), 1)
assert Poly(x*y**7 + 2*x**2*y**3).ET('grlex') == ((2, 3), 2)
def test_Poly_max_norm():
assert Poly(-1, x).max_norm() == 1
assert Poly( 0, x).max_norm() == 0
assert Poly( 1, x).max_norm() == 1
def test_Poly_l1_norm():
assert Poly(-1, x).l1_norm() == 1
assert Poly( 0, x).l1_norm() == 0
assert Poly( 1, x).l1_norm() == 1
def test_Poly_clear_denoms():
coeff, poly = Poly(x + 2, x).clear_denoms()
assert coeff == 1 and poly == Poly(
x + 2, x, domain='ZZ') and poly.get_domain() == ZZ
coeff, poly = Poly(x/2 + 1, x).clear_denoms()
assert coeff == 2 and poly == Poly(
x + 2, x, domain='QQ') and poly.get_domain() == QQ
coeff, poly = Poly(x/2 + 1, x).clear_denoms(convert=True)
assert coeff == 2 and poly == Poly(
x + 2, x, domain='ZZ') and poly.get_domain() == ZZ
coeff, poly = Poly(x/y + 1, x).clear_denoms(convert=True)
assert coeff == y and poly == Poly(
x + y, x, domain='ZZ[y]') and poly.get_domain() == ZZ[y]
coeff, poly = Poly(x/3 + sqrt(2), x, domain='EX').clear_denoms()
assert coeff == 3 and poly == Poly(
x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX
coeff, poly = Poly(
x/3 + sqrt(2), x, domain='EX').clear_denoms(convert=True)
assert coeff == 3 and poly == Poly(
x + 3*sqrt(2), x, domain='EX') and poly.get_domain() == EX
def test_Poly_rat_clear_denoms():
f = Poly(x**2/y + 1, x)
g = Poly(x**3 + y, x)
assert f.rat_clear_denoms(g) == \
(Poly(x**2 + y, x), Poly(y*x**3 + y**2, x))
f = f.set_domain(EX)
g = g.set_domain(EX)
assert f.rat_clear_denoms(g) == (f, g)
def test_issue_20427():
f = Poly(-117968192370600*18**(S(1)/3)/(217603955769048*(24201 +
253*sqrt(9165))**(S(1)/3) + 2273005839412*sqrt(9165)*(24201 +
253*sqrt(9165))**(S(1)/3)) - 15720318185*2**(S(2)/3)*3**(S(1)/3)*(24201
+ 253*sqrt(9165))**(S(2)/3)/(217603955769048*(24201 + 253*sqrt(9165))**
(S(1)/3) + 2273005839412*sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3))
+ 15720318185*12**(S(1)/3)*(24201 + 253*sqrt(9165))**(S(2)/3)/(
217603955769048*(24201 + 253*sqrt(9165))**(S(1)/3) + 2273005839412*
sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)) + 117968192370600*2**(
S(1)/3)*3**(S(2)/3)/(217603955769048*(24201 + 253*sqrt(9165))**(S(1)/3)
+ 2273005839412*sqrt(9165)*(24201 + 253*sqrt(9165))**(S(1)/3)), x)
assert f == Poly(0, x, domain='EX')
def test_Poly_integrate():
assert Poly(x + 1).integrate() == Poly(x**2/2 + x)
assert Poly(x + 1).integrate(x) == Poly(x**2/2 + x)
assert Poly(x + 1).integrate((x, 1)) == Poly(x**2/2 + x)
assert Poly(x*y + 1).integrate(x) == Poly(x**2*y/2 + x)
assert Poly(x*y + 1).integrate(y) == Poly(x*y**2/2 + y)
assert Poly(x*y + 1).integrate(x, x) == Poly(x**3*y/6 + x**2/2)
assert Poly(x*y + 1).integrate(y, y) == Poly(x*y**3/6 + y**2/2)
assert Poly(x*y + 1).integrate((x, 2)) == Poly(x**3*y/6 + x**2/2)
assert Poly(x*y + 1).integrate((y, 2)) == Poly(x*y**3/6 + y**2/2)
assert Poly(x*y + 1).integrate(x, y) == Poly(x**2*y**2/4 + x*y)
assert Poly(x*y + 1).integrate(y, x) == Poly(x**2*y**2/4 + x*y)
def test_Poly_diff():
assert Poly(x**2 + x).diff() == Poly(2*x + 1)
assert Poly(x**2 + x).diff(x) == Poly(2*x + 1)
assert Poly(x**2 + x).diff((x, 1)) == Poly(2*x + 1)
assert Poly(x**2*y**2 + x*y).diff(x) == Poly(2*x*y**2 + y)
assert Poly(x**2*y**2 + x*y).diff(y) == Poly(2*x**2*y + x)
assert Poly(x**2*y**2 + x*y).diff(x, x) == Poly(2*y**2, x, y)
assert Poly(x**2*y**2 + x*y).diff(y, y) == Poly(2*x**2, x, y)
assert Poly(x**2*y**2 + x*y).diff((x, 2)) == Poly(2*y**2, x, y)
assert Poly(x**2*y**2 + x*y).diff((y, 2)) == Poly(2*x**2, x, y)
assert Poly(x**2*y**2 + x*y).diff(x, y) == Poly(4*x*y + 1)
assert Poly(x**2*y**2 + x*y).diff(y, x) == Poly(4*x*y + 1)
def test_issue_9585():
assert diff(Poly(x**2 + x)) == Poly(2*x + 1)
assert diff(Poly(x**2 + x), x, evaluate=False) == \
Derivative(Poly(x**2 + x), x)
assert Derivative(Poly(x**2 + x), x).doit() == Poly(2*x + 1)
def test_Poly_eval():
assert Poly(0, x).eval(7) == 0
assert Poly(1, x).eval(7) == 1
assert Poly(x, x).eval(7) == 7
assert Poly(0, x).eval(0, 7) == 0
assert Poly(1, x).eval(0, 7) == 1
assert Poly(x, x).eval(0, 7) == 7
assert Poly(0, x).eval(x, 7) == 0
assert Poly(1, x).eval(x, 7) == 1
assert Poly(x, x).eval(x, 7) == 7
assert Poly(0, x).eval('x', 7) == 0
assert Poly(1, x).eval('x', 7) == 1
assert Poly(x, x).eval('x', 7) == 7
raises(PolynomialError, lambda: Poly(1, x).eval(1, 7))
raises(PolynomialError, lambda: Poly(1, x).eval(y, 7))
raises(PolynomialError, lambda: Poly(1, x).eval('y', 7))
assert Poly(123, x, y).eval(7) == Poly(123, y)
assert Poly(2*y, x, y).eval(7) == Poly(2*y, y)
assert Poly(x*y, x, y).eval(7) == Poly(7*y, y)
assert Poly(123, x, y).eval(x, 7) == Poly(123, y)
assert Poly(2*y, x, y).eval(x, 7) == Poly(2*y, y)
assert Poly(x*y, x, y).eval(x, 7) == Poly(7*y, y)
assert Poly(123, x, y).eval(y, 7) == Poly(123, x)
assert Poly(2*y, x, y).eval(y, 7) == Poly(14, x)
assert Poly(x*y, x, y).eval(y, 7) == Poly(7*x, x)
assert Poly(x*y + y, x, y).eval({x: 7}) == Poly(8*y, y)
assert Poly(x*y + y, x, y).eval({y: 7}) == Poly(7*x + 7, x)
assert Poly(x*y + y, x, y).eval({x: 6, y: 7}) == 49
assert Poly(x*y + y, x, y).eval({x: 7, y: 6}) == 48
assert Poly(x*y + y, x, y).eval((6, 7)) == 49
assert Poly(x*y + y, x, y).eval([6, 7]) == 49
assert Poly(x + 1, domain='ZZ').eval(S.Half) == Rational(3, 2)
assert Poly(x + 1, domain='ZZ').eval(sqrt(2)) == sqrt(2) + 1
raises(ValueError, lambda: Poly(x*y + y, x, y).eval((6, 7, 8)))
raises(DomainError, lambda: Poly(x + 1, domain='ZZ').eval(S.Half, auto=False))
# issue 6344
alpha = Symbol('alpha')
result = (2*alpha*z - 2*alpha + z**2 + 3)/(z**2 - 2*z + 1)
f = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, domain='ZZ[alpha]')
assert f.eval((z + 1)/(z - 1)) == result
g = Poly(x**2 + (alpha - 1)*x - alpha + 1, x, y, domain='ZZ[alpha]')
assert g.eval((z + 1)/(z - 1)) == Poly(result, y, domain='ZZ(alpha,z)')
def test_Poly___call__():
f = Poly(2*x*y + 3*x + y + 2*z)
assert f(2) == Poly(5*y + 2*z + 6)
assert f(2, 5) == Poly(2*z + 31)
assert f(2, 5, 7) == 45
def test_parallel_poly_from_expr():
assert parallel_poly_from_expr(
[x - 1, x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), x**2 - 1], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr([Poly(
x - 1, x), Poly(x**2 - 1, x)], x)[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([Poly(
x - 1, x), x**2 - 1], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([x - 1, Poly(
x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr([Poly(x - 1, x), Poly(
x**2 - 1, x)], x, y)[0] == [Poly(x - 1, x, y), Poly(x**2 - 1, x, y)]
assert parallel_poly_from_expr(
[x - 1, x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), x**2 - 1])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x - 1, Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[Poly(x - 1, x), Poly(x**2 - 1, x)])[0] == [Poly(x - 1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, x**2 - 1])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[1, Poly(x**2 - 1, x)])[0] == [Poly(1, x), Poly(x**2 - 1, x)]
assert parallel_poly_from_expr(
[x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[x**2 - 1, 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr(
[Poly(x**2 - 1, x), 1])[0] == [Poly(x**2 - 1, x), Poly(1, x)]
assert parallel_poly_from_expr([Poly(x, x, y), Poly(y, x, y)], x, y, order='lex')[0] == \
[Poly(x, x, y, domain='ZZ'), Poly(y, x, y, domain='ZZ')]
raises(PolificationFailed, lambda: parallel_poly_from_expr([0, 1]))
def test_pdiv():
f, g = x**2 - y**2, x - y
q, r = x + y, 0
F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ]
assert F.pdiv(G) == (Q, R)
assert F.prem(G) == R
assert F.pquo(G) == Q
assert F.pexquo(G) == Q
assert pdiv(f, g) == (q, r)
assert prem(f, g) == r
assert pquo(f, g) == q
assert pexquo(f, g) == q
assert pdiv(f, g, x, y) == (q, r)
assert prem(f, g, x, y) == r
assert pquo(f, g, x, y) == q
assert pexquo(f, g, x, y) == q
assert pdiv(f, g, (x, y)) == (q, r)
assert prem(f, g, (x, y)) == r
assert pquo(f, g, (x, y)) == q
assert pexquo(f, g, (x, y)) == q
assert pdiv(F, G) == (Q, R)
assert prem(F, G) == R
assert pquo(F, G) == Q
assert pexquo(F, G) == Q
assert pdiv(f, g, polys=True) == (Q, R)
assert prem(f, g, polys=True) == R
assert pquo(f, g, polys=True) == Q
assert pexquo(f, g, polys=True) == Q
assert pdiv(F, G, polys=False) == (q, r)
assert prem(F, G, polys=False) == r
assert pquo(F, G, polys=False) == q
assert pexquo(F, G, polys=False) == q
raises(ComputationFailed, lambda: pdiv(4, 2))
raises(ComputationFailed, lambda: prem(4, 2))
raises(ComputationFailed, lambda: pquo(4, 2))
raises(ComputationFailed, lambda: pexquo(4, 2))
def test_div():
f, g = x**2 - y**2, x - y
q, r = x + y, 0
F, G, Q, R = [ Poly(h, x, y) for h in (f, g, q, r) ]
assert F.div(G) == (Q, R)
assert F.rem(G) == R
assert F.quo(G) == Q
assert F.exquo(G) == Q
assert div(f, g) == (q, r)
assert rem(f, g) == r
assert quo(f, g) == q
assert exquo(f, g) == q
assert div(f, g, x, y) == (q, r)
assert rem(f, g, x, y) == r
assert quo(f, g, x, y) == q
assert exquo(f, g, x, y) == q
assert div(f, g, (x, y)) == (q, r)
assert rem(f, g, (x, y)) == r
assert quo(f, g, (x, y)) == q
assert exquo(f, g, (x, y)) == q
assert div(F, G) == (Q, R)
assert rem(F, G) == R
assert quo(F, G) == Q
assert exquo(F, G) == Q
assert div(f, g, polys=True) == (Q, R)
assert rem(f, g, polys=True) == R
assert quo(f, g, polys=True) == Q
assert exquo(f, g, polys=True) == Q
assert div(F, G, polys=False) == (q, r)
assert rem(F, G, polys=False) == r
assert quo(F, G, polys=False) == q
assert exquo(F, G, polys=False) == q
raises(ComputationFailed, lambda: div(4, 2))
raises(ComputationFailed, lambda: rem(4, 2))
raises(ComputationFailed, lambda: quo(4, 2))
raises(ComputationFailed, lambda: exquo(4, 2))
f, g = x**2 + 1, 2*x - 4
qz, rz = 0, x**2 + 1
qq, rq = x/2 + 1, 5
assert div(f, g) == (qq, rq)
assert div(f, g, auto=True) == (qq, rq)
assert div(f, g, auto=False) == (qz, rz)
assert div(f, g, domain=ZZ) == (qz, rz)
assert div(f, g, domain=QQ) == (qq, rq)
assert div(f, g, domain=ZZ, auto=True) == (qq, rq)
assert div(f, g, domain=ZZ, auto=False) == (qz, rz)
assert div(f, g, domain=QQ, auto=True) == (qq, rq)
assert div(f, g, domain=QQ, auto=False) == (qq, rq)
assert rem(f, g) == rq
assert rem(f, g, auto=True) == rq
assert rem(f, g, auto=False) == rz
assert rem(f, g, domain=ZZ) == rz
assert rem(f, g, domain=QQ) == rq
assert rem(f, g, domain=ZZ, auto=True) == rq
assert rem(f, g, domain=ZZ, auto=False) == rz
assert rem(f, g, domain=QQ, auto=True) == rq
assert rem(f, g, domain=QQ, auto=False) == rq
assert quo(f, g) == qq
assert quo(f, g, auto=True) == qq
assert quo(f, g, auto=False) == qz
assert quo(f, g, domain=ZZ) == qz
assert quo(f, g, domain=QQ) == qq
assert quo(f, g, domain=ZZ, auto=True) == qq
assert quo(f, g, domain=ZZ, auto=False) == qz
assert quo(f, g, domain=QQ, auto=True) == qq
assert quo(f, g, domain=QQ, auto=False) == qq
f, g, q = x**2, 2*x, x/2
assert exquo(f, g) == q
assert exquo(f, g, auto=True) == q
raises(ExactQuotientFailed, lambda: exquo(f, g, auto=False))
raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ))
assert exquo(f, g, domain=QQ) == q
assert exquo(f, g, domain=ZZ, auto=True) == q
raises(ExactQuotientFailed, lambda: exquo(f, g, domain=ZZ, auto=False))
assert exquo(f, g, domain=QQ, auto=True) == q
assert exquo(f, g, domain=QQ, auto=False) == q
f, g = Poly(x**2), Poly(x)
q, r = f.div(g)
assert q.get_domain().is_ZZ and r.get_domain().is_ZZ
r = f.rem(g)
assert r.get_domain().is_ZZ
q = f.quo(g)
assert q.get_domain().is_ZZ
q = f.exquo(g)
assert q.get_domain().is_ZZ
f, g = Poly(x+y, x), Poly(2*x+y, x)
q, r = f.div(g)
assert q.get_domain().is_Frac and r.get_domain().is_Frac
# https://github.com/sympy/sympy/issues/19579
p = Poly(2+3*I, x, domain=ZZ_I)
q = Poly(1-I, x, domain=ZZ_I)
assert p.div(q, auto=False) == \
(Poly(0, x, domain='ZZ_I'), Poly(2 + 3*I, x, domain='ZZ_I'))
assert p.div(q, auto=True) == \
(Poly(-S(1)/2 + 5*I/2, x, domain='QQ_I'), Poly(0, x, domain='QQ_I'))
def test_issue_7864():
q, r = div(a, .408248290463863*a)
assert abs(q - 2.44948974278318) < 1e-14
assert r == 0
def test_gcdex():
f, g = 2*x, x**2 - 16
s, t, h = x/32, Rational(-1, 16), 1
F, G, S, T, H = [ Poly(u, x, domain='QQ') for u in (f, g, s, t, h) ]
assert F.half_gcdex(G) == (S, H)
assert F.gcdex(G) == (S, T, H)
assert F.invert(G) == S
assert half_gcdex(f, g) == (s, h)
assert gcdex(f, g) == (s, t, h)
assert invert(f, g) == s
assert half_gcdex(f, g, x) == (s, h)
assert gcdex(f, g, x) == (s, t, h)
assert invert(f, g, x) == s
assert half_gcdex(f, g, (x,)) == (s, h)
assert gcdex(f, g, (x,)) == (s, t, h)
assert invert(f, g, (x,)) == s
assert half_gcdex(F, G) == (S, H)
assert gcdex(F, G) == (S, T, H)
assert invert(F, G) == S
assert half_gcdex(f, g, polys=True) == (S, H)
assert gcdex(f, g, polys=True) == (S, T, H)
assert invert(f, g, polys=True) == S
assert half_gcdex(F, G, polys=False) == (s, h)
assert gcdex(F, G, polys=False) == (s, t, h)
assert invert(F, G, polys=False) == s
assert half_gcdex(100, 2004) == (-20, 4)
assert gcdex(100, 2004) == (-20, 1, 4)
assert invert(3, 7) == 5
raises(DomainError, lambda: half_gcdex(x + 1, 2*x + 1, auto=False))
raises(DomainError, lambda: gcdex(x + 1, 2*x + 1, auto=False))
raises(DomainError, lambda: invert(x + 1, 2*x + 1, auto=False))
def test_revert():
f = Poly(1 - x**2/2 + x**4/24 - x**6/720)
g = Poly(61*x**6/720 + 5*x**4/24 + x**2/2 + 1)
assert f.revert(8) == g
def test_subresultants():
f, g, h = x**2 - 2*x + 1, x**2 - 1, 2*x - 2
F, G, H = Poly(f), Poly(g), Poly(h)
assert F.subresultants(G) == [F, G, H]
assert subresultants(f, g) == [f, g, h]
assert subresultants(f, g, x) == [f, g, h]
assert subresultants(f, g, (x,)) == [f, g, h]
assert subresultants(F, G) == [F, G, H]
assert subresultants(f, g, polys=True) == [F, G, H]
assert subresultants(F, G, polys=False) == [f, g, h]
raises(ComputationFailed, lambda: subresultants(4, 2))
def test_resultant():
f, g, h = x**2 - 2*x + 1, x**2 - 1, 0
F, G = Poly(f), Poly(g)
assert F.resultant(G) == h
assert resultant(f, g) == h
assert resultant(f, g, x) == h
assert resultant(f, g, (x,)) == h
assert resultant(F, G) == h
assert resultant(f, g, polys=True) == h
assert resultant(F, G, polys=False) == h
assert resultant(f, g, includePRS=True) == (h, [f, g, 2*x - 2])
f, g, h = x - a, x - b, a - b
F, G, H = Poly(f), Poly(g), Poly(h)
assert F.resultant(G) == H
assert resultant(f, g) == h
assert resultant(f, g, x) == h
assert resultant(f, g, (x,)) == h
assert resultant(F, G) == H
assert resultant(f, g, polys=True) == H
assert resultant(F, G, polys=False) == h
raises(ComputationFailed, lambda: resultant(4, 2))
def test_discriminant():
f, g = x**3 + 3*x**2 + 9*x - 13, -11664
F = Poly(f)
assert F.discriminant() == g
assert discriminant(f) == g
assert discriminant(f, x) == g
assert discriminant(f, (x,)) == g
assert discriminant(F) == g
assert discriminant(f, polys=True) == g
assert discriminant(F, polys=False) == g
f, g = a*x**2 + b*x + c, b**2 - 4*a*c
F, G = Poly(f), Poly(g)
assert F.discriminant() == G
assert discriminant(f) == g
assert discriminant(f, x, a, b, c) == g
assert discriminant(f, (x, a, b, c)) == g
assert discriminant(F) == G
assert discriminant(f, polys=True) == G
assert discriminant(F, polys=False) == g
raises(ComputationFailed, lambda: discriminant(4))
def test_dispersion():
# We test only the API here. For more mathematical
# tests see the dedicated test file.
fp = poly((x + 1)*(x + 2), x)
assert sorted(fp.dispersionset()) == [0, 1]
assert fp.dispersion() == 1
fp = poly(x**4 - 3*x**2 + 1, x)
gp = fp.shift(-3)
assert sorted(fp.dispersionset(gp)) == [2, 3, 4]
assert fp.dispersion(gp) == 4
def test_gcd_list():
F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2]
assert gcd_list(F) == x - 1
assert gcd_list(F, polys=True) == Poly(x - 1)
assert gcd_list([]) == 0
assert gcd_list([1, 2]) == 1
assert gcd_list([4, 6, 8]) == 2
assert gcd_list([x*(y + 42) - x*y - x*42]) == 0
gcd = gcd_list([], x)
assert gcd.is_Number and gcd is S.Zero
gcd = gcd_list([], x, polys=True)
assert gcd.is_Poly and gcd.is_zero
a = sqrt(2)
assert gcd_list([a, -a]) == gcd_list([-a, a]) == a
raises(ComputationFailed, lambda: gcd_list([], polys=True))
def test_lcm_list():
F = [x**3 - 1, x**2 - 1, x**2 - 3*x + 2]
assert lcm_list(F) == x**5 - x**4 - 2*x**3 - x**2 + x + 2
assert lcm_list(F, polys=True) == Poly(x**5 - x**4 - 2*x**3 - x**2 + x + 2)
assert lcm_list([]) == 1
assert lcm_list([1, 2]) == 2
assert lcm_list([4, 6, 8]) == 24
assert lcm_list([x*(y + 42) - x*y - x*42]) == 0
lcm = lcm_list([], x)
assert lcm.is_Number and lcm is S.One
lcm = lcm_list([], x, polys=True)
assert lcm.is_Poly and lcm.is_one
raises(ComputationFailed, lambda: lcm_list([], polys=True))
def test_gcd():
f, g = x**3 - 1, x**2 - 1
s, t = x**2 + x + 1, x + 1
h, r = x - 1, x**4 + x**3 - x - 1
F, G, S, T, H, R = [ Poly(u) for u in (f, g, s, t, h, r) ]
assert F.cofactors(G) == (H, S, T)
assert F.gcd(G) == H
assert F.lcm(G) == R
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == r
assert cofactors(f, g, x) == (h, s, t)
assert gcd(f, g, x) == h
assert lcm(f, g, x) == r
assert cofactors(f, g, (x,)) == (h, s, t)
assert gcd(f, g, (x,)) == h
assert lcm(f, g, (x,)) == r
assert cofactors(F, G) == (H, S, T)
assert gcd(F, G) == H
assert lcm(F, G) == R
assert cofactors(f, g, polys=True) == (H, S, T)
assert gcd(f, g, polys=True) == H
assert lcm(f, g, polys=True) == R
assert cofactors(F, G, polys=False) == (h, s, t)
assert gcd(F, G, polys=False) == h
assert lcm(F, G, polys=False) == r
f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0
h, s, t = g, 1.0*x + 1.0, 1.0
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == f
f, g = 1.0*x**2 - 1.0, 1.0*x - 1.0
h, s, t = g, 1.0*x + 1.0, 1.0
assert cofactors(f, g) == (h, s, t)
assert gcd(f, g) == h
assert lcm(f, g) == f
assert cofactors(8, 6) == (2, 4, 3)
assert gcd(8, 6) == 2
assert lcm(8, 6) == 24
f, g = x**2 - 3*x - 4, x**3 - 4*x**2 + x - 4
l = x**4 - 3*x**3 - 3*x**2 - 3*x - 4
h, s, t = x - 4, x + 1, x**2 + 1
assert cofactors(f, g, modulus=11) == (h, s, t)
assert gcd(f, g, modulus=11) == h
assert lcm(f, g, modulus=11) == l
f, g = x**2 + 8*x + 7, x**3 + 7*x**2 + x + 7
l = x**4 + 8*x**3 + 8*x**2 + 8*x + 7
h, s, t = x + 7, x + 1, x**2 + 1
assert cofactors(f, g, modulus=11, symmetric=False) == (h, s, t)
assert gcd(f, g, modulus=11, symmetric=False) == h
assert lcm(f, g, modulus=11, symmetric=False) == l
a, b = sqrt(2), -sqrt(2)
assert gcd(a, b) == gcd(b, a) == sqrt(2)
a, b = sqrt(-2), -sqrt(-2)
assert gcd(a, b) == gcd(b, a) == sqrt(2)
assert gcd(Poly(x - 2, x), Poly(I*x, x)) == Poly(1, x, domain=ZZ_I)
raises(TypeError, lambda: gcd(x))
raises(TypeError, lambda: lcm(x))
def test_gcd_numbers_vs_polys():
assert isinstance(gcd(3, 9), Integer)
assert isinstance(gcd(3*x, 9), Integer)
assert gcd(3, 9) == 3
assert gcd(3*x, 9) == 3
assert isinstance(gcd(Rational(3, 2), Rational(9, 4)), Rational)
assert isinstance(gcd(Rational(3, 2)*x, Rational(9, 4)), Rational)
assert gcd(Rational(3, 2), Rational(9, 4)) == Rational(3, 4)
assert gcd(Rational(3, 2)*x, Rational(9, 4)) == 1
assert isinstance(gcd(3.0, 9.0), Float)
assert isinstance(gcd(3.0*x, 9.0), Float)
assert gcd(3.0, 9.0) == 1.0
assert gcd(3.0*x, 9.0) == 1.0
# partial fix of 20597
assert gcd(Mul(2, 3, evaluate=False), 2) == 2
def test_terms_gcd():
assert terms_gcd(1) == 1
assert terms_gcd(1, x) == 1
assert terms_gcd(x - 1) == x - 1
assert terms_gcd(-x - 1) == -x - 1
assert terms_gcd(2*x + 3) == 2*x + 3
assert terms_gcd(6*x + 4) == Mul(2, 3*x + 2, evaluate=False)
assert terms_gcd(x**3*y + x*y**3) == x*y*(x**2 + y**2)
assert terms_gcd(2*x**3*y + 2*x*y**3) == 2*x*y*(x**2 + y**2)
assert terms_gcd(x**3*y/2 + x*y**3/2) == x*y/2*(x**2 + y**2)
assert terms_gcd(x**3*y + 2*x*y**3) == x*y*(x**2 + 2*y**2)
assert terms_gcd(2*x**3*y + 4*x*y**3) == 2*x*y*(x**2 + 2*y**2)
assert terms_gcd(2*x**3*y/3 + 4*x*y**3/5) == x*y*Rational(2, 15)*(5*x**2 + 6*y**2)
assert terms_gcd(2.0*x**3*y + 4.1*x*y**3) == x*y*(2.0*x**2 + 4.1*y**2)
assert _aresame(terms_gcd(2.0*x + 3), 2.0*x + 3)
assert terms_gcd((3 + 3*x)*(x + x*y), expand=False) == \
(3*x + 3)*(x*y + x)
assert terms_gcd((3 + 3*x)*(x + x*sin(3 + 3*y)), expand=False, deep=True) == \
3*x*(x + 1)*(sin(Mul(3, y + 1, evaluate=False)) + 1)
assert terms_gcd(sin(x + x*y), deep=True) == \
sin(x*(y + 1))
eq = Eq(2*x, 2*y + 2*z*y)
assert terms_gcd(eq) == Eq(2*x, 2*y*(z + 1))
assert terms_gcd(eq, deep=True) == Eq(2*x, 2*y*(z + 1))
raises(TypeError, lambda: terms_gcd(x < 2))
def test_trunc():
f, g = x**5 + 2*x**4 + 3*x**3 + 4*x**2 + 5*x + 6, x**5 - x**4 + x**2 - x
F, G = Poly(f), Poly(g)
assert F.trunc(3) == G
assert trunc(f, 3) == g
assert trunc(f, 3, x) == g
assert trunc(f, 3, (x,)) == g
assert trunc(F, 3) == G
assert trunc(f, 3, polys=True) == G
assert trunc(F, 3, polys=False) == g
f, g = 6*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1, -x**4 + x**3 - x + 1
F, G = Poly(f), Poly(g)
assert F.trunc(3) == G
assert trunc(f, 3) == g
assert trunc(f, 3, x) == g
assert trunc(f, 3, (x,)) == g
assert trunc(F, 3) == G
assert trunc(f, 3, polys=True) == G
assert trunc(F, 3, polys=False) == g
f = Poly(x**2 + 2*x + 3, modulus=5)
assert f.trunc(2) == Poly(x**2 + 1, modulus=5)
def test_monic():
f, g = 2*x - 1, x - S.Half
F, G = Poly(f, domain='QQ'), Poly(g)
assert F.monic() == G
assert monic(f) == g
assert monic(f, x) == g
assert monic(f, (x,)) == g
assert monic(F) == G
assert monic(f, polys=True) == G
assert monic(F, polys=False) == g
raises(ComputationFailed, lambda: monic(4))
assert monic(2*x**2 + 6*x + 4, auto=False) == x**2 + 3*x + 2
raises(ExactQuotientFailed, lambda: monic(2*x + 6*x + 1, auto=False))
assert monic(2.0*x**2 + 6.0*x + 4.0) == 1.0*x**2 + 3.0*x + 2.0
assert monic(2*x**2 + 3*x + 4, modulus=5) == x**2 - x + 2
def test_content():
f, F = 4*x + 2, Poly(4*x + 2)
assert F.content() == 2
assert content(f) == 2
raises(ComputationFailed, lambda: content(4))
f = Poly(2*x, modulus=3)
assert f.content() == 1
def test_primitive():
f, g = 4*x + 2, 2*x + 1
F, G = Poly(f), Poly(g)
assert F.primitive() == (2, G)
assert primitive(f) == (2, g)
assert primitive(f, x) == (2, g)
assert primitive(f, (x,)) == (2, g)
assert primitive(F) == (2, G)
assert primitive(f, polys=True) == (2, G)
assert primitive(F, polys=False) == (2, g)
raises(ComputationFailed, lambda: primitive(4))
f = Poly(2*x, modulus=3)
g = Poly(2.0*x, domain=RR)
assert f.primitive() == (1, f)
assert g.primitive() == (1.0, g)
assert primitive(S('-3*x/4 + y + 11/8')) == \
S('(1/8, -6*x + 8*y + 11)')
def test_compose():
f = x**12 + 20*x**10 + 150*x**8 + 500*x**6 + 625*x**4 - 2*x**3 - 10*x + 9
g = x**4 - 2*x + 9
h = x**3 + 5*x
F, G, H = map(Poly, (f, g, h))
assert G.compose(H) == F
assert compose(g, h) == f
assert compose(g, h, x) == f
assert compose(g, h, (x,)) == f
assert compose(G, H) == F
assert compose(g, h, polys=True) == F
assert compose(G, H, polys=False) == f
assert F.decompose() == [G, H]
assert decompose(f) == [g, h]
assert decompose(f, x) == [g, h]
assert decompose(f, (x,)) == [g, h]
assert decompose(F) == [G, H]
assert decompose(f, polys=True) == [G, H]
assert decompose(F, polys=False) == [g, h]
raises(ComputationFailed, lambda: compose(4, 2))
raises(ComputationFailed, lambda: decompose(4))
assert compose(x**2 - y**2, x - y, x, y) == x**2 - 2*x*y
assert compose(x**2 - y**2, x - y, y, x) == -y**2 + 2*x*y
def test_shift():
assert Poly(x**2 - 2*x + 1, x).shift(2) == Poly(x**2 + 2*x + 1, x)
def test_transform():
# Also test that 3-way unification is done correctly
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - 1)) == \
Poly(4, x) == \
cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - 1)))
assert Poly(x**2 - x/2 + 1, x).transform(Poly(x + 1), Poly(x - 1)) == \
Poly(3*x**2/2 + Rational(5, 2), x) == \
cancel((x - 1)**2*(x**2 - x/2 + 1).subs(x, (x + 1)/(x - 1)))
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + S.Half), Poly(x - 1)) == \
Poly(Rational(9, 4), x) == \
cancel((x - 1)**2*(x**2 - 2*x + 1).subs(x, (x + S.Half)/(x - 1)))
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1), Poly(x - S.Half)) == \
Poly(Rational(9, 4), x) == \
cancel((x - S.Half)**2*(x**2 - 2*x + 1).subs(x, (x + 1)/(x - S.Half)))
# Unify ZZ, QQ, and RR
assert Poly(x**2 - 2*x + 1, x).transform(Poly(x + 1.0), Poly(x - S.Half)) == \
Poly(Rational(9, 4), x, domain='RR') == \
cancel((x - S.Half)**2*(x**2 - 2*x + 1).subs(x, (x + 1.0)/(x - S.Half)))
raises(ValueError, lambda: Poly(x*y).transform(Poly(x + 1), Poly(x - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(y + 1), Poly(x - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(x + 1), Poly(y - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(x*y + 1), Poly(x - 1)))
raises(ValueError, lambda: Poly(x).transform(Poly(x + 1), Poly(x*y - 1)))
def test_sturm():
f, F = x, Poly(x, domain='QQ')
g, G = 1, Poly(1, x, domain='QQ')
assert F.sturm() == [F, G]
assert sturm(f) == [f, g]
assert sturm(f, x) == [f, g]
assert sturm(f, (x,)) == [f, g]
assert sturm(F) == [F, G]
assert sturm(f, polys=True) == [F, G]
assert sturm(F, polys=False) == [f, g]
raises(ComputationFailed, lambda: sturm(4))
raises(DomainError, lambda: sturm(f, auto=False))
f = Poly(S(1024)/(15625*pi**8)*x**5
- S(4096)/(625*pi**8)*x**4
+ S(32)/(15625*pi**4)*x**3
- S(128)/(625*pi**4)*x**2
+ Rational(1, 62500)*x
- Rational(1, 625), x, domain='ZZ(pi)')
assert sturm(f) == \
[Poly(x**3 - 100*x**2 + pi**4/64*x - 25*pi**4/16, x, domain='ZZ(pi)'),
Poly(3*x**2 - 200*x + pi**4/64, x, domain='ZZ(pi)'),
Poly((Rational(20000, 9) - pi**4/96)*x + 25*pi**4/18, x, domain='ZZ(pi)'),
Poly((-3686400000000*pi**4 - 11520000*pi**8 - 9*pi**12)/(26214400000000 - 245760000*pi**4 + 576*pi**8), x, domain='ZZ(pi)')]
def test_gff():
f = x**5 + 2*x**4 - x**3 - 2*x**2
assert Poly(f).gff_list() == [(Poly(x), 1), (Poly(x + 2), 4)]
assert gff_list(f) == [(x, 1), (x + 2, 4)]
raises(NotImplementedError, lambda: gff(f))
f = x*(x - 1)**3*(x - 2)**2*(x - 4)**2*(x - 5)
assert Poly(f).gff_list() == [(
Poly(x**2 - 5*x + 4), 1), (Poly(x**2 - 5*x + 4), 2), (Poly(x), 3)]
assert gff_list(f) == [(x**2 - 5*x + 4, 1), (x**2 - 5*x + 4, 2), (x, 3)]
raises(NotImplementedError, lambda: gff(f))
def test_norm():
a, b = sqrt(2), sqrt(3)
f = Poly(a*x + b*y, x, y, extension=(a, b))
assert f.norm() == Poly(4*x**4 - 12*x**2*y**2 + 9*y**4, x, y, domain='QQ')
def test_sqf_norm():
assert sqf_norm(x**2 - 2, extension=sqrt(3)) == \
(1, x**2 - 2*sqrt(3)*x + 1, x**4 - 10*x**2 + 1)
assert sqf_norm(x**2 - 3, extension=sqrt(2)) == \
(1, x**2 - 2*sqrt(2)*x - 1, x**4 - 10*x**2 + 1)
assert Poly(x**2 - 2, extension=sqrt(3)).sqf_norm() == \
(1, Poly(x**2 - 2*sqrt(3)*x + 1, x, extension=sqrt(3)),
Poly(x**4 - 10*x**2 + 1, x, domain='QQ'))
assert Poly(x**2 - 3, extension=sqrt(2)).sqf_norm() == \
(1, Poly(x**2 - 2*sqrt(2)*x - 1, x, extension=sqrt(2)),
Poly(x**4 - 10*x**2 + 1, x, domain='QQ'))
def test_sqf():
f = x**5 - x**3 - x**2 + 1
g = x**3 + 2*x**2 + 2*x + 1
h = x - 1
p = x**4 + x**3 - x - 1
F, G, H, P = map(Poly, (f, g, h, p))
assert F.sqf_part() == P
assert sqf_part(f) == p
assert sqf_part(f, x) == p
assert sqf_part(f, (x,)) == p
assert sqf_part(F) == P
assert sqf_part(f, polys=True) == P
assert sqf_part(F, polys=False) == p
assert F.sqf_list() == (1, [(G, 1), (H, 2)])
assert sqf_list(f) == (1, [(g, 1), (h, 2)])
assert sqf_list(f, x) == (1, [(g, 1), (h, 2)])
assert sqf_list(f, (x,)) == (1, [(g, 1), (h, 2)])
assert sqf_list(F) == (1, [(G, 1), (H, 2)])
assert sqf_list(f, polys=True) == (1, [(G, 1), (H, 2)])
assert sqf_list(F, polys=False) == (1, [(g, 1), (h, 2)])
assert F.sqf_list_include() == [(G, 1), (H, 2)]
raises(ComputationFailed, lambda: sqf_part(4))
assert sqf(1) == 1
assert sqf_list(1) == (1, [])
assert sqf((2*x**2 + 2)**7) == 128*(x**2 + 1)**7
assert sqf(f) == g*h**2
assert sqf(f, x) == g*h**2
assert sqf(f, (x,)) == g*h**2
d = x**2 + y**2
assert sqf(f/d) == (g*h**2)/d
assert sqf(f/d, x) == (g*h**2)/d
assert sqf(f/d, (x,)) == (g*h**2)/d
assert sqf(x - 1) == x - 1
assert sqf(-x - 1) == -x - 1
assert sqf(x - 1) == x - 1
assert sqf(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert sqf((6*x - 10)/(3*x - 6)) == Rational(2, 3)*((3*x - 5)/(x - 2))
assert sqf(Poly(x**2 - 2*x + 1)) == (x - 1)**2
f = 3 + x - x*(1 + x) + x**2
assert sqf(f) == 3
f = (x**2 + 2*x + 1)**20000000000
assert sqf(f) == (x + 1)**40000000000
assert sqf_list(f) == (1, [(x + 1, 40000000000)])
def test_factor():
f = x**5 - x**3 - x**2 + 1
u = x + 1
v = x - 1
w = x**2 + x + 1
F, U, V, W = map(Poly, (f, u, v, w))
assert F.factor_list() == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(f) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(f, x) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(f, (x,)) == (1, [(u, 1), (v, 2), (w, 1)])
assert factor_list(F) == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(f, polys=True) == (1, [(U, 1), (V, 2), (W, 1)])
assert factor_list(F, polys=False) == (1, [(u, 1), (v, 2), (w, 1)])
assert F.factor_list_include() == [(U, 1), (V, 2), (W, 1)]
assert factor_list(1) == (1, [])
assert factor_list(6) == (6, [])
assert factor_list(sqrt(3), x) == (sqrt(3), [])
assert factor_list((-1)**x, x) == (1, [(-1, x)])
assert factor_list((2*x)**y, x) == (1, [(2, y), (x, y)])
assert factor_list(sqrt(x*y), x) == (1, [(x*y, S.Half)])
assert factor(6) == 6 and factor(6).is_Integer
assert factor_list(3*x) == (3, [(x, 1)])
assert factor_list(3*x**2) == (3, [(x, 2)])
assert factor(3*x) == 3*x
assert factor(3*x**2) == 3*x**2
assert factor((2*x**2 + 2)**7) == 128*(x**2 + 1)**7
assert factor(f) == u*v**2*w
assert factor(f, x) == u*v**2*w
assert factor(f, (x,)) == u*v**2*w
g, p, q, r = x**2 - y**2, x - y, x + y, x**2 + 1
assert factor(f/g) == (u*v**2*w)/(p*q)
assert factor(f/g, x) == (u*v**2*w)/(p*q)
assert factor(f/g, (x,)) == (u*v**2*w)/(p*q)
p = Symbol('p', positive=True)
i = Symbol('i', integer=True)
r = Symbol('r', real=True)
assert factor(sqrt(x*y)).is_Pow is True
assert factor(sqrt(3*x**2 - 3)) == sqrt(3)*sqrt((x - 1)*(x + 1))
assert factor(sqrt(3*x**2 + 3)) == sqrt(3)*sqrt(x**2 + 1)
assert factor((y*x**2 - y)**i) == y**i*(x - 1)**i*(x + 1)**i
assert factor((y*x**2 + y)**i) == y**i*(x**2 + 1)**i
assert factor((y*x**2 - y)**t) == (y*(x - 1)*(x + 1))**t
assert factor((y*x**2 + y)**t) == (y*(x**2 + 1))**t
f = sqrt(expand((r**2 + 1)*(p + 1)*(p - 1)*(p - 2)**3))
g = sqrt((p - 2)**3*(p - 1))*sqrt(p + 1)*sqrt(r**2 + 1)
assert factor(f) == g
assert factor(g) == g
g = (x - 1)**5*(r**2 + 1)
f = sqrt(expand(g))
assert factor(f) == sqrt(g)
f = Poly(sin(1)*x + 1, x, domain=EX)
assert f.factor_list() == (1, [(f, 1)])
f = x**4 + 1
assert factor(f) == f
assert factor(f, extension=I) == (x**2 - I)*(x**2 + I)
assert factor(f, gaussian=True) == (x**2 - I)*(x**2 + I)
assert factor(
f, extension=sqrt(2)) == (x**2 + sqrt(2)*x + 1)*(x**2 - sqrt(2)*x + 1)
assert factor(x**2 + 4*I*x - 4) == (x + 2*I)**2
f = x**2 + 2*I*x - 4
assert factor(f) == f
f = 8192*x**2 + x*(22656 + 175232*I) - 921416 + 242313*I
f_zzi = I*(x*(64 - 64*I) + 773 + 596*I)**2
f_qqi = 8192*(x + S(177)/128 + 1369*I/128)**2
assert factor(f) == f_zzi
assert factor(f, domain=ZZ_I) == f_zzi
assert factor(f, domain=QQ_I) == f_qqi
f = x**2 + 2*sqrt(2)*x + 2
assert factor(f, extension=sqrt(2)) == (x + sqrt(2))**2
assert factor(f**3, extension=sqrt(2)) == (x + sqrt(2))**6
assert factor(x**2 - 2*y**2, extension=sqrt(2)) == \
(x + sqrt(2)*y)*(x - sqrt(2)*y)
assert factor(2*x**2 - 4*y**2, extension=sqrt(2)) == \
2*((x + sqrt(2)*y)*(x - sqrt(2)*y))
assert factor(x - 1) == x - 1
assert factor(-x - 1) == -x - 1
assert factor(x - 1) == x - 1
assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert factor(x**11 + x + 1, modulus=65537, symmetric=True) == \
(x**2 + x + 1)*(x**9 - x**8 + x**6 - x**5 + x**3 - x** 2 + 1)
assert factor(x**11 + x + 1, modulus=65537, symmetric=False) == \
(x**2 + x + 1)*(x**9 + 65536*x**8 + x**6 + 65536*x**5 +
x**3 + 65536*x** 2 + 1)
f = x/pi + x*sin(x)/pi
g = y/(pi**2 + 2*pi + 1) + y*sin(x)/(pi**2 + 2*pi + 1)
assert factor(f) == x*(sin(x) + 1)/pi
assert factor(g) == y*(sin(x) + 1)/(pi + 1)**2
assert factor(Eq(
x**2 + 2*x + 1, x**3 + 1)) == Eq((x + 1)**2, (x + 1)*(x**2 - x + 1))
f = (x**2 - 1)/(x**2 + 4*x + 4)
assert factor(f) == (x + 1)*(x - 1)/(x + 2)**2
assert factor(f, x) == (x + 1)*(x - 1)/(x + 2)**2
f = 3 + x - x*(1 + x) + x**2
assert factor(f) == 3
assert factor(f, x) == 3
assert factor(1/(x**2 + 2*x + 1/x) - 1) == -((1 - x + 2*x**2 +
x**3)/(1 + 2*x**2 + x**3))
assert factor(f, expand=False) == f
raises(PolynomialError, lambda: factor(f, x, expand=False))
raises(FlagError, lambda: factor(x**2 - 1, polys=True))
assert factor([x, Eq(x**2 - y**2, Tuple(x**2 - z**2, 1/x + 1/y))]) == \
[x, Eq((x - y)*(x + y), Tuple((x - z)*(x + z), (x + y)/x/y))]
assert not isinstance(
Poly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True
assert isinstance(
PurePoly(x**3 + x + 1).factor_list()[1][0][0], PurePoly) is True
assert factor(sqrt(-x)) == sqrt(-x)
# issue 5917
e = (-2*x*(-x + 1)*(x - 1)*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)*(x**2*(x -
1) - x*(x - 1) - x) - (-2*x**2*(x - 1)**2 - x*(-x + 1)*(-x*(-x + 1) +
x*(x - 1)))*(x**2*(x - 1)**4 - x*(-x*(-x + 1)*(x - 1) - x*(x - 1)**2)))
assert factor(e) == 0
# deep option
assert factor(sin(x**2 + x) + x, deep=True) == sin(x*(x + 1)) + x
assert factor(sin(x**2 + x)*x, deep=True) == sin(x*(x + 1))*x
assert factor(sqrt(x**2)) == sqrt(x**2)
# issue 13149
assert factor(expand((0.5*x+1)*(0.5*y+1))) == Mul(1.0, 0.5*x + 1.0,
0.5*y + 1.0, evaluate = False)
assert factor(expand((0.5*x+0.5)**2)) == 0.25*(1.0*x + 1.0)**2
eq = x**2*y**2 + 11*x**2*y + 30*x**2 + 7*x*y**2 + 77*x*y + 210*x + 12*y**2 + 132*y + 360
assert factor(eq, x) == (x + 3)*(x + 4)*(y**2 + 11*y + 30)
assert factor(eq, x, deep=True) == (x + 3)*(x + 4)*(y**2 + 11*y + 30)
assert factor(eq, y, deep=True) == (y + 5)*(y + 6)*(x**2 + 7*x + 12)
# fraction option
f = 5*x + 3*exp(2 - 7*x)
assert factor(f, deep=True) == factor(f, deep=True, fraction=True)
assert factor(f, deep=True, fraction=False) == 5*x + 3*exp(2)*exp(-7*x)
assert factor_list(x**3 - x*y**2, t, w, x) == (
1, [(x, 1), (x - y, 1), (x + y, 1)])
def test_factor_large():
f = (x**2 + 4*x + 4)**10000000*(x**2 + 1)*(x**2 + 2*x + 1)**1234567
g = ((x**2 + 2*x + 1)**3000*y**2 + (x**2 + 2*x + 1)**3000*2*y + (
x**2 + 2*x + 1)**3000)
assert factor(f) == (x + 2)**20000000*(x**2 + 1)*(x + 1)**2469134
assert factor(g) == (x + 1)**6000*(y + 1)**2
assert factor_list(
f) == (1, [(x + 1, 2469134), (x + 2, 20000000), (x**2 + 1, 1)])
assert factor_list(g) == (1, [(y + 1, 2), (x + 1, 6000)])
f = (x**2 - y**2)**200000*(x**7 + 1)
g = (x**2 + y**2)**200000*(x**7 + 1)
assert factor(f) == \
(x + 1)*(x - y)**200000*(x + y)**200000*(x**6 - x**5 +
x**4 - x**3 + x**2 - x + 1)
assert factor(g, gaussian=True) == \
(x + 1)*(x - I*y)**200000*(x + I*y)**200000*(x**6 - x**5 +
x**4 - x**3 + x**2 - x + 1)
assert factor_list(f) == \
(1, [(x + 1, 1), (x - y, 200000), (x + y, 200000), (x**6 -
x**5 + x**4 - x**3 + x**2 - x + 1, 1)])
assert factor_list(g, gaussian=True) == \
(1, [(x + 1, 1), (x - I*y, 200000), (x + I*y, 200000), (
x**6 - x**5 + x**4 - x**3 + x**2 - x + 1, 1)])
def test_factor_noeval():
assert factor(6*x - 10) == Mul(2, 3*x - 5, evaluate=False)
assert factor((6*x - 10)/(3*x - 6)) == Mul(Rational(2, 3), 3*x - 5, 1/(x - 2))
def test_intervals():
assert intervals(0) == []
assert intervals(1) == []
assert intervals(x, sqf=True) == [(0, 0)]
assert intervals(x) == [((0, 0), 1)]
assert intervals(x**128) == [((0, 0), 128)]
assert intervals([x**2, x**4]) == [((0, 0), {0: 2, 1: 4})]
f = Poly((x*Rational(2, 5) - Rational(17, 3))*(4*x + Rational(1, 257)))
assert f.intervals(sqf=True) == [(-1, 0), (14, 15)]
assert f.intervals() == [((-1, 0), 1), ((14, 15), 1)]
assert f.intervals(fast=True, sqf=True) == [(-1, 0), (14, 15)]
assert f.intervals(fast=True) == [((-1, 0), 1), ((14, 15), 1)]
assert f.intervals(eps=Rational(1, 10)) == f.intervals(eps=0.1) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert f.intervals(eps=Rational(1, 100)) == f.intervals(eps=0.01) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert f.intervals(eps=Rational(1, 1000)) == f.intervals(eps=0.001) == \
[((Rational(-1, 1002), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert f.intervals(eps=Rational(1, 10000)) == f.intervals(eps=0.0001) == \
[((Rational(-1, 1028), Rational(-1, 1028)), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
f = (x*Rational(2, 5) - Rational(17, 3))*(4*x + Rational(1, 257))
assert intervals(f, sqf=True) == [(-1, 0), (14, 15)]
assert intervals(f) == [((-1, 0), 1), ((14, 15), 1)]
assert intervals(f, eps=Rational(1, 10)) == intervals(f, eps=0.1) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert intervals(f, eps=Rational(1, 100)) == intervals(f, eps=0.01) == \
[((Rational(-1, 258), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert intervals(f, eps=Rational(1, 1000)) == intervals(f, eps=0.001) == \
[((Rational(-1, 1002), 0), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
assert intervals(f, eps=Rational(1, 10000)) == intervals(f, eps=0.0001) == \
[((Rational(-1, 1028), Rational(-1, 1028)), 1), ((Rational(85, 6), Rational(85, 6)), 1)]
f = Poly((x**2 - 2)*(x**2 - 3)**7*(x + 1)*(7*x + 3)**3)
assert f.intervals() == \
[((-2, Rational(-3, 2)), 7), ((Rational(-3, 2), -1), 1),
((-1, -1), 1), ((-1, 0), 3),
((1, Rational(3, 2)), 1), ((Rational(3, 2), 2), 7)]
assert intervals([x**5 - 200, x**5 - 201]) == \
[((Rational(75, 26), Rational(101, 35)), {0: 1}), ((Rational(309, 107), Rational(26, 9)), {1: 1})]
assert intervals([x**5 - 200, x**5 - 201], fast=True) == \
[((Rational(75, 26), Rational(101, 35)), {0: 1}), ((Rational(309, 107), Rational(26, 9)), {1: 1})]
assert intervals([x**2 - 200, x**2 - 201]) == \
[((Rational(-71, 5), Rational(-85, 6)), {1: 1}), ((Rational(-85, 6), -14), {0: 1}),
((14, Rational(85, 6)), {0: 1}), ((Rational(85, 6), Rational(71, 5)), {1: 1})]
assert intervals([x + 1, x + 2, x - 1, x + 1, 1, x - 1, x - 1, (x - 2)**2]) == \
[((-2, -2), {1: 1}), ((-1, -1), {0: 1, 3: 1}), ((1, 1), {2:
1, 5: 1, 6: 1}), ((2, 2), {7: 2})]
f, g, h = x**2 - 2, x**4 - 4*x**2 + 4, x - 1
assert intervals(f, inf=Rational(7, 4), sqf=True) == []
assert intervals(f, inf=Rational(7, 5), sqf=True) == [(Rational(7, 5), Rational(3, 2))]
assert intervals(f, sup=Rational(7, 4), sqf=True) == [(-2, -1), (1, Rational(3, 2))]
assert intervals(f, sup=Rational(7, 5), sqf=True) == [(-2, -1)]
assert intervals(g, inf=Rational(7, 4)) == []
assert intervals(g, inf=Rational(7, 5)) == [((Rational(7, 5), Rational(3, 2)), 2)]
assert intervals(g, sup=Rational(7, 4)) == [((-2, -1), 2), ((1, Rational(3, 2)), 2)]
assert intervals(g, sup=Rational(7, 5)) == [((-2, -1), 2)]
assert intervals([g, h], inf=Rational(7, 4)) == []
assert intervals([g, h], inf=Rational(7, 5)) == [((Rational(7, 5), Rational(3, 2)), {0: 2})]
assert intervals([g, h], sup=S(
7)/4) == [((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, Rational(3, 2)), {0: 2})]
assert intervals(
[g, h], sup=Rational(7, 5)) == [((-2, -1), {0: 2}), ((1, 1), {1: 1})]
assert intervals([x + 2, x**2 - 2]) == \
[((-2, -2), {0: 1}), ((-2, -1), {1: 1}), ((1, 2), {1: 1})]
assert intervals([x + 2, x**2 - 2], strict=True) == \
[((-2, -2), {0: 1}), ((Rational(-3, 2), -1), {1: 1}), ((1, 2), {1: 1})]
f = 7*z**4 - 19*z**3 + 20*z**2 + 17*z + 20
assert intervals(f) == []
real_part, complex_part = intervals(f, all=True, sqf=True)
assert real_part == []
assert all(re(a) < re(r) < re(b) and im(
a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f)))
assert complex_part == [(Rational(-40, 7) - I*40/7, 0),
(Rational(-40, 7), I*40/7),
(I*Rational(-40, 7), Rational(40, 7)),
(0, Rational(40, 7) + I*40/7)]
real_part, complex_part = intervals(f, all=True, sqf=True, eps=Rational(1, 10))
assert real_part == []
assert all(re(a) < re(r) < re(b) and im(
a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f)))
raises(ValueError, lambda: intervals(x**2 - 2, eps=10**-100000))
raises(ValueError, lambda: Poly(x**2 - 2).intervals(eps=10**-100000))
raises(
ValueError, lambda: intervals([x**2 - 2, x**2 - 3], eps=10**-100000))
def test_refine_root():
f = Poly(x**2 - 2)
assert f.refine_root(1, 2, steps=0) == (1, 2)
assert f.refine_root(-2, -1, steps=0) == (-2, -1)
assert f.refine_root(1, 2, steps=None) == (1, Rational(3, 2))
assert f.refine_root(-2, -1, steps=None) == (Rational(-3, 2), -1)
assert f.refine_root(1, 2, steps=1) == (1, Rational(3, 2))
assert f.refine_root(-2, -1, steps=1) == (Rational(-3, 2), -1)
assert f.refine_root(1, 2, steps=1, fast=True) == (1, Rational(3, 2))
assert f.refine_root(-2, -1, steps=1, fast=True) == (Rational(-3, 2), -1)
assert f.refine_root(1, 2, eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12))
assert f.refine_root(1, 2, eps=1e-2) == (Rational(24, 17), Rational(17, 12))
raises(PolynomialError, lambda: (f**2).refine_root(1, 2, check_sqf=True))
raises(RefinementFailed, lambda: (f**2).refine_root(1, 2))
raises(RefinementFailed, lambda: (f**2).refine_root(2, 3))
f = x**2 - 2
assert refine_root(f, 1, 2, steps=1) == (1, Rational(3, 2))
assert refine_root(f, -2, -1, steps=1) == (Rational(-3, 2), -1)
assert refine_root(f, 1, 2, steps=1, fast=True) == (1, Rational(3, 2))
assert refine_root(f, -2, -1, steps=1, fast=True) == (Rational(-3, 2), -1)
assert refine_root(f, 1, 2, eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12))
assert refine_root(f, 1, 2, eps=1e-2) == (Rational(24, 17), Rational(17, 12))
raises(PolynomialError, lambda: refine_root(1, 7, 8, eps=Rational(1, 100)))
raises(ValueError, lambda: Poly(f).refine_root(1, 2, eps=10**-100000))
raises(ValueError, lambda: refine_root(f, 1, 2, eps=10**-100000))
def test_count_roots():
assert count_roots(x**2 - 2) == 2
assert count_roots(x**2 - 2, inf=-oo) == 2
assert count_roots(x**2 - 2, sup=+oo) == 2
assert count_roots(x**2 - 2, inf=-oo, sup=+oo) == 2
assert count_roots(x**2 - 2, inf=-2) == 2
assert count_roots(x**2 - 2, inf=-1) == 1
assert count_roots(x**2 - 2, sup=1) == 1
assert count_roots(x**2 - 2, sup=2) == 2
assert count_roots(x**2 - 2, inf=-1, sup=1) == 0
assert count_roots(x**2 - 2, inf=-2, sup=2) == 2
assert count_roots(x**2 - 2, inf=-1, sup=1) == 0
assert count_roots(x**2 - 2, inf=-2, sup=2) == 2
assert count_roots(x**2 + 2) == 0
assert count_roots(x**2 + 2, inf=-2*I) == 2
assert count_roots(x**2 + 2, sup=+2*I) == 2
assert count_roots(x**2 + 2, inf=-2*I, sup=+2*I) == 2
assert count_roots(x**2 + 2, inf=0) == 0
assert count_roots(x**2 + 2, sup=0) == 0
assert count_roots(x**2 + 2, inf=-I) == 1
assert count_roots(x**2 + 2, sup=+I) == 1
assert count_roots(x**2 + 2, inf=+I/2, sup=+I) == 0
assert count_roots(x**2 + 2, inf=-I, sup=-I/2) == 0
raises(PolynomialError, lambda: count_roots(1))
def test_Poly_root():
f = Poly(2*x**3 - 7*x**2 + 4*x + 4)
assert f.root(0) == Rational(-1, 2)
assert f.root(1) == 2
assert f.root(2) == 2
raises(IndexError, lambda: f.root(3))
assert Poly(x**5 + x + 1).root(0) == rootof(x**3 - x**2 + 1, 0)
def test_real_roots():
assert real_roots(x) == [0]
assert real_roots(x, multiple=False) == [(0, 1)]
assert real_roots(x**3) == [0, 0, 0]
assert real_roots(x**3, multiple=False) == [(0, 3)]
assert real_roots(x*(x**3 + x + 3)) == [rootof(x**3 + x + 3, 0), 0]
assert real_roots(x*(x**3 + x + 3), multiple=False) == [(rootof(
x**3 + x + 3, 0), 1), (0, 1)]
assert real_roots(
x**3*(x**3 + x + 3)) == [rootof(x**3 + x + 3, 0), 0, 0, 0]
assert real_roots(x**3*(x**3 + x + 3), multiple=False) == [(rootof(
x**3 + x + 3, 0), 1), (0, 3)]
f = 2*x**3 - 7*x**2 + 4*x + 4
g = x**3 + x + 1
assert Poly(f).real_roots() == [Rational(-1, 2), 2, 2]
assert Poly(g).real_roots() == [rootof(g, 0)]
def test_all_roots():
f = 2*x**3 - 7*x**2 + 4*x + 4
g = x**3 + x + 1
assert Poly(f).all_roots() == [Rational(-1, 2), 2, 2]
assert Poly(g).all_roots() == [rootof(g, 0), rootof(g, 1), rootof(g, 2)]
def test_nroots():
assert Poly(0, x).nroots() == []
assert Poly(1, x).nroots() == []
assert Poly(x**2 - 1, x).nroots() == [-1.0, 1.0]
assert Poly(x**2 + 1, x).nroots() == [-1.0*I, 1.0*I]
roots = Poly(x**2 - 1, x).nroots()
assert roots == [-1.0, 1.0]
roots = Poly(x**2 + 1, x).nroots()
assert roots == [-1.0*I, 1.0*I]
roots = Poly(x**2/3 - Rational(1, 3), x).nroots()
assert roots == [-1.0, 1.0]
roots = Poly(x**2/3 + Rational(1, 3), x).nroots()
assert roots == [-1.0*I, 1.0*I]
assert Poly(x**2 + 2*I, x).nroots() == [-1.0 + 1.0*I, 1.0 - 1.0*I]
assert Poly(
x**2 + 2*I, x, extension=I).nroots() == [-1.0 + 1.0*I, 1.0 - 1.0*I]
assert Poly(0.2*x + 0.1).nroots() == [-0.5]
roots = nroots(x**5 + x + 1, n=5)
eps = Float("1e-5")
assert re(roots[0]).epsilon_eq(-0.75487, eps) is S.true
assert im(roots[0]) == 0.0
assert re(roots[1]) == -0.5
assert im(roots[1]).epsilon_eq(-0.86602, eps) is S.true
assert re(roots[2]) == -0.5
assert im(roots[2]).epsilon_eq(+0.86602, eps) is S.true
assert re(roots[3]).epsilon_eq(+0.87743, eps) is S.true
assert im(roots[3]).epsilon_eq(-0.74486, eps) is S.true
assert re(roots[4]).epsilon_eq(+0.87743, eps) is S.true
assert im(roots[4]).epsilon_eq(+0.74486, eps) is S.true
eps = Float("1e-6")
assert re(roots[0]).epsilon_eq(-0.75487, eps) is S.false
assert im(roots[0]) == 0.0
assert re(roots[1]) == -0.5
assert im(roots[1]).epsilon_eq(-0.86602, eps) is S.false
assert re(roots[2]) == -0.5
assert im(roots[2]).epsilon_eq(+0.86602, eps) is S.false
assert re(roots[3]).epsilon_eq(+0.87743, eps) is S.false
assert im(roots[3]).epsilon_eq(-0.74486, eps) is S.false
assert re(roots[4]).epsilon_eq(+0.87743, eps) is S.false
assert im(roots[4]).epsilon_eq(+0.74486, eps) is S.false
raises(DomainError, lambda: Poly(x + y, x).nroots())
raises(MultivariatePolynomialError, lambda: Poly(x + y).nroots())
assert nroots(x**2 - 1) == [-1.0, 1.0]
roots = nroots(x**2 - 1)
assert roots == [-1.0, 1.0]
assert nroots(x + I) == [-1.0*I]
assert nroots(x + 2*I) == [-2.0*I]
raises(PolynomialError, lambda: nroots(0))
# issue 8296
f = Poly(x**4 - 1)
assert f.nroots(2) == [w.n(2) for w in f.all_roots()]
assert str(Poly(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 +
39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 +
877969).nroots(2)) == ('[-1.7 - 1.9*I, -1.7 + 1.9*I, -1.7 '
'- 2.5*I, -1.7 + 2.5*I, -1.0*I, 1.0*I, -1.7*I, 1.7*I, -2.8*I, '
'2.8*I, -3.4*I, 3.4*I, 1.7 - 1.9*I, 1.7 + 1.9*I, 1.7 - 2.5*I, '
'1.7 + 2.5*I]')
assert str(Poly(1e-15*x**2 -1).nroots()) == ('[-31622776.6016838, 31622776.6016838]')
def test_ground_roots():
f = x**6 - 4*x**4 + 4*x**3 - x**2
assert Poly(f).ground_roots() == {S.One: 2, S.Zero: 2}
assert ground_roots(f) == {S.One: 2, S.Zero: 2}
def test_nth_power_roots_poly():
f = x**4 - x**2 + 1
f_2 = (x**2 - x + 1)**2
f_3 = (x**2 + 1)**2
f_4 = (x**2 + x + 1)**2
f_12 = (x - 1)**4
assert nth_power_roots_poly(f, 1) == f
raises(ValueError, lambda: nth_power_roots_poly(f, 0))
raises(ValueError, lambda: nth_power_roots_poly(f, x))
assert factor(nth_power_roots_poly(f, 2)) == f_2
assert factor(nth_power_roots_poly(f, 3)) == f_3
assert factor(nth_power_roots_poly(f, 4)) == f_4
assert factor(nth_power_roots_poly(f, 12)) == f_12
raises(MultivariatePolynomialError, lambda: nth_power_roots_poly(
x + y, 2, x, y))
def test_same_root():
f = Poly(x**4 + x**3 + x**2 + x + 1)
eq = f.same_root
r0 = exp(2 * I * pi / 5)
assert [i for i, r in enumerate(f.all_roots()) if eq(r, r0)] == [3]
raises(PolynomialError,
lambda: Poly(x + 1, domain=QQ).same_root(0, 0))
raises(DomainError,
lambda: Poly(x**2 + 1, domain=FF(7)).same_root(0, 0))
raises(DomainError,
lambda: Poly(x ** 2 + 1, domain=ZZ_I).same_root(0, 0))
raises(DomainError,
lambda: Poly(y * x**2 + 1, domain=ZZ[y]).same_root(0, 0))
raises(MultivariatePolynomialError,
lambda: Poly(x * y + 1, domain=ZZ).same_root(0, 0))
def test_torational_factor_list():
p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + sqrt(2))}))
assert _torational_factor_list(p, x) == (-2, [
(-x*(1 + sqrt(2))/2 + 1, 1),
(-x*(1 + sqrt(2)) - 1, 1),
(-x*(1 + sqrt(2)) + 1, 1)])
p = expand(((x**2-1)*(x-2)).subs({x:x*(1 + 2**Rational(1, 4))}))
assert _torational_factor_list(p, x) is None
def test_cancel():
assert cancel(0) == 0
assert cancel(7) == 7
assert cancel(x) == x
assert cancel(oo) is oo
assert cancel((2, 3)) == (1, 2, 3)
assert cancel((1, 0), x) == (1, 1, 0)
assert cancel((0, 1), x) == (1, 0, 1)
f, g, p, q = 4*x**2 - 4, 2*x - 2, 2*x + 2, 1
F, G, P, Q = [ Poly(u, x) for u in (f, g, p, q) ]
assert F.cancel(G) == (1, P, Q)
assert cancel((f, g)) == (1, p, q)
assert cancel((f, g), x) == (1, p, q)
assert cancel((f, g), (x,)) == (1, p, q)
assert cancel((F, G)) == (1, P, Q)
assert cancel((f, g), polys=True) == (1, P, Q)
assert cancel((F, G), polys=False) == (1, p, q)
f = (x**2 - 2)/(x + sqrt(2))
assert cancel(f) == f
assert cancel(f, greedy=False) == x - sqrt(2)
f = (x**2 - 2)/(x - sqrt(2))
assert cancel(f) == f
assert cancel(f, greedy=False) == x + sqrt(2)
assert cancel((x**2/4 - 1, x/2 - 1)) == (1, x + 2, 2)
# assert cancel((x**2/4 - 1, x/2 - 1)) == (S.Half, x + 2, 1)
assert cancel((x**2 - y)/(x - y)) == 1/(x - y)*(x**2 - y)
assert cancel((x**2 - y**2)/(x - y), x) == x + y
assert cancel((x**2 - y**2)/(x - y), y) == x + y
assert cancel((x**2 - y**2)/(x - y)) == x + y
assert cancel((x**3 - 1)/(x**2 - 1)) == (x**2 + x + 1)/(x + 1)
assert cancel((x**3/2 - S.Half)/(x**2 - 1)) == (x**2 + x + 1)/(2*x + 2)
assert cancel((exp(2*x) + 2*exp(x) + 1)/(exp(x) + 1)) == exp(x) + 1
f = Poly(x**2 - a**2, x)
g = Poly(x - a, x)
F = Poly(x + a, x, domain='ZZ[a]')
G = Poly(1, x, domain='ZZ[a]')
assert cancel((f, g)) == (1, F, G)
f = x**3 + (sqrt(2) - 2)*x**2 - (2*sqrt(2) + 3)*x - 3*sqrt(2)
g = x**2 - 2
assert cancel((f, g), extension=True) == (1, x**2 - 2*x - 3, x - sqrt(2))
f = Poly(-2*x + 3, x)
g = Poly(-x**9 + x**8 + x**6 - x**5 + 2*x**2 - 3*x + 1, x)
assert cancel((f, g)) == (1, -f, -g)
f = Poly(y, y, domain='ZZ(x)')
g = Poly(1, y, domain='ZZ[x]')
assert f.cancel(
g) == (1, Poly(y, y, domain='ZZ(x)'), Poly(1, y, domain='ZZ(x)'))
assert f.cancel(g, include=True) == (
Poly(y, y, domain='ZZ(x)'), Poly(1, y, domain='ZZ(x)'))
f = Poly(5*x*y + x, y, domain='ZZ(x)')
g = Poly(2*x**2*y, y, domain='ZZ(x)')
assert f.cancel(g, include=True) == (
Poly(5*y + 1, y, domain='ZZ(x)'), Poly(2*x*y, y, domain='ZZ(x)'))
f = -(-2*x - 4*y + 0.005*(z - y)**2)/((z - y)*(-z + y + 2))
assert cancel(f).is_Mul == True
P = tanh(x - 3.0)
Q = tanh(x + 3.0)
f = ((-2*P**2 + 2)*(-P**2 + 1)*Q**2/2 + (-2*P**2 + 2)*(-2*Q**2 + 2)*P*Q - (-2*P**2 + 2)*P**2*Q**2 + (-2*Q**2 + 2)*(-Q**2 + 1)*P**2/2 - (-2*Q**2 + 2)*P**2*Q**2)/(2*sqrt(P**2*Q**2 + 0.0001)) \
+ (-(-2*P**2 + 2)*P*Q**2/2 - (-2*Q**2 + 2)*P**2*Q/2)*((-2*P**2 + 2)*P*Q**2/2 + (-2*Q**2 + 2)*P**2*Q/2)/(2*(P**2*Q**2 + 0.0001)**Rational(3, 2))
assert cancel(f).is_Mul == True
# issue 7022
A = Symbol('A', commutative=False)
p1 = Piecewise((A*(x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True))
p2 = Piecewise((A*(x - 1), x > 1), (1/x, True))
assert cancel(p1) == p2
assert cancel(2*p1) == 2*p2
assert cancel(1 + p1) == 1 + p2
assert cancel((x**2 - 1)/(x + 1)*p1) == (x - 1)*p2
assert cancel((x**2 - 1)/(x + 1) + p1) == (x - 1) + p2
p3 = Piecewise(((x**2 - 1)/(x + 1), x > 1), ((x + 2)/(x**2 + 2*x), True))
p4 = Piecewise(((x - 1), x > 1), (1/x, True))
assert cancel(p3) == p4
assert cancel(2*p3) == 2*p4
assert cancel(1 + p3) == 1 + p4
assert cancel((x**2 - 1)/(x + 1)*p3) == (x - 1)*p4
assert cancel((x**2 - 1)/(x + 1) + p3) == (x - 1) + p4
# issue 4077
q = S('''(2*1*(x - 1/x)/(x*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x -
1/x)) - 2/x)) - 2*1*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x -
1/x)))*((-x + 1/x)*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x -
1/x)))/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) -
2/x) + 1)*((x - 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - 1/x)**2)) -
1/(x*(x - 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x
- 1/x)) - 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) -
1/(x**2*(x - 1/x)) - 2/x)/x - 1/x)*(((-x + 1/x)/((x*(x - 1/x)**2)) +
1/(x*(x - 1/x)))*((-(x - 1/x)/(x*(x - 1/x)) - 1/x)*((x - 1/x)/((x*(x -
1/x)**2)) - 1/(x*(x - 1/x)))/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) -
1/(x**2*(x - 1/x)) - 2/x) - 1 + (x - 1/x)/(x - 1/x))/((x*((x -
1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x -
1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) -
2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x
- 1/x)) - 2/x))) + ((x - 1/x)/((x*(x - 1/x))) + 1/x)/((x*(2*x - (-x +
1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x))) + 1/x)/(2*x +
2*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - 1/x)))*((-(x - 1/x)/(x*(x
- 1/x)) - 1/x)*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x - 1/x)))/(2*x -
(-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x) - 1 + (x -
1/x)/(x - 1/x))/((x*((x - 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x -
1/x)**2)) - 1/(x*(x - 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2)
- 1/(x**2*(x - 1/x)) - 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x
- 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x))) - 2*((x - 1/x)/((x*(x -
1/x))) + 1/x)/(x*(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x -
1/x)) - 2/x)) - 2/x) - ((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x -
1/x)))*((-x + 1/x)*((x - 1/x)/((x*(x - 1/x)**2)) - 1/(x*(x -
1/x)))/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) -
2/x) + 1)/(x*((x - 1/x)/((x - 1/x)**2) - ((x - 1/x)/((x*(x - 1/x)**2))
- 1/(x*(x - 1/x)))**2/(2*x - (-x + 1/x)/(x**2*(x - 1/x)**2) -
1/(x**2*(x - 1/x)) - 2/x) - 1/(x - 1/x))*(2*x - (-x + 1/x)/(x**2*(x -
1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x)) + (x - 1/x)/((x*(2*x - (-x +
1/x)/(x**2*(x - 1/x)**2) - 1/(x**2*(x - 1/x)) - 2/x))) - 1/x''',
evaluate=False)
assert cancel(q, _signsimp=False) is S.NaN
assert q.subs(x, 2) is S.NaN
assert signsimp(q) is S.NaN
# issue 9363
M = MatrixSymbol('M', 5, 5)
assert cancel(M[0,0] + 7) == M[0,0] + 7
expr = sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2] / z
assert cancel(expr) == (z*sin(M[1, 4] + M[2, 1] * 5 * M[4, 0]) - 5 * M[1, 2]) / z
assert cancel((x**2 + 1)/(x - I)) == x + I
def test_reduced():
f = 2*x**4 + y**2 - x**2 + y**3
G = [x**3 - x, y**3 - y]
Q = [2*x, 1]
r = x**2 + y**2 + y
assert reduced(f, G) == (Q, r)
assert reduced(f, G, x, y) == (Q, r)
H = groebner(G)
assert H.reduce(f) == (Q, r)
Q = [Poly(2*x, x, y), Poly(1, x, y)]
r = Poly(x**2 + y**2 + y, x, y)
assert _strict_eq(reduced(f, G, polys=True), (Q, r))
assert _strict_eq(reduced(f, G, x, y, polys=True), (Q, r))
H = groebner(G, polys=True)
assert _strict_eq(H.reduce(f), (Q, r))
f = 2*x**3 + y**3 + 3*y
G = groebner([x**2 + y**2 - 1, x*y - 2])
Q = [x**2 - x*y**3/2 + x*y/2 + y**6/4 - y**4/2 + y**2/4, -y**5/4 + y**3/2 + y*Rational(3, 4)]
r = 0
assert reduced(f, G) == (Q, r)
assert G.reduce(f) == (Q, r)
assert reduced(f, G, auto=False)[1] != 0
assert G.reduce(f, auto=False)[1] != 0
assert G.contains(f) is True
assert G.contains(f + 1) is False
assert reduced(1, [1], x) == ([1], 0)
raises(ComputationFailed, lambda: reduced(1, [1]))
def test_groebner():
assert groebner([], x, y, z) == []
assert groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex') == [1 + x**2, -1 + y**4]
assert groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex') == [-1 + y**4, z**3, 1 + x**2]
assert groebner([x**2 + 1, y**4*x + x**3], x, y, order='lex', polys=True) == \
[Poly(1 + x**2, x, y), Poly(-1 + y**4, x, y)]
assert groebner([x**2 + 1, y**4*x + x**3, x*y*z**3], x, y, z, order='grevlex', polys=True) == \
[Poly(-1 + y**4, x, y, z), Poly(z**3, x, y, z), Poly(1 + x**2, x, y, z)]
assert groebner([x**3 - 1, x**2 - 1]) == [x - 1]
assert groebner([Eq(x**3, 1), Eq(x**2, 1)]) == [x - 1]
F = [3*x**2 + y*z - 5*x - 1, 2*x + 3*x*y + y**2, x - 3*y + x*z - 2*z**2]
f = z**9 - x**2*y**3 - 3*x*y**2*z + 11*y*z**2 + x**2*z**2 - 5
G = groebner(F, x, y, z, modulus=7, symmetric=False)
assert G == [1 + x + y + 3*z + 2*z**2 + 2*z**3 + 6*z**4 + z**5,
1 + 3*y + y**2 + 6*z**2 + 3*z**3 + 3*z**4 + 3*z**5 + 4*z**6,
1 + 4*y + 4*z + y*z + 4*z**3 + z**4 + z**6,
6 + 6*z + z**2 + 4*z**3 + 3*z**4 + 6*z**5 + 3*z**6 + z**7]
Q, r = reduced(f, G, x, y, z, modulus=7, symmetric=False, polys=True)
assert sum([ q*g for q, g in zip(Q, G.polys)], r) == Poly(f, modulus=7)
F = [x*y - 2*y, 2*y**2 - x**2]
assert groebner(F, x, y, order='grevlex') == \
[y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y]
assert groebner(F, y, x, order='grevlex') == \
[x**3 - 2*x**2, -x**2 + 2*y**2, x*y - 2*y]
assert groebner(F, order='grevlex', field=True) == \
[y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y]
assert groebner([1], x) == [1]
assert groebner([x**2 + 2.0*y], x, y) == [1.0*x**2 + 2.0*y]
raises(ComputationFailed, lambda: groebner([1]))
assert groebner([x**2 - 1, x**3 + 1], method='buchberger') == [x + 1]
assert groebner([x**2 - 1, x**3 + 1], method='f5b') == [x + 1]
raises(ValueError, lambda: groebner([x, y], method='unknown'))
def test_fglm():
F = [a + b + c + d, a*b + a*d + b*c + b*d, a*b*c + a*b*d + a*c*d + b*c*d, a*b*c*d - 1]
G = groebner(F, a, b, c, d, order=grlex)
B = [
4*a + 3*d**9 - 4*d**5 - 3*d,
4*b + 4*c - 3*d**9 + 4*d**5 + 7*d,
4*c**2 + 3*d**10 - 4*d**6 - 3*d**2,
4*c*d**4 + 4*c - d**9 + 4*d**5 + 5*d,
d**12 - d**8 - d**4 + 1,
]
assert groebner(F, a, b, c, d, order=lex) == B
assert G.fglm(lex) == B
F = [9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9,
-72*t*x**7 - 252*t*x**6 + 192*t*x**5 + 1260*t*x**4 + 312*t*x**3 - 404*t*x**2 - 576*t*x + \
108*t - 72*x**7 - 256*x**6 + 192*x**5 + 1280*x**4 + 312*x**3 - 576*x + 96]
G = groebner(F, t, x, order=grlex)
B = [
203577793572507451707*t + 627982239411707112*x**7 - 666924143779443762*x**6 - \
10874593056632447619*x**5 + 5119998792707079562*x**4 + 72917161949456066376*x**3 + \
20362663855832380362*x**2 - 142079311455258371571*x + 183756699868981873194,
9*x**8 + 36*x**7 - 32*x**6 - 252*x**5 - 78*x**4 + 468*x**3 + 288*x**2 - 108*x + 9,
]
assert groebner(F, t, x, order=lex) == B
assert G.fglm(lex) == B
F = [x**2 - x - 3*y + 1, -2*x + y**2 + y - 1]
G = groebner(F, x, y, order=lex)
B = [
x**2 - x - 3*y + 1,
y**2 - 2*x + y - 1,
]
assert groebner(F, x, y, order=grlex) == B
assert G.fglm(grlex) == B
def test_is_zero_dimensional():
assert is_zero_dimensional([x, y], x, y) is True
assert is_zero_dimensional([x**3 + y**2], x, y) is False
assert is_zero_dimensional([x, y, z], x, y, z) is True
assert is_zero_dimensional([x, y, z], x, y, z, t) is False
F = [x*y - z, y*z - x, x*y - y]
assert is_zero_dimensional(F, x, y, z) is True
F = [x**2 - 2*x*z + 5, x*y**2 + y*z**3, 3*y**2 - 8*z**2]
assert is_zero_dimensional(F, x, y, z) is True
def test_GroebnerBasis():
F = [x*y - 2*y, 2*y**2 - x**2]
G = groebner(F, x, y, order='grevlex')
H = [y**3 - 2*y, x**2 - 2*y**2, x*y - 2*y]
P = [ Poly(h, x, y) for h in H ]
assert groebner(F + [0], x, y, order='grevlex') == G
assert isinstance(G, GroebnerBasis) is True
assert len(G) == 3
assert G[0] == H[0] and not G[0].is_Poly
assert G[1] == H[1] and not G[1].is_Poly
assert G[2] == H[2] and not G[2].is_Poly
assert G[1:] == H[1:] and not any(g.is_Poly for g in G[1:])
assert G[:2] == H[:2] and not any(g.is_Poly for g in G[1:])
assert G.exprs == H
assert G.polys == P
assert G.gens == (x, y)
assert G.domain == ZZ
assert G.order == grevlex
assert G == H
assert G == tuple(H)
assert G == P
assert G == tuple(P)
assert G != []
G = groebner(F, x, y, order='grevlex', polys=True)
assert G[0] == P[0] and G[0].is_Poly
assert G[1] == P[1] and G[1].is_Poly
assert G[2] == P[2] and G[2].is_Poly
assert G[1:] == P[1:] and all(g.is_Poly for g in G[1:])
assert G[:2] == P[:2] and all(g.is_Poly for g in G[1:])
def test_poly():
assert poly(x) == Poly(x, x)
assert poly(y) == Poly(y, y)
assert poly(x + y) == Poly(x + y, x, y)
assert poly(x + sin(x)) == Poly(x + sin(x), x, sin(x))
assert poly(x + y, wrt=y) == Poly(x + y, y, x)
assert poly(x + sin(x), wrt=sin(x)) == Poly(x + sin(x), sin(x), x)
assert poly(x*y + 2*x*z**2 + 17) == Poly(x*y + 2*x*z**2 + 17, x, y, z)
assert poly(2*(y + z)**2 - 1) == Poly(2*y**2 + 4*y*z + 2*z**2 - 1, y, z)
assert poly(
x*(y + z)**2 - 1) == Poly(x*y**2 + 2*x*y*z + x*z**2 - 1, x, y, z)
assert poly(2*x*(
y + z)**2 - 1) == Poly(2*x*y**2 + 4*x*y*z + 2*x*z**2 - 1, x, y, z)
assert poly(2*(
y + z)**2 - x - 1) == Poly(2*y**2 + 4*y*z + 2*z**2 - x - 1, x, y, z)
assert poly(x*(
y + z)**2 - x - 1) == Poly(x*y**2 + 2*x*y*z + x*z**2 - x - 1, x, y, z)
assert poly(2*x*(y + z)**2 - x - 1) == Poly(2*x*y**2 + 4*x*y*z + 2*
x*z**2 - x - 1, x, y, z)
assert poly(x*y + (x + y)**2 + (x + z)**2) == \
Poly(2*x*z + 3*x*y + y**2 + z**2 + 2*x**2, x, y, z)
assert poly(x*y*(x + y)*(x + z)**2) == \
Poly(x**3*y**2 + x*y**2*z**2 + y*x**2*z**2 + 2*z*x**2*
y**2 + 2*y*z*x**3 + y*x**4, x, y, z)
assert poly(Poly(x + y + z, y, x, z)) == Poly(x + y + z, y, x, z)
assert poly((x + y)**2, x) == Poly(x**2 + 2*x*y + y**2, x, domain=ZZ[y])
assert poly((x + y)**2, y) == Poly(x**2 + 2*x*y + y**2, y, domain=ZZ[x])
assert poly(1, x) == Poly(1, x)
raises(GeneratorsNeeded, lambda: poly(1))
# issue 6184
assert poly(x + y, x, y) == Poly(x + y, x, y)
assert poly(x + y, y, x) == Poly(x + y, y, x)
def test_keep_coeff():
u = Mul(2, x + 1, evaluate=False)
assert _keep_coeff(S.One, x) == x
assert _keep_coeff(S.NegativeOne, x) == -x
assert _keep_coeff(S(1.0), x) == 1.0*x
assert _keep_coeff(S(-1.0), x) == -1.0*x
assert _keep_coeff(S.One, 2*x) == 2*x
assert _keep_coeff(S(2), x/2) == x
assert _keep_coeff(S(2), sin(x)) == 2*sin(x)
assert _keep_coeff(S(2), x + 1) == u
assert _keep_coeff(x, 1/x) == 1
assert _keep_coeff(x + 1, S(2)) == u
assert _keep_coeff(S.Half, S.One) == S.Half
p = Pow(2, 3, evaluate=False)
assert _keep_coeff(S(-1), p) == Mul(-1, p, evaluate=False)
a = Add(2, p, evaluate=False)
assert _keep_coeff(S.Half, a, clear=True
) == Mul(S.Half, a, evaluate=False)
assert _keep_coeff(S.Half, a, clear=False
) == Add(1, Mul(S.Half, p, evaluate=False), evaluate=False)
def test_poly_matching_consistency():
# Test for this issue:
# https://github.com/sympy/sympy/issues/5514
assert I * Poly(x, x) == Poly(I*x, x)
assert Poly(x, x) * I == Poly(I*x, x)
def test_issue_5786():
assert expand(factor(expand(
(x - I*y)*(z - I*t)), extension=[I])) == -I*t*x - t*y + x*z - I*y*z
def test_noncommutative():
class foo(Expr):
is_commutative=False
e = x/(x + x*y)
c = 1/( 1 + y)
assert cancel(foo(e)) == foo(c)
assert cancel(e + foo(e)) == c + foo(c)
assert cancel(e*foo(c)) == c*foo(c)
def test_to_rational_coeffs():
assert to_rational_coeffs(
Poly(x**3 + y*x**2 + sqrt(y), x, domain='EX')) is None
# issue 21268
assert to_rational_coeffs(
Poly(y**3 + sqrt(2)*y**2*sin(x) + 1, y)) is None
assert to_rational_coeffs(Poly(x, y)) is None
assert to_rational_coeffs(Poly(sqrt(2)*y)) is None
def test_factor_terms():
# issue 7067
assert factor_list(x*(x + y)) == (1, [(x, 1), (x + y, 1)])
assert sqf_list(x*(x + y)) == (1, [(x**2 + x*y, 1)])
def test_as_list():
# issue 14496
assert Poly(x**3 + 2, x, domain='ZZ').as_list() == [1, 0, 0, 2]
assert Poly(x**2 + y + 1, x, y, domain='ZZ').as_list() == [[1], [], [1, 1]]
assert Poly(x**2 + y + 1, x, y, z, domain='ZZ').as_list() == \
[[[1]], [[]], [[1], [1]]]
def test_issue_11198():
assert factor_list(sqrt(2)*x) == (sqrt(2), [(x, 1)])
assert factor_list(sqrt(2)*sin(x), sin(x)) == (sqrt(2), [(sin(x), 1)])
def test_Poly_precision():
# Make sure Poly doesn't lose precision
p = Poly(pi.evalf(100)*x)
assert p.as_expr() == pi.evalf(100)*x
def test_issue_12400():
# Correction of check for negative exponents
assert poly(1/(1+sqrt(2)), x) == \
Poly(1/(1+sqrt(2)), x, domain='EX')
def test_issue_14364():
assert gcd(S(6)*(1 + sqrt(3))/5, S(3)*(1 + sqrt(3))/10) == Rational(3, 10) * (1 + sqrt(3))
assert gcd(sqrt(5)*Rational(4, 7), sqrt(5)*Rational(2, 3)) == sqrt(5)*Rational(2, 21)
assert lcm(Rational(2, 3)*sqrt(3), Rational(5, 6)*sqrt(3)) == S(10)*sqrt(3)/3
assert lcm(3*sqrt(3), 4/sqrt(3)) == 12*sqrt(3)
assert lcm(S(5)*(1 + 2**Rational(1, 3))/6, S(3)*(1 + 2**Rational(1, 3))/8) == Rational(15, 2) * (1 + 2**Rational(1, 3))
assert gcd(Rational(2, 3)*sqrt(3), Rational(5, 6)/sqrt(3)) == sqrt(3)/18
assert gcd(S(4)*sqrt(13)/7, S(3)*sqrt(13)/14) == sqrt(13)/14
# gcd_list and lcm_list
assert gcd([S(2)*sqrt(47)/7, S(6)*sqrt(47)/5, S(8)*sqrt(47)/5]) == sqrt(47)*Rational(2, 35)
assert gcd([S(6)*(1 + sqrt(7))/5, S(2)*(1 + sqrt(7))/7, S(4)*(1 + sqrt(7))/13]) == (1 + sqrt(7))*Rational(2, 455)
assert lcm((Rational(7, 2)/sqrt(15), Rational(5, 6)/sqrt(15), Rational(5, 8)/sqrt(15))) == Rational(35, 2)/sqrt(15)
assert lcm([S(5)*(2 + 2**Rational(5, 7))/6, S(7)*(2 + 2**Rational(5, 7))/2, S(13)*(2 + 2**Rational(5, 7))/4]) == Rational(455, 2) * (2 + 2**Rational(5, 7))
def test_issue_15669():
x = Symbol("x", positive=True)
expr = (16*x**3/(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**2 -
2*2**Rational(4, 5)*x*(-x**2 + sqrt(8*x**2 + (x**2 - 2)**2) + 2)**Rational(3, 5) + 10*x)
assert factor(expr, deep=True) == x*(x**2 + 2)
def test_issue_17988():
x = Symbol('x')
p = poly(x - 1)
with warns_deprecated_sympy():
M = Matrix([[poly(x + 1), poly(x + 1)]])
with warns(SymPyDeprecationWarning, test_stacklevel=False):
assert p * M == M * p == Matrix([[poly(x**2 - 1), poly(x**2 - 1)]])
def test_issue_18205():
assert cancel((2 + I)*(3 - I)) == 7 + I
assert cancel((2 + I)*(2 - I)) == 5
def test_issue_8695():
p = (x**2 + 1) * (x - 1)**2 * (x - 2)**3 * (x - 3)**3
result = (1, [(x**2 + 1, 1), (x - 1, 2), (x**2 - 5*x + 6, 3)])
assert sqf_list(p) == result
def test_issue_19113():
eq = sin(x)**3 - sin(x) + 1
raises(PolynomialError, lambda: refine_root(eq, 1, 2, 1e-2))
raises(PolynomialError, lambda: count_roots(eq, -1, 1))
raises(PolynomialError, lambda: real_roots(eq))
raises(PolynomialError, lambda: nroots(eq))
raises(PolynomialError, lambda: ground_roots(eq))
raises(PolynomialError, lambda: nth_power_roots_poly(eq, 2))
def test_issue_19360():
f = 2*x**2 - 2*sqrt(2)*x*y + y**2
assert factor(f, extension=sqrt(2)) == 2*(x - (sqrt(2)*y/2))**2
f = -I*t*x - t*y + x*z - I*y*z
assert factor(f, extension=I) == (x - I*y)*(-I*t + z)
def test_poly_copy_equals_original():
poly = Poly(x + y, x, y, z)
copy = poly.copy()
assert poly == copy, (
"Copied polynomial not equal to original.")
assert poly.gens == copy.gens, (
"Copied polynomial has different generators than original.")
def test_deserialized_poly_equals_original():
poly = Poly(x + y, x, y, z)
deserialized = pickle.loads(pickle.dumps(poly))
assert poly == deserialized, (
"Deserialized polynomial not equal to original.")
assert poly.gens == deserialized.gens, (
"Deserialized polynomial has different generators than original.")
def test_issue_20389():
result = degree(x * (x + 1) - x ** 2 - x, x)
assert result == -oo
def test_issue_20985():
from sympy.core.symbol import symbols
w, R = symbols('w R')
poly = Poly(1.0 + I*w/R, w, 1/R)
assert poly.degree() == S(1)
|
6ed2916aa7248efc863a10af21fc0bb510d51b6422c28e824e7a0f5bc8784bd1 | from sympy.polys.galoistools import (
gf_crt, gf_crt1, gf_crt2, gf_int,
gf_degree, gf_strip, gf_trunc, gf_normal,
gf_from_dict, gf_to_dict,
gf_from_int_poly, gf_to_int_poly,
gf_neg, gf_add_ground, gf_sub_ground, gf_mul_ground,
gf_add, gf_sub, gf_add_mul, gf_sub_mul, gf_mul, gf_sqr,
gf_div, gf_rem, gf_quo, gf_exquo,
gf_lshift, gf_rshift, gf_expand,
gf_pow, gf_pow_mod,
gf_gcdex, gf_gcd, gf_lcm, gf_cofactors,
gf_LC, gf_TC, gf_monic,
gf_eval, gf_multi_eval,
gf_compose, gf_compose_mod,
gf_trace_map,
gf_diff,
gf_irreducible, gf_irreducible_p,
gf_irred_p_ben_or, gf_irred_p_rabin,
gf_sqf_list, gf_sqf_part, gf_sqf_p,
gf_Qmatrix, gf_Qbasis,
gf_ddf_zassenhaus, gf_ddf_shoup,
gf_edf_zassenhaus, gf_edf_shoup,
gf_berlekamp,
gf_factor_sqf, gf_factor,
gf_value, linear_congruence, csolve_prime, gf_csolve,
gf_frobenius_map, gf_frobenius_monomial_base
)
from sympy.polys.polyerrors import (
ExactQuotientFailed,
)
from sympy.polys import polyconfig as config
from sympy.polys.domains import ZZ
from sympy.core.numbers import pi
from sympy.ntheory.generate import nextprime
from sympy.testing.pytest import raises
def test_gf_crt():
U = [49, 76, 65]
M = [99, 97, 95]
p = 912285
u = 639985
assert gf_crt(U, M, ZZ) == u
E = [9215, 9405, 9603]
S = [62, 24, 12]
assert gf_crt1(M, ZZ) == (p, E, S)
assert gf_crt2(U, M, p, E, S, ZZ) == u
def test_gf_int():
assert gf_int(0, 5) == 0
assert gf_int(1, 5) == 1
assert gf_int(2, 5) == 2
assert gf_int(3, 5) == -2
assert gf_int(4, 5) == -1
assert gf_int(5, 5) == 0
def test_gf_degree():
assert gf_degree([]) == -1
assert gf_degree([1]) == 0
assert gf_degree([1, 0]) == 1
assert gf_degree([1, 0, 0, 0, 1]) == 4
def test_gf_strip():
assert gf_strip([]) == []
assert gf_strip([0]) == []
assert gf_strip([0, 0, 0]) == []
assert gf_strip([1]) == [1]
assert gf_strip([0, 1]) == [1]
assert gf_strip([0, 0, 0, 1]) == [1]
assert gf_strip([1, 2, 0]) == [1, 2, 0]
assert gf_strip([0, 1, 2, 0]) == [1, 2, 0]
assert gf_strip([0, 0, 0, 1, 2, 0]) == [1, 2, 0]
def test_gf_trunc():
assert gf_trunc([], 11) == []
assert gf_trunc([1], 11) == [1]
assert gf_trunc([22], 11) == []
assert gf_trunc([12], 11) == [1]
assert gf_trunc([11, 22, 17, 1, 0], 11) == [6, 1, 0]
assert gf_trunc([12, 23, 17, 1, 0], 11) == [1, 1, 6, 1, 0]
def test_gf_normal():
assert gf_normal([11, 22, 17, 1, 0], 11, ZZ) == [6, 1, 0]
def test_gf_from_to_dict():
f = {11: 12, 6: 2, 0: 25}
F = {11: 1, 6: 2, 0: 3}
g = [1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3]
assert gf_from_dict(f, 11, ZZ) == g
assert gf_to_dict(g, 11) == F
f = {11: -5, 4: 0, 3: 1, 0: 12}
F = {11: -5, 3: 1, 0: 1}
g = [6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
assert gf_from_dict(f, 11, ZZ) == g
assert gf_to_dict(g, 11) == F
assert gf_to_dict([10], 11, symmetric=True) == {0: -1}
assert gf_to_dict([10], 11, symmetric=False) == {0: 10}
def test_gf_from_to_int_poly():
assert gf_from_int_poly([1, 0, 7, 2, 20], 5) == [1, 0, 2, 2, 0]
assert gf_to_int_poly([1, 0, 4, 2, 3], 5) == [1, 0, -1, 2, -2]
assert gf_to_int_poly([10], 11, symmetric=True) == [-1]
assert gf_to_int_poly([10], 11, symmetric=False) == [10]
def test_gf_LC():
assert gf_LC([], ZZ) == 0
assert gf_LC([1], ZZ) == 1
assert gf_LC([1, 2], ZZ) == 1
def test_gf_TC():
assert gf_TC([], ZZ) == 0
assert gf_TC([1], ZZ) == 1
assert gf_TC([1, 2], ZZ) == 2
def test_gf_monic():
assert gf_monic(ZZ.map([]), 11, ZZ) == (0, [])
assert gf_monic(ZZ.map([1]), 11, ZZ) == (1, [1])
assert gf_monic(ZZ.map([2]), 11, ZZ) == (2, [1])
assert gf_monic(ZZ.map([1, 2, 3, 4]), 11, ZZ) == (1, [1, 2, 3, 4])
assert gf_monic(ZZ.map([2, 3, 4, 5]), 11, ZZ) == (2, [1, 7, 2, 8])
def test_gf_arith():
assert gf_neg([], 11, ZZ) == []
assert gf_neg([1], 11, ZZ) == [10]
assert gf_neg([1, 2, 3], 11, ZZ) == [10, 9, 8]
assert gf_add_ground([], 0, 11, ZZ) == []
assert gf_sub_ground([], 0, 11, ZZ) == []
assert gf_add_ground([], 3, 11, ZZ) == [3]
assert gf_sub_ground([], 3, 11, ZZ) == [8]
assert gf_add_ground([1], 3, 11, ZZ) == [4]
assert gf_sub_ground([1], 3, 11, ZZ) == [9]
assert gf_add_ground([8], 3, 11, ZZ) == []
assert gf_sub_ground([3], 3, 11, ZZ) == []
assert gf_add_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 6]
assert gf_sub_ground([1, 2, 3], 3, 11, ZZ) == [1, 2, 0]
assert gf_mul_ground([], 0, 11, ZZ) == []
assert gf_mul_ground([], 1, 11, ZZ) == []
assert gf_mul_ground([1], 0, 11, ZZ) == []
assert gf_mul_ground([1], 1, 11, ZZ) == [1]
assert gf_mul_ground([1, 2, 3], 0, 11, ZZ) == []
assert gf_mul_ground([1, 2, 3], 1, 11, ZZ) == [1, 2, 3]
assert gf_mul_ground([1, 2, 3], 7, 11, ZZ) == [7, 3, 10]
assert gf_add([], [], 11, ZZ) == []
assert gf_add([1], [], 11, ZZ) == [1]
assert gf_add([], [1], 11, ZZ) == [1]
assert gf_add([1], [1], 11, ZZ) == [2]
assert gf_add([1], [2], 11, ZZ) == [3]
assert gf_add([1, 2], [1], 11, ZZ) == [1, 3]
assert gf_add([1], [1, 2], 11, ZZ) == [1, 3]
assert gf_add([1, 2, 3], [8, 9, 10], 11, ZZ) == [9, 0, 2]
assert gf_sub([], [], 11, ZZ) == []
assert gf_sub([1], [], 11, ZZ) == [1]
assert gf_sub([], [1], 11, ZZ) == [10]
assert gf_sub([1], [1], 11, ZZ) == []
assert gf_sub([1], [2], 11, ZZ) == [10]
assert gf_sub([1, 2], [1], 11, ZZ) == [1, 1]
assert gf_sub([1], [1, 2], 11, ZZ) == [10, 10]
assert gf_sub([3, 2, 1], [8, 9, 10], 11, ZZ) == [6, 4, 2]
assert gf_add_mul(
[1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [1, 2, 10, 8, 9]
assert gf_sub_mul(
[1, 5, 6], [7, 3], [8, 0, 6, 1], 11, ZZ) == [10, 9, 3, 2, 3]
assert gf_mul([], [], 11, ZZ) == []
assert gf_mul([], [1], 11, ZZ) == []
assert gf_mul([1], [], 11, ZZ) == []
assert gf_mul([1], [1], 11, ZZ) == [1]
assert gf_mul([5], [7], 11, ZZ) == [2]
assert gf_mul([3, 0, 0, 6, 1, 2], [4, 0, 1, 0], 11, ZZ) == [1, 0,
3, 2, 4, 3, 1, 2, 0]
assert gf_mul([4, 0, 1, 0], [3, 0, 0, 6, 1, 2], 11, ZZ) == [1, 0,
3, 2, 4, 3, 1, 2, 0]
assert gf_mul([2, 0, 0, 1, 7], [2, 0, 0, 1, 7], 11, ZZ) == [4, 0,
0, 4, 6, 0, 1, 3, 5]
assert gf_sqr([], 11, ZZ) == []
assert gf_sqr([2], 11, ZZ) == [4]
assert gf_sqr([1, 2], 11, ZZ) == [1, 4, 4]
assert gf_sqr([2, 0, 0, 1, 7], 11, ZZ) == [4, 0, 0, 4, 6, 0, 1, 3, 5]
def test_gf_division():
raises(ZeroDivisionError, lambda: gf_div([1, 2, 3], [], 11, ZZ))
raises(ZeroDivisionError, lambda: gf_rem([1, 2, 3], [], 11, ZZ))
raises(ZeroDivisionError, lambda: gf_quo([1, 2, 3], [], 11, ZZ))
raises(ZeroDivisionError, lambda: gf_quo([1, 2, 3], [], 11, ZZ))
assert gf_div([1], [1, 2, 3], 7, ZZ) == ([], [1])
assert gf_rem([1], [1, 2, 3], 7, ZZ) == [1]
assert gf_quo([1], [1, 2, 3], 7, ZZ) == []
f = ZZ.map([5, 4, 3, 2, 1, 0])
g = ZZ.map([1, 2, 3])
q = [5, 1, 0, 6]
r = [3, 3]
assert gf_div(f, g, 7, ZZ) == (q, r)
assert gf_rem(f, g, 7, ZZ) == r
assert gf_quo(f, g, 7, ZZ) == q
raises(ExactQuotientFailed, lambda: gf_exquo(f, g, 7, ZZ))
f = ZZ.map([5, 4, 3, 2, 1, 0])
g = ZZ.map([1, 2, 3, 0])
q = [5, 1, 0]
r = [6, 1, 0]
assert gf_div(f, g, 7, ZZ) == (q, r)
assert gf_rem(f, g, 7, ZZ) == r
assert gf_quo(f, g, 7, ZZ) == q
raises(ExactQuotientFailed, lambda: gf_exquo(f, g, 7, ZZ))
assert gf_quo(ZZ.map([1, 2, 1]), ZZ.map([1, 1]), 11, ZZ) == [1, 1]
def test_gf_shift():
f = [1, 2, 3, 4, 5]
assert gf_lshift([], 5, ZZ) == []
assert gf_rshift([], 5, ZZ) == ([], [])
assert gf_lshift(f, 1, ZZ) == [1, 2, 3, 4, 5, 0]
assert gf_lshift(f, 2, ZZ) == [1, 2, 3, 4, 5, 0, 0]
assert gf_rshift(f, 0, ZZ) == (f, [])
assert gf_rshift(f, 1, ZZ) == ([1, 2, 3, 4], [5])
assert gf_rshift(f, 3, ZZ) == ([1, 2], [3, 4, 5])
assert gf_rshift(f, 5, ZZ) == ([], f)
def test_gf_expand():
F = [([1, 1], 2), ([1, 2], 3)]
assert gf_expand(F, 11, ZZ) == [1, 8, 3, 5, 6, 8]
assert gf_expand((4, F), 11, ZZ) == [4, 10, 1, 9, 2, 10]
def test_gf_powering():
assert gf_pow([1, 0, 0, 1, 8], 0, 11, ZZ) == [1]
assert gf_pow([1, 0, 0, 1, 8], 1, 11, ZZ) == [1, 0, 0, 1, 8]
assert gf_pow([1, 0, 0, 1, 8], 2, 11, ZZ) == [1, 0, 0, 2, 5, 0, 1, 5, 9]
assert gf_pow([1, 0, 0, 1, 8], 5, 11, ZZ) == \
[1, 0, 0, 5, 7, 0, 10, 6, 2, 10, 9, 6, 10, 6, 6, 0, 5, 2, 5, 9, 10]
assert gf_pow([1, 0, 0, 1, 8], 8, 11, ZZ) == \
[1, 0, 0, 8, 9, 0, 6, 8, 10, 1, 2, 5, 10, 7, 7, 9, 1, 2, 0, 0, 6, 2,
5, 2, 5, 7, 7, 9, 10, 10, 7, 5, 5]
assert gf_pow([1, 0, 0, 1, 8], 45, 11, ZZ) == \
[ 1, 0, 0, 1, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0,
10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
6, 0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0,
10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0,
8, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, 9, 0, 0, 9, 6, 0, 0, 0, 0, 0, 0,
3, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0,
10, 0, 0, 10, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 5, 0, 0, 0, 0, 0, 0,
4, 0, 0, 4, 10]
assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 0, ZZ.map([2, 0, 7]), 11, ZZ) == [1]
assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 1, ZZ.map([2, 0, 7]), 11, ZZ) == [1, 1]
assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 2, ZZ.map([2, 0, 7]), 11, ZZ) == [2, 3]
assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 5, ZZ.map([2, 0, 7]), 11, ZZ) == [7, 8]
assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 8, ZZ.map([2, 0, 7]), 11, ZZ) == [1, 5]
assert gf_pow_mod(ZZ.map([1, 0, 0, 1, 8]), 45, ZZ.map([2, 0, 7]), 11, ZZ) == [5, 4]
def test_gf_gcdex():
assert gf_gcdex(ZZ.map([]), ZZ.map([]), 11, ZZ) == ([1], [], [])
assert gf_gcdex(ZZ.map([2]), ZZ.map([]), 11, ZZ) == ([6], [], [1])
assert gf_gcdex(ZZ.map([]), ZZ.map([2]), 11, ZZ) == ([], [6], [1])
assert gf_gcdex(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == ([], [6], [1])
assert gf_gcdex(ZZ.map([]), ZZ.map([3, 0]), 11, ZZ) == ([], [4], [1, 0])
assert gf_gcdex(ZZ.map([3, 0]), ZZ.map([]), 11, ZZ) == ([4], [], [1, 0])
assert gf_gcdex(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == ([], [4], [1, 0])
assert gf_gcdex(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == ([5, 6], [6], [1, 7])
def test_gf_gcd():
assert gf_gcd(ZZ.map([]), ZZ.map([]), 11, ZZ) == []
assert gf_gcd(ZZ.map([2]), ZZ.map([]), 11, ZZ) == [1]
assert gf_gcd(ZZ.map([]), ZZ.map([2]), 11, ZZ) == [1]
assert gf_gcd(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == [1]
assert gf_gcd(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == [1, 0]
assert gf_gcd(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == [1, 0]
assert gf_gcd(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == [1, 0]
assert gf_gcd(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == [1, 7]
def test_gf_lcm():
assert gf_lcm(ZZ.map([]), ZZ.map([]), 11, ZZ) == []
assert gf_lcm(ZZ.map([2]), ZZ.map([]), 11, ZZ) == []
assert gf_lcm(ZZ.map([]), ZZ.map([2]), 11, ZZ) == []
assert gf_lcm(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == [1]
assert gf_lcm(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == []
assert gf_lcm(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == []
assert gf_lcm(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == [1, 0]
assert gf_lcm(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == [1, 8, 8, 8, 7]
def test_gf_cofactors():
assert gf_cofactors(ZZ.map([]), ZZ.map([]), 11, ZZ) == ([], [], [])
assert gf_cofactors(ZZ.map([2]), ZZ.map([]), 11, ZZ) == ([1], [2], [])
assert gf_cofactors(ZZ.map([]), ZZ.map([2]), 11, ZZ) == ([1], [], [2])
assert gf_cofactors(ZZ.map([2]), ZZ.map([2]), 11, ZZ) == ([1], [2], [2])
assert gf_cofactors(ZZ.map([]), ZZ.map([1, 0]), 11, ZZ) == ([1, 0], [], [1])
assert gf_cofactors(ZZ.map([1, 0]), ZZ.map([]), 11, ZZ) == ([1, 0], [1], [])
assert gf_cofactors(ZZ.map([3, 0]), ZZ.map([3, 0]), 11, ZZ) == (
[1, 0], [3], [3])
assert gf_cofactors(ZZ.map([1, 8, 7]), ZZ.map([1, 7, 1, 7]), 11, ZZ) == (
([1, 7], [1, 1], [1, 0, 1]))
def test_gf_diff():
assert gf_diff([], 11, ZZ) == []
assert gf_diff([7], 11, ZZ) == []
assert gf_diff([7, 3], 11, ZZ) == [7]
assert gf_diff([7, 3, 1], 11, ZZ) == [3, 3]
assert gf_diff([1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], 11, ZZ) == []
def test_gf_eval():
assert gf_eval([], 4, 11, ZZ) == 0
assert gf_eval([], 27, 11, ZZ) == 0
assert gf_eval([7], 4, 11, ZZ) == 7
assert gf_eval([7], 27, 11, ZZ) == 7
assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 0, 11, ZZ) == 0
assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 4, 11, ZZ) == 9
assert gf_eval([1, 0, 3, 2, 4, 3, 1, 2, 0], 27, 11, ZZ) == 5
assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 0, 11, ZZ) == 5
assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 4, 11, ZZ) == 3
assert gf_eval([4, 0, 0, 4, 6, 0, 1, 3, 5], 27, 11, ZZ) == 9
assert gf_multi_eval([3, 2, 1], [0, 1, 2, 3], 11, ZZ) == [1, 6, 6, 1]
def test_gf_compose():
assert gf_compose([], [1, 0], 11, ZZ) == []
assert gf_compose_mod([], [1, 0], [1, 0], 11, ZZ) == []
assert gf_compose([1], [], 11, ZZ) == [1]
assert gf_compose([1, 0], [], 11, ZZ) == []
assert gf_compose([1, 0], [1, 0], 11, ZZ) == [1, 0]
f = ZZ.map([1, 1, 4, 9, 1])
g = ZZ.map([1, 1, 1])
h = ZZ.map([1, 0, 0, 2])
assert gf_compose(g, h, 11, ZZ) == [1, 0, 0, 5, 0, 0, 7]
assert gf_compose_mod(g, h, f, 11, ZZ) == [3, 9, 6, 10]
def test_gf_trace_map():
f = ZZ.map([1, 1, 4, 9, 1])
a = [1, 1, 1]
c = ZZ.map([1, 0])
b = gf_pow_mod(c, 11, f, 11, ZZ)
assert gf_trace_map(a, b, c, 0, f, 11, ZZ) == \
([1, 1, 1], [1, 1, 1])
assert gf_trace_map(a, b, c, 1, f, 11, ZZ) == \
([5, 2, 10, 3], [5, 3, 0, 4])
assert gf_trace_map(a, b, c, 2, f, 11, ZZ) == \
([5, 9, 5, 3], [10, 1, 5, 7])
assert gf_trace_map(a, b, c, 3, f, 11, ZZ) == \
([1, 10, 6, 0], [7])
assert gf_trace_map(a, b, c, 4, f, 11, ZZ) == \
([1, 1, 1], [1, 1, 8])
assert gf_trace_map(a, b, c, 5, f, 11, ZZ) == \
([5, 2, 10, 3], [5, 3, 0, 0])
assert gf_trace_map(a, b, c, 11, f, 11, ZZ) == \
([1, 10, 6, 0], [10])
def test_gf_irreducible():
assert gf_irreducible_p(gf_irreducible(1, 11, ZZ), 11, ZZ) is True
assert gf_irreducible_p(gf_irreducible(2, 11, ZZ), 11, ZZ) is True
assert gf_irreducible_p(gf_irreducible(3, 11, ZZ), 11, ZZ) is True
assert gf_irreducible_p(gf_irreducible(4, 11, ZZ), 11, ZZ) is True
assert gf_irreducible_p(gf_irreducible(5, 11, ZZ), 11, ZZ) is True
assert gf_irreducible_p(gf_irreducible(6, 11, ZZ), 11, ZZ) is True
assert gf_irreducible_p(gf_irreducible(7, 11, ZZ), 11, ZZ) is True
def test_gf_irreducible_p():
assert gf_irred_p_ben_or(ZZ.map([7]), 11, ZZ) is True
assert gf_irred_p_ben_or(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irred_p_ben_or(ZZ.map([7, 3, 1]), 11, ZZ) is False
assert gf_irred_p_rabin(ZZ.map([7]), 11, ZZ) is True
assert gf_irred_p_rabin(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irred_p_rabin(ZZ.map([7, 3, 1]), 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'ben-or')
assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'rabin')
assert gf_irreducible_p(ZZ.map([7]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3]), 11, ZZ) is True
assert gf_irreducible_p(ZZ.map([7, 3, 1]), 11, ZZ) is False
config.setup('GF_IRRED_METHOD', 'other')
raises(KeyError, lambda: gf_irreducible_p([7], 11, ZZ))
config.setup('GF_IRRED_METHOD')
f = ZZ.map([1, 9, 9, 13, 16, 15, 6, 7, 7, 7, 10])
g = ZZ.map([1, 7, 16, 7, 15, 13, 13, 11, 16, 10, 9])
h = gf_mul(f, g, 17, ZZ)
assert gf_irred_p_ben_or(f, 17, ZZ) is True
assert gf_irred_p_ben_or(g, 17, ZZ) is True
assert gf_irred_p_ben_or(h, 17, ZZ) is False
assert gf_irred_p_rabin(f, 17, ZZ) is True
assert gf_irred_p_rabin(g, 17, ZZ) is True
assert gf_irred_p_rabin(h, 17, ZZ) is False
def test_gf_squarefree():
assert gf_sqf_list([], 11, ZZ) == (0, [])
assert gf_sqf_list([1], 11, ZZ) == (1, [])
assert gf_sqf_list([1, 1], 11, ZZ) == (1, [([1, 1], 1)])
assert gf_sqf_p([], 11, ZZ) is True
assert gf_sqf_p([1], 11, ZZ) is True
assert gf_sqf_p([1, 1], 11, ZZ) is True
f = gf_from_dict({11: 1, 0: 1}, 11, ZZ)
assert gf_sqf_p(f, 11, ZZ) is False
assert gf_sqf_list(f, 11, ZZ) == \
(1, [([1, 1], 11)])
f = [1, 5, 8, 4]
assert gf_sqf_p(f, 11, ZZ) is False
assert gf_sqf_list(f, 11, ZZ) == \
(1, [([1, 1], 1),
([1, 2], 2)])
assert gf_sqf_part(f, 11, ZZ) == [1, 3, 2]
f = [1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0]
assert gf_sqf_list(f, 3, ZZ) == \
(1, [([1, 0], 1),
([1, 1], 3),
([1, 2], 6)])
def test_gf_frobenius_map():
f = ZZ.map([2, 0, 1, 0, 2, 2, 0, 2, 2, 2])
g = ZZ.map([1,1,0,2,0,1,0,2,0,1])
p = 3
b = gf_frobenius_monomial_base(g, p, ZZ)
h = gf_frobenius_map(f, g, b, p, ZZ)
h1 = gf_pow_mod(f, p, g, p, ZZ)
assert h == h1
def test_gf_berlekamp():
f = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11)
Q = [[1, 0, 0, 0, 0, 0],
[3, 5, 8, 8, 6, 5],
[3, 6, 6, 1, 10, 0],
[9, 4, 10, 3, 7, 9],
[7, 8, 10, 0, 0, 8],
[8, 10, 7, 8, 10, 8]]
V = [[1, 0, 0, 0, 0, 0],
[0, 1, 1, 1, 1, 0],
[0, 0, 7, 9, 0, 1]]
assert gf_Qmatrix(f, 11, ZZ) == Q
assert gf_Qbasis(Q, 11, ZZ) == V
assert gf_berlekamp(f, 11, ZZ) == \
[[1, 1], [1, 5, 3], [1, 2, 3, 4]]
f = ZZ.map([1, 0, 1, 0, 10, 10, 8, 2, 8])
Q = ZZ.map([[1, 0, 0, 0, 0, 0, 0, 0],
[2, 1, 7, 11, 10, 12, 5, 11],
[3, 6, 4, 3, 0, 4, 7, 2],
[4, 3, 6, 5, 1, 6, 2, 3],
[2, 11, 8, 8, 3, 1, 3, 11],
[6, 11, 8, 6, 2, 7, 10, 9],
[5, 11, 7, 10, 0, 11, 7, 12],
[3, 3, 12, 5, 0, 11, 9, 12]])
V = [[1, 0, 0, 0, 0, 0, 0, 0],
[0, 5, 5, 0, 9, 5, 1, 0],
[0, 9, 11, 9, 10, 12, 0, 1]]
assert gf_Qmatrix(f, 13, ZZ) == Q
assert gf_Qbasis(Q, 13, ZZ) == V
assert gf_berlekamp(f, 13, ZZ) == \
[[1, 3], [1, 8, 4, 12], [1, 2, 3, 4, 6]]
def test_gf_ddf():
f = gf_from_dict({15: ZZ(1), 0: ZZ(-1)}, 11, ZZ)
g = [([1, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], 2)]
assert gf_ddf_zassenhaus(f, 11, ZZ) == g
assert gf_ddf_shoup(f, 11, ZZ) == g
f = gf_from_dict({63: ZZ(1), 0: ZZ(1)}, 2, ZZ)
g = [([1, 1], 1),
([1, 1, 1], 2),
([1, 1, 1, 1, 1, 1, 1], 3),
([1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0,
0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1], 6)]
assert gf_ddf_zassenhaus(f, 2, ZZ) == g
assert gf_ddf_shoup(f, 2, ZZ) == g
f = gf_from_dict({6: ZZ(1), 5: ZZ(-1), 4: ZZ(1), 3: ZZ(1), 1: ZZ(-1)}, 3, ZZ)
g = [([1, 1, 0], 1),
([1, 1, 0, 1, 2], 2)]
assert gf_ddf_zassenhaus(f, 3, ZZ) == g
assert gf_ddf_shoup(f, 3, ZZ) == g
f = ZZ.map([1, 2, 5, 26, 677, 436, 791, 325, 456, 24, 577])
g = [([1, 701], 1),
([1, 110, 559, 532, 694, 151, 110, 70, 735, 122], 9)]
assert gf_ddf_zassenhaus(f, 809, ZZ) == g
assert gf_ddf_shoup(f, 809, ZZ) == g
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
g = [([1, 22730, 68144], 2),
([1, 64876, 83977, 10787, 12561, 68608, 52650, 88001, 84356], 4),
([1, 15347, 95022, 84569, 94508, 92335], 5)]
assert gf_ddf_zassenhaus(f, p, ZZ) == g
assert gf_ddf_shoup(f, p, ZZ) == g
def test_gf_edf():
f = ZZ.map([1, 1, 0, 1, 2])
g = ZZ.map([[1, 0, 1], [1, 1, 2]])
assert gf_edf_zassenhaus(f, 2, 3, ZZ) == g
assert gf_edf_shoup(f, 2, 3, ZZ) == g
def test_issue_23174():
f = ZZ.map([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])
g = ZZ.map([[1, 0, 0, 1, 1, 1, 0, 0, 1], [1, 1, 1, 0, 1, 0, 1, 1, 1]])
assert gf_edf_zassenhaus(f, 8, 2, ZZ) == g
def test_gf_factor():
assert gf_factor([], 11, ZZ) == (0, [])
assert gf_factor([1], 11, ZZ) == (1, [])
assert gf_factor([1, 1], 11, ZZ) == (1, [([1, 1], 1)])
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf([], 11, ZZ) == (0, [])
assert gf_factor_sqf([1], 11, ZZ) == (1, [])
assert gf_factor_sqf([1, 1], 11, ZZ) == (1, [[1, 1]])
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(ZZ.map([]), 11, ZZ) == (0, [])
assert gf_factor_sqf(ZZ.map([1]), 11, ZZ) == (1, [])
assert gf_factor_sqf(ZZ.map([1, 1]), 11, ZZ) == (1, [[1, 1]])
f, p = ZZ.map([1, 0, 0, 1, 0]), 2
g = (1, [([1, 0], 1),
([1, 1], 1),
([1, 1, 1], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 0],
[1, 1],
[1, 1, 1]])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
f, p = gf_from_int_poly([1, -3, 1, -3, -1, -3, 1], 11), 11
g = (1, [([1, 1], 1),
([1, 5, 3], 1),
([1, 2, 3, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 5, 8, 4], 11
g = (1, [([1, 1], 1), ([1, 2], 2)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = [1, 1, 10, 1, 0, 10, 10, 10, 0, 0], 11
g = (1, [([1, 0], 2), ([1, 9, 5], 1), ([1, 3, 0, 8, 5, 2], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: 1, 0: 1}, 11, ZZ), 11
g = (1, [([1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10], 1),
([1, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({32: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11
g = (8, [([1, 3], 1),
([1, 8], 1),
([1, 0, 9], 1),
([1, 2, 2], 1),
([1, 9, 2], 1),
([1, 0, 5, 0, 7], 1),
([1, 0, 6, 0, 7], 1),
([1, 0, 0, 0, 1, 0, 0, 0, 6], 1),
([1, 0, 0, 0, 10, 0, 0, 0, 6], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
f, p = gf_from_dict({63: ZZ(8), 0: ZZ(5)}, 11, ZZ), 11
g = (8, [([1, 7], 1),
([1, 4, 5], 1),
([1, 6, 8, 2], 1),
([1, 9, 9, 2], 1),
([1, 0, 0, 9, 0, 0, 4], 1),
([1, 2, 0, 8, 4, 6, 4], 1),
([1, 2, 3, 8, 0, 6, 4], 1),
([1, 2, 6, 0, 8, 4, 4], 1),
([1, 3, 3, 1, 6, 8, 4], 1),
([1, 5, 6, 0, 8, 6, 4], 1),
([1, 6, 2, 7, 9, 8, 4], 1),
([1, 10, 4, 7, 10, 7, 4], 1),
([1, 10, 10, 1, 4, 9, 4], 1)])
config.setup('GF_FACTOR_METHOD', 'berlekamp')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
# Gathen polynomials: x**n + x + 1 (mod p > 2**n * pi)
p = ZZ(nextprime(int((2**15 * pi).evalf())))
f = gf_from_dict({15: 1, 1: 1, 0: 1}, p, ZZ)
assert gf_sqf_p(f, p, ZZ) is True
g = (1, [([1, 22730, 68144], 1),
([1, 81553, 77449, 86810, 4724], 1),
([1, 86276, 56779, 14859, 31575], 1),
([1, 15347, 95022, 84569, 94508, 92335], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 22730, 68144],
[1, 81553, 77449, 86810, 4724],
[1, 86276, 56779, 14859, 31575],
[1, 15347, 95022, 84569, 94508, 92335]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
# Shoup polynomials: f = a_0 x**n + a_1 x**(n-1) + ... + a_n
# (mod p > 2**(n-2) * pi), where a_n = a_{n-1}**2 + 1, a_0 = 1
p = ZZ(nextprime(int((2**4 * pi).evalf())))
f = ZZ.map([1, 2, 5, 26, 41, 39, 38])
assert gf_sqf_p(f, p, ZZ) is True
g = (1, [([1, 44, 26], 1),
([1, 11, 25, 18, 30], 1)])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor(f, p, ZZ) == g
g = (1, [[1, 44, 26],
[1, 11, 25, 18, 30]])
config.setup('GF_FACTOR_METHOD', 'zassenhaus')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'shoup')
assert gf_factor_sqf(f, p, ZZ) == g
config.setup('GF_FACTOR_METHOD', 'other')
raises(KeyError, lambda: gf_factor([1, 1], 11, ZZ))
config.setup('GF_FACTOR_METHOD')
def test_gf_csolve():
assert gf_value([1, 7, 2, 4], 11) == 2204
assert linear_congruence(4, 3, 5) == [2]
assert linear_congruence(0, 3, 5) == []
assert linear_congruence(6, 1, 4) == []
assert linear_congruence(0, 5, 5) == [0, 1, 2, 3, 4]
assert linear_congruence(3, 12, 15) == [4, 9, 14]
assert linear_congruence(6, 0, 18) == [0, 3, 6, 9, 12, 15]
# with power = 1
assert csolve_prime([1, 3, 2, 17], 7) == [3]
assert csolve_prime([1, 3, 1, 5], 5) == [0, 1]
assert csolve_prime([3, 6, 9, 3], 3) == [0, 1, 2]
# with power > 1
assert csolve_prime(
[1, 1, 223], 3, 4) == [4, 13, 22, 31, 40, 49, 58, 67, 76]
assert csolve_prime([3, 5, 2, 25], 5, 3) == [16, 50, 99]
assert csolve_prime([3, 2, 2, 49], 7, 3) == [147, 190, 234]
assert gf_csolve([1, 1, 7], 189) == [13, 49, 76, 112, 139, 175]
assert gf_csolve([1, 3, 4, 1, 30], 60) == [10, 30]
assert gf_csolve([1, 1, 7], 15) == []
|
c7e9fe4fe0a89fef2985e94d6b4e413ecce40bf5025b50b4c803b0cfae7c23ad | """Tests for real and complex root isolation and refinement algorithms. """
from sympy.polys.rings import ring
from sympy.polys.domains import ZZ, QQ, ZZ_I, EX
from sympy.polys.polyerrors import DomainError, RefinementFailed, PolynomialError
from sympy.polys.rootisolation import (
dup_cauchy_upper_bound, dup_cauchy_lower_bound,
dup_mignotte_sep_bound_squared,
)
from sympy.testing.pytest import raises
def test_dup_sturm():
R, x = ring("x", QQ)
assert R.dup_sturm(5) == [1]
assert R.dup_sturm(x) == [x, 1]
f = x**3 - 2*x**2 + 3*x - 5
assert R.dup_sturm(f) == [f, 3*x**2 - 4*x + 3, -QQ(10,9)*x + QQ(13,3), -QQ(3303,100)]
def test_dup_cauchy_upper_bound():
raises(PolynomialError, lambda: dup_cauchy_upper_bound([], QQ))
raises(PolynomialError, lambda: dup_cauchy_upper_bound([QQ(1)], QQ))
raises(DomainError, lambda: dup_cauchy_upper_bound([ZZ_I(1), ZZ_I(1)], ZZ_I))
assert dup_cauchy_upper_bound([QQ(1), QQ(0), QQ(0)], QQ) == QQ.zero
assert dup_cauchy_upper_bound([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(3)
def test_dup_cauchy_lower_bound():
raises(PolynomialError, lambda: dup_cauchy_lower_bound([], QQ))
raises(PolynomialError, lambda: dup_cauchy_lower_bound([QQ(1)], QQ))
raises(PolynomialError, lambda: dup_cauchy_lower_bound([QQ(1), QQ(0), QQ(0)], QQ))
raises(DomainError, lambda: dup_cauchy_lower_bound([ZZ_I(1), ZZ_I(1)], ZZ_I))
assert dup_cauchy_lower_bound([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(2, 3)
def test_dup_mignotte_sep_bound_squared():
raises(PolynomialError, lambda: dup_mignotte_sep_bound_squared([], QQ))
raises(PolynomialError, lambda: dup_mignotte_sep_bound_squared([QQ(1)], QQ))
assert dup_mignotte_sep_bound_squared([QQ(1), QQ(0), QQ(-2)], QQ) == QQ(3, 5)
def test_dup_refine_real_root():
R, x = ring("x", ZZ)
f = x**2 - 2
assert R.dup_refine_real_root(f, QQ(1), QQ(1), steps=1) == (QQ(1), QQ(1))
assert R.dup_refine_real_root(f, QQ(1), QQ(1), steps=9) == (QQ(1), QQ(1))
raises(ValueError, lambda: R.dup_refine_real_root(f, QQ(-2), QQ(2)))
s, t = QQ(1, 1), QQ(2, 1)
assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(2, 1))
assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(1, 1), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(4, 3), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(10, 7))
s, t = QQ(1, 1), QQ(3, 2)
assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(4, 3), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(7, 5), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(10, 7))
assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(17, 12))
s, t = QQ(1, 1), QQ(5, 3)
assert R.dup_refine_real_root(f, s, t, steps=0) == (QQ(1, 1), QQ(5, 3))
assert R.dup_refine_real_root(f, s, t, steps=1) == (QQ(1, 1), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=2) == (QQ(7, 5), QQ(3, 2))
assert R.dup_refine_real_root(f, s, t, steps=3) == (QQ(7, 5), QQ(13, 9))
assert R.dup_refine_real_root(f, s, t, steps=4) == (QQ(7, 5), QQ(27, 19))
s, t = QQ(-1, 1), QQ(-2, 1)
assert R.dup_refine_real_root(f, s, t, steps=0) == (-QQ(2, 1), -QQ(1, 1))
assert R.dup_refine_real_root(f, s, t, steps=1) == (-QQ(3, 2), -QQ(1, 1))
assert R.dup_refine_real_root(f, s, t, steps=2) == (-QQ(3, 2), -QQ(4, 3))
assert R.dup_refine_real_root(f, s, t, steps=3) == (-QQ(3, 2), -QQ(7, 5))
assert R.dup_refine_real_root(f, s, t, steps=4) == (-QQ(10, 7), -QQ(7, 5))
raises(RefinementFailed, lambda: R.dup_refine_real_root(f, QQ(0), QQ(1)))
s, t, u, v, w = QQ(1), QQ(2), QQ(24, 17), QQ(17, 12), QQ(7, 5)
assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100)) == (u, v)
assert R.dup_refine_real_root(f, s, t, steps=6) == (u, v)
assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=5) == (w, v)
assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=6) == (u, v)
assert R.dup_refine_real_root(f, s, t, eps=QQ(1, 100), steps=7) == (u, v)
s, t, u, v = QQ(-2), QQ(-1), QQ(-3, 2), QQ(-4, 3)
assert R.dup_refine_real_root(f, s, t, disjoint=QQ(-5)) == (s, t)
assert R.dup_refine_real_root(f, s, t, disjoint=-v) == (s, t)
assert R.dup_refine_real_root(f, s, t, disjoint=v) == (u, v)
s, t, u, v = QQ(1), QQ(2), QQ(4, 3), QQ(3, 2)
assert R.dup_refine_real_root(f, s, t, disjoint=QQ(5)) == (s, t)
assert R.dup_refine_real_root(f, s, t, disjoint=-u) == (s, t)
assert R.dup_refine_real_root(f, s, t, disjoint=u) == (u, v)
def test_dup_isolate_real_roots_sqf():
R, x = ring("x", ZZ)
assert R.dup_isolate_real_roots_sqf(0) == []
assert R.dup_isolate_real_roots_sqf(5) == []
assert R.dup_isolate_real_roots_sqf(x**2 + x) == [(-1, -1), (0, 0)]
assert R.dup_isolate_real_roots_sqf(x**2 - x) == [( 0, 0), (1, 1)]
assert R.dup_isolate_real_roots_sqf(x**4 + x + 1) == []
I = [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**2 - 2) == I
assert R.dup_isolate_real_roots_sqf(-x**2 + 2) == I
assert R.dup_isolate_real_roots_sqf(x - 1) == \
[(1, 1)]
assert R.dup_isolate_real_roots_sqf(x**2 - 3*x + 2) == \
[(1, 1), (2, 2)]
assert R.dup_isolate_real_roots_sqf(x**3 - 6*x**2 + 11*x - 6) == \
[(1, 1), (2, 2), (3, 3)]
assert R.dup_isolate_real_roots_sqf(x**4 - 10*x**3 + 35*x**2 - 50*x + 24) == \
[(1, 1), (2, 2), (3, 3), (4, 4)]
assert R.dup_isolate_real_roots_sqf(x**5 - 15*x**4 + 85*x**3 - 225*x**2 + 274*x - 120) == \
[(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)]
assert R.dup_isolate_real_roots_sqf(x - 10) == \
[(10, 10)]
assert R.dup_isolate_real_roots_sqf(x**2 - 30*x + 200) == \
[(10, 10), (20, 20)]
assert R.dup_isolate_real_roots_sqf(x**3 - 60*x**2 + 1100*x - 6000) == \
[(10, 10), (20, 20), (30, 30)]
assert R.dup_isolate_real_roots_sqf(x**4 - 100*x**3 + 3500*x**2 - 50000*x + 240000) == \
[(10, 10), (20, 20), (30, 30), (40, 40)]
assert R.dup_isolate_real_roots_sqf(x**5 - 150*x**4 + 8500*x**3 - 225000*x**2 + 2740000*x - 12000000) == \
[(10, 10), (20, 20), (30, 30), (40, 40), (50, 50)]
assert R.dup_isolate_real_roots_sqf(x + 1) == \
[(-1, -1)]
assert R.dup_isolate_real_roots_sqf(x**2 + 3*x + 2) == \
[(-2, -2), (-1, -1)]
assert R.dup_isolate_real_roots_sqf(x**3 + 6*x**2 + 11*x + 6) == \
[(-3, -3), (-2, -2), (-1, -1)]
assert R.dup_isolate_real_roots_sqf(x**4 + 10*x**3 + 35*x**2 + 50*x + 24) == \
[(-4, -4), (-3, -3), (-2, -2), (-1, -1)]
assert R.dup_isolate_real_roots_sqf(x**5 + 15*x**4 + 85*x**3 + 225*x**2 + 274*x + 120) == \
[(-5, -5), (-4, -4), (-3, -3), (-2, -2), (-1, -1)]
assert R.dup_isolate_real_roots_sqf(x + 10) == \
[(-10, -10)]
assert R.dup_isolate_real_roots_sqf(x**2 + 30*x + 200) == \
[(-20, -20), (-10, -10)]
assert R.dup_isolate_real_roots_sqf(x**3 + 60*x**2 + 1100*x + 6000) == \
[(-30, -30), (-20, -20), (-10, -10)]
assert R.dup_isolate_real_roots_sqf(x**4 + 100*x**3 + 3500*x**2 + 50000*x + 240000) == \
[(-40, -40), (-30, -30), (-20, -20), (-10, -10)]
assert R.dup_isolate_real_roots_sqf(x**5 + 150*x**4 + 8500*x**3 + 225000*x**2 + 2740000*x + 12000000) == \
[(-50, -50), (-40, -40), (-30, -30), (-20, -20), (-10, -10)]
assert R.dup_isolate_real_roots_sqf(x**2 - 5) == [(-3, -2), (2, 3)]
assert R.dup_isolate_real_roots_sqf(x**3 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**4 - 5) == [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**5 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**6 - 5) == [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**7 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**8 - 5) == [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**9 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**2 - 1) == \
[(-1, -1), (1, 1)]
assert R.dup_isolate_real_roots_sqf(x**3 + 2*x**2 - x - 2) == \
[(-2, -2), (-1, -1), (1, 1)]
assert R.dup_isolate_real_roots_sqf(x**4 - 5*x**2 + 4) == \
[(-2, -2), (-1, -1), (1, 1), (2, 2)]
assert R.dup_isolate_real_roots_sqf(x**5 + 3*x**4 - 5*x**3 - 15*x**2 + 4*x + 12) == \
[(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2)]
assert R.dup_isolate_real_roots_sqf(x**6 - 14*x**4 + 49*x**2 - 36) == \
[(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2), (3, 3)]
assert R.dup_isolate_real_roots_sqf(2*x**7 + x**6 - 28*x**5 - 14*x**4 + 98*x**3 + 49*x**2 - 72*x - 36) == \
[(-3, -3), (-2, -2), (-1, -1), (-1, 0), (1, 1), (2, 2), (3, 3)]
assert R.dup_isolate_real_roots_sqf(4*x**8 - 57*x**6 + 210*x**4 - 193*x**2 + 36) == \
[(-3, -3), (-2, -2), (-1, -1), (-1, 0), (0, 1), (1, 1), (2, 2), (3, 3)]
f = 9*x**2 - 2
assert R.dup_isolate_real_roots_sqf(f) == \
[(-1, 0), (0, 1)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10)) == \
[(QQ(-1, 2), QQ(-3, 7)), (QQ(3, 7), QQ(1, 2))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \
[(QQ(-9, 19), QQ(-8, 17)), (QQ(8, 17), QQ(9, 19))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000)) == \
[(QQ(-33, 70), QQ(-8, 17)), (QQ(8, 17), QQ(33, 70))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10000)) == \
[(QQ(-33, 70), QQ(-107, 227)), (QQ(107, 227), QQ(33, 70))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \
[(QQ(-305, 647), QQ(-272, 577)), (QQ(272, 577), QQ(305, 647))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000000)) == \
[(QQ(-1121, 2378), QQ(-272, 577)), (QQ(272, 577), QQ(1121, 2378))]
f = 200100012*x**5 - 700390052*x**4 + 700490079*x**3 - 200240054*x**2 + 40017*x - 2
assert R.dup_isolate_real_roots_sqf(f) == \
[(QQ(0), QQ(1, 10002)), (QQ(1, 10002), QQ(1, 10002)),
(QQ(1, 2), QQ(1, 2)), (QQ(1), QQ(1)), (QQ(2), QQ(2))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \
[(QQ(1, 10003), QQ(1, 10003)), (QQ(1, 10002), QQ(1, 10002)),
(QQ(1, 2), QQ(1, 2)), (QQ(1), QQ(1)), (QQ(2), QQ(2))]
a, b, c, d = 10000090000001, 2000100003, 10000300007, 10000005000008
f = 20001600074001600021*x**4 \
+ 1700135866278935491773999857*x**3 \
- 2000179008931031182161141026995283662899200197*x**2 \
- 800027600594323913802305066986600025*x \
+ 100000950000540000725000008
assert R.dup_isolate_real_roots_sqf(f) == \
[(-a, -a), (-1, 0), (0, 1), (d, d)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000000000)) == \
[(-QQ(a), -QQ(a)), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (QQ(d), QQ(d))]
(u, v), B, C, (s, t) = R.dup_isolate_real_roots_sqf(f, fast=True)
assert u < -a < v and B == (-QQ(1), QQ(0)) and C == (QQ(0), QQ(1)) and s < d < t
assert R.dup_isolate_real_roots_sqf(f, fast=True, eps=QQ(1, 100000000000000000000000000000)) == \
[(-QQ(a), -QQ(a)), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (QQ(d), QQ(d))]
f = -10*x**4 + 8*x**3 + 80*x**2 - 32*x - 160
assert R.dup_isolate_real_roots_sqf(f) == \
[(-2, -2), (-2, -1), (2, 2), (2, 3)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \
[(-QQ(2), -QQ(2)), (-QQ(23, 14), -QQ(18, 11)), (QQ(2), QQ(2)), (QQ(39, 16), QQ(22, 9))]
f = x - 1
assert R.dup_isolate_real_roots_sqf(f, inf=2) == []
assert R.dup_isolate_real_roots_sqf(f, sup=0) == []
assert R.dup_isolate_real_roots_sqf(f) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, inf=1) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, sup=1) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, inf=1, sup=1) == [(1, 1)]
f = x**2 - 2
assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 4)) == []
assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 5)) == [(QQ(7, 5), QQ(3, 2))]
assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 5)) == [(-2, -1)]
assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 4)) == [(-2, -1), (1, QQ(3, 2))]
assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 4)) == []
assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 5)) == [(-QQ(3, 2), -QQ(7, 5))]
assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 5)) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 4)) == [(-QQ(3, 2), -1), (1, 2)]
I = [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(f, inf=-2) == I
assert R.dup_isolate_real_roots_sqf(f, sup=+2) == I
assert R.dup_isolate_real_roots_sqf(f, inf=-2, sup=2) == I
R, x = ring("x", QQ)
f = QQ(8, 5)*x**2 - QQ(87374, 3855)*x - QQ(17, 771)
assert R.dup_isolate_real_roots_sqf(f) == [(-1, 0), (14, 15)]
R, x = ring("x", EX)
raises(DomainError, lambda: R.dup_isolate_real_roots_sqf(x + 3))
def test_dup_isolate_real_roots():
R, x = ring("x", ZZ)
assert R.dup_isolate_real_roots(0) == []
assert R.dup_isolate_real_roots(3) == []
assert R.dup_isolate_real_roots(5*x) == [((0, 0), 1)]
assert R.dup_isolate_real_roots(7*x**4) == [((0, 0), 4)]
assert R.dup_isolate_real_roots(x**2 + x) == [((-1, -1), 1), ((0, 0), 1)]
assert R.dup_isolate_real_roots(x**2 - x) == [((0, 0), 1), ((1, 1), 1)]
assert R.dup_isolate_real_roots(x**4 + x + 1) == []
I = [((-2, -1), 1), ((1, 2), 1)]
assert R.dup_isolate_real_roots(x**2 - 2) == I
assert R.dup_isolate_real_roots(-x**2 + 2) == I
f = 16*x**14 - 96*x**13 + 24*x**12 + 936*x**11 - 1599*x**10 - 2880*x**9 + 9196*x**8 \
+ 552*x**7 - 21831*x**6 + 13968*x**5 + 21690*x**4 - 26784*x**3 - 2916*x**2 + 15552*x - 5832
g = R.dup_sqf_part(f)
assert R.dup_isolate_real_roots(f) == \
[((-QQ(2), -QQ(3, 2)), 2), ((-QQ(3, 2), -QQ(1, 1)), 3), ((QQ(1), QQ(3, 2)), 3),
((QQ(3, 2), QQ(3, 2)), 4), ((QQ(5, 3), QQ(2)), 2)]
assert R.dup_isolate_real_roots_sqf(g) == \
[(-QQ(2), -QQ(3, 2)), (-QQ(3, 2), -QQ(1, 1)), (QQ(1), QQ(3, 2)),
(QQ(3, 2), QQ(3, 2)), (QQ(3, 2), QQ(2))]
assert R.dup_isolate_real_roots(g) == \
[((-QQ(2), -QQ(3, 2)), 1), ((-QQ(3, 2), -QQ(1, 1)), 1), ((QQ(1), QQ(3, 2)), 1),
((QQ(3, 2), QQ(3, 2)), 1), ((QQ(3, 2), QQ(2)), 1)]
f = x - 1
assert R.dup_isolate_real_roots(f, inf=2) == []
assert R.dup_isolate_real_roots(f, sup=0) == []
assert R.dup_isolate_real_roots(f) == [((1, 1), 1)]
assert R.dup_isolate_real_roots(f, inf=1) == [((1, 1), 1)]
assert R.dup_isolate_real_roots(f, sup=1) == [((1, 1), 1)]
assert R.dup_isolate_real_roots(f, inf=1, sup=1) == [((1, 1), 1)]
f = x**4 - 4*x**2 + 4
assert R.dup_isolate_real_roots(f, inf=QQ(7, 4)) == []
assert R.dup_isolate_real_roots(f, inf=QQ(7, 5)) == [((QQ(7, 5), QQ(3, 2)), 2)]
assert R.dup_isolate_real_roots(f, sup=QQ(7, 5)) == [((-2, -1), 2)]
assert R.dup_isolate_real_roots(f, sup=QQ(7, 4)) == [((-2, -1), 2), ((1, QQ(3, 2)), 2)]
assert R.dup_isolate_real_roots(f, sup=-QQ(7, 4)) == []
assert R.dup_isolate_real_roots(f, sup=-QQ(7, 5)) == [((-QQ(3, 2), -QQ(7, 5)), 2)]
assert R.dup_isolate_real_roots(f, inf=-QQ(7, 5)) == [((1, 2), 2)]
assert R.dup_isolate_real_roots(f, inf=-QQ(7, 4)) == [((-QQ(3, 2), -1), 2), ((1, 2), 2)]
I = [((-2, -1), 2), ((1, 2), 2)]
assert R.dup_isolate_real_roots(f, inf=-2) == I
assert R.dup_isolate_real_roots(f, sup=+2) == I
assert R.dup_isolate_real_roots(f, inf=-2, sup=2) == I
f = x**11 - 3*x**10 - x**9 + 11*x**8 - 8*x**7 - 8*x**6 + 12*x**5 - 4*x**4
assert R.dup_isolate_real_roots(f, basis=False) == \
[((-2, -1), 2), ((0, 0), 4), ((1, 1), 3), ((1, 2), 2)]
assert R.dup_isolate_real_roots(f, basis=True) == \
[((-2, -1), 2, [1, 0, -2]), ((0, 0), 4, [1, 0]), ((1, 1), 3, [1, -1]), ((1, 2), 2, [1, 0, -2])]
f = (x**45 - 45*x**44 + 990*x**43 - 1)
g = (x**46 - 15180*x**43 + 9366819*x**40 - 53524680*x**39 + 260932815*x**38 - 1101716330*x**37 + 4076350421*x**36 - 13340783196*x**35 + 38910617655*x**34 - 101766230790*x**33 + 239877544005*x**32 - 511738760544*x**31 + 991493848554*x**30 - 1749695026860*x**29 + 2818953098830*x**28 - 4154246671960*x**27 + 5608233007146*x**26 - 6943526580276*x**25 + 7890371113950*x**24 - 8233430727600*x**23 + 7890371113950*x**22 - 6943526580276*x**21 + 5608233007146*x**20 - 4154246671960*x**19 + 2818953098830*x**18 - 1749695026860*x**17 + 991493848554*x**16 - 511738760544*x**15 + 239877544005*x**14 - 101766230790*x**13 + 38910617655*x**12 - 13340783196*x**11 + 4076350421*x**10 - 1101716330*x**9 + 260932815*x**8 - 53524680*x**7 + 9366819*x**6 - 1370754*x**5 + 163185*x**4 - 15180*x**3 + 1035*x**2 - 47*x + 1)
assert R.dup_isolate_real_roots(f*g) == \
[((0, QQ(1, 2)), 1), ((QQ(2, 3), QQ(3, 4)), 1), ((QQ(3, 4), 1), 1), ((6, 7), 1), ((24, 25), 1)]
R, x = ring("x", EX)
raises(DomainError, lambda: R.dup_isolate_real_roots(x + 3))
def test_dup_isolate_real_roots_list():
R, x = ring("x", ZZ)
assert R.dup_isolate_real_roots_list([x**2 + x, x]) == \
[((-1, -1), {0: 1}), ((0, 0), {0: 1, 1: 1})]
assert R.dup_isolate_real_roots_list([x**2 - x, x]) == \
[((0, 0), {0: 1, 1: 1}), ((1, 1), {0: 1})]
assert R.dup_isolate_real_roots_list([x + 1, x + 2, x - 1, x + 1, x - 1, x - 1]) == \
[((-QQ(2), -QQ(2)), {1: 1}), ((-QQ(1), -QQ(1)), {0: 1, 3: 1}), ((QQ(1), QQ(1)), {2: 1, 4: 1, 5: 1})]
assert R.dup_isolate_real_roots_list([x + 1, x + 2, x - 1, x + 1, x - 1, x + 2]) == \
[((-QQ(2), -QQ(2)), {1: 1, 5: 1}), ((-QQ(1), -QQ(1)), {0: 1, 3: 1}), ((QQ(1), QQ(1)), {2: 1, 4: 1})]
f, g = x**4 - 4*x**2 + 4, x - 1
assert R.dup_isolate_real_roots_list([f, g], inf=QQ(7, 4)) == []
assert R.dup_isolate_real_roots_list([f, g], inf=QQ(7, 5)) == \
[((QQ(7, 5), QQ(3, 2)), {0: 2})]
assert R.dup_isolate_real_roots_list([f, g], sup=QQ(7, 5)) == \
[((-2, -1), {0: 2}), ((1, 1), {1: 1})]
assert R.dup_isolate_real_roots_list([f, g], sup=QQ(7, 4)) == \
[((-2, -1), {0: 2}), ((1, 1), {1: 1}), ((1, QQ(3, 2)), {0: 2})]
assert R.dup_isolate_real_roots_list([f, g], sup=-QQ(7, 4)) == []
assert R.dup_isolate_real_roots_list([f, g], sup=-QQ(7, 5)) == \
[((-QQ(3, 2), -QQ(7, 5)), {0: 2})]
assert R.dup_isolate_real_roots_list([f, g], inf=-QQ(7, 5)) == \
[((1, 1), {1: 1}), ((1, 2), {0: 2})]
assert R.dup_isolate_real_roots_list([f, g], inf=-QQ(7, 4)) == \
[((-QQ(3, 2), -1), {0: 2}), ((1, 1), {1: 1}), ((1, 2), {0: 2})]
f, g = 2*x**2 - 1, x**2 - 2
assert R.dup_isolate_real_roots_list([f, g]) == \
[((-QQ(2), -QQ(1)), {1: 1}), ((-QQ(1), QQ(0)), {0: 1}),
((QQ(0), QQ(1)), {0: 1}), ((QQ(1), QQ(2)), {1: 1})]
assert R.dup_isolate_real_roots_list([f, g], strict=True) == \
[((-QQ(3, 2), -QQ(4, 3)), {1: 1}), ((-QQ(1), -QQ(2, 3)), {0: 1}),
((QQ(2, 3), QQ(1)), {0: 1}), ((QQ(4, 3), QQ(3, 2)), {1: 1})]
f, g = x**2 - 2, x**3 - x**2 - 2*x + 2
assert R.dup_isolate_real_roots_list([f, g]) == \
[((-QQ(2), -QQ(1)), {1: 1, 0: 1}), ((QQ(1), QQ(1)), {1: 1}), ((QQ(1), QQ(2)), {1: 1, 0: 1})]
f, g = x**3 - 2*x, x**5 - x**4 - 2*x**3 + 2*x**2
assert R.dup_isolate_real_roots_list([f, g]) == \
[((-QQ(2), -QQ(1)), {1: 1, 0: 1}), ((QQ(0), QQ(0)), {0: 1, 1: 2}),
((QQ(1), QQ(1)), {1: 1}), ((QQ(1), QQ(2)), {1: 1, 0: 1})]
f, g = x**9 - 3*x**8 - x**7 + 11*x**6 - 8*x**5 - 8*x**4 + 12*x**3 - 4*x**2, x**5 - 2*x**4 + 3*x**3 - 4*x**2 + 2*x
assert R.dup_isolate_real_roots_list([f, g], basis=False) == \
[((-2, -1), {0: 2}), ((0, 0), {0: 2, 1: 1}), ((1, 1), {0: 3, 1: 2}), ((1, 2), {0: 2})]
assert R.dup_isolate_real_roots_list([f, g], basis=True) == \
[((-2, -1), {0: 2}, [1, 0, -2]), ((0, 0), {0: 2, 1: 1}, [1, 0]),
((1, 1), {0: 3, 1: 2}, [1, -1]), ((1, 2), {0: 2}, [1, 0, -2])]
R, x = ring("x", EX)
raises(DomainError, lambda: R.dup_isolate_real_roots_list([x + 3]))
def test_dup_isolate_real_roots_list_QQ():
R, x = ring("x", ZZ)
f = x**5 - 200
g = x**5 - 201
assert R.dup_isolate_real_roots_list([f, g]) == \
[((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})]
R, x = ring("x", QQ)
f = -QQ(1, 200)*x**5 + 1
g = -QQ(1, 201)*x**5 + 1
assert R.dup_isolate_real_roots_list([f, g]) == \
[((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})]
def test_dup_count_real_roots():
R, x = ring("x", ZZ)
assert R.dup_count_real_roots(0) == 0
assert R.dup_count_real_roots(7) == 0
f = x - 1
assert R.dup_count_real_roots(f) == 1
assert R.dup_count_real_roots(f, inf=1) == 1
assert R.dup_count_real_roots(f, sup=0) == 0
assert R.dup_count_real_roots(f, sup=1) == 1
assert R.dup_count_real_roots(f, inf=0, sup=1) == 1
assert R.dup_count_real_roots(f, inf=0, sup=2) == 1
assert R.dup_count_real_roots(f, inf=1, sup=2) == 1
f = x**2 - 2
assert R.dup_count_real_roots(f) == 2
assert R.dup_count_real_roots(f, sup=0) == 1
assert R.dup_count_real_roots(f, inf=-1, sup=1) == 0
# parameters for test_dup_count_complex_roots_n(): n = 1..8
a, b = (-QQ(1), -QQ(1)), (QQ(1), QQ(1))
c, d = ( QQ(0), QQ(0)), (QQ(1), QQ(1))
def test_dup_count_complex_roots_1():
R, x = ring("x", ZZ)
# z-1
f = x - 1
assert R.dup_count_complex_roots(f, a, b) == 1
assert R.dup_count_complex_roots(f, c, d) == 1
# z+1
f = x + 1
assert R.dup_count_complex_roots(f, a, b) == 1
assert R.dup_count_complex_roots(f, c, d) == 0
def test_dup_count_complex_roots_2():
R, x = ring("x", ZZ)
# (z-1)*(z)
f = x**2 - x
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-1)*(-z)
f = -x**2 + x
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 2
# (z+1)*(z)
f = x**2 + x
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
# (z+1)*(-z)
f = -x**2 - x
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
def test_dup_count_complex_roots_3():
R, x = ring("x", ZZ)
# (z-1)*(z+1)
f = x**2 - 1
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-1)*(z+1)*(z)
f = x**3 - x
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-1)*(z+1)*(-z)
f = -x**3 + x
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 2
def test_dup_count_complex_roots_4():
R, x = ring("x", ZZ)
# (z-I)*(z+I)
f = x**2 + 1
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I)*(z+I)*(z)
f = x**3 + x
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I)*(z+I)*(-z)
f = -x**3 - x
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I)*(z+I)*(z-1)
f = x**3 - x**2 + x - 1
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I)*(z+I)*(z-1)*(z)
f = x**4 - x**3 + x**2 - x
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 3
# (z-I)*(z+I)*(z-1)*(-z)
f = -x**4 + x**3 - x**2 + x
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 3
# (z-I)*(z+I)*(z-1)*(z+1)
f = x**4 - 1
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I)*(z+I)*(z-1)*(z+1)*(z)
f = x**5 - x
assert R.dup_count_complex_roots(f, a, b) == 5
assert R.dup_count_complex_roots(f, c, d) == 3
# (z-I)*(z+I)*(z-1)*(z+1)*(-z)
f = -x**5 + x
assert R.dup_count_complex_roots(f, a, b) == 5
assert R.dup_count_complex_roots(f, c, d) == 3
def test_dup_count_complex_roots_5():
R, x = ring("x", ZZ)
# (z-I+1)*(z+I+1)
f = x**2 + 2*x + 2
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 0
# (z-I+1)*(z+I+1)*(z-1)
f = x**3 + x**2 - 2
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I+1)*(z+I+1)*(z-1)*z
f = x**4 + x**3 - 2*x
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I+1)*(z+I+1)*(z+1)
f = x**3 + 3*x**2 + 4*x + 2
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 0
# (z-I+1)*(z+I+1)*(z+1)*z
f = x**4 + 3*x**3 + 4*x**2 + 2*x
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I+1)*(z+I+1)*(z-1)*(z+1)
f = x**4 + 2*x**3 + x**2 - 2*x - 2
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I+1)*(z+I+1)*(z-1)*(z+1)*z
f = x**5 + 2*x**4 + x**3 - 2*x**2 - 2*x
assert R.dup_count_complex_roots(f, a, b) == 5
assert R.dup_count_complex_roots(f, c, d) == 2
def test_dup_count_complex_roots_6():
R, x = ring("x", ZZ)
# (z-I-1)*(z+I-1)
f = x**2 - 2*x + 2
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I-1)*(z+I-1)*(z-1)
f = x**3 - 3*x**2 + 4*x - 2
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I-1)*(z+I-1)*(z-1)*z
f = x**4 - 3*x**3 + 4*x**2 - 2*x
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 3
# (z-I-1)*(z+I-1)*(z+1)
f = x**3 - x**2 + 2
assert R.dup_count_complex_roots(f, a, b) == 3
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I-1)*(z+I-1)*(z+1)*z
f = x**4 - x**3 + 2*x
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I-1)*(z+I-1)*(z-1)*(z+1)
f = x**4 - 2*x**3 + x**2 + 2*x - 2
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I-1)*(z+I-1)*(z-1)*(z+1)*z
f = x**5 - 2*x**4 + x**3 + 2*x**2 - 2*x
assert R.dup_count_complex_roots(f, a, b) == 5
assert R.dup_count_complex_roots(f, c, d) == 3
def test_dup_count_complex_roots_7():
R, x = ring("x", ZZ)
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)
f = x**4 + 4
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-2)
f = x**5 - 2*x**4 + 4*x - 8
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z**2-2)
f = x**6 - 2*x**4 + 4*x**2 - 8
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)
f = x**5 - x**4 + 4*x - 4
assert R.dup_count_complex_roots(f, a, b) == 5
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*z
f = x**6 - x**5 + 4*x**2 - 4*x
assert R.dup_count_complex_roots(f, a, b) == 6
assert R.dup_count_complex_roots(f, c, d) == 3
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z+1)
f = x**5 + x**4 + 4*x + 4
assert R.dup_count_complex_roots(f, a, b) == 5
assert R.dup_count_complex_roots(f, c, d) == 1
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z+1)*z
f = x**6 + x**5 + 4*x**2 + 4*x
assert R.dup_count_complex_roots(f, a, b) == 6
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)
f = x**6 - x**4 + 4*x**2 - 4
assert R.dup_count_complex_roots(f, a, b) == 6
assert R.dup_count_complex_roots(f, c, d) == 2
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*z
f = x**7 - x**5 + 4*x**3 - 4*x
assert R.dup_count_complex_roots(f, a, b) == 7
assert R.dup_count_complex_roots(f, c, d) == 3
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)
f = x**8 + 3*x**4 - 4
assert R.dup_count_complex_roots(f, a, b) == 8
assert R.dup_count_complex_roots(f, c, d) == 3
def test_dup_count_complex_roots_8():
R, x = ring("x", ZZ)
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)*z
f = x**9 + 3*x**5 - 4*x
assert R.dup_count_complex_roots(f, a, b) == 9
assert R.dup_count_complex_roots(f, c, d) == 4
# (z-I-1)*(z+I-1)*(z-I+1)*(z+I+1)*(z-1)*(z+1)*(z-I)*(z+I)*(z**2-2)*z
f = x**11 - 2*x**9 + 3*x**7 - 6*x**5 - 4*x**3 + 8*x
assert R.dup_count_complex_roots(f, a, b) == 9
assert R.dup_count_complex_roots(f, c, d) == 4
def test_dup_count_complex_roots_implicit():
R, x = ring("x", ZZ)
# z*(z-1)*(z+1)*(z-I)*(z+I)
f = x**5 - x
assert R.dup_count_complex_roots(f) == 5
assert R.dup_count_complex_roots(f, sup=(0, 0)) == 3
assert R.dup_count_complex_roots(f, inf=(0, 0)) == 3
def test_dup_count_complex_roots_exclude():
R, x = ring("x", ZZ)
# z*(z-1)*(z+1)*(z-I)*(z+I)
f = x**5 - x
a, b = (-QQ(1), QQ(0)), (QQ(1), QQ(1))
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['S']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['N']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['S', 'N']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['E']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['E', 'W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['N', 'S', 'E', 'W']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SE']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S']) == 1
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S', 'N']) == 0
a, b = (QQ(0), QQ(0)), (QQ(1), QQ(1))
assert R.dup_count_complex_roots(f, a, b, exclude=True) == 1
def test_dup_isolate_complex_roots_sqf():
R, x = ring("x", ZZ)
f = x**2 - 2*x + 3
assert R.dup_isolate_complex_roots_sqf(f) == \
[((0, -6), (6, 0)), ((0, 0), (6, 6))]
assert [ r.as_tuple() for r in R.dup_isolate_complex_roots_sqf(f, blackbox=True) ] == \
[((0, -6), (6, 0)), ((0, 0), (6, 6))]
assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 10)) == \
[((QQ(15, 16), -QQ(3, 2)), (QQ(33, 32), -QQ(45, 32))),
((QQ(15, 16), QQ(45, 32)), (QQ(33, 32), QQ(3, 2)))]
assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 100)) == \
[((QQ(255, 256), -QQ(363, 256)), (QQ(513, 512), -QQ(723, 512))),
((QQ(255, 256), QQ(723, 512)), (QQ(513, 512), QQ(363, 256)))]
f = 7*x**4 - 19*x**3 + 20*x**2 + 17*x + 20
assert R.dup_isolate_complex_roots_sqf(f) == \
[((-QQ(40, 7), -QQ(40, 7)), (0, 0)), ((-QQ(40, 7), 0), (0, QQ(40, 7))),
((0, -QQ(40, 7)), (QQ(40, 7), 0)), ((0, 0), (QQ(40, 7), QQ(40, 7)))]
def test_dup_isolate_all_roots_sqf():
R, x = ring("x", ZZ)
f = 4*x**4 - x**3 + 2*x**2 + 5*x
assert R.dup_isolate_all_roots_sqf(f) == \
([(-1, 0), (0, 0)],
[((0, -QQ(5, 2)), (QQ(5, 2), 0)), ((0, 0), (QQ(5, 2), QQ(5, 2)))])
assert R.dup_isolate_all_roots_sqf(f, eps=QQ(1, 10)) == \
([(QQ(-7, 8), QQ(-6, 7)), (0, 0)],
[((QQ(35, 64), -QQ(35, 32)), (QQ(5, 8), -QQ(65, 64))), ((QQ(35, 64), QQ(65, 64)), (QQ(5, 8), QQ(35, 32)))])
def test_dup_isolate_all_roots():
R, x = ring("x", ZZ)
f = 4*x**4 - x**3 + 2*x**2 + 5*x
assert R.dup_isolate_all_roots(f) == \
([((-1, 0), 1), ((0, 0), 1)],
[(((0, -QQ(5, 2)), (QQ(5, 2), 0)), 1),
(((0, 0), (QQ(5, 2), QQ(5, 2))), 1)])
assert R.dup_isolate_all_roots(f, eps=QQ(1, 10)) == \
([((QQ(-7, 8), QQ(-6, 7)), 1), ((0, 0), 1)],
[(((QQ(35, 64), -QQ(35, 32)), (QQ(5, 8), -QQ(65, 64))), 1),
(((QQ(35, 64), QQ(65, 64)), (QQ(5, 8), QQ(35, 32))), 1)])
f = x**5 + x**4 - 2*x**3 - 2*x**2 + x + 1
raises(NotImplementedError, lambda: R.dup_isolate_all_roots(f))
|
5ae6ab0397d2f3ec7680f0a9642583731e1cc938402aaea82cc6a3440ae30afc | """Prime ideals in number fields. """
from sympy.core.expr import Expr
from sympy.polys.polytools import Poly
from sympy.polys.domains.finitefield import FF
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.domains.integerring import ZZ
from sympy.polys.matrices.domainmatrix import DomainMatrix
from sympy.polys.polyerrors import CoercionFailed, GeneratorsNeeded
from sympy.polys.polyutils import IntegerPowerable
from sympy.utilities.decorator import public
from .basis import round_two, nilradical_mod_p
from .exceptions import StructureError
from .modules import ModuleEndomorphism, find_min_poly
from .utilities import coeff_search, supplement_a_subspace
def _check_formal_conditions_for_maximal_order(submodule):
r"""
Several functions in this module accept an argument which is to be a
:py:class:`~.Submodule` representing the maximal order in a number field,
such as returned by the :py:func:`~sympy.polys.numberfields.basis.round_two`
algorithm.
We do not attempt to check that the given ``Submodule`` actually represents
a maximal order, but we do check a basic set of formal conditions that the
``Submodule`` must satisfy, at a minimum. The purpose is to catch an
obviously ill-formed argument.
"""
prefix = 'The submodule representing the maximal order should '
cond = None
if not submodule.is_power_basis_submodule():
cond = 'be a direct submodule of a power basis.'
elif not submodule.starts_with_unity():
cond = 'have 1 as its first generator.'
elif not submodule.is_sq_maxrank_HNF():
cond = 'have square matrix, of maximal rank, in Hermite Normal Form.'
if cond is not None:
raise StructureError(prefix + cond)
class PrimeIdeal(IntegerPowerable):
r"""
A prime ideal in a ring of algebraic integers.
"""
def __init__(self, ZK, p, alpha, f, e=None):
"""
Parameters
==========
ZK : :py:class:`~.Submodule`
The maximal order where this ideal lives.
p : int
The rational prime this ideal divides.
alpha : :py:class:`~.PowerBasisElement`
Such that the ideal is equal to ``p*ZK + alpha*ZK``.
f : int
The inertia degree.
e : int, ``None``, optional
The ramification index, if already known. If ``None``, we will
compute it here.
"""
_check_formal_conditions_for_maximal_order(ZK)
self.ZK = ZK
self.p = p
self.alpha = alpha
self.f = f
self._test_factor = None
self.e = e if e is not None else self.valuation(p * ZK)
def pretty(self, field_gen=None, just_gens=False):
"""
Print a representation of this prime ideal.
Examples
========
>>> from sympy import cyclotomic_poly, QQ
>>> from sympy.abc import x, zeta
>>> T = cyclotomic_poly(7, x)
>>> K = QQ.algebraic_field((T, zeta))
>>> P = K.primes_above(11)
>>> print(P[0].pretty())
[ (11, x**3 + 5*x**2 + 4*x - 1) e=1, f=3 ]
>>> print(P[0].pretty(field_gen=zeta))
[ (11, zeta**3 + 5*zeta**2 + 4*zeta - 1) e=1, f=3 ]
>>> print(P[0].pretty(field_gen=zeta, just_gens=True))
(11, zeta**3 + 5*zeta**2 + 4*zeta - 1)
Parameters
==========
field_gen : :py:class:`~.Symbol`, ``None``, optional (default=None)
The symbol to use for the generator of the field. This will appear
in our representation of ``self.alpha``. If ``None``, we use the
variable of the defining polynomial of ``self.ZK``.
just_gens : bool, optional (default=False)
If ``True``, just print the "(p, alpha)" part, showing "just the
generators" of the prime ideal. Otherwise, print a string of the
form "[ (p, alpha) e=..., f=... ]", giving the ramification index
and inertia degree, along with the generators.
"""
field_gen = field_gen or self.ZK.parent.T.gen
p, alpha, e, f = self.p, self.alpha, self.e, self.f
alpha_rep = str(alpha.numerator(x=field_gen).as_expr())
if alpha.denom > 1:
alpha_rep = f'({alpha_rep})/{alpha.denom}'
gens = f'({p}, {alpha_rep})'
if just_gens:
return gens
return f'[ {gens} e={e}, f={f} ]'
def __repr__(self):
return self.pretty()
def as_submodule(self):
r"""
Represent this prime ideal as a :py:class:`~.Submodule`.
Explanation
===========
The :py:class:`~.PrimeIdeal` class serves to bundle information about
a prime ideal, such as its inertia degree, ramification index, and
two-generator representation, as well as to offer helpful methods like
:py:meth:`~.PrimeIdeal.valuation` and
:py:meth:`~.PrimeIdeal.test_factor`.
However, in order to be added and multiplied by other ideals or
rational numbers, it must first be converted into a
:py:class:`~.Submodule`, which is a class that supports these
operations.
In many cases, the user need not perform this conversion deliberately,
since it is automatically performed by the arithmetic operator methods
:py:meth:`~.PrimeIdeal.__add__` and :py:meth:`~.PrimeIdeal.__mul__`.
Raising a :py:class:`~.PrimeIdeal` to a non-negative integer power is
also supported.
Examples
========
>>> from sympy import Poly, cyclotomic_poly, prime_decomp
>>> T = Poly(cyclotomic_poly(7))
>>> P0 = prime_decomp(7, T)[0]
>>> print(P0**6 == 7*P0.ZK)
True
Note that, on both sides of the equation above, we had a
:py:class:`~.Submodule`. In the next equation we recall that adding
ideals yields their GCD. This time, we need a deliberate conversion
to :py:class:`~.Submodule` on the right:
>>> print(P0 + 7*P0.ZK == P0.as_submodule())
True
Returns
=======
:py:class:`~.Submodule`
Will be equal to ``self.p * self.ZK + self.alpha * self.ZK``.
See Also
========
__add__
__mul__
"""
M = self.p * self.ZK + self.alpha * self.ZK
# Pre-set expensive boolean properties whose value we already know:
M._starts_with_unity = False
M._is_sq_maxrank_HNF = True
return M
def __eq__(self, other):
if isinstance(other, PrimeIdeal):
return self.as_submodule() == other.as_submodule()
return NotImplemented
def __add__(self, other):
"""
Convert to a :py:class:`~.Submodule` and add to another
:py:class:`~.Submodule`.
See Also
========
as_submodule
"""
return self.as_submodule() + other
__radd__ = __add__
def __mul__(self, other):
"""
Convert to a :py:class:`~.Submodule` and multiply by another
:py:class:`~.Submodule` or a rational number.
See Also
========
as_submodule
"""
return self.as_submodule() * other
__rmul__ = __mul__
def _zeroth_power(self):
return self.ZK
def _first_power(self):
return self
def test_factor(self):
r"""
Compute a test factor for this prime ideal.
Explanation
===========
Write $\mathfrak{p}$ for this prime ideal, $p$ for the rational prime
it divides. Then, for computing $\mathfrak{p}$-adic valuations it is
useful to have a number $\beta \in \mathbb{Z}_K$ such that
$p/\mathfrak{p} = p \mathbb{Z}_K + \beta \mathbb{Z}_K$.
Essentially, this is the same as the number $\Psi$ (or the "reagent")
from Kummer's 1847 paper (*Ueber die Zerlegung...*, Crelle vol. 35) in
which ideal divisors were invented.
"""
if self._test_factor is None:
self._test_factor = _compute_test_factor(self.p, [self.alpha], self.ZK)
return self._test_factor
def valuation(self, I):
r"""
Compute the $\mathfrak{p}$-adic valuation of integral ideal I at this
prime ideal.
Parameters
==========
I : :py:class:`~.Submodule`
See Also
========
prime_valuation
"""
return prime_valuation(I, self)
def reduce_poly(self, f, gen=None):
r"""
Reduce a univariate :py:class:`~.Poly` *f*, or an :py:class:`~.Expr`
expressing the same, modulo this :py:class:`~.PrimeIdeal`.
Explanation
===========
If our second generator $\alpha$ is zero, then we simply reduce the
coefficients of *f* mod the rational prime $p$ lying under this ideal.
Otherwise we first reduce *f* mod $\alpha$ (as a polynomial in the same
variable as *f*), and then mod $p$.
Examples
========
>>> from sympy import QQ, cyclotomic_poly, symbols
>>> zeta = symbols('zeta')
>>> Phi = cyclotomic_poly(7, zeta)
>>> k = QQ.algebraic_field((Phi, zeta))
>>> P = k.primes_above(11)
>>> frp = P[0]
>>> B = k.integral_basis(fmt='sympy')
>>> print([frp.reduce_poly(b, zeta) for b in B])
[1, zeta, zeta**2, -5*zeta**2 - 4*zeta + 1, -zeta**2 - zeta - 5,
4*zeta**2 - zeta - 1]
Parameters
==========
f : :py:class:`~.Poly`, :py:class:`~.Expr`
The univariate polynomial to be reduced.
gen : :py:class:`~.Symbol`, None, optional (default=None)
Symbol to use as the variable in the polynomials. If *f* is a
:py:class:`~.Poly` or a non-constant :py:class:`~.Expr`, this
replaces its variable. If *f* is a constant :py:class:`~.Expr`,
then *gen* must be supplied.
Returns
=======
:py:class:`~.Poly`, :py:class:`~.Expr`
Type is same as that of given *f*. If returning a
:py:class:`~.Poly`, its domain will be the finite field
$\mathbb{F}_p$.
Raises
======
GeneratorsNeeded
If *f* is a constant :py:class:`~.Expr` and *gen* is ``None``.
NotImplementedError
If *f* is other than :py:class:`~.Poly` or :py:class:`~.Expr`,
or is not univariate.
"""
if isinstance(f, Expr):
try:
g = Poly(f)
except GeneratorsNeeded as e:
if gen is None:
raise e from None
g = Poly(f, gen)
return self.reduce_poly(g).as_expr()
if isinstance(f, Poly) and f.is_univariate:
a = self.alpha.poly(f.gen)
if a != 0:
f = f.rem(a)
return f.set_modulus(self.p)
raise NotImplementedError
def _compute_test_factor(p, gens, ZK):
r"""
Compute the test factor for a :py:class:`~.PrimeIdeal` $\mathfrak{p}$.
Parameters
==========
p : int
The rational prime $\mathfrak{p}$ divides
gens : list of :py:class:`PowerBasisElement`
A complete set of generators for $\mathfrak{p}$ over *ZK*, EXCEPT that
an element equivalent to rational *p* can and should be omitted (since
it has no effect except to waste time).
ZK : :py:class:`~.Submodule`
The maximal order where the prime ideal $\mathfrak{p}$ lives.
Returns
=======
:py:class:`~.PowerBasisElement`
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
(See Proposition 4.8.15.)
"""
_check_formal_conditions_for_maximal_order(ZK)
E = ZK.endomorphism_ring()
matrices = [E.inner_endomorphism(g).matrix(modulus=p) for g in gens]
B = DomainMatrix.zeros((0, ZK.n), FF(p)).vstack(*matrices)
# A nonzero element of the nullspace of B will represent a
# lin comb over the omegas which (i) is not a multiple of p
# (since it is nonzero over FF(p)), while (ii) is such that
# its product with each g in gens _is_ a multiple of p (since
# B represents multiplication by these generators). Theory
# predicts that such an element must exist, so nullspace should
# be non-trivial.
x = B.nullspace()[0, :].transpose()
beta = ZK.parent(ZK.matrix * x, denom=ZK.denom)
return beta
@public
def prime_valuation(I, P):
r"""
Compute the *P*-adic valuation for an integral ideal *I*.
Examples
========
>>> from sympy import QQ
>>> from sympy.polys.numberfields import prime_valuation
>>> K = QQ.cyclotomic_field(5)
>>> P = K.primes_above(5)
>>> ZK = K.maximal_order()
>>> print(prime_valuation(25*ZK, P[0]))
8
Parameters
==========
I : :py:class:`~.Submodule`
An integral ideal whose valuation is desired.
P : :py:class:`~.PrimeIdeal`
The prime at which to compute the valuation.
Returns
=======
int
See Also
========
.PrimeIdeal.valuation
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
(See Algorithm 4.8.17.)
"""
p, ZK = P.p, P.ZK
n, W, d = ZK.n, ZK.matrix, ZK.denom
A = W.convert_to(QQ).inv() * I.matrix * d / I.denom
# Although A must have integer entries, given that I is an integral ideal,
# as a DomainMatrix it will still be over QQ, so we convert back:
A = A.convert_to(ZZ)
D = A.det()
if D % p != 0:
return 0
beta = P.test_factor()
f = d ** n // W.det()
need_complete_test = (f % p == 0)
v = 0
while True:
# Entering the loop, the cols of A represent lin combs of omegas.
# Turn them into lin combs of thetas:
A = W * A
# And then one column at a time...
for j in range(n):
c = ZK.parent(A[:, j], denom=d)
c *= beta
# ...turn back into lin combs of omegas, after multiplying by beta:
c = ZK.represent(c).flat()
for i in range(n):
A[i, j] = c[i]
if A[n - 1, n - 1].element % p != 0:
break
A = A / p
# As noted above, domain converts to QQ even when division goes evenly.
# So must convert back, even when we don't "need_complete_test".
if need_complete_test:
# In this case, having a non-integer entry is actually just our
# halting condition.
try:
A = A.convert_to(ZZ)
except CoercionFailed:
break
else:
# In this case theory says we should not have any non-integer entries.
A = A.convert_to(ZZ)
v += 1
return v
def _two_elt_rep(gens, ZK, p, f=None, Np=None):
r"""
Given a set of *ZK*-generators of a prime ideal, compute a set of just two
*ZK*-generators for the same ideal, one of which is *p* itself.
Parameters
==========
gens : list of :py:class:`PowerBasisElement`
Generators for the prime ideal over *ZK*, the ring of integers of the
field $K$.
ZK : :py:class:`~.Submodule`
The maximal order in $K$.
p : int
The rational prime divided by the prime ideal.
f : int, optional
The inertia degree of the prime ideal, if known.
Np : int, optional
The norm $p^f$ of the prime ideal, if known.
NOTE: There is no reason to supply both *f* and *Np*. Either one will
save us from having to compute the norm *Np* ourselves. If both are known,
*Np* is preferred since it saves one exponentiation.
Returns
=======
:py:class:`~.PowerBasisElement` representing a single algebraic integer
alpha such that the prime ideal is equal to ``p*ZK + alpha*ZK``.
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
(See Algorithm 4.7.10.)
"""
_check_formal_conditions_for_maximal_order(ZK)
pb = ZK.parent
T = pb.T
# Detect the special cases in which either (a) all generators are multiples
# of p, or (b) there are no generators (so `all` is vacuously true):
if all((g % p).equiv(0) for g in gens):
return pb.zero()
if Np is None:
if f is not None:
Np = p**f
else:
Np = abs(pb.submodule_from_gens(gens).matrix.det())
omega = ZK.basis_element_pullbacks()
beta = [p*om for om in omega[1:]] # note: we omit omega[0] == 1
beta += gens
search = coeff_search(len(beta), 1)
for c in search:
alpha = sum(ci*betai for ci, betai in zip(c, beta))
# Note: It may be tempting to reduce alpha mod p here, to try to work
# with smaller numbers, but must not do that, as it can result in an
# infinite loop! E.g. try factoring 2 in Q(sqrt(-7)).
n = alpha.norm(T) // Np
if n % p != 0:
# Now can reduce alpha mod p.
return alpha % p
def _prime_decomp_easy_case(p, ZK):
r"""
Compute the decomposition of rational prime *p* in the ring of integers
*ZK* (given as a :py:class:`~.Submodule`), in the "easy case", i.e. the
case where *p* does not divide the index of $\theta$ in *ZK*, where
$\theta$ is the generator of the ``PowerBasis`` of which *ZK* is a
``Submodule``.
"""
T = ZK.parent.T
T_bar = Poly(T, modulus=p)
lc, fl = T_bar.factor_list()
return [PrimeIdeal(ZK, p,
ZK.parent.element_from_poly(Poly(t, domain=ZZ)),
t.degree(), e)
for t, e in fl]
def _prime_decomp_compute_kernel(I, p, ZK):
r"""
Parameters
==========
I : :py:class:`~.Module`
An ideal of ``ZK/pZK``.
p : int
The rational prime being factored.
ZK : :py:class:`~.Submodule`
The maximal order.
Returns
=======
Pair ``(N, G)``, where:
``N`` is a :py:class:`~.Module` representing the kernel of the map
``a |--> a**p - a`` on ``(O/pO)/I``, guaranteed to be a module with
unity.
``G`` is a :py:class:`~.Module` representing a basis for the separable
algebra ``A = O/I`` (see Cohen).
"""
W = I.matrix
n, r = W.shape
# Want to take the Fp-basis given by the columns of I, adjoin (1, 0, ..., 0)
# (which we know is not already in there since I is a basis for a prime ideal)
# and then supplement this with additional columns to make an invertible n x n
# matrix. This will then represent a full basis for ZK, whose first r columns
# are pullbacks of the basis for I.
if r == 0:
B = W.eye(n, ZZ)
else:
B = W.hstack(W.eye(n, ZZ)[:, 0])
if B.shape[1] < n:
B = supplement_a_subspace(B.convert_to(FF(p))).convert_to(ZZ)
G = ZK.submodule_from_matrix(B)
# Must compute G's multiplication table _before_ discarding the first r
# columns. (See Step 9 in Alg 6.2.9 in Cohen, where the betas are actually
# needed in order to represent each product of gammas. However, once we've
# found the representations, then we can ignore the betas.)
G.compute_mult_tab()
G = G.discard_before(r)
phi = ModuleEndomorphism(G, lambda x: x**p - x)
N = phi.kernel(modulus=p)
assert N.starts_with_unity()
return N, G
def _prime_decomp_maximal_ideal(I, p, ZK):
r"""
We have reached the case where we have a maximal (hence prime) ideal *I*,
which we know because the quotient ``O/I`` is a field.
Parameters
==========
I : :py:class:`~.Module`
An ideal of ``O/pO``.
p : int
The rational prime being factored.
ZK : :py:class:`~.Submodule`
The maximal order.
Returns
=======
:py:class:`~.PrimeIdeal` instance representing this prime
"""
m, n = I.matrix.shape
f = m - n
G = ZK.matrix * I.matrix
gens = [ZK.parent(G[:, j], denom=ZK.denom) for j in range(G.shape[1])]
alpha = _two_elt_rep(gens, ZK, p, f=f)
return PrimeIdeal(ZK, p, alpha, f)
def _prime_decomp_split_ideal(I, p, N, G, ZK):
r"""
Perform the step in the prime decomposition algorithm where we have determined
the the quotient ``ZK/I`` is _not_ a field, and we want to perform a non-trivial
factorization of *I* by locating an idempotent element of ``ZK/I``.
"""
assert I.parent == ZK and G.parent is ZK and N.parent is G
# Since ZK/I is not a field, the kernel computed in the previous step contains
# more than just the prime field Fp, and our basis N for the nullspace therefore
# contains at least a second column (which represents an element outside Fp).
# Let alpha be such an element:
alpha = N(1).to_parent()
assert alpha.module is G
alpha_powers = []
m = find_min_poly(alpha, FF(p), powers=alpha_powers)
# TODO (future work):
# We don't actually need full factorization, so might use a faster method
# to just break off a single non-constant factor m1?
lc, fl = m.factor_list()
m1 = fl[0][0]
m2 = m.quo(m1)
U, V, g = m1.gcdex(m2)
# Sanity check: theory says m is squarefree, so m1, m2 should be coprime:
assert g == 1
E = list(reversed(Poly(U * m1, domain=ZZ).rep.rep))
eps1 = sum(E[i]*alpha_powers[i] for i in range(len(E)))
eps2 = 1 - eps1
idemps = [eps1, eps2]
factors = []
for eps in idemps:
e = eps.to_parent()
assert e.module is ZK
D = I.matrix.convert_to(FF(p)).hstack(*[
(e * om).column(domain=FF(p)) for om in ZK.basis_elements()
])
W = D.columnspace().convert_to(ZZ)
H = ZK.submodule_from_matrix(W)
factors.append(H)
return factors
@public
def prime_decomp(p, T=None, ZK=None, dK=None, radical=None):
r"""
Compute the decomposition of rational prime *p* in a number field.
Explanation
===========
Ordinarily this should be accessed through the
:py:meth:`~.AlgebraicField.primes_above` method of an
:py:class:`~.AlgebraicField`.
Examples
========
>>> from sympy import Poly, QQ
>>> from sympy.abc import x, theta
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> K = QQ.algebraic_field((T, theta))
>>> print(K.primes_above(2))
[[ (2, x**2 + 1) e=1, f=1 ], [ (2, (x**2 + 3*x + 2)/2) e=1, f=1 ],
[ (2, (3*x**2 + 3*x)/2) e=1, f=1 ]]
Parameters
==========
p : int
The rational prime whose decomposition is desired.
T : :py:class:`~.Poly`, optional
Monic irreducible polynomial defining the number field $K$ in which to
factor. NOTE: at least one of *T* or *ZK* must be provided.
ZK : :py:class:`~.Submodule`, optional
The maximal order for $K$, if already known.
NOTE: at least one of *T* or *ZK* must be provided.
dK : int, optional
The discriminant of the field $K$, if already known.
radical : :py:class:`~.Submodule`, optional
The nilradical mod *p* in the integers of $K$, if already known.
Returns
=======
List of :py:class:`~.PrimeIdeal` instances.
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
(See Algorithm 6.2.9.)
"""
if T is None and ZK is None:
raise ValueError('At least one of T or ZK must be provided.')
if ZK is not None:
_check_formal_conditions_for_maximal_order(ZK)
if T is None:
T = ZK.parent.T
radicals = {}
if dK is None or ZK is None:
ZK, dK = round_two(T, radicals=radicals)
dT = T.discriminant()
f_squared = dT // dK
if f_squared % p != 0:
return _prime_decomp_easy_case(p, ZK)
radical = radical or radicals.get(p) or nilradical_mod_p(ZK, p)
stack = [radical]
primes = []
while stack:
I = stack.pop()
N, G = _prime_decomp_compute_kernel(I, p, ZK)
if N.n == 1:
P = _prime_decomp_maximal_ideal(I, p, ZK)
primes.append(P)
else:
I1, I2 = _prime_decomp_split_ideal(I, p, N, G, ZK)
stack.extend([I1, I2])
return primes
|
66a381c3435541bbea7b3e59233d9bc1674c4c83f662b6b3254f55426de3fe2b | """Computing integral bases for number fields. """
from sympy.polys.polytools import Poly
from sympy.polys.domains.integerring import ZZ
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.polyerrors import CoercionFailed
from sympy.utilities.decorator import public
from .modules import ModuleEndomorphism, ModuleHomomorphism, PowerBasis
from .utilities import extract_fundamental_discriminant
def _apply_Dedekind_criterion(T, p):
r"""
Apply the "Dedekind criterion" to test whether the order needs to be
enlarged relative to a given prime *p*.
"""
x = T.gen
T_bar = Poly(T, modulus=p)
lc, fl = T_bar.factor_list()
assert lc == 1
g_bar = Poly(1, x, modulus=p)
for ti_bar, _ in fl:
g_bar *= ti_bar
h_bar = T_bar // g_bar
g = Poly(g_bar, domain=ZZ)
h = Poly(h_bar, domain=ZZ)
f = (g * h - T) // p
f_bar = Poly(f, modulus=p)
Z_bar = f_bar
for b in [g_bar, h_bar]:
Z_bar = Z_bar.gcd(b)
U_bar = T_bar // Z_bar
m = Z_bar.degree()
return U_bar, m
def nilradical_mod_p(H, p, q=None):
r"""
Compute the nilradical mod *p* for a given order *H*, and prime *p*.
Explanation
===========
This is the ideal $I$ in $H/pH$ consisting of all elements some positive
power of which is zero in this quotient ring, i.e. is a multiple of *p*.
Parameters
==========
H : :py:class:`~.Submodule`
The given order.
p : int
The rational prime.
q : int, optional
If known, the smallest power of *p* that is $>=$ the dimension of *H*.
If not provided, we compute it here.
Returns
=======
:py:class:`~.Module` representing the nilradical mod *p* in *H*.
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory*.
(See Lemma 6.1.6.)
"""
n = H.n
if q is None:
q = p
while q < n:
q *= p
phi = ModuleEndomorphism(H, lambda x: x**q)
return phi.kernel(modulus=p)
def _second_enlargement(H, p, q):
r"""
Perform the second enlargement in the Round Two algorithm.
"""
Ip = nilradical_mod_p(H, p, q=q)
B = H.parent.submodule_from_matrix(H.matrix * Ip.matrix, denom=H.denom)
C = B + p*H
E = C.endomorphism_ring()
phi = ModuleHomomorphism(H, E, lambda x: E.inner_endomorphism(x))
gamma = phi.kernel(modulus=p)
G = H.parent.submodule_from_matrix(H.matrix * gamma.matrix, denom=H.denom * p)
H1 = G + H
return H1, Ip
@public
def round_two(T, radicals=None):
r"""
Zassenhaus's "Round 2" algorithm.
Explanation
===========
Carry out Zassenhaus's "Round 2" algorithm on a monic irreducible
polynomial *T* over :ref:`ZZ`. This computes an integral basis and the
discriminant for the field $K = \mathbb{Q}[x]/(T(x))$.
Ordinarily this function need not be called directly, as one can instead
access the :py:meth:`~.AlgebraicField.maximal_order`,
:py:meth:`~.AlgebraicField.integral_basis`, and
:py:meth:`~.AlgebraicField.discriminant` methods of an
:py:class:`~.AlgebraicField`.
Examples
========
Working through an AlgebraicField:
>>> from sympy import Poly, QQ
>>> from sympy.abc import x
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> K = QQ.alg_field_from_poly(T, "theta")
>>> print(K.maximal_order())
Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2
>>> print(K.discriminant())
-503
>>> print(K.integral_basis(fmt='sympy'))
[1, theta, theta/2 + theta**2/2]
Calling directly:
>>> from sympy import Poly
>>> from sympy.abc import x
>>> from sympy.polys.numberfields.basis import round_two
>>> T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
>>> print(round_two(T))
(Submodule[[2, 0, 0], [0, 2, 0], [0, 1, 1]]/2, -503)
The nilradicals mod $p$ that are sometimes computed during the Round Two
algorithm may be useful in further calculations. Pass a dictionary under
`radicals` to receive these:
>>> T = Poly(x**3 + 3*x**2 + 5)
>>> rad = {}
>>> ZK, dK = round_two(T, radicals=rad)
>>> print(rad)
{3: Submodule[[-1, 1, 0], [-1, 0, 1]]}
Parameters
==========
T : :py:class:`~.Poly`
The irreducible monic polynomial over :ref:`ZZ` defining the number
field.
radicals : dict, optional
This is a way for any $p$-radicals (if computed) to be returned by
reference. If desired, pass an empty dictionary. If the algorithm
reaches the point where it computes the nilradical mod $p$ of the ring
of integers $Z_K$, then an $\mathbb{F}_p$-basis for this ideal will be
stored in this dictionary under the key ``p``. This can be useful for
other algorithms, such as prime decomposition.
Returns
=======
Pair ``(ZK, dK)``, where:
``ZK`` is a :py:class:`~sympy.polys.numberfields.modules.Submodule`
representing the maximal order.
``dK`` is the discriminant of the field $K = \mathbb{Q}[x]/(T(x))$.
See Also
========
.AlgebraicField.maximal_order
.AlgebraicField.integral_basis
.AlgebraicField.discriminant
References
==========
.. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.*
"""
if T.domain == QQ:
try:
T = Poly(T, domain=ZZ)
except CoercionFailed:
pass # Let the error be raised by the next clause.
if ( not T.is_univariate
or not T.is_irreducible
or not T.is_monic
or not T.domain == ZZ):
raise ValueError('Round 2 requires a monic irreducible univariate polynomial over ZZ.')
n = T.degree()
D = T.discriminant()
D_modulus = ZZ.from_sympy(abs(D))
# D must be 0 or 1 mod 4 (see Cohen Sec 4.4), which ensures we can write
# it in the form D = D_0 * F**2, where D_0 is 1 or a fundamental discriminant.
_, F = extract_fundamental_discriminant(D)
Ztheta = PowerBasis(T)
H = Ztheta.whole_submodule()
nilrad = None
while F:
# Next prime:
p, e = F.popitem()
U_bar, m = _apply_Dedekind_criterion(T, p)
if m == 0:
continue
# For a given prime p, the first enlargement of the order spanned by
# the current basis can be done in a simple way:
U = Ztheta.element_from_poly(Poly(U_bar, domain=ZZ))
# TODO:
# Theory says only first m columns of the U//p*H term below are needed.
# Could be slightly more efficient to use only those. Maybe `Submodule`
# class should support a slice operator?
H = H.add(U // p * H, hnf_modulus=D_modulus)
if e <= m:
continue
# A second, and possibly more, enlargements for p will be needed.
# These enlargements require a more involved procedure.
q = p
while q < n:
q *= p
H1, nilrad = _second_enlargement(H, p, q)
while H1 != H:
H = H1
H1, nilrad = _second_enlargement(H, p, q)
# Note: We do not store all nilradicals mod p, only the very last. This is
# because, unless computed against the entire integral basis, it might not
# be accurate. (In other words, if H was not already equal to ZK when we
# passed it to `_second_enlargement`, then we can't trust the nilradical
# so computed.) Example: if T(x) = x ** 3 + 15 * x ** 2 - 9 * x + 13, then
# F is divisible by 2, 3, and 7, and the nilradical mod 2 as computed above
# will not be accurate for the full, maximal order ZK.
if nilrad is not None and isinstance(radicals, dict):
radicals[p] = nilrad
ZK = H
# Pre-set expensive boolean properties which we already know to be true:
ZK._starts_with_unity = True
ZK._is_sq_maxrank_HNF = True
dK = (D * ZK.matrix.det() ** 2) // ZK.denom ** (2 * n)
return ZK, dK
|
f1faf7fda5eaf8b39f920c5393755aa959c48e765a0a7abf867090e76170eb3e | """Minimal polynomials for algebraic numbers."""
from functools import reduce
from sympy.core.add import Add
from sympy.core.exprtools import Factors
from sympy.core.function import expand_mul, expand_multinomial, _mexpand
from sympy.core.mul import Mul
from sympy.core.numbers import (I, Rational, pi, _illegal)
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.core.traversal import preorder_traversal
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.trigonometric import cos, sin, tan
from sympy.ntheory.factor_ import divisors
from sympy.utilities.iterables import subsets
from sympy.polys.domains import ZZ, QQ, FractionField
from sympy.polys.orthopolys import dup_chebyshevt
from sympy.polys.polyerrors import (
NotAlgebraic,
GeneratorsError,
)
from sympy.polys.polytools import (
Poly, PurePoly, invert, factor_list, groebner, resultant,
degree, poly_from_expr, parallel_poly_from_expr, lcm
)
from sympy.polys.polyutils import dict_from_expr, expr_from_dict
from sympy.polys.ring_series import rs_compose_add
from sympy.polys.rings import ring
from sympy.polys.rootoftools import CRootOf
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.utilities import (
numbered_symbols, public, sift
)
def _choose_factor(factors, x, v, dom=QQ, prec=200, bound=5):
"""
Return a factor having root ``v``
It is assumed that one of the factors has root ``v``.
"""
if isinstance(factors[0], tuple):
factors = [f[0] for f in factors]
if len(factors) == 1:
return factors[0]
prec1 = 10
points = {}
symbols = dom.symbols if hasattr(dom, 'symbols') else []
while prec1 <= prec:
# when dealing with non-Rational numbers we usually evaluate
# with `subs` argument but we only need a ballpark evaluation
xv = {x:v if not v.is_number else v.n(prec1)}
fe = [f.as_expr().xreplace(xv) for f in factors]
# assign integers [0, n) to symbols (if any)
for n in subsets(range(bound), k=len(symbols), repetition=True):
for s, i in zip(symbols, n):
points[s] = i
# evaluate the expression at these points
candidates = [(abs(f.subs(points).n(prec1)), i)
for i,f in enumerate(fe)]
# if we get invalid numbers (e.g. from division by zero)
# we try again
if any(i in _illegal for i, _ in candidates):
continue
# find the smallest two -- if they differ significantly
# then we assume we have found the factor that becomes
# 0 when v is substituted into it
can = sorted(candidates)
(a, ix), (b, _) = can[:2]
if b > a * 10**6: # XXX what to use?
return factors[ix]
prec1 *= 2
raise NotImplementedError("multiple candidates for the minimal polynomial of %s" % v)
def _is_sum_surds(p):
args = p.args if p.is_Add else [p]
for y in args:
if not ((y**2).is_Rational and y.is_extended_real):
return False
return True
def _separate_sq(p):
"""
helper function for ``_minimal_polynomial_sq``
It selects a rational ``g`` such that the polynomial ``p``
consists of a sum of terms whose surds squared have gcd equal to ``g``
and a sum of terms with surds squared prime with ``g``;
then it takes the field norm to eliminate ``sqrt(g)``
See simplify.simplify.split_surds and polytools.sqf_norm.
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import x
>>> from sympy.polys.numberfields.minpoly import _separate_sq
>>> p= -x + sqrt(2) + sqrt(3) + sqrt(7)
>>> p = _separate_sq(p); p
-x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8
>>> p = _separate_sq(p); p
-x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20
>>> p = _separate_sq(p); p
-x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400
"""
def is_sqrt(expr):
return expr.is_Pow and expr.exp is S.Half
# p = c1*sqrt(q1) + ... + cn*sqrt(qn) -> a = [(c1, q1), .., (cn, qn)]
a = []
for y in p.args:
if not y.is_Mul:
if is_sqrt(y):
a.append((S.One, y**2))
elif y.is_Atom:
a.append((y, S.One))
elif y.is_Pow and y.exp.is_integer:
a.append((y, S.One))
else:
raise NotImplementedError
else:
T, F = sift(y.args, is_sqrt, binary=True)
a.append((Mul(*F), Mul(*T)**2))
a.sort(key=lambda z: z[1])
if a[-1][1] is S.One:
# there are no surds
return p
surds = [z for y, z in a]
for i in range(len(surds)):
if surds[i] != 1:
break
from sympy.simplify.radsimp import _split_gcd
g, b1, b2 = _split_gcd(*surds[i:])
a1 = []
a2 = []
for y, z in a:
if z in b1:
a1.append(y*z**S.Half)
else:
a2.append(y*z**S.Half)
p1 = Add(*a1)
p2 = Add(*a2)
p = _mexpand(p1**2) - _mexpand(p2**2)
return p
def _minimal_polynomial_sq(p, n, x):
"""
Returns the minimal polynomial for the ``nth-root`` of a sum of surds
or ``None`` if it fails.
Parameters
==========
p : sum of surds
n : positive integer
x : variable of the returned polynomial
Examples
========
>>> from sympy.polys.numberfields.minpoly import _minimal_polynomial_sq
>>> from sympy import sqrt
>>> from sympy.abc import x
>>> q = 1 + sqrt(2) + sqrt(3)
>>> _minimal_polynomial_sq(q, 3, x)
x**12 - 4*x**9 - 4*x**6 + 16*x**3 - 8
"""
p = sympify(p)
n = sympify(n)
if not n.is_Integer or not n > 0 or not _is_sum_surds(p):
return None
pn = p**Rational(1, n)
# eliminate the square roots
p -= x
while 1:
p1 = _separate_sq(p)
if p1 is p:
p = p1.subs({x:x**n})
break
else:
p = p1
# _separate_sq eliminates field extensions in a minimal way, so that
# if n = 1 then `p = constant*(minimal_polynomial(p))`
# if n > 1 it contains the minimal polynomial as a factor.
if n == 1:
p1 = Poly(p)
if p.coeff(x**p1.degree(x)) < 0:
p = -p
p = p.primitive()[1]
return p
# by construction `p` has root `pn`
# the minimal polynomial is the factor vanishing in x = pn
factors = factor_list(p)[1]
result = _choose_factor(factors, x, pn)
return result
def _minpoly_op_algebraic_element(op, ex1, ex2, x, dom, mp1=None, mp2=None):
"""
return the minimal polynomial for ``op(ex1, ex2)``
Parameters
==========
op : operation ``Add`` or ``Mul``
ex1, ex2 : expressions for the algebraic elements
x : indeterminate of the polynomials
dom: ground domain
mp1, mp2 : minimal polynomials for ``ex1`` and ``ex2`` or None
Examples
========
>>> from sympy import sqrt, Add, Mul, QQ
>>> from sympy.polys.numberfields.minpoly import _minpoly_op_algebraic_element
>>> from sympy.abc import x, y
>>> p1 = sqrt(sqrt(2) + 1)
>>> p2 = sqrt(sqrt(2) - 1)
>>> _minpoly_op_algebraic_element(Mul, p1, p2, x, QQ)
x - 1
>>> q1 = sqrt(y)
>>> q2 = 1 / y
>>> _minpoly_op_algebraic_element(Add, q1, q2, x, QQ.frac_field(y))
x**2*y**2 - 2*x*y - y**3 + 1
References
==========
.. [1] https://en.wikipedia.org/wiki/Resultant
.. [2] I.M. Isaacs, Proc. Amer. Math. Soc. 25 (1970), 638
"Degrees of sums in a separable field extension".
"""
y = Dummy(str(x))
if mp1 is None:
mp1 = _minpoly_compose(ex1, x, dom)
if mp2 is None:
mp2 = _minpoly_compose(ex2, y, dom)
else:
mp2 = mp2.subs({x: y})
if op is Add:
# mp1a = mp1.subs({x: x - y})
if dom == QQ:
R, X = ring('X', QQ)
p1 = R(dict_from_expr(mp1)[0])
p2 = R(dict_from_expr(mp2)[0])
else:
(p1, p2), _ = parallel_poly_from_expr((mp1, x - y), x, y)
r = p1.compose(p2)
mp1a = r.as_expr()
elif op is Mul:
mp1a = _muly(mp1, x, y)
else:
raise NotImplementedError('option not available')
if op is Mul or dom != QQ:
r = resultant(mp1a, mp2, gens=[y, x])
else:
r = rs_compose_add(p1, p2)
r = expr_from_dict(r.as_expr_dict(), x)
deg1 = degree(mp1, x)
deg2 = degree(mp2, y)
if op is Mul and deg1 == 1 or deg2 == 1:
# if deg1 = 1, then mp1 = x - a; mp1a = x - y - a;
# r = mp2(x - a), so that `r` is irreducible
return r
r = Poly(r, x, domain=dom)
_, factors = r.factor_list()
res = _choose_factor(factors, x, op(ex1, ex2), dom)
return res.as_expr()
def _invertx(p, x):
"""
Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))``
"""
p1 = poly_from_expr(p, x)[0]
n = degree(p1)
a = [c * x**(n - i) for (i,), c in p1.terms()]
return Add(*a)
def _muly(p, x, y):
"""
Returns ``_mexpand(y**deg*p.subs({x:x / y}))``
"""
p1 = poly_from_expr(p, x)[0]
n = degree(p1)
a = [c * x**i * y**(n - i) for (i,), c in p1.terms()]
return Add(*a)
def _minpoly_pow(ex, pw, x, dom, mp=None):
"""
Returns ``minpoly(ex**pw, x)``
Parameters
==========
ex : algebraic element
pw : rational number
x : indeterminate of the polynomial
dom: ground domain
mp : minimal polynomial of ``p``
Examples
========
>>> from sympy import sqrt, QQ, Rational
>>> from sympy.polys.numberfields.minpoly import _minpoly_pow, minpoly
>>> from sympy.abc import x, y
>>> p = sqrt(1 + sqrt(2))
>>> _minpoly_pow(p, 2, x, QQ)
x**2 - 2*x - 1
>>> minpoly(p**2, x)
x**2 - 2*x - 1
>>> _minpoly_pow(y, Rational(1, 3), x, QQ.frac_field(y))
x**3 - y
>>> minpoly(y**Rational(1, 3), x)
x**3 - y
"""
pw = sympify(pw)
if not mp:
mp = _minpoly_compose(ex, x, dom)
if not pw.is_rational:
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
if pw < 0:
if mp == x:
raise ZeroDivisionError('%s is zero' % ex)
mp = _invertx(mp, x)
if pw == -1:
return mp
pw = -pw
ex = 1/ex
y = Dummy(str(x))
mp = mp.subs({x: y})
n, d = pw.as_numer_denom()
res = Poly(resultant(mp, x**d - y**n, gens=[y]), x, domain=dom)
_, factors = res.factor_list()
res = _choose_factor(factors, x, ex**pw, dom)
return res.as_expr()
def _minpoly_add(x, dom, *a):
"""
returns ``minpoly(Add(*a), dom, x)``
"""
mp = _minpoly_op_algebraic_element(Add, a[0], a[1], x, dom)
p = a[0] + a[1]
for px in a[2:]:
mp = _minpoly_op_algebraic_element(Add, p, px, x, dom, mp1=mp)
p = p + px
return mp
def _minpoly_mul(x, dom, *a):
"""
returns ``minpoly(Mul(*a), dom, x)``
"""
mp = _minpoly_op_algebraic_element(Mul, a[0], a[1], x, dom)
p = a[0] * a[1]
for px in a[2:]:
mp = _minpoly_op_algebraic_element(Mul, p, px, x, dom, mp1=mp)
p = p * px
return mp
def _minpoly_sin(ex, x):
"""
Returns the minimal polynomial of ``sin(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
n = c.q
q = sympify(n)
if q.is_prime:
# for a = pi*p/q with q odd prime, using chebyshevt
# write sin(q*a) = mp(sin(a))*sin(a);
# the roots of mp(x) are sin(pi*p/q) for p = 1,..., q - 1
a = dup_chebyshevt(n, ZZ)
return Add(*[x**(n - i - 1)*a[i] for i in range(n)])
if c.p == 1:
if q == 9:
return 64*x**6 - 96*x**4 + 36*x**2 - 3
if n % 2 == 1:
# for a = pi*p/q with q odd, use
# sin(q*a) = 0 to see that the minimal polynomial must be
# a factor of dup_chebyshevt(n, ZZ)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i)*a[i] for i in range(n + 1)]
r = Add(*a)
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
expr = ((1 - cos(2*c*pi))/2)**S.Half
res = _minpoly_compose(expr, x, QQ)
return res
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
def _minpoly_cos(ex, x):
"""
Returns the minimal polynomial of ``cos(ex)``
see http://mathworld.wolfram.com/TrigonometryAngles.html
"""
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
if c.p == 1:
if c.q == 7:
return 8*x**3 - 4*x**2 - 4*x + 1
if c.q == 9:
return 8*x**3 - 6*x - 1
elif c.p == 2:
q = sympify(c.q)
if q.is_prime:
s = _minpoly_sin(ex, x)
return _mexpand(s.subs({x:sqrt((1 - x)/2)}))
# for a = pi*p/q, cos(q*a) =T_q(cos(a)) = (-1)**p
n = int(c.q)
a = dup_chebyshevt(n, ZZ)
a = [x**(n - i)*a[i] for i in range(n + 1)]
r = Add(*a) - (-1)**c.p
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
def _minpoly_tan(ex, x):
"""
Returns the minimal polynomial of ``tan(ex)``
see https://github.com/sympy/sympy/issues/21430
"""
c, a = ex.args[0].as_coeff_Mul()
if a is pi:
if c.is_rational:
c = c * 2
n = int(c.q)
a = n if c.p % 2 == 0 else 1
terms = []
for k in range((c.p+1)%2, n+1, 2):
terms.append(a*x**k)
a = -(a*(n-k-1)*(n-k)) // ((k+1)*(k+2))
r = Add(*terms)
_, factors = factor_list(r)
res = _choose_factor(factors, x, ex)
return res
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
def _minpoly_exp(ex, x):
"""
Returns the minimal polynomial of ``exp(ex)``
"""
c, a = ex.args[0].as_coeff_Mul()
if a == I*pi:
if c.is_rational:
q = sympify(c.q)
if c.p == 1 or c.p == -1:
if q == 3:
return x**2 - x + 1
if q == 4:
return x**4 + 1
if q == 6:
return x**4 - x**2 + 1
if q == 8:
return x**8 + 1
if q == 9:
return x**6 - x**3 + 1
if q == 10:
return x**8 - x**6 + x**4 - x**2 + 1
if q.is_prime:
s = 0
for i in range(q):
s += (-x)**i
return s
# x**(2*q) = product(factors)
factors = [cyclotomic_poly(i, x) for i in divisors(2*q)]
mp = _choose_factor(factors, x, ex)
return mp
else:
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
def _minpoly_rootof(ex, x):
"""
Returns the minimal polynomial of a ``CRootOf`` object.
"""
p = ex.expr
p = p.subs({ex.poly.gens[0]:x})
_, factors = factor_list(p, x)
result = _choose_factor(factors, x, ex)
return result
def _minpoly_compose(ex, x, dom):
"""
Computes the minimal polynomial of an algebraic element
using operations on minimal polynomials
Examples
========
>>> from sympy import minimal_polynomial, sqrt, Rational
>>> from sympy.abc import x, y
>>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=True)
x**2 - 2*x - 1
>>> minimal_polynomial(sqrt(y) + 1/y, x, compose=True)
x**2*y**2 - 2*x*y - y**3 + 1
"""
if ex.is_Rational:
return ex.q*x - ex.p
if ex is I:
_, factors = factor_list(x**2 + 1, x, domain=dom)
return x**2 + 1 if len(factors) == 1 else x - I
if ex is S.GoldenRatio:
_, factors = factor_list(x**2 - x - 1, x, domain=dom)
if len(factors) == 1:
return x**2 - x - 1
else:
return _choose_factor(factors, x, (1 + sqrt(5))/2, dom=dom)
if ex is S.TribonacciConstant:
_, factors = factor_list(x**3 - x**2 - x - 1, x, domain=dom)
if len(factors) == 1:
return x**3 - x**2 - x - 1
else:
fac = (1 + cbrt(19 - 3*sqrt(33)) + cbrt(19 + 3*sqrt(33))) / 3
return _choose_factor(factors, x, fac, dom=dom)
if hasattr(dom, 'symbols') and ex in dom.symbols:
return x - ex
if dom.is_QQ and _is_sum_surds(ex):
# eliminate the square roots
ex -= x
while 1:
ex1 = _separate_sq(ex)
if ex1 is ex:
return ex
else:
ex = ex1
if ex.is_Add:
res = _minpoly_add(x, dom, *ex.args)
elif ex.is_Mul:
f = Factors(ex).factors
r = sift(f.items(), lambda itx: itx[0].is_Rational and itx[1].is_Rational)
if r[True] and dom == QQ:
ex1 = Mul(*[bx**ex for bx, ex in r[False] + r[None]])
r1 = dict(r[True])
dens = [y.q for y in r1.values()]
lcmdens = reduce(lcm, dens, 1)
neg1 = S.NegativeOne
expn1 = r1.pop(neg1, S.Zero)
nums = [base**(y.p*lcmdens // y.q) for base, y in r1.items()]
ex2 = Mul(*nums)
mp1 = minimal_polynomial(ex1, x)
# use the fact that in SymPy canonicalization products of integers
# raised to rational powers are organized in relatively prime
# bases, and that in ``base**(n/d)`` a perfect power is
# simplified with the root
# Powers of -1 have to be treated separately to preserve sign.
mp2 = ex2.q*x**lcmdens - ex2.p*neg1**(expn1*lcmdens)
ex2 = neg1**expn1 * ex2**Rational(1, lcmdens)
res = _minpoly_op_algebraic_element(Mul, ex1, ex2, x, dom, mp1=mp1, mp2=mp2)
else:
res = _minpoly_mul(x, dom, *ex.args)
elif ex.is_Pow:
res = _minpoly_pow(ex.base, ex.exp, x, dom)
elif ex.__class__ is sin:
res = _minpoly_sin(ex, x)
elif ex.__class__ is cos:
res = _minpoly_cos(ex, x)
elif ex.__class__ is tan:
res = _minpoly_tan(ex, x)
elif ex.__class__ is exp:
res = _minpoly_exp(ex, x)
elif ex.__class__ is CRootOf:
res = _minpoly_rootof(ex, x)
else:
raise NotAlgebraic("%s does not seem to be an algebraic element" % ex)
return res
@public
def minimal_polynomial(ex, x=None, compose=True, polys=False, domain=None):
"""
Computes the minimal polynomial of an algebraic element.
Parameters
==========
ex : Expr
Element or expression whose minimal polynomial is to be calculated.
x : Symbol, optional
Independent variable of the minimal polynomial
compose : boolean, optional (default=True)
Method to use for computing minimal polynomial. If ``compose=True``
(default) then ``_minpoly_compose`` is used, if ``compose=False`` then
groebner bases are used.
polys : boolean, optional (default=False)
If ``True`` returns a ``Poly`` object else an ``Expr`` object.
domain : Domain, optional
Ground domain
Notes
=====
By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
are computed, then the arithmetic operations on them are performed using the resultant
and factorization.
If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
The default algorithm stalls less frequently.
If no ground domain is given, it will be generated automatically from the expression.
Examples
========
>>> from sympy import minimal_polynomial, sqrt, solve, QQ
>>> from sympy.abc import x, y
>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y
"""
ex = sympify(ex)
if ex.is_number:
# not sure if it's always needed but try it for numbers (issue 8354)
ex = _mexpand(ex, recursive=True)
for expr in preorder_traversal(ex):
if expr.is_AlgebraicNumber:
compose = False
break
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not domain:
if ex.free_symbols:
domain = FractionField(QQ, list(ex.free_symbols))
else:
domain = QQ
if hasattr(domain, 'symbols') and x in domain.symbols:
raise GeneratorsError("the variable %s is an element of the ground "
"domain %s" % (x, domain))
if compose:
result = _minpoly_compose(ex, x, domain)
result = result.primitive()[1]
c = result.coeff(x**degree(result, x))
if c.is_negative:
result = expand_mul(-result)
return cls(result, x, field=True) if polys else result.collect(x)
if not domain.is_QQ:
raise NotImplementedError("groebner method only works for QQ")
result = _minpoly_groebner(ex, x, cls)
return cls(result, x, field=True) if polys else result.collect(x)
def _minpoly_groebner(ex, x, cls):
"""
Computes the minimal polynomial of an algebraic number
using Groebner bases
Examples
========
>>> from sympy import minimal_polynomial, sqrt, Rational
>>> from sympy.abc import x
>>> minimal_polynomial(sqrt(2) + 3*Rational(1, 3), x, compose=False)
x**2 - 2*x - 1
"""
generator = numbered_symbols('a', cls=Dummy)
mapping, symbols = {}, {}
def update_mapping(ex, exp, base=None):
a = next(generator)
symbols[ex] = a
if base is not None:
mapping[ex] = a**exp + base
else:
mapping[ex] = exp.as_expr(a)
return a
def bottom_up_scan(ex):
"""
Transform a given algebraic expression *ex* into a multivariate
polynomial, by introducing fresh variables with defining equations.
Explanation
===========
The critical elements of the algebraic expression *ex* are root
extractions, instances of :py:class:`~.AlgebraicNumber`, and negative
powers.
When we encounter a root extraction or an :py:class:`~.AlgebraicNumber`
we replace this expression with a fresh variable ``a_i``, and record
the defining polynomial for ``a_i``. For example, if ``a_0**(1/3)``
occurs, we will replace it with ``a_1``, and record the new defining
polynomial ``a_1**3 - a_0``.
When we encounter a negative power we transform it into a positive
power by algebraically inverting the base. This means computing the
minimal polynomial in ``x`` for the base, inverting ``x`` modulo this
poly (which generates a new polynomial) and then substituting the
original base expression for ``x`` in this last polynomial.
We return the transformed expression, and we record the defining
equations for new symbols using the ``update_mapping()`` function.
"""
if ex.is_Atom:
if ex is S.ImaginaryUnit:
if ex not in mapping:
return update_mapping(ex, 2, 1)
else:
return symbols[ex]
elif ex.is_Rational:
return ex
elif ex.is_Add:
return Add(*[ bottom_up_scan(g) for g in ex.args ])
elif ex.is_Mul:
return Mul(*[ bottom_up_scan(g) for g in ex.args ])
elif ex.is_Pow:
if ex.exp.is_Rational:
if ex.exp < 0:
minpoly_base = _minpoly_groebner(ex.base, x, cls)
inverse = invert(x, minpoly_base).as_expr()
base_inv = inverse.subs(x, ex.base).expand()
if ex.exp == -1:
return bottom_up_scan(base_inv)
else:
ex = base_inv**(-ex.exp)
if not ex.exp.is_Integer:
base, exp = (
ex.base**ex.exp.p).expand(), Rational(1, ex.exp.q)
else:
base, exp = ex.base, ex.exp
base = bottom_up_scan(base)
expr = base**exp
if expr not in mapping:
if exp.is_Integer:
return expr.expand()
else:
return update_mapping(expr, 1 / exp, -base)
else:
return symbols[expr]
elif ex.is_AlgebraicNumber:
if ex not in mapping:
return update_mapping(ex, ex.minpoly_of_element())
else:
return symbols[ex]
raise NotAlgebraic("%s does not seem to be an algebraic number" % ex)
def simpler_inverse(ex):
"""
Returns True if it is more likely that the minimal polynomial
algorithm works better with the inverse
"""
if ex.is_Pow:
if (1/ex.exp).is_integer and ex.exp < 0:
if ex.base.is_Add:
return True
if ex.is_Mul:
hit = True
for p in ex.args:
if p.is_Add:
return False
if p.is_Pow:
if p.base.is_Add and p.exp > 0:
return False
if hit:
return True
return False
inverted = False
ex = expand_multinomial(ex)
if ex.is_AlgebraicNumber:
return ex.minpoly_of_element().as_expr(x)
elif ex.is_Rational:
result = ex.q*x - ex.p
else:
inverted = simpler_inverse(ex)
if inverted:
ex = ex**-1
res = None
if ex.is_Pow and (1/ex.exp).is_Integer:
n = 1/ex.exp
res = _minimal_polynomial_sq(ex.base, n, x)
elif _is_sum_surds(ex):
res = _minimal_polynomial_sq(ex, S.One, x)
if res is not None:
result = res
if res is None:
bus = bottom_up_scan(ex)
F = [x - bus] + list(mapping.values())
G = groebner(F, list(symbols.values()) + [x], order='lex')
_, factors = factor_list(G[-1])
# by construction G[-1] has root `ex`
result = _choose_factor(factors, x, ex)
if inverted:
result = _invertx(result, x)
if result.coeff(x**degree(result, x)) < 0:
result = expand_mul(-result)
return result
@public
def minpoly(ex, x=None, compose=True, polys=False, domain=None):
"""This is a synonym for :py:func:`~.minimal_polynomial`."""
return minimal_polynomial(ex, x=x, compose=compose, polys=polys, domain=domain)
|
0055d01f3c0c3ce9de933befd55ee6f05debaa946b6d524ff9dcbba233f9b079 | r"""
Functions in ``polys.numberfields.subfield`` solve the "Subfield Problem" and
allied problems, for algebraic number fields.
Following Cohen (see [Cohen93]_ Section 4.5), we can define the main problem as
follows:
* **Subfield Problem:**
Given two number fields $\mathbb{Q}(\alpha)$, $\mathbb{Q}(\beta)$
via the minimal polynomials for their generators $\alpha$ and $\beta$, decide
whether one field is isomorphic to a subfield of the other.
From a solution to this problem flow solutions to the following problems as
well:
* **Primitive Element Problem:**
Given several algebraic numbers
$\alpha_1, \ldots, \alpha_m$, compute a single algebraic number $\theta$
such that $\mathbb{Q}(\alpha_1, \ldots, \alpha_m) = \mathbb{Q}(\theta)$.
* **Field Isomorphism Problem:**
Decide whether two number fields
$\mathbb{Q}(\alpha)$, $\mathbb{Q}(\beta)$ are isomorphic.
* **Field Membership Problem:**
Given two algebraic numbers $\alpha$,
$\beta$, decide whether $\alpha \in \mathbb{Q}(\beta)$, and if so write
$\alpha = f(\beta)$ for some $f(x) \in \mathbb{Q}[x]$.
"""
from sympy.core.add import Add
from sympy.core.numbers import AlgebraicNumber
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify, _sympify
from sympy.ntheory import sieve
from sympy.polys.densetools import dup_eval
from sympy.polys.domains import QQ
from sympy.polys.numberfields.minpoly import _choose_factor, minimal_polynomial
from sympy.polys.polyerrors import IsomorphismFailed
from sympy.polys.polytools import Poly, PurePoly, factor_list
from sympy.utilities import public
from mpmath import MPContext
def is_isomorphism_possible(a, b):
"""Necessary but not sufficient test for isomorphism. """
n = a.minpoly.degree()
m = b.minpoly.degree()
if m % n != 0:
return False
if n == m:
return True
da = a.minpoly.discriminant()
db = b.minpoly.discriminant()
i, k, half = 1, m//n, db//2
while True:
p = sieve[i]
P = p**k
if P > half:
break
if ((da % p) % 2) and not (db % P):
return False
i += 1
return True
def field_isomorphism_pslq(a, b):
"""Construct field isomorphism using PSLQ algorithm. """
if not a.root.is_real or not b.root.is_real:
raise NotImplementedError("PSLQ doesn't support complex coefficients")
f = a.minpoly
g = b.minpoly.replace(f.gen)
n, m, prev = 100, b.minpoly.degree(), None
ctx = MPContext()
for i in range(1, 5):
A = a.root.evalf(n)
B = b.root.evalf(n)
basis = [1, B] + [ B**i for i in range(2, m) ] + [-A]
ctx.dps = n
coeffs = ctx.pslq(basis, maxcoeff=10**10, maxsteps=1000)
if coeffs is None:
# PSLQ can't find an integer linear combination. Give up.
break
if coeffs != prev:
prev = coeffs
else:
# Increasing precision didn't produce anything new. Give up.
break
# We have
# c0 + c1*B + c2*B^2 + ... + cm-1*B^(m-1) - cm*A ~ 0.
# So bring cm*A to the other side, and divide through by cm,
# for an approximate representation of A as a polynomial in B.
# (We know cm != 0 since `b.minpoly` is irreducible.)
coeffs = [S(c)/coeffs[-1] for c in coeffs[:-1]]
# Throw away leading zeros.
while not coeffs[-1]:
coeffs.pop()
coeffs = list(reversed(coeffs))
h = Poly(coeffs, f.gen, domain='QQ')
# We only have A ~ h(B). We must check whether the relation is exact.
if f.compose(h).rem(g).is_zero:
# Now we know that h(b) is in fact equal to _some conjugate of_ a.
# But from the very precise approximation A ~ h(B) we can assume
# the conjugate is a itself.
return coeffs
else:
n *= 2
return None
def field_isomorphism_factor(a, b):
"""Construct field isomorphism via factorization. """
_, factors = factor_list(a.minpoly, extension=b)
for f, _ in factors:
if f.degree() == 1:
# Any linear factor f(x) represents some conjugate of a in QQ(b).
# We want to know whether this linear factor represents a itself.
# Let f = x - c
c = -f.rep.TC()
# Write c as polynomial in b
coeffs = c.to_sympy_list()
d, terms = len(coeffs) - 1, []
for i, coeff in enumerate(coeffs):
terms.append(coeff*b.root**(d - i))
r = Add(*terms)
# Check whether we got the number a
if a.minpoly.same_root(r, a):
return coeffs
# If none of the linear factors represented a in QQ(b), then in fact a is
# not an element of QQ(b).
return None
@public
def field_isomorphism(a, b, *, fast=True):
r"""
Find an embedding of one number field into another.
Explanation
===========
This function looks for an isomorphism from $\mathbb{Q}(a)$ onto some
subfield of $\mathbb{Q}(b)$. Thus, it solves the Subfield Problem.
Examples
========
>>> from sympy import sqrt, field_isomorphism, I
>>> print(field_isomorphism(3, sqrt(2))) # doctest: +SKIP
[3]
>>> print(field_isomorphism( I*sqrt(3), I*sqrt(3)/2)) # doctest: +SKIP
[2, 0]
Parameters
==========
a : :py:class:`~.Expr`
Any expression representing an algebraic number.
b : :py:class:`~.Expr`
Any expression representing an algebraic number.
fast : boolean, optional (default=True)
If ``True``, we first attempt a potentially faster way of computing the
isomorphism, falling back on a slower method if this fails. If
``False``, we go directly to the slower method, which is guaranteed to
return a result.
Returns
=======
List of rational numbers, or None
If $\mathbb{Q}(a)$ is not isomorphic to some subfield of
$\mathbb{Q}(b)$, then return ``None``. Otherwise, return a list of
rational numbers representing an element of $\mathbb{Q}(b)$ to which
$a$ may be mapped, in order to define a monomorphism, i.e. an
isomorphism from $\mathbb{Q}(a)$ to some subfield of $\mathbb{Q}(b)$.
The elements of the list are the coefficients of falling powers of $b$.
"""
a, b = sympify(a), sympify(b)
if not a.is_AlgebraicNumber:
a = AlgebraicNumber(a)
if not b.is_AlgebraicNumber:
b = AlgebraicNumber(b)
a = a.to_primitive_element()
b = b.to_primitive_element()
if a == b:
return a.coeffs()
n = a.minpoly.degree()
m = b.minpoly.degree()
if n == 1:
return [a.root]
if m % n != 0:
return None
if fast:
try:
result = field_isomorphism_pslq(a, b)
if result is not None:
return result
except NotImplementedError:
pass
return field_isomorphism_factor(a, b)
def _switch_domain(g, K):
# An algebraic relation f(a, b) = 0 over Q can also be written
# g(b) = 0 where g is in Q(a)[x] and h(a) = 0 where h is in Q(b)[x].
# This function transforms g into h where Q(b) = K.
frep = g.rep.inject()
hrep = frep.eject(K, front=True)
return g.new(hrep, g.gens[0])
def _linsolve(p):
# Compute root of linear polynomial.
c, d = p.rep.rep
return -d/c
@public
def primitive_element(extension, x=None, *, ex=False, polys=False):
r"""
Find a single generator for a number field given by several generators.
Explanation
===========
The basic problem is this: Given several algebraic numbers
$\alpha_1, \alpha_2, \ldots, \alpha_n$, find a single algebraic number
$\theta$ such that
$\mathbb{Q}(\alpha_1, \alpha_2, \ldots, \alpha_n) = \mathbb{Q}(\theta)$.
This function actually guarantees that $\theta$ will be a linear
combination of the $\alpha_i$, with non-negative integer coefficients.
Furthermore, if desired, this function will tell you how to express each
$\alpha_i$ as a $\mathbb{Q}$-linear combination of the powers of $\theta$.
Examples
========
>>> from sympy import primitive_element, sqrt, S, minpoly, simplify
>>> from sympy.abc import x
>>> f, lincomb, reps = primitive_element([sqrt(2), sqrt(3)], x, ex=True)
Then ``lincomb`` tells us the primitive element as a linear combination of
the given generators ``sqrt(2)`` and ``sqrt(3)``.
>>> print(lincomb)
[1, 1]
This means the primtiive element is $\sqrt{2} + \sqrt{3}$.
Meanwhile ``f`` is the minimal polynomial for this primitive element.
>>> print(f)
x**4 - 10*x**2 + 1
>>> print(minpoly(sqrt(2) + sqrt(3), x))
x**4 - 10*x**2 + 1
Finally, ``reps`` (which was returned only because we set keyword arg
``ex=True``) tells us how to recover each of the generators $\sqrt{2}$ and
$\sqrt{3}$ as $\mathbb{Q}$-linear combinations of the powers of the
primitive element $\sqrt{2} + \sqrt{3}$.
>>> print([S(r) for r in reps[0]])
[1/2, 0, -9/2, 0]
>>> theta = sqrt(2) + sqrt(3)
>>> print(simplify(theta**3/2 - 9*theta/2))
sqrt(2)
>>> print([S(r) for r in reps[1]])
[-1/2, 0, 11/2, 0]
>>> print(simplify(-theta**3/2 + 11*theta/2))
sqrt(3)
Parameters
==========
extension : list of :py:class:`~.Expr`
Each expression must represent an algebraic number $\alpha_i$.
x : :py:class:`~.Symbol`, optional (default=None)
The desired symbol to appear in the computed minimal polynomial for the
primitive element $\theta$. If ``None``, we use a dummy symbol.
ex : boolean, optional (default=False)
If and only if ``True``, compute the representation of each $\alpha_i$
as a $\mathbb{Q}$-linear combination over the powers of $\theta$.
polys : boolean, optional (default=False)
If ``True``, return the minimal polynomial as a :py:class:`~.Poly`.
Otherwise return it as an :py:class:`~.Expr`.
Returns
=======
Pair (f, coeffs) or triple (f, coeffs, reps), where:
``f`` is the minimal polynomial for the primitive element.
``coeffs`` gives the primitive element as a linear combination of the
given generators.
``reps`` is present if and only if argument ``ex=True`` was passed,
and is a list of lists of rational numbers. Each list gives the
coefficients of falling powers of the primitive element, to recover
one of the original, given generators.
"""
if not extension:
raise ValueError("Cannot compute primitive element for empty extension")
extension = [_sympify(ext) for ext in extension]
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not ex:
gen, coeffs = extension[0], [1]
g = minimal_polynomial(gen, x, polys=True)
for ext in extension[1:]:
if ext.is_Rational:
coeffs.append(0)
continue
_, factors = factor_list(g, extension=ext)
g = _choose_factor(factors, x, gen)
s, _, g = g.sqf_norm()
gen += s*ext
coeffs.append(s)
if not polys:
return g.as_expr(), coeffs
else:
return cls(g), coeffs
gen, coeffs = extension[0], [1]
f = minimal_polynomial(gen, x, polys=True)
K = QQ.algebraic_field((f, gen)) # incrementally constructed field
reps = [K.unit] # representations of extension elements in K
for ext in extension[1:]:
if ext.is_Rational:
coeffs.append(0) # rational ext is not included in the expression of a primitive element
reps.append(K.convert(ext)) # but it is included in reps
continue
p = minimal_polynomial(ext, x, polys=True)
L = QQ.algebraic_field((p, ext))
_, factors = factor_list(f, domain=L)
f = _choose_factor(factors, x, gen)
s, g, f = f.sqf_norm()
gen += s*ext
coeffs.append(s)
K = QQ.algebraic_field((f, gen))
h = _switch_domain(g, K)
erep = _linsolve(h.gcd(p)) # ext as element of K
ogen = K.unit - s*erep # old gen as element of K
reps = [dup_eval(_.rep, ogen, K) for _ in reps] + [erep]
if K.ext.root.is_Rational: # all extensions are rational
H = [K.convert(_).rep for _ in extension]
coeffs = [0]*len(extension)
f = cls(x, domain=QQ)
else:
H = [_.rep for _ in reps]
if not polys:
return f.as_expr(), coeffs, H
else:
return f, coeffs, H
@public
def to_number_field(extension, theta=None, *, gen=None):
r"""
Express one algebraic number in the field generated by another.
Explanation
===========
Given two algebraic numbers $\eta, \theta$, this function either expresses
$\eta$ as an element of $\mathbb{Q}(\theta)$, or else raises an exception
if $\eta \not\in \mathbb{Q}(\theta)$.
This function is essentially just a convenience, utilizing
:py:func:`~.field_isomorphism` (our solution of the Subfield Problem) to
solve this, the Field Membership Problem.
As an additional convenience, this function allows you to pass a list of
algebraic numbers $\alpha_1, \alpha_2, \ldots, \alpha_n$ instead of $\eta$.
It then computes $\eta$ for you, as a solution of the Primitive Element
Problem, using :py:func:`~.primitive_element` on the list of $\alpha_i$.
Examples
========
>>> from sympy import sqrt, to_number_field
>>> eta = sqrt(2)
>>> theta = sqrt(2) + sqrt(3)
>>> a = to_number_field(eta, theta)
>>> print(type(a))
<class 'sympy.core.numbers.AlgebraicNumber'>
>>> a.root
sqrt(2) + sqrt(3)
>>> print(a)
sqrt(2)
>>> a.coeffs()
[1/2, 0, -9/2, 0]
We get an :py:class:`~.AlgebraicNumber`, whose ``.root`` is $\theta$, whose
value is $\eta$, and whose ``.coeffs()`` show how to write $\eta$ as a
$\mathbb{Q}$-linear combination in falling powers of $\theta$.
Parameters
==========
extension : :py:class:`~.Expr` or list of :py:class:`~.Expr`
Either the algebraic number that is to be expressed in the other field,
or else a list of algebraic numbers, a primitive element for which is
to be expressed in the other field.
theta : :py:class:`~.Expr`, None, optional (default=None)
If an :py:class:`~.Expr` representing an algebraic number, behavior is
as described under **Explanation**. If ``None``, then this function
reduces to a shorthand for calling :py:func:`~.primitive_element` on
``extension`` and turning the computed primitive element into an
:py:class:`~.AlgebraicNumber`.
gen : :py:class:`~.Symbol`, None, optional (default=None)
If provided, this will be used as the generator symbol for the returned
:py:class:`~.AlgebraicNumber`.
Returns
=======
AlgebraicNumber
Belonging to $\mathbb{Q}(\theta)$ and equaling $\eta$.
Raises
======
IsomorphismFailed
If $\eta \not\in \mathbb{Q}(\theta)$.
See Also
========
field_isomorphism
primitive_element
"""
if hasattr(extension, '__iter__'):
extension = list(extension)
else:
extension = [extension]
if len(extension) == 1 and isinstance(extension[0], tuple):
return AlgebraicNumber(extension[0])
minpoly, coeffs = primitive_element(extension, gen, polys=True)
root = sum([ coeff*ext for coeff, ext in zip(coeffs, extension) ])
if theta is None:
return AlgebraicNumber((minpoly, root))
else:
theta = sympify(theta)
if not theta.is_AlgebraicNumber:
theta = AlgebraicNumber(theta, gen=gen)
coeffs = field_isomorphism(root, theta)
if coeffs is not None:
return AlgebraicNumber(theta, coeffs)
else:
raise IsomorphismFailed(
"%s is not in a subfield of %s" % (root, theta.root))
|
a0a89780213df03b1663ae13e6753ad90fe25eaac852d84730cf6ad12d533da7 | """Tests for classes defining properties of ground domains, e.g. ZZ, QQ, ZZ[x] ... """
from sympy.core.numbers import (E, Float, I, Integer, Rational, oo, pi, _illegal)
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import sin
from sympy.polys.polytools import Poly
from sympy.abc import x, y, z
from sympy.external.gmpy import HAS_GMPY
from sympy.polys.domains import (ZZ, QQ, RR, CC, FF, GF, EX, EXRAW, ZZ_gmpy,
ZZ_python, QQ_gmpy, QQ_python)
from sympy.polys.domains.algebraicfield import AlgebraicField
from sympy.polys.domains.gaussiandomains import ZZ_I, QQ_I
from sympy.polys.domains.polynomialring import PolynomialRing
from sympy.polys.domains.realfield import RealField
from sympy.polys.numberfields.subfield import field_isomorphism
from sympy.polys.rings import ring
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.polys.fields import field
from sympy.polys.agca.extensions import FiniteExtension
from sympy.polys.polyerrors import (
UnificationFailed,
GeneratorsError,
CoercionFailed,
NotInvertible,
DomainError)
from sympy.testing.pytest import raises
from itertools import product
ALG = QQ.algebraic_field(sqrt(2), sqrt(3))
def unify(K0, K1):
return K0.unify(K1)
def test_Domain_unify():
F3 = GF(3)
assert unify(F3, F3) == F3
assert unify(F3, ZZ) == ZZ
assert unify(F3, QQ) == QQ
assert unify(F3, ALG) == ALG
assert unify(F3, RR) == RR
assert unify(F3, CC) == CC
assert unify(F3, ZZ[x]) == ZZ[x]
assert unify(F3, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(F3, EX) == EX
assert unify(ZZ, F3) == ZZ
assert unify(ZZ, ZZ) == ZZ
assert unify(ZZ, QQ) == QQ
assert unify(ZZ, ALG) == ALG
assert unify(ZZ, RR) == RR
assert unify(ZZ, CC) == CC
assert unify(ZZ, ZZ[x]) == ZZ[x]
assert unify(ZZ, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ, EX) == EX
assert unify(QQ, F3) == QQ
assert unify(QQ, ZZ) == QQ
assert unify(QQ, QQ) == QQ
assert unify(QQ, ALG) == ALG
assert unify(QQ, RR) == RR
assert unify(QQ, CC) == CC
assert unify(QQ, ZZ[x]) == QQ[x]
assert unify(QQ, ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ, EX) == EX
assert unify(ZZ_I, F3) == ZZ_I
assert unify(ZZ_I, ZZ) == ZZ_I
assert unify(ZZ_I, ZZ_I) == ZZ_I
assert unify(ZZ_I, QQ) == QQ_I
assert unify(ZZ_I, ALG) == QQ.algebraic_field(I, sqrt(2), sqrt(3))
assert unify(ZZ_I, RR) == CC
assert unify(ZZ_I, CC) == CC
assert unify(ZZ_I, ZZ[x]) == ZZ_I[x]
assert unify(ZZ_I, ZZ_I[x]) == ZZ_I[x]
assert unify(ZZ_I, ZZ.frac_field(x)) == ZZ_I.frac_field(x)
assert unify(ZZ_I, ZZ_I.frac_field(x)) == ZZ_I.frac_field(x)
assert unify(ZZ_I, EX) == EX
assert unify(QQ_I, F3) == QQ_I
assert unify(QQ_I, ZZ) == QQ_I
assert unify(QQ_I, ZZ_I) == QQ_I
assert unify(QQ_I, QQ) == QQ_I
assert unify(QQ_I, ALG) == QQ.algebraic_field(I, sqrt(2), sqrt(3))
assert unify(QQ_I, RR) == CC
assert unify(QQ_I, CC) == CC
assert unify(QQ_I, ZZ[x]) == QQ_I[x]
assert unify(QQ_I, ZZ_I[x]) == QQ_I[x]
assert unify(QQ_I, QQ[x]) == QQ_I[x]
assert unify(QQ_I, QQ_I[x]) == QQ_I[x]
assert unify(QQ_I, ZZ.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, ZZ_I.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, QQ.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, QQ_I.frac_field(x)) == QQ_I.frac_field(x)
assert unify(QQ_I, EX) == EX
assert unify(RR, F3) == RR
assert unify(RR, ZZ) == RR
assert unify(RR, QQ) == RR
assert unify(RR, ALG) == RR
assert unify(RR, RR) == RR
assert unify(RR, CC) == CC
assert unify(RR, ZZ[x]) == RR[x]
assert unify(RR, ZZ.frac_field(x)) == RR.frac_field(x)
assert unify(RR, EX) == EX
assert RR[x].unify(ZZ.frac_field(y)) == RR.frac_field(x, y)
assert unify(CC, F3) == CC
assert unify(CC, ZZ) == CC
assert unify(CC, QQ) == CC
assert unify(CC, ALG) == CC
assert unify(CC, RR) == CC
assert unify(CC, CC) == CC
assert unify(CC, ZZ[x]) == CC[x]
assert unify(CC, ZZ.frac_field(x)) == CC.frac_field(x)
assert unify(CC, EX) == EX
assert unify(ZZ[x], F3) == ZZ[x]
assert unify(ZZ[x], ZZ) == ZZ[x]
assert unify(ZZ[x], QQ) == QQ[x]
assert unify(ZZ[x], ALG) == ALG[x]
assert unify(ZZ[x], RR) == RR[x]
assert unify(ZZ[x], CC) == CC[x]
assert unify(ZZ[x], ZZ[x]) == ZZ[x]
assert unify(ZZ[x], ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ[x], EX) == EX
assert unify(ZZ.frac_field(x), F3) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x), ALG) == ALG.frac_field(x)
assert unify(ZZ.frac_field(x), RR) == RR.frac_field(x)
assert unify(ZZ.frac_field(x), CC) == CC.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ[x]) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), EX) == EX
assert unify(EX, F3) == EX
assert unify(EX, ZZ) == EX
assert unify(EX, QQ) == EX
assert unify(EX, ALG) == EX
assert unify(EX, RR) == EX
assert unify(EX, CC) == EX
assert unify(EX, ZZ[x]) == EX
assert unify(EX, ZZ.frac_field(x)) == EX
assert unify(EX, EX) == EX
def test_Domain_unify_composite():
assert unify(ZZ.poly_ring(x), ZZ) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), QQ) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), ZZ) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), QQ) == QQ.poly_ring(x)
assert unify(ZZ, ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(QQ, ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(ZZ, QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ, QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(ZZ.poly_ring(x, y), ZZ) == ZZ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x, y), QQ) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), ZZ) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), QQ) == QQ.poly_ring(x, y)
assert unify(ZZ, ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert unify(QQ, ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(ZZ, QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(QQ, QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(ZZ.frac_field(x), ZZ) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), ZZ) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), QQ) == QQ.frac_field(x)
assert unify(ZZ, ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(QQ, ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ, QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ, QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x, y), ZZ) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), QQ) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), ZZ) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), QQ) == QQ.frac_field(x, y)
assert unify(ZZ, ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ, ZZ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ, QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(QQ, QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.poly_ring(x), ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert unify(ZZ.poly_ring(x), QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(QQ.poly_ring(x), QQ.poly_ring(x)) == QQ.poly_ring(x)
assert unify(ZZ.poly_ring(x, y), ZZ.poly_ring(x)) == ZZ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x, y), QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), ZZ.poly_ring(x)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x, y), QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x), ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x), QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x), ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(QQ.poly_ring(x), QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert unify(ZZ.poly_ring(x, y), ZZ.poly_ring(x, z)) == ZZ.poly_ring(x, y, z)
assert unify(ZZ.poly_ring(x, y), QQ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert unify(QQ.poly_ring(x, y), ZZ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert unify(QQ.poly_ring(x, y), QQ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), ZZ.frac_field(x)) == QQ.frac_field(x)
assert unify(QQ.frac_field(x), QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x, y), ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), QQ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), ZZ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), QQ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x), ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x), QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x), ZZ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(QQ.frac_field(x), QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(ZZ.frac_field(x, y), QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), ZZ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(ZZ.poly_ring(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(ZZ.poly_ring(x), QQ.frac_field(x)) == ZZ.frac_field(x)
assert unify(QQ.poly_ring(x), ZZ.frac_field(x)) == ZZ.frac_field(x)
assert unify(QQ.poly_ring(x), QQ.frac_field(x)) == QQ.frac_field(x)
assert unify(ZZ.poly_ring(x, y), ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(ZZ.poly_ring(x, y), QQ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x, y), ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x, y), QQ.frac_field(x)) == QQ.frac_field(x, y)
assert unify(ZZ.poly_ring(x), ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(ZZ.poly_ring(x), QQ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x), ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.poly_ring(x), QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.poly_ring(x, y), ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(ZZ.poly_ring(x, y), QQ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.poly_ring(x, y), ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.poly_ring(x, y), QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert unify(ZZ.frac_field(x), ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(ZZ.frac_field(x), QQ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(QQ.frac_field(x), ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert unify(QQ.frac_field(x), QQ.poly_ring(x)) == QQ.frac_field(x)
assert unify(ZZ.frac_field(x, y), ZZ.poly_ring(x)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), QQ.poly_ring(x)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), ZZ.poly_ring(x)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x, y), QQ.poly_ring(x)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x), ZZ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert unify(ZZ.frac_field(x), QQ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x), ZZ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert unify(QQ.frac_field(x), QQ.poly_ring(x, y)) == QQ.frac_field(x, y)
assert unify(ZZ.frac_field(x, y), ZZ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert unify(ZZ.frac_field(x, y), QQ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), ZZ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert unify(QQ.frac_field(x, y), QQ.poly_ring(x, z)) == QQ.frac_field(x, y, z)
def test_Domain_unify_algebraic():
sqrt5 = QQ.algebraic_field(sqrt(5))
sqrt7 = QQ.algebraic_field(sqrt(7))
sqrt57 = QQ.algebraic_field(sqrt(5), sqrt(7))
assert sqrt5.unify(sqrt7) == sqrt57
assert sqrt5.unify(sqrt5[x, y]) == sqrt5[x, y]
assert sqrt5[x, y].unify(sqrt5) == sqrt5[x, y]
assert sqrt5.unify(sqrt5.frac_field(x, y)) == sqrt5.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt5) == sqrt5.frac_field(x, y)
assert sqrt5.unify(sqrt7[x, y]) == sqrt57[x, y]
assert sqrt5[x, y].unify(sqrt7) == sqrt57[x, y]
assert sqrt5.unify(sqrt7.frac_field(x, y)) == sqrt57.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt7) == sqrt57.frac_field(x, y)
def test_Domain_unify_FiniteExtension():
KxZZ = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ))
KxQQ = FiniteExtension(Poly(x**2 - 2, x, domain=QQ))
KxZZy = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ[y]))
KxQQy = FiniteExtension(Poly(x**2 - 2, x, domain=QQ[y]))
assert KxZZ.unify(KxZZ) == KxZZ
assert KxQQ.unify(KxQQ) == KxQQ
assert KxZZy.unify(KxZZy) == KxZZy
assert KxQQy.unify(KxQQy) == KxQQy
assert KxZZ.unify(ZZ) == KxZZ
assert KxZZ.unify(QQ) == KxQQ
assert KxQQ.unify(ZZ) == KxQQ
assert KxQQ.unify(QQ) == KxQQ
assert KxZZ.unify(ZZ[y]) == KxZZy
assert KxZZ.unify(QQ[y]) == KxQQy
assert KxQQ.unify(ZZ[y]) == KxQQy
assert KxQQ.unify(QQ[y]) == KxQQy
assert KxZZy.unify(ZZ) == KxZZy
assert KxZZy.unify(QQ) == KxQQy
assert KxQQy.unify(ZZ) == KxQQy
assert KxQQy.unify(QQ) == KxQQy
assert KxZZy.unify(ZZ[y]) == KxZZy
assert KxZZy.unify(QQ[y]) == KxQQy
assert KxQQy.unify(ZZ[y]) == KxQQy
assert KxQQy.unify(QQ[y]) == KxQQy
K = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ[y]))
assert K.unify(ZZ) == K
assert K.unify(ZZ[x]) == K
assert K.unify(ZZ[y]) == K
assert K.unify(ZZ[x, y]) == K
Kz = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ[y, z]))
assert K.unify(ZZ[z]) == Kz
assert K.unify(ZZ[x, z]) == Kz
assert K.unify(ZZ[y, z]) == Kz
assert K.unify(ZZ[x, y, z]) == Kz
Kx = FiniteExtension(Poly(x**2 - 2, x, domain=ZZ))
Ky = FiniteExtension(Poly(y**2 - 2, y, domain=ZZ))
Kxy = FiniteExtension(Poly(y**2 - 2, y, domain=Kx))
assert Kx.unify(Kx) == Kx
assert Ky.unify(Ky) == Ky
assert Kx.unify(Ky) == Kxy
assert Ky.unify(Kx) == Kxy
def test_Domain_unify_with_symbols():
raises(UnificationFailed, lambda: ZZ[x, y].unify_with_symbols(ZZ, (y, z)))
raises(UnificationFailed, lambda: ZZ.unify_with_symbols(ZZ[x, y], (y, z)))
def test_Domain__contains__():
assert (0 in EX) is True
assert (0 in ZZ) is True
assert (0 in QQ) is True
assert (0 in RR) is True
assert (0 in CC) is True
assert (0 in ALG) is True
assert (0 in ZZ[x, y]) is True
assert (0 in QQ[x, y]) is True
assert (0 in RR[x, y]) is True
assert (-7 in EX) is True
assert (-7 in ZZ) is True
assert (-7 in QQ) is True
assert (-7 in RR) is True
assert (-7 in CC) is True
assert (-7 in ALG) is True
assert (-7 in ZZ[x, y]) is True
assert (-7 in QQ[x, y]) is True
assert (-7 in RR[x, y]) is True
assert (17 in EX) is True
assert (17 in ZZ) is True
assert (17 in QQ) is True
assert (17 in RR) is True
assert (17 in CC) is True
assert (17 in ALG) is True
assert (17 in ZZ[x, y]) is True
assert (17 in QQ[x, y]) is True
assert (17 in RR[x, y]) is True
assert (Rational(-1, 7) in EX) is True
assert (Rational(-1, 7) in ZZ) is False
assert (Rational(-1, 7) in QQ) is True
assert (Rational(-1, 7) in RR) is True
assert (Rational(-1, 7) in CC) is True
assert (Rational(-1, 7) in ALG) is True
assert (Rational(-1, 7) in ZZ[x, y]) is False
assert (Rational(-1, 7) in QQ[x, y]) is True
assert (Rational(-1, 7) in RR[x, y]) is True
assert (Rational(3, 5) in EX) is True
assert (Rational(3, 5) in ZZ) is False
assert (Rational(3, 5) in QQ) is True
assert (Rational(3, 5) in RR) is True
assert (Rational(3, 5) in CC) is True
assert (Rational(3, 5) in ALG) is True
assert (Rational(3, 5) in ZZ[x, y]) is False
assert (Rational(3, 5) in QQ[x, y]) is True
assert (Rational(3, 5) in RR[x, y]) is True
assert (3.0 in EX) is True
assert (3.0 in ZZ) is True
assert (3.0 in QQ) is True
assert (3.0 in RR) is True
assert (3.0 in CC) is True
assert (3.0 in ALG) is True
assert (3.0 in ZZ[x, y]) is True
assert (3.0 in QQ[x, y]) is True
assert (3.0 in RR[x, y]) is True
assert (3.14 in EX) is True
assert (3.14 in ZZ) is False
assert (3.14 in QQ) is True
assert (3.14 in RR) is True
assert (3.14 in CC) is True
assert (3.14 in ALG) is True
assert (3.14 in ZZ[x, y]) is False
assert (3.14 in QQ[x, y]) is True
assert (3.14 in RR[x, y]) is True
assert (oo in ALG) is False
assert (oo in ZZ[x, y]) is False
assert (oo in QQ[x, y]) is False
assert (-oo in ZZ) is False
assert (-oo in QQ) is False
assert (-oo in ALG) is False
assert (-oo in ZZ[x, y]) is False
assert (-oo in QQ[x, y]) is False
assert (sqrt(7) in EX) is True
assert (sqrt(7) in ZZ) is False
assert (sqrt(7) in QQ) is False
assert (sqrt(7) in RR) is True
assert (sqrt(7) in CC) is True
assert (sqrt(7) in ALG) is False
assert (sqrt(7) in ZZ[x, y]) is False
assert (sqrt(7) in QQ[x, y]) is False
assert (sqrt(7) in RR[x, y]) is True
assert (2*sqrt(3) + 1 in EX) is True
assert (2*sqrt(3) + 1 in ZZ) is False
assert (2*sqrt(3) + 1 in QQ) is False
assert (2*sqrt(3) + 1 in RR) is True
assert (2*sqrt(3) + 1 in CC) is True
assert (2*sqrt(3) + 1 in ALG) is True
assert (2*sqrt(3) + 1 in ZZ[x, y]) is False
assert (2*sqrt(3) + 1 in QQ[x, y]) is False
assert (2*sqrt(3) + 1 in RR[x, y]) is True
assert (sin(1) in EX) is True
assert (sin(1) in ZZ) is False
assert (sin(1) in QQ) is False
assert (sin(1) in RR) is True
assert (sin(1) in CC) is True
assert (sin(1) in ALG) is False
assert (sin(1) in ZZ[x, y]) is False
assert (sin(1) in QQ[x, y]) is False
assert (sin(1) in RR[x, y]) is True
assert (x**2 + 1 in EX) is True
assert (x**2 + 1 in ZZ) is False
assert (x**2 + 1 in QQ) is False
assert (x**2 + 1 in RR) is False
assert (x**2 + 1 in CC) is False
assert (x**2 + 1 in ALG) is False
assert (x**2 + 1 in ZZ[x]) is True
assert (x**2 + 1 in QQ[x]) is True
assert (x**2 + 1 in RR[x]) is True
assert (x**2 + 1 in ZZ[x, y]) is True
assert (x**2 + 1 in QQ[x, y]) is True
assert (x**2 + 1 in RR[x, y]) is True
assert (x**2 + y**2 in EX) is True
assert (x**2 + y**2 in ZZ) is False
assert (x**2 + y**2 in QQ) is False
assert (x**2 + y**2 in RR) is False
assert (x**2 + y**2 in CC) is False
assert (x**2 + y**2 in ALG) is False
assert (x**2 + y**2 in ZZ[x]) is False
assert (x**2 + y**2 in QQ[x]) is False
assert (x**2 + y**2 in RR[x]) is False
assert (x**2 + y**2 in ZZ[x, y]) is True
assert (x**2 + y**2 in QQ[x, y]) is True
assert (x**2 + y**2 in RR[x, y]) is True
assert (Rational(3, 2)*x/(y + 1) - z in QQ[x, y, z]) is False
def test_issue_14433():
assert (Rational(2, 3)*x in QQ.frac_field(1/x)) is True
assert (1/x in QQ.frac_field(x)) is True
assert ((x**2 + y**2) in QQ.frac_field(1/x, 1/y)) is True
assert ((x + y) in QQ.frac_field(1/x, y)) is True
assert ((x - y) in QQ.frac_field(x, 1/y)) is True
def test_Domain_get_ring():
assert ZZ.has_assoc_Ring is True
assert QQ.has_assoc_Ring is True
assert ZZ[x].has_assoc_Ring is True
assert QQ[x].has_assoc_Ring is True
assert ZZ[x, y].has_assoc_Ring is True
assert QQ[x, y].has_assoc_Ring is True
assert ZZ.frac_field(x).has_assoc_Ring is True
assert QQ.frac_field(x).has_assoc_Ring is True
assert ZZ.frac_field(x, y).has_assoc_Ring is True
assert QQ.frac_field(x, y).has_assoc_Ring is True
assert EX.has_assoc_Ring is False
assert RR.has_assoc_Ring is False
assert ALG.has_assoc_Ring is False
assert ZZ.get_ring() == ZZ
assert QQ.get_ring() == ZZ
assert ZZ[x].get_ring() == ZZ[x]
assert QQ[x].get_ring() == QQ[x]
assert ZZ[x, y].get_ring() == ZZ[x, y]
assert QQ[x, y].get_ring() == QQ[x, y]
assert ZZ.frac_field(x).get_ring() == ZZ[x]
assert QQ.frac_field(x).get_ring() == QQ[x]
assert ZZ.frac_field(x, y).get_ring() == ZZ[x, y]
assert QQ.frac_field(x, y).get_ring() == QQ[x, y]
assert EX.get_ring() == EX
assert RR.get_ring() == RR
# XXX: This should also be like RR
raises(DomainError, lambda: ALG.get_ring())
def test_Domain_get_field():
assert EX.has_assoc_Field is True
assert ZZ.has_assoc_Field is True
assert QQ.has_assoc_Field is True
assert RR.has_assoc_Field is True
assert ALG.has_assoc_Field is True
assert ZZ[x].has_assoc_Field is True
assert QQ[x].has_assoc_Field is True
assert ZZ[x, y].has_assoc_Field is True
assert QQ[x, y].has_assoc_Field is True
assert EX.get_field() == EX
assert ZZ.get_field() == QQ
assert QQ.get_field() == QQ
assert RR.get_field() == RR
assert ALG.get_field() == ALG
assert ZZ[x].get_field() == ZZ.frac_field(x)
assert QQ[x].get_field() == QQ.frac_field(x)
assert ZZ[x, y].get_field() == ZZ.frac_field(x, y)
assert QQ[x, y].get_field() == QQ.frac_field(x, y)
def test_Domain_get_exact():
assert EX.get_exact() == EX
assert ZZ.get_exact() == ZZ
assert QQ.get_exact() == QQ
assert RR.get_exact() == QQ
assert ALG.get_exact() == ALG
assert ZZ[x].get_exact() == ZZ[x]
assert QQ[x].get_exact() == QQ[x]
assert ZZ[x, y].get_exact() == ZZ[x, y]
assert QQ[x, y].get_exact() == QQ[x, y]
assert ZZ.frac_field(x).get_exact() == ZZ.frac_field(x)
assert QQ.frac_field(x).get_exact() == QQ.frac_field(x)
assert ZZ.frac_field(x, y).get_exact() == ZZ.frac_field(x, y)
assert QQ.frac_field(x, y).get_exact() == QQ.frac_field(x, y)
def test_Domain_is_unit():
nums = [-2, -1, 0, 1, 2]
invring = [False, True, False, True, False]
invfield = [True, True, False, True, True]
ZZx, QQx, QQxf = ZZ[x], QQ[x], QQ.frac_field(x)
assert [ZZ.is_unit(ZZ(n)) for n in nums] == invring
assert [QQ.is_unit(QQ(n)) for n in nums] == invfield
assert [ZZx.is_unit(ZZx(n)) for n in nums] == invring
assert [QQx.is_unit(QQx(n)) for n in nums] == invfield
assert [QQxf.is_unit(QQxf(n)) for n in nums] == invfield
assert ZZx.is_unit(ZZx(x)) is False
assert QQx.is_unit(QQx(x)) is False
assert QQxf.is_unit(QQxf(x)) is True
def test_Domain_convert():
def check_element(e1, e2, K1, K2, K3):
assert type(e1) is type(e2), '%s, %s: %s %s -> %s' % (e1, e2, K1, K2, K3)
assert e1 == e2, '%s, %s: %s %s -> %s' % (e1, e2, K1, K2, K3)
def check_domains(K1, K2):
K3 = K1.unify(K2)
check_element(K3.convert_from( K1.one, K1), K3.one, K1, K2, K3)
check_element(K3.convert_from( K2.one, K2), K3.one, K1, K2, K3)
check_element(K3.convert_from(K1.zero, K1), K3.zero, K1, K2, K3)
check_element(K3.convert_from(K2.zero, K2), K3.zero, K1, K2, K3)
def composite_domains(K):
domains = [
K,
K[y], K[z], K[y, z],
K.frac_field(y), K.frac_field(z), K.frac_field(y, z),
# XXX: These should be tested and made to work...
# K.old_poly_ring(y), K.old_frac_field(y),
]
return domains
QQ2 = QQ.algebraic_field(sqrt(2))
QQ3 = QQ.algebraic_field(sqrt(3))
doms = [ZZ, QQ, QQ2, QQ3, QQ_I, ZZ_I, RR, CC]
for i, K1 in enumerate(doms):
for K2 in doms[i:]:
for K3 in composite_domains(K1):
for K4 in composite_domains(K2):
check_domains(K3, K4)
assert QQ.convert(10e-52) == QQ(1684996666696915, 1684996666696914987166688442938726917102321526408785780068975640576)
R, xr = ring("x", ZZ)
assert ZZ.convert(xr - xr) == 0
assert ZZ.convert(xr - xr, R.to_domain()) == 0
assert CC.convert(ZZ_I(1, 2)) == CC(1, 2)
assert CC.convert(QQ_I(1, 2)) == CC(1, 2)
K1 = QQ.frac_field(x)
K2 = ZZ.frac_field(x)
K3 = QQ[x]
K4 = ZZ[x]
Ks = [K1, K2, K3, K4]
for Ka, Kb in product(Ks, Ks):
assert Ka.convert_from(Kb.from_sympy(x), Kb) == Ka.from_sympy(x)
assert K2.convert_from(QQ(1, 2), QQ) == K2(QQ(1, 2))
def test_GlobalPolynomialRing_convert():
K1 = QQ.old_poly_ring(x)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
assert K2.convert(x) == K2.convert(K1.convert(x), K1)
K1 = QQ.old_poly_ring(x, y)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
#assert K2.convert(x) == K2.convert(K1.convert(x), K1)
K1 = ZZ.old_poly_ring(x, y)
K2 = QQ[x]
assert K1.convert(x) == K1.convert(K2.convert(x), K2)
#assert K2.convert(x) == K2.convert(K1.convert(x), K1)
def test_PolynomialRing__init():
R, = ring("", ZZ)
assert ZZ.poly_ring() == R.to_domain()
def test_FractionField__init():
F, = field("", ZZ)
assert ZZ.frac_field() == F.to_domain()
def test_FractionField_convert():
K = QQ.frac_field(x)
assert K.convert(QQ(2, 3), QQ) == K.from_sympy(Rational(2, 3))
K = QQ.frac_field(x)
assert K.convert(ZZ(2), ZZ) == K.from_sympy(Integer(2))
def test_inject():
assert ZZ.inject(x, y, z) == ZZ[x, y, z]
assert ZZ[x].inject(y, z) == ZZ[x, y, z]
assert ZZ.frac_field(x).inject(y, z) == ZZ.frac_field(x, y, z)
raises(GeneratorsError, lambda: ZZ[x].inject(x))
def test_drop():
assert ZZ.drop(x) == ZZ
assert ZZ[x].drop(x) == ZZ
assert ZZ[x, y].drop(x) == ZZ[y]
assert ZZ.frac_field(x).drop(x) == ZZ
assert ZZ.frac_field(x, y).drop(x) == ZZ.frac_field(y)
assert ZZ[x][y].drop(y) == ZZ[x]
assert ZZ[x][y].drop(x) == ZZ[y]
assert ZZ.frac_field(x)[y].drop(x) == ZZ[y]
assert ZZ.frac_field(x)[y].drop(y) == ZZ.frac_field(x)
Ky = FiniteExtension(Poly(x**2-1, x, domain=ZZ[y]))
K = FiniteExtension(Poly(x**2-1, x, domain=ZZ))
assert Ky.drop(y) == K
raises(GeneratorsError, lambda: Ky.drop(x))
def test_Domain_map():
seq = ZZ.map([1, 2, 3, 4])
assert all(ZZ.of_type(elt) for elt in seq)
seq = ZZ.map([[1, 2, 3, 4]])
assert all(ZZ.of_type(elt) for elt in seq[0]) and len(seq) == 1
def test_Domain___eq__():
assert (ZZ[x, y] == ZZ[x, y]) is True
assert (QQ[x, y] == QQ[x, y]) is True
assert (ZZ[x, y] == QQ[x, y]) is False
assert (QQ[x, y] == ZZ[x, y]) is False
assert (ZZ.frac_field(x, y) == ZZ.frac_field(x, y)) is True
assert (QQ.frac_field(x, y) == QQ.frac_field(x, y)) is True
assert (ZZ.frac_field(x, y) == QQ.frac_field(x, y)) is False
assert (QQ.frac_field(x, y) == ZZ.frac_field(x, y)) is False
assert RealField()[x] == RR[x]
def test_Domain__algebraic_field():
alg = ZZ.algebraic_field(sqrt(2))
assert alg.ext.minpoly == Poly(x**2 - 2)
assert alg.dom == QQ
alg = QQ.algebraic_field(sqrt(2))
assert alg.ext.minpoly == Poly(x**2 - 2)
assert alg.dom == QQ
alg = alg.algebraic_field(sqrt(3))
assert alg.ext.minpoly == Poly(x**4 - 10*x**2 + 1)
assert alg.dom == QQ
def test_Domain_alg_field_from_poly():
f = Poly(x**2 - 2)
g = Poly(x**2 - 3)
h = Poly(x**4 - 10*x**2 + 1)
alg = ZZ.alg_field_from_poly(f)
assert alg.ext.minpoly == f
assert alg.dom == QQ
alg = QQ.alg_field_from_poly(f)
assert alg.ext.minpoly == f
assert alg.dom == QQ
alg = alg.alg_field_from_poly(g)
assert alg.ext.minpoly == h
assert alg.dom == QQ
def test_Domain_cyclotomic_field():
K = ZZ.cyclotomic_field(12)
assert K.ext.minpoly == Poly(cyclotomic_poly(12))
assert K.dom == QQ
F = QQ.cyclotomic_field(3)
assert F.ext.minpoly == Poly(cyclotomic_poly(3))
assert F.dom == QQ
E = F.cyclotomic_field(4)
assert field_isomorphism(E.ext, K.ext) is not None
assert E.dom == QQ
def test_PolynomialRing_from_FractionField():
F, x,y = field("x,y", ZZ)
R, X,Y = ring("x,y", ZZ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
F, x,y = field("x,y", QQ)
R, X,Y = ring("x,y", QQ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
def test_FractionField_from_PolynomialRing():
R, x,y = ring("x,y", QQ)
F, X,Y = field("x,y", ZZ)
f = 3*x**2 + 5*y**2
g = x**2/3 + y**2/5
assert F.to_domain().from_PolynomialRing(f, R.to_domain()) == 3*X**2 + 5*Y**2
assert F.to_domain().from_PolynomialRing(g, R.to_domain()) == (5*X**2 + 3*Y**2)/15
def test_FF_of_type():
assert FF(3).of_type(FF(3)(1)) is True
assert FF(5).of_type(FF(5)(3)) is True
assert FF(5).of_type(FF(7)(3)) is False
def test___eq__():
assert not QQ[x] == ZZ[x]
assert not QQ.frac_field(x) == ZZ.frac_field(x)
def test_RealField_from_sympy():
assert RR.convert(S.Zero) == RR.dtype(0)
assert RR.convert(S(0.0)) == RR.dtype(0.0)
assert RR.convert(S.One) == RR.dtype(1)
assert RR.convert(S(1.0)) == RR.dtype(1.0)
assert RR.convert(sin(1)) == RR.dtype(sin(1).evalf())
def test_not_in_any_domain():
check = list(_illegal) + [x] + [
float(i) for i in _illegal[:3]]
for dom in (ZZ, QQ, RR, CC, EX):
for i in check:
if i == x and dom == EX:
continue
assert i not in dom, (i, dom)
raises(CoercionFailed, lambda: dom.convert(i))
def test_ModularInteger():
F3 = FF(3)
a = F3(0)
assert isinstance(a, F3.dtype) and a == 0
a = F3(1)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)
assert isinstance(a, F3.dtype) and a == 2
a = F3(3)
assert isinstance(a, F3.dtype) and a == 0
a = F3(4)
assert isinstance(a, F3.dtype) and a == 1
a = F3(F3(0))
assert isinstance(a, F3.dtype) and a == 0
a = F3(F3(1))
assert isinstance(a, F3.dtype) and a == 1
a = F3(F3(2))
assert isinstance(a, F3.dtype) and a == 2
a = F3(F3(3))
assert isinstance(a, F3.dtype) and a == 0
a = F3(F3(4))
assert isinstance(a, F3.dtype) and a == 1
a = -F3(1)
assert isinstance(a, F3.dtype) and a == 2
a = -F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2 + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2) + F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 3 - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(3) - F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)*F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 2/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/2
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)/F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = 1 % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % 2
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(1) % F3(2)
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)**0
assert isinstance(a, F3.dtype) and a == 1
a = F3(2)**1
assert isinstance(a, F3.dtype) and a == 2
a = F3(2)**2
assert isinstance(a, F3.dtype) and a == 1
F7 = FF(7)
a = F7(3)**100000000000
assert isinstance(a, F7.dtype) and a == 4
a = F7(3)**-100000000000
assert isinstance(a, F7.dtype) and a == 2
a = F7(3)**S(2)
assert isinstance(a, F7.dtype) and a == 2
assert bool(F3(3)) is False
assert bool(F3(4)) is True
F5 = FF(5)
a = F5(1)**(-1)
assert isinstance(a, F5.dtype) and a == 1
a = F5(2)**(-1)
assert isinstance(a, F5.dtype) and a == 3
a = F5(3)**(-1)
assert isinstance(a, F5.dtype) and a == 2
a = F5(4)**(-1)
assert isinstance(a, F5.dtype) and a == 4
assert (F5(1) < F5(2)) is True
assert (F5(1) <= F5(2)) is True
assert (F5(1) > F5(2)) is False
assert (F5(1) >= F5(2)) is False
assert (F5(3) < F5(2)) is False
assert (F5(3) <= F5(2)) is False
assert (F5(3) > F5(2)) is True
assert (F5(3) >= F5(2)) is True
assert (F5(1) < F5(7)) is True
assert (F5(1) <= F5(7)) is True
assert (F5(1) > F5(7)) is False
assert (F5(1) >= F5(7)) is False
assert (F5(3) < F5(7)) is False
assert (F5(3) <= F5(7)) is False
assert (F5(3) > F5(7)) is True
assert (F5(3) >= F5(7)) is True
assert (F5(1) < 2) is True
assert (F5(1) <= 2) is True
assert (F5(1) > 2) is False
assert (F5(1) >= 2) is False
assert (F5(3) < 2) is False
assert (F5(3) <= 2) is False
assert (F5(3) > 2) is True
assert (F5(3) >= 2) is True
assert (F5(1) < 7) is True
assert (F5(1) <= 7) is True
assert (F5(1) > 7) is False
assert (F5(1) >= 7) is False
assert (F5(3) < 7) is False
assert (F5(3) <= 7) is False
assert (F5(3) > 7) is True
assert (F5(3) >= 7) is True
raises(NotInvertible, lambda: F5(0)**(-1))
raises(NotInvertible, lambda: F5(5)**(-1))
raises(ValueError, lambda: FF(0))
raises(ValueError, lambda: FF(2.1))
def test_QQ_int():
assert int(QQ(2**2000, 3**1250)) == 455431
assert int(QQ(2**100, 3)) == 422550200076076467165567735125
def test_RR_double():
assert RR(3.14) > 1e-50
assert RR(1e-13) > 1e-50
assert RR(1e-14) > 1e-50
assert RR(1e-15) > 1e-50
assert RR(1e-20) > 1e-50
assert RR(1e-40) > 1e-50
def test_RR_Float():
f1 = Float("1.01")
f2 = Float("1.0000000000000000000001")
assert f1._prec == 53
assert f2._prec == 80
assert RR(f1)-1 > 1e-50
assert RR(f2)-1 < 1e-50 # RR's precision is lower than f2's
RR2 = RealField(prec=f2._prec)
assert RR2(f1)-1 > 1e-50
assert RR2(f2)-1 > 1e-50 # RR's precision is equal to f2's
def test_CC_double():
assert CC(3.14).real > 1e-50
assert CC(1e-13).real > 1e-50
assert CC(1e-14).real > 1e-50
assert CC(1e-15).real > 1e-50
assert CC(1e-20).real > 1e-50
assert CC(1e-40).real > 1e-50
assert CC(3.14j).imag > 1e-50
assert CC(1e-13j).imag > 1e-50
assert CC(1e-14j).imag > 1e-50
assert CC(1e-15j).imag > 1e-50
assert CC(1e-20j).imag > 1e-50
assert CC(1e-40j).imag > 1e-50
def test_gaussian_domains():
I = S.ImaginaryUnit
a, b, c, d = [ZZ_I.convert(x) for x in (5, 2 + I, 3 - I, 5 - 5*I)]
assert ZZ_I.gcd(a, b) == b
assert ZZ_I.gcd(a, c) == b
assert ZZ_I.lcm(a, b) == a
assert ZZ_I.lcm(a, c) == d
assert ZZ_I(3, 4) != QQ_I(3, 4) # XXX is this right or should QQ->ZZ if possible?
assert ZZ_I(3, 0) != 3 # and should this go to Integer?
assert QQ_I(S(3)/4, 0) != S(3)/4 # and this to Rational?
assert ZZ_I(0, 0).quadrant() == 0
assert ZZ_I(-1, 0).quadrant() == 2
assert QQ_I.convert(QQ(3, 2)) == QQ_I(QQ(3, 2), QQ(0))
assert QQ_I.convert(QQ(3, 2), QQ) == QQ_I(QQ(3, 2), QQ(0))
for G in (QQ_I, ZZ_I):
q = G(3, 4)
assert str(q) == '3 + 4*I'
assert q.parent() == G
assert q._get_xy(pi) == (None, None)
assert q._get_xy(2) == (2, 0)
assert q._get_xy(2*I) == (0, 2)
assert hash(q) == hash((3, 4))
assert G(1, 2) == G(1, 2)
assert G(1, 2) != G(1, 3)
assert G(3, 0) == G(3)
assert q + q == G(6, 8)
assert q - q == G(0, 0)
assert 3 - q == -q + 3 == G(0, -4)
assert 3 + q == q + 3 == G(6, 4)
assert q * q == G(-7, 24)
assert 3 * q == q * 3 == G(9, 12)
assert q ** 0 == G(1, 0)
assert q ** 1 == q
assert q ** 2 == q * q == G(-7, 24)
assert q ** 3 == q * q * q == G(-117, 44)
assert 1 / q == q ** -1 == QQ_I(S(3)/25, - S(4)/25)
assert q / 1 == QQ_I(3, 4)
assert q / 2 == QQ_I(S(3)/2, 2)
assert q/3 == QQ_I(1, S(4)/3)
assert 3/q == QQ_I(S(9)/25, -S(12)/25)
i, r = divmod(q, 2)
assert 2*i + r == q
i, r = divmod(2, q)
assert q*i + r == G(2, 0)
raises(ZeroDivisionError, lambda: q % 0)
raises(ZeroDivisionError, lambda: q / 0)
raises(ZeroDivisionError, lambda: q // 0)
raises(ZeroDivisionError, lambda: divmod(q, 0))
raises(ZeroDivisionError, lambda: divmod(q, 0))
raises(TypeError, lambda: q + x)
raises(TypeError, lambda: q - x)
raises(TypeError, lambda: x + q)
raises(TypeError, lambda: x - q)
raises(TypeError, lambda: q * x)
raises(TypeError, lambda: x * q)
raises(TypeError, lambda: q / x)
raises(TypeError, lambda: x / q)
raises(TypeError, lambda: q // x)
raises(TypeError, lambda: x // q)
assert G.from_sympy(S(2)) == G(2, 0)
assert G.to_sympy(G(2, 0)) == S(2)
raises(CoercionFailed, lambda: G.from_sympy(pi))
PR = G.inject(x)
assert isinstance(PR, PolynomialRing)
assert PR.domain == G
assert len(PR.gens) == 1 and PR.gens[0].as_expr() == x
if G is QQ_I:
AF = G.as_AlgebraicField()
assert isinstance(AF, AlgebraicField)
assert AF.domain == QQ
assert AF.ext.args[0] == I
for qi in [G(-1, 0), G(1, 0), G(0, -1), G(0, 1)]:
assert G.is_negative(qi) is False
assert G.is_positive(qi) is False
assert G.is_nonnegative(qi) is False
assert G.is_nonpositive(qi) is False
domains = [ZZ_python(), QQ_python(), AlgebraicField(QQ, I)]
if HAS_GMPY:
domains += [ZZ_gmpy(), QQ_gmpy()]
for K in domains:
assert G.convert(K(2)) == G(2, 0)
assert G.convert(K(2), K) == G(2, 0)
for K in ZZ_I, QQ_I:
assert G.convert(K(1, 1)) == G(1, 1)
assert G.convert(K(1, 1), K) == G(1, 1)
if G == ZZ_I:
assert repr(q) == 'ZZ_I(3, 4)'
assert q//3 == G(1, 1)
assert 12//q == G(1, -2)
assert 12 % q == G(1, 2)
assert q % 2 == G(-1, 0)
assert i == G(0, 0)
assert r == G(2, 0)
assert G.get_ring() == G
assert G.get_field() == QQ_I
else:
assert repr(q) == 'QQ_I(3, 4)'
assert G.get_ring() == ZZ_I
assert G.get_field() == G
assert q//3 == G(1, S(4)/3)
assert 12//q == G(S(36)/25, -S(48)/25)
assert 12 % q == G(0, 0)
assert q % 2 == G(0, 0)
assert i == G(S(6)/25, -S(8)/25), (G,i)
assert r == G(0, 0)
q2 = G(S(3)/2, S(5)/3)
assert G.numer(q2) == ZZ_I(9, 10)
assert G.denom(q2) == ZZ_I(6)
def test_EX_EXRAW():
assert EXRAW.zero is S.Zero
assert EXRAW.one is S.One
assert EX(1) == EX.Expression(1)
assert EX(1).ex is S.One
assert EXRAW(1) is S.One
# EX has cancelling but EXRAW does not
assert 2*EX((x + y*x)/x) == EX(2 + 2*y) != 2*((x + y*x)/x)
assert 2*EXRAW((x + y*x)/x) == 2*((x + y*x)/x) != (1 + y)
assert EXRAW.convert_from(EX(1), EX) is EXRAW.one
assert EX.convert_from(EXRAW(1), EXRAW) == EX.one
assert EXRAW.from_sympy(S.One) is S.One
assert EXRAW.to_sympy(EXRAW.one) is S.One
raises(CoercionFailed, lambda: EXRAW.from_sympy([]))
assert EXRAW.get_field() == EXRAW
assert EXRAW.unify(EX) == EXRAW
assert EX.unify(EXRAW) == EXRAW
def test_canonical_unit():
for K in [ZZ, QQ, RR]: # CC?
assert K.canonical_unit(K(2)) == K(1)
assert K.canonical_unit(K(-2)) == K(-1)
for K in [ZZ_I, QQ_I]:
i = K.from_sympy(I)
assert K.canonical_unit(K(2)) == K(1)
assert K.canonical_unit(K(2)*i) == -i
assert K.canonical_unit(-K(2)) == K(-1)
assert K.canonical_unit(-K(2)*i) == i
K = ZZ[x]
assert K.canonical_unit(K(x + 1)) == K(1)
assert K.canonical_unit(K(-x + 1)) == K(-1)
K = ZZ_I[x]
assert K.canonical_unit(K.from_sympy(I*x)) == ZZ_I(0, -1)
K = ZZ_I.frac_field(x, y)
i = K.from_sympy(I)
assert i / i == K.one
assert (K.one + i)/(i - K.one) == -i
def test_issue_18278():
assert str(RR(2).parent()) == 'RR'
assert str(CC(2).parent()) == 'CC'
def test_Domain_is_negative():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_negative(a) == False
assert CC.is_negative(b) == False
def test_Domain_is_positive():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_positive(a) == False
assert CC.is_positive(b) == False
def test_Domain_is_nonnegative():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_nonnegative(a) == False
assert CC.is_nonnegative(b) == False
def test_Domain_is_nonpositive():
I = S.ImaginaryUnit
a, b = [CC.convert(x) for x in (2 + I, 5)]
assert CC.is_nonpositive(a) == False
assert CC.is_nonpositive(b) == False
def test_exponential_domain():
K = ZZ[E]
eK = K.from_sympy(E)
assert K.from_sympy(exp(3)) == eK ** 3
assert K.convert(exp(3)) == eK ** 3
|
f07306c7604d9667327bd032c812204c1dc04479e8c6e0ecf48c54f7f536fcd0 | from sympy.abc import x
from sympy.core import S
from sympy.core.numbers import AlgebraicNumber
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.polys import Poly, cyclotomic_poly
from sympy.polys.domains import QQ
from sympy.polys.matrices import DomainMatrix, DM
from sympy.polys.numberfields.basis import round_two
from sympy.testing.pytest import raises
def test_round_two():
# Poly must be monic, irreducible, and over ZZ:
raises(ValueError, lambda: round_two(Poly(3 * x ** 2 + 1)))
raises(ValueError, lambda: round_two(Poly(x ** 2 - 1)))
raises(ValueError, lambda: round_two(Poly(x ** 2 + QQ(1, 2))))
# Test on many fields:
cases = (
# A couple of cyclotomic fields:
(cyclotomic_poly(5), DomainMatrix.eye(4, QQ), 125),
(cyclotomic_poly(7), DomainMatrix.eye(6, QQ), -16807),
# A couple of quadratic fields (one 1 mod 4, one 3 mod 4):
(x ** 2 - 5, DM([[1, (1, 2)], [0, (1, 2)]], QQ), 5),
(x ** 2 - 7, DM([[1, 0], [0, 1]], QQ), 28),
# Dedekind's example of a field with 2 as essential disc divisor:
(x ** 3 + x ** 2 - 2 * x + 8, DM([[1, 0, 0], [0, 1, 0], [0, (1, 2), (1, 2)]], QQ).transpose(), -503),
# A bunch of cubics with various forms for F -- all of these require
# second or third enlargements. (Five of them require a third, while the rest require just a second.)
# F = 2^2
(x**3 + 3 * x**2 - 4 * x + 4, DM([((1, 2), (1, 4), (1, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -83),
# F = 2^2 * 3
(x**3 + 3 * x**2 + 3 * x - 3, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -108),
# F = 2^3
(x**3 + 5 * x**2 - x + 3, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -31),
# F = 2^2 * 5
(x**3 + 5 * x**2 - 5 * x - 5, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 1300),
# F = 3^2
(x**3 + 3 * x**2 + 5, DM([((1, 3), (1, 3), (1, 3)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -135),
# F = 3^3
(x**3 + 6 * x**2 + 3 * x - 1, DM([((1, 3), (1, 3), (1, 3)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 81),
# F = 2^2 * 3^2
(x**3 + 6 * x**2 + 4, DM([((1, 3), (2, 3), (1, 3)), (0, 1, 0), (0, 0, (1, 2))], QQ).transpose(), -108),
# F = 2^3 * 7
(x**3 + 7 * x**2 + 7 * x - 7, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), 49),
# F = 2^2 * 13
(x**3 + 7 * x**2 - x + 5, DM([((1, 2), 0, (1, 2)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -2028),
# F = 2^4
(x**3 + 7 * x**2 - 5 * x + 5, DM([((1, 4), 0, (3, 4)), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), -140),
# F = 5^2
(x**3 + 4 * x**2 - 3 * x + 7, DM([((1, 5), (4, 5), (4, 5)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -175),
# F = 7^2
(x**3 + 8 * x**2 + 5 * x - 1, DM([((1, 7), (6, 7), (2, 7)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), 49),
# F = 2 * 5 * 7
(x**3 + 8 * x**2 - 2 * x + 6, DM([(1, 0, 0), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -14700),
# F = 2^2 * 3 * 5
(x**3 + 6 * x**2 - 3 * x + 8, DM([(1, 0, 0), (0, (1, 4), (1, 4)), (0, 0, 1)], QQ).transpose(), -675),
# F = 2 * 3^2 * 7
(x**3 + 9 * x**2 + 6 * x - 8, DM([(1, 0, 0), (0, (1, 2), (1, 2)), (0, 0, 1)], QQ).transpose(), 3969),
# F = 2^2 * 3^2 * 7
(x**3 + 15 * x**2 - 9 * x + 13, DM([((1, 6), (1, 3), (1, 6)), (0, 1, 0), (0, 0, 1)], QQ).transpose(), -5292),
)
for f, B_exp, d_exp in cases:
K = QQ.alg_field_from_poly(f)
B = K.maximal_order().QQ_matrix
d = K.discriminant()
assert d == d_exp
# The computed basis need not equal the expected one, but their quotient
# must be unimodular:
assert (B.inv()*B_exp).det()**2 == 1
def test_AlgebraicField_integral_basis():
alpha = AlgebraicNumber(sqrt(5), alias='alpha')
k = QQ.algebraic_field(alpha)
B0 = k.integral_basis()
B1 = k.integral_basis(fmt='sympy')
B2 = k.integral_basis(fmt='alg')
assert B0 == [k([1]), k([S.Half, S.Half])]
assert B1 == [1, S.Half + alpha/2]
assert B2 == [alpha.field_element([1]),
alpha.field_element([S.Half, S.Half])]
|
fda09bbee92b6a1bfe4320b0353ec4a53602d6ee77c9494c85f1aa5791b35b76 | """Tests for the subfield problem and allied problems. """
from sympy.core.numbers import (AlgebraicNumber, I, pi, Rational)
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.external.gmpy import MPQ
from sympy.polys.numberfields.subfield import (
is_isomorphism_possible,
field_isomorphism_pslq,
field_isomorphism,
primitive_element,
to_number_field,
)
from sympy.polys.polyerrors import IsomorphismFailed
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.testing.pytest import raises
from sympy.abc import x
Q = Rational
def test_field_isomorphism_pslq():
a = AlgebraicNumber(I)
b = AlgebraicNumber(I*sqrt(3))
raises(NotImplementedError, lambda: field_isomorphism_pslq(a, b))
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
d = AlgebraicNumber(sqrt(2) + sqrt(3))
e = AlgebraicNumber(sqrt(2) + sqrt(3) + sqrt(7))
assert field_isomorphism_pslq(a, a) == [1, 0]
assert field_isomorphism_pslq(a, b) is None
assert field_isomorphism_pslq(a, c) is None
assert field_isomorphism_pslq(a, d) == [Q(1, 2), 0, -Q(9, 2), 0]
assert field_isomorphism_pslq(
a, e) == [Q(1, 80), 0, -Q(1, 2), 0, Q(59, 20), 0]
assert field_isomorphism_pslq(b, a) is None
assert field_isomorphism_pslq(b, b) == [1, 0]
assert field_isomorphism_pslq(b, c) is None
assert field_isomorphism_pslq(b, d) == [-Q(1, 2), 0, Q(11, 2), 0]
assert field_isomorphism_pslq(b, e) == [-Q(
3, 640), 0, Q(67, 320), 0, -Q(297, 160), 0, Q(313, 80), 0]
assert field_isomorphism_pslq(c, a) is None
assert field_isomorphism_pslq(c, b) is None
assert field_isomorphism_pslq(c, c) == [1, 0]
assert field_isomorphism_pslq(c, d) is None
assert field_isomorphism_pslq(c, e) == [Q(
3, 640), 0, -Q(71, 320), 0, Q(377, 160), 0, -Q(469, 80), 0]
assert field_isomorphism_pslq(d, a) is None
assert field_isomorphism_pslq(d, b) is None
assert field_isomorphism_pslq(d, c) is None
assert field_isomorphism_pslq(d, d) == [1, 0]
assert field_isomorphism_pslq(d, e) == [-Q(
3, 640), 0, Q(71, 320), 0, -Q(377, 160), 0, Q(549, 80), 0]
assert field_isomorphism_pslq(e, a) is None
assert field_isomorphism_pslq(e, b) is None
assert field_isomorphism_pslq(e, c) is None
assert field_isomorphism_pslq(e, d) is None
assert field_isomorphism_pslq(e, e) == [1, 0]
f = AlgebraicNumber(3*sqrt(2) + 8*sqrt(7) - 5)
assert field_isomorphism_pslq(
f, e) == [Q(3, 80), 0, -Q(139, 80), 0, Q(347, 20), 0, -Q(761, 20), -5]
def test_field_isomorphism():
assert field_isomorphism(3, sqrt(2)) == [3]
assert field_isomorphism( I*sqrt(3), I*sqrt(3)/2) == [ 2, 0]
assert field_isomorphism(-I*sqrt(3), I*sqrt(3)/2) == [-2, 0]
assert field_isomorphism( I*sqrt(3), -I*sqrt(3)/2) == [-2, 0]
assert field_isomorphism(-I*sqrt(3), -I*sqrt(3)/2) == [ 2, 0]
assert field_isomorphism( 2*I*sqrt(3)/7, 5*I*sqrt(3)/3) == [ Rational(6, 35), 0]
assert field_isomorphism(-2*I*sqrt(3)/7, 5*I*sqrt(3)/3) == [Rational(-6, 35), 0]
assert field_isomorphism( 2*I*sqrt(3)/7, -5*I*sqrt(3)/3) == [Rational(-6, 35), 0]
assert field_isomorphism(-2*I*sqrt(3)/7, -5*I*sqrt(3)/3) == [ Rational(6, 35), 0]
assert field_isomorphism(
2*I*sqrt(3)/7 + 27, 5*I*sqrt(3)/3) == [ Rational(6, 35), 27]
assert field_isomorphism(
-2*I*sqrt(3)/7 + 27, 5*I*sqrt(3)/3) == [Rational(-6, 35), 27]
assert field_isomorphism(
2*I*sqrt(3)/7 + 27, -5*I*sqrt(3)/3) == [Rational(-6, 35), 27]
assert field_isomorphism(
-2*I*sqrt(3)/7 + 27, -5*I*sqrt(3)/3) == [ Rational(6, 35), 27]
p = AlgebraicNumber( sqrt(2) + sqrt(3))
q = AlgebraicNumber(-sqrt(2) + sqrt(3))
r = AlgebraicNumber( sqrt(2) - sqrt(3))
s = AlgebraicNumber(-sqrt(2) - sqrt(3))
pos_coeffs = [ S.Half, S.Zero, Rational(-9, 2), S.Zero]
neg_coeffs = [Rational(-1, 2), S.Zero, Rational(9, 2), S.Zero]
a = AlgebraicNumber(sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
a = AlgebraicNumber(-sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
pos_coeffs = [ S.Half, S.Zero, Rational(-11, 2), S.Zero]
neg_coeffs = [Rational(-1, 2), S.Zero, Rational(11, 2), S.Zero]
a = AlgebraicNumber(sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(-sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
pos_coeffs = [ Rational(3, 2), S.Zero, Rational(-33, 2), -S(8)]
neg_coeffs = [Rational(-3, 2), S.Zero, Rational(33, 2), -S(8)]
a = AlgebraicNumber(3*sqrt(3) - 8)
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = AlgebraicNumber(3*sqrt(2) + 2*sqrt(3) + 1)
pos_1_coeffs = [ S.Half, S.Zero, Rational(-5, 2), S.One]
neg_5_coeffs = [Rational(-5, 2), S.Zero, Rational(49, 2), S.One]
pos_5_coeffs = [ Rational(5, 2), S.Zero, Rational(-49, 2), S.One]
neg_1_coeffs = [Rational(-1, 2), S.Zero, Rational(5, 2), S.One]
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
c = AlgebraicNumber(sqrt(7))
assert is_isomorphism_possible(a, b) is True
assert is_isomorphism_possible(b, a) is True
assert is_isomorphism_possible(c, p) is False
assert field_isomorphism(sqrt(2), sqrt(3), fast=True) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=True) is None
assert field_isomorphism(sqrt(2), sqrt(3), fast=False) is None
assert field_isomorphism(sqrt(3), sqrt(2), fast=False) is None
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(2 ** (S(1) / 3))
assert is_isomorphism_possible(a, b) is False
assert field_isomorphism(a, b) is None
def test_primitive_element():
assert primitive_element([sqrt(2)], x) == (x**2 - 2, [1])
assert primitive_element(
[sqrt(2), sqrt(3)], x) == (x**4 - 10*x**2 + 1, [1, 1])
assert primitive_element([sqrt(2)], x, polys=True) == (Poly(x**2 - 2, domain='QQ'), [1])
assert primitive_element([sqrt(
2), sqrt(3)], x, polys=True) == (Poly(x**4 - 10*x**2 + 1, domain='QQ'), [1, 1])
assert primitive_element(
[sqrt(2)], x, ex=True) == (x**2 - 2, [1], [[1, 0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True) == \
(x**4 - 10*x**2 + 1, [1, 1], [[Q(1, 2), 0, -Q(9, 2), 0], [-
Q(1, 2), 0, Q(11, 2), 0]])
assert primitive_element(
[sqrt(2)], x, ex=True, polys=True) == (Poly(x**2 - 2, domain='QQ'), [1], [[1, 0]])
assert primitive_element([sqrt(2), sqrt(3)], x, ex=True, polys=True) == \
(Poly(x**4 - 10*x**2 + 1, domain='QQ'), [1, 1], [[Q(1, 2), 0, -Q(9, 2),
0], [-Q(1, 2), 0, Q(11, 2), 0]])
assert primitive_element([sqrt(2)], polys=True) == (Poly(x**2 - 2), [1])
raises(ValueError, lambda: primitive_element([], x, ex=False))
raises(ValueError, lambda: primitive_element([], x, ex=True))
# Issue 14117
a, b = I*sqrt(2*sqrt(2) + 3), I*sqrt(-2*sqrt(2) + 3)
assert primitive_element([a, b, I], x) == (x**4 + 6*x**2 + 1, [1, 0, 0])
assert primitive_element([sqrt(2), 0], x) == (x**2 - 2, [1, 0])
assert primitive_element([0, sqrt(2)], x) == (x**2 - 2, [1, 1])
assert primitive_element([sqrt(2), 0], x, ex=True) == (x**2 - 2, [1, 0], [[MPQ(1,1), MPQ(0,1)], []])
assert primitive_element([0, sqrt(2)], x, ex=True) == (x**2 - 2, [1, 1], [[], [MPQ(1,1), MPQ(0,1)]])
def test_to_number_field():
assert to_number_field(sqrt(2)) == AlgebraicNumber(sqrt(2))
assert to_number_field(
[sqrt(2), sqrt(3)]) == AlgebraicNumber(sqrt(2) + sqrt(3))
a = AlgebraicNumber(sqrt(2) + sqrt(3), [S.Half, S.Zero, Rational(-9, 2), S.Zero])
assert to_number_field(sqrt(2), sqrt(2) + sqrt(3)) == a
assert to_number_field(sqrt(2), AlgebraicNumber(sqrt(2) + sqrt(3))) == a
raises(IsomorphismFailed, lambda: to_number_field(sqrt(2), sqrt(3)))
def test_issue_22561():
a = to_number_field(sqrt(2), sqrt(2) + sqrt(3))
b = to_number_field(sqrt(2), sqrt(2) + sqrt(5))
assert field_isomorphism(a, b) == [1, 0]
def test_issue_22736():
a = CRootOf(x**4 + x**3 + x**2 + x + 1, -1)
a._reset()
b = exp(2*I*pi/5)
assert field_isomorphism(a, b) == [1, 0]
|
8862b8750a713cd1c90234412f963c97b54a1a8aebf6ea3a18e3b9e9bdee8a5e | from sympy import QQ, ZZ, S
from sympy.abc import x, theta
from sympy.core.mul import prod
from sympy.ntheory import factorint
from sympy.ntheory.residue_ntheory import n_order
from sympy.polys import Poly, cyclotomic_poly
from sympy.polys.matrices import DomainMatrix
from sympy.polys.numberfields.basis import round_two
from sympy.polys.numberfields.exceptions import StructureError
from sympy.polys.numberfields.modules import PowerBasis
from sympy.polys.numberfields.primes import (
prime_decomp, _two_elt_rep,
_check_formal_conditions_for_maximal_order,
)
from sympy.polys.polyerrors import GeneratorsNeeded
from sympy.testing.pytest import raises
def test_check_formal_conditions_for_maximal_order():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
D = A.submodule_from_matrix(DomainMatrix.eye(4, ZZ)[:, :-1])
# Is a direct submodule of a power basis, but lacks 1 as first generator:
raises(StructureError, lambda: _check_formal_conditions_for_maximal_order(B))
# Is not a direct submodule of a power basis:
raises(StructureError, lambda: _check_formal_conditions_for_maximal_order(C))
# Is direct submod of pow basis, and starts with 1, but not sq/max rank/HNF:
raises(StructureError, lambda: _check_formal_conditions_for_maximal_order(D))
def test_two_elt_rep():
ell = 7
T = Poly(cyclotomic_poly(ell))
ZK, dK = round_two(T)
for p in [29, 13, 11, 5]:
P = prime_decomp(p, T)
for Pi in P:
# We have Pi in two-element representation, and, because we are
# looking at a cyclotomic field, this was computed by the "easy"
# method that just factors T mod p. We will now convert this to
# a set of Z-generators, then convert that back into a two-element
# rep. The latter need not be identical to the two-elt rep we
# already have, but it must have the same HNF.
H = p*ZK + Pi.alpha*ZK
gens = H.basis_element_pullbacks()
# Note: we could supply f = Pi.f, but prefer to test behavior without it.
b = _two_elt_rep(gens, ZK, p)
if b != Pi.alpha:
H2 = p*ZK + b*ZK
assert H2 == H
def test_valuation_at_prime_ideal():
p = 7
T = Poly(cyclotomic_poly(p))
ZK, dK = round_two(T)
P = prime_decomp(p, T, dK=dK, ZK=ZK)
assert len(P) == 1
P0 = P[0]
v = P0.valuation(p*ZK)
assert v == P0.e
# Test easy 0 case:
assert P0.valuation(5*ZK) == 0
def test_decomp_1():
# All prime decompositions in cyclotomic fields are in the "easy case,"
# since the index is unity.
# Here we check the ramified prime.
T = Poly(cyclotomic_poly(7))
raises(ValueError, lambda: prime_decomp(7))
P = prime_decomp(7, T)
assert len(P) == 1
P0 = P[0]
assert P0.e == 6
assert P0.f == 1
# Test powers:
assert P0**0 == P0.ZK
assert P0**1 == P0
assert P0**6 == 7 * P0.ZK
def test_decomp_2():
# More easy cyclotomic cases, but here we check unramified primes.
ell = 7
T = Poly(cyclotomic_poly(ell))
for p in [29, 13, 11, 5]:
f_exp = n_order(p, ell)
g_exp = (ell - 1) // f_exp
P = prime_decomp(p, T)
assert len(P) == g_exp
for Pi in P:
assert Pi.e == 1
assert Pi.f == f_exp
def test_decomp_3():
T = Poly(x ** 2 - 35)
rad = {}
ZK, dK = round_two(T, radicals=rad)
# 35 is 3 mod 4, so field disc is 4*5*7, and theory says each of the
# rational primes 2, 5, 7 should be the square of a prime ideal.
for p in [2, 5, 7]:
P = prime_decomp(p, T, dK=dK, ZK=ZK, radical=rad.get(p))
assert len(P) == 1
assert P[0].e == 2
assert P[0]**2 == p*ZK
def test_decomp_4():
T = Poly(x ** 2 - 21)
rad = {}
ZK, dK = round_two(T, radicals=rad)
# 21 is 1 mod 4, so field disc is 3*7, and theory says the
# rational primes 3, 7 should be the square of a prime ideal.
for p in [3, 7]:
P = prime_decomp(p, T, dK=dK, ZK=ZK, radical=rad.get(p))
assert len(P) == 1
assert P[0].e == 2
assert P[0]**2 == p*ZK
def test_decomp_5():
# Here is our first test of the "hard case" of prime decomposition.
# We work in a quadratic extension Q(sqrt(d)) where d is 1 mod 4, and
# we consider the factorization of the rational prime 2, which divides
# the index.
# Theory says the form of p's factorization depends on the residue of
# d mod 8, so we consider both cases, d = 1 mod 8 and d = 5 mod 8.
for d in [-7, -3]:
T = Poly(x ** 2 - d)
rad = {}
ZK, dK = round_two(T, radicals=rad)
p = 2
P = prime_decomp(p, T, dK=dK, ZK=ZK, radical=rad.get(p))
if d % 8 == 1:
assert len(P) == 2
assert all(P[i].e == 1 and P[i].f == 1 for i in range(2))
assert prod(Pi**Pi.e for Pi in P) == p * ZK
else:
assert d % 8 == 5
assert len(P) == 1
assert P[0].e == 1
assert P[0].f == 2
assert P[0].as_submodule() == p * ZK
def test_decomp_6():
# Another case where 2 divides the index. This is Dedekind's example of
# an essential discriminant divisor. (See Cohen, Excercise 6.10.)
T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
rad = {}
ZK, dK = round_two(T, radicals=rad)
p = 2
P = prime_decomp(p, T, dK=dK, ZK=ZK, radical=rad.get(p))
assert len(P) == 3
assert all(Pi.e == Pi.f == 1 for Pi in P)
assert prod(Pi**Pi.e for Pi in P) == p*ZK
def test_decomp_7():
# Try working through an AlgebraicField
T = Poly(x ** 3 + x ** 2 - 2 * x + 8)
K = QQ.alg_field_from_poly(T)
p = 2
P = K.primes_above(p)
ZK = K.maximal_order()
assert len(P) == 3
assert all(Pi.e == Pi.f == 1 for Pi in P)
assert prod(Pi**Pi.e for Pi in P) == p*ZK
def test_decomp_8():
# This time we consider various cubics, and try factoring all primes
# dividing the index.
cases = (
x ** 3 + 3 * x ** 2 - 4 * x + 4,
x ** 3 + 3 * x ** 2 + 3 * x - 3,
x ** 3 + 5 * x ** 2 - x + 3,
x ** 3 + 5 * x ** 2 - 5 * x - 5,
x ** 3 + 3 * x ** 2 + 5,
x ** 3 + 6 * x ** 2 + 3 * x - 1,
x ** 3 + 6 * x ** 2 + 4,
x ** 3 + 7 * x ** 2 + 7 * x - 7,
x ** 3 + 7 * x ** 2 - x + 5,
x ** 3 + 7 * x ** 2 - 5 * x + 5,
x ** 3 + 4 * x ** 2 - 3 * x + 7,
x ** 3 + 8 * x ** 2 + 5 * x - 1,
x ** 3 + 8 * x ** 2 - 2 * x + 6,
x ** 3 + 6 * x ** 2 - 3 * x + 8,
x ** 3 + 9 * x ** 2 + 6 * x - 8,
x ** 3 + 15 * x ** 2 - 9 * x + 13,
)
'''
def display(T, p, radical, P, I, J):
"""Useful for inspection, when running test manually."""
print('=' * 20)
print(T, p, radical)
for Pi in P:
print(f' ({Pi.pretty()})')
print("I: ", I)
print("J: ", J)
print(f'Equal: {I == J}')
'''
for g in cases:
T = Poly(g)
rad = {}
ZK, dK = round_two(T, radicals=rad)
dT = T.discriminant()
f_squared = dT // dK
F = factorint(f_squared)
for p in F:
radical = rad.get(p)
P = prime_decomp(p, T, dK=dK, ZK=ZK, radical=radical)
I = prod(Pi**Pi.e for Pi in P)
J = p * ZK
#display(T, p, radical, P, I, J)
assert I == J
def test_PrimeIdeal_eq():
# `==` should fail on objects of different types, so even a completely
# inert PrimeIdeal should test unequal to the rational prime it divides.
T = Poly(cyclotomic_poly(7))
P0 = prime_decomp(5, T)[0]
assert P0.f == 6
assert P0.as_submodule() == 5 * P0.ZK
assert P0 != 5
def test_PrimeIdeal_add():
T = Poly(cyclotomic_poly(7))
P0 = prime_decomp(7, T)[0]
# Adding ideals computes their GCD, so adding the ramified prime dividing
# 7 to 7 itself should reproduce this prime (as a submodule).
assert P0 + 7 * P0.ZK == P0.as_submodule()
def test_pretty_printing():
d = -7
T = Poly(x ** 2 - d)
rad = {}
ZK, dK = round_two(T, radicals=rad)
p = 2
P = prime_decomp(p, T, dK=dK, ZK=ZK, radical=rad.get(p))
assert repr(P[0]) == '[ (2, (3*x + 1)/2) e=1, f=1 ]'
assert P[0].pretty(field_gen=theta) == '[ (2, (3*theta + 1)/2) e=1, f=1 ]'
assert P[0].pretty(field_gen=theta, just_gens=True) == '(2, (3*theta + 1)/2)'
def test_PrimeIdeal_reduce_poly():
T = Poly(cyclotomic_poly(7, x))
k = QQ.algebraic_field((T, x))
P = k.primes_above(11)
frp = P[0]
B = k.integral_basis(fmt='sympy')
assert [frp.reduce_poly(b, x) for b in B] == [
1, x, x ** 2, -5 * x ** 2 - 4 * x + 1, -x ** 2 - x - 5,
4 * x ** 2 - x - 1]
Q = k.primes_above(19)
frq = Q[0]
assert frq.alpha.equiv(0)
assert frq.reduce_poly(20*x**2 + 10) == x**2 - 9
raises(GeneratorsNeeded, lambda: frp.reduce_poly(S(1)))
raises(NotImplementedError, lambda: frp.reduce_poly(1))
|
88a3b5d13bee6b55de9ba6b6d88587e120eba457ebf9b33ead7c74dfd51aed80 | import warnings
from sympy.testing.pytest import (raises, warns, ignore_warnings,
warns_deprecated_sympy, Failed)
from sympy.utilities.exceptions import sympy_deprecation_warning
# Test callables
def test_expected_exception_is_silent_callable():
def f():
raise ValueError()
raises(ValueError, f)
# Under pytest raises will raise Failed rather than AssertionError
def test_lack_of_exception_triggers_AssertionError_callable():
try:
raises(Exception, lambda: 1 + 1)
assert False
except Failed as e:
assert "DID NOT RAISE" in str(e)
def test_unexpected_exception_is_passed_through_callable():
def f():
raise ValueError("some error message")
try:
raises(TypeError, f)
assert False
except ValueError as e:
assert str(e) == "some error message"
# Test with statement
def test_expected_exception_is_silent_with():
with raises(ValueError):
raise ValueError()
def test_lack_of_exception_triggers_AssertionError_with():
try:
with raises(Exception):
1 + 1
assert False
except Failed as e:
assert "DID NOT RAISE" in str(e)
def test_unexpected_exception_is_passed_through_with():
try:
with raises(TypeError):
raise ValueError("some error message")
assert False
except ValueError as e:
assert str(e) == "some error message"
# Now we can use raises() instead of try/catch
# to test that a specific exception class is raised
def test_second_argument_should_be_callable_or_string():
raises(TypeError, lambda: raises("irrelevant", 42))
def test_warns_catches_warning():
with warnings.catch_warnings(record=True) as w:
with warns(UserWarning):
warnings.warn('this is the warning message')
assert len(w) == 0
def test_warns_raises_without_warning():
with raises(Failed):
with warns(UserWarning):
pass
def test_warns_hides_other_warnings():
with raises(RuntimeWarning):
with warns(UserWarning):
warnings.warn('this is the warning message', UserWarning)
warnings.warn('this is the other message', RuntimeWarning)
def test_warns_continues_after_warning():
with warnings.catch_warnings(record=True) as w:
finished = False
with warns(UserWarning):
warnings.warn('this is the warning message')
finished = True
assert finished
assert len(w) == 0
def test_warns_many_warnings():
with warns(UserWarning):
warnings.warn('this is the warning message', UserWarning)
warnings.warn('this is the other warning message', UserWarning)
def test_warns_match_matching():
with warnings.catch_warnings(record=True) as w:
with warns(UserWarning, match='this is the warning message'):
warnings.warn('this is the warning message', UserWarning)
assert len(w) == 0
def test_warns_match_non_matching():
with warnings.catch_warnings(record=True) as w:
with raises(Failed):
with warns(UserWarning, match='this is the warning message'):
warnings.warn('this is not the expected warning message', UserWarning)
assert len(w) == 0
def _warn_sympy_deprecation(stacklevel=3):
sympy_deprecation_warning(
"feature",
active_deprecations_target="active-deprecations",
deprecated_since_version="0.0.0",
stacklevel=stacklevel,
)
def test_warns_deprecated_sympy_catches_warning():
with warnings.catch_warnings(record=True) as w:
with warns_deprecated_sympy():
_warn_sympy_deprecation()
assert len(w) == 0
def test_warns_deprecated_sympy_raises_without_warning():
with raises(Failed):
with warns_deprecated_sympy():
pass
def test_warns_deprecated_sympy_wrong_stacklevel():
with raises(Failed):
with warns_deprecated_sympy():
_warn_sympy_deprecation(stacklevel=1)
def test_warns_deprecated_sympy_doesnt_hide_other_warnings():
# Unlike pytest's deprecated_call, we should not hide other warnings.
with raises(RuntimeWarning):
with warns_deprecated_sympy():
_warn_sympy_deprecation()
warnings.warn('this is the other message', RuntimeWarning)
def test_warns_deprecated_sympy_continues_after_warning():
with warnings.catch_warnings(record=True) as w:
finished = False
with warns_deprecated_sympy():
_warn_sympy_deprecation()
finished = True
assert finished
assert len(w) == 0
def test_ignore_ignores_warning():
with warnings.catch_warnings(record=True) as w:
with ignore_warnings(UserWarning):
warnings.warn('this is the warning message')
assert len(w) == 0
def test_ignore_does_not_raise_without_warning():
with warnings.catch_warnings(record=True) as w:
with ignore_warnings(UserWarning):
pass
assert len(w) == 0
def test_ignore_allows_other_warnings():
with warnings.catch_warnings(record=True) as w:
# This is needed when pytest is run as -Werror
# the setting is reverted at the end of the catch_Warnings block.
warnings.simplefilter("always")
with ignore_warnings(UserWarning):
warnings.warn('this is the warning message', UserWarning)
warnings.warn('this is the other message', RuntimeWarning)
assert len(w) == 1
assert isinstance(w[0].message, RuntimeWarning)
assert str(w[0].message) == 'this is the other message'
def test_ignore_continues_after_warning():
with warnings.catch_warnings(record=True) as w:
finished = False
with ignore_warnings(UserWarning):
warnings.warn('this is the warning message')
finished = True
assert finished
assert len(w) == 0
def test_ignore_many_warnings():
with warnings.catch_warnings(record=True) as w:
# This is needed when pytest is run as -Werror
# the setting is reverted at the end of the catch_Warnings block.
warnings.simplefilter("always")
with ignore_warnings(UserWarning):
warnings.warn('this is the warning message', UserWarning)
warnings.warn('this is the other message', RuntimeWarning)
warnings.warn('this is the warning message', UserWarning)
warnings.warn('this is the other message', RuntimeWarning)
warnings.warn('this is the other message', RuntimeWarning)
assert len(w) == 3
for wi in w:
assert isinstance(wi.message, RuntimeWarning)
assert str(wi.message) == 'this is the other message'
|
c10647b38c0336e2b87e568a28bb095ba5d132dad58ccbffe6b50fdd6b2980f0 | from sympy.testing.pytest import warns_deprecated_sympy
def test_deprecated_testing_randtest():
with warns_deprecated_sympy():
import sympy.testing.randtest # noqa:F401
|
328f3c2609b2dc2b92433db95921f7e049d94acfc9058ec4f8e28f3b64a384f7 | from sympy.core import Rational, S
from sympy.simplify import simplify, trigsimp
from sympy.core.function import (Derivative, Function, diff)
from sympy.core.numbers import pi
from sympy.core.symbol import symbols
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.integrals.integrals import Integral
from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
from sympy.vector.vector import Vector, BaseVector, VectorAdd, \
VectorMul, VectorZero
from sympy.vector.coordsysrect import CoordSys3D
from sympy.vector.vector import Cross, Dot, cross
from sympy.testing.pytest import raises
C = CoordSys3D('C')
i, j, k = C.base_vectors()
a, b, c = symbols('a b c')
def test_cross():
v1 = C.x * i + C.z * C.z * j
v2 = C.x * i + C.y * j + C.z * k
assert Cross(v1, v2) == Cross(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k)
assert Cross(v1, v2).doit() == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k
assert cross(v1, v2) == C.z**3*C.i + (-C.x*C.z)*C.j + (C.x*C.y - C.x*C.z**2)*C.k
assert Cross(v1, v2) == -Cross(v2, v1)
assert Cross(v1, v2) + Cross(v2, v1) == Vector.zero
def test_dot():
v1 = C.x * i + C.z * C.z * j
v2 = C.x * i + C.y * j + C.z * k
assert Dot(v1, v2) == Dot(C.x*C.i + C.z**2*C.j, C.x*C.i + C.y*C.j + C.z*C.k)
assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2
assert Dot(v1, v2).doit() == C.x**2 + C.y*C.z**2
assert Dot(v1, v2) == Dot(v2, v1)
def test_vector_sympy():
"""
Test whether the Vector framework confirms to the hashing
and equality testing properties of SymPy.
"""
v1 = 3*j
assert v1 == j*3
assert v1.components == {j: 3}
v2 = 3*i + 4*j + 5*k
v3 = 2*i + 4*j + i + 4*k + k
assert v3 == v2
assert v3.__hash__() == v2.__hash__()
def test_vector():
assert isinstance(i, BaseVector)
assert i != j
assert j != k
assert k != i
assert i - i == Vector.zero
assert i + Vector.zero == i
assert i - Vector.zero == i
assert Vector.zero != 0
assert -Vector.zero == Vector.zero
v1 = a*i + b*j + c*k
v2 = a**2*i + b**2*j + c**2*k
v3 = v1 + v2
v4 = 2 * v1
v5 = a * i
assert isinstance(v1, VectorAdd)
assert v1 - v1 == Vector.zero
assert v1 + Vector.zero == v1
assert v1.dot(i) == a
assert v1.dot(j) == b
assert v1.dot(k) == c
assert i.dot(v2) == a**2
assert j.dot(v2) == b**2
assert k.dot(v2) == c**2
assert v3.dot(i) == a**2 + a
assert v3.dot(j) == b**2 + b
assert v3.dot(k) == c**2 + c
assert v1 + v2 == v2 + v1
assert v1 - v2 == -1 * (v2 - v1)
assert a * v1 == v1 * a
assert isinstance(v5, VectorMul)
assert v5.base_vector == i
assert v5.measure_number == a
assert isinstance(v4, Vector)
assert isinstance(v4, VectorAdd)
assert isinstance(v4, Vector)
assert isinstance(Vector.zero, VectorZero)
assert isinstance(Vector.zero, Vector)
assert isinstance(v1 * 0, VectorZero)
assert v1.to_matrix(C) == Matrix([[a], [b], [c]])
assert i.components == {i: 1}
assert v5.components == {i: a}
assert v1.components == {i: a, j: b, k: c}
assert VectorAdd(v1, Vector.zero) == v1
assert VectorMul(a, v1) == v1*a
assert VectorMul(1, i) == i
assert VectorAdd(v1, Vector.zero) == v1
assert VectorMul(0, Vector.zero) == Vector.zero
raises(TypeError, lambda: v1.outer(1))
raises(TypeError, lambda: v1.dot(1))
def test_vector_magnitude_normalize():
assert Vector.zero.magnitude() == 0
assert Vector.zero.normalize() == Vector.zero
assert i.magnitude() == 1
assert j.magnitude() == 1
assert k.magnitude() == 1
assert i.normalize() == i
assert j.normalize() == j
assert k.normalize() == k
v1 = a * i
assert v1.normalize() == (a/sqrt(a**2))*i
assert v1.magnitude() == sqrt(a**2)
v2 = a*i + b*j + c*k
assert v2.magnitude() == sqrt(a**2 + b**2 + c**2)
assert v2.normalize() == v2 / v2.magnitude()
v3 = i + j
assert v3.normalize() == (sqrt(2)/2)*C.i + (sqrt(2)/2)*C.j
def test_vector_simplify():
A, s, k, m = symbols('A, s, k, m')
test1 = (1 / a + 1 / b) * i
assert (test1 & i) != (a + b) / (a * b)
test1 = simplify(test1)
assert (test1 & i) == (a + b) / (a * b)
assert test1.simplify() == simplify(test1)
test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * i
test2 = simplify(test2)
assert (test2 & i) == (A**2 * s**4 / (4 * pi * k * m**3))
test3 = ((4 + 4 * a - 2 * (2 + 2 * a)) / (2 + 2 * a)) * i
test3 = simplify(test3)
assert (test3 & i) == 0
test4 = ((-4 * a * b**2 - 2 * b**3 - 2 * a**2 * b) / (a + b)**2) * i
test4 = simplify(test4)
assert (test4 & i) == -2 * b
v = (sin(a)+cos(a))**2*i - j
assert trigsimp(v) == (2*sin(a + pi/4)**2)*i + (-1)*j
assert trigsimp(v) == v.trigsimp()
assert simplify(Vector.zero) == Vector.zero
def test_vector_dot():
assert i.dot(Vector.zero) == 0
assert Vector.zero.dot(i) == 0
assert i & Vector.zero == 0
assert i.dot(i) == 1
assert i.dot(j) == 0
assert i.dot(k) == 0
assert i & i == 1
assert i & j == 0
assert i & k == 0
assert j.dot(i) == 0
assert j.dot(j) == 1
assert j.dot(k) == 0
assert j & i == 0
assert j & j == 1
assert j & k == 0
assert k.dot(i) == 0
assert k.dot(j) == 0
assert k.dot(k) == 1
assert k & i == 0
assert k & j == 0
assert k & k == 1
raises(TypeError, lambda: k.dot(1))
def test_vector_cross():
assert i.cross(Vector.zero) == Vector.zero
assert Vector.zero.cross(i) == Vector.zero
assert i.cross(i) == Vector.zero
assert i.cross(j) == k
assert i.cross(k) == -j
assert i ^ i == Vector.zero
assert i ^ j == k
assert i ^ k == -j
assert j.cross(i) == -k
assert j.cross(j) == Vector.zero
assert j.cross(k) == i
assert j ^ i == -k
assert j ^ j == Vector.zero
assert j ^ k == i
assert k.cross(i) == j
assert k.cross(j) == -i
assert k.cross(k) == Vector.zero
assert k ^ i == j
assert k ^ j == -i
assert k ^ k == Vector.zero
assert k.cross(1) == Cross(k, 1)
def test_projection():
v1 = i + j + k
v2 = 3*i + 4*j
v3 = 0*i + 0*j
assert v1.projection(v1) == i + j + k
assert v1.projection(v2) == Rational(7, 3)*C.i + Rational(7, 3)*C.j + Rational(7, 3)*C.k
assert v1.projection(v1, scalar=True) == S.One
assert v1.projection(v2, scalar=True) == Rational(7, 3)
assert v3.projection(v1) == Vector.zero
assert v3.projection(v1, scalar=True) == S.Zero
def test_vector_diff_integrate():
f = Function('f')
v = f(a)*C.i + a**2*C.j - C.k
assert Derivative(v, a) == Derivative((f(a))*C.i +
a**2*C.j + (-1)*C.k, a)
assert (diff(v, a) == v.diff(a) == Derivative(v, a).doit() ==
(Derivative(f(a), a))*C.i + 2*a*C.j)
assert (Integral(v, a) == (Integral(f(a), a))*C.i +
(Integral(a**2, a))*C.j + (Integral(-1, a))*C.k)
def test_vector_args():
raises(ValueError, lambda: BaseVector(3, C))
raises(TypeError, lambda: BaseVector(0, Vector.zero))
def test_srepr():
from sympy.printing.repr import srepr
res = "CoordSys3D(Str('C'), Tuple(ImmutableDenseMatrix([[Integer(1), "\
"Integer(0), Integer(0)], [Integer(0), Integer(1), Integer(0)], "\
"[Integer(0), Integer(0), Integer(1)]]), VectorZero())).i"
assert srepr(C.i) == res
def test_scalar():
from sympy.vector import CoordSys3D
C = CoordSys3D('C')
v1 = 3*C.i + 4*C.j + 5*C.k
v2 = 3*C.i - 4*C.j + 5*C.k
assert v1.is_Vector is True
assert v1.is_scalar is False
assert (v1.dot(v2)).is_scalar is True
assert (v1.cross(v2)).is_scalar is False
|
abd9fd3881bf0a90f8290376d1dcf0459217a0cbfb552829f1f1e65679197ebd | from sympy.testing.pytest import raises
from sympy.vector.coordsysrect import CoordSys3D
from sympy.vector.scalar import BaseScalar
from sympy.core.function import expand
from sympy.core.numbers import pi
from sympy.core.symbol import symbols
from sympy.functions.elementary.hyperbolic import (cosh, sinh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, atan2, cos, sin)
from sympy.matrices.dense import zeros
from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix
from sympy.simplify.simplify import simplify
from sympy.vector.functions import express
from sympy.vector.point import Point
from sympy.vector.vector import Vector
from sympy.vector.orienters import (AxisOrienter, BodyOrienter,
SpaceOrienter, QuaternionOrienter)
x, y, z = symbols('x y z')
a, b, c, q = symbols('a b c q')
q1, q2, q3, q4 = symbols('q1 q2 q3 q4')
def test_func_args():
A = CoordSys3D('A')
assert A.x.func(*A.x.args) == A.x
expr = 3*A.x + 4*A.y
assert expr.func(*expr.args) == expr
assert A.i.func(*A.i.args) == A.i
v = A.x*A.i + A.y*A.j + A.z*A.k
assert v.func(*v.args) == v
assert A.origin.func(*A.origin.args) == A.origin
def test_coordsys3d_equivalence():
A = CoordSys3D('A')
A1 = CoordSys3D('A')
assert A1 == A
B = CoordSys3D('B')
assert A != B
def test_orienters():
A = CoordSys3D('A')
axis_orienter = AxisOrienter(a, A.k)
body_orienter = BodyOrienter(a, b, c, '123')
space_orienter = SpaceOrienter(a, b, c, '123')
q_orienter = QuaternionOrienter(q1, q2, q3, q4)
assert axis_orienter.rotation_matrix(A) == Matrix([
[ cos(a), sin(a), 0],
[-sin(a), cos(a), 0],
[ 0, 0, 1]])
assert body_orienter.rotation_matrix() == Matrix([
[ cos(b)*cos(c), sin(a)*sin(b)*cos(c) + sin(c)*cos(a),
sin(a)*sin(c) - sin(b)*cos(a)*cos(c)],
[-sin(c)*cos(b), -sin(a)*sin(b)*sin(c) + cos(a)*cos(c),
sin(a)*cos(c) + sin(b)*sin(c)*cos(a)],
[ sin(b), -sin(a)*cos(b),
cos(a)*cos(b)]])
assert space_orienter.rotation_matrix() == Matrix([
[cos(b)*cos(c), sin(c)*cos(b), -sin(b)],
[sin(a)*sin(b)*cos(c) - sin(c)*cos(a),
sin(a)*sin(b)*sin(c) + cos(a)*cos(c), sin(a)*cos(b)],
[sin(a)*sin(c) + sin(b)*cos(a)*cos(c), -sin(a)*cos(c) +
sin(b)*sin(c)*cos(a), cos(a)*cos(b)]])
assert q_orienter.rotation_matrix() == Matrix([
[q1**2 + q2**2 - q3**2 - q4**2, 2*q1*q4 + 2*q2*q3,
-2*q1*q3 + 2*q2*q4],
[-2*q1*q4 + 2*q2*q3, q1**2 - q2**2 + q3**2 - q4**2,
2*q1*q2 + 2*q3*q4],
[2*q1*q3 + 2*q2*q4,
-2*q1*q2 + 2*q3*q4, q1**2 - q2**2 - q3**2 + q4**2]])
def test_coordinate_vars():
"""
Tests the coordinate variables functionality with respect to
reorientation of coordinate systems.
"""
A = CoordSys3D('A')
# Note that the name given on the lhs is different from A.x._name
assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}') == A.x
assert BaseScalar(1, A, 'A_y', r'\mathbf{{y}_{A}}') == A.y
assert BaseScalar(2, A, 'A_z', r'\mathbf{{z}_{A}}') == A.z
assert BaseScalar(0, A, 'A_x', r'\mathbf{{x}_{A}}').__hash__() == A.x.__hash__()
assert isinstance(A.x, BaseScalar) and \
isinstance(A.y, BaseScalar) and \
isinstance(A.z, BaseScalar)
assert A.x*A.y == A.y*A.x
assert A.scalar_map(A) == {A.x: A.x, A.y: A.y, A.z: A.z}
assert A.x.system == A
assert A.x.diff(A.x) == 1
B = A.orient_new_axis('B', q, A.k)
assert B.scalar_map(A) == {B.z: A.z, B.y: -A.x*sin(q) + A.y*cos(q),
B.x: A.x*cos(q) + A.y*sin(q)}
assert A.scalar_map(B) == {A.x: B.x*cos(q) - B.y*sin(q),
A.y: B.x*sin(q) + B.y*cos(q), A.z: B.z}
assert express(B.x, A, variables=True) == A.x*cos(q) + A.y*sin(q)
assert express(B.y, A, variables=True) == -A.x*sin(q) + A.y*cos(q)
assert express(B.z, A, variables=True) == A.z
assert expand(express(B.x*B.y*B.z, A, variables=True)) == \
expand(A.z*(-A.x*sin(q) + A.y*cos(q))*(A.x*cos(q) + A.y*sin(q)))
assert express(B.x*B.i + B.y*B.j + B.z*B.k, A) == \
(B.x*cos(q) - B.y*sin(q))*A.i + (B.x*sin(q) + \
B.y*cos(q))*A.j + B.z*A.k
assert simplify(express(B.x*B.i + B.y*B.j + B.z*B.k, A, \
variables=True)) == \
A.x*A.i + A.y*A.j + A.z*A.k
assert express(A.x*A.i + A.y*A.j + A.z*A.k, B) == \
(A.x*cos(q) + A.y*sin(q))*B.i + \
(-A.x*sin(q) + A.y*cos(q))*B.j + A.z*B.k
assert simplify(express(A.x*A.i + A.y*A.j + A.z*A.k, B, \
variables=True)) == \
B.x*B.i + B.y*B.j + B.z*B.k
N = B.orient_new_axis('N', -q, B.k)
assert N.scalar_map(A) == \
{N.x: A.x, N.z: A.z, N.y: A.y}
C = A.orient_new_axis('C', q, A.i + A.j + A.k)
mapping = A.scalar_map(C)
assert mapping[A.x].equals(C.x*(2*cos(q) + 1)/3 +
C.y*(-2*sin(q + pi/6) + 1)/3 +
C.z*(-2*cos(q + pi/3) + 1)/3)
assert mapping[A.y].equals(C.x*(-2*cos(q + pi/3) + 1)/3 +
C.y*(2*cos(q) + 1)/3 +
C.z*(-2*sin(q + pi/6) + 1)/3)
assert mapping[A.z].equals(C.x*(-2*sin(q + pi/6) + 1)/3 +
C.y*(-2*cos(q + pi/3) + 1)/3 +
C.z*(2*cos(q) + 1)/3)
D = A.locate_new('D', a*A.i + b*A.j + c*A.k)
assert D.scalar_map(A) == {D.z: A.z - c, D.x: A.x - a, D.y: A.y - b}
E = A.orient_new_axis('E', a, A.k, a*A.i + b*A.j + c*A.k)
assert A.scalar_map(E) == {A.z: E.z + c,
A.x: E.x*cos(a) - E.y*sin(a) + a,
A.y: E.x*sin(a) + E.y*cos(a) + b}
assert E.scalar_map(A) == {E.x: (A.x - a)*cos(a) + (A.y - b)*sin(a),
E.y: (-A.x + a)*sin(a) + (A.y - b)*cos(a),
E.z: A.z - c}
F = A.locate_new('F', Vector.zero)
assert A.scalar_map(F) == {A.z: F.z, A.x: F.x, A.y: F.y}
def test_rotation_matrix():
N = CoordSys3D('N')
A = N.orient_new_axis('A', q1, N.k)
B = A.orient_new_axis('B', q2, A.i)
C = B.orient_new_axis('C', q3, B.j)
D = N.orient_new_axis('D', q4, N.j)
E = N.orient_new_space('E', q1, q2, q3, '123')
F = N.orient_new_quaternion('F', q1, q2, q3, q4)
G = N.orient_new_body('G', q1, q2, q3, '123')
assert N.rotation_matrix(C) == Matrix([
[- sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), - sin(q1) *
cos(q2), sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], \
[sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), \
cos(q1) * cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * \
cos(q3)], [- sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]])
test_mat = D.rotation_matrix(C) - Matrix(
[[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) +
sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) *
cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) * \
(- sin(q4) * cos(q2) + sin(q1) * sin(q2) * cos(q4))], \
[sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * \
cos(q2), sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)], \
[sin(q4) * cos(q1) * cos(q3) - sin(q3) * (cos(q2) * cos(q4) + \
sin(q1) * sin(q2) * \
sin(q4)), sin(q2) *
cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) * \
sin(q4) * cos(q1) + cos(q3) * (cos(q2) * cos(q4) + \
sin(q1) * sin(q2) * sin(q4))]])
assert test_mat.expand() == zeros(3, 3)
assert E.rotation_matrix(N) == Matrix(
[[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)],
[sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), \
sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2)], \
[sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), - \
sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2)]])
assert F.rotation_matrix(N) == Matrix([[
q1**2 + q2**2 - q3**2 - q4**2,
2*q1*q4 + 2*q2*q3, -2*q1*q3 + 2*q2*q4],[ -2*q1*q4 + 2*q2*q3,
q1**2 - q2**2 + q3**2 - q4**2, 2*q1*q2 + 2*q3*q4],
[2*q1*q3 + 2*q2*q4,
-2*q1*q2 + 2*q3*q4,
q1**2 - q2**2 - q3**2 + q4**2]])
assert G.rotation_matrix(N) == Matrix([[
cos(q2)*cos(q3), sin(q1)*sin(q2)*cos(q3) + sin(q3)*cos(q1),
sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3)], [
-sin(q3)*cos(q2), -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3),
sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],[
sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]])
def test_vector_with_orientation():
"""
Tests the effects of orientation of coordinate systems on
basic vector operations.
"""
N = CoordSys3D('N')
A = N.orient_new_axis('A', q1, N.k)
B = A.orient_new_axis('B', q2, A.i)
C = B.orient_new_axis('C', q3, B.j)
# Test to_matrix
v1 = a*N.i + b*N.j + c*N.k
assert v1.to_matrix(A) == Matrix([[ a*cos(q1) + b*sin(q1)],
[-a*sin(q1) + b*cos(q1)],
[ c]])
# Test dot
assert N.i.dot(A.i) == cos(q1)
assert N.i.dot(A.j) == -sin(q1)
assert N.i.dot(A.k) == 0
assert N.j.dot(A.i) == sin(q1)
assert N.j.dot(A.j) == cos(q1)
assert N.j.dot(A.k) == 0
assert N.k.dot(A.i) == 0
assert N.k.dot(A.j) == 0
assert N.k.dot(A.k) == 1
assert N.i.dot(A.i + A.j) == -sin(q1) + cos(q1) == \
(A.i + A.j).dot(N.i)
assert A.i.dot(C.i) == cos(q3)
assert A.i.dot(C.j) == 0
assert A.i.dot(C.k) == sin(q3)
assert A.j.dot(C.i) == sin(q2)*sin(q3)
assert A.j.dot(C.j) == cos(q2)
assert A.j.dot(C.k) == -sin(q2)*cos(q3)
assert A.k.dot(C.i) == -cos(q2)*sin(q3)
assert A.k.dot(C.j) == sin(q2)
assert A.k.dot(C.k) == cos(q2)*cos(q3)
# Test cross
assert N.i.cross(A.i) == sin(q1)*A.k
assert N.i.cross(A.j) == cos(q1)*A.k
assert N.i.cross(A.k) == -sin(q1)*A.i - cos(q1)*A.j
assert N.j.cross(A.i) == -cos(q1)*A.k
assert N.j.cross(A.j) == sin(q1)*A.k
assert N.j.cross(A.k) == cos(q1)*A.i - sin(q1)*A.j
assert N.k.cross(A.i) == A.j
assert N.k.cross(A.j) == -A.i
assert N.k.cross(A.k) == Vector.zero
assert N.i.cross(A.i) == sin(q1)*A.k
assert N.i.cross(A.j) == cos(q1)*A.k
assert N.i.cross(A.i + A.j) == sin(q1)*A.k + cos(q1)*A.k
assert (A.i + A.j).cross(N.i) == (-sin(q1) - cos(q1))*N.k
assert A.i.cross(C.i) == sin(q3)*C.j
assert A.i.cross(C.j) == -sin(q3)*C.i + cos(q3)*C.k
assert A.i.cross(C.k) == -cos(q3)*C.j
assert C.i.cross(A.i) == (-sin(q3)*cos(q2))*A.j + \
(-sin(q2)*sin(q3))*A.k
assert C.j.cross(A.i) == (sin(q2))*A.j + (-cos(q2))*A.k
assert express(C.k.cross(A.i), C).trigsimp() == cos(q3)*C.j
def test_orient_new_methods():
N = CoordSys3D('N')
orienter1 = AxisOrienter(q4, N.j)
orienter2 = SpaceOrienter(q1, q2, q3, '123')
orienter3 = QuaternionOrienter(q1, q2, q3, q4)
orienter4 = BodyOrienter(q1, q2, q3, '123')
D = N.orient_new('D', (orienter1, ))
E = N.orient_new('E', (orienter2, ))
F = N.orient_new('F', (orienter3, ))
G = N.orient_new('G', (orienter4, ))
assert D == N.orient_new_axis('D', q4, N.j)
assert E == N.orient_new_space('E', q1, q2, q3, '123')
assert F == N.orient_new_quaternion('F', q1, q2, q3, q4)
assert G == N.orient_new_body('G', q1, q2, q3, '123')
def test_locatenew_point():
"""
Tests Point class, and locate_new method in CoordSys3D.
"""
A = CoordSys3D('A')
assert isinstance(A.origin, Point)
v = a*A.i + b*A.j + c*A.k
C = A.locate_new('C', v)
assert C.origin.position_wrt(A) == \
C.position_wrt(A) == \
C.origin.position_wrt(A.origin) == v
assert A.origin.position_wrt(C) == \
A.position_wrt(C) == \
A.origin.position_wrt(C.origin) == -v
assert A.origin.express_coordinates(C) == (-a, -b, -c)
p = A.origin.locate_new('p', -v)
assert p.express_coordinates(A) == (-a, -b, -c)
assert p.position_wrt(C.origin) == p.position_wrt(C) == \
-2 * v
p1 = p.locate_new('p1', 2*v)
assert p1.position_wrt(C.origin) == Vector.zero
assert p1.express_coordinates(C) == (0, 0, 0)
p2 = p.locate_new('p2', A.i)
assert p1.position_wrt(p2) == 2*v - A.i
assert p2.express_coordinates(C) == (-2*a + 1, -2*b, -2*c)
def test_create_new():
a = CoordSys3D('a')
c = a.create_new('c', transformation='spherical')
assert c._parent == a
assert c.transformation_to_parent() == \
(c.r*sin(c.theta)*cos(c.phi), c.r*sin(c.theta)*sin(c.phi), c.r*cos(c.theta))
assert c.transformation_from_parent() == \
(sqrt(a.x**2 + a.y**2 + a.z**2), acos(a.z/sqrt(a.x**2 + a.y**2 + a.z**2)), atan2(a.y, a.x))
def test_evalf():
A = CoordSys3D('A')
v = 3*A.i + 4*A.j + a*A.k
assert v.n() == v.evalf()
assert v.evalf(subs={a:1}) == v.subs(a, 1).evalf()
def test_lame_coefficients():
a = CoordSys3D('a', 'spherical')
assert a.lame_coefficients() == (1, a.r, sin(a.theta)*a.r)
a = CoordSys3D('a')
assert a.lame_coefficients() == (1, 1, 1)
a = CoordSys3D('a', 'cartesian')
assert a.lame_coefficients() == (1, 1, 1)
a = CoordSys3D('a', 'cylindrical')
assert a.lame_coefficients() == (1, a.r, 1)
def test_transformation_equations():
x, y, z = symbols('x y z')
# Str
a = CoordSys3D('a', transformation='spherical',
variable_names=["r", "theta", "phi"])
r, theta, phi = a.base_scalars()
assert r == a.r
assert theta == a.theta
assert phi == a.phi
raises(AttributeError, lambda: a.x)
raises(AttributeError, lambda: a.y)
raises(AttributeError, lambda: a.z)
assert a.transformation_to_parent() == (
r*sin(theta)*cos(phi),
r*sin(theta)*sin(phi),
r*cos(theta)
)
assert a.lame_coefficients() == (1, r, r*sin(theta))
assert a.transformation_from_parent_function()(x, y, z) == (
sqrt(x ** 2 + y ** 2 + z ** 2),
acos((z) / sqrt(x**2 + y**2 + z**2)),
atan2(y, x)
)
a = CoordSys3D('a', transformation='cylindrical',
variable_names=["r", "theta", "z"])
r, theta, z = a.base_scalars()
assert a.transformation_to_parent() == (
r*cos(theta),
r*sin(theta),
z
)
assert a.lame_coefficients() == (1, a.r, 1)
assert a.transformation_from_parent_function()(x, y, z) == (sqrt(x**2 + y**2),
atan2(y, x), z)
a = CoordSys3D('a', 'cartesian')
assert a.transformation_to_parent() == (a.x, a.y, a.z)
assert a.lame_coefficients() == (1, 1, 1)
assert a.transformation_from_parent_function()(x, y, z) == (x, y, z)
# Variables and expressions
# Cartesian with equation tuple:
x, y, z = symbols('x y z')
a = CoordSys3D('a', ((x, y, z), (x, y, z)))
a._calculate_inv_trans_equations()
assert a.transformation_to_parent() == (a.x1, a.x2, a.x3)
assert a.lame_coefficients() == (1, 1, 1)
assert a.transformation_from_parent_function()(x, y, z) == (x, y, z)
r, theta, z = symbols("r theta z")
# Cylindrical with equation tuple:
a = CoordSys3D('a', [(r, theta, z), (r*cos(theta), r*sin(theta), z)],
variable_names=["r", "theta", "z"])
r, theta, z = a.base_scalars()
assert a.transformation_to_parent() == (
r*cos(theta), r*sin(theta), z
)
assert a.lame_coefficients() == (
sqrt(sin(theta)**2 + cos(theta)**2),
sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2),
1
) # ==> this should simplify to (1, r, 1), tests are too slow with `simplify`.
# Definitions with `lambda`:
# Cartesian with `lambda`
a = CoordSys3D('a', lambda x, y, z: (x, y, z))
assert a.transformation_to_parent() == (a.x1, a.x2, a.x3)
assert a.lame_coefficients() == (1, 1, 1)
a._calculate_inv_trans_equations()
assert a.transformation_from_parent_function()(x, y, z) == (x, y, z)
# Spherical with `lambda`
a = CoordSys3D('a', lambda r, theta, phi: (r*sin(theta)*cos(phi), r*sin(theta)*sin(phi), r*cos(theta)),
variable_names=["r", "theta", "phi"])
r, theta, phi = a.base_scalars()
assert a.transformation_to_parent() == (
r*sin(theta)*cos(phi), r*sin(phi)*sin(theta), r*cos(theta)
)
assert a.lame_coefficients() == (
sqrt(sin(phi)**2*sin(theta)**2 + sin(theta)**2*cos(phi)**2 + cos(theta)**2),
sqrt(r**2*sin(phi)**2*cos(theta)**2 + r**2*sin(theta)**2 + r**2*cos(phi)**2*cos(theta)**2),
sqrt(r**2*sin(phi)**2*sin(theta)**2 + r**2*sin(theta)**2*cos(phi)**2)
) # ==> this should simplify to (1, r, sin(theta)*r), `simplify` is too slow.
# Cylindrical with `lambda`
a = CoordSys3D('a', lambda r, theta, z:
(r*cos(theta), r*sin(theta), z),
variable_names=["r", "theta", "z"]
)
r, theta, z = a.base_scalars()
assert a.transformation_to_parent() == (r*cos(theta), r*sin(theta), z)
assert a.lame_coefficients() == (
sqrt(sin(theta)**2 + cos(theta)**2),
sqrt(r**2*sin(theta)**2 + r**2*cos(theta)**2),
1
) # ==> this should simplify to (1, a.x, 1)
raises(TypeError, lambda: CoordSys3D('a', transformation={
x: x*sin(y)*cos(z), y:x*sin(y)*sin(z), z: x*cos(y)}))
def test_check_orthogonality():
x, y, z = symbols('x y z')
u,v = symbols('u, v')
a = CoordSys3D('a', transformation=((x, y, z), (x*sin(y)*cos(z), x*sin(y)*sin(z), x*cos(y))))
assert a._check_orthogonality(a._transformation) is True
a = CoordSys3D('a', transformation=((x, y, z), (x * cos(y), x * sin(y), z)))
assert a._check_orthogonality(a._transformation) is True
a = CoordSys3D('a', transformation=((u, v, z), (cosh(u) * cos(v), sinh(u) * sin(v), z)))
assert a._check_orthogonality(a._transformation) is True
raises(ValueError, lambda: CoordSys3D('a', transformation=((x, y, z), (x, x, z))))
raises(ValueError, lambda: CoordSys3D('a', transformation=(
(x, y, z), (x*sin(y/2)*cos(z), x*sin(y)*sin(z), x*cos(y)))))
def test_rotation_trans_equations():
a = CoordSys3D('a')
from sympy.core.symbol import symbols
q0 = symbols('q0')
assert a._rotation_trans_equations(a._parent_rotation_matrix, a.base_scalars()) == (a.x, a.y, a.z)
assert a._rotation_trans_equations(a._inverse_rotation_matrix(), a.base_scalars()) == (a.x, a.y, a.z)
b = a.orient_new_axis('b', 0, -a.k)
assert b._rotation_trans_equations(b._parent_rotation_matrix, b.base_scalars()) == (b.x, b.y, b.z)
assert b._rotation_trans_equations(b._inverse_rotation_matrix(), b.base_scalars()) == (b.x, b.y, b.z)
c = a.orient_new_axis('c', q0, -a.k)
assert c._rotation_trans_equations(c._parent_rotation_matrix, c.base_scalars()) == \
(-sin(q0) * c.y + cos(q0) * c.x, sin(q0) * c.x + cos(q0) * c.y, c.z)
assert c._rotation_trans_equations(c._inverse_rotation_matrix(), c.base_scalars()) == \
(sin(q0) * c.y + cos(q0) * c.x, -sin(q0) * c.x + cos(q0) * c.y, c.z)
|
dc1796e79890e2a5dd7c34c5aa9b44dae7c93c92013b375f218571f389ff1300 | # -*- coding: utf-8 -*-
from sympy.core.function import Function
from sympy.integrals.integrals import Integral
from sympy.printing.latex import latex
from sympy.printing.pretty import pretty as xpretty
from sympy.vector import CoordSys3D, Del, Vector, express
from sympy.abc import a, b, c
from sympy.testing.pytest import XFAIL
def pretty(expr):
"""ASCII pretty-printing"""
return xpretty(expr, use_unicode=False, wrap_line=False)
def upretty(expr):
"""Unicode pretty-printing"""
return xpretty(expr, use_unicode=True, wrap_line=False)
# Initialize the basic and tedious vector/dyadic expressions
# needed for testing.
# Some of the pretty forms shown denote how the expressions just
# above them should look with pretty printing.
N = CoordSys3D('N')
C = N.orient_new_axis('C', a, N.k) # type: ignore
v = []
d = []
v.append(Vector.zero)
v.append(N.i) # type: ignore
v.append(-N.i) # type: ignore
v.append(N.i + N.j) # type: ignore
v.append(a*N.i) # type: ignore
v.append(a*N.i - b*N.j) # type: ignore
v.append((a**2 + N.x)*N.i + N.k) # type: ignore
v.append((a**2 + b)*N.i + 3*(C.y - c)*N.k) # type: ignore
f = Function('f')
v.append(N.j - (Integral(f(b)) - C.x**2)*N.k) # type: ignore
upretty_v_8 = """\
⎛ 2 ⌠ ⎞ \n\
j_N + ⎜x_C - ⎮ f(b) db⎟ k_N\n\
⎝ ⌡ ⎠ \
"""
pretty_v_8 = """\
j_N + / / \\\n\
| 2 | |\n\
|x_C - | f(b) db|\n\
| | |\n\
\\ / / \
"""
v.append(N.i + C.k) # type: ignore
v.append(express(N.i, C)) # type: ignore
v.append((a**2 + b)*N.i + (Integral(f(b)))*N.k) # type: ignore
upretty_v_11 = """\
⎛ 2 ⎞ ⎛⌠ ⎞ \n\
⎝a + b⎠ i_N + ⎜⎮ f(b) db⎟ k_N\n\
⎝⌡ ⎠ \
"""
pretty_v_11 = """\
/ 2 \\ + / / \\\n\
\\a + b/ i_N| | |\n\
| | f(b) db|\n\
| | |\n\
\\/ / \
"""
for x in v:
d.append(x | N.k) # type: ignore
s = 3*N.x**2*C.y # type: ignore
upretty_s = """\
2\n\
3⋅y_C⋅x_N \
"""
pretty_s = """\
2\n\
3*y_C*x_N \
"""
# This is the pretty form for ((a**2 + b)*N.i + 3*(C.y - c)*N.k) | N.k
upretty_d_7 = """\
⎛ 2 ⎞ \n\
⎝a + b⎠ (i_N|k_N) + (3⋅y_C - 3⋅c) (k_N|k_N)\
"""
pretty_d_7 = """\
/ 2 \\ (i_N|k_N) + (3*y_C - 3*c) (k_N|k_N)\n\
\\a + b/ \
"""
def test_str_printing():
assert str(v[0]) == '0'
assert str(v[1]) == 'N.i'
assert str(v[2]) == '(-1)*N.i'
assert str(v[3]) == 'N.i + N.j'
assert str(v[8]) == 'N.j + (C.x**2 - Integral(f(b), b))*N.k'
assert str(v[9]) == 'C.k + N.i'
assert str(s) == '3*C.y*N.x**2'
assert str(d[0]) == '0'
assert str(d[1]) == '(N.i|N.k)'
assert str(d[4]) == 'a*(N.i|N.k)'
assert str(d[5]) == 'a*(N.i|N.k) + (-b)*(N.j|N.k)'
assert str(d[8]) == ('(N.j|N.k) + (C.x**2 - ' +
'Integral(f(b), b))*(N.k|N.k)')
@XFAIL
def test_pretty_printing_ascii():
assert pretty(v[0]) == '0'
assert pretty(v[1]) == 'i_N'
assert pretty(v[5]) == '(a) i_N + (-b) j_N'
assert pretty(v[8]) == pretty_v_8
assert pretty(v[2]) == '(-1) i_N'
assert pretty(v[11]) == pretty_v_11
assert pretty(s) == pretty_s
assert pretty(d[0]) == '(0|0)'
assert pretty(d[5]) == '(a) (i_N|k_N) + (-b) (j_N|k_N)'
assert pretty(d[7]) == pretty_d_7
assert pretty(d[10]) == '(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)'
def test_pretty_print_unicode_v():
assert upretty(v[0]) == '0'
assert upretty(v[1]) == 'i_N'
assert upretty(v[5]) == '(a) i_N + (-b) j_N'
# Make sure the printing works in other objects
assert upretty(v[5].args) == '((a) i_N, (-b) j_N)'
assert upretty(v[8]) == upretty_v_8
assert upretty(v[2]) == '(-1) i_N'
assert upretty(v[11]) == upretty_v_11
assert upretty(s) == upretty_s
assert upretty(d[0]) == '(0|0)'
assert upretty(d[5]) == '(a) (i_N|k_N) + (-b) (j_N|k_N)'
assert upretty(d[7]) == upretty_d_7
assert upretty(d[10]) == '(cos(a)) (i_C|k_N) + (-sin(a)) (j_C|k_N)'
def test_latex_printing():
assert latex(v[0]) == '\\mathbf{\\hat{0}}'
assert latex(v[1]) == '\\mathbf{\\hat{i}_{N}}'
assert latex(v[2]) == '- \\mathbf{\\hat{i}_{N}}'
assert latex(v[5]) == ('\\left(a\\right)\\mathbf{\\hat{i}_{N}} + ' +
'\\left(- b\\right)\\mathbf{\\hat{j}_{N}}')
assert latex(v[6]) == ('\\left(\\mathbf{{x}_{N}} + a^{2}\\right)\\mathbf{\\hat{i}_' +
'{N}} + \\mathbf{\\hat{k}_{N}}')
assert latex(v[8]) == ('\\mathbf{\\hat{j}_{N}} + \\left(\\mathbf{{x}_' +
'{C}}^{2} - \\int f{\\left(b \\right)}\\,' +
' db\\right)\\mathbf{\\hat{k}_{N}}')
assert latex(s) == '3 \\mathbf{{y}_{C}} \\mathbf{{x}_{N}}^{2}'
assert latex(d[0]) == '(\\mathbf{\\hat{0}}|\\mathbf{\\hat{0}})'
assert latex(d[4]) == ('\\left(a\\right)\\left(\\mathbf{\\hat{i}_{N}}{\\middle|}' +
'\\mathbf{\\hat{k}_{N}}\\right)')
assert latex(d[9]) == ('\\left(\\mathbf{\\hat{k}_{C}}{\\middle|}' +
'\\mathbf{\\hat{k}_{N}}\\right) + \\left(' +
'\\mathbf{\\hat{i}_{N}}{\\middle|}\\mathbf{' +
'\\hat{k}_{N}}\\right)')
assert latex(d[11]) == ('\\left(a^{2} + b\\right)\\left(\\mathbf{\\hat{i}_{N}}' +
'{\\middle|}\\mathbf{\\hat{k}_{N}}\\right) + ' +
'\\left(\\int f{\\left(b \\right)}\\, db\\right)\\left(' +
'\\mathbf{\\hat{k}_{N}}{\\middle|}\\mathbf{' +
'\\hat{k}_{N}}\\right)')
def test_issue_23058():
from sympy import symbols, sin, cos, pi, UnevaluatedExpr
delop = Del()
CC_ = CoordSys3D("C")
y = CC_.y
xhat = CC_.i
t = symbols("t")
ten = symbols("10", positive=True)
eps, mu = 4*pi*ten**(-11), ten**(-5)
Bx = 2 * ten**(-4) * cos(ten**5 * t) * sin(ten**(-3) * y)
vecB = Bx * xhat
vecE = (1/eps) * Integral(delop.cross(vecB/mu).doit(), t)
vecE = vecE.doit()
vecB_str = """\
⎛ ⎛y_C⎞ ⎛ 5 ⎞⎞ \n\
⎜2⋅sin⎜───⎟⋅cos⎝10 ⋅t⎠⎟ i_C\n\
⎜ ⎜ 3⎟ ⎟ \n\
⎜ ⎝10 ⎠ ⎟ \n\
⎜─────────────────────⎟ \n\
⎜ 4 ⎟ \n\
⎝ 10 ⎠ \
"""
vecE_str = """\
⎛ 4 ⎛ 5 ⎞ ⎛y_C⎞ ⎞ \n\
⎜-10 ⋅sin⎝10 ⋅t⎠⋅cos⎜───⎟ ⎟ k_C\n\
⎜ ⎜ 3⎟ ⎟ \n\
⎜ ⎝10 ⎠ ⎟ \n\
⎜─────────────────────────⎟ \n\
⎝ 2⋅π ⎠ \
"""
assert upretty(vecB) == vecB_str
assert upretty(vecE) == vecE_str
ten = UnevaluatedExpr(10)
eps, mu = 4*pi*ten**(-11), ten**(-5)
Bx = 2 * ten**(-4) * cos(ten**5 * t) * sin(ten**(-3) * y)
vecB = Bx * xhat
vecB_str = """\
⎛ -4 ⎛ 5⎞ ⎛ -3⎞⎞ \n\
⎝2⋅10 ⋅cos⎝t⋅10 ⎠⋅sin⎝y_C⋅10 ⎠⎠ i_C \
"""
assert upretty(vecB) == vecB_str
def test_custom_names():
A = CoordSys3D('A', vector_names=['x', 'y', 'z'],
variable_names=['i', 'j', 'k'])
assert A.i.__str__() == 'A.i'
assert A.x.__str__() == 'A.x'
assert A.i._pretty_form == 'i_A'
assert A.x._pretty_form == 'x_A'
assert A.i._latex_form == r'\mathbf{{i}_{A}}'
assert A.x._latex_form == r"\mathbf{\hat{x}_{A}}"
|
712c7fc95b45a30495441d8ab86c9e35fb19e9cb0490798b9193765a641c434d | from sympy.core.numbers import (Float, Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, cos, sin)
from sympy.sets import EmptySet
from sympy.simplify.simplify import simplify
from sympy.functions.elementary.trigonometric import tan
from sympy.geometry import (Circle, GeometryError, Line, Point, Ray,
Segment, Triangle, intersection, Point3D, Line3D, Ray3D, Segment3D,
Point2D, Line2D)
from sympy.geometry.line import Undecidable
from sympy.geometry.polygon import _asa as asa
from sympy.utilities.iterables import cartes
from sympy.testing.pytest import raises, warns, warns_deprecated_sympy
x = Symbol('x', real=True)
y = Symbol('y', real=True)
z = Symbol('z', real=True)
k = Symbol('k', real=True)
x1 = Symbol('x1', real=True)
y1 = Symbol('y1', real=True)
t = Symbol('t', real=True)
a, b = symbols('a,b', real=True)
m = symbols('m', real=True)
def test_object_from_equation():
from sympy.abc import x, y, a, b
assert Line(3*x + y + 18) == Line2D(Point2D(0, -18), Point2D(1, -21))
assert Line(3*x + 5 * y + 1) == Line2D(
Point2D(0, Rational(-1, 5)), Point2D(1, Rational(-4, 5)))
assert Line(3*a + b + 18, x="a", y="b") == Line2D(
Point2D(0, -18), Point2D(1, -21))
assert Line(3*x + y) == Line2D(Point2D(0, 0), Point2D(1, -3))
assert Line(x + y) == Line2D(Point2D(0, 0), Point2D(1, -1))
assert Line(Eq(3*a + b, -18), x="a", y=b) == Line2D(
Point2D(0, -18), Point2D(1, -21))
# issue 22361
assert Line(x - 1) == Line2D(Point2D(1, 0), Point2D(1, 1))
assert Line(2*x - 2, y=x) == Line2D(Point2D(0, 1), Point2D(1, 1))
assert Line(y) == Line2D(Point2D(0, 0), Point2D(1, 0))
assert Line(2*y, x=y) == Line2D(Point2D(0, 0), Point2D(0, 1))
assert Line(y, x=y) == Line2D(Point2D(0, 0), Point2D(0, 1))
raises(ValueError, lambda: Line(x / y))
raises(ValueError, lambda: Line(a / b, x='a', y='b'))
raises(ValueError, lambda: Line(y / x))
raises(ValueError, lambda: Line(b / a, x='a', y='b'))
raises(ValueError, lambda: Line((x + 1)**2 + y))
def feq(a, b):
"""Test if two floating point values are 'equal'."""
t_float = Float("1.0E-10")
return -t_float < a - b < t_float
def test_angle_between():
a = Point(1, 2, 3, 4)
b = a.orthogonal_direction
o = a.origin
assert feq(Line.angle_between(Line(Point(0, 0), Point(1, 1)),
Line(Point(0, 0), Point(5, 0))).evalf(), pi.evalf() / 4)
assert Line(a, o).angle_between(Line(b, o)) == pi / 2
z = Point3D(0, 0, 0)
assert Line3D.angle_between(Line3D(z, Point3D(1, 1, 1)),
Line3D(z, Point3D(5, 0, 0))) == acos(sqrt(3) / 3)
# direction of points is used to determine angle
assert Line3D.angle_between(Line3D(z, Point3D(1, 1, 1)),
Line3D(Point3D(5, 0, 0), z)) == acos(-sqrt(3) / 3)
def test_closing_angle():
a = Ray((0, 0), angle=0)
b = Ray((1, 2), angle=pi/2)
assert a.closing_angle(b) == -pi/2
assert b.closing_angle(a) == pi/2
assert a.closing_angle(a) == 0
def test_smallest_angle():
a = Line(Point(1, 1), Point(1, 2))
b = Line(Point(1, 1),Point(2, 3))
assert a.smallest_angle_between(b) == acos(2*sqrt(5)/5)
def test_svg():
a = Line(Point(1, 1),Point(1, 2))
assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 1.00000000000000,1.00000000000000 L 1.00000000000000,2.00000000000000" marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>'
a = Segment(Point(1, 0),Point(1, 1))
assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 1.00000000000000,0 L 1.00000000000000,1.00000000000000" />'
a = Ray(Point(2, 3), Point(3, 5))
assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 2.00000000000000,3.00000000000000 L 3.00000000000000,5.00000000000000" marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>'
def test_arbitrary_point():
l1 = Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
l2 = Line(Point(x1, x1), Point(y1, y1))
assert l2.arbitrary_point() in l2
assert Ray((1, 1), angle=pi / 4).arbitrary_point() == \
Point(t + 1, t + 1)
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)
assert l1.perpendicular_segment(l1.arbitrary_point()) == l1.arbitrary_point()
assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]).arbitrary_point() == \
Point3D(t + 1, 2 * t + 1, 3 * t + 1)
assert Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1)).midpoint == \
Point3D(S.Half, S.Half, S.Half)
assert Segment3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1)).length == sqrt(3) * sqrt((x1 - y1) ** 2)
assert Segment3D((1, 1, 1), (2, 3, 4)).arbitrary_point() == \
Point3D(t + 1, 2 * t + 1, 3 * t + 1)
raises(ValueError, (lambda: Line((x, 1), (2, 3)).arbitrary_point(x)))
def test_are_concurrent_2d():
l1 = Line(Point(0, 0), Point(1, 1))
l2 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert Line.are_concurrent(l1) is False
assert Line.are_concurrent(l1, l2)
assert Line.are_concurrent(l1, l1, l1, l2)
assert Line.are_concurrent(l1, l2, Line(Point(5, x1), Point(Rational(-3, 5), x1)))
assert Line.are_concurrent(l1, Line(Point(0, 0), Point(-x1, x1)), l2) is False
def test_are_concurrent_3d():
p1 = Point3D(0, 0, 0)
l1 = Line(p1, Point3D(1, 1, 1))
parallel_1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
parallel_2 = Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))
assert Line3D.are_concurrent(l1) is False
assert Line3D.are_concurrent(l1, Line(Point3D(x1, x1, x1), Point3D(y1, y1, y1))) is False
assert Line3D.are_concurrent(l1, Line3D(p1, Point3D(x1, x1, x1)),
Line(Point3D(x1, x1, x1), Point3D(x1, 1 + x1, 1))) is True
assert Line3D.are_concurrent(parallel_1, parallel_2) is False
def test_arguments():
"""Functions accepting `Point` objects in `geometry`
should also accept tuples, lists, and generators and
automatically convert them to points."""
from sympy.utilities.iterables import subsets
singles2d = ((1, 2), [1, 3], Point(1, 5))
doubles2d = subsets(singles2d, 2)
l2d = Line(Point2D(1, 2), Point2D(2, 3))
singles3d = ((1, 2, 3), [1, 2, 4], Point(1, 2, 6))
doubles3d = subsets(singles3d, 2)
l3d = Line(Point3D(1, 2, 3), Point3D(1, 1, 2))
singles4d = ((1, 2, 3, 4), [1, 2, 3, 5], Point(1, 2, 3, 7))
doubles4d = subsets(singles4d, 2)
l4d = Line(Point(1, 2, 3, 4), Point(2, 2, 2, 2))
# test 2D
test_single = ['contains', 'distance', 'equals', 'parallel_line', 'perpendicular_line', 'perpendicular_segment',
'projection', 'intersection']
for p in doubles2d:
Line2D(*p)
for func in test_single:
for p in singles2d:
getattr(l2d, func)(p)
# test 3D
for p in doubles3d:
Line3D(*p)
for func in test_single:
for p in singles3d:
getattr(l3d, func)(p)
# test 4D
for p in doubles4d:
Line(*p)
for func in test_single:
for p in singles4d:
getattr(l4d, func)(p)
def test_basic_properties_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
p10 = Point(2000, 2000)
p_r3 = Ray(p1, p2).random_point()
p_r4 = Ray(p2, p1).random_point()
l1 = Line(p1, p2)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
l4 = Line(p1, Point(1, 0))
r1 = Ray(p1, Point(0, 1))
r2 = Ray(Point(0, 1), p1)
s1 = Segment(p1, p10)
p_s1 = s1.random_point()
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=oo).bounds == (1, 1, 1, 2)
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
assert Line(p1, p2).scale(2, 1) == Line(p1, Point(2, 1))
assert Line(p1, p2) == Line(p1, p2)
assert Line(p1, p2) != Line(p2, p1)
assert l1 != Line(Point(x1, x1), Point(y1, y1))
assert l1 != l3
assert Line(p1, p10) != Line(p10, p1)
assert Line(p1, p10) != p1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert s1 in Line(p1, p10)
assert Ray(Point(0, 0), Point(0, 1)) in Ray(Point(0, 0), Point(0, 2))
assert Ray(Point(0, 0), Point(0, 2)) in Ray(Point(0, 0), Point(0, 1))
assert Ray(Point(0, 0), Point(0, 2)).xdirection == S.Zero
assert Ray(Point(0, 0), Point(1, 2)).xdirection == S.Infinity
assert Ray(Point(0, 0), Point(-1, 2)).xdirection == S.NegativeInfinity
assert Ray(Point(0, 0), Point(2, 0)).ydirection == S.Zero
assert Ray(Point(0, 0), Point(2, 2)).ydirection == S.Infinity
assert Ray(Point(0, 0), Point(2, -2)).ydirection == S.NegativeInfinity
assert (r1 in s1) is False
assert Segment(p1, p2) in s1
assert Ray(Point(x1, x1), Point(x1, 1 + x1)) != Ray(p1, Point(-1, 5))
assert Segment(p1, p2).midpoint == Point(S.Half, S.Half)
assert Segment(p1, Point(-x1, x1)).length == sqrt(2 * (x1 ** 2))
assert l1.slope == 1
assert l3.slope is oo
assert l4.slope == 0
assert Line(p1, Point(0, 1)).slope is oo
assert Line(r1.source, r1.random_point()).slope == r1.slope
assert Line(r2.source, r2.random_point()).slope == r2.slope
assert Segment(Point(0, -1), Segment(p1, Point(0, 1)).random_point()).slope == Segment(p1, Point(0, 1)).slope
assert l4.coefficients == (0, 1, 0)
assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)
assert Line(p1, Point(0, 1)).coefficients == (1, 0, 0)
# issue 7963
r = Ray((0, 0), angle=x)
assert r.subs(x, 3 * pi / 4) == Ray((0, 0), (-1, 1))
assert r.subs(x, 5 * pi / 4) == Ray((0, 0), (-1, -1))
assert r.subs(x, -pi / 4) == Ray((0, 0), (1, -1))
assert r.subs(x, pi / 2) == Ray((0, 0), (0, 1))
assert r.subs(x, -pi / 2) == Ray((0, 0), (0, -1))
for ind in range(0, 5):
assert l3.random_point() in l3
assert p_r3.x >= p1.x and p_r3.y >= p1.y
assert p_r4.x <= p2.x and p_r4.y <= p2.y
assert p1.x <= p_s1.x <= p10.x and p1.y <= p_s1.y <= p10.y
assert hash(s1) != hash(Segment(p10, p1))
assert s1.plot_interval() == [t, 0, 1]
assert Line(p1, p10).plot_interval() == [t, -5, 5]
assert Ray((0, 0), angle=pi / 4).plot_interval() == [t, 0, 10]
def test_basic_properties_3d():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(x1, x1, x1)
p5 = Point3D(x1, 1 + x1, 1)
l1 = Line3D(p1, p2)
l3 = Line3D(p3, p5)
r1 = Ray3D(p1, Point3D(-1, 5, 0))
r3 = Ray3D(p1, p2)
s1 = Segment3D(p1, p2)
assert Line3D((1, 1, 1), direction_ratio=[2, 3, 4]) == Line3D(Point3D(1, 1, 1), Point3D(3, 4, 5))
assert Line3D((1, 1, 1), direction_ratio=[1, 5, 7]) == Line3D(Point3D(1, 1, 1), Point3D(2, 6, 8))
assert Line3D((1, 1, 1), direction_ratio=[1, 2, 3]) == Line3D(Point3D(1, 1, 1), Point3D(2, 3, 4))
assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).direction_cosine == [1, 0, 0]
assert Line3D(Line3D(p1, Point3D(0, 1, 0))) == Line3D(p1, Point3D(0, 1, 0))
assert Ray3D(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))) == Ray3D(p1, Point3D(1, 0, 0))
assert Line3D(p1, p2) != Line3D(p2, p1)
assert l1 != l3
assert l1 != Line3D(p3, Point3D(y1, y1, y1))
assert r3 != r1
assert Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) in Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2))
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)) in Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).xdirection == S.Infinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).ydirection == S.Infinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).zdirection == S.Infinity
assert Ray3D(Point3D(0, 0, 0), Point3D(-2, 2, 2)).xdirection == S.NegativeInfinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, -2, 2)).ydirection == S.NegativeInfinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, -2)).zdirection == S.NegativeInfinity
assert Ray3D(Point3D(0, 0, 0), Point3D(0, 2, 2)).xdirection == S.Zero
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 0, 2)).ydirection == S.Zero
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 0)).zdirection == S.Zero
assert p1 in l1
assert p1 not in l3
assert l1.direction_ratio == [1, 1, 1]
assert s1.midpoint == Point3D(S.Half, S.Half, S.Half)
# Test zdirection
assert Ray3D(p1, Point3D(0, 0, -1)).zdirection is S.NegativeInfinity
def test_contains():
p1 = Point(0, 0)
r = Ray(p1, Point(4, 4))
r1 = Ray3D(p1, Point3D(0, 0, -1))
r2 = Ray3D(p1, Point3D(0, 1, 0))
r3 = Ray3D(p1, Point3D(0, 0, 1))
l = Line(Point(0, 1), Point(3, 4))
# Segment contains
assert Point(0, (a + b) / 2) in Segment((0, a), (0, b))
assert Point((a + b) / 2, 0) in Segment((a, 0), (b, 0))
assert Point3D(0, 1, 0) in Segment3D((0, 1, 0), (0, 1, 0))
assert Point3D(1, 0, 0) in Segment3D((1, 0, 0), (1, 0, 0))
assert Segment3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).contains([]) is True
assert Segment3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).contains(
Segment3D(Point3D(2, 2, 2), Point3D(3, 2, 2))) is False
# Line contains
assert l.contains(Point(0, 1)) is True
assert l.contains((0, 1)) is True
assert l.contains((0, 0)) is False
# Ray contains
assert r.contains(p1) is True
assert r.contains((1, 1)) is True
assert r.contains((1, 3)) is False
assert r.contains(Segment((1, 1), (2, 2))) is True
assert r.contains(Segment((1, 2), (2, 5))) is False
assert r.contains(Ray((2, 2), (3, 3))) is True
assert r.contains(Ray((2, 2), (3, 5))) is False
assert r1.contains(Segment3D(p1, Point3D(0, 0, -10))) is True
assert r1.contains(Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))) is False
assert r2.contains(Point3D(0, 0, 0)) is True
assert r3.contains(Point3D(0, 0, 0)) is True
assert Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0)).contains([]) is False
assert Line3D((0, 0, 0), (x, y, z)).contains((2 * x, 2 * y, 2 * z))
with warns(UserWarning, test_stacklevel=False):
assert Line3D(p1, Point3D(0, 1, 0)).contains(Point(1.0, 1.0)) is False
with warns(UserWarning, test_stacklevel=False):
assert r3.contains(Point(1.0, 1.0)) is False
def test_contains_nonreal_symbols():
u, v, w, z = symbols('u, v, w, z')
l = Segment(Point(u, w), Point(v, z))
p = Point(u*Rational(2, 3) + v/3, w*Rational(2, 3) + z/3)
assert l.contains(p)
def test_distance_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
half = S.Half
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
r = Ray(p1, p2)
assert s1.distance(Point(0, 0)) == 0
assert s1.distance((0, 0)) == 0
assert s2.distance(Point(0, 0)) == 2 ** half / 2
assert s2.distance(Point(Rational(3) / 2, Rational(3) / 2)) == 2 ** half
assert Line(p1, p2).distance(Point(-1, 1)) == sqrt(2)
assert Line(p1, p2).distance(Point(1, -1)) == sqrt(2)
assert Line(p1, p2).distance(Point(2, 2)) == 0
assert Line(p1, p2).distance((-1, 1)) == sqrt(2)
assert Line((0, 0), (0, 1)).distance(p1) == 0
assert Line((0, 0), (0, 1)).distance(p2) == 1
assert Line((0, 0), (1, 0)).distance(p1) == 0
assert Line((0, 0), (1, 0)).distance(p2) == 1
assert r.distance(Point(-1, -1)) == sqrt(2)
assert r.distance(Point(1, 1)) == 0
assert r.distance(Point(-1, 1)) == sqrt(2)
assert Ray((1, 1), (2, 2)).distance(Point(1.5, 3)) == 3 * sqrt(2) / 4
assert r.distance((1, 1)) == 0
def test_dimension_normalization():
with warns(UserWarning, test_stacklevel=False):
assert Ray((1, 1), (2, 1, 2)) == Ray((1, 1, 0), (2, 1, 2))
def test_distance_3d():
p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
p3 = Point3D(Rational(3) / 2, Rational(3) / 2, Rational(3) / 2)
s1 = Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
s2 = Segment3D(Point3D(S.Half, S.Half, S.Half), Point3D(1, 0, 1))
r = Ray3D(p1, p2)
assert s1.distance(p1) == 0
assert s2.distance(p1) == sqrt(3) / 2
assert s2.distance(p3) == 2 * sqrt(6) / 3
assert s1.distance((0, 0, 0)) == 0
assert s2.distance((0, 0, 0)) == sqrt(3) / 2
assert s1.distance(p1) == 0
assert s2.distance(p1) == sqrt(3) / 2
assert s2.distance(p3) == 2 * sqrt(6) / 3
assert s1.distance((0, 0, 0)) == 0
assert s2.distance((0, 0, 0)) == sqrt(3) / 2
# Line to point
assert Line3D(p1, p2).distance(Point3D(-1, 1, 1)) == 2 * sqrt(6) / 3
assert Line3D(p1, p2).distance(Point3D(1, -1, 1)) == 2 * sqrt(6) / 3
assert Line3D(p1, p2).distance(Point3D(2, 2, 2)) == 0
assert Line3D(p1, p2).distance((2, 2, 2)) == 0
assert Line3D(p1, p2).distance((1, -1, 1)) == 2 * sqrt(6) / 3
assert Line3D((0, 0, 0), (0, 1, 0)).distance(p1) == 0
assert Line3D((0, 0, 0), (0, 1, 0)).distance(p2) == sqrt(2)
assert Line3D((0, 0, 0), (1, 0, 0)).distance(p1) == 0
assert Line3D((0, 0, 0), (1, 0, 0)).distance(p2) == sqrt(2)
# Ray to point
assert r.distance(Point3D(-1, -1, -1)) == sqrt(3)
assert r.distance(Point3D(1, 1, 1)) == 0
assert r.distance((-1, -1, -1)) == sqrt(3)
assert r.distance((1, 1, 1)) == 0
assert Ray3D((0, 0, 0), (1, 1, 2)).distance((-1, -1, 2)) == 4 * sqrt(3) / 3
assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, -3, -1)) == Rational(9) / 2
assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, 3, 1)) == sqrt(78) / 6
def test_equals():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l2 = Line((0, 5), slope=m)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert l1.perpendicular_line(p1.args).equals(Line(Point(0, 0), Point(1, -1)))
assert l1.perpendicular_line(p1).equals(Line(Point(0, 0), Point(1, -1)))
assert Line(Point(x1, x1), Point(y1, y1)).parallel_line(Point(-x1, x1)). \
equals(Line(Point(-x1, x1), Point(-y1, 2 * x1 - y1)))
assert l3.parallel_line(p1.args).equals(Line(Point(0, 0), Point(0, -1)))
assert l3.parallel_line(p1).equals(Line(Point(0, 0), Point(0, -1)))
assert (l2.distance(Point(2, 3)) - 2 * abs(m + 1) / sqrt(m ** 2 + 1)).equals(0)
assert Line3D(p1, Point3D(0, 1, 0)).equals(Point(1.0, 1.0)) is False
assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).equals(Line3D(Point3D(-5, 0, 0), Point3D(-1, 0, 0))) is True
assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).equals(Line3D(p1, Point3D(0, 1, 0))) is False
assert Ray3D(p1, Point3D(0, 0, -1)).equals(Point(1.0, 1.0)) is False
assert Ray3D(p1, Point3D(0, 0, -1)).equals(Ray3D(p1, Point3D(0, 0, -1))) is True
assert Line3D((0, 0), (t, t)).perpendicular_line(Point(0, 1, 0)).equals(
Line3D(Point3D(0, 1, 0), Point3D(S.Half, S.Half, 0)))
assert Line3D((0, 0), (t, t)).perpendicular_segment(Point(0, 1, 0)).equals(Segment3D((0, 1), (S.Half, S.Half)))
assert Line3D(p1, Point3D(0, 1, 0)).equals(Point(1.0, 1.0)) is False
def test_equation():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, Point(1, 0)).equation(x=x, y=y) == y
assert Line(p1, Point(0, 1)).equation() == x
assert Line(Point(2, 0), Point(2, 1)).equation() == x - 2
assert Line(p2, Point(2, 1)).equation() == y - 1
assert Line3D(Point(x1, x1, x1), Point(y1, y1, y1)
).equation() == (-x + y, -x + z)
assert Line3D(Point(1, 2, 3), Point(2, 3, 4)
).equation() == (-x + y - 1, -x + z - 2)
assert Line3D(Point(1, 2, 3), Point(1, 3, 4)
).equation() == (x - 1, -y + z - 1)
assert Line3D(Point(1, 2, 3), Point(2, 2, 4)
).equation() == (y - 2, -x + z - 2)
assert Line3D(Point(1, 2, 3), Point(2, 3, 3)
).equation() == (-x + y - 1, z - 3)
assert Line3D(Point(1, 2, 3), Point(1, 2, 4)
).equation() == (x - 1, y - 2)
assert Line3D(Point(1, 2, 3), Point(1, 3, 3)
).equation() == (x - 1, z - 3)
assert Line3D(Point(1, 2, 3), Point(2, 2, 3)
).equation() == (y - 2, z - 3)
with warns_deprecated_sympy():
assert Line3D(Point(1, 2, 3), Point(2, 2, 3)
).equation(k='k') == (y - 2, z - 3)
def test_intersection_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
l1 = Line(p1, p2)
l3 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(0, 0), Point(3, 4))
r4 = Ray(p1, p2)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
s1 = Segment(p1, p2)
s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
s3 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, p1) == [p1]
assert intersection(l1, Point(x1, 1 + x1)) == []
assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
assert intersection(l3, l3) == [l3]
assert intersection(l3, r2) == [r2]
assert intersection(l3, s3) == [s3]
assert intersection(s3, l3) == [s3]
assert intersection(Segment(Point(-10, 10), Point(10, 10)), Segment(Point(-5, -5), Point(-5, 5))) == []
assert intersection(r2, l3) == [r2]
assert intersection(r1, Ray(Point(2, 2), Point(0, 0))) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
assert intersection(r1, Segment(Point(0, 0), Point(2, 2))) == [Segment(Point(1, 1), Point(2, 2))]
assert r4.intersection(s2) == [s2]
assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
assert r4.intersection(Ray(p2, p1)) == [s1]
assert Ray(p2, p1).intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
assert Ray3D((0, 0), (3, 0)).intersection(Ray3D((1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
assert Ray3D((1, 0), (3, 0)).intersection(Ray3D((0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \
[Segment(Point(0, 0), Point(0, 1))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
assert Segment3D((1, 0), (2, 0)).intersection(
Segment3D((0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((3, 0), (4, 0))) == [Point3D((3, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((2, 0), (5, 0))) == [Segment3D((2, 0), (3, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((-2, 0), (0, 0))) == [Point3D(0, 0)]
assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
assert s1.intersection(Segment(Point(0.5, 0.5), Point(1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)]
assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
assert s1.intersection(s2) == [s2]
assert s2.intersection(s1) == [s2]
assert asa(120, 8, 52) == \
Triangle(
Point(0, 0),
Point(8, 0),
Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45)))
assert Line((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
assert Line((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))
assert s1.intersection(Ray((1, 1), (4, 4))) == [Point(1, 1)]
# This test is disabled because it hangs after rref changes which simplify
# intermediate results and return a different representation from when the
# test was written.
# # 16628 - this should be fast
# p0 = Point2D(Rational(249, 5), Rational(497999, 10000))
# p1 = Point2D((-58977084786*sqrt(405639795226) + 2030690077184193 +
# 20112207807*sqrt(630547164901) + 99600*sqrt(255775022850776494562626))
# /(2000*sqrt(255775022850776494562626) + 1991998000*sqrt(405639795226)
# + 1991998000*sqrt(630547164901) + 1622561172902000),
# (-498000*sqrt(255775022850776494562626) - 995999*sqrt(630547164901) +
# 90004251917891999 +
# 496005510002*sqrt(405639795226))/(10000*sqrt(255775022850776494562626)
# + 9959990000*sqrt(405639795226) + 9959990000*sqrt(630547164901) +
# 8112805864510000))
# p2 = Point2D(Rational(497, 10), Rational(-497, 10))
# p3 = Point2D(Rational(-497, 10), Rational(-497, 10))
# l = Line(p0, p1)
# s = Segment(p2, p3)
# n = (-52673223862*sqrt(405639795226) - 15764156209307469 -
# 9803028531*sqrt(630547164901) +
# 33200*sqrt(255775022850776494562626))
# d = sqrt(405639795226) + 315274080450 + 498000*sqrt(
# 630547164901) + sqrt(255775022850776494562626)
# assert intersection(l, s) == [
# Point2D(n/d*Rational(3, 2000), Rational(-497, 10))]
def test_line_intersection():
# see also test_issue_11238 in test_matrices.py
x0 = tan(pi*Rational(13, 45))
x1 = sqrt(3)
x2 = x0**2
x, y = [8*x0/(x0 + x1), (24*x0 - 8*x1*x2)/(x2 - 3)]
assert Line(Point(0, 0), Point(1, -sqrt(3))).contains(Point(x, y)) is True
def test_intersection_3d():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
l1 = Line3D(p1, p2)
l2 = Line3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
r1 = Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
r2 = Ray3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
s1 = Segment3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
assert intersection(l1, p1) == [p1]
assert intersection(l1, Point3D(x1, 1 + x1, 1)) == []
assert intersection(l1, l1.parallel_line(p1)) == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))]
assert intersection(l2, r2) == [r2]
assert intersection(l2, s1) == [s1]
assert intersection(r2, l2) == [r2]
assert intersection(r1, Ray3D(Point3D(1, 1, 1), Point3D(-1, -1, -1))) == [Point3D(1, 1, 1)]
assert intersection(r1, Segment3D(Point3D(0, 0, 0), Point3D(2, 2, 2))) == [
Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
assert intersection(Ray3D(Point3D(1, 0, 0), Point3D(-1, 0, 0)), Ray3D(Point3D(0, 1, 0), Point3D(0, -1, 0))) \
== [Point3D(0, 0, 0)]
assert intersection(r1, Ray3D(Point3D(2, 2, 2), Point3D(0, 0, 0))) == \
[Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
assert intersection(s1, r2) == [s1]
assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).intersection(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) == \
[Point3D(2, 2, 1)]
assert Line3D((0, 1, 2), (0, 2, 3)).intersection(Line3D((0, 1, 2), (0, 1, 1))) == [Point3D(0, 1, 2)]
assert Line3D((0, 0), (t, t)).intersection(Line3D((0, 1), (t, t))) == \
[Point3D(t, t)]
assert Ray3D(Point3D(0, 0, 0), Point3D(0, 4, 0)).intersection(Ray3D(Point3D(0, 1, 1), Point3D(0, -1, 1))) == []
def test_is_parallel():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(x1, x1, x1)
l2 = Line(Point(x1, x1), Point(y1, y1))
l2_1 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert Line.is_parallel(Line(Point(0, 0), Point(1, 1)), l2)
assert Line.is_parallel(l2, Line(Point(x1, x1), Point(x1, 1 + x1))) is False
assert Line.is_parallel(l2, l2.parallel_line(Point(-x1, x1)))
assert Line.is_parallel(l2_1, l2_1.parallel_line(Point(0, 0)))
assert Line3D(p1, p2).is_parallel(Line3D(p1, p2)) # same as in 2D
assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).is_parallel(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) is False
assert Line3D(p1, p2).parallel_line(p3) == Line3D(Point3D(x1, x1, x1),
Point3D(x1 + 1, x1 + 1, x1 + 1))
assert Line3D(p1, p2).parallel_line(p3.args) == \
Line3D(Point3D(x1, x1, x1), Point3D(x1 + 1, x1 + 1, x1 + 1))
assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).is_parallel(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) is False
def test_is_perpendicular():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l2 = Line(Point(x1, x1), Point(y1, y1))
l1_1 = Line(p1, Point(-x1, x1))
# 2D
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) is False
p = l1.random_point()
assert l1.perpendicular_segment(p) == p
# 3D
assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)),
Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))) is True
assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)),
Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))) is False
assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)),
Line3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1))) is False
def test_is_similar():
p1 = Point(2000, 2000)
p2 = p1.scale(2, 2)
r1 = Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0))
r2 = Ray(Point(0, 0), Point(0, 1))
s1 = Segment(Point(0, 0), p1)
assert s1.is_similar(Segment(p1, p2))
assert s1.is_similar(r2) is False
assert r1.is_similar(Line3D(Point3D(1, 1, 1), Point3D(1, 0, 0))) is True
assert r1.is_similar(Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))) is False
def test_length():
s2 = Segment3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1))
assert Line(Point(0, 0), Point(1, 1)).length is oo
assert s2.length == sqrt(3) * sqrt((x1 - y1) ** 2)
assert Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)).length is oo
def test_projection():
p1 = Point(0, 0)
p2 = Point3D(0, 0, 0)
p3 = Point(-x1, x1)
l1 = Line(p1, Point(1, 1))
l2 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
l3 = Line3D(p2, Point3D(1, 1, 1))
r1 = Ray(Point(1, 1), Point(2, 2))
s1 = Segment(Point2D(0, 0), Point2D(0, 1))
s2 = Segment(Point2D(1, 0), Point2D(2, 1/2))
assert Line(Point(x1, x1), Point(y1, y1)).projection(Point(y1, y1)) == Point(y1, y1)
assert Line(Point(x1, x1), Point(x1, 1 + x1)).projection(Point(1, 1)) == Point(x1, 1)
assert Segment(Point(-2, 2), Point(0, 4)).projection(r1) == Segment(Point(-1, 3), Point(0, 4))
assert Segment(Point(0, 4), Point(-2, 2)).projection(r1) == Segment(Point(0, 4), Point(-1, 3))
assert s2.projection(s1) == EmptySet
assert l1.projection(p3) == p1
assert l1.projection(Ray(p1, Point(-1, 5))) == Ray(Point(0, 0), Point(2, 2))
assert l1.projection(Ray(p1, Point(-1, 1))) == p1
assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1)
assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1)
assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2))
assert l3.projection(Ray3D(p2, Point3D(-1, 5, 0))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(4, 3), Rational(4, 3), Rational(4, 3)))
assert l3.projection(Ray3D(p2, Point3D(-1, 1, 1))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(1, 3), Rational(1, 3), Rational(1, 3)))
assert l2.projection(Point3D(5, 5, 0)) == Point3D(5, 0)
assert l2.projection(Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))).equals(l2)
def test_perpendicular_line():
# 3d - requires a particular orthogonal to be selected
p1, p2, p3 = Point(0, 0, 0), Point(2, 3, 4), Point(-2, 2, 0)
l = Line(p1, p2)
p = l.perpendicular_line(p3)
assert p.p1 == p3
assert p.p2 in l
# 2d - does not require special selection
p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
l = Line(p1, p2)
p = l.perpendicular_line(p3)
assert p.p1 == p3
# p is directed from l to p3
assert p.direction.unit == (p3 - l.projection(p3)).unit
def test_perpendicular_bisector():
s1 = Segment(Point(0, 0), Point(1, 1))
aline = Line(Point(S.Half, S.Half), Point(Rational(3, 2), Rational(-1, 2)))
on_line = Segment(Point(S.Half, S.Half), Point(Rational(3, 2), Rational(-1, 2))).midpoint
assert s1.perpendicular_bisector().equals(aline)
assert s1.perpendicular_bisector(on_line).equals(Segment(s1.midpoint, on_line))
assert s1.perpendicular_bisector(on_line + (1, 0)).equals(aline)
def test_raises():
d, e = symbols('a,b', real=True)
s = Segment((d, 0), (e, 0))
raises(TypeError, lambda: Line((1, 1), 1))
raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))
raises(Undecidable, lambda: Point(2 * d, 0) in s)
raises(ValueError, lambda: Ray3D(Point(1.0, 1.0)))
raises(ValueError, lambda: Line3D(Point3D(0, 0, 0), Point3D(0, 0, 0)))
raises(TypeError, lambda: Line3D((1, 1), 1))
raises(ValueError, lambda: Line3D(Point3D(0, 0, 0)))
raises(TypeError, lambda: Ray((1, 1), 1))
raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0))
.projection(Circle(Point(0, 0), 1)))
def test_ray_generation():
assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=4.05 * pi) == Ray(Point(1, 1),
Point(2, -sqrt(5) * sqrt(2 * sqrt(5) + 10) / 4 - sqrt(
2 * sqrt(5) + 10) / 4 + 2 + sqrt(5)))
assert Ray((1, 1), angle=4.02 * pi) == Ray(Point(1, 1),
Point(2, 1 + tan(4.02 * pi)))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + tan(5)))
assert Ray3D((1, 1, 1), direction_ratio=[4, 4, 4]) == Ray3D(Point3D(1, 1, 1), Point3D(5, 5, 5))
assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]) == Ray3D(Point3D(1, 1, 1), Point3D(2, 3, 4))
assert Ray3D((1, 1, 1), direction_ratio=[1, 1, 1]) == Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
def test_symbolic_intersect():
# Issue 7814.
circle = Circle(Point(x, 0), y)
line = Line(Point(k, z), slope=0)
assert line.intersection(circle) == [Point(x + sqrt((y - z) * (y + z)), z), Point(x - sqrt((y - z) * (y + z)), z)]
def test_issue_2941():
def _check():
for f, g in cartes(*[(Line, Ray, Segment)] * 2):
l1 = f(a, b)
l2 = g(c, d)
assert l1.intersection(l2) == l2.intersection(l1)
# intersect at end point
c, d = (-2, -2), (-2, 0)
a, b = (0, 0), (1, 1)
_check()
# midline intersection
c, d = (-2, -3), (-2, 0)
_check()
def test_parameter_value():
t = Symbol('t')
p1, p2 = Point(0, 1), Point(5, 6)
l = Line(p1, p2)
assert l.parameter_value((5, 6), t) == {t: 1}
raises(ValueError, lambda: l.parameter_value((0, 0), t))
def test_bisectors():
r1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
r2 = Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))
bisections = r1.bisectors(r2)
assert bisections == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 0)),
Line3D(Point3D(0, 0, 0), Point3D(1, -1, 0))]
ans = [Line3D(Point3D(0, 0, 0), Point3D(1, 0, 1)),
Line3D(Point3D(0, 0, 0), Point3D(-1, 0, 1))]
l1 = (0, 0, 0), (0, 0, 1)
l2 = (0, 0), (1, 0)
for a, b in cartes((Line, Segment, Ray), repeat=2):
assert a(*l1).bisectors(b(*l2)) == ans
def test_issue_8615():
a = Line3D(Point3D(6, 5, 0), Point3D(6, -6, 0))
b = Line3D(Point3D(6, -1, 19/10), Point3D(6, -1, 0))
assert a.intersection(b) == [Point3D(6, -1, 0)]
def test_issue_12598():
r1 = Ray(Point(0, 1), Point(0.98, 0.79).n(2))
r2 = Ray(Point(0, 0), Point(0.71, 0.71).n(2))
assert str(r1.intersection(r2)[0]) == 'Point2D(0.82, 0.82)'
l1 = Line((0, 0), (1, 1))
l2 = Segment((-1, 1), (0, -1)).n(2)
assert str(l1.intersection(l2)[0]) == 'Point2D(-0.33, -0.33)'
l2 = Segment((-1, 1), (-1/2, 1/2)).n(2)
assert not l1.intersection(l2)
|
69db29c50f447680904f0a7904563d612fa986e6843690042307fef62c366155 | from sympy.core.basic import Basic
from sympy.core.numbers import (I, Rational, pi)
from sympy.core.parameters import evaluate
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.geometry import Line, Point, Point2D, Point3D, Line3D, Plane
from sympy.geometry.entity import rotate, scale, translate, GeometryEntity
from sympy.matrices import Matrix
from sympy.utilities.iterables import subsets, permutations, cartes
from sympy.utilities.misc import Undecidable
from sympy.testing.pytest import raises, warns
def test_point():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
x1 = Symbol('x1', real=True)
x2 = Symbol('x2', real=True)
y1 = Symbol('y1', real=True)
y2 = Symbol('y2', real=True)
half = S.Half
p1 = Point(x1, x2)
p2 = Point(y1, y2)
p3 = Point(0, 0)
p4 = Point(1, 1)
p5 = Point(0, 1)
line = Line(Point(1, 0), slope=1)
assert p1 in p1
assert p1 not in p2
assert p2.y == y2
assert (p3 + p4) == p4
assert (p2 - p1) == Point(y1 - x1, y2 - x2)
assert -p2 == Point(-y1, -y2)
raises(TypeError, lambda: Point(1))
raises(ValueError, lambda: Point([1]))
raises(ValueError, lambda: Point(3, I))
raises(ValueError, lambda: Point(2*I, I))
raises(ValueError, lambda: Point(3 + I, I))
assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
assert Point.midpoint(p3, p4) == Point(half, half)
assert Point.midpoint(p1, p4) == Point(half + half*x1, half + half*x2)
assert Point.midpoint(p2, p2) == p2
assert p2.midpoint(p2) == p2
assert p1.origin == Point(0, 0)
assert Point.distance(p3, p4) == sqrt(2)
assert Point.distance(p1, p1) == 0
assert Point.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2)
raises(TypeError, lambda: Point.distance(p1, 0))
raises(TypeError, lambda: Point.distance(p1, GeometryEntity()))
# distance should be symmetric
assert p1.distance(line) == line.distance(p1)
assert p4.distance(line) == line.distance(p4)
assert Point.taxicab_distance(p4, p3) == 2
assert Point.canberra_distance(p4, p5) == 1
raises(ValueError, lambda: Point.canberra_distance(p3, p3))
p1_1 = Point(x1, x1)
p1_2 = Point(y2, y2)
p1_3 = Point(x1 + 1, x1)
assert Point.is_collinear(p3)
with warns(UserWarning, test_stacklevel=False):
assert Point.is_collinear(p3, Point(p3, dim=4))
assert p3.is_collinear()
assert Point.is_collinear(p3, p4)
assert Point.is_collinear(p3, p4, p1_1, p1_2)
assert Point.is_collinear(p3, p4, p1_1, p1_3) is False
assert Point.is_collinear(p3, p3, p4, p5) is False
raises(TypeError, lambda: Point.is_collinear(line))
raises(TypeError, lambda: p1_1.is_collinear(line))
assert p3.intersection(Point(0, 0)) == [p3]
assert p3.intersection(p4) == []
assert p3.intersection(line) == []
with warns(UserWarning, test_stacklevel=False):
assert Point.intersection(Point(0, 0, 0), Point(0, 0)) == [Point(0, 0, 0)]
x_pos = Symbol('x', positive=True)
p2_1 = Point(x_pos, 0)
p2_2 = Point(0, x_pos)
p2_3 = Point(-x_pos, 0)
p2_4 = Point(0, -x_pos)
p2_5 = Point(x_pos, 5)
assert Point.is_concyclic(p2_1)
assert Point.is_concyclic(p2_1, p2_2)
assert Point.is_concyclic(p2_1, p2_2, p2_3, p2_4)
for pts in permutations((p2_1, p2_2, p2_3, p2_5)):
assert Point.is_concyclic(*pts) is False
assert Point.is_concyclic(p4, p4 * 2, p4 * 3) is False
assert Point(0, 0).is_concyclic((1, 1), (2, 2), (2, 1)) is False
assert Point.is_concyclic(Point(0, 0, 0, 0), Point(1, 0, 0, 0), Point(1, 1, 0, 0), Point(1, 1, 1, 0)) is False
assert p1.is_scalar_multiple(p1)
assert p1.is_scalar_multiple(2*p1)
assert not p1.is_scalar_multiple(p2)
assert Point.is_scalar_multiple(Point(1, 1), (-1, -1))
assert Point.is_scalar_multiple(Point(0, 0), (0, -1))
# test when is_scalar_multiple can't be determined
raises(Undecidable, lambda: Point.is_scalar_multiple(Point(sympify("x1%y1"), sympify("x2%y2")), Point(0, 1)))
assert Point(0, 1).orthogonal_direction == Point(1, 0)
assert Point(1, 0).orthogonal_direction == Point(0, 1)
assert p1.is_zero is None
assert p3.is_zero
assert p4.is_zero is False
assert p1.is_nonzero is None
assert p3.is_nonzero is False
assert p4.is_nonzero
assert p4.scale(2, 3) == Point(2, 3)
assert p3.scale(2, 3) == p3
assert p4.rotate(pi, Point(0.5, 0.5)) == p3
assert p1.__radd__(p2) == p1.midpoint(p2).scale(2, 2)
assert (-p3).__rsub__(p4) == p3.midpoint(p4).scale(2, 2)
assert p4 * 5 == Point(5, 5)
assert p4 / 5 == Point(0.2, 0.2)
assert 5 * p4 == Point(5, 5)
raises(ValueError, lambda: Point(0, 0) + 10)
# Point differences should be simplified
assert Point(x*(x - 1), y) - Point(x**2 - x, y + 1) == Point(0, -1)
a, b = S.Half, Rational(1, 3)
assert Point(a, b).evalf(2) == \
Point(a.n(2), b.n(2), evaluate=False)
raises(ValueError, lambda: Point(1, 2) + 1)
# test project
assert Point.project((0, 1), (1, 0)) == Point(0, 0)
assert Point.project((1, 1), (1, 0)) == Point(1, 0)
raises(ValueError, lambda: Point.project(p1, Point(0, 0)))
# test transformations
p = Point(1, 0)
assert p.rotate(pi/2) == Point(0, 1)
assert p.rotate(pi/2, p) == p
p = Point(1, 1)
assert p.scale(2, 3) == Point(2, 3)
assert p.translate(1, 2) == Point(2, 3)
assert p.translate(1) == Point(2, 1)
assert p.translate(y=1) == Point(1, 2)
assert p.translate(*p.args) == Point(2, 2)
# Check invalid input for transform
raises(ValueError, lambda: p3.transform(p3))
raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
# test __contains__
assert 0 in Point(0, 0, 0, 0)
assert 1 not in Point(0, 0, 0, 0)
# test affine_rank
assert Point.affine_rank() == -1
def test_point3D():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
x1 = Symbol('x1', real=True)
x2 = Symbol('x2', real=True)
x3 = Symbol('x3', real=True)
y1 = Symbol('y1', real=True)
y2 = Symbol('y2', real=True)
y3 = Symbol('y3', real=True)
half = S.Half
p1 = Point3D(x1, x2, x3)
p2 = Point3D(y1, y2, y3)
p3 = Point3D(0, 0, 0)
p4 = Point3D(1, 1, 1)
p5 = Point3D(0, 1, 2)
assert p1 in p1
assert p1 not in p2
assert p2.y == y2
assert (p3 + p4) == p4
assert (p2 - p1) == Point3D(y1 - x1, y2 - x2, y3 - x3)
assert -p2 == Point3D(-y1, -y2, -y3)
assert Point(34.05, sqrt(3)) == Point(Rational(681, 20), sqrt(3))
assert Point3D.midpoint(p3, p4) == Point3D(half, half, half)
assert Point3D.midpoint(p1, p4) == Point3D(half + half*x1, half + half*x2,
half + half*x3)
assert Point3D.midpoint(p2, p2) == p2
assert p2.midpoint(p2) == p2
assert Point3D.distance(p3, p4) == sqrt(3)
assert Point3D.distance(p1, p1) == 0
assert Point3D.distance(p3, p2) == sqrt(p2.x**2 + p2.y**2 + p2.z**2)
p1_1 = Point3D(x1, x1, x1)
p1_2 = Point3D(y2, y2, y2)
p1_3 = Point3D(x1 + 1, x1, x1)
Point3D.are_collinear(p3)
assert Point3D.are_collinear(p3, p4)
assert Point3D.are_collinear(p3, p4, p1_1, p1_2)
assert Point3D.are_collinear(p3, p4, p1_1, p1_3) is False
assert Point3D.are_collinear(p3, p3, p4, p5) is False
assert p3.intersection(Point3D(0, 0, 0)) == [p3]
assert p3.intersection(p4) == []
assert p4 * 5 == Point3D(5, 5, 5)
assert p4 / 5 == Point3D(0.2, 0.2, 0.2)
assert 5 * p4 == Point3D(5, 5, 5)
raises(ValueError, lambda: Point3D(0, 0, 0) + 10)
# Test coordinate properties
assert p1.coordinates == (x1, x2, x3)
assert p2.coordinates == (y1, y2, y3)
assert p3.coordinates == (0, 0, 0)
assert p4.coordinates == (1, 1, 1)
assert p5.coordinates == (0, 1, 2)
assert p5.x == 0
assert p5.y == 1
assert p5.z == 2
# Point differences should be simplified
assert Point3D(x*(x - 1), y, 2) - Point3D(x**2 - x, y + 1, 1) == \
Point3D(0, -1, 1)
a, b, c = S.Half, Rational(1, 3), Rational(1, 4)
assert Point3D(a, b, c).evalf(2) == \
Point(a.n(2), b.n(2), c.n(2), evaluate=False)
raises(ValueError, lambda: Point3D(1, 2, 3) + 1)
# test transformations
p = Point3D(1, 1, 1)
assert p.scale(2, 3) == Point3D(2, 3, 1)
assert p.translate(1, 2) == Point3D(2, 3, 1)
assert p.translate(1) == Point3D(2, 1, 1)
assert p.translate(z=1) == Point3D(1, 1, 2)
assert p.translate(*p.args) == Point3D(2, 2, 2)
# Test __new__
assert Point3D(0.1, 0.2, evaluate=False, on_morph='ignore').args[0].is_Float
# Test length property returns correctly
assert p.length == 0
assert p1_1.length == 0
assert p1_2.length == 0
# Test are_colinear type error
raises(TypeError, lambda: Point3D.are_collinear(p, x))
# Test are_coplanar
assert Point.are_coplanar()
assert Point.are_coplanar((1, 2, 0), (1, 2, 0), (1, 3, 0))
assert Point.are_coplanar((1, 2, 0), (1, 2, 3))
with warns(UserWarning, test_stacklevel=False):
raises(ValueError, lambda: Point2D.are_coplanar((1, 2), (1, 2, 3)))
assert Point3D.are_coplanar((1, 2, 0), (1, 2, 3))
assert Point.are_coplanar((0, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 1)) is False
planar2 = Point3D(1, -1, 1)
planar3 = Point3D(-1, 1, 1)
assert Point3D.are_coplanar(p, planar2, planar3) == True
assert Point3D.are_coplanar(p, planar2, planar3, p3) == False
assert Point.are_coplanar(p, planar2)
planar2 = Point3D(1, 1, 2)
planar3 = Point3D(1, 1, 3)
assert Point3D.are_coplanar(p, planar2, planar3) # line, not plane
plane = Plane((1, 2, 1), (2, 1, 0), (3, 1, 2))
assert Point.are_coplanar(*[plane.projection(((-1)**i, i)) for i in range(4)])
# all 2D points are coplanar
assert Point.are_coplanar(Point(x, y), Point(x, x + y), Point(y, x + 2)) is True
# Test Intersection
assert planar2.intersection(Line3D(p, planar3)) == [Point3D(1, 1, 2)]
# Test Scale
assert planar2.scale(1, 1, 1) == planar2
assert planar2.scale(2, 2, 2, planar3) == Point3D(1, 1, 1)
assert planar2.scale(1, 1, 1, p3) == planar2
# Test Transform
identity = Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]])
assert p.transform(identity) == p
trans = Matrix([[1, 0, 0, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0, 0, 0, 1]])
assert p.transform(trans) == Point3D(2, 2, 2)
raises(ValueError, lambda: p.transform(p))
raises(ValueError, lambda: p.transform(Matrix([[1, 0], [0, 1]])))
# Test Equals
assert p.equals(x1) == False
# Test __sub__
p_4d = Point(0, 0, 0, 1)
with warns(UserWarning, test_stacklevel=False):
assert p - p_4d == Point(1, 1, 1, -1)
p_4d3d = Point(0, 0, 1, 0)
with warns(UserWarning, test_stacklevel=False):
assert p - p_4d3d == Point(1, 1, 0, 0)
def test_Point2D():
# Test Distance
p1 = Point2D(1, 5)
p2 = Point2D(4, 2.5)
p3 = (6, 3)
assert p1.distance(p2) == sqrt(61)/2
assert p2.distance(p3) == sqrt(17)/2
# Test coordinates
assert p1.x == 1
assert p1.y == 5
assert p2.x == 4
assert p2.y == 2.5
assert p1.coordinates == (1, 5)
assert p2.coordinates == (4, 2.5)
# test bounds
assert p1.bounds == (1, 5, 1, 5)
def test_issue_9214():
p1 = Point3D(4, -2, 6)
p2 = Point3D(1, 2, 3)
p3 = Point3D(7, 2, 3)
assert Point3D.are_collinear(p1, p2, p3) is False
def test_issue_11617():
p1 = Point3D(1,0,2)
p2 = Point2D(2,0)
with warns(UserWarning, test_stacklevel=False):
assert p1.distance(p2) == sqrt(5)
def test_transform():
p = Point(1, 1)
assert p.transform(rotate(pi/2)) == Point(-1, 1)
assert p.transform(scale(3, 2)) == Point(3, 2)
assert p.transform(translate(1, 2)) == Point(2, 3)
assert Point(1, 1).scale(2, 3, (4, 5)) == \
Point(-2, -7)
assert Point(1, 1).translate(4, 5) == \
Point(5, 6)
def test_concyclic_doctest_bug():
p1, p2 = Point(-1, 0), Point(1, 0)
p3, p4 = Point(0, 1), Point(-1, 2)
assert Point.is_concyclic(p1, p2, p3)
assert not Point.is_concyclic(p1, p2, p3, p4)
def test_arguments():
"""Functions accepting `Point` objects in `geometry`
should also accept tuples and lists and
automatically convert them to points."""
singles2d = ((1,2), [1,2], Point(1,2))
singles2d2 = ((1,3), [1,3], Point(1,3))
doubles2d = cartes(singles2d, singles2d2)
p2d = Point2D(1,2)
singles3d = ((1,2,3), [1,2,3], Point(1,2,3))
doubles3d = subsets(singles3d, 2)
p3d = Point3D(1,2,3)
singles4d = ((1,2,3,4), [1,2,3,4], Point(1,2,3,4))
doubles4d = subsets(singles4d, 2)
p4d = Point(1,2,3,4)
# test 2D
test_single = ['distance', 'is_scalar_multiple', 'taxicab_distance', 'midpoint', 'intersection', 'dot', 'equals', '__add__', '__sub__']
test_double = ['is_concyclic', 'is_collinear']
for p in singles2d:
Point2D(p)
for func in test_single:
for p in singles2d:
getattr(p2d, func)(p)
for func in test_double:
for p in doubles2d:
getattr(p2d, func)(*p)
# test 3D
test_double = ['is_collinear']
for p in singles3d:
Point3D(p)
for func in test_single:
for p in singles3d:
getattr(p3d, func)(p)
for func in test_double:
for p in doubles3d:
getattr(p3d, func)(*p)
# test 4D
test_double = ['is_collinear']
for p in singles4d:
Point(p)
for func in test_single:
for p in singles4d:
getattr(p4d, func)(p)
for func in test_double:
for p in doubles4d:
getattr(p4d, func)(*p)
# test evaluate=False for ops
x = Symbol('x')
a = Point(0, 1)
assert a + (0.1, x) == Point(0.1, 1 + x, evaluate=False)
a = Point(0, 1)
assert a/10.0 == Point(0, 0.1, evaluate=False)
a = Point(0, 1)
assert a*10.0 == Point(0.0, 10.0, evaluate=False)
# test evaluate=False when changing dimensions
u = Point(.1, .2, evaluate=False)
u4 = Point(u, dim=4, on_morph='ignore')
assert u4.args == (.1, .2, 0, 0)
assert all(i.is_Float for i in u4.args[:2])
# and even when *not* changing dimensions
assert all(i.is_Float for i in Point(u).args)
# never raise error if creating an origin
assert Point(dim=3, on_morph='error')
# raise error with unmatched dimension
raises(ValueError, lambda: Point(1, 1, dim=3, on_morph='error'))
# test unknown on_morph
raises(ValueError, lambda: Point(1, 1, dim=3, on_morph='unknown'))
# test invalid expressions
raises(TypeError, lambda: Point(Basic(), Basic()))
def test_unit():
assert Point(1, 1).unit == Point(sqrt(2)/2, sqrt(2)/2)
def test_dot():
raises(TypeError, lambda: Point(1, 2).dot(Line((0, 0), (1, 1))))
def test__normalize_dimension():
assert Point._normalize_dimension(Point(1, 2), Point(3, 4)) == [
Point(1, 2), Point(3, 4)]
assert Point._normalize_dimension(
Point(1, 2), Point(3, 4, 0), on_morph='ignore') == [
Point(1, 2, 0), Point(3, 4, 0)]
def test_issue_22684():
# Used to give an error
with evaluate(False):
Point(1, 2)
def test_direction_cosine():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
assert p1.direction_cosine(Point3D(1, 0, 0)) == [1, 0, 0]
assert p1.direction_cosine(Point3D(0, 1, 0)) == [0, 1, 0]
assert p1.direction_cosine(Point3D(0, 0, pi)) == [0, 0, 1]
assert p1.direction_cosine(Point3D(5, 0, 0)) == [1, 0, 0]
assert p1.direction_cosine(Point3D(0, sqrt(3), 0)) == [0, 1, 0]
assert p1.direction_cosine(Point3D(0, 0, 5)) == [0, 0, 1]
assert p1.direction_cosine(Point3D(2.4, 2.4, 0)) == [sqrt(2)/2, sqrt(2)/2, 0]
assert p1.direction_cosine(Point3D(1, 1, 1)) == [sqrt(3) / 3, sqrt(3) / 3, sqrt(3) / 3]
assert p1.direction_cosine(Point3D(-12, 0 -15)) == [-4*sqrt(41)/41, -5*sqrt(41)/41, 0]
assert p2.direction_cosine(Point3D(0, 0, 0)) == [-sqrt(3) / 3, -sqrt(3) / 3, -sqrt(3) / 3]
assert p2.direction_cosine(Point3D(1, 1, 12)) == [0, 0, 1]
assert p2.direction_cosine(Point3D(12, 1, 12)) == [sqrt(2) / 2, 0, sqrt(2) / 2]
|
efd8819108e95fe09d3b59ad31a5f7946bd57e4a290621612c2dd07d60ab7960 | from sympy.core.numbers import (Float, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, cos, sin)
from sympy.functions.elementary.trigonometric import tan
from sympy.geometry import (Circle, Ellipse, GeometryError, Point, Point2D,
Polygon, Ray, RegularPolygon, Segment, Triangle,
are_similar, convex_hull, intersection, Line, Ray2D)
from sympy.testing.pytest import raises, slow, warns
from sympy.core.random import verify_numerically
from sympy.geometry.polygon import rad, deg
from sympy.integrals.integrals import integrate
def feq(a, b):
"""Test if two floating point values are 'equal'."""
t_float = Float("1.0E-10")
return -t_float < a - b < t_float
@slow
def test_polygon():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
q = Symbol('q', real=True)
u = Symbol('u', real=True)
v = Symbol('v', real=True)
w = Symbol('w', real=True)
x1 = Symbol('x1', real=True)
half = S.Half
a, b, c = Point(0, 0), Point(2, 0), Point(3, 3)
t = Triangle(a, b, c)
assert Polygon(Point(0, 0)) == Point(0, 0)
assert Polygon(a, Point(1, 0), b, c) == t
assert Polygon(Point(1, 0), b, c, a) == t
assert Polygon(b, c, a, Point(1, 0)) == t
# 2 "remove folded" tests
assert Polygon(a, Point(3, 0), b, c) == t
assert Polygon(a, b, Point(3, -1), b, c) == t
# remove multiple collinear points
assert Polygon(Point(-4, 15), Point(-11, 15), Point(-15, 15),
Point(-15, 33/5), Point(-15, -87/10), Point(-15, -15),
Point(-42/5, -15), Point(-2, -15), Point(7, -15), Point(15, -15),
Point(15, -3), Point(15, 10), Point(15, 15)) == \
Polygon(Point(-15, -15), Point(15, -15), Point(15, 15), Point(-15, 15))
p1 = Polygon(
Point(0, 0), Point(3, -1),
Point(6, 0), Point(4, 5),
Point(2, 3), Point(0, 3))
p2 = Polygon(
Point(6, 0), Point(3, -1),
Point(0, 0), Point(0, 3),
Point(2, 3), Point(4, 5))
p3 = Polygon(
Point(0, 0), Point(3, 0),
Point(5, 2), Point(4, 4))
p4 = Polygon(
Point(0, 0), Point(4, 4),
Point(5, 2), Point(3, 0))
p5 = Polygon(
Point(0, 0), Point(4, 4),
Point(0, 4))
p6 = Polygon(
Point(-11, 1), Point(-9, 6.6),
Point(-4, -3), Point(-8.4, -8.7))
p7 = Polygon(
Point(x, y), Point(q, u),
Point(v, w))
p8 = Polygon(
Point(x, y), Point(v, w),
Point(q, u))
p9 = Polygon(
Point(0, 0), Point(4, 4),
Point(3, 0), Point(5, 2))
p10 = Polygon(
Point(0, 2), Point(2, 2),
Point(0, 0), Point(2, 0))
p11 = Polygon(Point(0, 0), 1, n=3)
p12 = Polygon(Point(0, 0), 1, 0, n=3)
r = Ray(Point(-9, 6.6), Point(-9, 5.5))
#
# General polygon
#
assert p1 == p2
assert len(p1.args) == 6
assert len(p1.sides) == 6
assert p1.perimeter == 5 + 2*sqrt(10) + sqrt(29) + sqrt(8)
assert p1.area == 22
assert not p1.is_convex()
assert Polygon((-1, 1), (2, -1), (2, 1), (-1, -1), (3, 0)
).is_convex() is False
# ensure convex for both CW and CCW point specification
assert p3.is_convex()
assert p4.is_convex()
dict5 = p5.angles
assert dict5[Point(0, 0)] == pi / 4
assert dict5[Point(0, 4)] == pi / 2
assert p5.encloses_point(Point(x, y)) is None
assert p5.encloses_point(Point(1, 3))
assert p5.encloses_point(Point(0, 0)) is False
assert p5.encloses_point(Point(4, 0)) is False
assert p1.encloses(Circle(Point(2.5, 2.5), 5)) is False
assert p1.encloses(Ellipse(Point(2.5, 2), 5, 6)) is False
assert p5.plot_interval('x') == [x, 0, 1]
assert p5.distance(
Polygon(Point(10, 10), Point(14, 14), Point(10, 14))) == 6 * sqrt(2)
assert p5.distance(
Polygon(Point(1, 8), Point(5, 8), Point(8, 12), Point(1, 12))) == 4
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
Polygon(Point(0, 0), Point(1, 0), Point(1, 1)).distance(
Polygon(Point(0, 0), Point(0, 1), Point(1, 1)))
assert hash(p5) == hash(Polygon(Point(0, 0), Point(4, 4), Point(0, 4)))
assert hash(p1) == hash(p2)
assert hash(p7) == hash(p8)
assert hash(p3) != hash(p9)
assert p5 == Polygon(Point(4, 4), Point(0, 4), Point(0, 0))
assert Polygon(Point(4, 4), Point(0, 4), Point(0, 0)) in p5
assert p5 != Point(0, 4)
assert Point(0, 1) in p5
assert p5.arbitrary_point('t').subs(Symbol('t', real=True), 0) == \
Point(0, 0)
raises(ValueError, lambda: Polygon(
Point(x, 0), Point(0, y), Point(x, y)).arbitrary_point('x'))
assert p6.intersection(r) == [Point(-9, Rational(-84, 13)), Point(-9, Rational(33, 5))]
assert p10.area == 0
assert p11 == RegularPolygon(Point(0, 0), 1, 3, 0)
assert p11 == p12
assert p11.vertices[0] == Point(1, 0)
assert p11.args[0] == Point(0, 0)
p11.spin(pi/2)
assert p11.vertices[0] == Point(0, 1)
#
# Regular polygon
#
p1 = RegularPolygon(Point(0, 0), 10, 5)
p2 = RegularPolygon(Point(0, 0), 5, 5)
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), Point(0,
1), Point(1, 1)))
raises(GeometryError, lambda: RegularPolygon(Point(0, 0), 1, 2))
raises(ValueError, lambda: RegularPolygon(Point(0, 0), 1, 2.5))
assert p1 != p2
assert p1.interior_angle == pi*Rational(3, 5)
assert p1.exterior_angle == pi*Rational(2, 5)
assert p2.apothem == 5*cos(pi/5)
assert p2.circumcenter == p1.circumcenter == Point(0, 0)
assert p1.circumradius == p1.radius == 10
assert p2.circumcircle == Circle(Point(0, 0), 5)
assert p2.incircle == Circle(Point(0, 0), p2.apothem)
assert p2.inradius == p2.apothem == (5 * (1 + sqrt(5)) / 4)
p2.spin(pi / 10)
dict1 = p2.angles
assert dict1[Point(0, 5)] == 3 * pi / 5
assert p1.is_convex()
assert p1.rotation == 0
assert p1.encloses_point(Point(0, 0))
assert p1.encloses_point(Point(11, 0)) is False
assert p2.encloses_point(Point(0, 4.9))
p1.spin(pi/3)
assert p1.rotation == pi/3
assert p1.vertices[0] == Point(5, 5*sqrt(3))
for var in p1.args:
if isinstance(var, Point):
assert var == Point(0, 0)
else:
assert var in (5, 10, pi / 3)
assert p1 != Point(0, 0)
assert p1 != p5
# while spin works in place (notice that rotation is 2pi/3 below)
# rotate returns a new object
p1_old = p1
assert p1.rotate(pi/3) == RegularPolygon(Point(0, 0), 10, 5, pi*Rational(2, 3))
assert p1 == p1_old
assert p1.area == (-250*sqrt(5) + 1250)/(4*tan(pi/5))
assert p1.length == 20*sqrt(-sqrt(5)/8 + Rational(5, 8))
assert p1.scale(2, 2) == \
RegularPolygon(p1.center, p1.radius*2, p1._n, p1.rotation)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3) == \
Polygon(Point(2, 0), Point(0, 3), Point(-2, 0), Point(0, -3))
assert repr(p1) == str(p1)
#
# Angles
#
angles = p4.angles
assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))
angles = p3.angles
assert feq(angles[Point(0, 0)].evalf(), Float("0.7853981633974483"))
assert feq(angles[Point(4, 4)].evalf(), Float("1.2490457723982544"))
assert feq(angles[Point(5, 2)].evalf(), Float("1.8925468811915388"))
assert feq(angles[Point(3, 0)].evalf(), Float("2.3561944901923449"))
#
# Triangle
#
p1 = Point(0, 0)
p2 = Point(5, 0)
p3 = Point(0, 5)
t1 = Triangle(p1, p2, p3)
t2 = Triangle(p1, p2, Point(Rational(5, 2), sqrt(Rational(75, 4))))
t3 = Triangle(p1, Point(x1, 0), Point(0, x1))
s1 = t1.sides
assert Triangle(p1, p2, p1) == Polygon(p1, p2, p1) == Segment(p1, p2)
raises(GeometryError, lambda: Triangle(Point(0, 0)))
# Basic stuff
assert Triangle(p1, p1, p1) == p1
assert Triangle(p2, p2*2, p2*3) == Segment(p2, p2*3)
assert t1.area == Rational(25, 2)
assert t1.is_right()
assert t2.is_right() is False
assert t3.is_right()
assert p1 in t1
assert t1.sides[0] in t1
assert Segment((0, 0), (1, 0)) in t1
assert Point(5, 5) not in t2
assert t1.is_convex()
assert feq(t1.angles[p1].evalf(), pi.evalf()/2)
assert t1.is_equilateral() is False
assert t2.is_equilateral()
assert t3.is_equilateral() is False
assert are_similar(t1, t2) is False
assert are_similar(t1, t3)
assert are_similar(t2, t3) is False
assert t1.is_similar(Point(0, 0)) is False
assert t1.is_similar(t2) is False
# Bisectors
bisectors = t1.bisectors()
assert bisectors[p1] == Segment(
p1, Point(Rational(5, 2), Rational(5, 2)))
assert t2.bisectors()[p2] == Segment(
Point(5, 0), Point(Rational(5, 4), 5*sqrt(3)/4))
p4 = Point(0, x1)
assert t3.bisectors()[p4] == Segment(p4, Point(x1*(sqrt(2) - 1), 0))
ic = (250 - 125*sqrt(2))/50
assert t1.incenter == Point(ic, ic)
# Inradius
assert t1.inradius == t1.incircle.radius == 5 - 5*sqrt(2)/2
assert t2.inradius == t2.incircle.radius == 5*sqrt(3)/6
assert t3.inradius == t3.incircle.radius == x1**2/((2 + sqrt(2))*Abs(x1))
# Exradius
assert t1.exradii[t1.sides[2]] == 5*sqrt(2)/2
# Excenters
assert t1.excenters[t1.sides[2]] == Point2D(25*sqrt(2), -5*sqrt(2)/2)
# Circumcircle
assert t1.circumcircle.center == Point(2.5, 2.5)
# Medians + Centroid
m = t1.medians
assert t1.centroid == Point(Rational(5, 3), Rational(5, 3))
assert m[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert t3.medians[p1] == Segment(p1, Point(x1/2, x1/2))
assert intersection(m[p1], m[p2], m[p3]) == [t1.centroid]
assert t1.medial == Triangle(Point(2.5, 0), Point(0, 2.5), Point(2.5, 2.5))
# Nine-point circle
assert t1.nine_point_circle == Circle(Point(2.5, 0),
Point(0, 2.5), Point(2.5, 2.5))
assert t1.nine_point_circle == Circle(Point(0, 0),
Point(0, 2.5), Point(2.5, 2.5))
# Perpendicular
altitudes = t1.altitudes
assert altitudes[p1] == Segment(p1, Point(Rational(5, 2), Rational(5, 2)))
assert altitudes[p2].equals(s1[0])
assert altitudes[p3] == s1[2]
assert t1.orthocenter == p1
t = S('''Triangle(
Point(100080156402737/5000000000000, 79782624633431/500000000000),
Point(39223884078253/2000000000000, 156345163124289/1000000000000),
Point(31241359188437/1250000000000, 338338270939941/1000000000000000))''')
assert t.orthocenter == S('''Point(-780660869050599840216997'''
'''79471538701955848721853/80368430960602242240789074233100000000000000,'''
'''20151573611150265741278060334545897615974257/16073686192120448448157'''
'''8148466200000000000)''')
# Ensure
assert len(intersection(*bisectors.values())) == 1
assert len(intersection(*altitudes.values())) == 1
assert len(intersection(*m.values())) == 1
# Distance
p1 = Polygon(
Point(0, 0), Point(1, 0),
Point(1, 1), Point(0, 1))
p2 = Polygon(
Point(0, Rational(5)/4), Point(1, Rational(5)/4),
Point(1, Rational(9)/4), Point(0, Rational(9)/4))
p3 = Polygon(
Point(1, 2), Point(2, 2),
Point(2, 1))
p4 = Polygon(
Point(1, 1), Point(Rational(6)/5, 1),
Point(1, Rational(6)/5))
pt1 = Point(half, half)
pt2 = Point(1, 1)
'''Polygon to Point'''
assert p1.distance(pt1) == half
assert p1.distance(pt2) == 0
assert p2.distance(pt1) == Rational(3)/4
assert p3.distance(pt2) == sqrt(2)/2
'''Polygon to Polygon'''
# p1.distance(p2) emits a warning
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
assert p1.distance(p2) == half/2
assert p1.distance(p3) == sqrt(2)/2
# p3.distance(p4) emits a warning
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output"):
assert p3.distance(p4) == (sqrt(2)/2 - sqrt(Rational(2)/25)/2)
def test_convex_hull():
p = [Point(-5, -1), Point(-2, 1), Point(-2, -1), Point(-1, -3), \
Point(0, 0), Point(1, 1), Point(2, 2), Point(2, -1), Point(3, 1), \
Point(4, -1), Point(6, 2)]
ch = Polygon(p[0], p[3], p[9], p[10], p[6], p[1])
#test handling of duplicate points
p.append(p[3])
#more than 3 collinear points
another_p = [Point(-45, -85), Point(-45, 85), Point(-45, 26), \
Point(-45, -24)]
ch2 = Segment(another_p[0], another_p[1])
assert convex_hull(*another_p) == ch2
assert convex_hull(*p) == ch
assert convex_hull(p[0]) == p[0]
assert convex_hull(p[0], p[1]) == Segment(p[0], p[1])
# no unique points
assert convex_hull(*[p[-1]]*3) == p[-1]
# collection of items
assert convex_hull(*[Point(0, 0), \
Segment(Point(1, 0), Point(1, 1)), \
RegularPolygon(Point(2, 0), 2, 4)]) == \
Polygon(Point(0, 0), Point(2, -2), Point(4, 0), Point(2, 2))
def test_encloses():
# square with a dimpled left side
s = Polygon(Point(0, 0), Point(1, 0), Point(1, 1), Point(0, 1), \
Point(S.Half, S.Half))
# the following is True if the polygon isn't treated as closing on itself
assert s.encloses(Point(0, S.Half)) is False
assert s.encloses(Point(S.Half, S.Half)) is False # it's a vertex
assert s.encloses(Point(Rational(3, 4), S.Half)) is True
def test_triangle_kwargs():
assert Triangle(sss=(3, 4, 5)) == \
Triangle(Point(0, 0), Point(3, 0), Point(3, 4))
assert Triangle(asa=(30, 2, 30)) == \
Triangle(Point(0, 0), Point(2, 0), Point(1, sqrt(3)/3))
assert Triangle(sas=(1, 45, 2)) == \
Triangle(Point(0, 0), Point(2, 0), Point(sqrt(2)/2, sqrt(2)/2))
assert Triangle(sss=(1, 2, 5)) is None
assert deg(rad(180)) == 180
def test_transform():
pts = [Point(0, 0), Point(S.Half, Rational(1, 4)), Point(1, 1)]
pts_out = [Point(-4, -10), Point(-3, Rational(-37, 4)), Point(-2, -7)]
assert Triangle(*pts).scale(2, 3, (4, 5)) == Triangle(*pts_out)
assert RegularPolygon((0, 0), 1, 4).scale(2, 3, (4, 5)) == \
Polygon(Point(-2, -10), Point(-4, -7), Point(-6, -10), Point(-4, -13))
# Checks for symmetric scaling
assert RegularPolygon((0, 0), 1, 4).scale(2, 2) == \
RegularPolygon(Point2D(0, 0), 2, 4, 0)
def test_reflect():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
b = Symbol('b')
m = Symbol('m')
l = Line((0, b), slope=m)
p = Point(x, y)
r = p.reflect(l)
dp = l.perpendicular_segment(p).length
dr = l.perpendicular_segment(r).length
assert verify_numerically(dp, dr)
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=oo)) \
== Triangle(Point(5, 0), Point(4, 0), Point(4, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=oo)) \
== Triangle(Point(-1, 0), Point(-2, 0), Point(-2, 2))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((0, 3), slope=0)) \
== Triangle(Point(1, 6), Point(2, 6), Point(2, 4))
assert Polygon((1, 0), (2, 0), (2, 2)).reflect(Line((3, 0), slope=0)) \
== Triangle(Point(1, 0), Point(2, 0), Point(2, -2))
def test_bisectors():
p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
p = Polygon(Point(0, 0), Point(2, 0), Point(1, 1), Point(0, 3))
q = Polygon(Point(1, 0), Point(2, 0), Point(3, 3), Point(-1, 5))
poly = Polygon(Point(3, 4), Point(0, 0), Point(8, 7), Point(-1, 1), Point(19, -19))
t = Triangle(p1, p2, p3)
assert t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
assert p.bisectors()[Point2D(0, 3)] == Ray2D(Point2D(0, 3), \
Point2D(sin(acos(2*sqrt(5)/5)/2), 3 - cos(acos(2*sqrt(5)/5)/2)))
assert q.bisectors()[Point2D(-1, 5)] == \
Ray2D(Point2D(-1, 5), Point2D(-1 + sqrt(29)*(5*sin(acos(9*sqrt(145)/145)/2) + \
2*cos(acos(9*sqrt(145)/145)/2))/29, sqrt(29)*(-5*cos(acos(9*sqrt(145)/145)/2) + \
2*sin(acos(9*sqrt(145)/145)/2))/29 + 5))
assert poly.bisectors()[Point2D(-1, 1)] == Ray2D(Point2D(-1, 1), \
Point2D(-1 + sin(acos(sqrt(26)/26)/2 + pi/4), 1 - sin(-acos(sqrt(26)/26)/2 + pi/4)))
def test_incenter():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).incenter \
== Point(1 - sqrt(2)/2, 1 - sqrt(2)/2)
def test_inradius():
assert Triangle(Point(0, 0), Point(4, 0), Point(0, 3)).inradius == 1
def test_incircle():
assert Triangle(Point(0, 0), Point(2, 0), Point(0, 2)).incircle \
== Circle(Point(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2))
def test_exradii():
t = Triangle(Point(0, 0), Point(6, 0), Point(0, 2))
assert t.exradii[t.sides[2]] == (-2 + sqrt(10))
def test_medians():
t = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
assert t.medians[Point(0, 0)] == Segment(Point(0, 0), Point(S.Half, S.Half))
def test_medial():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).medial \
== Triangle(Point(S.Half, 0), Point(S.Half, S.Half), Point(0, S.Half))
def test_nine_point_circle():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).nine_point_circle \
== Circle(Point2D(Rational(1, 4), Rational(1, 4)), sqrt(2)/4)
def test_eulerline():
assert Triangle(Point(0, 0), Point(1, 0), Point(0, 1)).eulerline \
== Line(Point2D(0, 0), Point2D(S.Half, S.Half))
assert Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3))).eulerline \
== Point2D(5, 5*sqrt(3)/3)
assert Triangle(Point(4, -6), Point(4, -1), Point(-3, 3)).eulerline \
== Line(Point2D(Rational(64, 7), 3), Point2D(Rational(-29, 14), Rational(-7, 2)))
def test_intersection():
poly1 = Triangle(Point(0, 0), Point(1, 0), Point(0, 1))
poly2 = Polygon(Point(0, 1), Point(-5, 0),
Point(0, -4), Point(0, Rational(1, 5)),
Point(S.Half, -0.1), Point(1, 0), Point(0, 1))
assert poly1.intersection(poly2) == [Point2D(Rational(1, 3), 0),
Segment(Point(0, Rational(1, 5)), Point(0, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly1) == [Point(Rational(1, 3), 0),
Segment(Point(0, 0), Point(0, Rational(1, 5))),
Segment(Point(1, 0), Point(0, 1))]
assert poly1.intersection(Point(0, 0)) == [Point(0, 0)]
assert poly1.intersection(Point(-12, -43)) == []
assert poly2.intersection(Line((-12, 0), (12, 0))) == [Point(-5, 0),
Point(0, 0), Point(Rational(1, 3), 0), Point(1, 0)]
assert poly2.intersection(Line((-12, 12), (12, 12))) == []
assert poly2.intersection(Ray((-3, 4), (1, 0))) == [Segment(Point(1, 0),
Point(0, 1))]
assert poly2.intersection(Circle((0, -1), 1)) == [Point(0, -2),
Point(0, 0)]
assert poly1.intersection(poly1) == [Segment(Point(0, 0), Point(1, 0)),
Segment(Point(0, 1), Point(0, 0)), Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(poly2) == [Segment(Point(-5, 0), Point(0, -4)),
Segment(Point(0, -4), Point(0, Rational(1, 5))),
Segment(Point(0, Rational(1, 5)), Point(S.Half, Rational(-1, 10))),
Segment(Point(0, 1), Point(-5, 0)),
Segment(Point(S.Half, Rational(-1, 10)), Point(1, 0)),
Segment(Point(1, 0), Point(0, 1))]
assert poly2.intersection(Triangle(Point(0, 1), Point(1, 0), Point(-1, 1))) \
== [Point(Rational(-5, 7), Rational(6, 7)), Segment(Point2D(0, 1), Point(1, 0))]
assert poly1.intersection(RegularPolygon((-12, -15), 3, 3)) == []
def test_parameter_value():
t = Symbol('t')
sq = Polygon((0, 0), (0, 1), (1, 1), (1, 0))
assert sq.parameter_value((0.5, 1), t) == {t: Rational(3, 8)}
q = Polygon((0, 0), (2, 1), (2, 4), (4, 0))
assert q.parameter_value((4, 0), t) == {t: -6 + 3*sqrt(5)} # ~= 0.708
raises(ValueError, lambda: sq.parameter_value((5, 6), t))
raises(ValueError, lambda: sq.parameter_value(Circle(Point(0, 0), 1), t))
def test_issue_12966():
poly = Polygon(Point(0, 0), Point(0, 10), Point(5, 10), Point(5, 5),
Point(10, 5), Point(10, 0))
t = Symbol('t')
pt = poly.arbitrary_point(t)
DELTA = 5/poly.perimeter
assert [pt.subs(t, DELTA*i) for i in range(int(1/DELTA))] == [
Point(0, 0), Point(0, 5), Point(0, 10), Point(5, 10),
Point(5, 5), Point(10, 5), Point(10, 0), Point(5, 0)]
def test_second_moment_of_area():
x, y = symbols('x, y')
# triangle
p1, p2, p3 = [(0, 0), (4, 0), (0, 2)]
p = (0, 0)
# equation of hypotenuse
eq_y = (1-x/4)*2
I_yy = integrate((x**2) * (integrate(1, (y, 0, eq_y))), (x, 0, 4))
I_xx = integrate(1 * (integrate(y**2, (y, 0, eq_y))), (x, 0, 4))
I_xy = integrate(x * (integrate(y, (y, 0, eq_y))), (x, 0, 4))
triangle = Polygon(p1, p2, p3)
assert (I_xx - triangle.second_moment_of_area(p)[0]) == 0
assert (I_yy - triangle.second_moment_of_area(p)[1]) == 0
assert (I_xy - triangle.second_moment_of_area(p)[2]) == 0
# rectangle
p1, p2, p3, p4=[(0, 0), (4, 0), (4, 2), (0, 2)]
I_yy = integrate((x**2) * integrate(1, (y, 0, 2)), (x, 0, 4))
I_xx = integrate(1 * integrate(y**2, (y, 0, 2)), (x, 0, 4))
I_xy = integrate(x * integrate(y, (y, 0, 2)), (x, 0, 4))
rectangle = Polygon(p1, p2, p3, p4)
assert (I_xx - rectangle.second_moment_of_area(p)[0]) == 0
assert (I_yy - rectangle.second_moment_of_area(p)[1]) == 0
assert (I_xy - rectangle.second_moment_of_area(p)[2]) == 0
r = RegularPolygon(Point(0, 0), 5, 3)
assert r.second_moment_of_area() == (1875*sqrt(3)/S(32), 1875*sqrt(3)/S(32), 0)
def test_first_moment():
a, b = symbols('a, b', positive=True)
# rectangle
p1 = Polygon((0, 0), (a, 0), (a, b), (0, b))
assert p1.first_moment_of_area() == (a*b**2/8, a**2*b/8)
assert p1.first_moment_of_area((a/3, b/4)) == (-3*a*b**2/32, -a**2*b/9)
p1 = Polygon((0, 0), (40, 0), (40, 30), (0, 30))
assert p1.first_moment_of_area() == (4500, 6000)
# triangle
p2 = Polygon((0, 0), (a, 0), (a/2, b))
assert p2.first_moment_of_area() == (4*a*b**2/81, a**2*b/24)
assert p2.first_moment_of_area((a/8, b/6)) == (-25*a*b**2/648, -5*a**2*b/768)
p2 = Polygon((0, 0), (12, 0), (12, 30))
assert p2.first_moment_of_area() == (S(1600)/3, -S(640)/3)
def test_section_modulus_and_polar_second_moment_of_area():
a, b = symbols('a, b', positive=True)
x, y = symbols('x, y')
rectangle = Polygon((0, b), (0, 0), (a, 0), (a, b))
assert rectangle.section_modulus(Point(x, y)) == (a*b**3/12/(-b/2 + y), a**3*b/12/(-a/2 + x))
assert rectangle.polar_second_moment_of_area() == a**3*b/12 + a*b**3/12
convex = RegularPolygon((0, 0), 1, 6)
assert convex.section_modulus() == (Rational(5, 8), sqrt(3)*Rational(5, 16))
assert convex.polar_second_moment_of_area() == 5*sqrt(3)/S(8)
concave = Polygon((0, 0), (1, 8), (3, 4), (4, 6), (7, 1))
assert concave.section_modulus() == (Rational(-6371, 429), Rational(-9778, 519))
assert concave.polar_second_moment_of_area() == Rational(-38669, 252)
def test_cut_section():
# concave polygon
p = Polygon((-1, -1), (1, Rational(5, 2)), (2, 1), (3, Rational(5, 2)), (4, 2), (5, 3), (-1, 3))
l = Line((0, 0), (Rational(9, 2), 3))
p1 = p.cut_section(l)[0]
p2 = p.cut_section(l)[1]
assert p1 == Polygon(
Point2D(Rational(-9, 13), Rational(-6, 13)), Point2D(1, Rational(5, 2)), Point2D(Rational(24, 13), Rational(16, 13)),
Point2D(Rational(12, 5), Rational(8, 5)), Point2D(3, Rational(5, 2)), Point2D(Rational(24, 7), Rational(16, 7)),
Point2D(Rational(9, 2), 3), Point2D(-1, 3), Point2D(-1, Rational(-2, 3)))
assert p2 == Polygon(Point2D(-1, -1), Point2D(Rational(-9, 13), Rational(-6, 13)), Point2D(Rational(24, 13), Rational(16, 13)),
Point2D(2, 1), Point2D(Rational(12, 5), Rational(8, 5)), Point2D(Rational(24, 7), Rational(16, 7)), Point2D(4, 2), Point2D(5, 3),
Point2D(Rational(9, 2), 3), Point2D(-1, Rational(-2, 3)))
# convex polygon
p = RegularPolygon(Point2D(0, 0), 6, 6)
s = p.cut_section(Line((0, 0), slope=1))
assert s[0] == Polygon(Point2D(-3*sqrt(3) + 9, -3*sqrt(3) + 9), Point2D(3, 3*sqrt(3)),
Point2D(-3, 3*sqrt(3)), Point2D(-6, 0), Point2D(-9 + 3*sqrt(3), -9 + 3*sqrt(3)))
assert s[1] == Polygon(Point2D(6, 0), Point2D(-3*sqrt(3) + 9, -3*sqrt(3) + 9),
Point2D(-9 + 3*sqrt(3), -9 + 3*sqrt(3)), Point2D(-3, -3*sqrt(3)), Point2D(3, -3*sqrt(3)))
# case where line does not intersects but coincides with the edge of polygon
a, b = 20, 10
t1, t2, t3, t4 = [(0, b), (0, 0), (a, 0), (a, b)]
p = Polygon(t1, t2, t3, t4)
p1, p2 = p.cut_section(Line((0, b), slope=0))
assert p1 == None
assert p2 == Polygon(Point2D(0, 10), Point2D(0, 0), Point2D(20, 0), Point2D(20, 10))
p3, p4 = p.cut_section(Line((0, 0), slope=0))
assert p3 == Polygon(Point2D(0, 10), Point2D(0, 0), Point2D(20, 0), Point2D(20, 10))
assert p4 == None
# case where the line does not intersect with a polygon at all
raises(ValueError, lambda: p.cut_section(Line((0, a), slope=0)))
def test_type_of_triangle():
# Isoceles triangle
p1 = Polygon(Point(0, 0), Point(5, 0), Point(2, 4))
assert p1.is_isosceles() == True
assert p1.is_scalene() == False
assert p1.is_equilateral() == False
# Scalene triangle
p2 = Polygon (Point(0, 0), Point(0, 2), Point(4, 0))
assert p2.is_isosceles() == False
assert p2.is_scalene() == True
assert p2.is_equilateral() == False
# Equilateral triagle
p3 = Polygon(Point(0, 0), Point(6, 0), Point(3, sqrt(27)))
assert p3.is_isosceles() == True
assert p3.is_scalene() == False
assert p3.is_equilateral() == True
def test_do_poly_distance():
# Non-intersecting polygons
square1 = Polygon (Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
triangle1 = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
assert square1._do_poly_distance(triangle1) == sqrt(2)/2
# Polygons which sides intersect
square2 = Polygon(Point(1, 0), Point(2, 0), Point(2, 1), Point(1, 1))
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output", test_stacklevel=False):
assert square1._do_poly_distance(square2) == 0
# Polygons which bodies intersect
triangle2 = Polygon(Point(0, -1), Point(2, -1), Point(S.Half, S.Half))
with warns(UserWarning, \
match="Polygons may intersect producing erroneous output", test_stacklevel=False):
assert triangle2._do_poly_distance(square1) == 0
|
0453bc408a18c7b221e6b3235719daaff1eec066aef969d18e478a2826589dfe | from sympy.core.numbers import (Rational, oo)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.geometry.ellipse import (Circle, Ellipse)
from sympy.geometry.line import (Line, Ray2D, Segment2D)
from sympy.geometry.parabola import Parabola
from sympy.geometry.point import (Point, Point2D)
from sympy.testing.pytest import raises
from sympy.abc import x, y
def test_parabola_geom():
a, b = symbols('a b')
p1 = Point(0, 0)
p2 = Point(3, 7)
p3 = Point(0, 4)
p4 = Point(6, 0)
p5 = Point(a, a)
d1 = Line(Point(4, 0), Point(4, 9))
d2 = Line(Point(7, 6), Point(3, 6))
d3 = Line(Point(4, 0), slope=oo)
d4 = Line(Point(7, 6), slope=0)
d5 = Line(Point(b, a), slope=oo)
d6 = Line(Point(a, b), slope=0)
half = S.Half
pa1 = Parabola(None, d2)
pa2 = Parabola(directrix=d1)
pa3 = Parabola(p1, d1)
pa4 = Parabola(p2, d2)
pa5 = Parabola(p2, d4)
pa6 = Parabola(p3, d2)
pa7 = Parabola(p2, d1)
pa8 = Parabola(p4, d1)
pa9 = Parabola(p4, d3)
pa10 = Parabola(p5, d5)
pa11 = Parabola(p5, d6)
d = Line(Point(3, 7), Point(2, 9))
pa12 = Parabola(Point(7, 8), d)
pa12r = Parabola(Point(7, 8).reflect(d), d)
raises(ValueError, lambda:
Parabola(Point(7, 8, 9), Line(Point(6, 7), Point(7, 7))))
raises(ValueError, lambda:
Parabola(Point(0, 2), Line(Point(7, 2), Point(6, 2))))
raises(ValueError, lambda: Parabola(Point(7, 8), Point(3, 8)))
# Basic Stuff
assert pa1.focus == Point(0, 0)
assert pa1.ambient_dimension == S(2)
assert pa2 == pa3
assert pa4 != pa7
assert pa6 != pa7
assert pa6.focus == Point2D(0, 4)
assert pa6.focal_length == 1
assert pa6.p_parameter == -1
assert pa6.vertex == Point2D(0, 5)
assert pa6.eccentricity == 1
assert pa7.focus == Point2D(3, 7)
assert pa7.focal_length == half
assert pa7.p_parameter == -half
assert pa7.vertex == Point2D(7*half, 7)
assert pa4.focal_length == half
assert pa4.p_parameter == half
assert pa4.vertex == Point2D(3, 13*half)
assert pa8.focal_length == 1
assert pa8.p_parameter == 1
assert pa8.vertex == Point2D(5, 0)
assert pa4.focal_length == pa5.focal_length
assert pa4.p_parameter == pa5.p_parameter
assert pa4.vertex == pa5.vertex
assert pa4.equation() == pa5.equation()
assert pa8.focal_length == pa9.focal_length
assert pa8.p_parameter == pa9.p_parameter
assert pa8.vertex == pa9.vertex
assert pa8.equation() == pa9.equation()
assert pa10.focal_length == pa11.focal_length == sqrt((a - b) ** 2) / 2 # if a, b real == abs(a - b)/2
assert pa11.vertex == Point(*pa10.vertex[::-1]) == Point(a,
a - sqrt((a - b)**2)*sign(a - b)/2) # change axis x->y, y->x on pa10
aos = pa12.axis_of_symmetry
assert aos == Line(Point(7, 8), Point(5, 7))
assert pa12.directrix == Line(Point(3, 7), Point(2, 9))
assert pa12.directrix.angle_between(aos) == S.Pi/2
assert pa12.eccentricity == 1
assert pa12.equation(x, y) == (x - 7)**2 + (y - 8)**2 - (-2*x - y + 13)**2/5
assert pa12.focal_length == 9*sqrt(5)/10
assert pa12.focus == Point(7, 8)
assert pa12.p_parameter == 9*sqrt(5)/10
assert pa12.vertex == Point2D(S(26)/5, S(71)/10)
assert pa12r.focal_length == 9*sqrt(5)/10
assert pa12r.focus == Point(-S(1)/5, S(22)/5)
assert pa12r.p_parameter == -9*sqrt(5)/10
assert pa12r.vertex == Point(S(8)/5, S(53)/10)
def test_parabola_intersection():
l1 = Line(Point(1, -2), Point(-1,-2))
l2 = Line(Point(1, 2), Point(-1,2))
l3 = Line(Point(1, 0), Point(-1,0))
p1 = Point(0,0)
p2 = Point(0, -2)
p3 = Point(120, -12)
parabola1 = Parabola(p1, l1)
# parabola with parabola
assert parabola1.intersection(parabola1) == [parabola1]
assert parabola1.intersection(Parabola(p1, l2)) == [Point2D(-2, 0), Point2D(2, 0)]
assert parabola1.intersection(Parabola(p2, l3)) == [Point2D(0, -1)]
assert parabola1.intersection(Parabola(Point(16, 0), l1)) == [Point2D(8, 15)]
assert parabola1.intersection(Parabola(Point(0, 16), l1)) == [Point2D(-6, 8), Point2D(6, 8)]
assert parabola1.intersection(Parabola(p3, l3)) == []
# parabola with point
assert parabola1.intersection(p1) == []
assert parabola1.intersection(Point2D(0, -1)) == [Point2D(0, -1)]
assert parabola1.intersection(Point2D(4, 3)) == [Point2D(4, 3)]
# parabola with line
assert parabola1.intersection(Line(Point2D(-7, 3), Point(12, 3))) == [Point2D(-4, 3), Point2D(4, 3)]
assert parabola1.intersection(Line(Point(-4, -1), Point(4, -1))) == [Point(0, -1)]
assert parabola1.intersection(Line(Point(2, 0), Point(0, -2))) == [Point2D(2, 0)]
raises(TypeError, lambda: parabola1.intersection(Line(Point(0, 0, 0), Point(1, 1, 1))))
# parabola with segment
assert parabola1.intersection(Segment2D((-4, -5), (4, 3))) == [Point2D(0, -1), Point2D(4, 3)]
assert parabola1.intersection(Segment2D((0, -5), (0, 6))) == [Point2D(0, -1)]
assert parabola1.intersection(Segment2D((-12, -65), (14, -68))) == []
# parabola with ray
assert parabola1.intersection(Ray2D((-4, -5), (4, 3))) == [Point2D(0, -1), Point2D(4, 3)]
assert parabola1.intersection(Ray2D((0, 7), (1, 14))) == [Point2D(14 + 2*sqrt(57), 105 + 14*sqrt(57))]
assert parabola1.intersection(Ray2D((0, 7), (0, 14))) == []
# parabola with ellipse/circle
assert parabola1.intersection(Circle(p1, 2)) == [Point2D(-2, 0), Point2D(2, 0)]
assert parabola1.intersection(Circle(p2, 1)) == [Point2D(0, -1), Point2D(0, -1)]
assert parabola1.intersection(Ellipse(p2, 2, 1)) == [Point2D(0, -1), Point2D(0, -1)]
assert parabola1.intersection(Ellipse(Point(0, 19), 5, 7)) == []
assert parabola1.intersection(Ellipse((0, 3), 12, 4)) == \
[Point2D(0, -1), Point2D(0, -1), Point2D(-4*sqrt(17)/3, Rational(59, 9)), Point2D(4*sqrt(17)/3, Rational(59, 9))]
# parabola with unsupported type
raises(TypeError, lambda: parabola1.intersection(2))
|
f719e9277d15ac5e44e35901c6c8802f4a5fe3fe77904312b16f2fce6ea31aa5 | from sympy.external import import_module
import os
cin = import_module('clang.cindex', import_kwargs = {'fromlist': ['cindex']})
"""
This module contains all the necessary Classes and Function used to Parse C and
C++ code into SymPy expression
The module serves as a backend for SymPyExpression to parse C code
It is also dependent on Clang's AST and SymPy's Codegen AST.
The module only supports the features currently supported by the Clang and
codegen AST which will be updated as the development of codegen AST and this
module progresses.
You might find unexpected bugs and exceptions while using the module, feel free
to report them to the SymPy Issue Tracker
Features Supported
==================
- Variable Declarations (integers and reals)
- Assignment (using integer & floating literal and function calls)
- Function Definitions and Declaration
- Function Calls
- Compound statements, Return statements
Notes
=====
The module is dependent on an external dependency which needs to be installed
to use the features of this module.
Clang: The C and C++ compiler which is used to extract an AST from the provided
C source code.
Refrences
=========
.. [1] https://github.com/sympy/sympy/issues
.. [2] https://clang.llvm.org/docs/
.. [3] https://clang.llvm.org/docs/IntroductionToTheClangAST.html
"""
if cin:
from sympy.codegen.ast import (Variable, Integer, Float,
FunctionPrototype, FunctionDefinition, FunctionCall,
none, Return, Assignment, intc, int8, int16, int64,
uint8, uint16, uint32, uint64, float32, float64, float80,
aug_assign, bool_, While, CodeBlock)
from sympy.codegen.cnodes import (PreDecrement, PostDecrement,
PreIncrement, PostIncrement)
from sympy.core import Add, Mod, Mul, Pow, Rel
from sympy.logic.boolalg import And, as_Boolean, Not, Or
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.logic.boolalg import (false, true)
import sys
import tempfile
class BaseParser:
"""Base Class for the C parser"""
def __init__(self):
"""Initializes the Base parser creating a Clang AST index"""
self.index = cin.Index.create()
def diagnostics(self, out):
"""Diagostics function for the Clang AST"""
for diag in self.tu.diagnostics:
print('%s %s (line %s, col %s) %s' % (
{
4: 'FATAL',
3: 'ERROR',
2: 'WARNING',
1: 'NOTE',
0: 'IGNORED',
}[diag.severity],
diag.location.file,
diag.location.line,
diag.location.column,
diag.spelling
), file=out)
class CCodeConverter(BaseParser):
"""The Code Convereter for Clang AST
The converter object takes the C source code or file as input and
converts them to SymPy Expressions.
"""
def __init__(self):
"""Initializes the code converter"""
super().__init__()
self._py_nodes = []
self._data_types = {
"void": {
cin.TypeKind.VOID: none
},
"bool": {
cin.TypeKind.BOOL: bool_
},
"int": {
cin.TypeKind.SCHAR: int8,
cin.TypeKind.SHORT: int16,
cin.TypeKind.INT: intc,
cin.TypeKind.LONG: int64,
cin.TypeKind.UCHAR: uint8,
cin.TypeKind.USHORT: uint16,
cin.TypeKind.UINT: uint32,
cin.TypeKind.ULONG: uint64
},
"float": {
cin.TypeKind.FLOAT: float32,
cin.TypeKind.DOUBLE: float64,
cin.TypeKind.LONGDOUBLE: float80
}
}
def parse(self, filenames, flags):
"""Function to parse a file with C source code
It takes the filename as an attribute and creates a Clang AST
Translation Unit parsing the file.
Then the transformation function is called on the transaltion unit,
whose reults are collected into a list which is returned by the
function.
Parameters
==========
filenames : string
Path to the C file to be parsed
flags: list
Arguments to be passed to Clang while parsing the C code
Returns
=======
py_nodes: list
A list of SymPy AST nodes
"""
filename = os.path.abspath(filenames)
self.tu = self.index.parse(
filename,
args=flags,
options=cin.TranslationUnit.PARSE_DETAILED_PROCESSING_RECORD
)
for child in self.tu.cursor.get_children():
if child.kind == cin.CursorKind.VAR_DECL:
self._py_nodes.append(self.transform(child))
elif (child.kind == cin.CursorKind.FUNCTION_DECL):
self._py_nodes.append(self.transform(child))
else:
pass
return self._py_nodes
def parse_str(self, source, flags):
"""Function to parse a string with C source code
It takes the source code as an attribute, stores it in a temporary
file and creates a Clang AST Translation Unit parsing the file.
Then the transformation function is called on the transaltion unit,
whose reults are collected into a list which is returned by the
function.
Parameters
==========
source : string
Path to the C file to be parsed
flags: list
Arguments to be passed to Clang while parsing the C code
Returns
=======
py_nodes: list
A list of SymPy AST nodes
"""
file = tempfile.NamedTemporaryFile(mode = 'w+', suffix = '.cpp')
file.write(source)
file.seek(0)
self.tu = self.index.parse(
file.name,
args=flags,
options=cin.TranslationUnit.PARSE_DETAILED_PROCESSING_RECORD
)
file.close()
for child in self.tu.cursor.get_children():
if child.kind == cin.CursorKind.VAR_DECL:
self._py_nodes.append(self.transform(child))
elif (child.kind == cin.CursorKind.FUNCTION_DECL):
self._py_nodes.append(self.transform(child))
else:
pass
return self._py_nodes
def transform(self, node):
"""Transformation Function for Clang AST nodes
It determines the kind of node and calls the respective
transformation function for that node.
Raises
======
NotImplementedError : if the transformation for the provided node
is not implemented
"""
try:
handler = getattr(self, 'transform_%s' % node.kind.name.lower())
except AttributeError:
print(
"Ignoring node of type %s (%s)" % (
node.kind,
' '.join(
t.spelling for t in node.get_tokens())
),
file=sys.stderr
)
handler = None
if handler:
result = handler(node)
return result
def transform_var_decl(self, node):
"""Transformation Function for Variable Declaration
Used to create nodes for variable declarations and assignments with
values or function call for the respective nodes in the clang AST
Returns
=======
A variable node as Declaration, with the initial value if given
Raises
======
NotImplementedError : if called for data types not currently
implemented
Notes
=====
The function currently supports following data types:
Boolean:
bool, _Bool
Integer:
8-bit: signed char and unsigned char
16-bit: short, short int, signed short,
signed short int, unsigned short, unsigned short int
32-bit: int, signed int, unsigned int
64-bit: long, long int, signed long,
signed long int, unsigned long, unsigned long int
Floating point:
Single Precision: float
Double Precision: double
Extended Precision: long double
"""
if node.type.kind in self._data_types["int"]:
type = self._data_types["int"][node.type.kind]
elif node.type.kind in self._data_types["float"]:
type = self._data_types["float"][node.type.kind]
elif node.type.kind in self._data_types["bool"]:
type = self._data_types["bool"][node.type.kind]
else:
raise NotImplementedError("Only bool, int "
"and float are supported")
try:
children = node.get_children()
child = next(children)
#ignoring namespace and type details for the variable
while child.kind == cin.CursorKind.NAMESPACE_REF:
child = next(children)
while child.kind == cin.CursorKind.TYPE_REF:
child = next(children)
val = self.transform(child)
supported_rhs = [
cin.CursorKind.INTEGER_LITERAL,
cin.CursorKind.FLOATING_LITERAL,
cin.CursorKind.UNEXPOSED_EXPR,
cin.CursorKind.BINARY_OPERATOR,
cin.CursorKind.PAREN_EXPR,
cin.CursorKind.UNARY_OPERATOR,
cin.CursorKind.CXX_BOOL_LITERAL_EXPR
]
if child.kind in supported_rhs:
if isinstance(val, str):
value = Symbol(val)
elif isinstance(val, bool):
if node.type.kind in self._data_types["int"]:
value = Integer(0) if val == False else Integer(1)
elif node.type.kind in self._data_types["float"]:
value = Float(0.0) if val == False else Float(1.0)
elif node.type.kind in self._data_types["bool"]:
value = sympify(val)
elif isinstance(val, (Integer, int, Float, float)):
if node.type.kind in self._data_types["int"]:
value = Integer(val)
elif node.type.kind in self._data_types["float"]:
value = Float(val)
elif node.type.kind in self._data_types["bool"]:
value = sympify(bool(val))
else:
value = val
return Variable(
node.spelling
).as_Declaration(
type = type,
value = value
)
elif child.kind == cin.CursorKind.CALL_EXPR:
return Variable(
node.spelling
).as_Declaration(
value = val
)
else:
raise NotImplementedError("Given "
"variable declaration \"{}\" "
"is not possible to parse yet!"
.format(" ".join(
t.spelling for t in node.get_tokens()
)
))
except StopIteration:
return Variable(
node.spelling
).as_Declaration(
type = type
)
def transform_function_decl(self, node):
"""Transformation Function For Function Declaration
Used to create nodes for function declarations and definitions for
the respective nodes in the clang AST
Returns
=======
function : Codegen AST node
- FunctionPrototype node if function body is not present
- FunctionDefinition node if the function body is present
"""
if node.result_type.kind in self._data_types["int"]:
ret_type = self._data_types["int"][node.result_type.kind]
elif node.result_type.kind in self._data_types["float"]:
ret_type = self._data_types["float"][node.result_type.kind]
elif node.result_type.kind in self._data_types["bool"]:
ret_type = self._data_types["bool"][node.result_type.kind]
elif node.result_type.kind in self._data_types["void"]:
ret_type = self._data_types["void"][node.result_type.kind]
else:
raise NotImplementedError("Only void, bool, int "
"and float are supported")
body = []
param = []
try:
children = node.get_children()
child = next(children)
# If the node has any children, the first children will be the
# return type and namespace for the function declaration. These
# nodes can be ignored.
while child.kind == cin.CursorKind.NAMESPACE_REF:
child = next(children)
while child.kind == cin.CursorKind.TYPE_REF:
child = next(children)
# Subsequent nodes will be the parameters for the function.
try:
while True:
decl = self.transform(child)
if (child.kind == cin.CursorKind.PARM_DECL):
param.append(decl)
elif (child.kind == cin.CursorKind.COMPOUND_STMT):
for val in decl:
body.append(val)
else:
body.append(decl)
child = next(children)
except StopIteration:
pass
except StopIteration:
pass
if body == []:
function = FunctionPrototype(
return_type = ret_type,
name = node.spelling,
parameters = param
)
else:
function = FunctionDefinition(
return_type = ret_type,
name = node.spelling,
parameters = param,
body = body
)
return function
def transform_parm_decl(self, node):
"""Transformation function for Parameter Declaration
Used to create parameter nodes for the required functions for the
respective nodes in the clang AST
Returns
=======
param : Codegen AST Node
Variable node with the value and type of the variable
Raises
======
ValueError if multiple children encountered in the parameter node
"""
if node.type.kind in self._data_types["int"]:
type = self._data_types["int"][node.type.kind]
elif node.type.kind in self._data_types["float"]:
type = self._data_types["float"][node.type.kind]
elif node.type.kind in self._data_types["bool"]:
type = self._data_types["bool"][node.type.kind]
else:
raise NotImplementedError("Only bool, int "
"and float are supported")
try:
children = node.get_children()
child = next(children)
# Any namespace nodes can be ignored
while child.kind in [cin.CursorKind.NAMESPACE_REF,
cin.CursorKind.TYPE_REF,
cin.CursorKind.TEMPLATE_REF]:
child = next(children)
# If there is a child, it is the default value of the parameter.
lit = self.transform(child)
if node.type.kind in self._data_types["int"]:
val = Integer(lit)
elif node.type.kind in self._data_types["float"]:
val = Float(lit)
elif node.type.kind in self._data_types["bool"]:
val = sympify(bool(lit))
else:
raise NotImplementedError("Only bool, int "
"and float are supported")
param = Variable(
node.spelling
).as_Declaration(
type = type,
value = val
)
except StopIteration:
param = Variable(
node.spelling
).as_Declaration(
type = type
)
try:
self.transform(next(children))
raise ValueError("Can't handle multiple children on parameter")
except StopIteration:
pass
return param
def transform_integer_literal(self, node):
"""Transformation function for integer literal
Used to get the value and type of the given integer literal.
Returns
=======
val : list
List with two arguments type and Value
type contains the type of the integer
value contains the value stored in the variable
Notes
=====
Only Base Integer type supported for now
"""
try:
value = next(node.get_tokens()).spelling
except StopIteration:
# No tokens
value = node.literal
return int(value)
def transform_floating_literal(self, node):
"""Transformation function for floating literal
Used to get the value and type of the given floating literal.
Returns
=======
val : list
List with two arguments type and Value
type contains the type of float
value contains the value stored in the variable
Notes
=====
Only Base Float type supported for now
"""
try:
value = next(node.get_tokens()).spelling
except (StopIteration, ValueError):
# No tokens
value = node.literal
return float(value)
def transform_string_literal(self, node):
#TODO: No string type in AST
#type =
#try:
# value = next(node.get_tokens()).spelling
#except (StopIteration, ValueError):
# No tokens
# value = node.literal
#val = [type, value]
#return val
pass
def transform_character_literal(self, node):
"""Transformation function for character literal
Used to get the value of the given character literal.
Returns
=======
val : int
val contains the ascii value of the character literal
Notes
=====
Only for cases where character is assigned to a integer value,
since character literal is not in SymPy AST
"""
try:
value = next(node.get_tokens()).spelling
except (StopIteration, ValueError):
# No tokens
value = node.literal
return ord(str(value[1]))
def transform_cxx_bool_literal_expr(self, node):
"""Transformation function for boolean literal
Used to get the value of the given boolean literal.
Returns
=======
value : bool
value contains the boolean value of the variable
"""
try:
value = next(node.get_tokens()).spelling
except (StopIteration, ValueError):
value = node.literal
return True if value == 'true' else False
def transform_unexposed_decl(self,node):
"""Transformation function for unexposed declarations"""
pass
def transform_unexposed_expr(self, node):
"""Transformation function for unexposed expression
Unexposed expressions are used to wrap float, double literals and
expressions
Returns
=======
expr : Codegen AST Node
the result from the wrapped expression
None : NoneType
No childs are found for the node
Raises
======
ValueError if the expression contains multiple children
"""
# Ignore unexposed nodes; pass whatever is the first
# (and should be only) child unaltered.
try:
children = node.get_children()
expr = self.transform(next(children))
except StopIteration:
return None
try:
next(children)
raise ValueError("Unexposed expression has > 1 children.")
except StopIteration:
pass
return expr
def transform_decl_ref_expr(self, node):
"""Returns the name of the declaration reference"""
return node.spelling
def transform_call_expr(self, node):
"""Transformation function for a call expression
Used to create function call nodes for the function calls present
in the C code
Returns
=======
FunctionCall : Codegen AST Node
FunctionCall node with parameters if any parameters are present
"""
param = []
children = node.get_children()
child = next(children)
while child.kind == cin.CursorKind.NAMESPACE_REF:
child = next(children)
while child.kind == cin.CursorKind.TYPE_REF:
child = next(children)
first_child = self.transform(child)
try:
for child in children:
arg = self.transform(child)
if (child.kind == cin.CursorKind.INTEGER_LITERAL):
param.append(Integer(arg))
elif (child.kind == cin.CursorKind.FLOATING_LITERAL):
param.append(Float(arg))
else:
param.append(arg)
return FunctionCall(first_child, param)
except StopIteration:
return FunctionCall(first_child)
def transform_return_stmt(self, node):
"""Returns the Return Node for a return statement"""
return Return(next(node.get_children()).spelling)
def transform_compound_stmt(self, node):
"""Transformation function for compond statemets
Returns
=======
expr : list
list of Nodes for the expressions present in the statement
None : NoneType
if the compound statement is empty
"""
try:
expr = []
children = node.get_children()
for child in children:
expr.append(self.transform(child))
except StopIteration:
return None
return expr
def transform_decl_stmt(self, node):
"""Transformation function for declaration statements
These statements are used to wrap different kinds of declararions
like variable or function declaration
The function calls the transformer function for the child of the
given node
Returns
=======
statement : Codegen AST Node
contains the node returned by the children node for the type of
declaration
Raises
======
ValueError if multiple children present
"""
try:
children = node.get_children()
statement = self.transform(next(children))
except StopIteration:
pass
try:
self.transform(next(children))
raise ValueError("Don't know how to handle multiple statements")
except StopIteration:
pass
return statement
def transform_paren_expr(self, node):
"""Transformation function for Parenthesized expressions
Returns the result from its children nodes
"""
return self.transform(next(node.get_children()))
def transform_compound_assignment_operator(self, node):
"""Transformation function for handling shorthand operators
Returns
=======
augmented_assignment_expression: Codegen AST node
shorthand assignment expression represented as Codegen AST
Raises
======
NotImplementedError
If the shorthand operator for bitwise operators
(~=, ^=, &=, |=, <<=, >>=) is encountered
"""
return self.transform_binary_operator(node)
def transform_unary_operator(self, node):
"""Transformation function for handling unary operators
Returns
=======
unary_expression: Codegen AST node
simplified unary expression represented as Codegen AST
Raises
======
NotImplementedError
If dereferencing operator(*), address operator(&) or
bitwise NOT operator(~) is encountered
"""
# supported operators list
operators_list = ['+', '-', '++', '--', '!']
tokens = [token for token in node.get_tokens()]
# it can be either pre increment/decrement or any other operator from the list
if tokens[0].spelling in operators_list:
child = self.transform(next(node.get_children()))
# (decl_ref) e.g.; int a = ++b; or simply ++b;
if isinstance(child, str):
if tokens[0].spelling == '+':
return Symbol(child)
if tokens[0].spelling == '-':
return Mul(Symbol(child), -1)
if tokens[0].spelling == '++':
return PreIncrement(Symbol(child))
if tokens[0].spelling == '--':
return PreDecrement(Symbol(child))
if tokens[0].spelling == '!':
return Not(Symbol(child))
# e.g.; int a = -1; or int b = -(1 + 2);
else:
if tokens[0].spelling == '+':
return child
if tokens[0].spelling == '-':
return Mul(child, -1)
if tokens[0].spelling == '!':
return Not(sympify(bool(child)))
# it can be either post increment/decrement
# since variable name is obtained in token[0].spelling
elif tokens[1].spelling in ['++', '--']:
child = self.transform(next(node.get_children()))
if tokens[1].spelling == '++':
return PostIncrement(Symbol(child))
if tokens[1].spelling == '--':
return PostDecrement(Symbol(child))
else:
raise NotImplementedError("Dereferencing operator, "
"Address operator and bitwise NOT operator "
"have not been implemented yet!")
def transform_binary_operator(self, node):
"""Transformation function for handling binary operators
Returns
=======
binary_expression: Codegen AST node
simplified binary expression represented as Codegen AST
Raises
======
NotImplementedError
If a bitwise operator or
unary operator(which is a child of any binary
operator in Clang AST) is encountered
"""
# get all the tokens of assignment
# and store it in the tokens list
tokens = [token for token in node.get_tokens()]
# supported operators list
operators_list = ['+', '-', '*', '/', '%','=',
'>', '>=', '<', '<=', '==', '!=', '&&', '||', '+=', '-=',
'*=', '/=', '%=']
# this stack will contain variable content
# and type of variable in the rhs
combined_variables_stack = []
# this stack will contain operators
# to be processed in the rhs
operators_stack = []
# iterate through every token
for token in tokens:
# token is either '(', ')' or
# any of the supported operators from the operator list
if token.kind == cin.TokenKind.PUNCTUATION:
# push '(' to the operators stack
if token.spelling == '(':
operators_stack.append('(')
elif token.spelling == ')':
# keep adding the expression to the
# combined variables stack unless
# '(' is found
while (operators_stack
and operators_stack[-1] != '('):
if len(combined_variables_stack) < 2:
raise NotImplementedError(
"Unary operators as a part of "
"binary operators is not "
"supported yet!")
rhs = combined_variables_stack.pop()
lhs = combined_variables_stack.pop()
operator = operators_stack.pop()
combined_variables_stack.append(
self.perform_operation(
lhs, rhs, operator))
# pop '('
operators_stack.pop()
# token is an operator (supported)
elif token.spelling in operators_list:
while (operators_stack
and self.priority_of(token.spelling)
<= self.priority_of(
operators_stack[-1])):
if len(combined_variables_stack) < 2:
raise NotImplementedError(
"Unary operators as a part of "
"binary operators is not "
"supported yet!")
rhs = combined_variables_stack.pop()
lhs = combined_variables_stack.pop()
operator = operators_stack.pop()
combined_variables_stack.append(
self.perform_operation(
lhs, rhs, operator))
# push current operator
operators_stack.append(token.spelling)
# token is a bitwise operator
elif token.spelling in ['&', '|', '^', '<<', '>>']:
raise NotImplementedError(
"Bitwise operator has not been "
"implemented yet!")
# token is a shorthand bitwise operator
elif token.spelling in ['&=', '|=', '^=', '<<=',
'>>=']:
raise NotImplementedError(
"Shorthand bitwise operator has not been "
"implemented yet!")
else:
raise NotImplementedError(
"Given token {} is not implemented yet!"
.format(token.spelling))
# token is an identifier(variable)
elif token.kind == cin.TokenKind.IDENTIFIER:
combined_variables_stack.append(
[token.spelling, 'identifier'])
# token is a literal
elif token.kind == cin.TokenKind.LITERAL:
combined_variables_stack.append(
[token.spelling, 'literal'])
# token is a keyword, either true or false
elif (token.kind == cin.TokenKind.KEYWORD
and token.spelling in ['true', 'false']):
combined_variables_stack.append(
[token.spelling, 'boolean'])
else:
raise NotImplementedError(
"Given token {} is not implemented yet!"
.format(token.spelling))
# process remaining operators
while operators_stack:
if len(combined_variables_stack) < 2:
raise NotImplementedError(
"Unary operators as a part of "
"binary operators is not "
"supported yet!")
rhs = combined_variables_stack.pop()
lhs = combined_variables_stack.pop()
operator = operators_stack.pop()
combined_variables_stack.append(
self.perform_operation(lhs, rhs, operator))
return combined_variables_stack[-1][0]
def priority_of(self, op):
"""To get the priority of given operator"""
if op in ['=', '+=', '-=', '*=', '/=', '%=']:
return 1
if op in ['&&', '||']:
return 2
if op in ['<', '<=', '>', '>=', '==', '!=']:
return 3
if op in ['+', '-']:
return 4
if op in ['*', '/', '%']:
return 5
return 0
def perform_operation(self, lhs, rhs, op):
"""Performs operation supported by the SymPy core
Returns
=======
combined_variable: list
contains variable content and type of variable
"""
lhs_value = self.get_expr_for_operand(lhs)
rhs_value = self.get_expr_for_operand(rhs)
if op == '+':
return [Add(lhs_value, rhs_value), 'expr']
if op == '-':
return [Add(lhs_value, -rhs_value), 'expr']
if op == '*':
return [Mul(lhs_value, rhs_value), 'expr']
if op == '/':
return [Mul(lhs_value, Pow(rhs_value, Integer(-1))), 'expr']
if op == '%':
return [Mod(lhs_value, rhs_value), 'expr']
if op in ['<', '<=', '>', '>=', '==', '!=']:
return [Rel(lhs_value, rhs_value, op), 'expr']
if op == '&&':
return [And(as_Boolean(lhs_value), as_Boolean(rhs_value)), 'expr']
if op == '||':
return [Or(as_Boolean(lhs_value), as_Boolean(rhs_value)), 'expr']
if op == '=':
return [Assignment(Variable(lhs_value), rhs_value), 'expr']
if op in ['+=', '-=', '*=', '/=', '%=']:
return [aug_assign(Variable(lhs_value), op[0], rhs_value), 'expr']
def get_expr_for_operand(self, combined_variable):
"""Gives out SymPy Codegen AST node
AST node returned is corresponding to
combined variable passed.Combined variable contains
variable content and type of variable
"""
if combined_variable[1] == 'identifier':
return Symbol(combined_variable[0])
if combined_variable[1] == 'literal':
if '.' in combined_variable[0]:
return Float(float(combined_variable[0]))
else:
return Integer(int(combined_variable[0]))
if combined_variable[1] == 'expr':
return combined_variable[0]
if combined_variable[1] == 'boolean':
return true if combined_variable[0] == 'true' else false
def transform_null_stmt(self, node):
"""Handles Null Statement and returns None"""
return none
def transform_while_stmt(self, node):
"""Transformation function for handling while statement
Returns
=======
while statement : Codegen AST Node
contains the while statement node having condition and
statement block
"""
children = node.get_children()
condition = self.transform(next(children))
statements = self.transform(next(children))
if isinstance(statements, list):
statement_block = CodeBlock(*statements)
else:
statement_block = CodeBlock(statements)
return While(condition, statement_block)
else:
class CCodeConverter(): # type: ignore
def __init__(self, *args, **kwargs):
raise ImportError("Module not Installed")
def parse_c(source):
"""Function for converting a C source code
The function reads the source code present in the given file and parses it
to give out SymPy Expressions
Returns
=======
src : list
List of Python expression strings
"""
converter = CCodeConverter()
if os.path.exists(source):
src = converter.parse(source, flags = [])
else:
src = converter.parse_str(source, flags = [])
return src
|
0f400a56f29bc7f9e69b763e7e33f6419c224569269fe0d192e0e990d86dff26 | from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.parsing.ast_parser import parse_expr
from sympy.testing.pytest import raises
from sympy.core.sympify import SympifyError
import warnings
def test_parse_expr():
a, b = symbols('a, b')
# tests issue_16393
assert parse_expr('a + b', {}) == a + b
raises(SympifyError, lambda: parse_expr('a + ', {}))
# tests Transform.visit_Num
assert parse_expr('1 + 2', {}) == S(3)
assert parse_expr('1 + 2.0', {}) == S(3.0)
# tests Transform.visit_Name
assert parse_expr('Rational(1, 2)', {}) == S(1)/2
assert parse_expr('a', {'a': a}) == a
# tests issue_23092
with warnings.catch_warnings():
warnings.simplefilter('error')
assert parse_expr('6 * 7', {}) == S(42)
# Note: The test below can be removed when support for Python 3.7 is
# dropped. This test exists to ensure that the visit_Num function
# exists for Python 3.7, because in 3.7, Python didn't use the
# visit_Constant to create AST Nodes yet.
test_expr = parse_expr('1 / 3', {})
assert test_expr == S(1)/3 # sanity check
assert isinstance(test_expr, Rational)
|
a722ba739ae14911ce22e870b71257d76e00e588cd18bc7dd31a7d379e0de0f9 | # -*- coding: utf-8 -*-
import sys
import builtins
import types
from sympy.assumptions import Q
from sympy.core import Symbol, Function, Float, Rational, Integer, I, Mul, Pow, Eq
from sympy.functions import exp, factorial, factorial2, sin, Min, Max
from sympy.logic import And
from sympy.series import Limit
from sympy.testing.pytest import raises, skip
from sympy.parsing.sympy_parser import (
parse_expr, standard_transformations, rationalize, TokenError,
split_symbols, implicit_multiplication, convert_equals_signs,
convert_xor, function_exponentiation, lambda_notation, auto_symbol,
repeated_decimals, implicit_multiplication_application,
auto_number, factorial_notation, implicit_application,
_transformation, T
)
def test_sympy_parser():
x = Symbol('x')
inputs = {
'2*x': 2 * x,
'3.00': Float(3),
'22/7': Rational(22, 7),
'2+3j': 2 + 3*I,
'exp(x)': exp(x),
'x!': factorial(x),
'x!!': factorial2(x),
'(x + 1)! - 1': factorial(x + 1) - 1,
'3.[3]': Rational(10, 3),
'.0[3]': Rational(1, 30),
'3.2[3]': Rational(97, 30),
'1.3[12]': Rational(433, 330),
'1 + 3.[3]': Rational(13, 3),
'1 + .0[3]': Rational(31, 30),
'1 + 3.2[3]': Rational(127, 30),
'.[0011]': Rational(1, 909),
'0.1[00102] + 1': Rational(366697, 333330),
'1.[0191]': Rational(10190, 9999),
'10!': 3628800,
'-(2)': -Integer(2),
'[-1, -2, 3]': [Integer(-1), Integer(-2), Integer(3)],
'Symbol("x").free_symbols': x.free_symbols,
"S('S(3).n(n=3)')": 3.00,
'factorint(12, visual=True)': Mul(
Pow(2, 2, evaluate=False),
Pow(3, 1, evaluate=False),
evaluate=False),
'Limit(sin(x), x, 0, dir="-")': Limit(sin(x), x, 0, dir='-'),
'Q.even(x)': Q.even(x),
}
for text, result in inputs.items():
assert parse_expr(text) == result
raises(TypeError, lambda:
parse_expr('x', standard_transformations))
raises(TypeError, lambda:
parse_expr('x', transformations=lambda x,y: 1))
raises(TypeError, lambda:
parse_expr('x', transformations=(lambda x,y: 1,)))
raises(TypeError, lambda: parse_expr('x', transformations=((),)))
raises(TypeError, lambda: parse_expr('x', {}, [], []))
raises(TypeError, lambda: parse_expr('x', [], [], {}))
raises(TypeError, lambda: parse_expr('x', [], [], {}))
def test_rationalize():
inputs = {
'0.123': Rational(123, 1000)
}
transformations = standard_transformations + (rationalize,)
for text, result in inputs.items():
assert parse_expr(text, transformations=transformations) == result
def test_factorial_fail():
inputs = ['x!!!', 'x!!!!', '(!)']
for text in inputs:
try:
parse_expr(text)
assert False
except TokenError:
assert True
def test_repeated_fail():
inputs = ['1[1]', '.1e1[1]', '0x1[1]', '1.1j[1]', '1.1[1 + 1]',
'0.1[[1]]', '0x1.1[1]']
# All are valid Python, so only raise TypeError for invalid indexing
for text in inputs:
raises(TypeError, lambda: parse_expr(text))
inputs = ['0.1[', '0.1[1', '0.1[]']
for text in inputs:
raises((TokenError, SyntaxError), lambda: parse_expr(text))
def test_repeated_dot_only():
assert parse_expr('.[1]') == Rational(1, 9)
assert parse_expr('1 + .[1]') == Rational(10, 9)
def test_local_dict():
local_dict = {
'my_function': lambda x: x + 2
}
inputs = {
'my_function(2)': Integer(4)
}
for text, result in inputs.items():
assert parse_expr(text, local_dict=local_dict) == result
def test_local_dict_split_implmult():
t = standard_transformations + (split_symbols, implicit_multiplication,)
w = Symbol('w', real=True)
y = Symbol('y')
assert parse_expr('yx', local_dict={'x':w}, transformations=t) == y*w
def test_local_dict_symbol_to_fcn():
x = Symbol('x')
d = {'foo': Function('bar')}
assert parse_expr('foo(x)', local_dict=d) == d['foo'](x)
d = {'foo': Symbol('baz')}
raises(TypeError, lambda: parse_expr('foo(x)', local_dict=d))
def test_global_dict():
global_dict = {
'Symbol': Symbol
}
inputs = {
'Q & S': And(Symbol('Q'), Symbol('S'))
}
for text, result in inputs.items():
assert parse_expr(text, global_dict=global_dict) == result
def test_no_globals():
# Replicate creating the default global_dict:
default_globals = {}
exec('from sympy import *', default_globals)
builtins_dict = vars(builtins)
for name, obj in builtins_dict.items():
if isinstance(obj, types.BuiltinFunctionType):
default_globals[name] = obj
default_globals['max'] = Max
default_globals['min'] = Min
# Need to include Symbol or parse_expr will not work:
default_globals.pop('Symbol')
global_dict = {'Symbol':Symbol}
for name in default_globals:
obj = parse_expr(name, global_dict=global_dict)
assert obj == Symbol(name)
def test_issue_2515():
raises(TokenError, lambda: parse_expr('(()'))
raises(TokenError, lambda: parse_expr('"""'))
def test_issue_7663():
x = Symbol('x')
e = '2*(x+1)'
assert parse_expr(e, evaluate=0) == parse_expr(e, evaluate=False)
assert parse_expr(e, evaluate=0).equals(2*(x+1))
def test_recursive_evaluate_false_10560():
inputs = {
'4*-3' : '4*-3',
'-4*3' : '(-4)*3',
"-2*x*y": '(-2)*x*y',
"x*-4*x": "x*(-4)*x"
}
for text, result in inputs.items():
assert parse_expr(text, evaluate=False) == parse_expr(result, evaluate=False)
def test_function_evaluate_false():
inputs = [
'Abs(0)', 'im(0)', 're(0)', 'sign(0)', 'arg(0)', 'conjugate(0)',
'acos(0)', 'acot(0)', 'acsc(0)', 'asec(0)', 'asin(0)', 'atan(0)',
'acosh(0)', 'acoth(0)', 'acsch(0)', 'asech(0)', 'asinh(0)', 'atanh(0)',
'cos(0)', 'cot(0)', 'csc(0)', 'sec(0)', 'sin(0)', 'tan(0)',
'cosh(0)', 'coth(0)', 'csch(0)', 'sech(0)', 'sinh(0)', 'tanh(0)',
'exp(0)', 'log(0)', 'sqrt(0)',
]
for case in inputs:
expr = parse_expr(case, evaluate=False)
assert case == str(expr) != str(expr.doit())
assert str(parse_expr('ln(0)', evaluate=False)) == 'log(0)'
assert str(parse_expr('cbrt(0)', evaluate=False)) == '0**(1/3)'
def test_issue_10773():
inputs = {
'-10/5': '(-10)/5',
'-10/-5' : '(-10)/(-5)',
}
for text, result in inputs.items():
assert parse_expr(text, evaluate=False) == parse_expr(result, evaluate=False)
def test_split_symbols():
transformations = standard_transformations + \
(split_symbols, implicit_multiplication,)
x = Symbol('x')
y = Symbol('y')
xy = Symbol('xy')
assert parse_expr("xy") == xy
assert parse_expr("xy", transformations=transformations) == x*y
def test_split_symbols_function():
transformations = standard_transformations + \
(split_symbols, implicit_multiplication,)
x = Symbol('x')
y = Symbol('y')
a = Symbol('a')
f = Function('f')
assert parse_expr("ay(x+1)", transformations=transformations) == a*y*(x+1)
assert parse_expr("af(x+1)", transformations=transformations,
local_dict={'f':f}) == a*f(x+1)
def test_functional_exponent():
t = standard_transformations + (convert_xor, function_exponentiation)
x = Symbol('x')
y = Symbol('y')
a = Symbol('a')
yfcn = Function('y')
assert parse_expr("sin^2(x)", transformations=t) == (sin(x))**2
assert parse_expr("sin^y(x)", transformations=t) == (sin(x))**y
assert parse_expr("exp^y(x)", transformations=t) == (exp(x))**y
assert parse_expr("E^y(x)", transformations=t) == exp(yfcn(x))
assert parse_expr("a^y(x)", transformations=t) == a**(yfcn(x))
def test_match_parentheses_implicit_multiplication():
transformations = standard_transformations + \
(implicit_multiplication,)
raises(TokenError, lambda: parse_expr('(1,2),(3,4]',transformations=transformations))
def test_convert_equals_signs():
transformations = standard_transformations + \
(convert_equals_signs, )
x = Symbol('x')
y = Symbol('y')
assert parse_expr("1*2=x", transformations=transformations) == Eq(2, x)
assert parse_expr("y = x", transformations=transformations) == Eq(y, x)
assert parse_expr("(2*y = x) = False",
transformations=transformations) == Eq(Eq(2*y, x), False)
def test_parse_function_issue_3539():
x = Symbol('x')
f = Function('f')
assert parse_expr('f(x)') == f(x)
def test_split_symbols_numeric():
transformations = (
standard_transformations +
(implicit_multiplication_application,))
n = Symbol('n')
expr1 = parse_expr('2**n * 3**n')
expr2 = parse_expr('2**n3**n', transformations=transformations)
assert expr1 == expr2 == 2**n*3**n
expr1 = parse_expr('n12n34', transformations=transformations)
assert expr1 == n*12*n*34
def test_unicode_names():
assert parse_expr('α') == Symbol('α')
def test_python3_features():
# Make sure the tokenizer can handle Python 3-only features
if sys.version_info < (3, 7):
skip("test_python3_features requires Python 3.7 or newer")
assert parse_expr("123_456") == 123456
assert parse_expr("1.2[3_4]") == parse_expr("1.2[34]") == Rational(611, 495)
assert parse_expr("1.2[012_012]") == parse_expr("1.2[012012]") == Rational(400, 333)
assert parse_expr('.[3_4]') == parse_expr('.[34]') == Rational(34, 99)
assert parse_expr('.1[3_4]') == parse_expr('.1[34]') == Rational(133, 990)
assert parse_expr('123_123.123_123[3_4]') == parse_expr('123123.123123[34]') == Rational(12189189189211, 99000000)
def test_issue_19501():
x = Symbol('x')
eq = parse_expr('E**x(1+x)', local_dict={'x': x}, transformations=(
standard_transformations +
(implicit_multiplication_application,)))
assert eq.free_symbols == {x}
def test_parsing_definitions():
from sympy.abc import x
assert len(_transformation) == 12 # if this changes, extend below
assert _transformation[0] == lambda_notation
assert _transformation[1] == auto_symbol
assert _transformation[2] == repeated_decimals
assert _transformation[3] == auto_number
assert _transformation[4] == factorial_notation
assert _transformation[5] == implicit_multiplication_application
assert _transformation[6] == convert_xor
assert _transformation[7] == implicit_application
assert _transformation[8] == implicit_multiplication
assert _transformation[9] == convert_equals_signs
assert _transformation[10] == function_exponentiation
assert _transformation[11] == rationalize
assert T[:5] == T[0,1,2,3,4] == standard_transformations
t = _transformation
assert T[-1, 0] == (t[len(t) - 1], t[0])
assert T[:5, 8] == standard_transformations + (t[8],)
assert parse_expr('0.3x^2', transformations='all') == 3*x**2/10
assert parse_expr('sin 3x', transformations='implicit') == sin(3*x)
def test_builtins():
cases = [
('abs(x)', 'Abs(x)'),
('max(x, y)', 'Max(x, y)'),
('min(x, y)', 'Min(x, y)'),
('pow(x, y)', 'Pow(x, y)'),
]
for built_in_func_call, sympy_func_call in cases:
assert parse_expr(built_in_func_call) == parse_expr(sympy_func_call)
assert str(parse_expr('pow(38, -1, 97)')) == '23'
def test_issue_22822():
raises(ValueError, lambda: parse_expr('x', {'': 1}))
data = {'some_parameter': None}
assert parse_expr('some_parameter is None', data) is True
|
9c2602f62c2f328f11f41a0d28399e7a110bc226141cdbf440390332d1b19915 | from sympy import sin, Function, symbols, Dummy, Lambda, cos
from sympy.parsing.mathematica import mathematica, MathematicaParser
from sympy.core.sympify import sympify
from sympy.abc import n, w, x, y, z
from sympy.testing.pytest import raises
def test_mathematica():
d = {
'- 6x': '-6*x',
'Sin[x]^2': 'sin(x)**2',
'2(x-1)': '2*(x-1)',
'3y+8': '3*y+8',
'ArcSin[2x+9(4-x)^2]/x': 'asin(2*x+9*(4-x)**2)/x',
'x+y': 'x+y',
'355/113': '355/113',
'2.718281828': '2.718281828',
'Sin[12]': 'sin(12)',
'Exp[Log[4]]': 'exp(log(4))',
'(x+1)(x+3)': '(x+1)*(x+3)',
'Cos[ArcCos[3.6]]': 'cos(acos(3.6))',
'Cos[x]==Sin[y]': 'Eq(cos(x), sin(y))',
'2*Sin[x+y]': '2*sin(x+y)',
'Sin[x]+Cos[y]': 'sin(x)+cos(y)',
'Sin[Cos[x]]': 'sin(cos(x))',
'2*Sqrt[x+y]': '2*sqrt(x+y)', # Test case from the issue 4259
'+Sqrt[2]': 'sqrt(2)',
'-Sqrt[2]': '-sqrt(2)',
'-1/Sqrt[2]': '-1/sqrt(2)',
'-(1/Sqrt[3])': '-(1/sqrt(3))',
'1/(2*Sqrt[5])': '1/(2*sqrt(5))',
'Mod[5,3]': 'Mod(5,3)',
'-Mod[5,3]': '-Mod(5,3)',
'(x+1)y': '(x+1)*y',
'x(y+1)': 'x*(y+1)',
'Sin[x]Cos[y]': 'sin(x)*cos(y)',
'Sin[x]^2Cos[y]^2': 'sin(x)**2*cos(y)**2',
'Cos[x]^2(1 - Cos[y]^2)': 'cos(x)**2*(1-cos(y)**2)',
'x y': 'x*y',
'x y': 'x*y',
'2 x': '2*x',
'x 8': 'x*8',
'2 8': '2*8',
'4.x': '4.*x',
'4. 3': '4.*3',
'4. 3.': '4.*3.',
'1 2 3': '1*2*3',
' - 2 * Sqrt[ 2 3 * ( 1 + 5 ) ] ': '-2*sqrt(2*3*(1+5))',
'Log[2,4]': 'log(4,2)',
'Log[Log[2,4],4]': 'log(4,log(4,2))',
'Exp[Sqrt[2]^2Log[2, 8]]': 'exp(sqrt(2)**2*log(8,2))',
'ArcSin[Cos[0]]': 'asin(cos(0))',
'Log2[16]': 'log(16,2)',
'Max[1,-2,3,-4]': 'Max(1,-2,3,-4)',
'Min[1,-2,3]': 'Min(1,-2,3)',
'Exp[I Pi/2]': 'exp(I*pi/2)',
'ArcTan[x,y]': 'atan2(y,x)',
'Pochhammer[x,y]': 'rf(x,y)',
'ExpIntegralEi[x]': 'Ei(x)',
'SinIntegral[x]': 'Si(x)',
'CosIntegral[x]': 'Ci(x)',
'AiryAi[x]': 'airyai(x)',
'AiryAiPrime[5]': 'airyaiprime(5)',
'AiryBi[x]': 'airybi(x)',
'AiryBiPrime[7]': 'airybiprime(7)',
'LogIntegral[4]': ' li(4)',
'PrimePi[7]': 'primepi(7)',
'Prime[5]': 'prime(5)',
'PrimeQ[5]': 'isprime(5)'
}
for e in d:
assert mathematica(e) == sympify(d[e])
# The parsed form of this expression should not evaluate the Lambda object:
assert mathematica("Sin[#]^2 + Cos[#]^2 &[x]") == sin(x)**2 + cos(x)**2
d1, d2, d3 = symbols("d1:4", cls=Dummy)
assert mathematica("Sin[#] + Cos[#3] &").dummy_eq(Lambda((d1, d2, d3), sin(d1) + cos(d3)))
assert mathematica("Sin[#^2] &").dummy_eq(Lambda(d1, sin(d1**2)))
assert mathematica("Function[x, x^3]") == Lambda(x, x**3)
assert mathematica("Function[{x, y}, x^2 + y^2]") == Lambda((x, y), x**2+y**2)
def test_parser_mathematica_tokenizer():
parser = MathematicaParser()
chain = lambda expr: parser._from_tokens_to_fullformlist(parser._from_mathematica_to_tokens(expr))
# Basic patterns
assert chain("x") == "x"
assert chain("42") == "42"
assert chain(".2") == ".2"
assert chain("+x") == "x"
assert chain("-1") == "-1"
assert chain("- 3") == "-3"
assert chain("+Sin[x]") == ["Sin", "x"]
assert chain("-Sin[x]") == ["Times", "-1", ["Sin", "x"]]
assert chain("x(a+1)") == ["Times", "x", ["Plus", "a", "1"]]
assert chain("(x)") == "x"
assert chain("(+x)") == "x"
assert chain("-a") == ["Times", "-1", "a"]
assert chain("(-x)") == ["Times", "-1", "x"]
assert chain("(x + y)") == ["Plus", "x", "y"]
assert chain("3 + 4") == ["Plus", "3", "4"]
assert chain("a - 3") == ["Plus", "a", "-3"]
assert chain("a - b") == ["Plus", "a", ["Times", "-1", "b"]]
assert chain("7 * 8") == ["Times", "7", "8"]
assert chain("a + b*c") == ["Plus", "a", ["Times", "b", "c"]]
assert chain("a + b* c* d + 2 * e") == ["Plus", "a", ["Times", "b", "c", "d"], ["Times", "2", "e"]]
assert chain("a / b") == ["Times", "a", ["Power", "b", "-1"]]
# Missing asterisk (*) patterns:
assert chain("x y") == ["Times", "x", "y"]
assert chain("3 4") == ["Times", "3", "4"]
assert chain("a[b] c") == ["Times", ["a", "b"], "c"]
assert chain("(x) (y)") == ["Times", "x", "y"]
assert chain("3 (a)") == ["Times", "3", "a"]
assert chain("(a) b") == ["Times", "a", "b"]
assert chain("4.2") == "4.2"
assert chain("4 2") == ["Times", "4", "2"]
assert chain("4 2") == ["Times", "4", "2"]
assert chain("3 . 4") == ["Dot", "3", "4"]
assert chain("4. 2") == ["Times", "4.", "2"]
assert chain("x.y") == ["Dot", "x", "y"]
assert chain("4.y") == ["Times", "4.", "y"]
assert chain("4 .y") == ["Dot", "4", "y"]
assert chain("x.4") == ["Times", "x", ".4"]
assert chain("x0.3") == ["Times", "x0", ".3"]
assert chain("x. 4") == ["Dot", "x", "4"]
# Comments
assert chain("a (* +b *) + c") == ["Plus", "a", "c"]
assert chain("a (* + b *) + (**)c (* +d *) + e") == ["Plus", "a", "c", "e"]
assert chain("""a + (*
+ b
*) c + (* d
*) e
""") == ["Plus", "a", "c", "e"]
# Operators couples + and -, * and / are mutually associative:
# (i.e. expression gets flattened when mixing these operators)
assert chain("a*b/c") == ["Times", "a", "b", ["Power", "c", "-1"]]
assert chain("a/b*c") == ["Times", "a", ["Power", "b", "-1"], "c"]
assert chain("a+b-c") == ["Plus", "a", "b", ["Times", "-1", "c"]]
assert chain("a-b+c") == ["Plus", "a", ["Times", "-1", "b"], "c"]
assert chain("-a + b -c ") == ["Plus", ["Times", "-1", "a"], "b", ["Times", "-1", "c"]]
assert chain("a/b/c*d") == ["Times", "a", ["Power", "b", "-1"], ["Power", "c", "-1"], "d"]
assert chain("a/b/c") == ["Times", "a", ["Power", "b", "-1"], ["Power", "c", "-1"]]
assert chain("a-b-c") == ["Plus", "a", ["Times", "-1", "b"], ["Times", "-1", "c"]]
assert chain("1/a") == ["Times", "1", ["Power", "a", "-1"]]
assert chain("1/a/b") == ["Times", "1", ["Power", "a", "-1"], ["Power", "b", "-1"]]
assert chain("-1/a*b") == ["Times", "-1", ["Power", "a", "-1"], "b"]
# Enclosures of various kinds, i.e. ( ) [ ] [[ ]] { }
assert chain("(a + b) + c") == ["Plus", ["Plus", "a", "b"], "c"]
assert chain(" a + (b + c) + d ") == ["Plus", "a", ["Plus", "b", "c"], "d"]
assert chain("a * (b + c)") == ["Times", "a", ["Plus", "b", "c"]]
assert chain("a b (c d)") == ["Times", "a", "b", ["Times", "c", "d"]]
assert chain("{a, b, 2, c}") == ["List", "a", "b", "2", "c"]
assert chain("{a, {b, c}}") == ["List", "a", ["List", "b", "c"]]
assert chain("{{a}}") == ["List", ["List", "a"]]
assert chain("a[b, c]") == ["a", "b", "c"]
assert chain("a[[b, c]]") == ["Part", "a", "b", "c"]
assert chain("a[b[c]]") == ["a", ["b", "c"]]
assert chain("a[[b, c[[d, {e,f}]]]]") == ["Part", "a", "b", ["Part", "c", "d", ["List", "e", "f"]]]
assert chain("a[b[[c,d]]]") == ["a", ["Part", "b", "c", "d"]]
assert chain("a[[b[c]]]") == ["Part", "a", ["b", "c"]]
assert chain("a[[b[[c]]]]") == ["Part", "a", ["Part", "b", "c"]]
assert chain("a[[b[c[[d]]]]]") == ["Part", "a", ["b", ["Part", "c", "d"]]]
assert chain("a[b[[c[d]]]]") == ["a", ["Part", "b", ["c", "d"]]]
assert chain("x[[a+1, b+2, c+3]]") == ["Part", "x", ["Plus", "a", "1"], ["Plus", "b", "2"], ["Plus", "c", "3"]]
assert chain("x[a+1, b+2, c+3]") == ["x", ["Plus", "a", "1"], ["Plus", "b", "2"], ["Plus", "c", "3"]]
assert chain("{a+1, b+2, c+3}") == ["List", ["Plus", "a", "1"], ["Plus", "b", "2"], ["Plus", "c", "3"]]
# Flat operator:
assert chain("a*b*c*d*e") == ["Times", "a", "b", "c", "d", "e"]
assert chain("a +b + c+ d+e") == ["Plus", "a", "b", "c", "d", "e"]
# Right priority operator:
assert chain("a^b") == ["Power", "a", "b"]
assert chain("a^b^c") == ["Power", "a", ["Power", "b", "c"]]
assert chain("a^b^c^d") == ["Power", "a", ["Power", "b", ["Power", "c", "d"]]]
# Left priority operator:
assert chain("a/.b") == ["ReplaceAll", "a", "b"]
assert chain("a/.b/.c/.d") == ["ReplaceAll", ["ReplaceAll", ["ReplaceAll", "a", "b"], "c"], "d"]
assert chain("a//b") == ["a", "b"]
assert chain("a//b//c") == [["a", "b"], "c"]
assert chain("a//b//c//d") == [[["a", "b"], "c"], "d"]
# Compound expressions
assert chain("a;b") == ["CompoundExpression", "a", "b"]
assert chain("a;") == ["CompoundExpression", "a", "Null"]
assert chain("a;b;") == ["CompoundExpression", "a", "b", "Null"]
assert chain("a[b;c]") == ["a", ["CompoundExpression", "b", "c"]]
assert chain("a[b,c;d,e]") == ["a", "b", ["CompoundExpression", "c", "d"], "e"]
assert chain("a[b,c;,d]") == ["a", "b", ["CompoundExpression", "c", "Null"], "d"]
# New lines
assert chain("a\nb\n") == ["CompoundExpression", "a", "b"]
assert chain("a\n\nb\n (c \nd) \n") == ["CompoundExpression", "a", "b", ["Times", "c", "d"]]
assert chain("\na; b\nc") == ["CompoundExpression", "a", "b", "c"]
assert chain("a + \nb\n") == ["Plus", "a", "b"]
assert chain("a\nb; c; d\n e; (f \n g); h + \n i") == ["CompoundExpression", "a", "b", "c", "d", "e", ["Times", "f", "g"], ["Plus", "h", "i"]]
assert chain("\n{\na\nb; c; d\n e (f \n g); h + \n i\n\n}\n") == ["List", ["CompoundExpression", ["Times", "a", "b"], "c", ["Times", "d", "e", ["Times", "f", "g"]], ["Plus", "h", "i"]]]
# Patterns
assert chain("y_") == ["Pattern", "y", ["Blank"]]
assert chain("y_.") == ["Optional", ["Pattern", "y", ["Blank"]]]
assert chain("y__") == ["Pattern", "y", ["BlankSequence"]]
assert chain("y___") == ["Pattern", "y", ["BlankNullSequence"]]
assert chain("a[b_.,c_]") == ["a", ["Optional", ["Pattern", "b", ["Blank"]]], ["Pattern", "c", ["Blank"]]]
assert chain("b_. c") == ["Times", ["Optional", ["Pattern", "b", ["Blank"]]], "c"]
# Slots for lambda functions
assert chain("#") == ["Slot", "1"]
assert chain("#3") == ["Slot", "3"]
assert chain("#n") == ["Slot", "n"]
assert chain("##") == ["SlotSequence", "1"]
assert chain("##a") == ["SlotSequence", "a"]
# Lambda functions
assert chain("x&") == ["Function", "x"]
assert chain("#&") == ["Function", ["Slot", "1"]]
assert chain("#+3&") == ["Function", ["Plus", ["Slot", "1"], "3"]]
assert chain("#1 + #2&") == ["Function", ["Plus", ["Slot", "1"], ["Slot", "2"]]]
assert chain("# + #&") == ["Function", ["Plus", ["Slot", "1"], ["Slot", "1"]]]
assert chain("#&[x]") == [["Function", ["Slot", "1"]], "x"]
assert chain("#1 + #2 & [x, y]") == [["Function", ["Plus", ["Slot", "1"], ["Slot", "2"]]], "x", "y"]
assert chain("#1^2#2^3&") == ["Function", ["Times", ["Power", ["Slot", "1"], "2"], ["Power", ["Slot", "2"], "3"]]]
# Invalid expressions:
raises(SyntaxError, lambda: chain("(,"))
raises(SyntaxError, lambda: chain("()"))
raises(SyntaxError, lambda: chain("a (* b"))
def test_parser_mathematica_exp_alt():
parser = MathematicaParser()
convert_chain2 = lambda expr: parser._from_fullformlist_to_fullformsympy(parser._from_fullform_to_fullformlist(expr))
convert_chain3 = lambda expr: parser._from_fullformsympy_to_sympy(convert_chain2(expr))
Sin, Times, Plus, Power = symbols("Sin Times Plus Power", cls=Function)
full_form1 = "Sin[Times[x, y]]"
full_form2 = "Plus[Times[x, y], z]"
full_form3 = "Sin[Times[x, Plus[y, z], Power[w, n]]]]"
assert parser._from_fullform_to_fullformlist(full_form1) == ["Sin", ["Times", "x", "y"]]
assert parser._from_fullform_to_fullformlist(full_form2) == ["Plus", ["Times", "x", "y"], "z"]
assert parser._from_fullform_to_fullformlist(full_form3) == ["Sin", ["Times", "x", ["Plus", "y", "z"], ["Power", "w", "n"]]]
assert convert_chain2(full_form1) == Sin(Times(x, y))
assert convert_chain2(full_form2) == Plus(Times(x, y), z)
assert convert_chain2(full_form3) == Sin(Times(x, Plus(y, z), Power(w, n)))
assert convert_chain3(full_form1) == sin(x*y)
assert convert_chain3(full_form2) == x*y + z
assert convert_chain3(full_form3) == sin(x*(y + z)*w**n)
|
d0e4ca5e9dd54f985b88f3547403abd76ccf87830dee7be0c10f4effb4ef0a5f | from typing import Type
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.expr import Expr
from sympy.core.function import expand
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.symbol import Symbol
from sympy.polys.polyroots import roots
from sympy.polys.polytools import (cancel, degree)
from sympy.core.containers import Tuple
from sympy.core.evalf import EvalfMixin
from sympy.core.logic import fuzzy_and
from sympy.core.numbers import Integer, ComplexInfinity
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify, _sympify
from sympy.polys import Poly, rootof
from sympy.series import limit
from sympy.matrices import ImmutableMatrix, eye
from sympy.matrices.expressions import MatMul, MatAdd
from mpmath.libmp.libmpf import prec_to_dps
__all__ = ['TransferFunction', 'Series', 'MIMOSeries', 'Parallel', 'MIMOParallel',
'Feedback', 'MIMOFeedback', 'TransferFunctionMatrix']
def _roots(poly, var):
""" like roots, but works on higher-order polynomials. """
r = roots(poly, var, multiple=True)
n = degree(poly)
if len(r) != n:
r = [rootof(poly, var, k) for k in range(n)]
return r
class LinearTimeInvariant(Basic, EvalfMixin):
"""A common class for all the Linear Time-Invariant Dynamical Systems."""
_clstype: Type
# Users should not directly interact with this class.
def __new__(cls, *system, **kwargs):
if cls is LinearTimeInvariant:
raise NotImplementedError('The LTICommon class is not meant to be used directly.')
return super(LinearTimeInvariant, cls).__new__(cls, *system, **kwargs)
@classmethod
def _check_args(cls, args):
if not args:
raise ValueError("Atleast 1 argument must be passed.")
if not all(isinstance(arg, cls._clstype) for arg in args):
raise TypeError(f"All arguments must be of type {cls._clstype}.")
var_set = {arg.var for arg in args}
if len(var_set) != 1:
raise ValueError("All transfer functions should use the same complex variable"
f" of the Laplace transform. {len(var_set)} different values found.")
@property
def is_SISO(self):
"""Returns `True` if the passed LTI system is SISO else returns False."""
return self._is_SISO
class SISOLinearTimeInvariant(LinearTimeInvariant):
"""A common class for all the SISO Linear Time-Invariant Dynamical Systems."""
# Users should not directly interact with this class.
_is_SISO = True
class MIMOLinearTimeInvariant(LinearTimeInvariant):
"""A common class for all the MIMO Linear Time-Invariant Dynamical Systems."""
# Users should not directly interact with this class.
_is_SISO = False
SISOLinearTimeInvariant._clstype = SISOLinearTimeInvariant
MIMOLinearTimeInvariant._clstype = MIMOLinearTimeInvariant
def _check_other_SISO(func):
def wrapper(*args, **kwargs):
if not isinstance(args[-1], SISOLinearTimeInvariant):
return NotImplemented
else:
return func(*args, **kwargs)
return wrapper
def _check_other_MIMO(func):
def wrapper(*args, **kwargs):
if not isinstance(args[-1], MIMOLinearTimeInvariant):
return NotImplemented
else:
return func(*args, **kwargs)
return wrapper
class TransferFunction(SISOLinearTimeInvariant):
r"""
A class for representing LTI (Linear, time-invariant) systems that can be strictly described
by ratio of polynomials in the Laplace transform complex variable. The arguments
are ``num``, ``den``, and ``var``, where ``num`` and ``den`` are numerator and
denominator polynomials of the ``TransferFunction`` respectively, and the third argument is
a complex variable of the Laplace transform used by these polynomials of the transfer function.
``num`` and ``den`` can be either polynomials or numbers, whereas ``var``
has to be a :py:class:`~.Symbol`.
Explanation
===========
Generally, a dynamical system representing a physical model can be described in terms of Linear
Ordinary Differential Equations like -
$\small{b_{m}y^{\left(m\right)}+b_{m-1}y^{\left(m-1\right)}+\dots+b_{1}y^{\left(1\right)}+b_{0}y=
a_{n}x^{\left(n\right)}+a_{n-1}x^{\left(n-1\right)}+\dots+a_{1}x^{\left(1\right)}+a_{0}x}$
Here, $x$ is the input signal and $y$ is the output signal and superscript on both is the order of derivative
(not exponent). Derivative is taken with respect to the independent variable, $t$. Also, generally $m$ is greater
than $n$.
It is not feasible to analyse the properties of such systems in their native form therefore, we use
mathematical tools like Laplace transform to get a better perspective. Taking the Laplace transform
of both the sides in the equation (at zero initial conditions), we get -
$\small{\mathcal{L}[b_{m}y^{\left(m\right)}+b_{m-1}y^{\left(m-1\right)}+\dots+b_{1}y^{\left(1\right)}+b_{0}y]=
\mathcal{L}[a_{n}x^{\left(n\right)}+a_{n-1}x^{\left(n-1\right)}+\dots+a_{1}x^{\left(1\right)}+a_{0}x]}$
Using the linearity property of Laplace transform and also considering zero initial conditions
(i.e. $\small{y(0^{-}) = 0}$, $\small{y'(0^{-}) = 0}$ and so on), the equation
above gets translated to -
$\small{b_{m}\mathcal{L}[y^{\left(m\right)}]+\dots+b_{1}\mathcal{L}[y^{\left(1\right)}]+b_{0}\mathcal{L}[y]=
a_{n}\mathcal{L}[x^{\left(n\right)}]+\dots+a_{1}\mathcal{L}[x^{\left(1\right)}]+a_{0}\mathcal{L}[x]}$
Now, applying Derivative property of Laplace transform,
$\small{b_{m}s^{m}\mathcal{L}[y]+\dots+b_{1}s\mathcal{L}[y]+b_{0}\mathcal{L}[y]=
a_{n}s^{n}\mathcal{L}[x]+\dots+a_{1}s\mathcal{L}[x]+a_{0}\mathcal{L}[x]}$
Here, the superscript on $s$ is **exponent**. Note that the zero initial conditions assumption, mentioned above, is very important
and cannot be ignored otherwise the dynamical system cannot be considered time-independent and the simplified equation above
cannot be reached.
Collecting $\mathcal{L}[y]$ and $\mathcal{L}[x]$ terms from both the sides and taking the ratio
$\frac{ \mathcal{L}\left\{y\right\} }{ \mathcal{L}\left\{x\right\} }$, we get the typical rational form of transfer
function.
The numerator of the transfer function is, therefore, the Laplace transform of the output signal
(The signals are represented as functions of time) and similarly, the denominator
of the transfer function is the Laplace transform of the input signal. It is also a convention
to denote the input and output signal's Laplace transform with capital alphabets like shown below.
$H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} }$
$s$, also known as complex frequency, is a complex variable in the Laplace domain. It corresponds to the
equivalent variable $t$, in the time domain. Transfer functions are sometimes also referred to as the Laplace
transform of the system's impulse response. Transfer function, $H$, is represented as a rational
function in $s$ like,
$H(s) =\ \frac{a_{n}s^{n}+a_{n-1}s^{n-1}+\dots+a_{1}s+a_{0}}{b_{m}s^{m}+b_{m-1}s^{m-1}+\dots+b_{1}s+b_{0}}$
Parameters
==========
num : Expr, Number
The numerator polynomial of the transfer function.
den : Expr, Number
The denominator polynomial of the transfer function.
var : Symbol
Complex variable of the Laplace transform used by the
polynomials of the transfer function.
Raises
======
TypeError
When ``var`` is not a Symbol or when ``num`` or ``den`` is not a
number or a polynomial.
ValueError
When ``den`` is zero.
Examples
========
>>> from sympy.abc import s, p, a
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction(s + a, s**2 + s + 1, s)
>>> tf1
TransferFunction(a + s, s**2 + s + 1, s)
>>> tf1.num
a + s
>>> tf1.den
s**2 + s + 1
>>> tf1.var
s
>>> tf1.args
(a + s, s**2 + s + 1, s)
Any complex variable can be used for ``var``.
>>> tf2 = TransferFunction(a*p**3 - a*p**2 + s*p, p + a**2, p)
>>> tf2
TransferFunction(a*p**3 - a*p**2 + p*s, a**2 + p, p)
>>> tf3 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p)
>>> tf3
TransferFunction((p - 1)*(p + 3), (p - 1)*(p + 5), p)
To negate a transfer function the ``-`` operator can be prepended:
>>> tf4 = TransferFunction(-a + s, p**2 + s, p)
>>> -tf4
TransferFunction(a - s, p**2 + s, p)
>>> tf5 = TransferFunction(s**4 - 2*s**3 + 5*s + 4, s + 4, s)
>>> -tf5
TransferFunction(-s**4 + 2*s**3 - 5*s - 4, s + 4, s)
You can use a Float or an Integer (or other constants) as numerator and denominator:
>>> tf6 = TransferFunction(1/2, 4, s)
>>> tf6.num
0.500000000000000
>>> tf6.den
4
>>> tf6.var
s
>>> tf6.args
(0.5, 4, s)
You can take the integer power of a transfer function using the ``**`` operator:
>>> tf7 = TransferFunction(s + a, s - a, s)
>>> tf7**3
TransferFunction((a + s)**3, (-a + s)**3, s)
>>> tf7**0
TransferFunction(1, 1, s)
>>> tf8 = TransferFunction(p + 4, p - 3, p)
>>> tf8**-1
TransferFunction(p - 3, p + 4, p)
Addition, subtraction, and multiplication of transfer functions can form
unevaluated ``Series`` or ``Parallel`` objects.
>>> tf9 = TransferFunction(s + 1, s**2 + s + 1, s)
>>> tf10 = TransferFunction(s - p, s + 3, s)
>>> tf11 = TransferFunction(4*s**2 + 2*s - 4, s - 1, s)
>>> tf12 = TransferFunction(1 - s, s**2 + 4, s)
>>> tf9 + tf10
Parallel(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-p + s, s + 3, s))
>>> tf10 - tf11
Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(-4*s**2 - 2*s + 4, s - 1, s))
>>> tf9 * tf10
Series(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-p + s, s + 3, s))
>>> tf10 - (tf9 + tf12)
Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(-s - 1, s**2 + s + 1, s), TransferFunction(s - 1, s**2 + 4, s))
>>> tf10 - (tf9 * tf12)
Parallel(TransferFunction(-p + s, s + 3, s), Series(TransferFunction(-1, 1, s), TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(1 - s, s**2 + 4, s)))
>>> tf11 * tf10 * tf9
Series(TransferFunction(4*s**2 + 2*s - 4, s - 1, s), TransferFunction(-p + s, s + 3, s), TransferFunction(s + 1, s**2 + s + 1, s))
>>> tf9 * tf11 + tf10 * tf12
Parallel(Series(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(4*s**2 + 2*s - 4, s - 1, s)), Series(TransferFunction(-p + s, s + 3, s), TransferFunction(1 - s, s**2 + 4, s)))
>>> (tf9 + tf12) * (tf10 + tf11)
Series(Parallel(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(1 - s, s**2 + 4, s)), Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(4*s**2 + 2*s - 4, s - 1, s)))
These unevaluated ``Series`` or ``Parallel`` objects can convert into the
resultant transfer function using ``.doit()`` method or by ``.rewrite(TransferFunction)``.
>>> ((tf9 + tf10) * tf12).doit()
TransferFunction((1 - s)*((-p + s)*(s**2 + s + 1) + (s + 1)*(s + 3)), (s + 3)*(s**2 + 4)*(s**2 + s + 1), s)
>>> (tf9 * tf10 - tf11 * tf12).rewrite(TransferFunction)
TransferFunction(-(1 - s)*(s + 3)*(s**2 + s + 1)*(4*s**2 + 2*s - 4) + (-p + s)*(s - 1)*(s + 1)*(s**2 + 4), (s - 1)*(s + 3)*(s**2 + 4)*(s**2 + s + 1), s)
See Also
========
Feedback, Series, Parallel
References
==========
.. [1] https://en.wikipedia.org/wiki/Transfer_function
.. [2] https://en.wikipedia.org/wiki/Laplace_transform
"""
def __new__(cls, num, den, var):
num, den = _sympify(num), _sympify(den)
if not isinstance(var, Symbol):
raise TypeError("Variable input must be a Symbol.")
if den == 0:
raise ValueError("TransferFunction cannot have a zero denominator.")
if (((isinstance(num, Expr) and num.has(Symbol)) or num.is_number) and
((isinstance(den, Expr) and den.has(Symbol)) or den.is_number)):
obj = super(TransferFunction, cls).__new__(cls, num, den, var)
obj._num = num
obj._den = den
obj._var = var
return obj
else:
raise TypeError("Unsupported type for numerator or denominator of TransferFunction.")
@classmethod
def from_rational_expression(cls, expr, var=None):
r"""
Creates a new ``TransferFunction`` efficiently from a rational expression.
Parameters
==========
expr : Expr, Number
The rational expression representing the ``TransferFunction``.
var : Symbol, optional
Complex variable of the Laplace transform used by the
polynomials of the transfer function.
Raises
======
ValueError
When ``expr`` is of type ``Number`` and optional parameter ``var``
is not passed.
When ``expr`` has more than one variables and an optional parameter
``var`` is not passed.
ZeroDivisionError
When denominator of ``expr`` is zero or it has ``ComplexInfinity``
in its numerator.
Examples
========
>>> from sympy.abc import s, p, a
>>> from sympy.physics.control.lti import TransferFunction
>>> expr1 = (s + 5)/(3*s**2 + 2*s + 1)
>>> tf1 = TransferFunction.from_rational_expression(expr1)
>>> tf1
TransferFunction(s + 5, 3*s**2 + 2*s + 1, s)
>>> expr2 = (a*p**3 - a*p**2 + s*p)/(p + a**2) # Expr with more than one variables
>>> tf2 = TransferFunction.from_rational_expression(expr2, p)
>>> tf2
TransferFunction(a*p**3 - a*p**2 + p*s, a**2 + p, p)
In case of conflict between two or more variables in a expression, SymPy will
raise a ``ValueError``, if ``var`` is not passed by the user.
>>> tf = TransferFunction.from_rational_expression((a + a*s)/(s**2 + s + 1))
Traceback (most recent call last):
...
ValueError: Conflicting values found for positional argument `var` ({a, s}). Specify it manually.
This can be corrected by specifying the ``var`` parameter manually.
>>> tf = TransferFunction.from_rational_expression((a + a*s)/(s**2 + s + 1), s)
>>> tf
TransferFunction(a*s + a, s**2 + s + 1, s)
``var`` also need to be specified when ``expr`` is a ``Number``
>>> tf3 = TransferFunction.from_rational_expression(10, s)
>>> tf3
TransferFunction(10, 1, s)
"""
expr = _sympify(expr)
if var is None:
_free_symbols = expr.free_symbols
_len_free_symbols = len(_free_symbols)
if _len_free_symbols == 1:
var = list(_free_symbols)[0]
elif _len_free_symbols == 0:
raise ValueError("Positional argument `var` not found in the TransferFunction defined. Specify it manually.")
else:
raise ValueError("Conflicting values found for positional argument `var` ({}). Specify it manually.".format(_free_symbols))
_num, _den = expr.as_numer_denom()
if _den == 0 or _num.has(ComplexInfinity):
raise ZeroDivisionError("TransferFunction cannot have a zero denominator.")
return cls(_num, _den, var)
@property
def num(self):
"""
Returns the numerator polynomial of the transfer function.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction
>>> G1 = TransferFunction(s**2 + p*s + 3, s - 4, s)
>>> G1.num
p*s + s**2 + 3
>>> G2 = TransferFunction((p + 5)*(p - 3), (p - 3)*(p + 1), p)
>>> G2.num
(p - 3)*(p + 5)
"""
return self._num
@property
def den(self):
"""
Returns the denominator polynomial of the transfer function.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction
>>> G1 = TransferFunction(s + 4, p**3 - 2*p + 4, s)
>>> G1.den
p**3 - 2*p + 4
>>> G2 = TransferFunction(3, 4, s)
>>> G2.den
4
"""
return self._den
@property
def var(self):
"""
Returns the complex variable of the Laplace transform used by the polynomials of
the transfer function.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction
>>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p)
>>> G1.var
p
>>> G2 = TransferFunction(0, s - 5, s)
>>> G2.var
s
"""
return self._var
def _eval_subs(self, old, new):
arg_num = self.num.subs(old, new)
arg_den = self.den.subs(old, new)
argnew = TransferFunction(arg_num, arg_den, self.var)
return self if old == self.var else argnew
def _eval_evalf(self, prec):
return TransferFunction(
self.num._eval_evalf(prec),
self.den._eval_evalf(prec),
self.var)
def _eval_simplify(self, **kwargs):
tf = cancel(Mul(self.num, 1/self.den, evaluate=False), expand=False).as_numer_denom()
num_, den_ = tf[0], tf[1]
return TransferFunction(num_, den_, self.var)
def expand(self):
"""
Returns the transfer function with numerator and denominator
in expanded form.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction
>>> G1 = TransferFunction((a - s)**2, (s**2 + a)**2, s)
>>> G1.expand()
TransferFunction(a**2 - 2*a*s + s**2, a**2 + 2*a*s**2 + s**4, s)
>>> G2 = TransferFunction((p + 3*b)*(p - b), (p - b)*(p + 2*b), p)
>>> G2.expand()
TransferFunction(-3*b**2 + 2*b*p + p**2, -2*b**2 + b*p + p**2, p)
"""
return TransferFunction(expand(self.num), expand(self.den), self.var)
def dc_gain(self):
"""
Computes the gain of the response as the frequency approaches zero.
The DC gain is infinite for systems with pure integrators.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction(s + 3, s**2 - 9, s)
>>> tf1.dc_gain()
-1/3
>>> tf2 = TransferFunction(p**2, p - 3 + p**3, p)
>>> tf2.dc_gain()
0
>>> tf3 = TransferFunction(a*p**2 - b, s + b, s)
>>> tf3.dc_gain()
(a*p**2 - b)/b
>>> tf4 = TransferFunction(1, s, s)
>>> tf4.dc_gain()
oo
"""
m = Mul(self.num, Pow(self.den, -1, evaluate=False), evaluate=False)
return limit(m, self.var, 0)
def poles(self):
"""
Returns the poles of a transfer function.
Examples
========
>>> from sympy.abc import s, p, a
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p)
>>> tf1.poles()
[-5, 1]
>>> tf2 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s)
>>> tf2.poles()
[I, I, -I, -I]
>>> tf3 = TransferFunction(s**2, a*s + p, s)
>>> tf3.poles()
[-p/a]
"""
return _roots(Poly(self.den, self.var), self.var)
def zeros(self):
"""
Returns the zeros of a transfer function.
Examples
========
>>> from sympy.abc import s, p, a
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p)
>>> tf1.zeros()
[-3, 1]
>>> tf2 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s)
>>> tf2.zeros()
[1, 1]
>>> tf3 = TransferFunction(s**2, a*s + p, s)
>>> tf3.zeros()
[0, 0]
"""
return _roots(Poly(self.num, self.var), self.var)
def is_stable(self):
"""
Returns True if the transfer function is asymptotically stable; else False.
This would not check the marginal or conditional stability of the system.
Examples
========
>>> from sympy.abc import s, p, a
>>> from sympy import symbols
>>> from sympy.physics.control.lti import TransferFunction
>>> q, r = symbols('q, r', negative=True)
>>> tf1 = TransferFunction((1 - s)**2, (s + 1)**2, s)
>>> tf1.is_stable()
True
>>> tf2 = TransferFunction((1 - p)**2, (s**2 + 1)**2, s)
>>> tf2.is_stable()
False
>>> tf3 = TransferFunction(4, q*s - r, s)
>>> tf3.is_stable()
False
>>> tf4 = TransferFunction(p + 1, a*p - s**2, p)
>>> tf4.is_stable() is None # Not enough info about the symbols to determine stability
True
"""
return fuzzy_and(pole.as_real_imag()[0].is_negative for pole in self.poles())
def __add__(self, other):
if isinstance(other, (TransferFunction, Series)):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
return Parallel(self, other)
elif isinstance(other, Parallel):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
arg_list = list(other.args)
return Parallel(self, *arg_list)
else:
raise ValueError("TransferFunction cannot be added with {}.".
format(type(other)))
def __radd__(self, other):
return self + other
def __sub__(self, other):
if isinstance(other, (TransferFunction, Series)):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
return Parallel(self, -other)
elif isinstance(other, Parallel):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
arg_list = [-i for i in list(other.args)]
return Parallel(self, *arg_list)
else:
raise ValueError("{} cannot be subtracted from a TransferFunction."
.format(type(other)))
def __rsub__(self, other):
return -self + other
def __mul__(self, other):
if isinstance(other, (TransferFunction, Parallel)):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
return Series(self, other)
elif isinstance(other, Series):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
arg_list = list(other.args)
return Series(self, *arg_list)
else:
raise ValueError("TransferFunction cannot be multiplied with {}."
.format(type(other)))
__rmul__ = __mul__
def __truediv__(self, other):
if (isinstance(other, Parallel) and len(other.args) == 2 and isinstance(other.args[0], TransferFunction)
and isinstance(other.args[1], (Series, TransferFunction))):
if not self.var == other.var:
raise ValueError("Both TransferFunction and Parallel should use the"
" same complex variable of the Laplace transform.")
if other.args[1] == self:
# plant and controller with unit feedback.
return Feedback(self, other.args[0])
other_arg_list = list(other.args[1].args) if isinstance(other.args[1], Series) else other.args[1]
if other_arg_list == other.args[1]:
return Feedback(self, other_arg_list)
elif self in other_arg_list:
other_arg_list.remove(self)
else:
return Feedback(self, Series(*other_arg_list))
if len(other_arg_list) == 1:
return Feedback(self, *other_arg_list)
else:
return Feedback(self, Series(*other_arg_list))
else:
raise ValueError("TransferFunction cannot be divided by {}.".
format(type(other)))
__rtruediv__ = __truediv__
def __pow__(self, p):
p = sympify(p)
if not isinstance(p, Integer):
raise ValueError("Exponent must be an Integer.")
if p == 0:
return TransferFunction(1, 1, self.var)
elif p > 0:
num_, den_ = self.num**p, self.den**p
else:
p = abs(p)
num_, den_ = self.den**p, self.num**p
return TransferFunction(num_, den_, self.var)
def __neg__(self):
return TransferFunction(-self.num, self.den, self.var)
@property
def is_proper(self):
"""
Returns True if degree of the numerator polynomial is less than
or equal to degree of the denominator polynomial, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s)
>>> tf1.is_proper
False
>>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*p + 2, p)
>>> tf2.is_proper
True
"""
return degree(self.num, self.var) <= degree(self.den, self.var)
@property
def is_strictly_proper(self):
"""
Returns True if degree of the numerator polynomial is strictly less
than degree of the denominator polynomial, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf1.is_strictly_proper
False
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> tf2.is_strictly_proper
True
"""
return degree(self.num, self.var) < degree(self.den, self.var)
@property
def is_biproper(self):
"""
Returns True if degree of the numerator polynomial is equal to
degree of the denominator polynomial, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf1.is_biproper
True
>>> tf2 = TransferFunction(p**2, p + a, p)
>>> tf2.is_biproper
False
"""
return degree(self.num, self.var) == degree(self.den, self.var)
def to_expr(self):
"""
Converts a ``TransferFunction`` object to SymPy Expr.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy import Expr
>>> tf1 = TransferFunction(s, a*s**2 + 1, s)
>>> tf1.to_expr()
s/(a*s**2 + 1)
>>> isinstance(_, Expr)
True
>>> tf2 = TransferFunction(1, (p + 3*b)*(b - p), p)
>>> tf2.to_expr()
1/((b - p)*(3*b + p))
>>> tf3 = TransferFunction((s - 2)*(s - 3), (s - 1)*(s - 2)*(s - 3), s)
>>> tf3.to_expr()
((s - 3)*(s - 2))/(((s - 3)*(s - 2)*(s - 1)))
"""
if self.num != 1:
return Mul(self.num, Pow(self.den, -1, evaluate=False), evaluate=False)
else:
return Pow(self.den, -1, evaluate=False)
def _flatten_args(args, _cls):
temp_args = []
for arg in args:
if isinstance(arg, _cls):
temp_args.extend(arg.args)
else:
temp_args.append(arg)
return tuple(temp_args)
def _dummify_args(_arg, var):
dummy_dict = {}
dummy_arg_list = []
for arg in _arg:
_s = Dummy()
dummy_dict[_s] = var
dummy_arg = arg.subs({var: _s})
dummy_arg_list.append(dummy_arg)
return dummy_arg_list, dummy_dict
class Series(SISOLinearTimeInvariant):
r"""
A class for representing a series configuration of SISO systems.
Parameters
==========
args : SISOLinearTimeInvariant
SISO systems in a series configuration.
evaluate : Boolean, Keyword
When passed ``True``, returns the equivalent
``Series(*args).doit()``. Set to ``False`` by default.
Raises
======
ValueError
When no argument is passed.
``var`` attribute is not same for every system.
TypeError
Any of the passed ``*args`` has unsupported type
A combination of SISO and MIMO systems is
passed. There should be homogeneity in the
type of systems passed, SISO in this case.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Series, Parallel
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> tf3 = TransferFunction(p**2, p + s, s)
>>> S1 = Series(tf1, tf2)
>>> S1
Series(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s))
>>> S1.var
s
>>> S2 = Series(tf2, Parallel(tf3, -tf1))
>>> S2
Series(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), Parallel(TransferFunction(p**2, p + s, s), TransferFunction(-a*p**2 - b*s, -p + s, s)))
>>> S2.var
s
>>> S3 = Series(Parallel(tf1, tf2), Parallel(tf2, tf3))
>>> S3
Series(Parallel(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)), Parallel(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), TransferFunction(p**2, p + s, s)))
>>> S3.var
s
You can get the resultant transfer function by using ``.doit()`` method:
>>> S3 = Series(tf1, tf2, -tf3)
>>> S3.doit()
TransferFunction(-p**2*(s**3 - 2)*(a*p**2 + b*s), (-p + s)*(p + s)*(s**4 + 5*s + 6), s)
>>> S4 = Series(tf2, Parallel(tf1, -tf3))
>>> S4.doit()
TransferFunction((s**3 - 2)*(-p**2*(-p + s) + (p + s)*(a*p**2 + b*s)), (-p + s)*(p + s)*(s**4 + 5*s + 6), s)
Notes
=====
All the transfer functions should use the same complex variable
``var`` of the Laplace transform.
See Also
========
MIMOSeries, Parallel, TransferFunction, Feedback
"""
def __new__(cls, *args, evaluate=False):
args = _flatten_args(args, Series)
cls._check_args(args)
obj = super().__new__(cls, *args)
return obj.doit() if evaluate else obj
@property
def var(self):
"""
Returns the complex variable used by all the transfer functions.
Examples
========
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, Series, Parallel
>>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p)
>>> G2 = TransferFunction(p, 4 - p, p)
>>> G3 = TransferFunction(0, p**4 - 1, p)
>>> Series(G1, G2).var
p
>>> Series(-G3, Parallel(G1, G2)).var
p
"""
return self.args[0].var
def doit(self, **kwargs):
"""
Returns the resultant transfer function obtained after evaluating
the transfer functions in series configuration.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Series
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> Series(tf2, tf1).doit()
TransferFunction((s**3 - 2)*(a*p**2 + b*s), (-p + s)*(s**4 + 5*s + 6), s)
>>> Series(-tf1, -tf2).doit()
TransferFunction((2 - s**3)*(-a*p**2 - b*s), (-p + s)*(s**4 + 5*s + 6), s)
"""
_num_arg = (arg.doit().num for arg in self.args)
_den_arg = (arg.doit().den for arg in self.args)
res_num = Mul(*_num_arg, evaluate=True)
res_den = Mul(*_den_arg, evaluate=True)
return TransferFunction(res_num, res_den, self.var)
def _eval_rewrite_as_TransferFunction(self, *args, **kwargs):
return self.doit()
@_check_other_SISO
def __add__(self, other):
if isinstance(other, Parallel):
arg_list = list(other.args)
return Parallel(self, *arg_list)
return Parallel(self, other)
__radd__ = __add__
@_check_other_SISO
def __sub__(self, other):
return self + (-other)
def __rsub__(self, other):
return -self + other
@_check_other_SISO
def __mul__(self, other):
arg_list = list(self.args)
return Series(*arg_list, other)
def __truediv__(self, other):
if (isinstance(other, Parallel) and len(other.args) == 2
and isinstance(other.args[0], TransferFunction) and isinstance(other.args[1], Series)):
if not self.var == other.var:
raise ValueError("All the transfer functions should use the same complex variable "
"of the Laplace transform.")
self_arg_list = set(list(self.args))
other_arg_list = set(list(other.args[1].args))
res = list(self_arg_list ^ other_arg_list)
if len(res) == 0:
return Feedback(self, other.args[0])
elif len(res) == 1:
return Feedback(self, *res)
else:
return Feedback(self, Series(*res))
else:
raise ValueError("This transfer function expression is invalid.")
def __neg__(self):
return Series(TransferFunction(-1, 1, self.var), self)
def to_expr(self):
"""Returns the equivalent ``Expr`` object."""
return Mul(*(arg.to_expr() for arg in self.args), evaluate=False)
@property
def is_proper(self):
"""
Returns True if degree of the numerator polynomial of the resultant transfer
function is less than or equal to degree of the denominator polynomial of
the same, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Series
>>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s)
>>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*s + 2, s)
>>> tf3 = TransferFunction(s, s**2 + s + 1, s)
>>> S1 = Series(-tf2, tf1)
>>> S1.is_proper
False
>>> S2 = Series(tf1, tf2, tf3)
>>> S2.is_proper
True
"""
return self.doit().is_proper
@property
def is_strictly_proper(self):
"""
Returns True if degree of the numerator polynomial of the resultant transfer
function is strictly less than degree of the denominator polynomial of
the same, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Series
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**2 + 5*s + 6, s)
>>> tf3 = TransferFunction(1, s**2 + s + 1, s)
>>> S1 = Series(tf1, tf2)
>>> S1.is_strictly_proper
False
>>> S2 = Series(tf1, tf2, tf3)
>>> S2.is_strictly_proper
True
"""
return self.doit().is_strictly_proper
@property
def is_biproper(self):
r"""
Returns True if degree of the numerator polynomial of the resultant transfer
function is equal to degree of the denominator polynomial of
the same, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Series
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(p, s**2, s)
>>> tf3 = TransferFunction(s**2, 1, s)
>>> S1 = Series(tf1, -tf2)
>>> S1.is_biproper
False
>>> S2 = Series(tf2, tf3)
>>> S2.is_biproper
True
"""
return self.doit().is_biproper
def _mat_mul_compatible(*args):
"""To check whether shapes are compatible for matrix mul."""
return all(args[i].num_outputs == args[i+1].num_inputs for i in range(len(args)-1))
class MIMOSeries(MIMOLinearTimeInvariant):
r"""
A class for representing a series configuration of MIMO systems.
Parameters
==========
args : MIMOLinearTimeInvariant
MIMO systems in a series configuration.
evaluate : Boolean, Keyword
When passed ``True``, returns the equivalent
``MIMOSeries(*args).doit()``. Set to ``False`` by default.
Raises
======
ValueError
When no argument is passed.
``var`` attribute is not same for every system.
``num_outputs`` of the MIMO system is not equal to the
``num_inputs`` of its adjacent MIMO system. (Matrix
multiplication constraint, basically)
TypeError
Any of the passed ``*args`` has unsupported type
A combination of SISO and MIMO systems is
passed. There should be homogeneity in the
type of systems passed, MIMO in this case.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import MIMOSeries, TransferFunctionMatrix
>>> from sympy import Matrix, pprint
>>> mat_a = Matrix([[5*s], [5]]) # 2 Outputs 1 Input
>>> mat_b = Matrix([[5, 1/(6*s**2)]]) # 1 Output 2 Inputs
>>> mat_c = Matrix([[1, s], [5/s, 1]]) # 2 Outputs 2 Inputs
>>> tfm_a = TransferFunctionMatrix.from_Matrix(mat_a, s)
>>> tfm_b = TransferFunctionMatrix.from_Matrix(mat_b, s)
>>> tfm_c = TransferFunctionMatrix.from_Matrix(mat_c, s)
>>> MIMOSeries(tfm_c, tfm_b, tfm_a)
MIMOSeries(TransferFunctionMatrix(((TransferFunction(1, 1, s), TransferFunction(s, 1, s)), (TransferFunction(5, s, s), TransferFunction(1, 1, s)))), TransferFunctionMatrix(((TransferFunction(5, 1, s), TransferFunction(1, 6*s**2, s)),)), TransferFunctionMatrix(((TransferFunction(5*s, 1, s),), (TransferFunction(5, 1, s),))))
>>> pprint(_, use_unicode=False) # For Better Visualization
[5*s] [1 s]
[---] [5 1 ] [- -]
[ 1 ] [- ----] [1 1]
[ ] *[1 2] *[ ]
[ 5 ] [ 6*s ]{t} [5 1]
[ - ] [- -]
[ 1 ]{t} [s 1]{t}
>>> MIMOSeries(tfm_c, tfm_b, tfm_a).doit()
TransferFunctionMatrix(((TransferFunction(150*s**4 + 25*s, 6*s**3, s), TransferFunction(150*s**4 + 5*s, 6*s**2, s)), (TransferFunction(150*s**3 + 25, 6*s**3, s), TransferFunction(150*s**3 + 5, 6*s**2, s))))
>>> pprint(_, use_unicode=False) # (2 Inputs -A-> 2 Outputs) -> (2 Inputs -B-> 1 Output) -> (1 Input -C-> 2 Outputs) is equivalent to (2 Inputs -Series Equivalent-> 2 Outputs).
[ 4 4 ]
[150*s + 25*s 150*s + 5*s]
[------------- ------------]
[ 3 2 ]
[ 6*s 6*s ]
[ ]
[ 3 3 ]
[ 150*s + 25 150*s + 5 ]
[ ----------- ---------- ]
[ 3 2 ]
[ 6*s 6*s ]{t}
Notes
=====
All the transfer function matrices should use the same complex variable ``var`` of the Laplace transform.
``MIMOSeries(A, B)`` is not equivalent to ``A*B``. It is always in the reverse order, that is ``B*A``.
See Also
========
Series, MIMOParallel
"""
def __new__(cls, *args, evaluate=False):
cls._check_args(args)
if _mat_mul_compatible(*args):
obj = super().__new__(cls, *args)
else:
raise ValueError("Number of input signals do not match the number"
" of output signals of adjacent systems for some args.")
return obj.doit() if evaluate else obj
@property
def var(self):
"""
Returns the complex variable used by all the transfer functions.
Examples
========
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, MIMOSeries, TransferFunctionMatrix
>>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p)
>>> G2 = TransferFunction(p, 4 - p, p)
>>> G3 = TransferFunction(0, p**4 - 1, p)
>>> tfm_1 = TransferFunctionMatrix([[G1, G2, G3]])
>>> tfm_2 = TransferFunctionMatrix([[G1], [G2], [G3]])
>>> MIMOSeries(tfm_2, tfm_1).var
p
"""
return self.args[0].var
@property
def num_inputs(self):
"""Returns the number of input signals of the series system."""
return self.args[0].num_inputs
@property
def num_outputs(self):
"""Returns the number of output signals of the series system."""
return self.args[-1].num_outputs
@property
def shape(self):
"""Returns the shape of the equivalent MIMO system."""
return self.num_outputs, self.num_inputs
def doit(self, cancel=False, **kwargs):
"""
Returns the resultant transfer function matrix obtained after evaluating
the MIMO systems arranged in a series configuration.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, MIMOSeries, TransferFunctionMatrix
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf2]])
>>> tfm2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf1]])
>>> MIMOSeries(tfm2, tfm1).doit()
TransferFunctionMatrix(((TransferFunction(2*(-p + s)*(s**3 - 2)*(a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)**2*(s**4 + 5*s + 6)**2, s), TransferFunction((-p + s)**2*(s**3 - 2)*(a*p**2 + b*s) + (-p + s)*(a*p**2 + b*s)**2*(s**4 + 5*s + 6), (-p + s)**3*(s**4 + 5*s + 6), s)), (TransferFunction((-p + s)*(s**3 - 2)**2*(s**4 + 5*s + 6) + (s**3 - 2)*(a*p**2 + b*s)*(s**4 + 5*s + 6)**2, (-p + s)*(s**4 + 5*s + 6)**3, s), TransferFunction(2*(s**3 - 2)*(a*p**2 + b*s), (-p + s)*(s**4 + 5*s + 6), s))))
"""
_arg = (arg.doit()._expr_mat for arg in reversed(self.args))
if cancel:
res = MatMul(*_arg, evaluate=True)
return TransferFunctionMatrix.from_Matrix(res, self.var)
_dummy_args, _dummy_dict = _dummify_args(_arg, self.var)
res = MatMul(*_dummy_args, evaluate=True)
temp_tfm = TransferFunctionMatrix.from_Matrix(res, self.var)
return temp_tfm.subs(_dummy_dict)
def _eval_rewrite_as_TransferFunctionMatrix(self, *args, **kwargs):
return self.doit()
@_check_other_MIMO
def __add__(self, other):
if isinstance(other, MIMOParallel):
arg_list = list(other.args)
return MIMOParallel(self, *arg_list)
return MIMOParallel(self, other)
__radd__ = __add__
@_check_other_MIMO
def __sub__(self, other):
return self + (-other)
def __rsub__(self, other):
return -self + other
@_check_other_MIMO
def __mul__(self, other):
if isinstance(other, MIMOSeries):
self_arg_list = list(self.args)
other_arg_list = list(other.args)
return MIMOSeries(*other_arg_list, *self_arg_list) # A*B = MIMOSeries(B, A)
arg_list = list(self.args)
return MIMOSeries(other, *arg_list)
def __neg__(self):
arg_list = list(self.args)
arg_list[0] = -arg_list[0]
return MIMOSeries(*arg_list)
class Parallel(SISOLinearTimeInvariant):
r"""
A class for representing a parallel configuration of SISO systems.
Parameters
==========
args : SISOLinearTimeInvariant
SISO systems in a parallel arrangement.
evaluate : Boolean, Keyword
When passed ``True``, returns the equivalent
``Parallel(*args).doit()``. Set to ``False`` by default.
Raises
======
ValueError
When no argument is passed.
``var`` attribute is not same for every system.
TypeError
Any of the passed ``*args`` has unsupported type
A combination of SISO and MIMO systems is
passed. There should be homogeneity in the
type of systems passed.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Parallel, Series
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> tf3 = TransferFunction(p**2, p + s, s)
>>> P1 = Parallel(tf1, tf2)
>>> P1
Parallel(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s))
>>> P1.var
s
>>> P2 = Parallel(tf2, Series(tf3, -tf1))
>>> P2
Parallel(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), Series(TransferFunction(p**2, p + s, s), TransferFunction(-a*p**2 - b*s, -p + s, s)))
>>> P2.var
s
>>> P3 = Parallel(Series(tf1, tf2), Series(tf2, tf3))
>>> P3
Parallel(Series(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)), Series(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), TransferFunction(p**2, p + s, s)))
>>> P3.var
s
You can get the resultant transfer function by using ``.doit()`` method:
>>> Parallel(tf1, tf2, -tf3).doit()
TransferFunction(-p**2*(-p + s)*(s**4 + 5*s + 6) + (-p + s)*(p + s)*(s**3 - 2) + (p + s)*(a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(p + s)*(s**4 + 5*s + 6), s)
>>> Parallel(tf2, Series(tf1, -tf3)).doit()
TransferFunction(-p**2*(a*p**2 + b*s)*(s**4 + 5*s + 6) + (-p + s)*(p + s)*(s**3 - 2), (-p + s)*(p + s)*(s**4 + 5*s + 6), s)
Notes
=====
All the transfer functions should use the same complex variable
``var`` of the Laplace transform.
See Also
========
Series, TransferFunction, Feedback
"""
def __new__(cls, *args, evaluate=False):
args = _flatten_args(args, Parallel)
cls._check_args(args)
obj = super().__new__(cls, *args)
return obj.doit() if evaluate else obj
@property
def var(self):
"""
Returns the complex variable used by all the transfer functions.
Examples
========
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, Parallel, Series
>>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p)
>>> G2 = TransferFunction(p, 4 - p, p)
>>> G3 = TransferFunction(0, p**4 - 1, p)
>>> Parallel(G1, G2).var
p
>>> Parallel(-G3, Series(G1, G2)).var
p
"""
return self.args[0].var
def doit(self, **kwargs):
"""
Returns the resultant transfer function obtained after evaluating
the transfer functions in parallel configuration.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Parallel
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> Parallel(tf2, tf1).doit()
TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)
>>> Parallel(-tf1, -tf2).doit()
TransferFunction((2 - s**3)*(-p + s) + (-a*p**2 - b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)
"""
_arg = (arg.doit().to_expr() for arg in self.args)
res = Add(*_arg).as_numer_denom()
return TransferFunction(*res, self.var)
def _eval_rewrite_as_TransferFunction(self, *args, **kwargs):
return self.doit()
@_check_other_SISO
def __add__(self, other):
self_arg_list = list(self.args)
return Parallel(*self_arg_list, other)
__radd__ = __add__
@_check_other_SISO
def __sub__(self, other):
return self + (-other)
def __rsub__(self, other):
return -self + other
@_check_other_SISO
def __mul__(self, other):
if isinstance(other, Series):
arg_list = list(other.args)
return Series(self, *arg_list)
return Series(self, other)
def __neg__(self):
return Series(TransferFunction(-1, 1, self.var), self)
def to_expr(self):
"""Returns the equivalent ``Expr`` object."""
return Add(*(arg.to_expr() for arg in self.args), evaluate=False)
@property
def is_proper(self):
"""
Returns True if degree of the numerator polynomial of the resultant transfer
function is less than or equal to degree of the denominator polynomial of
the same, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Parallel
>>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s)
>>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*s + 2, s)
>>> tf3 = TransferFunction(s, s**2 + s + 1, s)
>>> P1 = Parallel(-tf2, tf1)
>>> P1.is_proper
False
>>> P2 = Parallel(tf2, tf3)
>>> P2.is_proper
True
"""
return self.doit().is_proper
@property
def is_strictly_proper(self):
"""
Returns True if degree of the numerator polynomial of the resultant transfer
function is strictly less than degree of the denominator polynomial of
the same, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Parallel
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> tf3 = TransferFunction(s, s**2 + s + 1, s)
>>> P1 = Parallel(tf1, tf2)
>>> P1.is_strictly_proper
False
>>> P2 = Parallel(tf2, tf3)
>>> P2.is_strictly_proper
True
"""
return self.doit().is_strictly_proper
@property
def is_biproper(self):
"""
Returns True if degree of the numerator polynomial of the resultant transfer
function is equal to degree of the denominator polynomial of
the same, else False.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, Parallel
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(p**2, p + s, s)
>>> tf3 = TransferFunction(s, s**2 + s + 1, s)
>>> P1 = Parallel(tf1, -tf2)
>>> P1.is_biproper
True
>>> P2 = Parallel(tf2, tf3)
>>> P2.is_biproper
False
"""
return self.doit().is_biproper
class MIMOParallel(MIMOLinearTimeInvariant):
r"""
A class for representing a parallel configuration of MIMO systems.
Parameters
==========
args : MIMOLinearTimeInvariant
MIMO Systems in a parallel arrangement.
evaluate : Boolean, Keyword
When passed ``True``, returns the equivalent
``MIMOParallel(*args).doit()``. Set to ``False`` by default.
Raises
======
ValueError
When no argument is passed.
``var`` attribute is not same for every system.
All MIMO systems passed do not have same shape.
TypeError
Any of the passed ``*args`` has unsupported type
A combination of SISO and MIMO systems is
passed. There should be homogeneity in the
type of systems passed, MIMO in this case.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunctionMatrix, MIMOParallel
>>> from sympy import Matrix, pprint
>>> expr_1 = 1/s
>>> expr_2 = s/(s**2-1)
>>> expr_3 = (2 + s)/(s**2 - 1)
>>> expr_4 = 5
>>> tfm_a = TransferFunctionMatrix.from_Matrix(Matrix([[expr_1, expr_2], [expr_3, expr_4]]), s)
>>> tfm_b = TransferFunctionMatrix.from_Matrix(Matrix([[expr_2, expr_1], [expr_4, expr_3]]), s)
>>> tfm_c = TransferFunctionMatrix.from_Matrix(Matrix([[expr_3, expr_4], [expr_1, expr_2]]), s)
>>> MIMOParallel(tfm_a, tfm_b, tfm_c)
MIMOParallel(TransferFunctionMatrix(((TransferFunction(1, s, s), TransferFunction(s, s**2 - 1, s)), (TransferFunction(s + 2, s**2 - 1, s), TransferFunction(5, 1, s)))), TransferFunctionMatrix(((TransferFunction(s, s**2 - 1, s), TransferFunction(1, s, s)), (TransferFunction(5, 1, s), TransferFunction(s + 2, s**2 - 1, s)))), TransferFunctionMatrix(((TransferFunction(s + 2, s**2 - 1, s), TransferFunction(5, 1, s)), (TransferFunction(1, s, s), TransferFunction(s, s**2 - 1, s)))))
>>> pprint(_, use_unicode=False) # For Better Visualization
[ 1 s ] [ s 1 ] [s + 2 5 ]
[ - ------] [------ - ] [------ - ]
[ s 2 ] [ 2 s ] [ 2 1 ]
[ s - 1] [s - 1 ] [s - 1 ]
[ ] + [ ] + [ ]
[s + 2 5 ] [ 5 s + 2 ] [ 1 s ]
[------ - ] [ - ------] [ - ------]
[ 2 1 ] [ 1 2 ] [ s 2 ]
[s - 1 ]{t} [ s - 1]{t} [ s - 1]{t}
>>> MIMOParallel(tfm_a, tfm_b, tfm_c).doit()
TransferFunctionMatrix(((TransferFunction(s**2 + s*(2*s + 2) - 1, s*(s**2 - 1), s), TransferFunction(2*s**2 + 5*s*(s**2 - 1) - 1, s*(s**2 - 1), s)), (TransferFunction(s**2 + s*(s + 2) + 5*s*(s**2 - 1) - 1, s*(s**2 - 1), s), TransferFunction(5*s**2 + 2*s - 3, s**2 - 1, s))))
>>> pprint(_, use_unicode=False)
[ 2 2 / 2 \ ]
[ s + s*(2*s + 2) - 1 2*s + 5*s*\s - 1/ - 1]
[ -------------------- -----------------------]
[ / 2 \ / 2 \ ]
[ s*\s - 1/ s*\s - 1/ ]
[ ]
[ 2 / 2 \ 2 ]
[s + s*(s + 2) + 5*s*\s - 1/ - 1 5*s + 2*s - 3 ]
[--------------------------------- -------------- ]
[ / 2 \ 2 ]
[ s*\s - 1/ s - 1 ]{t}
Notes
=====
All the transfer function matrices should use the same complex variable
``var`` of the Laplace transform.
See Also
========
Parallel, MIMOSeries
"""
def __new__(cls, *args, evaluate=False):
args = _flatten_args(args, MIMOParallel)
cls._check_args(args)
if any(arg.shape != args[0].shape for arg in args):
raise TypeError("Shape of all the args is not equal.")
obj = super().__new__(cls, *args)
return obj.doit() if evaluate else obj
@property
def var(self):
"""
Returns the complex variable used by all the systems.
Examples
========
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOParallel
>>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p)
>>> G2 = TransferFunction(p, 4 - p, p)
>>> G3 = TransferFunction(0, p**4 - 1, p)
>>> G4 = TransferFunction(p**2, p**2 - 1, p)
>>> tfm_a = TransferFunctionMatrix([[G1, G2], [G3, G4]])
>>> tfm_b = TransferFunctionMatrix([[G2, G1], [G4, G3]])
>>> MIMOParallel(tfm_a, tfm_b).var
p
"""
return self.args[0].var
@property
def num_inputs(self):
"""Returns the number of input signals of the parallel system."""
return self.args[0].num_inputs
@property
def num_outputs(self):
"""Returns the number of output signals of the parallel system."""
return self.args[0].num_outputs
@property
def shape(self):
"""Returns the shape of the equivalent MIMO system."""
return self.num_outputs, self.num_inputs
def doit(self, **kwargs):
"""
Returns the resultant transfer function matrix obtained after evaluating
the MIMO systems arranged in a parallel configuration.
Examples
========
>>> from sympy.abc import s, p, a, b
>>> from sympy.physics.control.lti import TransferFunction, MIMOParallel, TransferFunctionMatrix
>>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s)
>>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)
>>> tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
>>> tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
>>> MIMOParallel(tfm_1, tfm_2).doit()
TransferFunctionMatrix(((TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s), TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)), (TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s), TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s))))
"""
_arg = (arg.doit()._expr_mat for arg in self.args)
res = MatAdd(*_arg, evaluate=True)
return TransferFunctionMatrix.from_Matrix(res, self.var)
def _eval_rewrite_as_TransferFunctionMatrix(self, *args, **kwargs):
return self.doit()
@_check_other_MIMO
def __add__(self, other):
self_arg_list = list(self.args)
return MIMOParallel(*self_arg_list, other)
__radd__ = __add__
@_check_other_MIMO
def __sub__(self, other):
return self + (-other)
def __rsub__(self, other):
return -self + other
@_check_other_MIMO
def __mul__(self, other):
if isinstance(other, MIMOSeries):
arg_list = list(other.args)
return MIMOSeries(*arg_list, self)
return MIMOSeries(other, self)
def __neg__(self):
arg_list = [-arg for arg in list(self.args)]
return MIMOParallel(*arg_list)
class Feedback(SISOLinearTimeInvariant):
r"""
A class for representing closed-loop feedback interconnection between two
SISO input/output systems.
The first argument, ``sys1``, is the feedforward part of the closed-loop
system or in simple words, the dynamical model representing the process
to be controlled. The second argument, ``sys2``, is the feedback system
and controls the fed back signal to ``sys1``. Both ``sys1`` and ``sys2``
can either be ``Series`` or ``TransferFunction`` objects.
Parameters
==========
sys1 : Series, TransferFunction
The feedforward path system.
sys2 : Series, TransferFunction, optional
The feedback path system (often a feedback controller).
It is the model sitting on the feedback path.
If not specified explicitly, the sys2 is
assumed to be unit (1.0) transfer function.
sign : int, optional
The sign of feedback. Can either be ``1``
(for positive feedback) or ``-1`` (for negative feedback).
Default value is `-1`.
Raises
======
ValueError
When ``sys1`` and ``sys2`` are not using the
same complex variable of the Laplace transform.
When a combination of ``sys1`` and ``sys2`` yields
zero denominator.
TypeError
When either ``sys1`` or ``sys2`` is not a ``Series`` or a
``TransferFunction`` object.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, Feedback
>>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> controller = TransferFunction(5*s - 10, s + 7, s)
>>> F1 = Feedback(plant, controller)
>>> F1
Feedback(TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s), TransferFunction(5*s - 10, s + 7, s), -1)
>>> F1.var
s
>>> F1.args
(TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s), TransferFunction(5*s - 10, s + 7, s), -1)
You can get the feedforward and feedback path systems by using ``.sys1`` and ``.sys2`` respectively.
>>> F1.sys1
TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> F1.sys2
TransferFunction(5*s - 10, s + 7, s)
You can get the resultant closed loop transfer function obtained by negative feedback
interconnection using ``.doit()`` method.
>>> F1.doit()
TransferFunction((s + 7)*(s**2 - 4*s + 2)*(3*s**2 + 7*s - 3), ((s + 7)*(s**2 - 4*s + 2) + (5*s - 10)*(3*s**2 + 7*s - 3))*(s**2 - 4*s + 2), s)
>>> G = TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s)
>>> C = TransferFunction(5*s + 10, s + 10, s)
>>> F2 = Feedback(G*C, TransferFunction(1, 1, s))
>>> F2.doit()
TransferFunction((s + 10)*(5*s + 10)*(s**2 + 2*s + 3)*(2*s**2 + 5*s + 1), (s + 10)*((s + 10)*(s**2 + 2*s + 3) + (5*s + 10)*(2*s**2 + 5*s + 1))*(s**2 + 2*s + 3), s)
To negate a ``Feedback`` object, the ``-`` operator can be prepended:
>>> -F1
Feedback(TransferFunction(-3*s**2 - 7*s + 3, s**2 - 4*s + 2, s), TransferFunction(10 - 5*s, s + 7, s), -1)
>>> -F2
Feedback(Series(TransferFunction(-1, 1, s), TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s), TransferFunction(5*s + 10, s + 10, s)), TransferFunction(-1, 1, s), -1)
See Also
========
MIMOFeedback, Series, Parallel
"""
def __new__(cls, sys1, sys2=None, sign=-1):
if not sys2:
sys2 = TransferFunction(1, 1, sys1.var)
if not (isinstance(sys1, (TransferFunction, Series))
and isinstance(sys2, (TransferFunction, Series))):
raise TypeError("Unsupported type for `sys1` or `sys2` of Feedback.")
if sign not in [-1, 1]:
raise ValueError("Unsupported type for feedback. `sign` arg should "
"either be 1 (positive feedback loop) or -1 (negative feedback loop).")
if Mul(sys1.to_expr(), sys2.to_expr()).simplify() == sign:
raise ValueError("The equivalent system will have zero denominator.")
if sys1.var != sys2.var:
raise ValueError("Both `sys1` and `sys2` should be using the"
" same complex variable.")
return super().__new__(cls, sys1, sys2, _sympify(sign))
@property
def sys1(self):
"""
Returns the feedforward system of the feedback interconnection.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction, Feedback
>>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> controller = TransferFunction(5*s - 10, s + 7, s)
>>> F1 = Feedback(plant, controller)
>>> F1.sys1
TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p)
>>> C = TransferFunction(5*p + 10, p + 10, p)
>>> P = TransferFunction(1 - s, p + 2, p)
>>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P)
>>> F2.sys1
TransferFunction(1, 1, p)
"""
return self.args[0]
@property
def sys2(self):
"""
Returns the feedback controller of the feedback interconnection.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction, Feedback
>>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> controller = TransferFunction(5*s - 10, s + 7, s)
>>> F1 = Feedback(plant, controller)
>>> F1.sys2
TransferFunction(5*s - 10, s + 7, s)
>>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p)
>>> C = TransferFunction(5*p + 10, p + 10, p)
>>> P = TransferFunction(1 - s, p + 2, p)
>>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P)
>>> F2.sys2
Series(TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p), TransferFunction(5*p + 10, p + 10, p), TransferFunction(1 - s, p + 2, p))
"""
return self.args[1]
@property
def var(self):
"""
Returns the complex variable of the Laplace transform used by all
the transfer functions involved in the feedback interconnection.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction, Feedback
>>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> controller = TransferFunction(5*s - 10, s + 7, s)
>>> F1 = Feedback(plant, controller)
>>> F1.var
s
>>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p)
>>> C = TransferFunction(5*p + 10, p + 10, p)
>>> P = TransferFunction(1 - s, p + 2, p)
>>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P)
>>> F2.var
p
"""
return self.sys1.var
@property
def sign(self):
"""
Returns the type of MIMO Feedback model. ``1``
for Positive and ``-1`` for Negative.
"""
return self.args[2]
@property
def sensitivity(self):
"""
Returns the sensitivity function of the feedback loop.
Sensitivity of a Feedback system is the ratio
of change in the open loop gain to the change in
the closed loop gain.
.. note::
This method would not return the complementary
sensitivity function.
Examples
========
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, Feedback
>>> C = TransferFunction(5*p + 10, p + 10, p)
>>> P = TransferFunction(1 - p, p + 2, p)
>>> F_1 = Feedback(P, C)
>>> F_1.sensitivity
1/((1 - p)*(5*p + 10)/((p + 2)*(p + 10)) + 1)
"""
return 1/(1 - self.sign*self.sys1.to_expr()*self.sys2.to_expr())
def doit(self, cancel=False, expand=False, **kwargs):
"""
Returns the resultant transfer function obtained by the
feedback interconnection.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, Feedback
>>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s)
>>> controller = TransferFunction(5*s - 10, s + 7, s)
>>> F1 = Feedback(plant, controller)
>>> F1.doit()
TransferFunction((s + 7)*(s**2 - 4*s + 2)*(3*s**2 + 7*s - 3), ((s + 7)*(s**2 - 4*s + 2) + (5*s - 10)*(3*s**2 + 7*s - 3))*(s**2 - 4*s + 2), s)
>>> G = TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s)
>>> F2 = Feedback(G, TransferFunction(1, 1, s))
>>> F2.doit()
TransferFunction((s**2 + 2*s + 3)*(2*s**2 + 5*s + 1), (s**2 + 2*s + 3)*(3*s**2 + 7*s + 4), s)
Use kwarg ``expand=True`` to expand the resultant transfer function.
Use ``cancel=True`` to cancel out the common terms in numerator and
denominator.
>>> F2.doit(cancel=True, expand=True)
TransferFunction(2*s**2 + 5*s + 1, 3*s**2 + 7*s + 4, s)
>>> F2.doit(expand=True)
TransferFunction(2*s**4 + 9*s**3 + 17*s**2 + 17*s + 3, 3*s**4 + 13*s**3 + 27*s**2 + 29*s + 12, s)
"""
arg_list = list(self.sys1.args) if isinstance(self.sys1, Series) else [self.sys1]
# F_n and F_d are resultant TFs of num and den of Feedback.
F_n, unit = self.sys1.doit(), TransferFunction(1, 1, self.sys1.var)
if self.sign == -1:
F_d = Parallel(unit, Series(self.sys2, *arg_list)).doit()
else:
F_d = Parallel(unit, -Series(self.sys2, *arg_list)).doit()
_resultant_tf = TransferFunction(F_n.num * F_d.den, F_n.den * F_d.num, F_n.var)
if cancel:
_resultant_tf = _resultant_tf.simplify()
if expand:
_resultant_tf = _resultant_tf.expand()
return _resultant_tf
def _eval_rewrite_as_TransferFunction(self, num, den, sign, **kwargs):
return self.doit()
def __neg__(self):
return Feedback(-self.sys1, -self.sys2, self.sign)
def _is_invertible(a, b, sign):
"""
Checks whether a given pair of MIMO
systems passed is invertible or not.
"""
_mat = eye(a.num_outputs) - sign*(a.doit()._expr_mat)*(b.doit()._expr_mat)
_det = _mat.det()
return _det != 0
class MIMOFeedback(MIMOLinearTimeInvariant):
r"""
A class for representing closed-loop feedback interconnection between two
MIMO input/output systems.
Parameters
==========
sys1 : MIMOSeries, TransferFunctionMatrix
The MIMO system placed on the feedforward path.
sys2 : MIMOSeries, TransferFunctionMatrix
The system placed on the feedback path
(often a feedback controller).
sign : int, optional
The sign of feedback. Can either be ``1``
(for positive feedback) or ``-1`` (for negative feedback).
Default value is `-1`.
Raises
======
ValueError
When ``sys1`` and ``sys2`` are not using the
same complex variable of the Laplace transform.
Forward path model should have an equal number of inputs/outputs
to the feedback path outputs/inputs.
When product of ``sys1`` and ``sys2`` is not a square matrix.
When the equivalent MIMO system is not invertible.
TypeError
When either ``sys1`` or ``sys2`` is not a ``MIMOSeries`` or a
``TransferFunctionMatrix`` object.
Examples
========
>>> from sympy import Matrix, pprint
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunctionMatrix, MIMOFeedback
>>> plant_mat = Matrix([[1, 1/s], [0, 1]])
>>> controller_mat = Matrix([[10, 0], [0, 10]]) # Constant Gain
>>> plant = TransferFunctionMatrix.from_Matrix(plant_mat, s)
>>> controller = TransferFunctionMatrix.from_Matrix(controller_mat, s)
>>> feedback = MIMOFeedback(plant, controller) # Negative Feedback (default)
>>> pprint(feedback, use_unicode=False)
/ [1 1] [10 0 ] \-1 [1 1]
| [- -] [-- - ] | [- -]
| [1 s] [1 1 ] | [1 s]
|I + [ ] *[ ] | * [ ]
| [0 1] [0 10] | [0 1]
| [- -] [- --] | [- -]
\ [1 1]{t} [1 1 ]{t}/ [1 1]{t}
To get the equivalent system matrix, use either ``doit`` or ``rewrite`` method.
>>> pprint(feedback.doit(), use_unicode=False)
[1 1 ]
[-- -----]
[11 121*s]
[ ]
[0 1 ]
[- -- ]
[1 11 ]{t}
To negate the ``MIMOFeedback`` object, use ``-`` operator.
>>> neg_feedback = -feedback
>>> pprint(neg_feedback.doit(), use_unicode=False)
[-1 -1 ]
[--- -----]
[ 11 121*s]
[ ]
[ 0 -1 ]
[ - --- ]
[ 1 11 ]{t}
See Also
========
Feedback, MIMOSeries, MIMOParallel
"""
def __new__(cls, sys1, sys2, sign=-1):
if not (isinstance(sys1, (TransferFunctionMatrix, MIMOSeries))
and isinstance(sys2, (TransferFunctionMatrix, MIMOSeries))):
raise TypeError("Unsupported type for `sys1` or `sys2` of MIMO Feedback.")
if sys1.num_inputs != sys2.num_outputs or \
sys1.num_outputs != sys2.num_inputs:
raise ValueError("Product of `sys1` and `sys2` "
"must yield a square matrix.")
if sign not in [-1, 1]:
raise ValueError("Unsupported type for feedback. `sign` arg should "
"either be 1 (positive feedback loop) or -1 (negative feedback loop).")
if not _is_invertible(sys1, sys2, sign):
raise ValueError("Non-Invertible system inputted.")
if sys1.var != sys2.var:
raise ValueError("Both `sys1` and `sys2` should be using the"
" same complex variable.")
return super().__new__(cls, sys1, sys2, _sympify(sign))
@property
def sys1(self):
r"""
Returns the system placed on the feedforward path of the MIMO feedback interconnection.
Examples
========
>>> from sympy import pprint
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback
>>> tf1 = TransferFunction(s**2 + s + 1, s**2 - s + 1, s)
>>> tf2 = TransferFunction(1, s, s)
>>> tf3 = TransferFunction(1, 1, s)
>>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
>>> sys2 = TransferFunctionMatrix([[tf3, tf3], [tf3, tf2]])
>>> F_1 = MIMOFeedback(sys1, sys2, 1)
>>> F_1.sys1
TransferFunctionMatrix(((TransferFunction(s**2 + s + 1, s**2 - s + 1, s), TransferFunction(1, s, s)), (TransferFunction(1, s, s), TransferFunction(s**2 + s + 1, s**2 - s + 1, s))))
>>> pprint(_, use_unicode=False)
[ 2 ]
[s + s + 1 1 ]
[---------- - ]
[ 2 s ]
[s - s + 1 ]
[ ]
[ 2 ]
[ 1 s + s + 1]
[ - ----------]
[ s 2 ]
[ s - s + 1]{t}
"""
return self.args[0]
@property
def sys2(self):
r"""
Returns the feedback controller of the MIMO feedback interconnection.
Examples
========
>>> from sympy import pprint
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback
>>> tf1 = TransferFunction(s**2, s**3 - s + 1, s)
>>> tf2 = TransferFunction(1, s, s)
>>> tf3 = TransferFunction(1, 1, s)
>>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
>>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]])
>>> F_1 = MIMOFeedback(sys1, sys2)
>>> F_1.sys2
TransferFunctionMatrix(((TransferFunction(s**2, s**3 - s + 1, s), TransferFunction(1, 1, s)), (TransferFunction(1, 1, s), TransferFunction(1, s, s))))
>>> pprint(_, use_unicode=False)
[ 2 ]
[ s 1]
[---------- -]
[ 3 1]
[s - s + 1 ]
[ ]
[ 1 1]
[ - -]
[ 1 s]{t}
"""
return self.args[1]
@property
def var(self):
r"""
Returns the complex variable of the Laplace transform used by all
the transfer functions involved in the MIMO feedback loop.
Examples
========
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback
>>> tf1 = TransferFunction(p, 1 - p, p)
>>> tf2 = TransferFunction(1, p, p)
>>> tf3 = TransferFunction(1, 1, p)
>>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
>>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]])
>>> F_1 = MIMOFeedback(sys1, sys2, 1) # Positive feedback
>>> F_1.var
p
"""
return self.sys1.var
@property
def sign(self):
r"""
Returns the type of feedback interconnection of two models. ``1``
for Positive and ``-1`` for Negative.
"""
return self.args[2]
@property
def sensitivity(self):
r"""
Returns the sensitivity function matrix of the feedback loop.
Sensitivity of a closed-loop system is the ratio of change
in the open loop gain to the change in the closed loop gain.
.. note::
This method would not return the complementary
sensitivity function.
Examples
========
>>> from sympy import pprint
>>> from sympy.abc import p
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback
>>> tf1 = TransferFunction(p, 1 - p, p)
>>> tf2 = TransferFunction(1, p, p)
>>> tf3 = TransferFunction(1, 1, p)
>>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
>>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]])
>>> F_1 = MIMOFeedback(sys1, sys2, 1) # Positive feedback
>>> F_2 = MIMOFeedback(sys1, sys2) # Negative feedback
>>> pprint(F_1.sensitivity, use_unicode=False)
[ 4 3 2 5 4 2 ]
[- p + 3*p - 4*p + 3*p - 1 p - 2*p + 3*p - 3*p + 1 ]
[---------------------------- -----------------------------]
[ 4 3 2 5 4 3 2 ]
[ p + 3*p - 8*p + 8*p - 3 p + 3*p - 8*p + 8*p - 3*p]
[ ]
[ 4 3 2 3 2 ]
[ p - p - p + p 3*p - 6*p + 4*p - 1 ]
[ -------------------------- -------------------------- ]
[ 4 3 2 4 3 2 ]
[ p + 3*p - 8*p + 8*p - 3 p + 3*p - 8*p + 8*p - 3 ]
>>> pprint(F_2.sensitivity, use_unicode=False)
[ 4 3 2 5 4 2 ]
[p - 3*p + 2*p + p - 1 p - 2*p + 3*p - 3*p + 1]
[------------------------ --------------------------]
[ 4 3 5 4 2 ]
[ p - 3*p + 2*p - 1 p - 3*p + 2*p - p ]
[ ]
[ 4 3 2 4 3 ]
[ p - p - p + p 2*p - 3*p + 2*p - 1 ]
[ ------------------- --------------------- ]
[ 4 3 4 3 ]
[ p - 3*p + 2*p - 1 p - 3*p + 2*p - 1 ]
"""
_sys1_mat = self.sys1.doit()._expr_mat
_sys2_mat = self.sys2.doit()._expr_mat
return (eye(self.sys1.num_inputs) - \
self.sign*_sys1_mat*_sys2_mat).inv()
def doit(self, cancel=True, expand=False, **kwargs):
r"""
Returns the resultant transfer function matrix obtained by the
feedback interconnection.
Examples
========
>>> from sympy import pprint
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback
>>> tf1 = TransferFunction(s, 1 - s, s)
>>> tf2 = TransferFunction(1, s, s)
>>> tf3 = TransferFunction(5, 1, s)
>>> tf4 = TransferFunction(s - 1, s, s)
>>> tf5 = TransferFunction(0, 1, s)
>>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
>>> sys2 = TransferFunctionMatrix([[tf3, tf5], [tf5, tf5]])
>>> F_1 = MIMOFeedback(sys1, sys2, 1)
>>> pprint(F_1, use_unicode=False)
/ [ s 1 ] [5 0] \-1 [ s 1 ]
| [----- - ] [- -] | [----- - ]
| [1 - s s ] [1 1] | [1 - s s ]
|I - [ ] *[ ] | * [ ]
| [ 5 s - 1] [0 0] | [ 5 s - 1]
| [ - -----] [- -] | [ - -----]
\ [ 1 s ]{t} [1 1]{t}/ [ 1 s ]{t}
>>> pprint(F_1.doit(), use_unicode=False)
[ -s s - 1 ]
[------- ----------- ]
[6*s - 1 s*(6*s - 1) ]
[ ]
[5*s - 5 (s - 1)*(6*s + 24)]
[------- ------------------]
[6*s - 1 s*(6*s - 1) ]{t}
If the user wants the resultant ``TransferFunctionMatrix`` object without
canceling the common factors then the ``cancel`` kwarg should be passed ``False``.
>>> pprint(F_1.doit(cancel=False), use_unicode=False)
[ 25*s*(1 - s) 25 - 25*s ]
[ -------------------- -------------- ]
[ 25*(1 - 6*s)*(1 - s) 25*s*(1 - 6*s) ]
[ ]
[s*(25*s - 25) + 5*(1 - s)*(6*s - 1) s*(s - 1)*(6*s - 1) + s*(25*s - 25)]
[----------------------------------- -----------------------------------]
[ (1 - s)*(6*s - 1) 2 ]
[ s *(6*s - 1) ]{t}
If the user wants the expanded form of the resultant transfer function matrix,
the ``expand`` kwarg should be passed as ``True``.
>>> pprint(F_1.doit(expand=True), use_unicode=False)
[ -s s - 1 ]
[------- -------- ]
[6*s - 1 2 ]
[ 6*s - s ]
[ ]
[ 2 ]
[5*s - 5 6*s + 18*s - 24]
[------- ----------------]
[6*s - 1 2 ]
[ 6*s - s ]{t}
"""
_mat = self.sensitivity * self.sys1.doit()._expr_mat
_resultant_tfm = _to_TFM(_mat, self.var)
if cancel:
_resultant_tfm = _resultant_tfm.simplify()
if expand:
_resultant_tfm = _resultant_tfm.expand()
return _resultant_tfm
def _eval_rewrite_as_TransferFunctionMatrix(self, sys1, sys2, sign, **kwargs):
return self.doit()
def __neg__(self):
return MIMOFeedback(-self.sys1, -self.sys2, self.sign)
def _to_TFM(mat, var):
"""Private method to convert ImmutableMatrix to TransferFunctionMatrix efficiently"""
to_tf = lambda expr: TransferFunction.from_rational_expression(expr, var)
arg = [[to_tf(expr) for expr in row] for row in mat.tolist()]
return TransferFunctionMatrix(arg)
class TransferFunctionMatrix(MIMOLinearTimeInvariant):
r"""
A class for representing the MIMO (multiple-input and multiple-output)
generalization of the SISO (single-input and single-output) transfer function.
It is a matrix of transfer functions (``TransferFunction``, SISO-``Series`` or SISO-``Parallel``).
There is only one argument, ``arg`` which is also the compulsory argument.
``arg`` is expected to be strictly of the type list of lists
which holds the transfer functions or reducible to transfer functions.
Parameters
==========
arg : Nested ``List`` (strictly).
Users are expected to input a nested list of ``TransferFunction``, ``Series``
and/or ``Parallel`` objects.
Examples
========
.. note::
``pprint()`` can be used for better visualization of ``TransferFunctionMatrix`` objects.
>>> from sympy.abc import s, p, a
>>> from sympy import pprint
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, Series, Parallel
>>> tf_1 = TransferFunction(s + a, s**2 + s + 1, s)
>>> tf_2 = TransferFunction(p**4 - 3*p + 2, s + p, s)
>>> tf_3 = TransferFunction(3, s + 2, s)
>>> tf_4 = TransferFunction(-a + p, 9*s - 9, s)
>>> tfm_1 = TransferFunctionMatrix([[tf_1], [tf_2], [tf_3]])
>>> tfm_1
TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(3, s + 2, s),)))
>>> tfm_1.var
s
>>> tfm_1.num_inputs
1
>>> tfm_1.num_outputs
3
>>> tfm_1.shape
(3, 1)
>>> tfm_1.args
(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(3, s + 2, s),)),)
>>> tfm_2 = TransferFunctionMatrix([[tf_1, -tf_3], [tf_2, -tf_1], [tf_3, -tf_2]])
>>> tfm_2
TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(p**4 - 3*p + 2, p + s, s), TransferFunction(-a - s, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-p**4 + 3*p - 2, p + s, s))))
>>> pprint(tfm_2, use_unicode=False) # pretty-printing for better visualization
[ a + s -3 ]
[ ---------- ----- ]
[ 2 s + 2 ]
[ s + s + 1 ]
[ ]
[ 4 ]
[p - 3*p + 2 -a - s ]
[------------ ---------- ]
[ p + s 2 ]
[ s + s + 1 ]
[ ]
[ 4 ]
[ 3 - p + 3*p - 2]
[ ----- --------------]
[ s + 2 p + s ]{t}
TransferFunctionMatrix can be transposed, if user wants to switch the input and output transfer functions
>>> tfm_2.transpose()
TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(p**4 - 3*p + 2, p + s, s), TransferFunction(3, s + 2, s)), (TransferFunction(-3, s + 2, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(-p**4 + 3*p - 2, p + s, s))))
>>> pprint(_, use_unicode=False)
[ 4 ]
[ a + s p - 3*p + 2 3 ]
[---------- ------------ ----- ]
[ 2 p + s s + 2 ]
[s + s + 1 ]
[ ]
[ 4 ]
[ -3 -a - s - p + 3*p - 2]
[ ----- ---------- --------------]
[ s + 2 2 p + s ]
[ s + s + 1 ]{t}
>>> tf_5 = TransferFunction(5, s, s)
>>> tf_6 = TransferFunction(5*s, (2 + s**2), s)
>>> tf_7 = TransferFunction(5, (s*(2 + s**2)), s)
>>> tf_8 = TransferFunction(5, 1, s)
>>> tfm_3 = TransferFunctionMatrix([[tf_5, tf_6], [tf_7, tf_8]])
>>> tfm_3
TransferFunctionMatrix(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)), (TransferFunction(5, s*(s**2 + 2), s), TransferFunction(5, 1, s))))
>>> pprint(tfm_3, use_unicode=False)
[ 5 5*s ]
[ - ------]
[ s 2 ]
[ s + 2]
[ ]
[ 5 5 ]
[---------- - ]
[ / 2 \ 1 ]
[s*\s + 2/ ]{t}
>>> tfm_3.var
s
>>> tfm_3.shape
(2, 2)
>>> tfm_3.num_outputs
2
>>> tfm_3.num_inputs
2
>>> tfm_3.args
(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)), (TransferFunction(5, s*(s**2 + 2), s), TransferFunction(5, 1, s))),)
To access the ``TransferFunction`` at any index in the ``TransferFunctionMatrix``, use the index notation.
>>> tfm_3[1, 0] # gives the TransferFunction present at 2nd Row and 1st Col. Similar to that in Matrix classes
TransferFunction(5, s*(s**2 + 2), s)
>>> tfm_3[0, 0] # gives the TransferFunction present at 1st Row and 1st Col.
TransferFunction(5, s, s)
>>> tfm_3[:, 0] # gives the first column
TransferFunctionMatrix(((TransferFunction(5, s, s),), (TransferFunction(5, s*(s**2 + 2), s),)))
>>> pprint(_, use_unicode=False)
[ 5 ]
[ - ]
[ s ]
[ ]
[ 5 ]
[----------]
[ / 2 \]
[s*\s + 2/]{t}
>>> tfm_3[0, :] # gives the first row
TransferFunctionMatrix(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)),))
>>> pprint(_, use_unicode=False)
[5 5*s ]
[- ------]
[s 2 ]
[ s + 2]{t}
To negate a transfer function matrix, ``-`` operator can be prepended:
>>> tfm_4 = TransferFunctionMatrix([[tf_2], [-tf_1], [tf_3]])
>>> -tfm_4
TransferFunctionMatrix(((TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(-3, s + 2, s),)))
>>> tfm_5 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, -tf_1]])
>>> -tfm_5
TransferFunctionMatrix(((TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)), (TransferFunction(-3, s + 2, s), TransferFunction(a + s, s**2 + s + 1, s))))
``subs()`` returns the ``TransferFunctionMatrix`` object with the value substituted in the expression. This will not
mutate your original ``TransferFunctionMatrix``.
>>> tfm_2.subs(p, 2) # substituting p everywhere in tfm_2 with 2.
TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(12, s + 2, s), TransferFunction(-a - s, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-12, s + 2, s))))
>>> pprint(_, use_unicode=False)
[ a + s -3 ]
[---------- ----- ]
[ 2 s + 2 ]
[s + s + 1 ]
[ ]
[ 12 -a - s ]
[ ----- ----------]
[ s + 2 2 ]
[ s + s + 1]
[ ]
[ 3 -12 ]
[ ----- ----- ]
[ s + 2 s + 2 ]{t}
>>> pprint(tfm_2, use_unicode=False) # State of tfm_2 is unchanged after substitution
[ a + s -3 ]
[ ---------- ----- ]
[ 2 s + 2 ]
[ s + s + 1 ]
[ ]
[ 4 ]
[p - 3*p + 2 -a - s ]
[------------ ---------- ]
[ p + s 2 ]
[ s + s + 1 ]
[ ]
[ 4 ]
[ 3 - p + 3*p - 2]
[ ----- --------------]
[ s + 2 p + s ]{t}
``subs()`` also supports multiple substitutions.
>>> tfm_2.subs({p: 2, a: 1}) # substituting p with 2 and a with 1
TransferFunctionMatrix(((TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(12, s + 2, s), TransferFunction(-s - 1, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-12, s + 2, s))))
>>> pprint(_, use_unicode=False)
[ s + 1 -3 ]
[---------- ----- ]
[ 2 s + 2 ]
[s + s + 1 ]
[ ]
[ 12 -s - 1 ]
[ ----- ----------]
[ s + 2 2 ]
[ s + s + 1]
[ ]
[ 3 -12 ]
[ ----- ----- ]
[ s + 2 s + 2 ]{t}
Users can reduce the ``Series`` and ``Parallel`` elements of the matrix to ``TransferFunction`` by using
``doit()``.
>>> tfm_6 = TransferFunctionMatrix([[Series(tf_3, tf_4), Parallel(tf_3, tf_4)]])
>>> tfm_6
TransferFunctionMatrix(((Series(TransferFunction(3, s + 2, s), TransferFunction(-a + p, 9*s - 9, s)), Parallel(TransferFunction(3, s + 2, s), TransferFunction(-a + p, 9*s - 9, s))),))
>>> pprint(tfm_6, use_unicode=False)
[ -a + p 3 -a + p 3 ]
[-------*----- ------- + -----]
[9*s - 9 s + 2 9*s - 9 s + 2]{t}
>>> tfm_6.doit()
TransferFunctionMatrix(((TransferFunction(-3*a + 3*p, (s + 2)*(9*s - 9), s), TransferFunction(27*s + (-a + p)*(s + 2) - 27, (s + 2)*(9*s - 9), s)),))
>>> pprint(_, use_unicode=False)
[ -3*a + 3*p 27*s + (-a + p)*(s + 2) - 27]
[----------------- ----------------------------]
[(s + 2)*(9*s - 9) (s + 2)*(9*s - 9) ]{t}
>>> tf_9 = TransferFunction(1, s, s)
>>> tf_10 = TransferFunction(1, s**2, s)
>>> tfm_7 = TransferFunctionMatrix([[Series(tf_9, tf_10), tf_9], [tf_10, Parallel(tf_9, tf_10)]])
>>> tfm_7
TransferFunctionMatrix(((Series(TransferFunction(1, s, s), TransferFunction(1, s**2, s)), TransferFunction(1, s, s)), (TransferFunction(1, s**2, s), Parallel(TransferFunction(1, s, s), TransferFunction(1, s**2, s)))))
>>> pprint(tfm_7, use_unicode=False)
[ 1 1 ]
[---- - ]
[ 2 s ]
[s*s ]
[ ]
[ 1 1 1]
[ -- -- + -]
[ 2 2 s]
[ s s ]{t}
>>> tfm_7.doit()
TransferFunctionMatrix(((TransferFunction(1, s**3, s), TransferFunction(1, s, s)), (TransferFunction(1, s**2, s), TransferFunction(s**2 + s, s**3, s))))
>>> pprint(_, use_unicode=False)
[1 1 ]
[-- - ]
[ 3 s ]
[s ]
[ ]
[ 2 ]
[1 s + s]
[-- ------]
[ 2 3 ]
[s s ]{t}
Addition, subtraction, and multiplication of transfer function matrices can form
unevaluated ``Series`` or ``Parallel`` objects.
- For addition and subtraction:
All the transfer function matrices must have the same shape.
- For multiplication (C = A * B):
The number of inputs of the first transfer function matrix (A) must be equal to the
number of outputs of the second transfer function matrix (B).
Also, use pretty-printing (``pprint``) to analyse better.
>>> tfm_8 = TransferFunctionMatrix([[tf_3], [tf_2], [-tf_1]])
>>> tfm_9 = TransferFunctionMatrix([[-tf_3]])
>>> tfm_10 = TransferFunctionMatrix([[tf_1], [tf_2], [tf_4]])
>>> tfm_11 = TransferFunctionMatrix([[tf_4], [-tf_1]])
>>> tfm_12 = TransferFunctionMatrix([[tf_4, -tf_1, tf_3], [-tf_2, -tf_4, -tf_3]])
>>> tfm_8 + tfm_10
MIMOParallel(TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a + p, 9*s - 9, s),))))
>>> pprint(_, use_unicode=False)
[ 3 ] [ a + s ]
[ ----- ] [ ---------- ]
[ s + 2 ] [ 2 ]
[ ] [ s + s + 1 ]
[ 4 ] [ ]
[p - 3*p + 2] [ 4 ]
[------------] + [p - 3*p + 2]
[ p + s ] [------------]
[ ] [ p + s ]
[ -a - s ] [ ]
[ ---------- ] [ -a + p ]
[ 2 ] [ ------- ]
[ s + s + 1 ]{t} [ 9*s - 9 ]{t}
>>> -tfm_10 - tfm_8
MIMOParallel(TransferFunctionMatrix(((TransferFunction(-a - s, s**2 + s + 1, s),), (TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a - p, 9*s - 9, s),))), TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),), (TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a + s, s**2 + s + 1, s),))))
>>> pprint(_, use_unicode=False)
[ -a - s ] [ -3 ]
[ ---------- ] [ ----- ]
[ 2 ] [ s + 2 ]
[ s + s + 1 ] [ ]
[ ] [ 4 ]
[ 4 ] [- p + 3*p - 2]
[- p + 3*p - 2] + [--------------]
[--------------] [ p + s ]
[ p + s ] [ ]
[ ] [ a + s ]
[ a - p ] [ ---------- ]
[ ------- ] [ 2 ]
[ 9*s - 9 ]{t} [ s + s + 1 ]{t}
>>> tfm_12 * tfm_8
MIMOSeries(TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(-a + p, 9*s - 9, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(3, s + 2, s)), (TransferFunction(-p**4 + 3*p - 2, p + s, s), TransferFunction(a - p, 9*s - 9, s), TransferFunction(-3, s + 2, s)))))
>>> pprint(_, use_unicode=False)
[ 3 ]
[ ----- ]
[ -a + p -a - s 3 ] [ s + 2 ]
[ ------- ---------- -----] [ ]
[ 9*s - 9 2 s + 2] [ 4 ]
[ s + s + 1 ] [p - 3*p + 2]
[ ] *[------------]
[ 4 ] [ p + s ]
[- p + 3*p - 2 a - p -3 ] [ ]
[-------------- ------- -----] [ -a - s ]
[ p + s 9*s - 9 s + 2]{t} [ ---------- ]
[ 2 ]
[ s + s + 1 ]{t}
>>> tfm_12 * tfm_8 * tfm_9
MIMOSeries(TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),),)), TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(-a + p, 9*s - 9, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(3, s + 2, s)), (TransferFunction(-p**4 + 3*p - 2, p + s, s), TransferFunction(a - p, 9*s - 9, s), TransferFunction(-3, s + 2, s)))))
>>> pprint(_, use_unicode=False)
[ 3 ]
[ ----- ]
[ -a + p -a - s 3 ] [ s + 2 ]
[ ------- ---------- -----] [ ]
[ 9*s - 9 2 s + 2] [ 4 ]
[ s + s + 1 ] [p - 3*p + 2] [ -3 ]
[ ] *[------------] *[-----]
[ 4 ] [ p + s ] [s + 2]{t}
[- p + 3*p - 2 a - p -3 ] [ ]
[-------------- ------- -----] [ -a - s ]
[ p + s 9*s - 9 s + 2]{t} [ ---------- ]
[ 2 ]
[ s + s + 1 ]{t}
>>> tfm_10 + tfm_8*tfm_9
MIMOParallel(TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a + p, 9*s - 9, s),))), MIMOSeries(TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),),)), TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),)))))
>>> pprint(_, use_unicode=False)
[ a + s ] [ 3 ]
[ ---------- ] [ ----- ]
[ 2 ] [ s + 2 ]
[ s + s + 1 ] [ ]
[ ] [ 4 ]
[ 4 ] [p - 3*p + 2] [ -3 ]
[p - 3*p + 2] + [------------] *[-----]
[------------] [ p + s ] [s + 2]{t}
[ p + s ] [ ]
[ ] [ -a - s ]
[ -a + p ] [ ---------- ]
[ ------- ] [ 2 ]
[ 9*s - 9 ]{t} [ s + s + 1 ]{t}
These unevaluated ``Series`` or ``Parallel`` objects can convert into the
resultant transfer function matrix using ``.doit()`` method or by
``.rewrite(TransferFunctionMatrix)``.
>>> (-tfm_8 + tfm_10 + tfm_8*tfm_9).doit()
TransferFunctionMatrix(((TransferFunction((a + s)*(s + 2)**3 - 3*(s + 2)**2*(s**2 + s + 1) - 9*(s + 2)*(s**2 + s + 1), (s + 2)**3*(s**2 + s + 1), s),), (TransferFunction((p + s)*(-3*p**4 + 9*p - 6), (p + s)**2*(s + 2), s),), (TransferFunction((-a + p)*(s + 2)*(s**2 + s + 1)**2 + (a + s)*(s + 2)*(9*s - 9)*(s**2 + s + 1) + (3*a + 3*s)*(9*s - 9)*(s**2 + s + 1), (s + 2)*(9*s - 9)*(s**2 + s + 1)**2, s),)))
>>> (-tfm_12 * -tfm_8 * -tfm_9).rewrite(TransferFunctionMatrix)
TransferFunctionMatrix(((TransferFunction(3*(-3*a + 3*p)*(p + s)*(s + 2)*(s**2 + s + 1)**2 + 3*(-3*a - 3*s)*(p + s)*(s + 2)*(9*s - 9)*(s**2 + s + 1) + 3*(a + s)*(s + 2)**2*(9*s - 9)*(-p**4 + 3*p - 2)*(s**2 + s + 1), (p + s)*(s + 2)**3*(9*s - 9)*(s**2 + s + 1)**2, s),), (TransferFunction(3*(-a + p)*(p + s)*(s + 2)**2*(-p**4 + 3*p - 2)*(s**2 + s + 1) + 3*(3*a + 3*s)*(p + s)**2*(s + 2)*(9*s - 9) + 3*(p + s)*(s + 2)*(9*s - 9)*(-3*p**4 + 9*p - 6)*(s**2 + s + 1), (p + s)**2*(s + 2)**3*(9*s - 9)*(s**2 + s + 1), s),)))
See Also
========
TransferFunction, MIMOSeries, MIMOParallel, Feedback
"""
def __new__(cls, arg):
expr_mat_arg = []
try:
var = arg[0][0].var
except TypeError:
raise ValueError("`arg` param in TransferFunctionMatrix should "
"strictly be a nested list containing TransferFunction objects.")
for row_index, row in enumerate(arg):
temp = []
for col_index, element in enumerate(row):
if not isinstance(element, SISOLinearTimeInvariant):
raise TypeError("Each element is expected to be of type `SISOLinearTimeInvariant`.")
if var != element.var:
raise ValueError("Conflicting value(s) found for `var`. All TransferFunction instances in "
"TransferFunctionMatrix should use the same complex variable in Laplace domain.")
temp.append(element.to_expr())
expr_mat_arg.append(temp)
if isinstance(arg, (tuple, list, Tuple)):
# Making nested Tuple (sympy.core.containers.Tuple) from nested list or nested Python tuple
arg = Tuple(*(Tuple(*r, sympify=False) for r in arg), sympify=False)
obj = super(TransferFunctionMatrix, cls).__new__(cls, arg)
obj._expr_mat = ImmutableMatrix(expr_mat_arg)
return obj
@classmethod
def from_Matrix(cls, matrix, var):
"""
Creates a new ``TransferFunctionMatrix`` efficiently from a SymPy Matrix of ``Expr`` objects.
Parameters
==========
matrix : ``ImmutableMatrix`` having ``Expr``/``Number`` elements.
var : Symbol
Complex variable of the Laplace transform which will be used by the
all the ``TransferFunction`` objects in the ``TransferFunctionMatrix``.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunctionMatrix
>>> from sympy import Matrix, pprint
>>> M = Matrix([[s, 1/s], [1/(s+1), s]])
>>> M_tf = TransferFunctionMatrix.from_Matrix(M, s)
>>> pprint(M_tf, use_unicode=False)
[ s 1]
[ - -]
[ 1 s]
[ ]
[ 1 s]
[----- -]
[s + 1 1]{t}
>>> M_tf.elem_poles()
[[[], [0]], [[-1], []]]
>>> M_tf.elem_zeros()
[[[0], []], [[], [0]]]
"""
return _to_TFM(matrix, var)
@property
def var(self):
"""
Returns the complex variable used by all the transfer functions or
``Series``/``Parallel`` objects in a transfer function matrix.
Examples
========
>>> from sympy.abc import p, s
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, Series, Parallel
>>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p)
>>> G2 = TransferFunction(p, 4 - p, p)
>>> G3 = TransferFunction(0, p**4 - 1, p)
>>> G4 = TransferFunction(s + 1, s**2 + s + 1, s)
>>> S1 = Series(G1, G2)
>>> S2 = Series(-G3, Parallel(G2, -G1))
>>> tfm1 = TransferFunctionMatrix([[G1], [G2], [G3]])
>>> tfm1.var
p
>>> tfm2 = TransferFunctionMatrix([[-S1, -S2], [S1, S2]])
>>> tfm2.var
p
>>> tfm3 = TransferFunctionMatrix([[G4]])
>>> tfm3.var
s
"""
return self.args[0][0][0].var
@property
def num_inputs(self):
"""
Returns the number of inputs of the system.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix
>>> G1 = TransferFunction(s + 3, s**2 - 3, s)
>>> G2 = TransferFunction(4, s**2, s)
>>> G3 = TransferFunction(p**2 + s**2, p - 3, s)
>>> tfm_1 = TransferFunctionMatrix([[G2, -G1, G3], [-G2, -G1, -G3]])
>>> tfm_1.num_inputs
3
See Also
========
num_outputs
"""
return self._expr_mat.shape[1]
@property
def num_outputs(self):
"""
Returns the number of outputs of the system.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunctionMatrix
>>> from sympy import Matrix
>>> M_1 = Matrix([[s], [1/s]])
>>> TFM = TransferFunctionMatrix.from_Matrix(M_1, s)
>>> print(TFM)
TransferFunctionMatrix(((TransferFunction(s, 1, s),), (TransferFunction(1, s, s),)))
>>> TFM.num_outputs
2
See Also
========
num_inputs
"""
return self._expr_mat.shape[0]
@property
def shape(self):
"""
Returns the shape of the transfer function matrix, that is, ``(# of outputs, # of inputs)``.
Examples
========
>>> from sympy.abc import s, p
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix
>>> tf1 = TransferFunction(p**2 - 1, s**4 + s**3 - p, p)
>>> tf2 = TransferFunction(1 - p, p**2 - 3*p + 7, p)
>>> tf3 = TransferFunction(3, 4, p)
>>> tfm1 = TransferFunctionMatrix([[tf1, -tf2]])
>>> tfm1.shape
(1, 2)
>>> tfm2 = TransferFunctionMatrix([[-tf2, tf3], [tf1, -tf1]])
>>> tfm2.shape
(2, 2)
"""
return self._expr_mat.shape
def __neg__(self):
neg = -self._expr_mat
return _to_TFM(neg, self.var)
@_check_other_MIMO
def __add__(self, other):
if not isinstance(other, MIMOParallel):
return MIMOParallel(self, other)
other_arg_list = list(other.args)
return MIMOParallel(self, *other_arg_list)
@_check_other_MIMO
def __sub__(self, other):
return self + (-other)
@_check_other_MIMO
def __mul__(self, other):
if not isinstance(other, MIMOSeries):
return MIMOSeries(other, self)
other_arg_list = list(other.args)
return MIMOSeries(*other_arg_list, self)
def __getitem__(self, key):
trunc = self._expr_mat.__getitem__(key)
if isinstance(trunc, ImmutableMatrix):
return _to_TFM(trunc, self.var)
return TransferFunction.from_rational_expression(trunc, self.var)
def transpose(self):
"""Returns the transpose of the ``TransferFunctionMatrix`` (switched input and output layers)."""
transposed_mat = self._expr_mat.transpose()
return _to_TFM(transposed_mat, self.var)
def elem_poles(self):
"""
Returns the poles of each element of the ``TransferFunctionMatrix``.
.. note::
Actual poles of a MIMO system are NOT the poles of individual elements.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix
>>> tf_1 = TransferFunction(3, (s + 1), s)
>>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s)
>>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s)
>>> tf_4 = TransferFunction(s + 2, s**2 + 5*s - 10, s)
>>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]])
>>> tfm_1
TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s + 2, s**2 + 5*s - 10, s))))
>>> tfm_1.elem_poles()
[[[-1], [-2, -1]], [[-2, -1], [-5/2 + sqrt(65)/2, -sqrt(65)/2 - 5/2]]]
See Also
========
elem_zeros
"""
return [[element.poles() for element in row] for row in self.doit().args[0]]
def elem_zeros(self):
"""
Returns the zeros of each element of the ``TransferFunctionMatrix``.
.. note::
Actual zeros of a MIMO system are NOT the zeros of individual elements.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix
>>> tf_1 = TransferFunction(3, (s + 1), s)
>>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s)
>>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s)
>>> tf_4 = TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s)
>>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]])
>>> tfm_1
TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s))))
>>> tfm_1.elem_zeros()
[[[], [-6]], [[-3], [4, 5]]]
See Also
========
elem_poles
"""
return [[element.zeros() for element in row] for row in self.doit().args[0]]
def _flat(self):
"""Returns flattened list of args in TransferFunctionMatrix"""
return [elem for tup in self.args[0] for elem in tup]
def _eval_evalf(self, prec):
"""Calls evalf() on each transfer function in the transfer function matrix"""
dps = prec_to_dps(prec)
mat = self._expr_mat.applyfunc(lambda a: a.evalf(n=dps))
return _to_TFM(mat, self.var)
def _eval_simplify(self, **kwargs):
"""Simplifies the transfer function matrix"""
simp_mat = self._expr_mat.applyfunc(lambda a: cancel(a, expand=False))
return _to_TFM(simp_mat, self.var)
def expand(self, **hints):
"""Expands the transfer function matrix"""
expand_mat = self._expr_mat.expand(**hints)
return _to_TFM(expand_mat, self.var)
|
a9519a034752f0fa1b2e4dbe4a190b072c51689f1c393a109d4043fccedaa949 | from sympy.core.numbers import I, pi
from sympy.functions.elementary.exponential import (exp, log)
from sympy.polys.partfrac import apart
from sympy.core.symbol import Dummy
from sympy.external import import_module
from sympy.functions import arg, Abs
from sympy.integrals.transforms import _fast_inverse_laplace
from sympy.physics.control.lti import SISOLinearTimeInvariant
from sympy.plotting.plot import LineOver1DRangeSeries
from sympy.polys.polytools import Poly
from sympy.printing.latex import latex
__all__ = ['pole_zero_numerical_data', 'pole_zero_plot',
'step_response_numerical_data', 'step_response_plot',
'impulse_response_numerical_data', 'impulse_response_plot',
'ramp_response_numerical_data', 'ramp_response_plot',
'bode_magnitude_numerical_data', 'bode_phase_numerical_data',
'bode_magnitude_plot', 'bode_phase_plot', 'bode_plot']
matplotlib = import_module(
'matplotlib', import_kwargs={'fromlist': ['pyplot']},
catch=(RuntimeError,))
numpy = import_module('numpy')
if matplotlib:
plt = matplotlib.pyplot
if numpy:
np = numpy # Matplotlib already has numpy as a compulsory dependency. No need to install it separately.
def _check_system(system):
"""Function to check whether the dynamical system passed for plots is
compatible or not."""
if not isinstance(system, SISOLinearTimeInvariant):
raise NotImplementedError("Only SISO LTI systems are currently supported.")
sys = system.to_expr()
len_free_symbols = len(sys.free_symbols)
if len_free_symbols > 1:
raise ValueError("Extra degree of freedom found. Make sure"
" that there are no free symbols in the dynamical system other"
" than the variable of Laplace transform.")
if sys.has(exp):
# Should test that exp is not part of a constant, in which case
# no exception is required, compare exp(s) with s*exp(1)
raise NotImplementedError("Time delay terms are not supported.")
def pole_zero_numerical_data(system):
"""
Returns the numerical data of poles and zeros of the system.
It is internally used by ``pole_zero_plot`` to get the data
for plotting poles and zeros. Users can use this data to further
analyse the dynamics of the system or plot using a different
backend/plotting-module.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the pole-zero data is to be computed.
Returns
=======
tuple : (zeros, poles)
zeros = Zeros of the system. NumPy array of complex numbers.
poles = Poles of the system. NumPy array of complex numbers.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import pole_zero_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> pole_zero_numerical_data(tf1) # doctest: +SKIP
([-0.+1.j 0.-1.j], [-2. +0.j -0.5+0.8660254j -0.5-0.8660254j -1. +0.j ])
See Also
========
pole_zero_plot
"""
_check_system(system)
system = system.doit() # Get the equivalent TransferFunction object.
num_poly = Poly(system.num, system.var).all_coeffs()
den_poly = Poly(system.den, system.var).all_coeffs()
num_poly = np.array(num_poly, dtype=np.complex128)
den_poly = np.array(den_poly, dtype=np.complex128)
zeros = np.roots(num_poly)
poles = np.roots(den_poly)
return zeros, poles
def pole_zero_plot(system, pole_color='blue', pole_markersize=10,
zero_color='orange', zero_markersize=7, grid=True, show_axes=True,
show=True, **kwargs):
r"""
Returns the Pole-Zero plot (also known as PZ Plot or PZ Map) of a system.
A Pole-Zero plot is a graphical representation of a system's poles and
zeros. It is plotted on a complex plane, with circular markers representing
the system's zeros and 'x' shaped markers representing the system's poles.
Parameters
==========
system : SISOLinearTimeInvariant type systems
The system for which the pole-zero plot is to be computed.
pole_color : str, tuple, optional
The color of the pole points on the plot. Default color
is blue. The color can be provided as a matplotlib color string,
or a 3-tuple of floats each in the 0-1 range.
pole_markersize : Number, optional
The size of the markers used to mark the poles in the plot.
Default pole markersize is 10.
zero_color : str, tuple, optional
The color of the zero points on the plot. Default color
is orange. The color can be provided as a matplotlib color string,
or a 3-tuple of floats each in the 0-1 range.
zero_markersize : Number, optional
The size of the markers used to mark the zeros in the plot.
Default zero markersize is 7.
grid : boolean, optional
If ``True``, the plot will have a grid. Defaults to True.
show_axes : boolean, optional
If ``True``, the coordinate axes will be shown. Defaults to False.
show : boolean, optional
If ``True``, the plot will be displayed otherwise
the equivalent matplotlib ``plot`` object will be returned.
Defaults to True.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import pole_zero_plot
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> pole_zero_plot(tf1) # doctest: +SKIP
See Also
========
pole_zero_numerical_data
References
==========
.. [1] https://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot
"""
zeros, poles = pole_zero_numerical_data(system)
zero_real = np.real(zeros)
zero_imag = np.imag(zeros)
pole_real = np.real(poles)
pole_imag = np.imag(poles)
plt.plot(pole_real, pole_imag, 'x', mfc='none',
markersize=pole_markersize, color=pole_color)
plt.plot(zero_real, zero_imag, 'o', markersize=zero_markersize,
color=zero_color)
plt.xlabel('Real Axis')
plt.ylabel('Imaginary Axis')
plt.title(f'Poles and Zeros of ${latex(system)}$', pad=20)
if grid:
plt.grid()
if show_axes:
plt.axhline(0, color='black')
plt.axvline(0, color='black')
if show:
plt.show()
return
return plt
def step_response_numerical_data(system, prec=8, lower_limit=0,
upper_limit=10, **kwargs):
"""
Returns the numerical values of the points in the step response plot
of a SISO continuous-time system. By default, adaptive sampling
is used. If the user wants to instead get an uniformly
sampled response, then ``adaptive`` kwarg should be passed ``False``
and ``nb_of_points`` must be passed as additional kwargs.
Refer to the parameters of class :class:`sympy.plotting.plot.LineOver1DRangeSeries`
for more details.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the unit step response data is to be computed.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
lower_limit : Number, optional
The lower limit of the plot range. Defaults to 0.
upper_limit : Number, optional
The upper limit of the plot range. Defaults to 10.
kwargs :
Additional keyword arguments are passed to the underlying
:class:`sympy.plotting.plot.LineOver1DRangeSeries` class.
Returns
=======
tuple : (x, y)
x = Time-axis values of the points in the step response. NumPy array.
y = Amplitude-axis values of the points in the step response. NumPy array.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
When ``lower_limit`` parameter is less than 0.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import step_response_numerical_data
>>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
>>> step_response_numerical_data(tf1) # doctest: +SKIP
([0.0, 0.025413462339411542, 0.0484508722725343, ... , 9.670250533855183, 9.844291913708725, 10.0],
[0.0, 0.023844582399907256, 0.042894276802320226, ..., 6.828770759094287e-12, 6.456457160755703e-12])
See Also
========
step_response_plot
"""
if lower_limit < 0:
raise ValueError("Lower limit of time must be greater "
"than or equal to zero.")
_check_system(system)
_x = Dummy("x")
expr = system.to_expr()/(system.var)
expr = apart(expr, system.var, full=True)
_y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec)
return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit),
**kwargs).get_points()
def step_response_plot(system, color='b', prec=8, lower_limit=0,
upper_limit=10, show_axes=False, grid=True, show=True, **kwargs):
r"""
Returns the unit step response of a continuous-time system. It is
the response of the system when the input signal is a step function.
Parameters
==========
system : SISOLinearTimeInvariant type
The LTI SISO system for which the Step Response is to be computed.
color : str, tuple, optional
The color of the line. Default is Blue.
show : boolean, optional
If ``True``, the plot will be displayed otherwise
the equivalent matplotlib ``plot`` object will be returned.
Defaults to True.
lower_limit : Number, optional
The lower limit of the plot range. Defaults to 0.
upper_limit : Number, optional
The upper limit of the plot range. Defaults to 10.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
show_axes : boolean, optional
If ``True``, the coordinate axes will be shown. Defaults to False.
grid : boolean, optional
If ``True``, the plot will have a grid. Defaults to True.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import step_response_plot
>>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s)
>>> step_response_plot(tf1) # doctest: +SKIP
See Also
========
impulse_response_plot, ramp_response_plot
References
==========
.. [1] https://www.mathworks.com/help/control/ref/lti.step.html
"""
x, y = step_response_numerical_data(system, prec=prec,
lower_limit=lower_limit, upper_limit=upper_limit, **kwargs)
plt.plot(x, y, color=color)
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.title(f'Unit Step Response of ${latex(system)}$', pad=20)
if grid:
plt.grid()
if show_axes:
plt.axhline(0, color='black')
plt.axvline(0, color='black')
if show:
plt.show()
return
return plt
def impulse_response_numerical_data(system, prec=8, lower_limit=0,
upper_limit=10, **kwargs):
"""
Returns the numerical values of the points in the impulse response plot
of a SISO continuous-time system. By default, adaptive sampling
is used. If the user wants to instead get an uniformly
sampled response, then ``adaptive`` kwarg should be passed ``False``
and ``nb_of_points`` must be passed as additional kwargs.
Refer to the parameters of class :class:`sympy.plotting.plot.LineOver1DRangeSeries`
for more details.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the impulse response data is to be computed.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
lower_limit : Number, optional
The lower limit of the plot range. Defaults to 0.
upper_limit : Number, optional
The upper limit of the plot range. Defaults to 10.
kwargs :
Additional keyword arguments are passed to the underlying
:class:`sympy.plotting.plot.LineOver1DRangeSeries` class.
Returns
=======
tuple : (x, y)
x = Time-axis values of the points in the impulse response. NumPy array.
y = Amplitude-axis values of the points in the impulse response. NumPy array.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
When ``lower_limit`` parameter is less than 0.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import impulse_response_numerical_data
>>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
>>> impulse_response_numerical_data(tf1) # doctest: +SKIP
([0.0, 0.06616480200395854,... , 9.854500743565858, 10.0],
[0.9999999799999999, 0.7042848373025861,...,7.170748906965121e-13, -5.1901263495547205e-12])
See Also
========
impulse_response_plot
"""
if lower_limit < 0:
raise ValueError("Lower limit of time must be greater "
"than or equal to zero.")
_check_system(system)
_x = Dummy("x")
expr = system.to_expr()
expr = apart(expr, system.var, full=True)
_y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec)
return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit),
**kwargs).get_points()
def impulse_response_plot(system, color='b', prec=8, lower_limit=0,
upper_limit=10, show_axes=False, grid=True, show=True, **kwargs):
r"""
Returns the unit impulse response (Input is the Dirac-Delta Function) of a
continuous-time system.
Parameters
==========
system : SISOLinearTimeInvariant type
The LTI SISO system for which the Impulse Response is to be computed.
color : str, tuple, optional
The color of the line. Default is Blue.
show : boolean, optional
If ``True``, the plot will be displayed otherwise
the equivalent matplotlib ``plot`` object will be returned.
Defaults to True.
lower_limit : Number, optional
The lower limit of the plot range. Defaults to 0.
upper_limit : Number, optional
The upper limit of the plot range. Defaults to 10.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
show_axes : boolean, optional
If ``True``, the coordinate axes will be shown. Defaults to False.
grid : boolean, optional
If ``True``, the plot will have a grid. Defaults to True.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import impulse_response_plot
>>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s)
>>> impulse_response_plot(tf1) # doctest: +SKIP
See Also
========
step_response_plot, ramp_response_plot
References
==========
.. [1] https://www.mathworks.com/help/control/ref/lti.impulse.html
"""
x, y = impulse_response_numerical_data(system, prec=prec,
lower_limit=lower_limit, upper_limit=upper_limit, **kwargs)
plt.plot(x, y, color=color)
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.title(f'Impulse Response of ${latex(system)}$', pad=20)
if grid:
plt.grid()
if show_axes:
plt.axhline(0, color='black')
plt.axvline(0, color='black')
if show:
plt.show()
return
return plt
def ramp_response_numerical_data(system, slope=1, prec=8,
lower_limit=0, upper_limit=10, **kwargs):
"""
Returns the numerical values of the points in the ramp response plot
of a SISO continuous-time system. By default, adaptive sampling
is used. If the user wants to instead get an uniformly
sampled response, then ``adaptive`` kwarg should be passed ``False``
and ``nb_of_points`` must be passed as additional kwargs.
Refer to the parameters of class :class:`sympy.plotting.plot.LineOver1DRangeSeries`
for more details.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the ramp response data is to be computed.
slope : Number, optional
The slope of the input ramp function. Defaults to 1.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
lower_limit : Number, optional
The lower limit of the plot range. Defaults to 0.
upper_limit : Number, optional
The upper limit of the plot range. Defaults to 10.
kwargs :
Additional keyword arguments are passed to the underlying
:class:`sympy.plotting.plot.LineOver1DRangeSeries` class.
Returns
=======
tuple : (x, y)
x = Time-axis values of the points in the ramp response plot. NumPy array.
y = Amplitude-axis values of the points in the ramp response plot. NumPy array.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
When ``lower_limit`` parameter is less than 0.
When ``slope`` is negative.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import ramp_response_numerical_data
>>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
>>> ramp_response_numerical_data(tf1) # doctest: +SKIP
(([0.0, 0.12166980856813935,..., 9.861246379582118, 10.0],
[1.4504508011325967e-09, 0.006046440489058766,..., 0.12499999999568202, 0.12499999999661349]))
See Also
========
ramp_response_plot
"""
if slope < 0:
raise ValueError("Slope must be greater than or equal"
" to zero.")
if lower_limit < 0:
raise ValueError("Lower limit of time must be greater "
"than or equal to zero.")
_check_system(system)
_x = Dummy("x")
expr = (slope*system.to_expr())/((system.var)**2)
expr = apart(expr, system.var, full=True)
_y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec)
return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit),
**kwargs).get_points()
def ramp_response_plot(system, slope=1, color='b', prec=8, lower_limit=0,
upper_limit=10, show_axes=False, grid=True, show=True, **kwargs):
r"""
Returns the ramp response of a continuous-time system.
Ramp function is defined as the straight line
passing through origin ($f(x) = mx$). The slope of
the ramp function can be varied by the user and
the default value is 1.
Parameters
==========
system : SISOLinearTimeInvariant type
The LTI SISO system for which the Ramp Response is to be computed.
slope : Number, optional
The slope of the input ramp function. Defaults to 1.
color : str, tuple, optional
The color of the line. Default is Blue.
show : boolean, optional
If ``True``, the plot will be displayed otherwise
the equivalent matplotlib ``plot`` object will be returned.
Defaults to True.
lower_limit : Number, optional
The lower limit of the plot range. Defaults to 0.
upper_limit : Number, optional
The upper limit of the plot range. Defaults to 10.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
show_axes : boolean, optional
If ``True``, the coordinate axes will be shown. Defaults to False.
grid : boolean, optional
If ``True``, the plot will have a grid. Defaults to True.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import ramp_response_plot
>>> tf1 = TransferFunction(s, (s+4)*(s+8), s)
>>> ramp_response_plot(tf1, upper_limit=2) # doctest: +SKIP
See Also
========
step_response_plot, ramp_response_plot
References
==========
.. [1] https://en.wikipedia.org/wiki/Ramp_function
"""
x, y = ramp_response_numerical_data(system, slope=slope, prec=prec,
lower_limit=lower_limit, upper_limit=upper_limit, **kwargs)
plt.plot(x, y, color=color)
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.title(f'Ramp Response of ${latex(system)}$ [Slope = {slope}]', pad=20)
if grid:
plt.grid()
if show_axes:
plt.axhline(0, color='black')
plt.axvline(0, color='black')
if show:
plt.show()
return
return plt
def bode_magnitude_numerical_data(system, initial_exp=-5, final_exp=5, freq_unit='rad/sec', **kwargs):
"""
Returns the numerical data of the Bode magnitude plot of the system.
It is internally used by ``bode_magnitude_plot`` to get the data
for plotting Bode magnitude plot. Users can use this data to further
analyse the dynamics of the system or plot using a different
backend/plotting-module.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the data is to be computed.
initial_exp : Number, optional
The initial exponent of 10 of the semilog plot. Defaults to -5.
final_exp : Number, optional
The final exponent of 10 of the semilog plot. Defaults to 5.
freq_unit : string, optional
User can choose between ``'rad/sec'`` (radians/second) and ``'Hz'`` (Hertz) as frequency units.
Returns
=======
tuple : (x, y)
x = x-axis values of the Bode magnitude plot.
y = y-axis values of the Bode magnitude plot.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
When incorrect frequency units are given as input.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import bode_magnitude_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> bode_magnitude_numerical_data(tf1) # doctest: +SKIP
([1e-05, 1.5148378120533502e-05,..., 68437.36188804005, 100000.0],
[-6.020599914256786, -6.0205999155219505,..., -193.4117304087953, -200.00000000260573])
See Also
========
bode_magnitude_plot, bode_phase_numerical_data
"""
_check_system(system)
expr = system.to_expr()
freq_units = ('rad/sec', 'Hz')
if freq_unit not in freq_units:
raise ValueError('Only "rad/sec" and "Hz" are accepted frequency units.')
_w = Dummy("w", real=True)
if freq_unit == 'Hz':
repl = I*_w*2*pi
else:
repl = I*_w
w_expr = expr.subs({system.var: repl})
mag = 20*log(Abs(w_expr), 10)
x, y = LineOver1DRangeSeries(mag,
(_w, 10**initial_exp, 10**final_exp), xscale='log', **kwargs).get_points()
return x, y
def bode_magnitude_plot(system, initial_exp=-5, final_exp=5,
color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', **kwargs):
r"""
Returns the Bode magnitude plot of a continuous-time system.
See ``bode_plot`` for all the parameters.
"""
x, y = bode_magnitude_numerical_data(system, initial_exp=initial_exp,
final_exp=final_exp, freq_unit=freq_unit)
plt.plot(x, y, color=color, **kwargs)
plt.xscale('log')
plt.xlabel('Frequency (%s) [Log Scale]' % freq_unit)
plt.ylabel('Magnitude (dB)')
plt.title(f'Bode Plot (Magnitude) of ${latex(system)}$', pad=20)
if grid:
plt.grid(True)
if show_axes:
plt.axhline(0, color='black')
plt.axvline(0, color='black')
if show:
plt.show()
return
return plt
def bode_phase_numerical_data(system, initial_exp=-5, final_exp=5, freq_unit='rad/sec', phase_unit='rad', **kwargs):
"""
Returns the numerical data of the Bode phase plot of the system.
It is internally used by ``bode_phase_plot`` to get the data
for plotting Bode phase plot. Users can use this data to further
analyse the dynamics of the system or plot using a different
backend/plotting-module.
Parameters
==========
system : SISOLinearTimeInvariant
The system for which the Bode phase plot data is to be computed.
initial_exp : Number, optional
The initial exponent of 10 of the semilog plot. Defaults to -5.
final_exp : Number, optional
The final exponent of 10 of the semilog plot. Defaults to 5.
freq_unit : string, optional
User can choose between ``'rad/sec'`` (radians/second) and '``'Hz'`` (Hertz) as frequency units.
phase_unit : string, optional
User can choose between ``'rad'`` (radians) and ``'deg'`` (degree) as phase units.
Returns
=======
tuple : (x, y)
x = x-axis values of the Bode phase plot.
y = y-axis values of the Bode phase plot.
Raises
======
NotImplementedError
When a SISO LTI system is not passed.
When time delay terms are present in the system.
ValueError
When more than one free symbol is present in the system.
The only variable in the transfer function should be
the variable of the Laplace transform.
When incorrect frequency or phase units are given as input.
Examples
========
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import bode_phase_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> bode_phase_numerical_data(tf1) # doctest: +SKIP
([1e-05, 1.4472354033813751e-05, 2.035581932165858e-05,..., 47577.3248186011, 67884.09326036123, 100000.0],
[-2.5000000000291665e-05, -3.6180885085e-05, -5.08895483066e-05,...,-3.1415085799262523, -3.14155265358979])
See Also
========
bode_magnitude_plot, bode_phase_numerical_data
"""
_check_system(system)
expr = system.to_expr()
freq_units = ('rad/sec', 'Hz')
phase_units = ('rad', 'deg')
if freq_unit not in freq_units:
raise ValueError('Only "rad/sec" and "Hz" are accepted frequency units.')
if phase_unit not in phase_units:
raise ValueError('Only "rad" and "deg" are accepted phase units.')
_w = Dummy("w", real=True)
if freq_unit == 'Hz':
repl = I*_w*2*pi
else:
repl = I*_w
w_expr = expr.subs({system.var: repl})
if phase_unit == 'deg':
phase = arg(w_expr)*180/pi
else:
phase = arg(w_expr)
x, y = LineOver1DRangeSeries(phase,
(_w, 10**initial_exp, 10**final_exp), xscale='log', **kwargs).get_points()
return x, y
def bode_phase_plot(system, initial_exp=-5, final_exp=5,
color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', phase_unit='rad', **kwargs):
r"""
Returns the Bode phase plot of a continuous-time system.
See ``bode_plot`` for all the parameters.
"""
x, y = bode_phase_numerical_data(system, initial_exp=initial_exp,
final_exp=final_exp, freq_unit=freq_unit, phase_unit=phase_unit)
plt.plot(x, y, color=color, **kwargs)
plt.xscale('log')
plt.xlabel('Frequency (%s) [Log Scale]' % freq_unit)
plt.ylabel('Phase (%s)' % phase_unit)
plt.title(f'Bode Plot (Phase) of ${latex(system)}$', pad=20)
if grid:
plt.grid(True)
if show_axes:
plt.axhline(0, color='black')
plt.axvline(0, color='black')
if show:
plt.show()
return
return plt
def bode_plot(system, initial_exp=-5, final_exp=5,
grid=True, show_axes=False, show=True, freq_unit='rad/sec', phase_unit='rad', **kwargs):
r"""
Returns the Bode phase and magnitude plots of a continuous-time system.
Parameters
==========
system : SISOLinearTimeInvariant type
The LTI SISO system for which the Bode Plot is to be computed.
initial_exp : Number, optional
The initial exponent of 10 of the semilog plot. Defaults to -5.
final_exp : Number, optional
The final exponent of 10 of the semilog plot. Defaults to 5.
show : boolean, optional
If ``True``, the plot will be displayed otherwise
the equivalent matplotlib ``plot`` object will be returned.
Defaults to True.
prec : int, optional
The decimal point precision for the point coordinate values.
Defaults to 8.
grid : boolean, optional
If ``True``, the plot will have a grid. Defaults to True.
show_axes : boolean, optional
If ``True``, the coordinate axes will be shown. Defaults to False.
freq_unit : string, optional
User can choose between ``'rad/sec'`` (radians/second) and ``'Hz'`` (Hertz) as frequency units.
phase_unit : string, optional
User can choose between ``'rad'`` (radians) and ``'deg'`` (degree) as phase units.
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import bode_plot
>>> tf1 = TransferFunction(1*s**2 + 0.1*s + 7.5, 1*s**4 + 0.12*s**3 + 9*s**2, s)
>>> bode_plot(tf1, initial_exp=0.2, final_exp=0.7) # doctest: +SKIP
See Also
========
bode_magnitude_plot, bode_phase_plot
"""
plt.subplot(211)
mag = bode_magnitude_plot(system, initial_exp=initial_exp, final_exp=final_exp,
show=False, grid=grid, show_axes=show_axes,
freq_unit=freq_unit, **kwargs)
mag.title(f'Bode Plot of ${latex(system)}$', pad=20)
mag.xlabel(None)
plt.subplot(212)
bode_phase_plot(system, initial_exp=initial_exp, final_exp=final_exp,
show=False, grid=grid, show_axes=show_axes, freq_unit=freq_unit, phase_unit=phase_unit, **kwargs).title(None)
if show:
plt.show()
return
return plt
|
9c74f69b89ac4b983240b60f228d97e97705ef2d294d1658254631646d180949 | """Shor's algorithm and helper functions.
Todo:
* Get the CMod gate working again using the new Gate API.
* Fix everything.
* Update docstrings and reformat.
"""
import math
import random
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.core.numbers import igcd
from sympy.ntheory import continued_fraction_periodic as continued_fraction
from sympy.utilities.iterables import variations
from sympy.physics.quantum.gate import Gate
from sympy.physics.quantum.qubit import Qubit, measure_partial_oneshot
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.qft import QFT
from sympy.physics.quantum.qexpr import QuantumError
class OrderFindingException(QuantumError):
pass
class CMod(Gate):
"""A controlled mod gate.
This is black box controlled Mod function for use by shor's algorithm.
TODO: implement a decompose property that returns how to do this in terms
of elementary gates
"""
@classmethod
def _eval_args(cls, args):
# t = args[0]
# a = args[1]
# N = args[2]
raise NotImplementedError('The CMod gate has not been completed.')
@property
def t(self):
"""Size of 1/2 input register. First 1/2 holds output."""
return self.label[0]
@property
def a(self):
"""Base of the controlled mod function."""
return self.label[1]
@property
def N(self):
"""N is the type of modular arithmetic we are doing."""
return self.label[2]
def _apply_operator_Qubit(self, qubits, **options):
"""
This directly calculates the controlled mod of the second half of
the register and puts it in the second
This will look pretty when we get Tensor Symbolically working
"""
n = 1
k = 0
# Determine the value stored in high memory.
for i in range(self.t):
k += n*qubits[self.t + i]
n *= 2
# The value to go in low memory will be out.
out = int(self.a**k % self.N)
# Create array for new qbit-ket which will have high memory unaffected
outarray = list(qubits.args[0][:self.t])
# Place out in low memory
for i in reversed(range(self.t)):
outarray.append((out >> i) & 1)
return Qubit(*outarray)
def shor(N):
"""This function implements Shor's factoring algorithm on the Integer N
The algorithm starts by picking a random number (a) and seeing if it is
coprime with N. If it is not, then the gcd of the two numbers is a factor
and we are done. Otherwise, it begins the period_finding subroutine which
finds the period of a in modulo N arithmetic. This period, if even, can
be used to calculate factors by taking a**(r/2)-1 and a**(r/2)+1.
These values are returned.
"""
a = random.randrange(N - 2) + 2
if igcd(N, a) != 1:
return igcd(N, a)
r = period_find(a, N)
if r % 2 == 1:
shor(N)
answer = (igcd(a**(r/2) - 1, N), igcd(a**(r/2) + 1, N))
return answer
def getr(x, y, N):
fraction = continued_fraction(x, y)
# Now convert into r
total = ratioize(fraction, N)
return total
def ratioize(list, N):
if list[0] > N:
return S.Zero
if len(list) == 1:
return list[0]
return list[0] + ratioize(list[1:], N)
def period_find(a, N):
"""Finds the period of a in modulo N arithmetic
This is quantum part of Shor's algorithm. It takes two registers,
puts first in superposition of states with Hadamards so: ``|k>|0>``
with k being all possible choices. It then does a controlled mod and
a QFT to determine the order of a.
"""
epsilon = .5
# picks out t's such that maintains accuracy within epsilon
t = int(2*math.ceil(log(N, 2)))
# make the first half of register be 0's |000...000>
start = [0 for x in range(t)]
# Put second half into superposition of states so we have |1>x|0> + |2>x|0> + ... |k>x>|0> + ... + |2**n-1>x|0>
factor = 1/sqrt(2**t)
qubits = 0
for arr in variations(range(2), t, repetition=True):
qbitArray = list(arr) + start
qubits = qubits + Qubit(*qbitArray)
circuit = (factor*qubits).expand()
# Controlled second half of register so that we have:
# |1>x|a**1 %N> + |2>x|a**2 %N> + ... + |k>x|a**k %N >+ ... + |2**n-1=k>x|a**k % n>
circuit = CMod(t, a, N)*circuit
# will measure first half of register giving one of the a**k%N's
circuit = qapply(circuit)
for i in range(t):
circuit = measure_partial_oneshot(circuit, i)
# Now apply Inverse Quantum Fourier Transform on the second half of the register
circuit = qapply(QFT(t, t*2).decompose()*circuit, floatingPoint=True)
for i in range(t):
circuit = measure_partial_oneshot(circuit, i + t)
if isinstance(circuit, Qubit):
register = circuit
elif isinstance(circuit, Mul):
register = circuit.args[-1]
else:
register = circuit.args[-1].args[-1]
n = 1
answer = 0
for i in range(len(register)/2):
answer += n*register[i + t]
n = n << 1
if answer == 0:
raise OrderFindingException(
"Order finder returned 0. Happens with chance %f" % epsilon)
#turn answer into r using continued fractions
g = getr(answer, 2**t, N)
return g
|
0f98a930144287636cc48cb817a64b40017863ce3edab28a0be733095676d83b | """Utilities to deal with sympy.Matrix, numpy and scipy.sparse."""
from sympy.core.expr import Expr
from sympy.core.numbers import I
from sympy.core.singleton import S
from sympy.matrices.matrices import MatrixBase
from sympy.matrices import eye, zeros
from sympy.external import import_module
__all__ = [
'numpy_ndarray',
'scipy_sparse_matrix',
'sympy_to_numpy',
'sympy_to_scipy_sparse',
'numpy_to_sympy',
'scipy_sparse_to_sympy',
'flatten_scalar',
'matrix_dagger',
'to_sympy',
'to_numpy',
'to_scipy_sparse',
'matrix_tensor_product',
'matrix_zeros'
]
# Conditionally define the base classes for numpy and scipy.sparse arrays
# for use in isinstance tests.
np = import_module('numpy')
if not np:
class numpy_ndarray:
pass
else:
numpy_ndarray = np.ndarray # type: ignore
scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
if not scipy:
class scipy_sparse_matrix:
pass
sparse = None
else:
sparse = scipy.sparse
scipy_sparse_matrix = sparse.spmatrix # type: ignore
def sympy_to_numpy(m, **options):
"""Convert a SymPy Matrix/complex number to a numpy matrix or scalar."""
if not np:
raise ImportError
dtype = options.get('dtype', 'complex')
if isinstance(m, MatrixBase):
return np.array(m.tolist(), dtype=dtype)
elif isinstance(m, Expr):
if m.is_Number or m.is_NumberSymbol or m == I:
return complex(m)
raise TypeError('Expected MatrixBase or complex scalar, got: %r' % m)
def sympy_to_scipy_sparse(m, **options):
"""Convert a SymPy Matrix/complex number to a numpy matrix or scalar."""
if not np or not sparse:
raise ImportError
dtype = options.get('dtype', 'complex')
if isinstance(m, MatrixBase):
return sparse.csr_matrix(np.array(m.tolist(), dtype=dtype))
elif isinstance(m, Expr):
if m.is_Number or m.is_NumberSymbol or m == I:
return complex(m)
raise TypeError('Expected MatrixBase or complex scalar, got: %r' % m)
def scipy_sparse_to_sympy(m, **options):
"""Convert a scipy.sparse matrix to a SymPy matrix."""
return MatrixBase(m.todense())
def numpy_to_sympy(m, **options):
"""Convert a numpy matrix to a SymPy matrix."""
return MatrixBase(m)
def to_sympy(m, **options):
"""Convert a numpy/scipy.sparse matrix to a SymPy matrix."""
if isinstance(m, MatrixBase):
return m
elif isinstance(m, numpy_ndarray):
return numpy_to_sympy(m)
elif isinstance(m, scipy_sparse_matrix):
return scipy_sparse_to_sympy(m)
elif isinstance(m, Expr):
return m
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % m)
def to_numpy(m, **options):
"""Convert a sympy/scipy.sparse matrix to a numpy matrix."""
dtype = options.get('dtype', 'complex')
if isinstance(m, (MatrixBase, Expr)):
return sympy_to_numpy(m, dtype=dtype)
elif isinstance(m, numpy_ndarray):
return m
elif isinstance(m, scipy_sparse_matrix):
return m.todense()
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % m)
def to_scipy_sparse(m, **options):
"""Convert a sympy/numpy matrix to a scipy.sparse matrix."""
dtype = options.get('dtype', 'complex')
if isinstance(m, (MatrixBase, Expr)):
return sympy_to_scipy_sparse(m, dtype=dtype)
elif isinstance(m, numpy_ndarray):
if not sparse:
raise ImportError
return sparse.csr_matrix(m)
elif isinstance(m, scipy_sparse_matrix):
return m
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % m)
def flatten_scalar(e):
"""Flatten a 1x1 matrix to a scalar, return larger matrices unchanged."""
if isinstance(e, MatrixBase):
if e.shape == (1, 1):
e = e[0]
if isinstance(e, (numpy_ndarray, scipy_sparse_matrix)):
if e.shape == (1, 1):
e = complex(e[0, 0])
return e
def matrix_dagger(e):
"""Return the dagger of a sympy/numpy/scipy.sparse matrix."""
if isinstance(e, MatrixBase):
return e.H
elif isinstance(e, (numpy_ndarray, scipy_sparse_matrix)):
return e.conjugate().transpose()
raise TypeError('Expected sympy/numpy/scipy.sparse matrix, got: %r' % e)
# TODO: Move this into sympy.matricies.
def _sympy_tensor_product(*matrices):
"""Compute the kronecker product of a sequence of SymPy Matrices.
"""
from sympy.matrices.expressions.kronecker import matrix_kronecker_product
return matrix_kronecker_product(*matrices)
def _numpy_tensor_product(*product):
"""numpy version of tensor product of multiple arguments."""
if not np:
raise ImportError
answer = product[0]
for item in product[1:]:
answer = np.kron(answer, item)
return answer
def _scipy_sparse_tensor_product(*product):
"""scipy.sparse version of tensor product of multiple arguments."""
if not sparse:
raise ImportError
answer = product[0]
for item in product[1:]:
answer = sparse.kron(answer, item)
# The final matrices will just be multiplied, so csr is a good final
# sparse format.
return sparse.csr_matrix(answer)
def matrix_tensor_product(*product):
"""Compute the matrix tensor product of sympy/numpy/scipy.sparse matrices."""
if isinstance(product[0], MatrixBase):
return _sympy_tensor_product(*product)
elif isinstance(product[0], numpy_ndarray):
return _numpy_tensor_product(*product)
elif isinstance(product[0], scipy_sparse_matrix):
return _scipy_sparse_tensor_product(*product)
def _numpy_eye(n):
"""numpy version of complex eye."""
if not np:
raise ImportError
return np.array(np.eye(n, dtype='complex'))
def _scipy_sparse_eye(n):
"""scipy.sparse version of complex eye."""
if not sparse:
raise ImportError
return sparse.eye(n, n, dtype='complex')
def matrix_eye(n, **options):
"""Get the version of eye and tensor_product for a given format."""
format = options.get('format', 'sympy')
if format == 'sympy':
return eye(n)
elif format == 'numpy':
return _numpy_eye(n)
elif format == 'scipy.sparse':
return _scipy_sparse_eye(n)
raise NotImplementedError('Invalid format: %r' % format)
def _numpy_zeros(m, n, **options):
"""numpy version of zeros."""
dtype = options.get('dtype', 'float64')
if not np:
raise ImportError
return np.zeros((m, n), dtype=dtype)
def _scipy_sparse_zeros(m, n, **options):
"""scipy.sparse version of zeros."""
spmatrix = options.get('spmatrix', 'csr')
dtype = options.get('dtype', 'float64')
if not sparse:
raise ImportError
if spmatrix == 'lil':
return sparse.lil_matrix((m, n), dtype=dtype)
elif spmatrix == 'csr':
return sparse.csr_matrix((m, n), dtype=dtype)
def matrix_zeros(m, n, **options):
""""Get a zeros matrix for a given format."""
format = options.get('format', 'sympy')
if format == 'sympy':
return zeros(m, n)
elif format == 'numpy':
return _numpy_zeros(m, n, **options)
elif format == 'scipy.sparse':
return _scipy_sparse_zeros(m, n, **options)
raise NotImplementedError('Invaild format: %r' % format)
def _numpy_matrix_to_zero(e):
"""Convert a numpy zero matrix to the zero scalar."""
if not np:
raise ImportError
test = np.zeros_like(e)
if np.allclose(e, test):
return 0.0
else:
return e
def _scipy_sparse_matrix_to_zero(e):
"""Convert a scipy.sparse zero matrix to the zero scalar."""
if not np:
raise ImportError
edense = e.todense()
test = np.zeros_like(edense)
if np.allclose(edense, test):
return 0.0
else:
return e
def matrix_to_zero(e):
"""Convert a zero matrix to the scalar zero."""
if isinstance(e, MatrixBase):
if zeros(*e.shape) == e:
e = S.Zero
elif isinstance(e, numpy_ndarray):
e = _numpy_matrix_to_zero(e)
elif isinstance(e, scipy_sparse_matrix):
e = _scipy_sparse_matrix_to_zero(e)
return e
|
12858674d84521bbe0274ca5d97b053d1eaa80f694e535b0db34867d4f8d0b16 | """An implementation of gates that act on qubits.
Gates are unitary operators that act on the space of qubits.
Medium Term Todo:
* Optimize Gate._apply_operators_Qubit to remove the creation of many
intermediate Qubit objects.
* Add commutation relationships to all operators and use this in gate_sort.
* Fix gate_sort and gate_simp.
* Get multi-target UGates plotting properly.
* Get UGate to work with either sympy/numpy matrices and output either
format. This should also use the matrix slots.
"""
from itertools import chain
import random
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.mul import Mul
from sympy.core.numbers import (I, Integer)
from sympy.core.power import Pow
from sympy.core.numbers import Number
from sympy.core.singleton import S as _S
from sympy.core.sorting import default_sort_key
from sympy.core.sympify import _sympify
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.printing.pretty.stringpict import prettyForm, stringPict
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.hilbert import ComplexSpace
from sympy.physics.quantum.operator import (UnitaryOperator, Operator,
HermitianOperator)
from sympy.physics.quantum.matrixutils import matrix_tensor_product, matrix_eye
from sympy.physics.quantum.matrixcache import matrix_cache
from sympy.matrices.matrices import MatrixBase
from sympy.utilities.iterables import is_sequence
__all__ = [
'Gate',
'CGate',
'UGate',
'OneQubitGate',
'TwoQubitGate',
'IdentityGate',
'HadamardGate',
'XGate',
'YGate',
'ZGate',
'TGate',
'PhaseGate',
'SwapGate',
'CNotGate',
# Aliased gate names
'CNOT',
'SWAP',
'H',
'X',
'Y',
'Z',
'T',
'S',
'Phase',
'normalized',
'gate_sort',
'gate_simp',
'random_circuit',
'CPHASE',
'CGateS',
]
#-----------------------------------------------------------------------------
# Gate Super-Classes
#-----------------------------------------------------------------------------
_normalized = True
def _max(*args, **kwargs):
if "key" not in kwargs:
kwargs["key"] = default_sort_key
return max(*args, **kwargs)
def _min(*args, **kwargs):
if "key" not in kwargs:
kwargs["key"] = default_sort_key
return min(*args, **kwargs)
def normalized(normalize):
r"""Set flag controlling normalization of Hadamard gates by `1/\sqrt{2}`.
This is a global setting that can be used to simplify the look of various
expressions, by leaving off the leading `1/\sqrt{2}` of the Hadamard gate.
Parameters
----------
normalize : bool
Should the Hadamard gate include the `1/\sqrt{2}` normalization factor?
When True, the Hadamard gate will have the `1/\sqrt{2}`. When False, the
Hadamard gate will not have this factor.
"""
global _normalized
_normalized = normalize
def _validate_targets_controls(tandc):
tandc = list(tandc)
# Check for integers
for bit in tandc:
if not bit.is_Integer and not bit.is_Symbol:
raise TypeError('Integer expected, got: %r' % tandc[bit])
# Detect duplicates
if len(list(set(tandc))) != len(tandc):
raise QuantumError(
'Target/control qubits in a gate cannot be duplicated'
)
class Gate(UnitaryOperator):
"""Non-controlled unitary gate operator that acts on qubits.
This is a general abstract gate that needs to be subclassed to do anything
useful.
Parameters
----------
label : tuple, int
A list of the target qubits (as ints) that the gate will apply to.
Examples
========
"""
_label_separator = ','
gate_name = 'G'
gate_name_latex = 'G'
#-------------------------------------------------------------------------
# Initialization/creation
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
args = Tuple(*UnitaryOperator._eval_args(args))
_validate_targets_controls(args)
return args
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**(_max(args) + 1)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def nqubits(self):
"""The total number of qubits this gate acts on.
For controlled gate subclasses this includes both target and control
qubits, so that, for examples the CNOT gate acts on 2 qubits.
"""
return len(self.targets)
@property
def min_qubits(self):
"""The minimum number of qubits this gate needs to act on."""
return _max(self.targets) + 1
@property
def targets(self):
"""A tuple of target qubits."""
return self.label
@property
def gate_name_plot(self):
return r'$%s$' % self.gate_name_latex
#-------------------------------------------------------------------------
# Gate methods
#-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'):
"""The matrix represenation of the target part of the gate.
Parameters
----------
format : str
The format string ('sympy','numpy', etc.)
"""
raise NotImplementedError(
'get_target_matrix is not implemented in Gate.')
#-------------------------------------------------------------------------
# Apply
#-------------------------------------------------------------------------
def _apply_operator_IntQubit(self, qubits, **options):
"""Redirect an apply from IntQubit to Qubit"""
return self._apply_operator_Qubit(qubits, **options)
def _apply_operator_Qubit(self, qubits, **options):
"""Apply this gate to a Qubit."""
# Check number of qubits this gate acts on.
if qubits.nqubits < self.min_qubits:
raise QuantumError(
'Gate needs a minimum of %r qubits to act on, got: %r' %
(self.min_qubits, qubits.nqubits)
)
# If the controls are not met, just return
if isinstance(self, CGate):
if not self.eval_controls(qubits):
return qubits
targets = self.targets
target_matrix = self.get_target_matrix(format='sympy')
# Find which column of the target matrix this applies to.
column_index = 0
n = 1
for target in targets:
column_index += n*qubits[target]
n = n << 1
column = target_matrix[:, int(column_index)]
# Now apply each column element to the qubit.
result = 0
for index in range(column.rows):
# TODO: This can be optimized to reduce the number of Qubit
# creations. We should simply manipulate the raw list of qubit
# values and then build the new Qubit object once.
# Make a copy of the incoming qubits.
new_qubit = qubits.__class__(*qubits.args)
# Flip the bits that need to be flipped.
for bit in range(len(targets)):
if new_qubit[targets[bit]] != (index >> bit) & 1:
new_qubit = new_qubit.flip(targets[bit])
# The value in that row and column times the flipped-bit qubit
# is the result for that part.
result += column[index]*new_qubit
return result
#-------------------------------------------------------------------------
# Represent
#-------------------------------------------------------------------------
def _represent_default_basis(self, **options):
return self._represent_ZGate(None, **options)
def _represent_ZGate(self, basis, **options):
format = options.get('format', 'sympy')
nqubits = options.get('nqubits', 0)
if nqubits == 0:
raise QuantumError(
'The number of qubits must be given as nqubits.')
# Make sure we have enough qubits for the gate.
if nqubits < self.min_qubits:
raise QuantumError(
'The number of qubits %r is too small for the gate.' % nqubits
)
target_matrix = self.get_target_matrix(format)
targets = self.targets
if isinstance(self, CGate):
controls = self.controls
else:
controls = []
m = represent_zbasis(
controls, targets, target_matrix, nqubits, format
)
return m
#-------------------------------------------------------------------------
# Print methods
#-------------------------------------------------------------------------
def _sympystr(self, printer, *args):
label = self._print_label(printer, *args)
return '%s(%s)' % (self.gate_name, label)
def _pretty(self, printer, *args):
a = stringPict(self.gate_name)
b = self._print_label_pretty(printer, *args)
return self._print_subscript_pretty(a, b)
def _latex(self, printer, *args):
label = self._print_label(printer, *args)
return '%s_{%s}' % (self.gate_name_latex, label)
def plot_gate(self, axes, gate_idx, gate_grid, wire_grid):
raise NotImplementedError('plot_gate is not implemented.')
class CGate(Gate):
"""A general unitary gate with control qubits.
A general control gate applies a target gate to a set of targets if all
of the control qubits have a particular values (set by
``CGate.control_value``).
Parameters
----------
label : tuple
The label in this case has the form (controls, gate), where controls
is a tuple/list of control qubits (as ints) and gate is a ``Gate``
instance that is the target operator.
Examples
========
"""
gate_name = 'C'
gate_name_latex = 'C'
# The values this class controls for.
control_value = _S.One
simplify_cgate = False
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
# _eval_args has the right logic for the controls argument.
controls = args[0]
gate = args[1]
if not is_sequence(controls):
controls = (controls,)
controls = UnitaryOperator._eval_args(controls)
_validate_targets_controls(chain(controls, gate.targets))
return (Tuple(*controls), gate)
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**_max(_max(args[0]) + 1, args[1].min_qubits)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def nqubits(self):
"""The total number of qubits this gate acts on.
For controlled gate subclasses this includes both target and control
qubits, so that, for examples the CNOT gate acts on 2 qubits.
"""
return len(self.targets) + len(self.controls)
@property
def min_qubits(self):
"""The minimum number of qubits this gate needs to act on."""
return _max(_max(self.controls), _max(self.targets)) + 1
@property
def targets(self):
"""A tuple of target qubits."""
return self.gate.targets
@property
def controls(self):
"""A tuple of control qubits."""
return tuple(self.label[0])
@property
def gate(self):
"""The non-controlled gate that will be applied to the targets."""
return self.label[1]
#-------------------------------------------------------------------------
# Gate methods
#-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'):
return self.gate.get_target_matrix(format)
def eval_controls(self, qubit):
"""Return True/False to indicate if the controls are satisfied."""
return all(qubit[bit] == self.control_value for bit in self.controls)
def decompose(self, **options):
"""Decompose the controlled gate into CNOT and single qubits gates."""
if len(self.controls) == 1:
c = self.controls[0]
t = self.gate.targets[0]
if isinstance(self.gate, YGate):
g1 = PhaseGate(t)
g2 = CNotGate(c, t)
g3 = PhaseGate(t)
g4 = ZGate(t)
return g1*g2*g3*g4
if isinstance(self.gate, ZGate):
g1 = HadamardGate(t)
g2 = CNotGate(c, t)
g3 = HadamardGate(t)
return g1*g2*g3
else:
return self
#-------------------------------------------------------------------------
# Print methods
#-------------------------------------------------------------------------
def _print_label(self, printer, *args):
controls = self._print_sequence(self.controls, ',', printer, *args)
gate = printer._print(self.gate, *args)
return '(%s),%s' % (controls, gate)
def _pretty(self, printer, *args):
controls = self._print_sequence_pretty(
self.controls, ',', printer, *args)
gate = printer._print(self.gate)
gate_name = stringPict(self.gate_name)
first = self._print_subscript_pretty(gate_name, controls)
gate = self._print_parens_pretty(gate)
final = prettyForm(*first.right(gate))
return final
def _latex(self, printer, *args):
controls = self._print_sequence(self.controls, ',', printer, *args)
gate = printer._print(self.gate, *args)
return r'%s_{%s}{\left(%s\right)}' % \
(self.gate_name_latex, controls, gate)
def plot_gate(self, circ_plot, gate_idx):
"""
Plot the controlled gate. If *simplify_cgate* is true, simplify
C-X and C-Z gates into their more familiar forms.
"""
min_wire = int(_min(chain(self.controls, self.targets)))
max_wire = int(_max(chain(self.controls, self.targets)))
circ_plot.control_line(gate_idx, min_wire, max_wire)
for c in self.controls:
circ_plot.control_point(gate_idx, int(c))
if self.simplify_cgate:
if self.gate.gate_name == 'X':
self.gate.plot_gate_plus(circ_plot, gate_idx)
elif self.gate.gate_name == 'Z':
circ_plot.control_point(gate_idx, self.targets[0])
else:
self.gate.plot_gate(circ_plot, gate_idx)
else:
self.gate.plot_gate(circ_plot, gate_idx)
#-------------------------------------------------------------------------
# Miscellaneous
#-------------------------------------------------------------------------
def _eval_dagger(self):
if isinstance(self.gate, HermitianOperator):
return self
else:
return Gate._eval_dagger(self)
def _eval_inverse(self):
if isinstance(self.gate, HermitianOperator):
return self
else:
return Gate._eval_inverse(self)
def _eval_power(self, exp):
if isinstance(self.gate, HermitianOperator):
if exp == -1:
return Gate._eval_power(self, exp)
elif abs(exp) % 2 == 0:
return self*(Gate._eval_inverse(self))
else:
return self
else:
return Gate._eval_power(self, exp)
class CGateS(CGate):
"""Version of CGate that allows gate simplifications.
I.e. cnot looks like an oplus, cphase has dots, etc.
"""
simplify_cgate=True
class UGate(Gate):
"""General gate specified by a set of targets and a target matrix.
Parameters
----------
label : tuple
A tuple of the form (targets, U), where targets is a tuple of the
target qubits and U is a unitary matrix with dimension of
len(targets).
"""
gate_name = 'U'
gate_name_latex = 'U'
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
targets = args[0]
if not is_sequence(targets):
targets = (targets,)
targets = Gate._eval_args(targets)
_validate_targets_controls(targets)
mat = args[1]
if not isinstance(mat, MatrixBase):
raise TypeError('Matrix expected, got: %r' % mat)
#make sure this matrix is of a Basic type
mat = _sympify(mat)
dim = 2**len(targets)
if not all(dim == shape for shape in mat.shape):
raise IndexError(
'Number of targets must match the matrix size: %r %r' %
(targets, mat)
)
return (targets, mat)
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**(_max(args[0]) + 1)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def targets(self):
"""A tuple of target qubits."""
return tuple(self.label[0])
#-------------------------------------------------------------------------
# Gate methods
#-------------------------------------------------------------------------
def get_target_matrix(self, format='sympy'):
"""The matrix rep. of the target part of the gate.
Parameters
----------
format : str
The format string ('sympy','numpy', etc.)
"""
return self.label[1]
#-------------------------------------------------------------------------
# Print methods
#-------------------------------------------------------------------------
def _pretty(self, printer, *args):
targets = self._print_sequence_pretty(
self.targets, ',', printer, *args)
gate_name = stringPict(self.gate_name)
return self._print_subscript_pretty(gate_name, targets)
def _latex(self, printer, *args):
targets = self._print_sequence(self.targets, ',', printer, *args)
return r'%s_{%s}' % (self.gate_name_latex, targets)
def plot_gate(self, circ_plot, gate_idx):
circ_plot.one_qubit_box(
self.gate_name_plot,
gate_idx, int(self.targets[0])
)
class OneQubitGate(Gate):
"""A single qubit unitary gate base class."""
nqubits = _S.One
def plot_gate(self, circ_plot, gate_idx):
circ_plot.one_qubit_box(
self.gate_name_plot,
gate_idx, int(self.targets[0])
)
def _eval_commutator(self, other, **hints):
if isinstance(other, OneQubitGate):
if self.targets != other.targets or self.__class__ == other.__class__:
return _S.Zero
return Operator._eval_commutator(self, other, **hints)
def _eval_anticommutator(self, other, **hints):
if isinstance(other, OneQubitGate):
if self.targets != other.targets or self.__class__ == other.__class__:
return Integer(2)*self*other
return Operator._eval_anticommutator(self, other, **hints)
class TwoQubitGate(Gate):
"""A two qubit unitary gate base class."""
nqubits = Integer(2)
#-----------------------------------------------------------------------------
# Single Qubit Gates
#-----------------------------------------------------------------------------
class IdentityGate(OneQubitGate):
"""The single qubit identity gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = '1'
gate_name_latex = '1'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('eye2', format)
def _eval_commutator(self, other, **hints):
return _S.Zero
def _eval_anticommutator(self, other, **hints):
return Integer(2)*other
class HadamardGate(HermitianOperator, OneQubitGate):
"""The single qubit Hadamard gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
>>> from sympy import sqrt
>>> from sympy.physics.quantum.qubit import Qubit
>>> from sympy.physics.quantum.gate import HadamardGate
>>> from sympy.physics.quantum.qapply import qapply
>>> qapply(HadamardGate(0)*Qubit('1'))
sqrt(2)*|0>/2 - sqrt(2)*|1>/2
>>> # Hadamard on bell state, applied on 2 qubits.
>>> psi = 1/sqrt(2)*(Qubit('00')+Qubit('11'))
>>> qapply(HadamardGate(0)*HadamardGate(1)*psi)
sqrt(2)*|00>/2 + sqrt(2)*|11>/2
"""
gate_name = 'H'
gate_name_latex = 'H'
def get_target_matrix(self, format='sympy'):
if _normalized:
return matrix_cache.get_matrix('H', format)
else:
return matrix_cache.get_matrix('Hsqrt2', format)
def _eval_commutator_XGate(self, other, **hints):
return I*sqrt(2)*YGate(self.targets[0])
def _eval_commutator_YGate(self, other, **hints):
return I*sqrt(2)*(ZGate(self.targets[0]) - XGate(self.targets[0]))
def _eval_commutator_ZGate(self, other, **hints):
return -I*sqrt(2)*YGate(self.targets[0])
def _eval_anticommutator_XGate(self, other, **hints):
return sqrt(2)*IdentityGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return _S.Zero
def _eval_anticommutator_ZGate(self, other, **hints):
return sqrt(2)*IdentityGate(self.targets[0])
class XGate(HermitianOperator, OneQubitGate):
"""The single qubit X, or NOT, gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = 'X'
gate_name_latex = 'X'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('X', format)
def plot_gate(self, circ_plot, gate_idx):
OneQubitGate.plot_gate(self,circ_plot,gate_idx)
def plot_gate_plus(self, circ_plot, gate_idx):
circ_plot.not_point(
gate_idx, int(self.label[0])
)
def _eval_commutator_YGate(self, other, **hints):
return Integer(2)*I*ZGate(self.targets[0])
def _eval_anticommutator_XGate(self, other, **hints):
return Integer(2)*IdentityGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return _S.Zero
def _eval_anticommutator_ZGate(self, other, **hints):
return _S.Zero
class YGate(HermitianOperator, OneQubitGate):
"""The single qubit Y gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = 'Y'
gate_name_latex = 'Y'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('Y', format)
def _eval_commutator_ZGate(self, other, **hints):
return Integer(2)*I*XGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return Integer(2)*IdentityGate(self.targets[0])
def _eval_anticommutator_ZGate(self, other, **hints):
return _S.Zero
class ZGate(HermitianOperator, OneQubitGate):
"""The single qubit Z gate.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = 'Z'
gate_name_latex = 'Z'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('Z', format)
def _eval_commutator_XGate(self, other, **hints):
return Integer(2)*I*YGate(self.targets[0])
def _eval_anticommutator_YGate(self, other, **hints):
return _S.Zero
class PhaseGate(OneQubitGate):
"""The single qubit phase, or S, gate.
This gate rotates the phase of the state by pi/2 if the state is ``|1>`` and
does nothing if the state is ``|0>``.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = 'S'
gate_name_latex = 'S'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('S', format)
def _eval_commutator_ZGate(self, other, **hints):
return _S.Zero
def _eval_commutator_TGate(self, other, **hints):
return _S.Zero
class TGate(OneQubitGate):
"""The single qubit pi/8 gate.
This gate rotates the phase of the state by pi/4 if the state is ``|1>`` and
does nothing if the state is ``|0>``.
Parameters
----------
target : int
The target qubit this gate will apply to.
Examples
========
"""
gate_name = 'T'
gate_name_latex = 'T'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('T', format)
def _eval_commutator_ZGate(self, other, **hints):
return _S.Zero
def _eval_commutator_PhaseGate(self, other, **hints):
return _S.Zero
# Aliases for gate names.
H = HadamardGate
X = XGate
Y = YGate
Z = ZGate
T = TGate
Phase = S = PhaseGate
#-----------------------------------------------------------------------------
# 2 Qubit Gates
#-----------------------------------------------------------------------------
class CNotGate(HermitianOperator, CGate, TwoQubitGate):
"""Two qubit controlled-NOT.
This gate performs the NOT or X gate on the target qubit if the control
qubits all have the value 1.
Parameters
----------
label : tuple
A tuple of the form (control, target).
Examples
========
>>> from sympy.physics.quantum.gate import CNOT
>>> from sympy.physics.quantum.qapply import qapply
>>> from sympy.physics.quantum.qubit import Qubit
>>> c = CNOT(1,0)
>>> qapply(c*Qubit('10')) # note that qubits are indexed from right to left
|11>
"""
gate_name = 'CNOT'
gate_name_latex = r'\text{CNOT}'
simplify_cgate = True
#-------------------------------------------------------------------------
# Initialization
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
args = Gate._eval_args(args)
return args
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**(_max(args) + 1)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def min_qubits(self):
"""The minimum number of qubits this gate needs to act on."""
return _max(self.label) + 1
@property
def targets(self):
"""A tuple of target qubits."""
return (self.label[1],)
@property
def controls(self):
"""A tuple of control qubits."""
return (self.label[0],)
@property
def gate(self):
"""The non-controlled gate that will be applied to the targets."""
return XGate(self.label[1])
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
# The default printing of Gate works better than those of CGate, so we
# go around the overridden methods in CGate.
def _print_label(self, printer, *args):
return Gate._print_label(self, printer, *args)
def _pretty(self, printer, *args):
return Gate._pretty(self, printer, *args)
def _latex(self, printer, *args):
return Gate._latex(self, printer, *args)
#-------------------------------------------------------------------------
# Commutator/AntiCommutator
#-------------------------------------------------------------------------
def _eval_commutator_ZGate(self, other, **hints):
"""[CNOT(i, j), Z(i)] == 0."""
if self.controls[0] == other.targets[0]:
return _S.Zero
else:
raise NotImplementedError('Commutator not implemented: %r' % other)
def _eval_commutator_TGate(self, other, **hints):
"""[CNOT(i, j), T(i)] == 0."""
return self._eval_commutator_ZGate(other, **hints)
def _eval_commutator_PhaseGate(self, other, **hints):
"""[CNOT(i, j), S(i)] == 0."""
return self._eval_commutator_ZGate(other, **hints)
def _eval_commutator_XGate(self, other, **hints):
"""[CNOT(i, j), X(j)] == 0."""
if self.targets[0] == other.targets[0]:
return _S.Zero
else:
raise NotImplementedError('Commutator not implemented: %r' % other)
def _eval_commutator_CNotGate(self, other, **hints):
"""[CNOT(i, j), CNOT(i,k)] == 0."""
if self.controls[0] == other.controls[0]:
return _S.Zero
else:
raise NotImplementedError('Commutator not implemented: %r' % other)
class SwapGate(TwoQubitGate):
"""Two qubit SWAP gate.
This gate swap the values of the two qubits.
Parameters
----------
label : tuple
A tuple of the form (target1, target2).
Examples
========
"""
gate_name = 'SWAP'
gate_name_latex = r'\text{SWAP}'
def get_target_matrix(self, format='sympy'):
return matrix_cache.get_matrix('SWAP', format)
def decompose(self, **options):
"""Decompose the SWAP gate into CNOT gates."""
i, j = self.targets[0], self.targets[1]
g1 = CNotGate(i, j)
g2 = CNotGate(j, i)
return g1*g2*g1
def plot_gate(self, circ_plot, gate_idx):
min_wire = int(_min(self.targets))
max_wire = int(_max(self.targets))
circ_plot.control_line(gate_idx, min_wire, max_wire)
circ_plot.swap_point(gate_idx, min_wire)
circ_plot.swap_point(gate_idx, max_wire)
def _represent_ZGate(self, basis, **options):
"""Represent the SWAP gate in the computational basis.
The following representation is used to compute this:
SWAP = |1><1|x|1><1| + |0><0|x|0><0| + |1><0|x|0><1| + |0><1|x|1><0|
"""
format = options.get('format', 'sympy')
targets = [int(t) for t in self.targets]
min_target = _min(targets)
max_target = _max(targets)
nqubits = options.get('nqubits', self.min_qubits)
op01 = matrix_cache.get_matrix('op01', format)
op10 = matrix_cache.get_matrix('op10', format)
op11 = matrix_cache.get_matrix('op11', format)
op00 = matrix_cache.get_matrix('op00', format)
eye2 = matrix_cache.get_matrix('eye2', format)
result = None
for i, j in ((op01, op10), (op10, op01), (op00, op00), (op11, op11)):
product = nqubits*[eye2]
product[nqubits - min_target - 1] = i
product[nqubits - max_target - 1] = j
new_result = matrix_tensor_product(*product)
if result is None:
result = new_result
else:
result = result + new_result
return result
# Aliases for gate names.
CNOT = CNotGate
SWAP = SwapGate
def CPHASE(a,b): return CGateS((a,),Z(b))
#-----------------------------------------------------------------------------
# Represent
#-----------------------------------------------------------------------------
def represent_zbasis(controls, targets, target_matrix, nqubits, format='sympy'):
"""Represent a gate with controls, targets and target_matrix.
This function does the low-level work of representing gates as matrices
in the standard computational basis (ZGate). Currently, we support two
main cases:
1. One target qubit and no control qubits.
2. One target qubits and multiple control qubits.
For the base of multiple controls, we use the following expression [1]:
1_{2**n} + (|1><1|)^{(n-1)} x (target-matrix - 1_{2})
Parameters
----------
controls : list, tuple
A sequence of control qubits.
targets : list, tuple
A sequence of target qubits.
target_matrix : sympy.Matrix, numpy.matrix, scipy.sparse
The matrix form of the transformation to be performed on the target
qubits. The format of this matrix must match that passed into
the `format` argument.
nqubits : int
The total number of qubits used for the representation.
format : str
The format of the final matrix ('sympy', 'numpy', 'scipy.sparse').
Examples
========
References
----------
[1] http://www.johnlapeyre.com/qinf/qinf_html/node6.html.
"""
controls = [int(x) for x in controls]
targets = [int(x) for x in targets]
nqubits = int(nqubits)
# This checks for the format as well.
op11 = matrix_cache.get_matrix('op11', format)
eye2 = matrix_cache.get_matrix('eye2', format)
# Plain single qubit case
if len(controls) == 0 and len(targets) == 1:
product = []
bit = targets[0]
# Fill product with [I1,Gate,I2] such that the unitaries,
# I, cause the gate to be applied to the correct Qubit
if bit != nqubits - 1:
product.append(matrix_eye(2**(nqubits - bit - 1), format=format))
product.append(target_matrix)
if bit != 0:
product.append(matrix_eye(2**bit, format=format))
return matrix_tensor_product(*product)
# Single target, multiple controls.
elif len(targets) == 1 and len(controls) >= 1:
target = targets[0]
# Build the non-trivial part.
product2 = []
for i in range(nqubits):
product2.append(matrix_eye(2, format=format))
for control in controls:
product2[nqubits - 1 - control] = op11
product2[nqubits - 1 - target] = target_matrix - eye2
return matrix_eye(2**nqubits, format=format) + \
matrix_tensor_product(*product2)
# Multi-target, multi-control is not yet implemented.
else:
raise NotImplementedError(
'The representation of multi-target, multi-control gates '
'is not implemented.'
)
#-----------------------------------------------------------------------------
# Gate manipulation functions.
#-----------------------------------------------------------------------------
def gate_simp(circuit):
"""Simplifies gates symbolically
It first sorts gates using gate_sort. It then applies basic
simplification rules to the circuit, e.g., XGate**2 = Identity
"""
# Bubble sort out gates that commute.
circuit = gate_sort(circuit)
# Do simplifications by subing a simplification into the first element
# which can be simplified. We recursively call gate_simp with new circuit
# as input more simplifications exist.
if isinstance(circuit, Add):
return sum(gate_simp(t) for t in circuit.args)
elif isinstance(circuit, Mul):
circuit_args = circuit.args
elif isinstance(circuit, Pow):
b, e = circuit.as_base_exp()
circuit_args = (gate_simp(b)**e,)
else:
return circuit
# Iterate through each element in circuit, simplify if possible.
for i in range(len(circuit_args)):
# H,X,Y or Z squared is 1.
# T**2 = S, S**2 = Z
if isinstance(circuit_args[i], Pow):
if isinstance(circuit_args[i].base,
(HadamardGate, XGate, YGate, ZGate)) \
and isinstance(circuit_args[i].exp, Number):
# Build a new circuit taking replacing the
# H,X,Y,Z squared with one.
newargs = (circuit_args[:i] +
(circuit_args[i].base**(circuit_args[i].exp % 2),) +
circuit_args[i + 1:])
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
elif isinstance(circuit_args[i].base, PhaseGate):
# Build a new circuit taking old circuit but splicing
# in simplification.
newargs = circuit_args[:i]
# Replace PhaseGate**2 with ZGate.
newargs = newargs + (ZGate(circuit_args[i].base.args[0])**
(Integer(circuit_args[i].exp/2)), circuit_args[i].base**
(circuit_args[i].exp % 2))
# Append the last elements.
newargs = newargs + circuit_args[i + 1:]
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
elif isinstance(circuit_args[i].base, TGate):
# Build a new circuit taking all the old elements.
newargs = circuit_args[:i]
# Put an Phasegate in place of any TGate**2.
newargs = newargs + (PhaseGate(circuit_args[i].base.args[0])**
Integer(circuit_args[i].exp/2), circuit_args[i].base**
(circuit_args[i].exp % 2))
# Append the last elements.
newargs = newargs + circuit_args[i + 1:]
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
return circuit
def gate_sort(circuit):
"""Sorts the gates while keeping track of commutation relations
This function uses a bubble sort to rearrange the order of gate
application. Keeps track of Quantum computations special commutation
relations (e.g. things that apply to the same Qubit do not commute with
each other)
circuit is the Mul of gates that are to be sorted.
"""
# Make sure we have an Add or Mul.
if isinstance(circuit, Add):
return sum(gate_sort(t) for t in circuit.args)
if isinstance(circuit, Pow):
return gate_sort(circuit.base)**circuit.exp
elif isinstance(circuit, Gate):
return circuit
if not isinstance(circuit, Mul):
return circuit
changes = True
while changes:
changes = False
circ_array = circuit.args
for i in range(len(circ_array) - 1):
# Go through each element and switch ones that are in wrong order
if isinstance(circ_array[i], (Gate, Pow)) and \
isinstance(circ_array[i + 1], (Gate, Pow)):
# If we have a Pow object, look at only the base
first_base, first_exp = circ_array[i].as_base_exp()
second_base, second_exp = circ_array[i + 1].as_base_exp()
# Use SymPy's hash based sorting. This is not mathematical
# sorting, but is rather based on comparing hashes of objects.
# See Basic.compare for details.
if first_base.compare(second_base) > 0:
if Commutator(first_base, second_base).doit() == 0:
new_args = (circuit.args[:i] + (circuit.args[i + 1],) +
(circuit.args[i],) + circuit.args[i + 2:])
circuit = Mul(*new_args)
changes = True
break
if AntiCommutator(first_base, second_base).doit() == 0:
new_args = (circuit.args[:i] + (circuit.args[i + 1],) +
(circuit.args[i],) + circuit.args[i + 2:])
sign = _S.NegativeOne**(first_exp*second_exp)
circuit = sign*Mul(*new_args)
changes = True
break
return circuit
#-----------------------------------------------------------------------------
# Utility functions
#-----------------------------------------------------------------------------
def random_circuit(ngates, nqubits, gate_space=(X, Y, Z, S, T, H, CNOT, SWAP)):
"""Return a random circuit of ngates and nqubits.
This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP)
gates.
Parameters
----------
ngates : int
The number of gates in the circuit.
nqubits : int
The number of qubits in the circuit.
gate_space : tuple
A tuple of the gate classes that will be used in the circuit.
Repeating gate classes multiple times in this tuple will increase
the frequency they appear in the random circuit.
"""
qubit_space = range(nqubits)
result = []
for i in range(ngates):
g = random.choice(gate_space)
if g == CNotGate or g == SwapGate:
qubits = random.sample(qubit_space, 2)
g = g(*qubits)
else:
qubit = random.choice(qubit_space)
g = g(qubit)
result.append(g)
return Mul(*result)
def zx_basis_transform(self, format='sympy'):
"""Transformation matrix from Z to X basis."""
return matrix_cache.get_matrix('ZX', format)
def zy_basis_transform(self, format='sympy'):
"""Transformation matrix from Z to Y basis."""
return matrix_cache.get_matrix('ZY', format)
|
6ff9849b7abe8196c8579c79565ea9cdfc89cadaa5a7f6bdf31a5fb1ed807e02 | """Logic for representing operators in state in various bases.
TODO:
* Get represent working with continuous hilbert spaces.
* Document default basis functionality.
"""
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.mul import Mul
from sympy.core.numbers import I
from sympy.core.power import Pow
from sympy.integrals.integrals import integrate
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.commutator import Commutator
from sympy.physics.quantum.anticommutator import AntiCommutator
from sympy.physics.quantum.innerproduct import InnerProduct
from sympy.physics.quantum.qexpr import QExpr
from sympy.physics.quantum.tensorproduct import TensorProduct
from sympy.physics.quantum.matrixutils import flatten_scalar
from sympy.physics.quantum.state import KetBase, BraBase, StateBase
from sympy.physics.quantum.operator import Operator, OuterProduct
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.operatorset import operators_to_state, state_to_operators
__all__ = [
'represent',
'rep_innerproduct',
'rep_expectation',
'integrate_result',
'get_basis',
'enumerate_states'
]
#-----------------------------------------------------------------------------
# Represent
#-----------------------------------------------------------------------------
def _sympy_to_scalar(e):
"""Convert from a SymPy scalar to a Python scalar."""
if isinstance(e, Expr):
if e.is_Integer:
return int(e)
elif e.is_Float:
return float(e)
elif e.is_Rational:
return float(e)
elif e.is_Number or e.is_NumberSymbol or e == I:
return complex(e)
raise TypeError('Expected number, got: %r' % e)
def represent(expr, **options):
"""Represent the quantum expression in the given basis.
In quantum mechanics abstract states and operators can be represented in
various basis sets. Under this operation the follow transforms happen:
* Ket -> column vector or function
* Bra -> row vector of function
* Operator -> matrix or differential operator
This function is the top-level interface for this action.
This function walks the SymPy expression tree looking for ``QExpr``
instances that have a ``_represent`` method. This method is then called
and the object is replaced by the representation returned by this method.
By default, the ``_represent`` method will dispatch to other methods
that handle the representation logic for a particular basis set. The
naming convention for these methods is the following::
def _represent_FooBasis(self, e, basis, **options)
This function will have the logic for representing instances of its class
in the basis set having a class named ``FooBasis``.
Parameters
==========
expr : Expr
The expression to represent.
basis : Operator, basis set
An object that contains the information about the basis set. If an
operator is used, the basis is assumed to be the orthonormal
eigenvectors of that operator. In general though, the basis argument
can be any object that contains the basis set information.
options : dict
Key/value pairs of options that are passed to the underlying method
that finds the representation. These options can be used to
control how the representation is done. For example, this is where
the size of the basis set would be set.
Returns
=======
e : Expr
The SymPy expression of the represented quantum expression.
Examples
========
Here we subclass ``Operator`` and ``Ket`` to create the z-spin operator
and its spin 1/2 up eigenstate. By defining the ``_represent_SzOp``
method, the ket can be represented in the z-spin basis.
>>> from sympy.physics.quantum import Operator, represent, Ket
>>> from sympy import Matrix
>>> class SzUpKet(Ket):
... def _represent_SzOp(self, basis, **options):
... return Matrix([1,0])
...
>>> class SzOp(Operator):
... pass
...
>>> sz = SzOp('Sz')
>>> up = SzUpKet('up')
>>> represent(up, basis=sz)
Matrix([
[1],
[0]])
Here we see an example of representations in a continuous
basis. We see that the result of representing various combinations
of cartesian position operators and kets give us continuous
expressions involving DiracDelta functions.
>>> from sympy.physics.quantum.cartesian import XOp, XKet, XBra
>>> X = XOp()
>>> x = XKet()
>>> y = XBra('y')
>>> represent(X*x)
x*DiracDelta(x - x_2)
>>> represent(X*x*y)
x*DiracDelta(x - x_3)*DiracDelta(x_1 - y)
"""
format = options.get('format', 'sympy')
if format == 'numpy':
import numpy as np
if isinstance(expr, QExpr) and not isinstance(expr, OuterProduct):
options['replace_none'] = False
temp_basis = get_basis(expr, **options)
if temp_basis is not None:
options['basis'] = temp_basis
try:
return expr._represent(**options)
except NotImplementedError as strerr:
#If no _represent_FOO method exists, map to the
#appropriate basis state and try
#the other methods of representation
options['replace_none'] = True
if isinstance(expr, (KetBase, BraBase)):
try:
return rep_innerproduct(expr, **options)
except NotImplementedError:
raise NotImplementedError(strerr)
elif isinstance(expr, Operator):
try:
return rep_expectation(expr, **options)
except NotImplementedError:
raise NotImplementedError(strerr)
else:
raise NotImplementedError(strerr)
elif isinstance(expr, Add):
result = represent(expr.args[0], **options)
for args in expr.args[1:]:
# scipy.sparse doesn't support += so we use plain = here.
result = result + represent(args, **options)
return result
elif isinstance(expr, Pow):
base, exp = expr.as_base_exp()
if format in ('numpy', 'scipy.sparse'):
exp = _sympy_to_scalar(exp)
base = represent(base, **options)
# scipy.sparse doesn't support negative exponents
# and warns when inverting a matrix in csr format.
if format == 'scipy.sparse' and exp < 0:
from scipy.sparse.linalg import inv
exp = - exp
base = inv(base.tocsc()).tocsr()
if format == 'numpy':
return np.linalg.matrix_power(base, exp)
return base ** exp
elif isinstance(expr, TensorProduct):
new_args = [represent(arg, **options) for arg in expr.args]
return TensorProduct(*new_args)
elif isinstance(expr, Dagger):
return Dagger(represent(expr.args[0], **options))
elif isinstance(expr, Commutator):
A = expr.args[0]
B = expr.args[1]
return represent(Mul(A, B) - Mul(B, A), **options)
elif isinstance(expr, AntiCommutator):
A = expr.args[0]
B = expr.args[1]
return represent(Mul(A, B) + Mul(B, A), **options)
elif isinstance(expr, InnerProduct):
return represent(Mul(expr.bra, expr.ket), **options)
elif not isinstance(expr, (Mul, OuterProduct)):
# For numpy and scipy.sparse, we can only handle numerical prefactors.
if format in ('numpy', 'scipy.sparse'):
return _sympy_to_scalar(expr)
return expr
if not isinstance(expr, (Mul, OuterProduct)):
raise TypeError('Mul expected, got: %r' % expr)
if "index" in options:
options["index"] += 1
else:
options["index"] = 1
if "unities" not in options:
options["unities"] = []
result = represent(expr.args[-1], **options)
last_arg = expr.args[-1]
for arg in reversed(expr.args[:-1]):
if isinstance(last_arg, Operator):
options["index"] += 1
options["unities"].append(options["index"])
elif isinstance(last_arg, BraBase) and isinstance(arg, KetBase):
options["index"] += 1
elif isinstance(last_arg, KetBase) and isinstance(arg, Operator):
options["unities"].append(options["index"])
elif isinstance(last_arg, KetBase) and isinstance(arg, BraBase):
options["unities"].append(options["index"])
next_arg = represent(arg, **options)
if format == 'numpy' and isinstance(next_arg, np.ndarray):
# Must use np.matmult to "matrix multiply" two np.ndarray
result = np.matmul(next_arg, result)
else:
result = next_arg*result
last_arg = arg
# All three matrix formats create 1 by 1 matrices when inner products of
# vectors are taken. In these cases, we simply return a scalar.
result = flatten_scalar(result)
result = integrate_result(expr, result, **options)
return result
def rep_innerproduct(expr, **options):
"""
Returns an innerproduct like representation (e.g. ``<x'|x>``) for the
given state.
Attempts to calculate inner product with a bra from the specified
basis. Should only be passed an instance of KetBase or BraBase
Parameters
==========
expr : KetBase or BraBase
The expression to be represented
Examples
========
>>> from sympy.physics.quantum.represent import rep_innerproduct
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> rep_innerproduct(XKet())
DiracDelta(x - x_1)
>>> rep_innerproduct(XKet(), basis=PxOp())
sqrt(2)*exp(-I*px_1*x/hbar)/(2*sqrt(hbar)*sqrt(pi))
>>> rep_innerproduct(PxKet(), basis=XOp())
sqrt(2)*exp(I*px*x_1/hbar)/(2*sqrt(hbar)*sqrt(pi))
"""
if not isinstance(expr, (KetBase, BraBase)):
raise TypeError("expr passed is not a Bra or Ket")
basis = get_basis(expr, **options)
if not isinstance(basis, StateBase):
raise NotImplementedError("Can't form this representation!")
if "index" not in options:
options["index"] = 1
basis_kets = enumerate_states(basis, options["index"], 2)
if isinstance(expr, BraBase):
bra = expr
ket = (basis_kets[1] if basis_kets[0].dual == expr else basis_kets[0])
else:
bra = (basis_kets[1].dual if basis_kets[0]
== expr else basis_kets[0].dual)
ket = expr
prod = InnerProduct(bra, ket)
result = prod.doit()
format = options.get('format', 'sympy')
return expr._format_represent(result, format)
def rep_expectation(expr, **options):
"""
Returns an ``<x'|A|x>`` type representation for the given operator.
Parameters
==========
expr : Operator
Operator to be represented in the specified basis
Examples
========
>>> from sympy.physics.quantum.cartesian import XOp, PxOp, PxKet
>>> from sympy.physics.quantum.represent import rep_expectation
>>> rep_expectation(XOp())
x_1*DiracDelta(x_1 - x_2)
>>> rep_expectation(XOp(), basis=PxOp())
<px_2|*X*|px_1>
>>> rep_expectation(XOp(), basis=PxKet())
<px_2|*X*|px_1>
"""
if "index" not in options:
options["index"] = 1
if not isinstance(expr, Operator):
raise TypeError("The passed expression is not an operator")
basis_state = get_basis(expr, **options)
if basis_state is None or not isinstance(basis_state, StateBase):
raise NotImplementedError("Could not get basis kets for this operator")
basis_kets = enumerate_states(basis_state, options["index"], 2)
bra = basis_kets[1].dual
ket = basis_kets[0]
return qapply(bra*expr*ket)
def integrate_result(orig_expr, result, **options):
"""
Returns the result of integrating over any unities ``(|x><x|)`` in
the given expression. Intended for integrating over the result of
representations in continuous bases.
This function integrates over any unities that may have been
inserted into the quantum expression and returns the result.
It uses the interval of the Hilbert space of the basis state
passed to it in order to figure out the limits of integration.
The unities option must be
specified for this to work.
Note: This is mostly used internally by represent(). Examples are
given merely to show the use cases.
Parameters
==========
orig_expr : quantum expression
The original expression which was to be represented
result: Expr
The resulting representation that we wish to integrate over
Examples
========
>>> from sympy import symbols, DiracDelta
>>> from sympy.physics.quantum.represent import integrate_result
>>> from sympy.physics.quantum.cartesian import XOp, XKet
>>> x_ket = XKet()
>>> X_op = XOp()
>>> x, x_1, x_2 = symbols('x, x_1, x_2')
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2))
x*DiracDelta(x - x_1)*DiracDelta(x_1 - x_2)
>>> integrate_result(X_op*x_ket, x*DiracDelta(x-x_1)*DiracDelta(x_1-x_2),
... unities=[1])
x*DiracDelta(x - x_2)
"""
if not isinstance(result, Expr):
return result
options['replace_none'] = True
if "basis" not in options:
arg = orig_expr.args[-1]
options["basis"] = get_basis(arg, **options)
elif not isinstance(options["basis"], StateBase):
options["basis"] = get_basis(orig_expr, **options)
basis = options.pop("basis", None)
if basis is None:
return result
unities = options.pop("unities", [])
if len(unities) == 0:
return result
kets = enumerate_states(basis, unities)
coords = [k.label[0] for k in kets]
for coord in coords:
if coord in result.free_symbols:
#TODO: Add support for sets of operators
basis_op = state_to_operators(basis)
start = basis_op.hilbert_space.interval.start
end = basis_op.hilbert_space.interval.end
result = integrate(result, (coord, start, end))
return result
def get_basis(expr, *, basis=None, replace_none=True, **options):
"""
Returns a basis state instance corresponding to the basis specified in
options=s. If no basis is specified, the function tries to form a default
basis state of the given expression.
There are three behaviors:
1. The basis specified in options is already an instance of StateBase. If
this is the case, it is simply returned. If the class is specified but
not an instance, a default instance is returned.
2. The basis specified is an operator or set of operators. If this
is the case, the operator_to_state mapping method is used.
3. No basis is specified. If expr is a state, then a default instance of
its class is returned. If expr is an operator, then it is mapped to the
corresponding state. If it is neither, then we cannot obtain the basis
state.
If the basis cannot be mapped, then it is not changed.
This will be called from within represent, and represent will
only pass QExpr's.
TODO (?): Support for Muls and other types of expressions?
Parameters
==========
expr : Operator or StateBase
Expression whose basis is sought
Examples
========
>>> from sympy.physics.quantum.represent import get_basis
>>> from sympy.physics.quantum.cartesian import XOp, XKet, PxOp, PxKet
>>> x = XKet()
>>> X = XOp()
>>> get_basis(x)
|x>
>>> get_basis(X)
|x>
>>> get_basis(x, basis=PxOp())
|px>
>>> get_basis(x, basis=PxKet)
|px>
"""
if basis is None and not replace_none:
return None
if basis is None:
if isinstance(expr, KetBase):
return _make_default(expr.__class__)
elif isinstance(expr, BraBase):
return _make_default(expr.dual_class())
elif isinstance(expr, Operator):
state_inst = operators_to_state(expr)
return (state_inst if state_inst is not None else None)
else:
return None
elif (isinstance(basis, Operator) or
(not isinstance(basis, StateBase) and issubclass(basis, Operator))):
state = operators_to_state(basis)
if state is None:
return None
elif isinstance(state, StateBase):
return state
else:
return _make_default(state)
elif isinstance(basis, StateBase):
return basis
elif issubclass(basis, StateBase):
return _make_default(basis)
else:
return None
def _make_default(expr):
# XXX: Catching TypeError like this is a bad way of distinguishing
# instances from classes. The logic using this function should be
# rewritten somehow.
try:
expr = expr()
except TypeError:
return expr
return expr
def enumerate_states(*args, **options):
"""
Returns instances of the given state with dummy indices appended
Operates in two different modes:
1. Two arguments are passed to it. The first is the base state which is to
be indexed, and the second argument is a list of indices to append.
2. Three arguments are passed. The first is again the base state to be
indexed. The second is the start index for counting. The final argument
is the number of kets you wish to receive.
Tries to call state._enumerate_state. If this fails, returns an empty list
Parameters
==========
args : list
See list of operation modes above for explanation
Examples
========
>>> from sympy.physics.quantum.cartesian import XBra, XKet
>>> from sympy.physics.quantum.represent import enumerate_states
>>> test = XKet('foo')
>>> enumerate_states(test, 1, 3)
[|foo_1>, |foo_2>, |foo_3>]
>>> test2 = XBra('bar')
>>> enumerate_states(test2, [4, 5, 10])
[<bar_4|, <bar_5|, <bar_10|]
"""
state = args[0]
if len(args) not in (2, 3):
raise NotImplementedError("Wrong number of arguments!")
if not isinstance(state, StateBase):
raise TypeError("First argument is not a state!")
if len(args) == 3:
num_states = args[2]
options['start_index'] = args[1]
else:
num_states = len(args[1])
options['index_list'] = args[1]
try:
ret = state._enumerate_state(num_states, **options)
except NotImplementedError:
ret = []
return ret
|
3b2573185c9c830ca7f19bec08d62cb83986f34d49bab7cac252b15431c8042a | """Qubits for quantum computing.
Todo:
* Finish implementing measurement logic. This should include POVM.
* Update docstrings.
* Update tests.
"""
import math
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.numbers import Integer
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.functions.elementary.complexes import conjugate
from sympy.functions.elementary.exponential import log
from sympy.core.basic import _sympify
from sympy.external.gmpy import SYMPY_INTS
from sympy.matrices import Matrix, zeros
from sympy.printing.pretty.stringpict import prettyForm
from sympy.physics.quantum.hilbert import ComplexSpace
from sympy.physics.quantum.state import Ket, Bra, State
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.matrixutils import (
numpy_ndarray, scipy_sparse_matrix
)
from mpmath.libmp.libintmath import bitcount
__all__ = [
'Qubit',
'QubitBra',
'IntQubit',
'IntQubitBra',
'qubit_to_matrix',
'matrix_to_qubit',
'matrix_to_density',
'measure_all',
'measure_partial',
'measure_partial_oneshot',
'measure_all_oneshot'
]
#-----------------------------------------------------------------------------
# Qubit Classes
#-----------------------------------------------------------------------------
class QubitState(State):
"""Base class for Qubit and QubitBra."""
#-------------------------------------------------------------------------
# Initialization/creation
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
# If we are passed a QubitState or subclass, we just take its qubit
# values directly.
if len(args) == 1 and isinstance(args[0], QubitState):
return args[0].qubit_values
# Turn strings into tuple of strings
if len(args) == 1 and isinstance(args[0], str):
args = tuple( S.Zero if qb == "0" else S.One for qb in args[0])
else:
args = tuple( S.Zero if qb == "0" else S.One if qb == "1" else qb for qb in args)
args = tuple(_sympify(arg) for arg in args)
# Validate input (must have 0 or 1 input)
for element in args:
if element not in (S.Zero, S.One):
raise ValueError(
"Qubit values must be 0 or 1, got: %r" % element)
return args
@classmethod
def _eval_hilbert_space(cls, args):
return ComplexSpace(2)**len(args)
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def dimension(self):
"""The number of Qubits in the state."""
return len(self.qubit_values)
@property
def nqubits(self):
return self.dimension
@property
def qubit_values(self):
"""Returns the values of the qubits as a tuple."""
return self.label
#-------------------------------------------------------------------------
# Special methods
#-------------------------------------------------------------------------
def __len__(self):
return self.dimension
def __getitem__(self, bit):
return self.qubit_values[int(self.dimension - bit - 1)]
#-------------------------------------------------------------------------
# Utility methods
#-------------------------------------------------------------------------
def flip(self, *bits):
"""Flip the bit(s) given."""
newargs = list(self.qubit_values)
for i in bits:
bit = int(self.dimension - i - 1)
if newargs[bit] == 1:
newargs[bit] = 0
else:
newargs[bit] = 1
return self.__class__(*tuple(newargs))
class Qubit(QubitState, Ket):
"""A multi-qubit ket in the computational (z) basis.
We use the normal convention that the least significant qubit is on the
right, so ``|00001>`` has a 1 in the least significant qubit.
Parameters
==========
values : list, str
The qubit values as a list of ints ([0,0,0,1,1,]) or a string ('011').
Examples
========
Create a qubit in a couple of different ways and look at their attributes:
>>> from sympy.physics.quantum.qubit import Qubit
>>> Qubit(0,0,0)
|000>
>>> q = Qubit('0101')
>>> q
|0101>
>>> q.nqubits
4
>>> len(q)
4
>>> q.dimension
4
>>> q.qubit_values
(0, 1, 0, 1)
We can flip the value of an individual qubit:
>>> q.flip(1)
|0111>
We can take the dagger of a Qubit to get a bra:
>>> from sympy.physics.quantum.dagger import Dagger
>>> Dagger(q)
<0101|
>>> type(Dagger(q))
<class 'sympy.physics.quantum.qubit.QubitBra'>
Inner products work as expected:
>>> ip = Dagger(q)*q
>>> ip
<0101|0101>
>>> ip.doit()
1
"""
@classmethod
def dual_class(self):
return QubitBra
def _eval_innerproduct_QubitBra(self, bra, **hints):
if self.label == bra.label:
return S.One
else:
return S.Zero
def _represent_default_basis(self, **options):
return self._represent_ZGate(None, **options)
def _represent_ZGate(self, basis, **options):
"""Represent this qubits in the computational basis (ZGate).
"""
_format = options.get('format', 'sympy')
n = 1
definite_state = 0
for it in reversed(self.qubit_values):
definite_state += n*it
n = n*2
result = [0]*(2**self.dimension)
result[int(definite_state)] = 1
if _format == 'sympy':
return Matrix(result)
elif _format == 'numpy':
import numpy as np
return np.array(result, dtype='complex').transpose()
elif _format == 'scipy.sparse':
from scipy import sparse
return sparse.csr_matrix(result, dtype='complex').transpose()
def _eval_trace(self, bra, **kwargs):
indices = kwargs.get('indices', [])
#sort index list to begin trace from most-significant
#qubit
sorted_idx = list(indices)
if len(sorted_idx) == 0:
sorted_idx = list(range(0, self.nqubits))
sorted_idx.sort()
#trace out for each of index
new_mat = self*bra
for i in range(len(sorted_idx) - 1, -1, -1):
# start from tracing out from leftmost qubit
new_mat = self._reduced_density(new_mat, int(sorted_idx[i]))
if (len(sorted_idx) == self.nqubits):
#in case full trace was requested
return new_mat[0]
else:
return matrix_to_density(new_mat)
def _reduced_density(self, matrix, qubit, **options):
"""Compute the reduced density matrix by tracing out one qubit.
The qubit argument should be of type Python int, since it is used
in bit operations
"""
def find_index_that_is_projected(j, k, qubit):
bit_mask = 2**qubit - 1
return ((j >> qubit) << (1 + qubit)) + (j & bit_mask) + (k << qubit)
old_matrix = represent(matrix, **options)
old_size = old_matrix.cols
#we expect the old_size to be even
new_size = old_size//2
new_matrix = Matrix().zeros(new_size)
for i in range(new_size):
for j in range(new_size):
for k in range(2):
col = find_index_that_is_projected(j, k, qubit)
row = find_index_that_is_projected(i, k, qubit)
new_matrix[i, j] += old_matrix[row, col]
return new_matrix
class QubitBra(QubitState, Bra):
"""A multi-qubit bra in the computational (z) basis.
We use the normal convention that the least significant qubit is on the
right, so ``|00001>`` has a 1 in the least significant qubit.
Parameters
==========
values : list, str
The qubit values as a list of ints ([0,0,0,1,1,]) or a string ('011').
See also
========
Qubit: Examples using qubits
"""
@classmethod
def dual_class(self):
return Qubit
class IntQubitState(QubitState):
"""A base class for qubits that work with binary representations."""
@classmethod
def _eval_args(cls, args, nqubits=None):
# The case of a QubitState instance
if len(args) == 1 and isinstance(args[0], QubitState):
return QubitState._eval_args(args)
# otherwise, args should be integer
elif not all(isinstance(a, (int, Integer)) for a in args):
raise ValueError('values must be integers, got (%s)' % (tuple(type(a) for a in args),))
# use nqubits if specified
if nqubits is not None:
if not isinstance(nqubits, (int, Integer)):
raise ValueError('nqubits must be an integer, got (%s)' % type(nqubits))
if len(args) != 1:
raise ValueError(
'too many positional arguments (%s). should be (number, nqubits=n)' % (args,))
return cls._eval_args_with_nqubits(args[0], nqubits)
# For a single argument, we construct the binary representation of
# that integer with the minimal number of bits.
if len(args) == 1 and args[0] > 1:
#rvalues is the minimum number of bits needed to express the number
rvalues = reversed(range(bitcount(abs(args[0]))))
qubit_values = [(args[0] >> i) & 1 for i in rvalues]
return QubitState._eval_args(qubit_values)
# For two numbers, the second number is the number of bits
# on which it is expressed, so IntQubit(0,5) == |00000>.
elif len(args) == 2 and args[1] > 1:
return cls._eval_args_with_nqubits(args[0], args[1])
else:
return QubitState._eval_args(args)
@classmethod
def _eval_args_with_nqubits(cls, number, nqubits):
need = bitcount(abs(number))
if nqubits < need:
raise ValueError(
'cannot represent %s with %s bits' % (number, nqubits))
qubit_values = [(number >> i) & 1 for i in reversed(range(nqubits))]
return QubitState._eval_args(qubit_values)
def as_int(self):
"""Return the numerical value of the qubit."""
number = 0
n = 1
for i in reversed(self.qubit_values):
number += n*i
n = n << 1
return number
def _print_label(self, printer, *args):
return str(self.as_int())
def _print_label_pretty(self, printer, *args):
label = self._print_label(printer, *args)
return prettyForm(label)
_print_label_repr = _print_label
_print_label_latex = _print_label
class IntQubit(IntQubitState, Qubit):
"""A qubit ket that store integers as binary numbers in qubit values.
The differences between this class and ``Qubit`` are:
* The form of the constructor.
* The qubit values are printed as their corresponding integer, rather
than the raw qubit values. The internal storage format of the qubit
values in the same as ``Qubit``.
Parameters
==========
values : int, tuple
If a single argument, the integer we want to represent in the qubit
values. This integer will be represented using the fewest possible
number of qubits.
If a pair of integers and the second value is more than one, the first
integer gives the integer to represent in binary form and the second
integer gives the number of qubits to use.
List of zeros and ones is also accepted to generate qubit by bit pattern.
nqubits : int
The integer that represents the number of qubits.
This number should be passed with keyword ``nqubits=N``.
You can use this in order to avoid ambiguity of Qubit-style tuple of bits.
Please see the example below for more details.
Examples
========
Create a qubit for the integer 5:
>>> from sympy.physics.quantum.qubit import IntQubit
>>> from sympy.physics.quantum.qubit import Qubit
>>> q = IntQubit(5)
>>> q
|5>
We can also create an ``IntQubit`` by passing a ``Qubit`` instance.
>>> q = IntQubit(Qubit('101'))
>>> q
|5>
>>> q.as_int()
5
>>> q.nqubits
3
>>> q.qubit_values
(1, 0, 1)
We can go back to the regular qubit form.
>>> Qubit(q)
|101>
Please note that ``IntQubit`` also accepts a ``Qubit``-style list of bits.
So, the code below yields qubits 3, not a single bit ``1``.
>>> IntQubit(1, 1)
|3>
To avoid ambiguity, use ``nqubits`` parameter.
Use of this keyword is recommended especially when you provide the values by variables.
>>> IntQubit(1, nqubits=1)
|1>
>>> a = 1
>>> IntQubit(a, nqubits=1)
|1>
"""
@classmethod
def dual_class(self):
return IntQubitBra
def _eval_innerproduct_IntQubitBra(self, bra, **hints):
return Qubit._eval_innerproduct_QubitBra(self, bra)
class IntQubitBra(IntQubitState, QubitBra):
"""A qubit bra that store integers as binary numbers in qubit values."""
@classmethod
def dual_class(self):
return IntQubit
#-----------------------------------------------------------------------------
# Qubit <---> Matrix conversion functions
#-----------------------------------------------------------------------------
def matrix_to_qubit(matrix):
"""Convert from the matrix repr. to a sum of Qubit objects.
Parameters
----------
matrix : Matrix, numpy.matrix, scipy.sparse
The matrix to build the Qubit representation of. This works with
SymPy matrices, numpy matrices and scipy.sparse sparse matrices.
Examples
========
Represent a state and then go back to its qubit form:
>>> from sympy.physics.quantum.qubit import matrix_to_qubit, Qubit
>>> from sympy.physics.quantum.represent import represent
>>> q = Qubit('01')
>>> matrix_to_qubit(represent(q))
|01>
"""
# Determine the format based on the type of the input matrix
format = 'sympy'
if isinstance(matrix, numpy_ndarray):
format = 'numpy'
if isinstance(matrix, scipy_sparse_matrix):
format = 'scipy.sparse'
# Make sure it is of correct dimensions for a Qubit-matrix representation.
# This logic should work with sympy, numpy or scipy.sparse matrices.
if matrix.shape[0] == 1:
mlistlen = matrix.shape[1]
nqubits = log(mlistlen, 2)
ket = False
cls = QubitBra
elif matrix.shape[1] == 1:
mlistlen = matrix.shape[0]
nqubits = log(mlistlen, 2)
ket = True
cls = Qubit
else:
raise QuantumError(
'Matrix must be a row/column vector, got %r' % matrix
)
if not isinstance(nqubits, Integer):
raise QuantumError('Matrix must be a row/column vector of size '
'2**nqubits, got: %r' % matrix)
# Go through each item in matrix, if element is non-zero, make it into a
# Qubit item times the element.
result = 0
for i in range(mlistlen):
if ket:
element = matrix[i, 0]
else:
element = matrix[0, i]
if format in ('numpy', 'scipy.sparse'):
element = complex(element)
if element != 0.0:
# Form Qubit array; 0 in bit-locations where i is 0, 1 in
# bit-locations where i is 1
qubit_array = [int(i & (1 << x) != 0) for x in range(nqubits)]
qubit_array.reverse()
result = result + element*cls(*qubit_array)
# If SymPy simplified by pulling out a constant coefficient, undo that.
if isinstance(result, (Mul, Add, Pow)):
result = result.expand()
return result
def matrix_to_density(mat):
"""
Works by finding the eigenvectors and eigenvalues of the matrix.
We know we can decompose rho by doing:
sum(EigenVal*|Eigenvect><Eigenvect|)
"""
from sympy.physics.quantum.density import Density
eigen = mat.eigenvects()
args = [[matrix_to_qubit(Matrix(
[vector, ])), x[0]] for x in eigen for vector in x[2] if x[0] != 0]
if (len(args) == 0):
return S.Zero
else:
return Density(*args)
def qubit_to_matrix(qubit, format='sympy'):
"""Converts an Add/Mul of Qubit objects into it's matrix representation
This function is the inverse of ``matrix_to_qubit`` and is a shorthand
for ``represent(qubit)``.
"""
return represent(qubit, format=format)
#-----------------------------------------------------------------------------
# Measurement
#-----------------------------------------------------------------------------
def measure_all(qubit, format='sympy', normalize=True):
"""Perform an ensemble measurement of all qubits.
Parameters
==========
qubit : Qubit, Add
The qubit to measure. This can be any Qubit or a linear combination
of them.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
=======
result : list
A list that consists of primitive states and their probabilities.
Examples
========
>>> from sympy.physics.quantum.qubit import Qubit, measure_all
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_all(q)
[(|00>, 1/4), (|01>, 1/4), (|10>, 1/4), (|11>, 1/4)]
"""
m = qubit_to_matrix(qubit, format)
if format == 'sympy':
results = []
if normalize:
m = m.normalized()
size = max(m.shape) # Max of shape to account for bra or ket
nqubits = int(math.log(size)/math.log(2))
for i in range(size):
if m[i] != 0.0:
results.append(
(Qubit(IntQubit(i, nqubits=nqubits)), m[i]*conjugate(m[i]))
)
return results
else:
raise NotImplementedError(
"This function cannot handle non-SymPy matrix formats yet"
)
def measure_partial(qubit, bits, format='sympy', normalize=True):
"""Perform a partial ensemble measure on the specified qubits.
Parameters
==========
qubits : Qubit
The qubit to measure. This can be any Qubit or a linear combination
of them.
bits : tuple
The qubits to measure.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
=======
result : list
A list that consists of primitive states and their probabilities.
Examples
========
>>> from sympy.physics.quantum.qubit import Qubit, measure_partial
>>> from sympy.physics.quantum.gate import H
>>> from sympy.physics.quantum.qapply import qapply
>>> c = H(0)*H(1)*Qubit('00')
>>> c
H(0)*H(1)*|00>
>>> q = qapply(c)
>>> measure_partial(q, (0,))
[(sqrt(2)*|00>/2 + sqrt(2)*|10>/2, 1/2), (sqrt(2)*|01>/2 + sqrt(2)*|11>/2, 1/2)]
"""
m = qubit_to_matrix(qubit, format)
if isinstance(bits, (SYMPY_INTS, Integer)):
bits = (int(bits),)
if format == 'sympy':
if normalize:
m = m.normalized()
possible_outcomes = _get_possible_outcomes(m, bits)
# Form output from function.
output = []
for outcome in possible_outcomes:
# Calculate probability of finding the specified bits with
# given values.
prob_of_outcome = 0
prob_of_outcome += (outcome.H*outcome)[0]
# If the output has a chance, append it to output with found
# probability.
if prob_of_outcome != 0:
if normalize:
next_matrix = matrix_to_qubit(outcome.normalized())
else:
next_matrix = matrix_to_qubit(outcome)
output.append((
next_matrix,
prob_of_outcome
))
return output
else:
raise NotImplementedError(
"This function cannot handle non-SymPy matrix formats yet"
)
def measure_partial_oneshot(qubit, bits, format='sympy'):
"""Perform a partial oneshot measurement on the specified qubits.
A oneshot measurement is equivalent to performing a measurement on a
quantum system. This type of measurement does not return the probabilities
like an ensemble measurement does, but rather returns *one* of the
possible resulting states. The exact state that is returned is determined
by picking a state randomly according to the ensemble probabilities.
Parameters
----------
qubits : Qubit
The qubit to measure. This can be any Qubit or a linear combination
of them.
bits : tuple
The qubits to measure.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
-------
result : Qubit
The qubit that the system collapsed to upon measurement.
"""
import random
m = qubit_to_matrix(qubit, format)
if format == 'sympy':
m = m.normalized()
possible_outcomes = _get_possible_outcomes(m, bits)
# Form output from function
random_number = random.random()
total_prob = 0
for outcome in possible_outcomes:
# Calculate probability of finding the specified bits
# with given values
total_prob += (outcome.H*outcome)[0]
if total_prob >= random_number:
return matrix_to_qubit(outcome.normalized())
else:
raise NotImplementedError(
"This function cannot handle non-SymPy matrix formats yet"
)
def _get_possible_outcomes(m, bits):
"""Get the possible states that can be produced in a measurement.
Parameters
----------
m : Matrix
The matrix representing the state of the system.
bits : tuple, list
Which bits will be measured.
Returns
-------
result : list
The list of possible states which can occur given this measurement.
These are un-normalized so we can derive the probability of finding
this state by taking the inner product with itself
"""
# This is filled with loads of dirty binary tricks...You have been warned
size = max(m.shape) # Max of shape to account for bra or ket
nqubits = int(math.log(size, 2) + .1) # Number of qubits possible
# Make the output states and put in output_matrices, nothing in them now.
# Each state will represent a possible outcome of the measurement
# Thus, output_matrices[0] is the matrix which we get when all measured
# bits return 0. and output_matrices[1] is the matrix for only the 0th
# bit being true
output_matrices = []
for i in range(1 << len(bits)):
output_matrices.append(zeros(2**nqubits, 1))
# Bitmasks will help sort how to determine possible outcomes.
# When the bit mask is and-ed with a matrix-index,
# it will determine which state that index belongs to
bit_masks = []
for bit in bits:
bit_masks.append(1 << bit)
# Make possible outcome states
for i in range(2**nqubits):
trueness = 0 # This tells us to which output_matrix this value belongs
# Find trueness
for j in range(len(bit_masks)):
if i & bit_masks[j]:
trueness += j + 1
# Put the value in the correct output matrix
output_matrices[trueness][i] = m[i]
return output_matrices
def measure_all_oneshot(qubit, format='sympy'):
"""Perform a oneshot ensemble measurement on all qubits.
A oneshot measurement is equivalent to performing a measurement on a
quantum system. This type of measurement does not return the probabilities
like an ensemble measurement does, but rather returns *one* of the
possible resulting states. The exact state that is returned is determined
by picking a state randomly according to the ensemble probabilities.
Parameters
----------
qubits : Qubit
The qubit to measure. This can be any Qubit or a linear combination
of them.
format : str
The format of the intermediate matrices to use. Possible values are
('sympy','numpy','scipy.sparse'). Currently only 'sympy' is
implemented.
Returns
-------
result : Qubit
The qubit that the system collapsed to upon measurement.
"""
import random
m = qubit_to_matrix(qubit)
if format == 'sympy':
m = m.normalized()
random_number = random.random()
total = 0
result = 0
for i in m:
total += i*i.conjugate()
if total > random_number:
break
result += 1
return Qubit(IntQubit(result, int(math.log(max(m.shape), 2) + .1)))
else:
raise NotImplementedError(
"This function cannot handle non-SymPy matrix formats yet"
)
|
84772cb707d41927a313345deb3de03e49db1ef658deaabce7d4c14ef919a1ba | from sympy.core.expr import Expr
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.matrices.dense import Matrix
from sympy.printing.pretty.stringpict import prettyForm
from sympy.core.containers import Tuple
from sympy.utilities.iterables import is_sequence
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.matrixutils import (
numpy_ndarray, scipy_sparse_matrix,
to_sympy, to_numpy, to_scipy_sparse
)
__all__ = [
'QuantumError',
'QExpr'
]
#-----------------------------------------------------------------------------
# Error handling
#-----------------------------------------------------------------------------
class QuantumError(Exception):
pass
def _qsympify_sequence(seq):
"""Convert elements of a sequence to standard form.
This is like sympify, but it performs special logic for arguments passed
to QExpr. The following conversions are done:
* (list, tuple, Tuple) => _qsympify_sequence each element and convert
sequence to a Tuple.
* basestring => Symbol
* Matrix => Matrix
* other => sympify
Strings are passed to Symbol, not sympify to make sure that variables like
'pi' are kept as Symbols, not the SymPy built-in number subclasses.
Examples
========
>>> from sympy.physics.quantum.qexpr import _qsympify_sequence
>>> _qsympify_sequence((1,2,[3,4,[1,]]))
(1, 2, (3, 4, (1,)))
"""
return tuple(__qsympify_sequence_helper(seq))
def __qsympify_sequence_helper(seq):
"""
Helper function for _qsympify_sequence
This function does the actual work.
"""
#base case. If not a list, do Sympification
if not is_sequence(seq):
if isinstance(seq, Matrix):
return seq
elif isinstance(seq, str):
return Symbol(seq)
else:
return sympify(seq)
# base condition, when seq is QExpr and also
# is iterable.
if isinstance(seq, QExpr):
return seq
#if list, recurse on each item in the list
result = [__qsympify_sequence_helper(item) for item in seq]
return Tuple(*result)
#-----------------------------------------------------------------------------
# Basic Quantum Expression from which all objects descend
#-----------------------------------------------------------------------------
class QExpr(Expr):
"""A base class for all quantum object like operators and states."""
# In sympy, slots are for instance attributes that are computed
# dynamically by the __new__ method. They are not part of args, but they
# derive from args.
# The Hilbert space a quantum Object belongs to.
__slots__ = ('hilbert_space', )
is_commutative = False
# The separator used in printing the label.
_label_separator = ''
@property
def free_symbols(self):
return {self}
def __new__(cls, *args, **kwargs):
"""Construct a new quantum object.
Parameters
==========
args : tuple
The list of numbers or parameters that uniquely specify the
quantum object. For a state, this will be its symbol or its
set of quantum numbers.
Examples
========
>>> from sympy.physics.quantum.qexpr import QExpr
>>> q = QExpr(0)
>>> q
0
>>> q.label
(0,)
>>> q.hilbert_space
H
>>> q.args
(0,)
>>> q.is_commutative
False
"""
# First compute args and call Expr.__new__ to create the instance
args = cls._eval_args(args, **kwargs)
if len(args) == 0:
args = cls._eval_args(tuple(cls.default_args()), **kwargs)
inst = Expr.__new__(cls, *args)
# Now set the slots on the instance
inst.hilbert_space = cls._eval_hilbert_space(args)
return inst
@classmethod
def _new_rawargs(cls, hilbert_space, *args, **old_assumptions):
"""Create new instance of this class with hilbert_space and args.
This is used to bypass the more complex logic in the ``__new__``
method in cases where you already have the exact ``hilbert_space``
and ``args``. This should be used when you are positive these
arguments are valid, in their final, proper form and want to optimize
the creation of the object.
"""
obj = Expr.__new__(cls, *args, **old_assumptions)
obj.hilbert_space = hilbert_space
return obj
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def label(self):
"""The label is the unique set of identifiers for the object.
Usually, this will include all of the information about the state
*except* the time (in the case of time-dependent objects).
This must be a tuple, rather than a Tuple.
"""
if len(self.args) == 0: # If there is no label specified, return the default
return self._eval_args(list(self.default_args()))
else:
return self.args
@property
def is_symbolic(self):
return True
@classmethod
def default_args(self):
"""If no arguments are specified, then this will return a default set
of arguments to be run through the constructor.
NOTE: Any classes that override this MUST return a tuple of arguments.
Should be overridden by subclasses to specify the default arguments for kets and operators
"""
raise NotImplementedError("No default arguments for this class!")
#-------------------------------------------------------------------------
# _eval_* methods
#-------------------------------------------------------------------------
def _eval_adjoint(self):
obj = Expr._eval_adjoint(self)
if obj is None:
obj = Expr.__new__(Dagger, self)
if isinstance(obj, QExpr):
obj.hilbert_space = self.hilbert_space
return obj
@classmethod
def _eval_args(cls, args):
"""Process the args passed to the __new__ method.
This simply runs args through _qsympify_sequence.
"""
return _qsympify_sequence(args)
@classmethod
def _eval_hilbert_space(cls, args):
"""Compute the Hilbert space instance from the args.
"""
from sympy.physics.quantum.hilbert import HilbertSpace
return HilbertSpace()
#-------------------------------------------------------------------------
# Printing
#-------------------------------------------------------------------------
# Utilities for printing: these operate on raw SymPy objects
def _print_sequence(self, seq, sep, printer, *args):
result = []
for item in seq:
result.append(printer._print(item, *args))
return sep.join(result)
def _print_sequence_pretty(self, seq, sep, printer, *args):
pform = printer._print(seq[0], *args)
for item in seq[1:]:
pform = prettyForm(*pform.right(sep))
pform = prettyForm(*pform.right(printer._print(item, *args)))
return pform
# Utilities for printing: these operate prettyForm objects
def _print_subscript_pretty(self, a, b):
top = prettyForm(*b.left(' '*a.width()))
bot = prettyForm(*a.right(' '*b.width()))
return prettyForm(binding=prettyForm.POW, *bot.below(top))
def _print_superscript_pretty(self, a, b):
return a**b
def _print_parens_pretty(self, pform, left='(', right=')'):
return prettyForm(*pform.parens(left=left, right=right))
# Printing of labels (i.e. args)
def _print_label(self, printer, *args):
"""Prints the label of the QExpr
This method prints self.label, using self._label_separator to separate
the elements. This method should not be overridden, instead, override
_print_contents to change printing behavior.
"""
return self._print_sequence(
self.label, self._label_separator, printer, *args
)
def _print_label_repr(self, printer, *args):
return self._print_sequence(
self.label, ',', printer, *args
)
def _print_label_pretty(self, printer, *args):
return self._print_sequence_pretty(
self.label, self._label_separator, printer, *args
)
def _print_label_latex(self, printer, *args):
return self._print_sequence(
self.label, self._label_separator, printer, *args
)
# Printing of contents (default to label)
def _print_contents(self, printer, *args):
"""Printer for contents of QExpr
Handles the printing of any unique identifying contents of a QExpr to
print as its contents, such as any variables or quantum numbers. The
default is to print the label, which is almost always the args. This
should not include printing of any brackets or parenteses.
"""
return self._print_label(printer, *args)
def _print_contents_pretty(self, printer, *args):
return self._print_label_pretty(printer, *args)
def _print_contents_latex(self, printer, *args):
return self._print_label_latex(printer, *args)
# Main printing methods
def _sympystr(self, printer, *args):
"""Default printing behavior of QExpr objects
Handles the default printing of a QExpr. To add other things to the
printing of the object, such as an operator name to operators or
brackets to states, the class should override the _print/_pretty/_latex
functions directly and make calls to _print_contents where appropriate.
This allows things like InnerProduct to easily control its printing the
printing of contents.
"""
return self._print_contents(printer, *args)
def _sympyrepr(self, printer, *args):
classname = self.__class__.__name__
label = self._print_label_repr(printer, *args)
return '%s(%s)' % (classname, label)
def _pretty(self, printer, *args):
pform = self._print_contents_pretty(printer, *args)
return pform
def _latex(self, printer, *args):
return self._print_contents_latex(printer, *args)
#-------------------------------------------------------------------------
# Methods from Basic and Expr
#-------------------------------------------------------------------------
def doit(self, **kw_args):
return self
#-------------------------------------------------------------------------
# Represent
#-------------------------------------------------------------------------
def _represent_default_basis(self, **options):
raise NotImplementedError('This object does not have a default basis')
def _represent(self, *, basis=None, **options):
"""Represent this object in a given basis.
This method dispatches to the actual methods that perform the
representation. Subclases of QExpr should define various methods to
determine how the object will be represented in various bases. The
format of these methods is::
def _represent_BasisName(self, basis, **options):
Thus to define how a quantum object is represented in the basis of
the operator Position, you would define::
def _represent_Position(self, basis, **options):
Usually, basis object will be instances of Operator subclasses, but
there is a chance we will relax this in the future to accommodate other
types of basis sets that are not associated with an operator.
If the ``format`` option is given it can be ("sympy", "numpy",
"scipy.sparse"). This will ensure that any matrices that result from
representing the object are returned in the appropriate matrix format.
Parameters
==========
basis : Operator
The Operator whose basis functions will be used as the basis for
representation.
options : dict
A dictionary of key/value pairs that give options and hints for
the representation, such as the number of basis functions to
be used.
"""
if basis is None:
result = self._represent_default_basis(**options)
else:
result = dispatch_method(self, '_represent', basis, **options)
# If we get a matrix representation, convert it to the right format.
format = options.get('format', 'sympy')
result = self._format_represent(result, format)
return result
def _format_represent(self, result, format):
if format == 'sympy' and not isinstance(result, Matrix):
return to_sympy(result)
elif format == 'numpy' and not isinstance(result, numpy_ndarray):
return to_numpy(result)
elif format == 'scipy.sparse' and \
not isinstance(result, scipy_sparse_matrix):
return to_scipy_sparse(result)
return result
def split_commutative_parts(e):
"""Split into commutative and non-commutative parts."""
c_part, nc_part = e.args_cnc()
c_part = list(c_part)
return c_part, nc_part
def split_qexpr_parts(e):
"""Split an expression into Expr and noncommutative QExpr parts."""
expr_part = []
qexpr_part = []
for arg in e.args:
if not isinstance(arg, QExpr):
expr_part.append(arg)
else:
qexpr_part.append(arg)
return expr_part, qexpr_part
def dispatch_method(self, basename, arg, **options):
"""Dispatch a method to the proper handlers."""
method_name = '%s_%s' % (basename, arg.__class__.__name__)
if hasattr(self, method_name):
f = getattr(self, method_name)
# This can raise and we will allow it to propagate.
result = f(arg, **options)
if result is not None:
return result
raise NotImplementedError(
"%s.%s cannot handle: %r" %
(self.__class__.__name__, basename, arg)
)
|
0aed843d3396a5fcb160c54e7460597b3f12059da2d901149879d388f3d28be3 | """Grover's algorithm and helper functions.
Todo:
* W gate construction (or perhaps -W gate based on Mermin's book)
* Generalize the algorithm for an unknown function that returns 1 on multiple
qubit states, not just one.
* Implement _represent_ZGate in OracleGate
"""
from sympy.core.numbers import pi
from sympy.core.sympify import sympify
from sympy.core.basic import Atom
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.matrices.dense import eye
from sympy.core.numbers import NegativeOne
from sympy.physics.quantum.qapply import qapply
from sympy.physics.quantum.qexpr import QuantumError
from sympy.physics.quantum.hilbert import ComplexSpace
from sympy.physics.quantum.operator import UnitaryOperator
from sympy.physics.quantum.gate import Gate
from sympy.physics.quantum.qubit import IntQubit
__all__ = [
'OracleGate',
'WGate',
'superposition_basis',
'grover_iteration',
'apply_grover'
]
def superposition_basis(nqubits):
"""Creates an equal superposition of the computational basis.
Parameters
==========
nqubits : int
The number of qubits.
Returns
=======
state : Qubit
An equal superposition of the computational basis with nqubits.
Examples
========
Create an equal superposition of 2 qubits::
>>> from sympy.physics.quantum.grover import superposition_basis
>>> superposition_basis(2)
|0>/2 + |1>/2 + |2>/2 + |3>/2
"""
amp = 1/sqrt(2**nqubits)
return sum([amp*IntQubit(n, nqubits=nqubits) for n in range(2**nqubits)])
class OracleGateFunction(Atom):
"""Wrapper for python functions used in `OracleGate`s"""
def __new__(cls, function):
if not callable(function):
raise TypeError('Callable expected, got: %r' % function)
obj = Atom.__new__(cls)
obj.function = function
return obj
def _hashable_content(self):
return type(self), self.function
def __call__(self, *args):
return self.function(*args)
class OracleGate(Gate):
"""A black box gate.
The gate marks the desired qubits of an unknown function by flipping
the sign of the qubits. The unknown function returns true when it
finds its desired qubits and false otherwise.
Parameters
==========
qubits : int
Number of qubits.
oracle : callable
A callable function that returns a boolean on a computational basis.
Examples
========
Apply an Oracle gate that flips the sign of ``|2>`` on different qubits::
>>> from sympy.physics.quantum.qubit import IntQubit
>>> from sympy.physics.quantum.qapply import qapply
>>> from sympy.physics.quantum.grover import OracleGate
>>> f = lambda qubits: qubits == IntQubit(2)
>>> v = OracleGate(2, f)
>>> qapply(v*IntQubit(2))
-|2>
>>> qapply(v*IntQubit(3))
|3>
"""
gate_name = 'V'
gate_name_latex = 'V'
#-------------------------------------------------------------------------
# Initialization/creation
#-------------------------------------------------------------------------
@classmethod
def _eval_args(cls, args):
if len(args) != 2:
raise QuantumError(
'Insufficient/excessive arguments to Oracle. Please ' +
'supply the number of qubits and an unknown function.'
)
sub_args = (args[0],)
sub_args = UnitaryOperator._eval_args(sub_args)
if not sub_args[0].is_Integer:
raise TypeError('Integer expected, got: %r' % sub_args[0])
function = args[1]
if not isinstance(function, OracleGateFunction):
function = OracleGateFunction(function)
return (sub_args[0], function)
@classmethod
def _eval_hilbert_space(cls, args):
"""This returns the smallest possible Hilbert space."""
return ComplexSpace(2)**args[0]
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def search_function(self):
"""The unknown function that helps find the sought after qubits."""
return self.label[1]
@property
def targets(self):
"""A tuple of target qubits."""
return sympify(tuple(range(self.args[0])))
#-------------------------------------------------------------------------
# Apply
#-------------------------------------------------------------------------
def _apply_operator_Qubit(self, qubits, **options):
"""Apply this operator to a Qubit subclass.
Parameters
==========
qubits : Qubit
The qubit subclass to apply this operator to.
Returns
=======
state : Expr
The resulting quantum state.
"""
if qubits.nqubits != self.nqubits:
raise QuantumError(
'OracleGate operates on %r qubits, got: %r'
% (self.nqubits, qubits.nqubits)
)
# If function returns 1 on qubits
# return the negative of the qubits (flip the sign)
if self.search_function(qubits):
return -qubits
else:
return qubits
#-------------------------------------------------------------------------
# Represent
#-------------------------------------------------------------------------
def _represent_ZGate(self, basis, **options):
"""
Represent the OracleGate in the computational basis.
"""
nbasis = 2**self.nqubits # compute it only once
matrixOracle = eye(nbasis)
# Flip the sign given the output of the oracle function
for i in range(nbasis):
if self.search_function(IntQubit(i, nqubits=self.nqubits)):
matrixOracle[i, i] = NegativeOne()
return matrixOracle
class WGate(Gate):
"""General n qubit W Gate in Grover's algorithm.
The gate performs the operation ``2|phi><phi| - 1`` on some qubits.
``|phi> = (tensor product of n Hadamards)*(|0> with n qubits)``
Parameters
==========
nqubits : int
The number of qubits to operate on
"""
gate_name = 'W'
gate_name_latex = 'W'
@classmethod
def _eval_args(cls, args):
if len(args) != 1:
raise QuantumError(
'Insufficient/excessive arguments to W gate. Please ' +
'supply the number of qubits to operate on.'
)
args = UnitaryOperator._eval_args(args)
if not args[0].is_Integer:
raise TypeError('Integer expected, got: %r' % args[0])
return args
#-------------------------------------------------------------------------
# Properties
#-------------------------------------------------------------------------
@property
def targets(self):
return sympify(tuple(reversed(range(self.args[0]))))
#-------------------------------------------------------------------------
# Apply
#-------------------------------------------------------------------------
def _apply_operator_Qubit(self, qubits, **options):
"""
qubits: a set of qubits (Qubit)
Returns: quantum object (quantum expression - QExpr)
"""
if qubits.nqubits != self.nqubits:
raise QuantumError(
'WGate operates on %r qubits, got: %r'
% (self.nqubits, qubits.nqubits)
)
# See 'Quantum Computer Science' by David Mermin p.92 -> W|a> result
# Return (2/(sqrt(2^n)))|phi> - |a> where |a> is the current basis
# state and phi is the superposition of basis states (see function
# create_computational_basis above)
basis_states = superposition_basis(self.nqubits)
change_to_basis = (2/sqrt(2**self.nqubits))*basis_states
return change_to_basis - qubits
def grover_iteration(qstate, oracle):
"""Applies one application of the Oracle and W Gate, WV.
Parameters
==========
qstate : Qubit
A superposition of qubits.
oracle : OracleGate
The black box operator that flips the sign of the desired basis qubits.
Returns
=======
Qubit : The qubits after applying the Oracle and W gate.
Examples
========
Perform one iteration of grover's algorithm to see a phase change::
>>> from sympy.physics.quantum.qapply import qapply
>>> from sympy.physics.quantum.qubit import IntQubit
>>> from sympy.physics.quantum.grover import OracleGate
>>> from sympy.physics.quantum.grover import superposition_basis
>>> from sympy.physics.quantum.grover import grover_iteration
>>> numqubits = 2
>>> basis_states = superposition_basis(numqubits)
>>> f = lambda qubits: qubits == IntQubit(2)
>>> v = OracleGate(numqubits, f)
>>> qapply(grover_iteration(basis_states, v))
|2>
"""
wgate = WGate(oracle.nqubits)
return wgate*oracle*qstate
def apply_grover(oracle, nqubits, iterations=None):
"""Applies grover's algorithm.
Parameters
==========
oracle : callable
The unknown callable function that returns true when applied to the
desired qubits and false otherwise.
Returns
=======
state : Expr
The resulting state after Grover's algorithm has been iterated.
Examples
========
Apply grover's algorithm to an even superposition of 2 qubits::
>>> from sympy.physics.quantum.qapply import qapply
>>> from sympy.physics.quantum.qubit import IntQubit
>>> from sympy.physics.quantum.grover import apply_grover
>>> f = lambda qubits: qubits == IntQubit(2)
>>> qapply(apply_grover(f, 2))
|2>
"""
if nqubits <= 0:
raise QuantumError(
'Grover\'s algorithm needs nqubits > 0, received %r qubits'
% nqubits
)
if iterations is None:
iterations = floor(sqrt(2**nqubits)*(pi/4))
v = OracleGate(nqubits, oracle)
iterated = superposition_basis(nqubits)
for iter in range(iterations):
iterated = grover_iteration(iterated, v)
iterated = qapply(iterated)
return iterated
|
75b94143719e71a6009f80515d2e624665fa980983784daf2137f94514f8dbad | """Matplotlib based plotting of quantum circuits.
Todo:
* Optimize printing of large circuits.
* Get this to work with single gates.
* Do a better job checking the form of circuits to make sure it is a Mul of
Gates.
* Get multi-target gates plotting.
* Get initial and final states to plot.
* Get measurements to plot. Might need to rethink measurement as a gate
issue.
* Get scale and figsize to be handled in a better way.
* Write some tests/examples!
"""
from typing import List, Dict as tDict
from sympy.core.mul import Mul
from sympy.external import import_module
from sympy.physics.quantum.gate import Gate, OneQubitGate, CGate, CGateS
from sympy.core.core import BasicMeta
from sympy.core.assumptions import ManagedProperties
__all__ = [
'CircuitPlot',
'circuit_plot',
'labeller',
'Mz',
'Mx',
'CreateOneQubitGate',
'CreateCGate',
]
np = import_module('numpy')
matplotlib = import_module(
'matplotlib', import_kwargs={'fromlist': ['pyplot']},
catch=(RuntimeError,)) # This is raised in environments that have no display.
if np and matplotlib:
pyplot = matplotlib.pyplot
Line2D = matplotlib.lines.Line2D
Circle = matplotlib.patches.Circle
#from matplotlib import rc
#rc('text',usetex=True)
class CircuitPlot:
"""A class for managing a circuit plot."""
scale = 1.0
fontsize = 20.0
linewidth = 1.0
control_radius = 0.05
not_radius = 0.15
swap_delta = 0.05
labels = [] # type: List[str]
inits = {} # type: tDict[str, str]
label_buffer = 0.5
def __init__(self, c, nqubits, **kwargs):
if not np or not matplotlib:
raise ImportError('numpy or matplotlib not available.')
self.circuit = c
self.ngates = len(self.circuit.args)
self.nqubits = nqubits
self.update(kwargs)
self._create_grid()
self._create_figure()
self._plot_wires()
self._plot_gates()
self._finish()
def update(self, kwargs):
"""Load the kwargs into the instance dict."""
self.__dict__.update(kwargs)
def _create_grid(self):
"""Create the grid of wires."""
scale = self.scale
wire_grid = np.arange(0.0, self.nqubits*scale, scale, dtype=float)
gate_grid = np.arange(0.0, self.ngates*scale, scale, dtype=float)
self._wire_grid = wire_grid
self._gate_grid = gate_grid
def _create_figure(self):
"""Create the main matplotlib figure."""
self._figure = pyplot.figure(
figsize=(self.ngates*self.scale, self.nqubits*self.scale),
facecolor='w',
edgecolor='w'
)
ax = self._figure.add_subplot(
1, 1, 1,
frameon=True
)
ax.set_axis_off()
offset = 0.5*self.scale
ax.set_xlim(self._gate_grid[0] - offset, self._gate_grid[-1] + offset)
ax.set_ylim(self._wire_grid[0] - offset, self._wire_grid[-1] + offset)
ax.set_aspect('equal')
self._axes = ax
def _plot_wires(self):
"""Plot the wires of the circuit diagram."""
xstart = self._gate_grid[0]
xstop = self._gate_grid[-1]
xdata = (xstart - self.scale, xstop + self.scale)
for i in range(self.nqubits):
ydata = (self._wire_grid[i], self._wire_grid[i])
line = Line2D(
xdata, ydata,
color='k',
lw=self.linewidth
)
self._axes.add_line(line)
if self.labels:
init_label_buffer = 0
if self.inits.get(self.labels[i]): init_label_buffer = 0.25
self._axes.text(
xdata[0]-self.label_buffer-init_label_buffer,ydata[0],
render_label(self.labels[i],self.inits),
size=self.fontsize,
color='k',ha='center',va='center')
self._plot_measured_wires()
def _plot_measured_wires(self):
ismeasured = self._measurements()
xstop = self._gate_grid[-1]
dy = 0.04 # amount to shift wires when doubled
# Plot doubled wires after they are measured
for im in ismeasured:
xdata = (self._gate_grid[ismeasured[im]],xstop+self.scale)
ydata = (self._wire_grid[im]+dy,self._wire_grid[im]+dy)
line = Line2D(
xdata, ydata,
color='k',
lw=self.linewidth
)
self._axes.add_line(line)
# Also double any controlled lines off these wires
for i,g in enumerate(self._gates()):
if isinstance(g, (CGate, CGateS)):
wires = g.controls + g.targets
for wire in wires:
if wire in ismeasured and \
self._gate_grid[i] > self._gate_grid[ismeasured[wire]]:
ydata = min(wires), max(wires)
xdata = self._gate_grid[i]-dy, self._gate_grid[i]-dy
line = Line2D(
xdata, ydata,
color='k',
lw=self.linewidth
)
self._axes.add_line(line)
def _gates(self):
"""Create a list of all gates in the circuit plot."""
gates = []
if isinstance(self.circuit, Mul):
for g in reversed(self.circuit.args):
if isinstance(g, Gate):
gates.append(g)
elif isinstance(self.circuit, Gate):
gates.append(self.circuit)
return gates
def _plot_gates(self):
"""Iterate through the gates and plot each of them."""
for i, gate in enumerate(self._gates()):
gate.plot_gate(self, i)
def _measurements(self):
"""Return a dict ``{i:j}`` where i is the index of the wire that has
been measured, and j is the gate where the wire is measured.
"""
ismeasured = {}
for i,g in enumerate(self._gates()):
if getattr(g,'measurement',False):
for target in g.targets:
if target in ismeasured:
if ismeasured[target] > i:
ismeasured[target] = i
else:
ismeasured[target] = i
return ismeasured
def _finish(self):
# Disable clipping to make panning work well for large circuits.
for o in self._figure.findobj():
o.set_clip_on(False)
def one_qubit_box(self, t, gate_idx, wire_idx):
"""Draw a box for a single qubit gate."""
x = self._gate_grid[gate_idx]
y = self._wire_grid[wire_idx]
self._axes.text(
x, y, t,
color='k',
ha='center',
va='center',
bbox=dict(ec='k', fc='w', fill=True, lw=self.linewidth),
size=self.fontsize
)
def two_qubit_box(self, t, gate_idx, wire_idx):
"""Draw a box for a two qubit gate. Does not work yet.
"""
# x = self._gate_grid[gate_idx]
# y = self._wire_grid[wire_idx]+0.5
print(self._gate_grid)
print(self._wire_grid)
# unused:
# obj = self._axes.text(
# x, y, t,
# color='k',
# ha='center',
# va='center',
# bbox=dict(ec='k', fc='w', fill=True, lw=self.linewidth),
# size=self.fontsize
# )
def control_line(self, gate_idx, min_wire, max_wire):
"""Draw a vertical control line."""
xdata = (self._gate_grid[gate_idx], self._gate_grid[gate_idx])
ydata = (self._wire_grid[min_wire], self._wire_grid[max_wire])
line = Line2D(
xdata, ydata,
color='k',
lw=self.linewidth
)
self._axes.add_line(line)
def control_point(self, gate_idx, wire_idx):
"""Draw a control point."""
x = self._gate_grid[gate_idx]
y = self._wire_grid[wire_idx]
radius = self.control_radius
c = Circle(
(x, y),
radius*self.scale,
ec='k',
fc='k',
fill=True,
lw=self.linewidth
)
self._axes.add_patch(c)
def not_point(self, gate_idx, wire_idx):
"""Draw a NOT gates as the circle with plus in the middle."""
x = self._gate_grid[gate_idx]
y = self._wire_grid[wire_idx]
radius = self.not_radius
c = Circle(
(x, y),
radius,
ec='k',
fc='w',
fill=False,
lw=self.linewidth
)
self._axes.add_patch(c)
l = Line2D(
(x, x), (y - radius, y + radius),
color='k',
lw=self.linewidth
)
self._axes.add_line(l)
def swap_point(self, gate_idx, wire_idx):
"""Draw a swap point as a cross."""
x = self._gate_grid[gate_idx]
y = self._wire_grid[wire_idx]
d = self.swap_delta
l1 = Line2D(
(x - d, x + d),
(y - d, y + d),
color='k',
lw=self.linewidth
)
l2 = Line2D(
(x - d, x + d),
(y + d, y - d),
color='k',
lw=self.linewidth
)
self._axes.add_line(l1)
self._axes.add_line(l2)
def circuit_plot(c, nqubits, **kwargs):
"""Draw the circuit diagram for the circuit with nqubits.
Parameters
==========
c : circuit
The circuit to plot. Should be a product of Gate instances.
nqubits : int
The number of qubits to include in the circuit. Must be at least
as big as the largest ``min_qubits`` of the gates.
"""
return CircuitPlot(c, nqubits, **kwargs)
def render_label(label, inits={}):
"""Slightly more flexible way to render labels.
>>> from sympy.physics.quantum.circuitplot import render_label
>>> render_label('q0')
'$\\\\left|q0\\\\right\\\\rangle$'
>>> render_label('q0', {'q0':'0'})
'$\\\\left|q0\\\\right\\\\rangle=\\\\left|0\\\\right\\\\rangle$'
"""
init = inits.get(label)
if init:
return r'$\left|%s\right\rangle=\left|%s\right\rangle$' % (label, init)
return r'$\left|%s\right\rangle$' % label
def labeller(n, symbol='q'):
"""Autogenerate labels for wires of quantum circuits.
Parameters
==========
n : int
number of qubits in the circuit.
symbol : string
A character string to precede all gate labels. E.g. 'q_0', 'q_1', etc.
>>> from sympy.physics.quantum.circuitplot import labeller
>>> labeller(2)
['q_1', 'q_0']
>>> labeller(3,'j')
['j_2', 'j_1', 'j_0']
"""
return ['%s_%d' % (symbol,n-i-1) for i in range(n)]
class Mz(OneQubitGate):
"""Mock-up of a z measurement gate.
This is in circuitplot rather than gate.py because it's not a real
gate, it just draws one.
"""
measurement = True
gate_name='Mz'
gate_name_latex='M_z'
class Mx(OneQubitGate):
"""Mock-up of an x measurement gate.
This is in circuitplot rather than gate.py because it's not a real
gate, it just draws one.
"""
measurement = True
gate_name='Mx'
gate_name_latex='M_x'
class CreateOneQubitGate(ManagedProperties):
def __new__(mcl, name, latexname=None):
if not latexname:
latexname = name
return BasicMeta.__new__(mcl, name + "Gate", (OneQubitGate,),
{'gate_name': name, 'gate_name_latex': latexname})
def CreateCGate(name, latexname=None):
"""Use a lexical closure to make a controlled gate.
"""
if not latexname:
latexname = name
onequbitgate = CreateOneQubitGate(name, latexname)
def ControlledGate(ctrls,target):
return CGate(tuple(ctrls),onequbitgate(target))
return ControlledGate
|
d928fba1e267e6efcc24a011cbabb0f04412a2d85940866e3ce6dc612f68550b | from sympy.core.backend import sympify
from sympy.physics.vector import Point, ReferenceFrame, Dyadic
from sympy.utilities.exceptions import sympy_deprecation_warning
__all__ = ['RigidBody']
class RigidBody:
"""An idealized rigid body.
Explanation
===========
This is essentially a container which holds the various components which
describe a rigid body: a name, mass, center of mass, reference frame, and
inertia.
All of these need to be supplied on creation, but can be changed
afterwards.
Attributes
==========
name : string
The body's name.
masscenter : Point
The point which represents the center of mass of the rigid body.
frame : ReferenceFrame
The ReferenceFrame which the rigid body is fixed in.
mass : Sympifyable
The body's mass.
inertia : (Dyadic, Point)
The body's inertia about a point; stored in a tuple as shown above.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
>>> from sympy.physics.mechanics import outer
>>> m = Symbol('m')
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer (A.x, A.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, A, m, inertia_tuple)
>>> # Or you could change them afterwards
>>> m2 = Symbol('m2')
>>> B.mass = m2
"""
def __init__(self, name, masscenter, frame, mass, inertia):
if not isinstance(name, str):
raise TypeError('Supply a valid name.')
self._name = name
self.masscenter = masscenter
self.mass = mass
self.frame = frame
self.inertia = inertia
self.potential_energy = 0
def __str__(self):
return self._name
def __repr__(self):
return self.__str__()
@property
def frame(self):
return self._frame
@frame.setter
def frame(self, F):
if not isinstance(F, ReferenceFrame):
raise TypeError("RigdBody frame must be a ReferenceFrame object.")
self._frame = F
@property
def masscenter(self):
return self._masscenter
@masscenter.setter
def masscenter(self, p):
if not isinstance(p, Point):
raise TypeError("RigidBody center of mass must be a Point object.")
self._masscenter = p
@property
def mass(self):
return self._mass
@mass.setter
def mass(self, m):
self._mass = sympify(m)
@property
def inertia(self):
return (self._inertia, self._inertia_point)
@inertia.setter
def inertia(self, I):
if not isinstance(I[0], Dyadic):
raise TypeError("RigidBody inertia must be a Dyadic object.")
if not isinstance(I[1], Point):
raise TypeError("RigidBody inertia must be about a Point.")
self._inertia = I[0]
self._inertia_point = I[1]
# have I S/O, want I S/S*
# I S/O = I S/S* + I S*/O; I S/S* = I S/O - I S*/O
# I_S/S* = I_S/O - I_S*/O
from sympy.physics.mechanics.functions import inertia_of_point_mass
I_Ss_O = inertia_of_point_mass(self.mass,
self.masscenter.pos_from(I[1]),
self.frame)
self._central_inertia = I[0] - I_Ss_O
@property
def central_inertia(self):
"""The body's central inertia dyadic."""
return self._central_inertia
def linear_momentum(self, frame):
""" Linear momentum of the rigid body.
Explanation
===========
The linear momentum L, of a rigid body B, with respect to frame N is
given by
L = M * v*
where M is the mass of the rigid body and v* is the velocity of
the mass center of B in the frame, N.
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> M, v = dynamicsymbols('M v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (N.x, N.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, N, M, Inertia_tuple)
>>> B.linear_momentum(N)
M*v*N.x
"""
return self.mass * self.masscenter.vel(frame)
def angular_momentum(self, point, frame):
"""Returns the angular momentum of the rigid body about a point in the
given frame.
Explanation
===========
The angular momentum H of a rigid body B about some point O in a frame
N is given by:
H = I . w + r x Mv
where I is the central inertia dyadic of B, w is the angular velocity
of body B in the frame, N, r is the position vector from point O to the
mass center of B, and v is the velocity of the mass center in the
frame, N.
Parameters
==========
point : Point
The point about which angular momentum is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> M, v, r, omega = dynamicsymbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, 1 * N.x)
>>> I = outer(b.x, b.x)
>>> B = RigidBody('B', P, b, M, (I, P))
>>> B.angular_momentum(P, N)
omega*b.x
"""
I = self.central_inertia
w = self.frame.ang_vel_in(frame)
m = self.mass
r = self.masscenter.pos_from(point)
v = self.masscenter.vel(frame)
return I.dot(w) + r.cross(m * v)
def kinetic_energy(self, frame):
"""Kinetic energy of the rigid body.
Explanation
===========
The kinetic energy, T, of a rigid body, B, is given by
'T = 1/2 (I omega^2 + m v^2)'
where I and m are the central inertia dyadic and mass of rigid body B,
respectively, omega is the body's angular velocity and v is the
velocity of the body's mass center in the supplied ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The RigidBody's angular velocity and the velocity of it's mass
center are typically defined with respect to an inertial frame but
any relevant frame in which the velocities are known can be supplied.
Examples
========
>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody
>>> from sympy import symbols
>>> M, v, r, omega = symbols('M v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (b.x, b.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, inertia_tuple)
>>> B.kinetic_energy(N)
M*v**2/2 + omega**2/2
"""
rotational_KE = (self.frame.ang_vel_in(frame) & (self.central_inertia &
self.frame.ang_vel_in(frame)) / sympify(2))
translational_KE = (self.mass * (self.masscenter.vel(frame) &
self.masscenter.vel(frame)) / sympify(2))
return rotational_KE + translational_KE
@property
def potential_energy(self):
"""The potential energy of the RigidBody.
Examples
========
>>> from sympy.physics.mechanics import RigidBody, Point, outer, ReferenceFrame
>>> from sympy import symbols
>>> M, g, h = symbols('M g h')
>>> b = ReferenceFrame('b')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
>>> B.potential_energy
M*g*h
"""
return self._pe
@potential_energy.setter
def potential_energy(self, scalar):
"""Used to set the potential energy of this RigidBody.
Parameters
==========
scalar: Sympifyable
The potential energy (a scalar) of the RigidBody.
Examples
========
>>> from sympy.physics.mechanics import Point, outer
>>> from sympy.physics.mechanics import RigidBody, ReferenceFrame
>>> from sympy import symbols
>>> b = ReferenceFrame('b')
>>> M, g, h = symbols('M g h')
>>> P = Point('P')
>>> I = outer (b.x, b.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, M, Inertia_tuple)
>>> B.potential_energy = M * g * h
"""
self._pe = sympify(scalar)
def set_potential_energy(self, scalar):
sympy_deprecation_warning(
"""
The sympy.physics.mechanics.RigidBody.set_potential_energy()
method is deprecated. Instead use
B.potential_energy = scalar
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-set-potential-energy",
)
self.potential_energy = scalar
# XXX: To be consistent with the parallel_axis method in Particle this
# should have a frame argument...
def parallel_axis(self, point):
"""Returns the inertia dyadic of the body with respect to another
point.
Parameters
==========
point : sympy.physics.vector.Point
The point to express the inertia dyadic about.
Returns
=======
inertia : sympy.physics.vector.Dyadic
The inertia dyadic of the rigid body expressed about the provided
point.
"""
# circular import issue
from sympy.physics.mechanics.functions import inertia
a, b, c = self.masscenter.pos_from(point).to_matrix(self.frame)
I = self.mass * inertia(self.frame, b**2 + c**2, c**2 + a**2, a**2 +
b**2, -a * b, -b * c, -a * c)
return self.central_inertia + I
|
810e676798878174a33f4806924d21afcc54a0ec21e58ef2ef2f13ebe9e039f7 | from sympy.utilities import dict_merge
from sympy.utilities.iterables import iterable
from sympy.physics.vector import (Dyadic, Vector, ReferenceFrame,
Point, dynamicsymbols)
from sympy.physics.vector.printing import (vprint, vsprint, vpprint, vlatex,
init_vprinting)
from sympy.physics.mechanics.particle import Particle
from sympy.physics.mechanics.rigidbody import RigidBody
from sympy.simplify.simplify import simplify
from sympy.core.backend import (Matrix, sympify, Mul, Derivative, sin, cos,
tan, AppliedUndef, S)
__all__ = ['inertia',
'inertia_of_point_mass',
'linear_momentum',
'angular_momentum',
'kinetic_energy',
'potential_energy',
'Lagrangian',
'mechanics_printing',
'mprint',
'msprint',
'mpprint',
'mlatex',
'msubs',
'find_dynamicsymbols']
# These are functions that we've moved and renamed during extracting the
# basic vector calculus code from the mechanics packages.
mprint = vprint
msprint = vsprint
mpprint = vpprint
mlatex = vlatex
def mechanics_printing(**kwargs):
"""
Initializes time derivative printing for all SymPy objects in
mechanics module.
"""
init_vprinting(**kwargs)
mechanics_printing.__doc__ = init_vprinting.__doc__
def inertia(frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0):
"""Simple way to create inertia Dyadic object.
Explanation
===========
If you do not know what a Dyadic is, just treat this like the inertia
tensor. Then, do the easy thing and define it in a body-fixed frame.
Parameters
==========
frame : ReferenceFrame
The frame the inertia is defined in
ixx : Sympifyable
the xx element in the inertia dyadic
iyy : Sympifyable
the yy element in the inertia dyadic
izz : Sympifyable
the zz element in the inertia dyadic
ixy : Sympifyable
the xy element in the inertia dyadic
iyz : Sympifyable
the yz element in the inertia dyadic
izx : Sympifyable
the zx element in the inertia dyadic
Examples
========
>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')
>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Need to define the inertia in a frame')
ol = sympify(ixx) * (frame.x | frame.x)
ol += sympify(ixy) * (frame.x | frame.y)
ol += sympify(izx) * (frame.x | frame.z)
ol += sympify(ixy) * (frame.y | frame.x)
ol += sympify(iyy) * (frame.y | frame.y)
ol += sympify(iyz) * (frame.y | frame.z)
ol += sympify(izx) * (frame.z | frame.x)
ol += sympify(iyz) * (frame.z | frame.y)
ol += sympify(izz) * (frame.z | frame.z)
return ol
def inertia_of_point_mass(mass, pos_vec, frame):
"""Inertia dyadic of a point mass relative to point O.
Parameters
==========
mass : Sympifyable
Mass of the point mass
pos_vec : Vector
Position from point O to point mass
frame : ReferenceFrame
Reference frame to express the dyadic in
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass
>>> N = ReferenceFrame('N')
>>> r, m = symbols('r m')
>>> px = r * N.x
>>> inertia_of_point_mass(m, px, N)
m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)
"""
return mass * (((frame.x | frame.x) + (frame.y | frame.y) +
(frame.z | frame.z)) * (pos_vec & pos_vec) -
(pos_vec | pos_vec))
def linear_momentum(frame, *body):
"""Linear momentum of the system.
Explanation
===========
This function returns the linear momentum of a system of Particle's and/or
RigidBody's. The linear momentum of a system is equal to the vector sum of
the linear momentum of its constituents. Consider a system, S, comprised of
a rigid body, A, and a particle, P. The linear momentum of the system, L,
is equal to the vector sum of the linear momentum of the particle, L1, and
the linear momentum of the rigid body, L2, i.e.
L = L1 + L2
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose linear momentum is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, linear_momentum
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = Point('Ac')
>>> Ac.set_vel(N, 25 * N.y)
>>> I = outer(N.x, N.x)
>>> A = RigidBody('A', Ac, N, 20, (I, Ac))
>>> linear_momentum(N, A, Pa)
10*N.x + 500*N.y
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please specify a valid ReferenceFrame')
else:
linear_momentum_sys = Vector(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
linear_momentum_sys += e.linear_momentum(frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return linear_momentum_sys
def angular_momentum(point, frame, *body):
"""Angular momentum of a system.
Explanation
===========
This function returns the angular momentum of a system of Particle's and/or
RigidBody's. The angular momentum of such a system is equal to the vector
sum of the angular momentum of its constituents. Consider a system, S,
comprised of a rigid body, A, and a particle, P. The angular momentum of
the system, H, is equal to the vector sum of the angular momentum of the
particle, H1, and the angular momentum of the rigid body, H2, i.e.
H = H1 + H2
Parameters
==========
point : Point
The point about which angular momentum of the system is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose angular momentum is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, angular_momentum
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> angular_momentum(O, N, Pa, A)
10*N.z
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please enter a valid ReferenceFrame')
if not isinstance(point, Point):
raise TypeError('Please specify a valid Point')
else:
angular_momentum_sys = Vector(0)
for e in body:
if isinstance(e, (RigidBody, Particle)):
angular_momentum_sys += e.angular_momentum(point, frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return angular_momentum_sys
def kinetic_energy(frame, *body):
"""Kinetic energy of a multibody system.
Explanation
===========
This function returns the kinetic energy of a system of Particle's and/or
RigidBody's. The kinetic energy of such a system is equal to the sum of
the kinetic energies of its constituents. Consider a system, S, comprising
a rigid body, A, and a particle, P. The kinetic energy of the system, T,
is equal to the vector sum of the kinetic energy of the particle, T1, and
the kinetic energy of the rigid body, T2, i.e.
T = T1 + T2
Kinetic energy is a scalar.
Parameters
==========
frame : ReferenceFrame
The frame in which the velocity or angular velocity of the body is
defined.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose kinetic energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, kinetic_energy
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> kinetic_energy(N, Pa, A)
350
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please enter a valid ReferenceFrame')
ke_sys = S.Zero
for e in body:
if isinstance(e, (RigidBody, Particle)):
ke_sys += e.kinetic_energy(frame)
else:
raise TypeError('*body must have only Particle or RigidBody')
return ke_sys
def potential_energy(*body):
"""Potential energy of a multibody system.
Explanation
===========
This function returns the potential energy of a system of Particle's and/or
RigidBody's. The potential energy of such a system is equal to the sum of
the potential energy of its constituents. Consider a system, S, comprising
a rigid body, A, and a particle, P. The potential energy of the system, V,
is equal to the vector sum of the potential energy of the particle, V1, and
the potential energy of the rigid body, V2, i.e.
V = V1 + V2
Potential energy is a scalar.
Parameters
==========
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose potential energy is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, potential_energy
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> Pa = Particle('Pa', P, m)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> a = ReferenceFrame('a')
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> potential_energy(Pa, A)
M*g*h + g*h*m
"""
pe_sys = S.Zero
for e in body:
if isinstance(e, (RigidBody, Particle)):
pe_sys += e.potential_energy
else:
raise TypeError('*body must have only Particle or RigidBody')
return pe_sys
def gravity(acceleration, *bodies):
"""
Returns a list of gravity forces given the acceleration
due to gravity and any number of particles or rigidbodies.
Example
=======
>>> from sympy.physics.mechanics import ReferenceFrame, Point, Particle, outer, RigidBody
>>> from sympy.physics.mechanics.functions import gravity
>>> from sympy import symbols
>>> N = ReferenceFrame('N')
>>> m, M, g = symbols('m M g')
>>> F1, F2 = symbols('F1 F2')
>>> po = Point('po')
>>> pa = Particle('pa', po, m)
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer(A.x, A.x)
>>> B = RigidBody('B', P, A, M, (I, P))
>>> forceList = [(po, F1), (P, F2)]
>>> forceList.extend(gravity(g*N.y, pa, B))
>>> forceList
[(po, F1), (P, F2), (po, g*m*N.y), (P, M*g*N.y)]
"""
gravity_force = []
if not bodies:
raise TypeError("No bodies(instances of Particle or Rigidbody) were passed.")
for e in bodies:
point = getattr(e, 'masscenter', None)
if point is None:
point = e.point
gravity_force.append((point, e.mass*acceleration))
return gravity_force
def center_of_mass(point, *bodies):
"""
Returns the position vector from the given point to the center of mass
of the given bodies(particles or rigidbodies).
Example
=======
>>> from sympy import symbols, S
>>> from sympy.physics.vector import Point
>>> from sympy.physics.mechanics import Particle, ReferenceFrame, RigidBody, outer
>>> from sympy.physics.mechanics.functions import center_of_mass
>>> a = ReferenceFrame('a')
>>> m = symbols('m', real=True)
>>> p1 = Particle('p1', Point('p1_pt'), S(1))
>>> p2 = Particle('p2', Point('p2_pt'), S(2))
>>> p3 = Particle('p3', Point('p3_pt'), S(3))
>>> p4 = Particle('p4', Point('p4_pt'), m)
>>> b_f = ReferenceFrame('b_f')
>>> b_cm = Point('b_cm')
>>> mb = symbols('mb')
>>> b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
>>> p2.point.set_pos(p1.point, a.x)
>>> p3.point.set_pos(p1.point, a.x + a.y)
>>> p4.point.set_pos(p1.point, a.y)
>>> b.masscenter.set_pos(p1.point, a.y + a.z)
>>> point_o=Point('o')
>>> point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
>>> expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
>>> point_o.pos_from(p1.point)
5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
"""
if not bodies:
raise TypeError("No bodies(instances of Particle or Rigidbody) were passed.")
total_mass = 0
vec = Vector(0)
for i in bodies:
total_mass += i.mass
masscenter = getattr(i, 'masscenter', None)
if masscenter is None:
masscenter = i.point
vec += i.mass*masscenter.pos_from(point)
return vec/total_mass
def Lagrangian(frame, *body):
"""Lagrangian of a multibody system.
Explanation
===========
This function returns the Lagrangian of a system of Particle's and/or
RigidBody's. The Lagrangian of such a system is equal to the difference
between the kinetic energies and potential energies of its constituents. If
T and V are the kinetic and potential energies of a system then it's
Lagrangian, L, is defined as
L = T - V
The Lagrangian is a scalar.
Parameters
==========
frame : ReferenceFrame
The frame in which the velocity or angular velocity of the body is
defined to determine the kinetic energy.
body1, body2, body3... : Particle and/or RigidBody
The body (or bodies) whose Lagrangian is required.
Examples
========
>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, Lagrangian
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> Lagrangian(N, Pa, A)
-M*g*h - g*h*m + 350
"""
if not isinstance(frame, ReferenceFrame):
raise TypeError('Please supply a valid ReferenceFrame')
for e in body:
if not isinstance(e, (RigidBody, Particle)):
raise TypeError('*body must have only Particle or RigidBody')
return kinetic_energy(frame, *body) - potential_energy(*body)
def find_dynamicsymbols(expression, exclude=None, reference_frame=None):
"""Find all dynamicsymbols in expression.
Explanation
===========
If the optional ``exclude`` kwarg is used, only dynamicsymbols
not in the iterable ``exclude`` are returned.
If we intend to apply this function on a vector, the optional
``reference_frame`` is also used to inform about the corresponding frame
with respect to which the dynamic symbols of the given vector is to be
determined.
Parameters
==========
expression : SymPy expression
exclude : iterable of dynamicsymbols, optional
reference_frame : ReferenceFrame, optional
The frame with respect to which the dynamic symbols of the
given vector is to be determined.
Examples
========
>>> from sympy.physics.mechanics import dynamicsymbols, find_dynamicsymbols
>>> from sympy.physics.mechanics import ReferenceFrame
>>> x, y = dynamicsymbols('x, y')
>>> expr = x + x.diff()*y
>>> find_dynamicsymbols(expr)
{x(t), y(t), Derivative(x(t), t)}
>>> find_dynamicsymbols(expr, exclude=[x, y])
{Derivative(x(t), t)}
>>> a, b, c = dynamicsymbols('a, b, c')
>>> A = ReferenceFrame('A')
>>> v = a * A.x + b * A.y + c * A.z
>>> find_dynamicsymbols(v, reference_frame=A)
{a(t), b(t), c(t)}
"""
t_set = {dynamicsymbols._t}
if exclude:
if iterable(exclude):
exclude_set = set(exclude)
else:
raise TypeError("exclude kwarg must be iterable")
else:
exclude_set = set()
if isinstance(expression, Vector):
if reference_frame is None:
raise ValueError("You must provide reference_frame when passing a "
"vector expression, got %s." % reference_frame)
else:
expression = expression.to_matrix(reference_frame)
return {i for i in expression.atoms(AppliedUndef, Derivative) if
i.free_symbols == t_set} - exclude_set
def msubs(expr, *sub_dicts, smart=False, **kwargs):
"""A custom subs for use on expressions derived in physics.mechanics.
Traverses the expression tree once, performing the subs found in sub_dicts.
Terms inside ``Derivative`` expressions are ignored:
Examples
========
>>> from sympy.physics.mechanics import dynamicsymbols, msubs
>>> x = dynamicsymbols('x')
>>> msubs(x.diff() + x, {x: 1})
Derivative(x(t), t) + 1
Note that sub_dicts can be a single dictionary, or several dictionaries:
>>> x, y, z = dynamicsymbols('x, y, z')
>>> sub1 = {x: 1, y: 2}
>>> sub2 = {z: 3, x.diff(): 4}
>>> msubs(x.diff() + x + y + z, sub1, sub2)
10
If smart=True (default False), also checks for conditions that may result
in ``nan``, but if simplified would yield a valid expression. For example:
>>> from sympy import sin, tan
>>> (sin(x)/tan(x)).subs(x, 0)
nan
>>> msubs(sin(x)/tan(x), {x: 0}, smart=True)
1
It does this by first replacing all ``tan`` with ``sin/cos``. Then each
node is traversed. If the node is a fraction, subs is first evaluated on
the denominator. If this results in 0, simplification of the entire
fraction is attempted. Using this selective simplification, only
subexpressions that result in 1/0 are targeted, resulting in faster
performance.
"""
sub_dict = dict_merge(*sub_dicts)
if smart:
func = _smart_subs
elif hasattr(expr, 'msubs'):
return expr.msubs(sub_dict)
else:
func = lambda expr, sub_dict: _crawl(expr, _sub_func, sub_dict)
if isinstance(expr, (Matrix, Vector, Dyadic)):
return expr.applyfunc(lambda x: func(x, sub_dict))
else:
return func(expr, sub_dict)
def _crawl(expr, func, *args, **kwargs):
"""Crawl the expression tree, and apply func to every node."""
val = func(expr, *args, **kwargs)
if val is not None:
return val
new_args = (_crawl(arg, func, *args, **kwargs) for arg in expr.args)
return expr.func(*new_args)
def _sub_func(expr, sub_dict):
"""Perform direct matching substitution, ignoring derivatives."""
if expr in sub_dict:
return sub_dict[expr]
elif not expr.args or expr.is_Derivative:
return expr
def _tan_repl_func(expr):
"""Replace tan with sin/cos."""
if isinstance(expr, tan):
return sin(*expr.args) / cos(*expr.args)
elif not expr.args or expr.is_Derivative:
return expr
def _smart_subs(expr, sub_dict):
"""Performs subs, checking for conditions that may result in `nan` or
`oo`, and attempts to simplify them out.
The expression tree is traversed twice, and the following steps are
performed on each expression node:
- First traverse:
Replace all `tan` with `sin/cos`.
- Second traverse:
If node is a fraction, check if the denominator evaluates to 0.
If so, attempt to simplify it out. Then if node is in sub_dict,
sub in the corresponding value."""
expr = _crawl(expr, _tan_repl_func)
def _recurser(expr, sub_dict):
# Decompose the expression into num, den
num, den = _fraction_decomp(expr)
if den != 1:
# If there is a non trivial denominator, we need to handle it
denom_subbed = _recurser(den, sub_dict)
if denom_subbed.evalf() == 0:
# If denom is 0 after this, attempt to simplify the bad expr
expr = simplify(expr)
else:
# Expression won't result in nan, find numerator
num_subbed = _recurser(num, sub_dict)
return num_subbed / denom_subbed
# We have to crawl the tree manually, because `expr` may have been
# modified in the simplify step. First, perform subs as normal:
val = _sub_func(expr, sub_dict)
if val is not None:
return val
new_args = (_recurser(arg, sub_dict) for arg in expr.args)
return expr.func(*new_args)
return _recurser(expr, sub_dict)
def _fraction_decomp(expr):
"""Return num, den such that expr = num/den"""
if not isinstance(expr, Mul):
return expr, 1
num = []
den = []
for a in expr.args:
if a.is_Pow and a.args[1] < 0:
den.append(1 / a)
else:
num.append(a)
if not den:
return expr, 1
num = Mul(*num)
den = Mul(*den)
return num, den
def _f_list_parser(fl, ref_frame):
"""Parses the provided forcelist composed of items
of the form (obj, force).
Returns a tuple containing:
vel_list: The velocity (ang_vel for Frames, vel for Points) in
the provided reference frame.
f_list: The forces.
Used internally in the KanesMethod and LagrangesMethod classes.
"""
def flist_iter():
for pair in fl:
obj, force = pair
if isinstance(obj, ReferenceFrame):
yield obj.ang_vel_in(ref_frame), force
elif isinstance(obj, Point):
yield obj.vel(ref_frame), force
else:
raise TypeError('First entry in each forcelist pair must '
'be a point or frame.')
if not fl:
vel_list, f_list = (), ()
else:
unzip = lambda l: list(zip(*l)) if l[0] else [(), ()]
vel_list, f_list = unzip(list(flist_iter()))
return vel_list, f_list
|
b045f9d8e40ffa927fa48970f2856fd35a103de1e23dfd0b0a6d4632e227d58a | __all__ = ['Linearizer']
from sympy.core.backend import Matrix, eye, zeros
from sympy.core.symbol import Dummy
from sympy.utilities.iterables import flatten
from sympy.physics.vector import dynamicsymbols
from sympy.physics.mechanics.functions import msubs
from collections import namedtuple
from collections.abc import Iterable
class Linearizer:
"""This object holds the general model form for a dynamic system.
This model is used for computing the linearized form of the system,
while properly dealing with constraints leading to dependent
coordinates and speeds.
Attributes
==========
f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a : Matrix
Matrices holding the general system form.
q, u, r : Matrix
Matrices holding the generalized coordinates, speeds, and
input vectors.
q_i, u_i : Matrix
Matrices of the independent generalized coordinates and speeds.
q_d, u_d : Matrix
Matrices of the dependent generalized coordinates and speeds.
perm_mat : Matrix
Permutation matrix such that [q_ind, u_ind]^T = perm_mat*[q, u]^T
"""
def __init__(self, f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a, q, u,
q_i=None, q_d=None, u_i=None, u_d=None, r=None, lams=None):
"""
Parameters
==========
f_0, f_1, f_2, f_3, f_4, f_c, f_v, f_a : array_like
System of equations holding the general system form.
Supply empty array or Matrix if the parameter
does not exist.
q : array_like
The generalized coordinates.
u : array_like
The generalized speeds
q_i, u_i : array_like, optional
The independent generalized coordinates and speeds.
q_d, u_d : array_like, optional
The dependent generalized coordinates and speeds.
r : array_like, optional
The input variables.
lams : array_like, optional
The lagrange multipliers
"""
# Generalized equation form
self.f_0 = Matrix(f_0)
self.f_1 = Matrix(f_1)
self.f_2 = Matrix(f_2)
self.f_3 = Matrix(f_3)
self.f_4 = Matrix(f_4)
self.f_c = Matrix(f_c)
self.f_v = Matrix(f_v)
self.f_a = Matrix(f_a)
# Generalized equation variables
self.q = Matrix(q)
self.u = Matrix(u)
none_handler = lambda x: Matrix(x) if x else Matrix()
self.q_i = none_handler(q_i)
self.q_d = none_handler(q_d)
self.u_i = none_handler(u_i)
self.u_d = none_handler(u_d)
self.r = none_handler(r)
self.lams = none_handler(lams)
# Derivatives of generalized equation variables
self._qd = self.q.diff(dynamicsymbols._t)
self._ud = self.u.diff(dynamicsymbols._t)
# If the user doesn't actually use generalized variables, and the
# qd and u vectors have any intersecting variables, this can cause
# problems. We'll fix this with some hackery, and Dummy variables
dup_vars = set(self._qd).intersection(self.u)
self._qd_dup = Matrix([var if var not in dup_vars else Dummy()
for var in self._qd])
# Derive dimesion terms
l = len(self.f_c)
m = len(self.f_v)
n = len(self.q)
o = len(self.u)
s = len(self.r)
k = len(self.lams)
dims = namedtuple('dims', ['l', 'm', 'n', 'o', 's', 'k'])
self._dims = dims(l, m, n, o, s, k)
self._Pq = None
self._Pqi = None
self._Pqd = None
self._Pu = None
self._Pui = None
self._Pud = None
self._C_0 = None
self._C_1 = None
self._C_2 = None
self.perm_mat = None
self._setup_done = False
def _setup(self):
# Calculations here only need to be run once. They are moved out of
# the __init__ method to increase the speed of Linearizer creation.
self._form_permutation_matrices()
self._form_block_matrices()
self._form_coefficient_matrices()
self._setup_done = True
def _form_permutation_matrices(self):
"""Form the permutation matrices Pq and Pu."""
# Extract dimension variables
l, m, n, o, s, k = self._dims
# Compute permutation matrices
if n != 0:
self._Pq = permutation_matrix(self.q, Matrix([self.q_i, self.q_d]))
if l > 0:
self._Pqi = self._Pq[:, :-l]
self._Pqd = self._Pq[:, -l:]
else:
self._Pqi = self._Pq
self._Pqd = Matrix()
if o != 0:
self._Pu = permutation_matrix(self.u, Matrix([self.u_i, self.u_d]))
if m > 0:
self._Pui = self._Pu[:, :-m]
self._Pud = self._Pu[:, -m:]
else:
self._Pui = self._Pu
self._Pud = Matrix()
# Compute combination permutation matrix for computing A and B
P_col1 = Matrix([self._Pqi, zeros(o + k, n - l)])
P_col2 = Matrix([zeros(n, o - m), self._Pui, zeros(k, o - m)])
if P_col1:
if P_col2:
self.perm_mat = P_col1.row_join(P_col2)
else:
self.perm_mat = P_col1
else:
self.perm_mat = P_col2
def _form_coefficient_matrices(self):
"""Form the coefficient matrices C_0, C_1, and C_2."""
# Extract dimension variables
l, m, n, o, s, k = self._dims
# Build up the coefficient matrices C_0, C_1, and C_2
# If there are configuration constraints (l > 0), form C_0 as normal.
# If not, C_0 is I_(nxn). Note that this works even if n=0
if l > 0:
f_c_jac_q = self.f_c.jacobian(self.q)
self._C_0 = (eye(n) - self._Pqd * (f_c_jac_q *
self._Pqd).LUsolve(f_c_jac_q)) * self._Pqi
else:
self._C_0 = eye(n)
# If there are motion constraints (m > 0), form C_1 and C_2 as normal.
# If not, C_1 is 0, and C_2 is I_(oxo). Note that this works even if
# o = 0.
if m > 0:
f_v_jac_u = self.f_v.jacobian(self.u)
temp = f_v_jac_u * self._Pud
if n != 0:
f_v_jac_q = self.f_v.jacobian(self.q)
self._C_1 = -self._Pud * temp.LUsolve(f_v_jac_q)
else:
self._C_1 = zeros(o, n)
self._C_2 = (eye(o) - self._Pud *
temp.LUsolve(f_v_jac_u)) * self._Pui
else:
self._C_1 = zeros(o, n)
self._C_2 = eye(o)
def _form_block_matrices(self):
"""Form the block matrices for composing M, A, and B."""
# Extract dimension variables
l, m, n, o, s, k = self._dims
# Block Matrix Definitions. These are only defined if under certain
# conditions. If undefined, an empty matrix is used instead
if n != 0:
self._M_qq = self.f_0.jacobian(self._qd)
self._A_qq = -(self.f_0 + self.f_1).jacobian(self.q)
else:
self._M_qq = Matrix()
self._A_qq = Matrix()
if n != 0 and m != 0:
self._M_uqc = self.f_a.jacobian(self._qd_dup)
self._A_uqc = -self.f_a.jacobian(self.q)
else:
self._M_uqc = Matrix()
self._A_uqc = Matrix()
if n != 0 and o - m + k != 0:
self._M_uqd = self.f_3.jacobian(self._qd_dup)
self._A_uqd = -(self.f_2 + self.f_3 + self.f_4).jacobian(self.q)
else:
self._M_uqd = Matrix()
self._A_uqd = Matrix()
if o != 0 and m != 0:
self._M_uuc = self.f_a.jacobian(self._ud)
self._A_uuc = -self.f_a.jacobian(self.u)
else:
self._M_uuc = Matrix()
self._A_uuc = Matrix()
if o != 0 and o - m + k != 0:
self._M_uud = self.f_2.jacobian(self._ud)
self._A_uud = -(self.f_2 + self.f_3).jacobian(self.u)
else:
self._M_uud = Matrix()
self._A_uud = Matrix()
if o != 0 and n != 0:
self._A_qu = -self.f_1.jacobian(self.u)
else:
self._A_qu = Matrix()
if k != 0 and o - m + k != 0:
self._M_uld = self.f_4.jacobian(self.lams)
else:
self._M_uld = Matrix()
if s != 0 and o - m + k != 0:
self._B_u = -self.f_3.jacobian(self.r)
else:
self._B_u = Matrix()
def linearize(self, op_point=None, A_and_B=False, simplify=False):
"""Linearize the system about the operating point. Note that
q_op, u_op, qd_op, ud_op must satisfy the equations of motion.
These may be either symbolic or numeric.
Parameters
==========
op_point : dict or iterable of dicts, optional
Dictionary or iterable of dictionaries containing the operating
point conditions. These will be substituted in to the linearized
system before the linearization is complete. Leave blank if you
want a completely symbolic form. Note that any reduction in
symbols (whether substituted for numbers or expressions with a
common parameter) will result in faster runtime.
A_and_B : bool, optional
If A_and_B=False (default), (M, A, B) is returned for forming
[M]*[q, u]^T = [A]*[q_ind, u_ind]^T + [B]r. If A_and_B=True,
(A, B) is returned for forming dx = [A]x + [B]r, where
x = [q_ind, u_ind]^T.
simplify : bool, optional
Determines if returned values are simplified before return.
For large expressions this may be time consuming. Default is False.
Potential Issues
================
Note that the process of solving with A_and_B=True is
computationally intensive if there are many symbolic parameters.
For this reason, it may be more desirable to use the default
A_and_B=False, returning M, A, and B. More values may then be
substituted in to these matrices later on. The state space form can
then be found as A = P.T*M.LUsolve(A), B = P.T*M.LUsolve(B), where
P = Linearizer.perm_mat.
"""
# Run the setup if needed:
if not self._setup_done:
self._setup()
# Compose dict of operating conditions
if isinstance(op_point, dict):
op_point_dict = op_point
elif isinstance(op_point, Iterable):
op_point_dict = {}
for op in op_point:
op_point_dict.update(op)
else:
op_point_dict = {}
# Extract dimension variables
l, m, n, o, s, k = self._dims
# Rename terms to shorten expressions
M_qq = self._M_qq
M_uqc = self._M_uqc
M_uqd = self._M_uqd
M_uuc = self._M_uuc
M_uud = self._M_uud
M_uld = self._M_uld
A_qq = self._A_qq
A_uqc = self._A_uqc
A_uqd = self._A_uqd
A_qu = self._A_qu
A_uuc = self._A_uuc
A_uud = self._A_uud
B_u = self._B_u
C_0 = self._C_0
C_1 = self._C_1
C_2 = self._C_2
# Build up Mass Matrix
# |M_qq 0_nxo 0_nxk|
# M = |M_uqc M_uuc 0_mxk|
# |M_uqd M_uud M_uld|
if o != 0:
col2 = Matrix([zeros(n, o), M_uuc, M_uud])
if k != 0:
col3 = Matrix([zeros(n + m, k), M_uld])
if n != 0:
col1 = Matrix([M_qq, M_uqc, M_uqd])
if o != 0 and k != 0:
M = col1.row_join(col2).row_join(col3)
elif o != 0:
M = col1.row_join(col2)
else:
M = col1
elif k != 0:
M = col2.row_join(col3)
else:
M = col2
M_eq = msubs(M, op_point_dict)
# Build up state coefficient matrix A
# |(A_qq + A_qu*C_1)*C_0 A_qu*C_2|
# A = |(A_uqc + A_uuc*C_1)*C_0 A_uuc*C_2|
# |(A_uqd + A_uud*C_1)*C_0 A_uud*C_2|
# Col 1 is only defined if n != 0
if n != 0:
r1c1 = A_qq
if o != 0:
r1c1 += (A_qu * C_1)
r1c1 = r1c1 * C_0
if m != 0:
r2c1 = A_uqc
if o != 0:
r2c1 += (A_uuc * C_1)
r2c1 = r2c1 * C_0
else:
r2c1 = Matrix()
if o - m + k != 0:
r3c1 = A_uqd
if o != 0:
r3c1 += (A_uud * C_1)
r3c1 = r3c1 * C_0
else:
r3c1 = Matrix()
col1 = Matrix([r1c1, r2c1, r3c1])
else:
col1 = Matrix()
# Col 2 is only defined if o != 0
if o != 0:
if n != 0:
r1c2 = A_qu * C_2
else:
r1c2 = Matrix()
if m != 0:
r2c2 = A_uuc * C_2
else:
r2c2 = Matrix()
if o - m + k != 0:
r3c2 = A_uud * C_2
else:
r3c2 = Matrix()
col2 = Matrix([r1c2, r2c2, r3c2])
else:
col2 = Matrix()
if col1:
if col2:
Amat = col1.row_join(col2)
else:
Amat = col1
else:
Amat = col2
Amat_eq = msubs(Amat, op_point_dict)
# Build up the B matrix if there are forcing variables
# |0_(n + m)xs|
# B = |B_u |
if s != 0 and o - m + k != 0:
Bmat = zeros(n + m, s).col_join(B_u)
Bmat_eq = msubs(Bmat, op_point_dict)
else:
Bmat_eq = Matrix()
# kwarg A_and_B indicates to return A, B for forming the equation
# dx = [A]x + [B]r, where x = [q_indnd, u_indnd]^T,
if A_and_B:
A_cont = self.perm_mat.T * M_eq.LUsolve(Amat_eq)
if Bmat_eq:
B_cont = self.perm_mat.T * M_eq.LUsolve(Bmat_eq)
else:
# Bmat = Matrix([]), so no need to sub
B_cont = Bmat_eq
if simplify:
A_cont.simplify()
B_cont.simplify()
return A_cont, B_cont
# Otherwise return M, A, B for forming the equation
# [M]dx = [A]x + [B]r, where x = [q, u]^T
else:
if simplify:
M_eq.simplify()
Amat_eq.simplify()
Bmat_eq.simplify()
return M_eq, Amat_eq, Bmat_eq
def permutation_matrix(orig_vec, per_vec):
"""Compute the permutation matrix to change order of
orig_vec into order of per_vec.
Parameters
==========
orig_vec : array_like
Symbols in original ordering.
per_vec : array_like
Symbols in new ordering.
Returns
=======
p_matrix : Matrix
Permutation matrix such that orig_vec == (p_matrix * per_vec).
"""
if not isinstance(orig_vec, (list, tuple)):
orig_vec = flatten(orig_vec)
if not isinstance(per_vec, (list, tuple)):
per_vec = flatten(per_vec)
if set(orig_vec) != set(per_vec):
raise ValueError("orig_vec and per_vec must be the same length, " +
"and contain the same symbols.")
ind_list = [orig_vec.index(i) for i in per_vec]
p_matrix = zeros(len(orig_vec))
for i, j in enumerate(ind_list):
p_matrix[i, j] = 1
return p_matrix
|
84f751aa37e2dacf15276fe1d02665fc5973a2a23a83476882d03eeb0a223ba0 | from sympy.core.backend import sympify
from sympy.physics.vector import Point
from sympy.utilities.exceptions import sympy_deprecation_warning
__all__ = ['Particle']
class Particle:
"""A particle.
Explanation
===========
Particles have a non-zero mass and lack spatial extension; they take up no
space.
Values need to be supplied on initialization, but can be changed later.
Parameters
==========
name : str
Name of particle
point : Point
A physics/mechanics Point which represents the position, velocity, and
acceleration of this Particle
mass : sympifyable
A SymPy expression representing the Particle's mass
Examples
========
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> po = Point('po')
>>> m = Symbol('m')
>>> pa = Particle('pa', po, m)
>>> # Or you could change these later
>>> pa.mass = m
>>> pa.point = po
"""
def __init__(self, name, point, mass):
if not isinstance(name, str):
raise TypeError('Supply a valid name.')
self._name = name
self.mass = mass
self.point = point
self.potential_energy = 0
def __str__(self):
return self._name
def __repr__(self):
return self.__str__()
@property
def mass(self):
"""Mass of the particle."""
return self._mass
@mass.setter
def mass(self, value):
self._mass = sympify(value)
@property
def point(self):
"""Point of the particle."""
return self._point
@point.setter
def point(self, p):
if not isinstance(p, Point):
raise TypeError("Particle point attribute must be a Point object.")
self._point = p
def linear_momentum(self, frame):
"""Linear momentum of the particle.
Explanation
===========
The linear momentum L, of a particle P, with respect to frame N is
given by
L = m * v
where m is the mass of the particle, and v is the velocity of the
particle in the frame N.
Parameters
==========
frame : ReferenceFrame
The frame in which linear momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v = dynamicsymbols('m v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> A = Particle('A', P, m)
>>> P.set_vel(N, v * N.x)
>>> A.linear_momentum(N)
m*v*N.x
"""
return self.mass * self.point.vel(frame)
def angular_momentum(self, point, frame):
"""Angular momentum of the particle about the point.
Explanation
===========
The angular momentum H, about some point O of a particle, P, is given
by:
H = r x m * v
where r is the position vector from point O to the particle P, m is
the mass of the particle, and v is the velocity of the particle in
the inertial frame, N.
Parameters
==========
point : Point
The point about which angular momentum of the particle is desired.
frame : ReferenceFrame
The frame in which angular momentum is desired.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v, r = dynamicsymbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> A = O.locatenew('A', r * N.x)
>>> P = Particle('P', A, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.angular_momentum(O, N)
m*r*v*N.z
"""
return self.point.pos_from(point) ^ (self.mass * self.point.vel(frame))
def kinetic_energy(self, frame):
"""Kinetic energy of the particle.
Explanation
===========
The kinetic energy, T, of a particle, P, is given by
'T = 1/2 m v^2'
where m is the mass of particle P, and v is the velocity of the
particle in the supplied ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The Particle's velocity is typically defined with respect to
an inertial frame but any relevant frame in which the velocity is
known can be supplied.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy import symbols
>>> m, v, r = symbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.kinetic_energy(N)
m*v**2/2
"""
return (self.mass / sympify(2) * self.point.vel(frame) &
self.point.vel(frame))
@property
def potential_energy(self):
"""The potential energy of the Particle.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
>>> P.potential_energy
g*h*m
"""
return self._pe
@potential_energy.setter
def potential_energy(self, scalar):
"""Used to set the potential energy of the Particle.
Parameters
==========
scalar : Sympifyable
The potential energy (a scalar) of the Particle.
Examples
========
>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
"""
self._pe = sympify(scalar)
def set_potential_energy(self, scalar):
sympy_deprecation_warning(
"""
The sympy.physics.mechanics.Particle.set_potential_energy()
method is deprecated. Instead use
P.potential_energy = scalar
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-set-potential-energy",
)
self.potential_energy = scalar
def parallel_axis(self, point, frame):
"""Returns an inertia dyadic of the particle with respect to another
point and frame.
Parameters
==========
point : sympy.physics.vector.Point
The point to express the inertia dyadic about.
frame : sympy.physics.vector.ReferenceFrame
The reference frame used to construct the dyadic.
Returns
=======
inertia : sympy.physics.vector.Dyadic
The inertia dyadic of the particle expressed about the provided
point and frame.
"""
# circular import issue
from sympy.physics.mechanics import inertia_of_point_mass
return inertia_of_point_mass(self.mass, self.point.pos_from(point),
frame)
|
a902742cd86e01e62c67db36365dcc0a6c9da9e84414c647667befe436d9d9e8 | # isort:skip_file
"""
Dimensional analysis and unit systems.
This module defines dimension/unit systems and physical quantities. It is
based on a group-theoretical construction where dimensions are represented as
vectors (coefficients being the exponents), and units are defined as a dimension
to which we added a scale.
Quantities are built from a factor and a unit, and are the basic objects that
one will use when doing computations.
All objects except systems and prefixes can be used in SymPy expressions.
Note that as part of a CAS, various objects do not combine automatically
under operations.
Details about the implementation can be found in the documentation, and we
will not repeat all the explanations we gave there concerning our approach.
Ideas about future developments can be found on the `Github wiki
<https://github.com/sympy/sympy/wiki/Unit-systems>`_, and you should consult
this page if you are willing to help.
Useful functions:
- ``find_unit``: easily lookup pre-defined units.
- ``convert_to(expr, newunit)``: converts an expression into the same
expression expressed in another unit.
"""
from .dimensions import Dimension, DimensionSystem
from .unitsystem import UnitSystem
from .util import convert_to
from .quantities import Quantity
from .definitions.dimension_definitions import (
amount_of_substance, acceleration, action, area,
capacitance, charge, conductance, current, energy,
force, frequency, impedance, inductance, length,
luminous_intensity, magnetic_density,
magnetic_flux, mass, momentum, power, pressure, temperature, time,
velocity, voltage, volume
)
Unit = Quantity
speed = velocity
luminosity = luminous_intensity
magnetic_flux_density = magnetic_density
amount = amount_of_substance
from .prefixes import (
# 10-power based:
yotta,
zetta,
exa,
peta,
tera,
giga,
mega,
kilo,
hecto,
deca,
deci,
centi,
milli,
micro,
nano,
pico,
femto,
atto,
zepto,
yocto,
# 2-power based:
kibi,
mebi,
gibi,
tebi,
pebi,
exbi,
)
from .definitions import (
percent, percents,
permille,
rad, radian, radians,
deg, degree, degrees,
sr, steradian, steradians,
mil, angular_mil, angular_mils,
m, meter, meters,
kg, kilogram, kilograms,
s, second, seconds,
A, ampere, amperes,
K, kelvin, kelvins,
mol, mole, moles,
cd, candela, candelas,
g, gram, grams,
mg, milligram, milligrams,
ug, microgram, micrograms,
t, tonne, metric_ton,
newton, newtons, N,
joule, joules, J,
watt, watts, W,
pascal, pascals, Pa, pa,
hertz, hz, Hz,
coulomb, coulombs, C,
volt, volts, v, V,
ohm, ohms,
siemens, S, mho, mhos,
farad, farads, F,
henry, henrys, H,
tesla, teslas, T,
weber, webers, Wb, wb,
optical_power, dioptre, D,
lux, lx,
katal, kat,
gray, Gy,
becquerel, Bq,
km, kilometer, kilometers,
dm, decimeter, decimeters,
cm, centimeter, centimeters,
mm, millimeter, millimeters,
um, micrometer, micrometers, micron, microns,
nm, nanometer, nanometers,
pm, picometer, picometers,
ft, foot, feet,
inch, inches,
yd, yard, yards,
mi, mile, miles,
nmi, nautical_mile, nautical_miles,
ha, hectare,
l, L, liter, liters,
dl, dL, deciliter, deciliters,
cl, cL, centiliter, centiliters,
ml, mL, milliliter, milliliters,
ms, millisecond, milliseconds,
us, microsecond, microseconds,
ns, nanosecond, nanoseconds,
ps, picosecond, picoseconds,
minute, minutes,
h, hour, hours,
day, days,
anomalistic_year, anomalistic_years,
sidereal_year, sidereal_years,
tropical_year, tropical_years,
common_year, common_years,
julian_year, julian_years,
draconic_year, draconic_years,
gaussian_year, gaussian_years,
full_moon_cycle, full_moon_cycles,
year, years,
G, gravitational_constant,
c, speed_of_light,
elementary_charge,
hbar,
planck,
eV, electronvolt, electronvolts,
avogadro_number,
avogadro, avogadro_constant,
boltzmann, boltzmann_constant,
stefan, stefan_boltzmann_constant,
R, molar_gas_constant,
faraday_constant,
josephson_constant,
von_klitzing_constant,
Da, dalton, amu, amus, atomic_mass_unit, atomic_mass_constant,
gee, gees, acceleration_due_to_gravity,
u0, magnetic_constant, vacuum_permeability,
e0, electric_constant, vacuum_permittivity,
Z0, vacuum_impedance,
coulomb_constant, electric_force_constant,
atmosphere, atmospheres, atm,
kPa,
bar, bars,
pound, pounds,
psi,
dHg0,
mmHg, torr,
mmu, mmus, milli_mass_unit,
quart, quarts,
ly, lightyear, lightyears,
au, astronomical_unit, astronomical_units,
planck_mass,
planck_time,
planck_temperature,
planck_length,
planck_charge,
planck_area,
planck_volume,
planck_momentum,
planck_energy,
planck_force,
planck_power,
planck_density,
planck_energy_density,
planck_intensity,
planck_angular_frequency,
planck_pressure,
planck_current,
planck_voltage,
planck_impedance,
planck_acceleration,
bit, bits,
byte,
kibibyte, kibibytes,
mebibyte, mebibytes,
gibibyte, gibibytes,
tebibyte, tebibytes,
pebibyte, pebibytes,
exbibyte, exbibytes,
)
from .systems import (
mks, mksa, si
)
def find_unit(quantity, unit_system="SI"):
"""
Return a list of matching units or dimension names.
- If ``quantity`` is a string -- units/dimensions containing the string
`quantity`.
- If ``quantity`` is a unit or dimension -- units having matching base
units or dimensions.
Examples
========
>>> from sympy.physics import units as u
>>> u.find_unit('charge')
['C', 'coulomb', 'coulombs', 'planck_charge', 'elementary_charge']
>>> u.find_unit(u.charge)
['C', 'coulomb', 'coulombs', 'planck_charge', 'elementary_charge']
>>> u.find_unit("ampere")
['ampere', 'amperes']
>>> u.find_unit('volt')
['volt', 'volts', 'electronvolt', 'electronvolts', 'planck_voltage']
>>> u.find_unit(u.inch**3)[:9]
['L', 'l', 'cL', 'cl', 'dL', 'dl', 'mL', 'ml', 'liter']
"""
unit_system = UnitSystem.get_unit_system(unit_system)
import sympy.physics.units as u
rv = []
if isinstance(quantity, str):
rv = [i for i in dir(u) if quantity in i and isinstance(getattr(u, i), Quantity)]
dim = getattr(u, quantity)
if isinstance(dim, Dimension):
rv.extend(find_unit(dim))
else:
for i in sorted(dir(u)):
other = getattr(u, i)
if not isinstance(other, Quantity):
continue
if isinstance(quantity, Quantity):
if quantity.dimension == other.dimension:
rv.append(str(i))
elif isinstance(quantity, Dimension):
if other.dimension == quantity:
rv.append(str(i))
elif other.dimension == Dimension(unit_system.get_dimensional_expr(quantity)):
rv.append(str(i))
return sorted(set(rv), key=lambda x: (len(x), x))
# NOTE: the old units module had additional variables:
# 'density', 'illuminance', 'resistance'.
# They were not dimensions, but units (old Unit class).
__all__ = [
'Dimension', 'DimensionSystem',
'UnitSystem',
'convert_to',
'Quantity',
'amount_of_substance', 'acceleration', 'action', 'area',
'capacitance', 'charge', 'conductance', 'current', 'energy',
'force', 'frequency', 'impedance', 'inductance', 'length',
'luminous_intensity', 'magnetic_density',
'magnetic_flux', 'mass', 'momentum', 'power', 'pressure', 'temperature', 'time',
'velocity', 'voltage', 'volume',
'Unit',
'speed',
'luminosity',
'magnetic_flux_density',
'amount',
'yotta',
'zetta',
'exa',
'peta',
'tera',
'giga',
'mega',
'kilo',
'hecto',
'deca',
'deci',
'centi',
'milli',
'micro',
'nano',
'pico',
'femto',
'atto',
'zepto',
'yocto',
'kibi',
'mebi',
'gibi',
'tebi',
'pebi',
'exbi',
'percent', 'percents',
'permille',
'rad', 'radian', 'radians',
'deg', 'degree', 'degrees',
'sr', 'steradian', 'steradians',
'mil', 'angular_mil', 'angular_mils',
'm', 'meter', 'meters',
'kg', 'kilogram', 'kilograms',
's', 'second', 'seconds',
'A', 'ampere', 'amperes',
'K', 'kelvin', 'kelvins',
'mol', 'mole', 'moles',
'cd', 'candela', 'candelas',
'g', 'gram', 'grams',
'mg', 'milligram', 'milligrams',
'ug', 'microgram', 'micrograms',
't', 'tonne', 'metric_ton',
'newton', 'newtons', 'N',
'joule', 'joules', 'J',
'watt', 'watts', 'W',
'pascal', 'pascals', 'Pa', 'pa',
'hertz', 'hz', 'Hz',
'coulomb', 'coulombs', 'C',
'volt', 'volts', 'v', 'V',
'ohm', 'ohms',
'siemens', 'S', 'mho', 'mhos',
'farad', 'farads', 'F',
'henry', 'henrys', 'H',
'tesla', 'teslas', 'T',
'weber', 'webers', 'Wb', 'wb',
'optical_power', 'dioptre', 'D',
'lux', 'lx',
'katal', 'kat',
'gray', 'Gy',
'becquerel', 'Bq',
'km', 'kilometer', 'kilometers',
'dm', 'decimeter', 'decimeters',
'cm', 'centimeter', 'centimeters',
'mm', 'millimeter', 'millimeters',
'um', 'micrometer', 'micrometers', 'micron', 'microns',
'nm', 'nanometer', 'nanometers',
'pm', 'picometer', 'picometers',
'ft', 'foot', 'feet',
'inch', 'inches',
'yd', 'yard', 'yards',
'mi', 'mile', 'miles',
'nmi', 'nautical_mile', 'nautical_miles',
'ha', 'hectare',
'l', 'L', 'liter', 'liters',
'dl', 'dL', 'deciliter', 'deciliters',
'cl', 'cL', 'centiliter', 'centiliters',
'ml', 'mL', 'milliliter', 'milliliters',
'ms', 'millisecond', 'milliseconds',
'us', 'microsecond', 'microseconds',
'ns', 'nanosecond', 'nanoseconds',
'ps', 'picosecond', 'picoseconds',
'minute', 'minutes',
'h', 'hour', 'hours',
'day', 'days',
'anomalistic_year', 'anomalistic_years',
'sidereal_year', 'sidereal_years',
'tropical_year', 'tropical_years',
'common_year', 'common_years',
'julian_year', 'julian_years',
'draconic_year', 'draconic_years',
'gaussian_year', 'gaussian_years',
'full_moon_cycle', 'full_moon_cycles',
'year', 'years',
'G', 'gravitational_constant',
'c', 'speed_of_light',
'elementary_charge',
'hbar',
'planck',
'eV', 'electronvolt', 'electronvolts',
'avogadro_number',
'avogadro', 'avogadro_constant',
'boltzmann', 'boltzmann_constant',
'stefan', 'stefan_boltzmann_constant',
'R', 'molar_gas_constant',
'faraday_constant',
'josephson_constant',
'von_klitzing_constant',
'Da', 'dalton', 'amu', 'amus', 'atomic_mass_unit', 'atomic_mass_constant',
'gee', 'gees', 'acceleration_due_to_gravity',
'u0', 'magnetic_constant', 'vacuum_permeability',
'e0', 'electric_constant', 'vacuum_permittivity',
'Z0', 'vacuum_impedance',
'coulomb_constant', 'electric_force_constant',
'atmosphere', 'atmospheres', 'atm',
'kPa',
'bar', 'bars',
'pound', 'pounds',
'psi',
'dHg0',
'mmHg', 'torr',
'mmu', 'mmus', 'milli_mass_unit',
'quart', 'quarts',
'ly', 'lightyear', 'lightyears',
'au', 'astronomical_unit', 'astronomical_units',
'planck_mass',
'planck_time',
'planck_temperature',
'planck_length',
'planck_charge',
'planck_area',
'planck_volume',
'planck_momentum',
'planck_energy',
'planck_force',
'planck_power',
'planck_density',
'planck_energy_density',
'planck_intensity',
'planck_angular_frequency',
'planck_pressure',
'planck_current',
'planck_voltage',
'planck_impedance',
'planck_acceleration',
'bit', 'bits',
'byte',
'kibibyte', 'kibibytes',
'mebibyte', 'mebibytes',
'gibibyte', 'gibibytes',
'tebibyte', 'tebibytes',
'pebibyte', 'pebibytes',
'exbibyte', 'exbibytes',
'mks', 'mksa', 'si',
]
|
ce87a8dca0d930fc76c65254498c418da3052d4fc3d1d8c4a5f80b83f45849c3 | """
Unit system for physical quantities; include definition of constants.
"""
from typing import Dict as tDict
from sympy.core.add import Add
from sympy.core.function import (Derivative, Function)
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.physics.units.dimensions import _QuantityMapper
from .dimensions import Dimension
class UnitSystem(_QuantityMapper):
"""
UnitSystem represents a coherent set of units.
A unit system is basically a dimension system with notions of scales. Many
of the methods are defined in the same way.
It is much better if all base units have a symbol.
"""
_unit_systems = {} # type: tDict[str, UnitSystem]
def __init__(self, base_units, units=(), name="", descr="", dimension_system=None):
UnitSystem._unit_systems[name] = self
self.name = name
self.descr = descr
self._base_units = base_units
self._dimension_system = dimension_system
self._units = tuple(set(base_units) | set(units))
self._base_units = tuple(base_units)
super().__init__()
def __str__(self):
"""
Return the name of the system.
If it does not exist, then it makes a list of symbols (or names) of
the base dimensions.
"""
if self.name != "":
return self.name
else:
return "UnitSystem((%s))" % ", ".join(
str(d) for d in self._base_units)
def __repr__(self):
return '<UnitSystem: %s>' % repr(self._base_units)
def extend(self, base, units=(), name="", description="", dimension_system=None):
"""Extend the current system into a new one.
Take the base and normal units of the current system to merge
them to the base and normal units given in argument.
If not provided, name and description are overridden by empty strings.
"""
base = self._base_units + tuple(base)
units = self._units + tuple(units)
return UnitSystem(base, units, name, description, dimension_system)
def get_dimension_system(self):
return self._dimension_system
def get_quantity_dimension(self, unit):
qdm = self.get_dimension_system()._quantity_dimension_map
if unit in qdm:
return qdm[unit]
return super().get_quantity_dimension(unit)
def get_quantity_scale_factor(self, unit):
qsfm = self.get_dimension_system()._quantity_scale_factors
if unit in qsfm:
return qsfm[unit]
return super().get_quantity_scale_factor(unit)
@staticmethod
def get_unit_system(unit_system):
if isinstance(unit_system, UnitSystem):
return unit_system
if unit_system not in UnitSystem._unit_systems:
raise ValueError(
"Unit system is not supported. Currently"
"supported unit systems are {}".format(
", ".join(sorted(UnitSystem._unit_systems))
)
)
return UnitSystem._unit_systems[unit_system]
@staticmethod
def get_default_unit_system():
return UnitSystem._unit_systems["SI"]
@property
def dim(self):
"""
Give the dimension of the system.
That is return the number of units forming the basis.
"""
return len(self._base_units)
@property
def is_consistent(self):
"""
Check if the underlying dimension system is consistent.
"""
# test is performed in DimensionSystem
return self.get_dimension_system().is_consistent
def get_dimensional_expr(self, expr):
from sympy.physics.units import Quantity
if isinstance(expr, Mul):
return Mul(*[self.get_dimensional_expr(i) for i in expr.args])
elif isinstance(expr, Pow):
return self.get_dimensional_expr(expr.base) ** expr.exp
elif isinstance(expr, Add):
return self.get_dimensional_expr(expr.args[0])
elif isinstance(expr, Derivative):
dim = self.get_dimensional_expr(expr.expr)
for independent, count in expr.variable_count:
dim /= self.get_dimensional_expr(independent)**count
return dim
elif isinstance(expr, Function):
args = [self.get_dimensional_expr(arg) for arg in expr.args]
if all(i == 1 for i in args):
return S.One
return expr.func(*args)
elif isinstance(expr, Quantity):
return self.get_quantity_dimension(expr).name
return S.One
def _collect_factor_and_dimension(self, expr):
"""
Return tuple with scale factor expression and dimension expression.
"""
from sympy.physics.units import Quantity
if isinstance(expr, Quantity):
return expr.scale_factor, expr.dimension
elif isinstance(expr, Mul):
factor = 1
dimension = Dimension(1)
for arg in expr.args:
arg_factor, arg_dim = self._collect_factor_and_dimension(arg)
factor *= arg_factor
dimension *= arg_dim
return factor, dimension
elif isinstance(expr, Pow):
factor, dim = self._collect_factor_and_dimension(expr.base)
exp_factor, exp_dim = self._collect_factor_and_dimension(expr.exp)
if self.get_dimension_system().is_dimensionless(exp_dim):
exp_dim = 1
return factor ** exp_factor, dim ** (exp_factor * exp_dim)
elif isinstance(expr, Add):
factor, dim = self._collect_factor_and_dimension(expr.args[0])
for addend in expr.args[1:]:
addend_factor, addend_dim = \
self._collect_factor_and_dimension(addend)
if dim != addend_dim:
raise ValueError(
'Dimension of "{}" is {}, '
'but it should be {}'.format(
addend, addend_dim, dim))
factor += addend_factor
return factor, dim
elif isinstance(expr, Derivative):
factor, dim = self._collect_factor_and_dimension(expr.args[0])
for independent, count in expr.variable_count:
ifactor, idim = self._collect_factor_and_dimension(independent)
factor /= ifactor**count
dim /= idim**count
return factor, dim
elif isinstance(expr, Function):
fds = [self._collect_factor_and_dimension(
arg) for arg in expr.args]
return (expr.func(*(f[0] for f in fds)),
expr.func(*(d[1] for d in fds)))
elif isinstance(expr, Dimension):
return S.One, expr
else:
return expr, Dimension(1)
|
a70f2efbed0058ca5d55b825cf336585d78361e8da56839211cac6c7856c7840 | """
Definition of physical dimensions.
Unit systems will be constructed on top of these dimensions.
Most of the examples in the doc use MKS system and are presented from the
computer point of view: from a human point, adding length to time is not legal
in MKS but it is in natural system; for a computer in natural system there is
no time dimension (but a velocity dimension instead) - in the basis - so the
question of adding time to length has no meaning.
"""
from typing import Dict as tDict
import collections
from functools import reduce
from sympy.core.basic import Basic
from sympy.core.containers import (Dict, Tuple)
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.matrices.dense import Matrix
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.core.expr import Expr
from sympy.core.power import Pow
class _QuantityMapper:
_quantity_scale_factors_global = {} # type: tDict[Expr, Expr]
_quantity_dimensional_equivalence_map_global = {} # type: tDict[Expr, Expr]
_quantity_dimension_global = {} # type: tDict[Expr, Expr]
def __init__(self, *args, **kwargs):
self._quantity_dimension_map = {}
self._quantity_scale_factors = {}
def set_quantity_dimension(self, unit, dimension):
from sympy.physics.units import Quantity
dimension = sympify(dimension)
if not isinstance(dimension, Dimension):
if dimension == 1:
dimension = Dimension(1)
else:
raise ValueError("expected dimension or 1")
elif isinstance(dimension, Quantity):
dimension = self.get_quantity_dimension(dimension)
self._quantity_dimension_map[unit] = dimension
def set_quantity_scale_factor(self, unit, scale_factor):
from sympy.physics.units import Quantity
from sympy.physics.units.prefixes import Prefix
scale_factor = sympify(scale_factor)
# replace all prefixes by their ratio to canonical units:
scale_factor = scale_factor.replace(
lambda x: isinstance(x, Prefix),
lambda x: x.scale_factor
)
# replace all quantities by their ratio to canonical units:
scale_factor = scale_factor.replace(
lambda x: isinstance(x, Quantity),
lambda x: self.get_quantity_scale_factor(x)
)
self._quantity_scale_factors[unit] = scale_factor
def get_quantity_dimension(self, unit):
from sympy.physics.units import Quantity
# First look-up the local dimension map, then the global one:
if unit in self._quantity_dimension_map:
return self._quantity_dimension_map[unit]
if unit in self._quantity_dimension_global:
return self._quantity_dimension_global[unit]
if unit in self._quantity_dimensional_equivalence_map_global:
dep_unit = self._quantity_dimensional_equivalence_map_global[unit]
if isinstance(dep_unit, Quantity):
return self.get_quantity_dimension(dep_unit)
else:
return Dimension(self.get_dimensional_expr(dep_unit))
if isinstance(unit, Quantity):
return Dimension(unit.name)
else:
return Dimension(1)
def get_quantity_scale_factor(self, unit):
if unit in self._quantity_scale_factors:
return self._quantity_scale_factors[unit]
if unit in self._quantity_scale_factors_global:
mul_factor, other_unit = self._quantity_scale_factors_global[unit]
return mul_factor*self.get_quantity_scale_factor(other_unit)
return S.One
class Dimension(Expr):
"""
This class represent the dimension of a physical quantities.
The ``Dimension`` constructor takes as parameters a name and an optional
symbol.
For example, in classical mechanics we know that time is different from
temperature and dimensions make this difference (but they do not provide
any measure of these quantites.
>>> from sympy.physics.units import Dimension
>>> length = Dimension('length')
>>> length
Dimension(length)
>>> time = Dimension('time')
>>> time
Dimension(time)
Dimensions can be composed using multiplication, division and
exponentiation (by a number) to give new dimensions. Addition and
subtraction is defined only when the two objects are the same dimension.
>>> velocity = length / time
>>> velocity
Dimension(length/time)
It is possible to use a dimension system object to get the dimensionsal
dependencies of a dimension, for example the dimension system used by the
SI units convention can be used:
>>> from sympy.physics.units.systems.si import dimsys_SI
>>> dimsys_SI.get_dimensional_dependencies(velocity)
{'length': 1, 'time': -1}
>>> length + length
Dimension(length)
>>> l2 = length**2
>>> l2
Dimension(length**2)
>>> dimsys_SI.get_dimensional_dependencies(l2)
{'length': 2}
"""
_op_priority = 13.0
# XXX: This doesn't seem to be used anywhere...
_dimensional_dependencies = dict() # type: ignore
is_commutative = True
is_number = False
# make sqrt(M**2) --> M
is_positive = True
is_real = True
def __new__(cls, name, symbol=None):
if isinstance(name, str):
name = Symbol(name)
else:
name = sympify(name)
if not isinstance(name, Expr):
raise TypeError("Dimension name needs to be a valid math expression")
if isinstance(symbol, str):
symbol = Symbol(symbol)
elif symbol is not None:
assert isinstance(symbol, Symbol)
if symbol is not None:
obj = Expr.__new__(cls, name, symbol)
else:
obj = Expr.__new__(cls, name)
obj._name = name
obj._symbol = symbol
return obj
@property
def name(self):
return self._name
@property
def symbol(self):
return self._symbol
def __hash__(self):
return Expr.__hash__(self)
def __eq__(self, other):
if isinstance(other, Dimension):
return self.name == other.name
return False
def __str__(self):
"""
Display the string representation of the dimension.
"""
if self.symbol is None:
return "Dimension(%s)" % (self.name)
else:
return "Dimension(%s, %s)" % (self.name, self.symbol)
def __repr__(self):
return self.__str__()
def __neg__(self):
return self
def __add__(self, other):
from sympy.physics.units.quantities import Quantity
other = sympify(other)
if isinstance(other, Basic):
if other.has(Quantity):
raise TypeError("cannot sum dimension and quantity")
if isinstance(other, Dimension) and self == other:
return self
return super().__add__(other)
return self
def __radd__(self, other):
return self.__add__(other)
def __sub__(self, other):
# there is no notion of ordering (or magnitude) among dimension,
# subtraction is equivalent to addition when the operation is legal
return self + other
def __rsub__(self, other):
# there is no notion of ordering (or magnitude) among dimension,
# subtraction is equivalent to addition when the operation is legal
return self + other
def __pow__(self, other):
return self._eval_power(other)
def _eval_power(self, other):
other = sympify(other)
return Dimension(self.name**other)
def __mul__(self, other):
from sympy.physics.units.quantities import Quantity
if isinstance(other, Basic):
if other.has(Quantity):
raise TypeError("cannot sum dimension and quantity")
if isinstance(other, Dimension):
return Dimension(self.name*other.name)
if not other.free_symbols: # other.is_number cannot be used
return self
return super().__mul__(other)
return self
def __rmul__(self, other):
return self.__mul__(other)
def __truediv__(self, other):
return self*Pow(other, -1)
def __rtruediv__(self, other):
return other * pow(self, -1)
@classmethod
def _from_dimensional_dependencies(cls, dependencies):
return reduce(lambda x, y: x * y, (
Dimension(d)**e for d, e in dependencies.items()
), 1)
def has_integer_powers(self, dim_sys):
"""
Check if the dimension object has only integer powers.
All the dimension powers should be integers, but rational powers may
appear in intermediate steps. This method may be used to check that the
final result is well-defined.
"""
return all(dpow.is_Integer for dpow in dim_sys.get_dimensional_dependencies(self).values())
# Create dimensions according to the base units in MKSA.
# For other unit systems, they can be derived by transforming the base
# dimensional dependency dictionary.
class DimensionSystem(Basic, _QuantityMapper):
r"""
DimensionSystem represents a coherent set of dimensions.
The constructor takes three parameters:
- base dimensions;
- derived dimensions: these are defined in terms of the base dimensions
(for example velocity is defined from the division of length by time);
- dependency of dimensions: how the derived dimensions depend
on the base dimensions.
Optionally either the ``derived_dims`` or the ``dimensional_dependencies``
may be omitted.
"""
def __new__(cls, base_dims, derived_dims=(), dimensional_dependencies={}):
dimensional_dependencies = dict(dimensional_dependencies)
def parse_dim(dim):
if isinstance(dim, str):
dim = Dimension(Symbol(dim))
elif isinstance(dim, Dimension):
pass
elif isinstance(dim, Symbol):
dim = Dimension(dim)
else:
raise TypeError("%s wrong type" % dim)
return dim
base_dims = [parse_dim(i) for i in base_dims]
derived_dims = [parse_dim(i) for i in derived_dims]
for dim in base_dims:
dim = dim.name
if (dim in dimensional_dependencies
and (len(dimensional_dependencies[dim]) != 1 or
dimensional_dependencies[dim].get(dim, None) != 1)):
raise IndexError("Repeated value in base dimensions")
dimensional_dependencies[dim] = Dict({dim: 1})
def parse_dim_name(dim):
if isinstance(dim, Dimension):
return dim.name
elif isinstance(dim, str):
return Symbol(dim)
elif isinstance(dim, Symbol):
return dim
else:
raise TypeError("unrecognized type %s for %s" % (type(dim), dim))
for dim in dimensional_dependencies.keys():
dim = parse_dim(dim)
if (dim not in derived_dims) and (dim not in base_dims):
derived_dims.append(dim)
def parse_dict(d):
return Dict({parse_dim_name(i): j for i, j in d.items()})
# Make sure everything is a SymPy type:
dimensional_dependencies = {parse_dim_name(i): parse_dict(j) for i, j in
dimensional_dependencies.items()}
for dim in derived_dims:
if dim in base_dims:
raise ValueError("Dimension %s both in base and derived" % dim)
if dim.name not in dimensional_dependencies:
# TODO: should this raise a warning?
dimensional_dependencies[dim.name] = Dict({dim.name: 1})
base_dims.sort(key=default_sort_key)
derived_dims.sort(key=default_sort_key)
base_dims = Tuple(*base_dims)
derived_dims = Tuple(*derived_dims)
dimensional_dependencies = Dict({i: Dict(j) for i, j in dimensional_dependencies.items()})
obj = Basic.__new__(cls, base_dims, derived_dims, dimensional_dependencies)
return obj
@property
def base_dims(self):
return self.args[0]
@property
def derived_dims(self):
return self.args[1]
@property
def dimensional_dependencies(self):
return self.args[2]
def _get_dimensional_dependencies_for_name(self, name):
if isinstance(name, Dimension):
name = name.name
if isinstance(name, str):
name = Symbol(name)
if name.is_Symbol:
# Dimensions not included in the dependencies are considered
# as base dimensions:
return dict(self.dimensional_dependencies.get(name, {name: 1}))
if name.is_number or name.is_NumberSymbol:
return {}
get_for_name = self._get_dimensional_dependencies_for_name
if name.is_Mul:
ret = collections.defaultdict(int)
dicts = [get_for_name(i) for i in name.args]
for d in dicts:
for k, v in d.items():
ret[k] += v
return {k: v for (k, v) in ret.items() if v != 0}
if name.is_Add:
dicts = [get_for_name(i) for i in name.args]
if all(d == dicts[0] for d in dicts[1:]):
return dicts[0]
raise TypeError("Only equivalent dimensions can be added or subtracted.")
if name.is_Pow:
dim_base = get_for_name(name.base)
dim_exp = get_for_name(name.exp)
if dim_exp == {} or name.exp.is_Symbol:
return {k: v*name.exp for (k, v) in dim_base.items()}
else:
raise TypeError("The exponent for the power operator must be a Symbol or dimensionless.")
if name.is_Function:
args = (Dimension._from_dimensional_dependencies(
get_for_name(arg)) for arg in name.args)
result = name.func(*args)
dicts = [get_for_name(i) for i in name.args]
if isinstance(result, Dimension):
return self.get_dimensional_dependencies(result)
elif result.func == name.func:
if isinstance(name, TrigonometricFunction):
if dicts[0] in ({}, {Symbol('angle'): 1}):
return {}
else:
raise TypeError("The input argument for the function {} must be dimensionless or have dimensions of angle.".format(name.func))
else:
if all( (item == {} for item in dicts) ):
return {}
else:
raise TypeError("The input arguments for the function {} must be dimensionless.".format(name.func))
else:
return get_for_name(result)
raise TypeError("Type {} not implemented for get_dimensional_dependencies".format(type(name)))
def get_dimensional_dependencies(self, name, mark_dimensionless=False):
dimdep = self._get_dimensional_dependencies_for_name(name)
if mark_dimensionless and dimdep == {}:
return {'dimensionless': 1}
return {str(i): j for i, j in dimdep.items()}
def equivalent_dims(self, dim1, dim2):
deps1 = self.get_dimensional_dependencies(dim1)
deps2 = self.get_dimensional_dependencies(dim2)
return deps1 == deps2
def extend(self, new_base_dims, new_derived_dims=(), new_dim_deps=None):
deps = dict(self.dimensional_dependencies)
if new_dim_deps:
deps.update(new_dim_deps)
new_dim_sys = DimensionSystem(
tuple(self.base_dims) + tuple(new_base_dims),
tuple(self.derived_dims) + tuple(new_derived_dims),
deps
)
new_dim_sys._quantity_dimension_map.update(self._quantity_dimension_map)
new_dim_sys._quantity_scale_factors.update(self._quantity_scale_factors)
return new_dim_sys
def is_dimensionless(self, dimension):
"""
Check if the dimension object really has a dimension.
A dimension should have at least one component with non-zero power.
"""
if dimension.name == 1:
return True
return self.get_dimensional_dependencies(dimension) == {}
@property
def list_can_dims(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
List all canonical dimension names.
"""
dimset = set()
for i in self.base_dims:
dimset.update(set(self.get_dimensional_dependencies(i).keys()))
return tuple(sorted(dimset, key=str))
@property
def inv_can_transf_matrix(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Compute the inverse transformation matrix from the base to the
canonical dimension basis.
It corresponds to the matrix where columns are the vector of base
dimensions in canonical basis.
This matrix will almost never be used because dimensions are always
defined with respect to the canonical basis, so no work has to be done
to get them in this basis. Nonetheless if this matrix is not square
(or not invertible) it means that we have chosen a bad basis.
"""
matrix = reduce(lambda x, y: x.row_join(y),
[self.dim_can_vector(d) for d in self.base_dims])
return matrix
@property
def can_transf_matrix(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Return the canonical transformation matrix from the canonical to the
base dimension basis.
It is the inverse of the matrix computed with inv_can_transf_matrix().
"""
#TODO: the inversion will fail if the system is inconsistent, for
# example if the matrix is not a square
return reduce(lambda x, y: x.row_join(y),
[self.dim_can_vector(d) for d in sorted(self.base_dims, key=str)]
).inv()
def dim_can_vector(self, dim):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Dimensional representation in terms of the canonical base dimensions.
"""
vec = []
for d in self.list_can_dims:
vec.append(self.get_dimensional_dependencies(dim).get(d, 0))
return Matrix(vec)
def dim_vector(self, dim):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Vector representation in terms of the base dimensions.
"""
return self.can_transf_matrix * Matrix(self.dim_can_vector(dim))
def print_dim_base(self, dim):
"""
Give the string expression of a dimension in term of the basis symbols.
"""
dims = self.dim_vector(dim)
symbols = [i.symbol if i.symbol is not None else i.name for i in self.base_dims]
res = S.One
for (s, p) in zip(symbols, dims):
res *= s**p
return res
@property
def dim(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Give the dimension of the system.
That is return the number of dimensions forming the basis.
"""
return len(self.base_dims)
@property
def is_consistent(self):
"""
Useless method, kept for compatibility with previous versions.
DO NOT USE.
Check if the system is well defined.
"""
# not enough or too many base dimensions compared to independent
# dimensions
# in vector language: the set of vectors do not form a basis
return self.inv_can_transf_matrix.is_square
|
825c042aee17afc5701a31d27e7d3e7c956059dd527f9b90fabd5f9e748c3818 | """
Module defining unit prefixe class and some constants.
Constant dict for SI and binary prefixes are defined as PREFIXES and
BIN_PREFIXES.
"""
from sympy.core.expr import Expr
from sympy.core.sympify import sympify
class Prefix(Expr):
"""
This class represent prefixes, with their name, symbol and factor.
Prefixes are used to create derived units from a given unit. They should
always be encapsulated into units.
The factor is constructed from a base (default is 10) to some power, and
it gives the total multiple or fraction. For example the kilometer km
is constructed from the meter (factor 1) and the kilo (10 to the power 3,
i.e. 1000). The base can be changed to allow e.g. binary prefixes.
A prefix multiplied by something will always return the product of this
other object times the factor, except if the other object:
- is a prefix and they can be combined into a new prefix;
- defines multiplication with prefixes (which is the case for the Unit
class).
"""
_op_priority = 13.0
is_commutative = True
def __new__(cls, name, abbrev, exponent, base=sympify(10)):
name = sympify(name)
abbrev = sympify(abbrev)
exponent = sympify(exponent)
base = sympify(base)
obj = Expr.__new__(cls, name, abbrev, exponent, base)
obj._name = name
obj._abbrev = abbrev
obj._scale_factor = base**exponent
obj._exponent = exponent
obj._base = base
return obj
@property
def name(self):
return self._name
@property
def abbrev(self):
return self._abbrev
@property
def scale_factor(self):
return self._scale_factor
@property
def base(self):
return self._base
def __str__(self):
# TODO: add proper printers and tests:
if self.base == 10:
return "Prefix(%r, %r, %r)" % (
str(self.name), str(self.abbrev), self._exponent)
else:
return "Prefix(%r, %r, %r, %r)" % (
str(self.name), str(self.abbrev), self._exponent, self.base)
__repr__ = __str__
def __mul__(self, other):
from sympy.physics.units import Quantity
if not isinstance(other, (Quantity, Prefix)):
return super().__mul__(other)
fact = self.scale_factor * other.scale_factor
if fact == 1:
return 1
elif isinstance(other, Prefix):
# simplify prefix
for p in PREFIXES:
if PREFIXES[p].scale_factor == fact:
return PREFIXES[p]
return fact
return self.scale_factor * other
def __truediv__(self, other):
if not hasattr(other, "scale_factor"):
return super().__truediv__(other)
fact = self.scale_factor / other.scale_factor
if fact == 1:
return 1
elif isinstance(other, Prefix):
for p in PREFIXES:
if PREFIXES[p].scale_factor == fact:
return PREFIXES[p]
return fact
return self.scale_factor / other
def __rtruediv__(self, other):
if other == 1:
for p in PREFIXES:
if PREFIXES[p].scale_factor == 1 / self.scale_factor:
return PREFIXES[p]
return other / self.scale_factor
def prefix_unit(unit, prefixes):
"""
Return a list of all units formed by unit and the given prefixes.
You can use the predefined PREFIXES or BIN_PREFIXES, but you can also
pass as argument a subdict of them if you do not want all prefixed units.
>>> from sympy.physics.units.prefixes import (PREFIXES,
... prefix_unit)
>>> from sympy.physics.units import m
>>> pref = {"m": PREFIXES["m"], "c": PREFIXES["c"], "d": PREFIXES["d"]}
>>> prefix_unit(m, pref) # doctest: +SKIP
[millimeter, centimeter, decimeter]
"""
from sympy.physics.units.quantities import Quantity
from sympy.physics.units import UnitSystem
prefixed_units = []
for prefix_abbr, prefix in prefixes.items():
quantity = Quantity(
"%s%s" % (prefix.name, unit.name),
abbrev=("%s%s" % (prefix.abbrev, unit.abbrev))
)
UnitSystem._quantity_dimensional_equivalence_map_global[quantity] = unit
UnitSystem._quantity_scale_factors_global[quantity] = (prefix.scale_factor, unit)
prefixed_units.append(quantity)
return prefixed_units
yotta = Prefix('yotta', 'Y', 24)
zetta = Prefix('zetta', 'Z', 21)
exa = Prefix('exa', 'E', 18)
peta = Prefix('peta', 'P', 15)
tera = Prefix('tera', 'T', 12)
giga = Prefix('giga', 'G', 9)
mega = Prefix('mega', 'M', 6)
kilo = Prefix('kilo', 'k', 3)
hecto = Prefix('hecto', 'h', 2)
deca = Prefix('deca', 'da', 1)
deci = Prefix('deci', 'd', -1)
centi = Prefix('centi', 'c', -2)
milli = Prefix('milli', 'm', -3)
micro = Prefix('micro', 'mu', -6)
nano = Prefix('nano', 'n', -9)
pico = Prefix('pico', 'p', -12)
femto = Prefix('femto', 'f', -15)
atto = Prefix('atto', 'a', -18)
zepto = Prefix('zepto', 'z', -21)
yocto = Prefix('yocto', 'y', -24)
# http://physics.nist.gov/cuu/Units/prefixes.html
PREFIXES = {
'Y': yotta,
'Z': zetta,
'E': exa,
'P': peta,
'T': tera,
'G': giga,
'M': mega,
'k': kilo,
'h': hecto,
'da': deca,
'd': deci,
'c': centi,
'm': milli,
'mu': micro,
'n': nano,
'p': pico,
'f': femto,
'a': atto,
'z': zepto,
'y': yocto,
}
kibi = Prefix('kibi', 'Y', 10, 2)
mebi = Prefix('mebi', 'Y', 20, 2)
gibi = Prefix('gibi', 'Y', 30, 2)
tebi = Prefix('tebi', 'Y', 40, 2)
pebi = Prefix('pebi', 'Y', 50, 2)
exbi = Prefix('exbi', 'Y', 60, 2)
# http://physics.nist.gov/cuu/Units/binary.html
BIN_PREFIXES = {
'Ki': kibi,
'Mi': mebi,
'Gi': gibi,
'Ti': tebi,
'Pi': pebi,
'Ei': exbi,
}
|
28f9c9ca3beb96528aa3b1796809ca9b2d86529968ee17e770f1754c28d78b02 | """
Physical quantities.
"""
from sympy.core.expr import AtomicExpr
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.physics.units.dimensions import _QuantityMapper
from sympy.physics.units.prefixes import Prefix
from sympy.utilities.exceptions import (sympy_deprecation_warning,
SymPyDeprecationWarning,
ignore_warnings)
class Quantity(AtomicExpr):
"""
Physical quantity: can be a unit of measure, a constant or a generic quantity.
"""
is_commutative = True
is_real = True
is_number = False
is_nonzero = True
_diff_wrt = True
def __new__(cls, name, abbrev=None, dimension=None, scale_factor=None,
latex_repr=None, pretty_unicode_repr=None,
pretty_ascii_repr=None, mathml_presentation_repr=None,
**assumptions):
if not isinstance(name, Symbol):
name = Symbol(name)
# For Quantity(name, dim, scale, abbrev) to work like in the
# old version of SymPy:
if not isinstance(abbrev, str) and not \
isinstance(abbrev, Symbol):
dimension, scale_factor, abbrev = abbrev, dimension, scale_factor
if dimension is not None:
sympy_deprecation_warning(
"""
The 'dimension' argument to to Quantity() is deprecated.
Instead use the unit_system.set_quantity_dimension() method.
""",
deprecated_since_version="1.3",
active_deprecations_target="deprecated-quantity-dimension-scale-factor"
)
if scale_factor is not None:
sympy_deprecation_warning(
"""
The 'scale_factor' argument to to Quantity() is deprecated.
Instead use the unit_system.set_quantity_scale_factors()
method.
""",
deprecated_since_version="1.3",
active_deprecations_target="deprecated-quantity-dimension-scale-factor"
)
if abbrev is None:
abbrev = name
elif isinstance(abbrev, str):
abbrev = Symbol(abbrev)
obj = AtomicExpr.__new__(cls, name, abbrev)
obj._name = name
obj._abbrev = abbrev
obj._latex_repr = latex_repr
obj._unicode_repr = pretty_unicode_repr
obj._ascii_repr = pretty_ascii_repr
obj._mathml_repr = mathml_presentation_repr
if dimension is not None:
# TODO: remove after deprecation:
with ignore_warnings(SymPyDeprecationWarning):
obj.set_dimension(dimension)
if scale_factor is not None:
# TODO: remove after deprecation:
with ignore_warnings(SymPyDeprecationWarning):
obj.set_scale_factor(scale_factor)
return obj
def set_dimension(self, dimension, unit_system="SI"):
sympy_deprecation_warning(
f"""
Quantity.set_dimension() is deprecated. Use either
unit_system.set_quantity_dimension() or
{self}.set_global_dimension() instead.
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-quantity-methods",
)
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_unit_system(unit_system)
unit_system.set_quantity_dimension(self, dimension)
def set_scale_factor(self, scale_factor, unit_system="SI"):
sympy_deprecation_warning(
f"""
Quantity.set_scale_factor() is deprecated. Use either
unit_system.set_quantity_scale_factors() or
{self}.set_global_relative_scale_factor() instead.
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-quantity-methods",
)
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_unit_system(unit_system)
unit_system.set_quantity_scale_factor(self, scale_factor)
def set_global_dimension(self, dimension):
_QuantityMapper._quantity_dimension_global[self] = dimension
def set_global_relative_scale_factor(self, scale_factor, reference_quantity):
"""
Setting a scale factor that is valid across all unit system.
"""
from sympy.physics.units import UnitSystem
scale_factor = sympify(scale_factor)
# replace all prefixes by their ratio to canonical units:
scale_factor = scale_factor.replace(
lambda x: isinstance(x, Prefix),
lambda x: x.scale_factor
)
scale_factor = sympify(scale_factor)
UnitSystem._quantity_scale_factors_global[self] = (scale_factor, reference_quantity)
UnitSystem._quantity_dimensional_equivalence_map_global[self] = reference_quantity
@property
def name(self):
return self._name
@property
def dimension(self):
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_default_unit_system()
return unit_system.get_quantity_dimension(self)
@property
def abbrev(self):
"""
Symbol representing the unit name.
Prepend the abbreviation with the prefix symbol if it is defines.
"""
return self._abbrev
@property
def scale_factor(self):
"""
Overall magnitude of the quantity as compared to the canonical units.
"""
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_default_unit_system()
return unit_system.get_quantity_scale_factor(self)
def _eval_is_positive(self):
return True
def _eval_is_constant(self):
return True
def _eval_Abs(self):
return self
def _eval_subs(self, old, new):
if isinstance(new, Quantity) and self != old:
return self
@staticmethod
def get_dimensional_expr(expr, unit_system="SI"):
sympy_deprecation_warning(
"""
Quantity.get_dimensional_expr() is deprecated. It is now
associated with UnitSystem objects. The dimensional relations
depend on the unit system used. Use
unit_system.get_dimensional_expr() instead.
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-quantity-methods",
)
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_unit_system(unit_system)
return unit_system.get_dimensional_expr(expr)
@staticmethod
def _collect_factor_and_dimension(expr, unit_system="SI"):
"""Return tuple with scale factor expression and dimension expression."""
sympy_deprecation_warning(
"""
Quantity._collect_factor_and_dimension() is deprecated. This
method has been moved to the UnitSystem class. Use
unit_system._collect_factor_and_dimension(expr) instead.
""",
deprecated_since_version="1.5",
active_deprecations_target="deprecated-quantity-methods",
)
from sympy.physics.units import UnitSystem
unit_system = UnitSystem.get_unit_system(unit_system)
return unit_system._collect_factor_and_dimension(expr)
def _latex(self, printer):
if self._latex_repr:
return self._latex_repr
else:
return r'\text{{{}}}'.format(self.args[1] \
if len(self.args) >= 2 else self.args[0])
def convert_to(self, other, unit_system="SI"):
"""
Convert the quantity to another quantity of same dimensions.
Examples
========
>>> from sympy.physics.units import speed_of_light, meter, second
>>> speed_of_light
speed_of_light
>>> speed_of_light.convert_to(meter/second)
299792458*meter/second
>>> from sympy.physics.units import liter
>>> liter.convert_to(meter**3)
meter**3/1000
"""
from .util import convert_to
return convert_to(self, other, unit_system)
@property
def free_symbols(self):
"""Return free symbols from quantity."""
return set()
|
8909b265089bfbd39aeffb2ec701e9949f475b5f7c2cf1a6a6d0e5770a91976d | from sympy.core.function import Derivative
from sympy.core.function import UndefinedFunction, AppliedUndef
from sympy.core.symbol import Symbol
from sympy.interactive.printing import init_printing
from sympy.printing.latex import LatexPrinter
from sympy.printing.pretty.pretty import PrettyPrinter
from sympy.printing.pretty.pretty_symbology import center_accent
from sympy.printing.str import StrPrinter
from sympy.printing.precedence import PRECEDENCE
__all__ = ['vprint', 'vsstrrepr', 'vsprint', 'vpprint', 'vlatex',
'init_vprinting']
class VectorStrPrinter(StrPrinter):
"""String Printer for vector expressions. """
def _print_Derivative(self, e):
from sympy.physics.vector.functions import dynamicsymbols
t = dynamicsymbols._t
if (bool(sum([i == t for i in e.variables])) &
isinstance(type(e.args[0]), UndefinedFunction)):
ol = str(e.args[0].func)
for i, v in enumerate(e.variables):
ol += dynamicsymbols._str
return ol
else:
return StrPrinter().doprint(e)
def _print_Function(self, e):
from sympy.physics.vector.functions import dynamicsymbols
t = dynamicsymbols._t
if isinstance(type(e), UndefinedFunction):
return StrPrinter().doprint(e).replace("(%s)" % t, '')
return e.func.__name__ + "(%s)" % self.stringify(e.args, ", ")
class VectorStrReprPrinter(VectorStrPrinter):
"""String repr printer for vector expressions."""
def _print_str(self, s):
return repr(s)
class VectorLatexPrinter(LatexPrinter):
"""Latex Printer for vector expressions. """
def _print_Function(self, expr, exp=None):
from sympy.physics.vector.functions import dynamicsymbols
func = expr.func.__name__
t = dynamicsymbols._t
if (hasattr(self, '_print_' + func) and not
isinstance(type(expr), UndefinedFunction)):
return getattr(self, '_print_' + func)(expr, exp)
elif isinstance(type(expr), UndefinedFunction) and (expr.args == (t,)):
# treat this function like a symbol
expr = Symbol(func)
if exp is not None:
# copied from LatexPrinter._helper_print_standard_power, which
# we can't call because we only have exp as a string.
base = self.parenthesize(expr, PRECEDENCE['Pow'])
base = self.parenthesize_super(base)
return r"%s^{%s}" % (base, exp)
else:
return super()._print(expr)
else:
return super()._print_Function(expr, exp)
def _print_Derivative(self, der_expr):
from sympy.physics.vector.functions import dynamicsymbols
# make sure it is in the right form
der_expr = der_expr.doit()
if not isinstance(der_expr, Derivative):
return r"\left(%s\right)" % self.doprint(der_expr)
# check if expr is a dynamicsymbol
t = dynamicsymbols._t
expr = der_expr.expr
red = expr.atoms(AppliedUndef)
syms = der_expr.variables
test1 = not all(True for i in red if i.free_symbols == {t})
test2 = not all(t == i for i in syms)
if test1 or test2:
return super()._print_Derivative(der_expr)
# done checking
dots = len(syms)
base = self._print_Function(expr)
base_split = base.split('_', 1)
base = base_split[0]
if dots == 1:
base = r"\dot{%s}" % base
elif dots == 2:
base = r"\ddot{%s}" % base
elif dots == 3:
base = r"\dddot{%s}" % base
elif dots == 4:
base = r"\ddddot{%s}" % base
else: # Fallback to standard printing
return super()._print_Derivative(der_expr)
if len(base_split) != 1:
base += '_' + base_split[1]
return base
class VectorPrettyPrinter(PrettyPrinter):
"""Pretty Printer for vectorialexpressions. """
def _print_Derivative(self, deriv):
from sympy.physics.vector.functions import dynamicsymbols
# XXX use U('PARTIAL DIFFERENTIAL') here ?
t = dynamicsymbols._t
dot_i = 0
syms = list(reversed(deriv.variables))
while len(syms) > 0:
if syms[-1] == t:
syms.pop()
dot_i += 1
else:
return super()._print_Derivative(deriv)
if not (isinstance(type(deriv.expr), UndefinedFunction) and
(deriv.expr.args == (t,))):
return super()._print_Derivative(deriv)
else:
pform = self._print_Function(deriv.expr)
# the following condition would happen with some sort of non-standard
# dynamic symbol I guess, so we'll just print the SymPy way
if len(pform.picture) > 1:
return super()._print_Derivative(deriv)
# There are only special symbols up to fourth-order derivatives
if dot_i >= 5:
return super()._print_Derivative(deriv)
# Deal with special symbols
dots = {0: "",
1: "\N{COMBINING DOT ABOVE}",
2: "\N{COMBINING DIAERESIS}",
3: "\N{COMBINING THREE DOTS ABOVE}",
4: "\N{COMBINING FOUR DOTS ABOVE}"}
d = pform.__dict__
# if unicode is false then calculate number of apostrophes needed and
# add to output
if not self._use_unicode:
apostrophes = ""
for i in range(0, dot_i):
apostrophes += "'"
d['picture'][0] += apostrophes + "(t)"
else:
d['picture'] = [center_accent(d['picture'][0], dots[dot_i])]
return pform
def _print_Function(self, e):
from sympy.physics.vector.functions import dynamicsymbols
t = dynamicsymbols._t
# XXX works only for applied functions
func = e.func
args = e.args
func_name = func.__name__
pform = self._print_Symbol(Symbol(func_name))
# If this function is an Undefined function of t, it is probably a
# dynamic symbol, so we'll skip the (t). The rest of the code is
# identical to the normal PrettyPrinter code
if not (isinstance(func, UndefinedFunction) and (args == (t,))):
return super()._print_Function(e)
return pform
def vprint(expr, **settings):
r"""Function for printing of expressions generated in the
sympy.physics vector package.
Extends SymPy's StrPrinter, takes the same setting accepted by SymPy's
:func:`~.sstr`, and is equivalent to ``print(sstr(foo))``.
Parameters
==========
expr : valid SymPy object
SymPy expression to print.
settings : args
Same as the settings accepted by SymPy's sstr().
Examples
========
>>> from sympy.physics.vector import vprint, dynamicsymbols
>>> u1 = dynamicsymbols('u1')
>>> print(u1)
u1(t)
>>> vprint(u1)
u1
"""
outstr = vsprint(expr, **settings)
import builtins
if (outstr != 'None'):
builtins._ = outstr
print(outstr)
def vsstrrepr(expr, **settings):
"""Function for displaying expression representation's with vector
printing enabled.
Parameters
==========
expr : valid SymPy object
SymPy expression to print.
settings : args
Same as the settings accepted by SymPy's sstrrepr().
"""
p = VectorStrReprPrinter(settings)
return p.doprint(expr)
def vsprint(expr, **settings):
r"""Function for displaying expressions generated in the
sympy.physics vector package.
Returns the output of vprint() as a string.
Parameters
==========
expr : valid SymPy object
SymPy expression to print
settings : args
Same as the settings accepted by SymPy's sstr().
Examples
========
>>> from sympy.physics.vector import vsprint, dynamicsymbols
>>> u1, u2 = dynamicsymbols('u1 u2')
>>> u2d = dynamicsymbols('u2', level=1)
>>> print("%s = %s" % (u1, u2 + u2d))
u1(t) = u2(t) + Derivative(u2(t), t)
>>> print("%s = %s" % (vsprint(u1), vsprint(u2 + u2d)))
u1 = u2 + u2'
"""
string_printer = VectorStrPrinter(settings)
return string_printer.doprint(expr)
def vpprint(expr, **settings):
r"""Function for pretty printing of expressions generated in the
sympy.physics vector package.
Mainly used for expressions not inside a vector; the output of running
scripts and generating equations of motion. Takes the same options as
SymPy's :func:`~.pretty_print`; see that function for more information.
Parameters
==========
expr : valid SymPy object
SymPy expression to pretty print
settings : args
Same as those accepted by SymPy's pretty_print.
"""
pp = VectorPrettyPrinter(settings)
# Note that this is copied from sympy.printing.pretty.pretty_print:
# XXX: this is an ugly hack, but at least it works
use_unicode = pp._settings['use_unicode']
from sympy.printing.pretty.pretty_symbology import pretty_use_unicode
uflag = pretty_use_unicode(use_unicode)
try:
return pp.doprint(expr)
finally:
pretty_use_unicode(uflag)
def vlatex(expr, **settings):
r"""Function for printing latex representation of sympy.physics.vector
objects.
For latex representation of Vectors, Dyadics, and dynamicsymbols. Takes the
same options as SymPy's :func:`~.latex`; see that function for more
information;
Parameters
==========
expr : valid SymPy object
SymPy expression to represent in LaTeX form
settings : args
Same as latex()
Examples
========
>>> from sympy.physics.vector import vlatex, ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> q1, q2 = dynamicsymbols('q1 q2')
>>> q1d, q2d = dynamicsymbols('q1 q2', 1)
>>> q1dd, q2dd = dynamicsymbols('q1 q2', 2)
>>> vlatex(N.x + N.y)
'\\mathbf{\\hat{n}_x} + \\mathbf{\\hat{n}_y}'
>>> vlatex(q1 + q2)
'q_{1} + q_{2}'
>>> vlatex(q1d)
'\\dot{q}_{1}'
>>> vlatex(q1 * q2d)
'q_{1} \\dot{q}_{2}'
>>> vlatex(q1dd * q1 / q1d)
'\\frac{q_{1} \\ddot{q}_{1}}{\\dot{q}_{1}}'
"""
latex_printer = VectorLatexPrinter(settings)
return latex_printer.doprint(expr)
def init_vprinting(**kwargs):
"""Initializes time derivative printing for all SymPy objects, i.e. any
functions of time will be displayed in a more compact notation. The main
benefit of this is for printing of time derivatives; instead of
displaying as ``Derivative(f(t),t)``, it will display ``f'``. This is
only actually needed for when derivatives are present and are not in a
physics.vector.Vector or physics.vector.Dyadic object. This function is a
light wrapper to :func:`~.init_printing`. Any keyword
arguments for it are valid here.
{0}
Examples
========
>>> from sympy import Function, symbols
>>> t, x = symbols('t, x')
>>> omega = Function('omega')
>>> omega(x).diff()
Derivative(omega(x), x)
>>> omega(t).diff()
Derivative(omega(t), t)
Now use the string printer:
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> omega(x).diff()
Derivative(omega(x), x)
>>> omega(t).diff()
omega'
"""
kwargs['str_printer'] = vsstrrepr
kwargs['pretty_printer'] = vpprint
kwargs['latex_printer'] = vlatex
init_printing(**kwargs)
params = init_printing.__doc__.split('Examples\n ========')[0] # type: ignore
init_vprinting.__doc__ = init_vprinting.__doc__.format(params) # type: ignore
|
6ad38feac4502d00cd02e1c0ed43e6230e5f7dc56ded1a2f506feb666dde8c99 | from sympy.core.backend import (S, sympify, expand, sqrt, Add, zeros, acos,
ImmutableMatrix as Matrix, _simplify_matrix)
from sympy.simplify.trigsimp import trigsimp
from sympy.printing.defaults import Printable
from sympy.utilities.misc import filldedent
from sympy.core.evalf import EvalfMixin
from mpmath.libmp.libmpf import prec_to_dps
__all__ = ['Vector']
class Vector(Printable, EvalfMixin):
"""The class used to define vectors.
It along with ReferenceFrame are the building blocks of describing a
classical mechanics system in PyDy and sympy.physics.vector.
Attributes
==========
simp : Boolean
Let certain methods use trigsimp on their outputs
"""
simp = False
is_number = False
def __init__(self, inlist):
"""This is the constructor for the Vector class. You should not be
calling this, it should only be used by other functions. You should be
treating Vectors like you would with if you were doing the math by
hand, and getting the first 3 from the standard basis vectors from a
ReferenceFrame.
The only exception is to create a zero vector:
zv = Vector(0)
"""
self.args = []
if inlist == 0:
inlist = []
if isinstance(inlist, dict):
d = inlist
else:
d = {}
for inp in inlist:
if inp[1] in d:
d[inp[1]] += inp[0]
else:
d[inp[1]] = inp[0]
for k, v in d.items():
if v != Matrix([0, 0, 0]):
self.args.append((v, k))
@property
def func(self):
"""Returns the class Vector. """
return Vector
def __hash__(self):
return hash(tuple(self.args))
def __add__(self, other):
"""The add operator for Vector. """
if other == 0:
return self
other = _check_vector(other)
return Vector(self.args + other.args)
def __and__(self, other):
"""Dot product of two vectors.
Returns a scalar, the dot product of the two Vectors
Parameters
==========
other : Vector
The Vector which we are dotting with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dot
>>> from sympy import symbols
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> dot(N.x, N.x)
1
>>> dot(N.x, N.y)
0
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> dot(N.y, A.y)
cos(q1)
"""
from sympy.physics.vector.dyadic import Dyadic
if isinstance(other, Dyadic):
return NotImplemented
other = _check_vector(other)
out = S.Zero
for i, v1 in enumerate(self.args):
for j, v2 in enumerate(other.args):
out += ((v2[0].T)
* (v2[1].dcm(v1[1]))
* (v1[0]))[0]
if Vector.simp:
return trigsimp(sympify(out), recursive=True)
else:
return sympify(out)
def __truediv__(self, other):
"""This uses mul and inputs self and 1 divided by other. """
return self.__mul__(sympify(1) / other)
def __eq__(self, other):
"""Tests for equality.
It is very import to note that this is only as good as the SymPy
equality test; False does not always mean they are not equivalent
Vectors.
If other is 0, and self is empty, returns True.
If other is 0 and self is not empty, returns False.
If none of the above, only accepts other as a Vector.
"""
if other == 0:
other = Vector(0)
try:
other = _check_vector(other)
except TypeError:
return False
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False
frame = self.args[0][1]
for v in frame:
if expand((self - other) & v) != 0:
return False
return True
def __mul__(self, other):
"""Multiplies the Vector by a sympifyable expression.
Parameters
==========
other : Sympifyable
The scalar to multiply this Vector with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> b = Symbol('b')
>>> V = 10 * b * N.x
>>> print(V)
10*b*N.x
"""
newlist = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1])
return Vector(newlist)
def __ne__(self, other):
return not self == other
def __neg__(self):
return self * -1
def __or__(self, other):
"""Outer product between two Vectors.
A rank increasing operation, which returns a Dyadic from two Vectors
Parameters
==========
other : Vector
The Vector to take the outer product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)
"""
from sympy.physics.vector.dyadic import Dyadic
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
for i2, v2 in enumerate(other.args):
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol
def _latex(self, printer):
"""Latex Printing method. """
ar = self.args # just to shorten things
if len(ar) == 0:
return str(0)
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].latex_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].latex_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = printer._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + ar[i][1].latex_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def _pretty(self, printer):
"""Pretty Printing method. """
from sympy.printing.pretty.stringpict import prettyForm
e = self
class Fake:
def render(self, *args, **kwargs):
ar = e.args # just to shorten things
if len(ar) == 0:
return str(0)
pforms = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
pform = printer._print(ar[i][1].pretty_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
pform = printer._print(ar[i][1].pretty_vecs[j])
pform = prettyForm(*pform.left(" - "))
bin = prettyForm.NEG
pform = prettyForm(binding=bin, *pform)
elif ar[i][0][j] != 0:
# If the basis vector coeff is not 1 or -1,
# we might wrap it in parentheses, for readability.
pform = printer._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
tmp = pform.parens()
pform = prettyForm(tmp[0], tmp[1])
pform = prettyForm(*pform.right(
" ", ar[i][1].pretty_vecs[j]))
else:
continue
pforms.append(pform)
pform = prettyForm.__add__(*pforms)
kwargs["wrap_line"] = kwargs.get("wrap_line")
kwargs["num_columns"] = kwargs.get("num_columns")
out_str = pform.render(*args, **kwargs)
mlines = [line.rstrip() for line in out_str.split("\n")]
return "\n".join(mlines)
return Fake()
def __ror__(self, other):
"""Outer product between two Vectors.
A rank increasing operation, which returns a Dyadic from two Vectors
Parameters
==========
other : Vector
The Vector to take the outer product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> outer(N.x, N.x)
(N.x|N.x)
"""
from sympy.physics.vector.dyadic import Dyadic
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(other.args):
for i2, v2 in enumerate(self.args):
# it looks this way because if we are in the same frame and
# use the enumerate function on the same frame in a nested
# fashion, then bad things happen
ol += Dyadic([(v[0][0] * v2[0][0], v[1].x, v2[1].x)])
ol += Dyadic([(v[0][0] * v2[0][1], v[1].x, v2[1].y)])
ol += Dyadic([(v[0][0] * v2[0][2], v[1].x, v2[1].z)])
ol += Dyadic([(v[0][1] * v2[0][0], v[1].y, v2[1].x)])
ol += Dyadic([(v[0][1] * v2[0][1], v[1].y, v2[1].y)])
ol += Dyadic([(v[0][1] * v2[0][2], v[1].y, v2[1].z)])
ol += Dyadic([(v[0][2] * v2[0][0], v[1].z, v2[1].x)])
ol += Dyadic([(v[0][2] * v2[0][1], v[1].z, v2[1].y)])
ol += Dyadic([(v[0][2] * v2[0][2], v[1].z, v2[1].z)])
return ol
def __rsub__(self, other):
return (-1 * self) + other
def _sympystr(self, printer, order=True):
"""Printing method. """
if not order or len(self.args) == 1:
ar = list(self.args)
elif len(self.args) == 0:
return printer._print(0)
else:
d = {v[1]: v[0] for v in self.args}
keys = sorted(d.keys(), key=lambda x: x.index)
ar = []
for key in keys:
ar.append((d[key], key))
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
for j in 0, 1, 2:
# if the coef of the basis vector is 1, we skip the 1
if ar[i][0][j] == 1:
ol.append(' + ' + ar[i][1].str_vecs[j])
# if the coef of the basis vector is -1, we skip the 1
elif ar[i][0][j] == -1:
ol.append(' - ' + ar[i][1].str_vecs[j])
elif ar[i][0][j] != 0:
# If the coefficient of the basis vector is not 1 or -1;
# also, we might wrap it in parentheses, for readability.
arg_str = printer._print(ar[i][0][j])
if isinstance(ar[i][0][j], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*' + ar[i][1].str_vecs[j])
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def __sub__(self, other):
"""The subtraction operator. """
return self.__add__(other * -1)
def __xor__(self, other):
"""The cross product operator for two Vectors.
Returns a Vector, expressed in the same ReferenceFrames as self.
Parameters
==========
other : Vector
The Vector which we are crossing with
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame, cross
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> cross(N.x, N.y)
N.z
>>> A = ReferenceFrame('A')
>>> A.orient_axis(N, q1, N.x)
>>> cross(A.x, N.y)
N.z
>>> cross(N.y, A.x)
- sin(q1)*A.y - cos(q1)*A.z
"""
from sympy.physics.vector.dyadic import Dyadic
if isinstance(other, Dyadic):
return NotImplemented
other = _check_vector(other)
if other.args == []:
return Vector(0)
def _det(mat):
"""This is needed as a little method for to find the determinant
of a list in python; needs to work for a 3x3 list.
SymPy's Matrix will not take in Vector, so need a custom function.
You should not be calling this.
"""
return (mat[0][0] * (mat[1][1] * mat[2][2] - mat[1][2] * mat[2][1])
+ mat[0][1] * (mat[1][2] * mat[2][0] - mat[1][0] *
mat[2][2]) + mat[0][2] * (mat[1][0] * mat[2][1] -
mat[1][1] * mat[2][0]))
outlist = []
ar = other.args # For brevity
for i, v in enumerate(ar):
tempx = v[1].x
tempy = v[1].y
tempz = v[1].z
tempm = ([[tempx, tempy, tempz],
[self & tempx, self & tempy, self & tempz],
[Vector([ar[i]]) & tempx, Vector([ar[i]]) & tempy,
Vector([ar[i]]) & tempz]])
outlist += _det(tempm).args
return Vector(outlist)
__radd__ = __add__
__rand__ = __and__
__rmul__ = __mul__
def separate(self):
"""
The constituents of this vector in different reference frames,
as per its definition.
Returns a dict mapping each ReferenceFrame to the corresponding
constituent Vector.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> R1 = ReferenceFrame('R1')
>>> R2 = ReferenceFrame('R2')
>>> v = R1.x + R2.x
>>> v.separate() == {R1: R1.x, R2: R2.x}
True
"""
components = {}
for x in self.args:
components[x[1]] = Vector([x])
return components
def dot(self, other):
return self & other
dot.__doc__ = __and__.__doc__
def cross(self, other):
return self ^ other
cross.__doc__ = __xor__.__doc__
def outer(self, other):
return self | other
outer.__doc__ = __or__.__doc__
def diff(self, var, frame, var_in_dcm=True):
"""Returns the partial derivative of the vector with respect to a
variable in the provided reference frame.
Parameters
==========
var : Symbol
What the partial derivative is taken with respect to.
frame : ReferenceFrame
The reference frame that the partial derivative is taken in.
var_in_dcm : boolean
If true, the differentiation algorithm assumes that the variable
may be present in any of the direction cosine matrices that relate
the frame to the frames of any component of the vector. But if it
is known that the variable is not present in the direction cosine
matrices, false can be set to skip full reexpression in the desired
frame.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.vector import dynamicsymbols, ReferenceFrame
>>> from sympy.physics.vector import Vector
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> Vector.simp = True
>>> t = Symbol('t')
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.diff(t, N)
- sin(q1)*q1'*N.x - cos(q1)*q1'*N.z
>>> A.x.diff(t, N).express(A)
- q1'*A.z
>>> B = ReferenceFrame('B')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> v = u1 * A.x + u2 * B.y
>>> v.diff(u2, N, var_in_dcm=False)
B.y
"""
from sympy.physics.vector.frame import _check_frame
var = sympify(var)
_check_frame(frame)
inlist = []
for vector_component in self.args:
measure_number = vector_component[0]
component_frame = vector_component[1]
if component_frame == frame:
inlist += [(measure_number.diff(var), frame)]
else:
# If the direction cosine matrix relating the component frame
# with the derivative frame does not contain the variable.
if not var_in_dcm or (frame.dcm(component_frame).diff(var) ==
zeros(3, 3)):
inlist += [(measure_number.diff(var), component_frame)]
else: # else express in the frame
reexp_vec_comp = Vector([vector_component]).express(frame)
deriv = reexp_vec_comp.args[0][0].diff(var)
inlist += Vector([(deriv, frame)]).args
return Vector(inlist)
def express(self, otherframe, variables=False):
"""
Returns a Vector equivalent to this one, expressed in otherframe.
Uses the global express method.
Parameters
==========
otherframe : ReferenceFrame
The frame for this Vector to be described in
variables : boolean
If True, the coordinate symbols(if present) in this Vector
are re-expressed in terms otherframe
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> q1 = dynamicsymbols('q1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.y])
>>> A.x.express(N)
cos(q1)*N.x - sin(q1)*N.z
"""
from sympy.physics.vector import express
return express(self, otherframe, variables=variables)
def to_matrix(self, reference_frame):
"""Returns the matrix form of the vector with respect to the given
frame.
Parameters
----------
reference_frame : ReferenceFrame
The reference frame that the rows of the matrix correspond to.
Returns
-------
matrix : ImmutableMatrix, shape(3,1)
The matrix that gives the 1D vector.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> a, b, c = symbols('a, b, c')
>>> N = ReferenceFrame('N')
>>> vector = a * N.x + b * N.y + c * N.z
>>> vector.to_matrix(N)
Matrix([
[a],
[b],
[c]])
>>> beta = symbols('beta')
>>> A = N.orientnew('A', 'Axis', (beta, N.x))
>>> vector.to_matrix(A)
Matrix([
[ a],
[ b*cos(beta) + c*sin(beta)],
[-b*sin(beta) + c*cos(beta)]])
"""
return Matrix([self.dot(unit_vec) for unit_vec in
reference_frame]).reshape(3, 1)
def doit(self, **hints):
"""Calls .doit() on each term in the Vector"""
d = {}
for v in self.args:
d[v[1]] = v[0].applyfunc(lambda x: x.doit(**hints))
return Vector(d)
def dt(self, otherframe):
"""
Returns a Vector which is the time derivative of
the self Vector, taken in frame otherframe.
Calls the global time_derivative method
Parameters
==========
otherframe : ReferenceFrame
The frame to calculate the time derivative in
"""
from sympy.physics.vector import time_derivative
return time_derivative(self, otherframe)
def simplify(self):
"""Returns a simplified Vector."""
d = {}
for v in self.args:
d[v[1]] = _simplify_matrix(v[0])
return Vector(d)
def subs(self, *args, **kwargs):
"""Substitution on the Vector.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = N.x * s
>>> a.subs({s: 2})
2*N.x
"""
d = {}
for v in self.args:
d[v[1]] = v[0].subs(*args, **kwargs)
return Vector(d)
def magnitude(self):
"""Returns the magnitude (Euclidean norm) of self.
Warnings
========
Python ignores the leading negative sign so that might
give wrong results.
``-A.x.magnitude()`` would be treated as ``-(A.x.magnitude())``,
instead of ``(-A.x).magnitude()``.
"""
return sqrt(self & self)
def normalize(self):
"""Returns a Vector of magnitude 1, codirectional with self."""
return Vector(self.args + []) / self.magnitude()
def applyfunc(self, f):
"""Apply a function to each component of a vector."""
if not callable(f):
raise TypeError("`f` must be callable.")
d = {}
for v in self.args:
d[v[1]] = v[0].applyfunc(f)
return Vector(d)
def angle_between(self, vec):
"""
Returns the smallest angle between Vector 'vec' and self.
Parameter
=========
vec : Vector
The Vector between which angle is needed.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> A = ReferenceFrame("A")
>>> v1 = A.x
>>> v2 = A.y
>>> v1.angle_between(v2)
pi/2
>>> v3 = A.x + A.y + A.z
>>> v1.angle_between(v3)
acos(sqrt(3)/3)
Warnings
========
Python ignores the leading negative sign so that might give wrong
results. ``-A.x.angle_between()`` would be treated as
``-(A.x.angle_between())``, instead of ``(-A.x).angle_between()``.
"""
vec1 = self.normalize()
vec2 = vec.normalize()
angle = acos(vec1.dot(vec2))
return angle
def free_symbols(self, reference_frame):
"""Returns the free symbols in the measure numbers of the vector
expressed in the given reference frame.
Parameters
==========
reference_frame : ReferenceFrame
The frame with respect to which the free symbols of the given
vector is to be determined.
Returns
=======
set of Symbol
set of symbols present in the measure numbers of
``reference_frame``.
"""
return self.to_matrix(reference_frame).free_symbols
def free_dynamicsymbols(self, reference_frame):
"""Returns the free dynamic symbols (functions of time ``t``) in the
measure numbers of the vector expressed in the given reference frame.
Parameters
==========
reference_frame : ReferenceFrame
The frame with respect to which the free dynamic symbols of the
given vector is to be determined.
Returns
=======
set
Set of functions of time ``t``, e.g.
``Function('f')(me.dynamicsymbols._t)``.
"""
# TODO : Circular dependency if imported at top. Should move
# find_dynamicsymbols into physics.vector.functions.
from sympy.physics.mechanics.functions import find_dynamicsymbols
return find_dynamicsymbols(self, reference_frame=reference_frame)
def _eval_evalf(self, prec):
if not self.args:
return self
new_args = []
dps = prec_to_dps(prec)
for mat, frame in self.args:
new_args.append([mat.evalf(n=dps), frame])
return Vector(new_args)
def xreplace(self, rule):
"""Replace occurrences of objects within the measure numbers of the
vector.
Parameters
==========
rule : dict-like
Expresses a replacement rule.
Returns
=======
Vector
Result of the replacement.
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.physics.vector import ReferenceFrame
>>> A = ReferenceFrame('A')
>>> x, y, z = symbols('x y z')
>>> ((1 + x*y) * A.x).xreplace({x: pi})
(pi*y + 1)*A.x
>>> ((1 + x*y) * A.x).xreplace({x: pi, y: 2})
(1 + 2*pi)*A.x
Replacements occur only if an entire node in the expression tree is
matched:
>>> ((x*y + z) * A.x).xreplace({x*y: pi})
(z + pi)*A.x
>>> ((x*y*z) * A.x).xreplace({x*y: pi})
x*y*z*A.x
"""
new_args = []
for mat, frame in self.args:
mat = mat.xreplace(rule)
new_args.append([mat, frame])
return Vector(new_args)
class VectorTypeError(TypeError):
def __init__(self, other, want):
msg = filldedent("Expected an instance of %s, but received object "
"'%s' of %s." % (type(want), other, type(other)))
super().__init__(msg)
def _check_vector(other):
if not isinstance(other, Vector):
raise TypeError('A Vector must be supplied')
return other
|
76e69b5cfc982433add6aa3be1e3c63b8fda9dbef0234265f15bc47fe0f5f104 | from .vector import Vector, _check_vector
from .frame import _check_frame
from warnings import warn
__all__ = ['Point']
class Point:
"""This object represents a point in a dynamic system.
It stores the: position, velocity, and acceleration of a point.
The position is a vector defined as the vector distance from a parent
point to this point.
Parameters
==========
name : string
The display name of the Point
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Point('P')
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> O.set_vel(N, u1 * N.x + u2 * N.y + u3 * N.z)
>>> O.acc(N)
u1'*N.x + u2'*N.y + u3'*N.z
``symbols()`` can be used to create multiple Points in a single step, for
example:
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> from sympy import symbols
>>> N = ReferenceFrame('N')
>>> u1, u2 = dynamicsymbols('u1 u2')
>>> A, B = symbols('A B', cls=Point)
>>> type(A)
<class 'sympy.physics.vector.point.Point'>
>>> A.set_vel(N, u1 * N.x + u2 * N.y)
>>> B.set_vel(N, u2 * N.x + u1 * N.y)
>>> A.acc(N) - B.acc(N)
(u1' - u2')*N.x + (-u1' + u2')*N.y
"""
def __init__(self, name):
"""Initialization of a Point object. """
self.name = name
self._pos_dict = {}
self._vel_dict = {}
self._acc_dict = {}
self._pdlist = [self._pos_dict, self._vel_dict, self._acc_dict]
def __str__(self):
return self.name
__repr__ = __str__
def _check_point(self, other):
if not isinstance(other, Point):
raise TypeError('A Point must be supplied')
def _pdict_list(self, other, num):
"""Returns a list of points that gives the shortest path with respect
to position, velocity, or acceleration from this point to the provided
point.
Parameters
==========
other : Point
A point that may be related to this point by position, velocity, or
acceleration.
num : integer
0 for searching the position tree, 1 for searching the velocity
tree, and 2 for searching the acceleration tree.
Returns
=======
list of Points
A sequence of points from self to other.
Notes
=====
It is not clear if num = 1 or num = 2 actually works because the keys
to ``_vel_dict`` and ``_acc_dict`` are :class:`ReferenceFrame` objects
which do not have the ``_pdlist`` attribute.
"""
outlist = [[self]]
oldlist = [[]]
while outlist != oldlist:
oldlist = outlist[:]
for i, v in enumerate(outlist):
templist = v[-1]._pdlist[num].keys()
for i2, v2 in enumerate(templist):
if not v.__contains__(v2):
littletemplist = v + [v2]
if not outlist.__contains__(littletemplist):
outlist.append(littletemplist)
for i, v in enumerate(oldlist):
if v[-1] != other:
outlist.remove(v)
outlist.sort(key=len)
if len(outlist) != 0:
return outlist[0]
raise ValueError('No Connecting Path found between ' + other.name +
' and ' + self.name)
def a1pt_theory(self, otherpoint, outframe, interframe):
"""Sets the acceleration of this point with the 1-point theory.
The 1-point theory for point acceleration looks like this:
^N a^P = ^B a^P + ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B
x r^OP) + 2 ^N omega^B x ^B v^P
where O is a point fixed in B, P is a point moving in B, and B is
rotating in frame N.
Parameters
==========
otherpoint : Point
The first point of the 1-point theory (O)
outframe : ReferenceFrame
The frame we want this point's acceleration defined in (N)
fixedframe : ReferenceFrame
The intermediate frame in this calculation (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.a1pt_theory(O, N, B)
(-25*q + q'')*B.x + q2''*B.y - 10*q'*B.z
"""
_check_frame(outframe)
_check_frame(interframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v = self.vel(interframe)
a1 = otherpoint.acc(outframe)
a2 = self.acc(interframe)
omega = interframe.ang_vel_in(outframe)
alpha = interframe.ang_acc_in(outframe)
self.set_acc(outframe, a2 + 2 * (omega ^ v) + a1 + (alpha ^ dist) +
(omega ^ (omega ^ dist)))
return self.acc(outframe)
def a2pt_theory(self, otherpoint, outframe, fixedframe):
"""Sets the acceleration of this point with the 2-point theory.
The 2-point theory for point acceleration looks like this:
^N a^P = ^N a^O + ^N alpha^B x r^OP + ^N omega^B x (^N omega^B x r^OP)
where O and P are both points fixed in frame B, which is rotating in
frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's acceleration defined in (N)
fixedframe : ReferenceFrame
The frame in which both points are fixed (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.a2pt_theory(O, N, B)
- 10*q'**2*B.x + 10*q''*B.y
"""
_check_frame(outframe)
_check_frame(fixedframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
a = otherpoint.acc(outframe)
omega = fixedframe.ang_vel_in(outframe)
alpha = fixedframe.ang_acc_in(outframe)
self.set_acc(outframe, a + (alpha ^ dist) + (omega ^ (omega ^ dist)))
return self.acc(outframe)
def acc(self, frame):
"""The acceleration Vector of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which the returned acceleration vector will be defined
in.
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x
"""
_check_frame(frame)
if not (frame in self._acc_dict):
if self._vel_dict[frame] != 0:
return (self._vel_dict[frame]).dt(frame)
else:
return Vector(0)
return self._acc_dict[frame]
def locatenew(self, name, value):
"""Creates a new point with a position defined from this point.
Parameters
==========
name : str
The name for the new point
value : Vector
The position of the new point relative to this point
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> N = ReferenceFrame('N')
>>> P1 = Point('P1')
>>> P2 = P1.locatenew('P2', 10 * N.x)
"""
if not isinstance(name, str):
raise TypeError('Must supply a valid name')
if value == 0:
value = Vector(0)
value = _check_vector(value)
p = Point(name)
p.set_pos(self, value)
self.set_pos(p, -value)
return p
def pos_from(self, otherpoint):
"""Returns a Vector distance between this Point and the other Point.
Parameters
==========
otherpoint : Point
The otherpoint we are locating this one relative to
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x
"""
outvec = Vector(0)
plist = self._pdict_list(otherpoint, 0)
for i in range(len(plist) - 1):
outvec += plist[i]._pos_dict[plist[i + 1]]
return outvec
def set_acc(self, frame, value):
"""Used to set the acceleration of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which this point's acceleration is defined
value : Vector
The vector value of this point's acceleration in the frame
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_acc(N, 10 * N.x)
>>> p1.acc(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(frame)
self._acc_dict.update({frame: value})
def set_pos(self, otherpoint, value):
"""Used to set the position of this point w.r.t. another point.
Parameters
==========
otherpoint : Point
The other point which this point's location is defined relative to
value : Vector
The vector which defines the location of this point
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p2 = Point('p2')
>>> p1.set_pos(p2, 10 * N.x)
>>> p1.pos_from(p2)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
self._check_point(otherpoint)
self._pos_dict.update({otherpoint: value})
otherpoint._pos_dict.update({self: -value})
def set_vel(self, frame, value):
"""Sets the velocity Vector of this Point in a ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which this point's velocity is defined
value : Vector
The vector value of this point's velocity in the frame
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(frame)
self._vel_dict.update({frame: value})
def v1pt_theory(self, otherpoint, outframe, interframe):
"""Sets the velocity of this point with the 1-point theory.
The 1-point theory for point velocity looks like this:
^N v^P = ^B v^P + ^N v^O + ^N omega^B x r^OP
where O is a point fixed in B, P is a point moving in B, and B is
rotating in frame N.
Parameters
==========
otherpoint : Point
The first point of the 1-point theory (O)
outframe : ReferenceFrame
The frame we want this point's velocity defined in (N)
interframe : ReferenceFrame
The intermediate frame in this calculation (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> q = dynamicsymbols('q')
>>> q2 = dynamicsymbols('q2')
>>> qd = dynamicsymbols('q', 1)
>>> q2d = dynamicsymbols('q2', 1)
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.set_ang_vel(N, 5 * B.y)
>>> O = Point('O')
>>> P = O.locatenew('P', q * B.x)
>>> P.set_vel(B, qd * B.x + q2d * B.y)
>>> O.set_vel(N, 0)
>>> P.v1pt_theory(O, N, B)
q'*B.x + q2'*B.y - 5*q*B.z
"""
_check_frame(outframe)
_check_frame(interframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v1 = self.vel(interframe)
v2 = otherpoint.vel(outframe)
omega = interframe.ang_vel_in(outframe)
self.set_vel(outframe, v1 + v2 + (omega ^ dist))
return self.vel(outframe)
def v2pt_theory(self, otherpoint, outframe, fixedframe):
"""Sets the velocity of this point with the 2-point theory.
The 2-point theory for point velocity looks like this:
^N v^P = ^N v^O + ^N omega^B x r^OP
where O and P are both points fixed in frame B, which is rotating in
frame N.
Parameters
==========
otherpoint : Point
The first point of the 2-point theory (O)
outframe : ReferenceFrame
The frame we want this point's velocity defined in (N)
fixedframe : ReferenceFrame
The frame in which both points are fixed (B)
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> q = dynamicsymbols('q')
>>> qd = dynamicsymbols('q', 1)
>>> N = ReferenceFrame('N')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> O = Point('O')
>>> P = O.locatenew('P', 10 * B.x)
>>> O.set_vel(N, 5 * N.x)
>>> P.v2pt_theory(O, N, B)
5*N.x + 10*q'*B.y
"""
_check_frame(outframe)
_check_frame(fixedframe)
self._check_point(otherpoint)
dist = self.pos_from(otherpoint)
v = otherpoint.vel(outframe)
omega = fixedframe.ang_vel_in(outframe)
self.set_vel(outframe, v + (omega ^ dist))
return self.vel(outframe)
def vel(self, frame):
"""The velocity Vector of this Point in the ReferenceFrame.
Parameters
==========
frame : ReferenceFrame
The frame in which the returned velocity vector will be defined in
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> p1 = Point('p1')
>>> p1.set_vel(N, 10 * N.x)
>>> p1.vel(N)
10*N.x
Velocities will be automatically calculated if possible, otherwise a
``ValueError`` will be returned. If it is possible to calculate
multiple different velocities from the relative points, the points
defined most directly relative to this point will be used. In the case
of inconsistent relative positions of points, incorrect velocities may
be returned. It is up to the user to define prior relative positions
and velocities of points in a self-consistent way.
>>> p = Point('p')
>>> q = dynamicsymbols('q')
>>> p.set_vel(N, 10 * N.x)
>>> p2 = Point('p2')
>>> p2.set_pos(p, q*N.x)
>>> p2.vel(N)
(Derivative(q(t), t) + 10)*N.x
"""
_check_frame(frame)
if not (frame in self._vel_dict):
valid_neighbor_found = False
is_cyclic = False
visited = []
queue = [self]
candidate_neighbor = []
while queue: # BFS to find nearest point
node = queue.pop(0)
if node not in visited:
visited.append(node)
for neighbor, neighbor_pos in node._pos_dict.items():
if neighbor in visited:
continue
try:
# Checks if pos vector is valid
neighbor_pos.express(frame)
except ValueError:
continue
if neighbor in queue:
is_cyclic = True
try:
# Checks if point has its vel defined in req frame
neighbor_velocity = neighbor._vel_dict[frame]
except KeyError:
queue.append(neighbor)
continue
candidate_neighbor.append(neighbor)
if not valid_neighbor_found:
vel = None
for f in self.pos_from(neighbor).args:
if f[1] in self._vel_dict.keys():
if self._vel_dict[f[1]] != 0:
vel = self._vel_dict[f[1]]
break
if vel is None:
vel = self.pos_from(neighbor).dt(frame)
self.set_vel(frame, vel + neighbor_velocity)
valid_neighbor_found = True
if is_cyclic:
warn('Kinematic loops are defined among the positions of '
'points. This is likely not desired and may cause errors '
'in your calculations.')
if len(candidate_neighbor) > 1:
warn('Velocity automatically calculated based on point ' +
candidate_neighbor[0].name +
' but it is also possible from points(s):' +
str(candidate_neighbor[1:]) +
'. Velocities from these points are not necessarily the '
'same. This may cause errors in your calculations.')
if valid_neighbor_found:
return self._vel_dict[frame]
else:
raise ValueError('Velocity of point ' + self.name +
' has not been'
' defined in ReferenceFrame ' + frame.name)
return self._vel_dict[frame]
def partial_velocity(self, frame, *gen_speeds):
"""Returns the partial velocities of the linear velocity vector of this
point in the given frame with respect to one or more provided
generalized speeds.
Parameters
==========
frame : ReferenceFrame
The frame with which the velocity is defined in.
gen_speeds : functions of time
The generalized speeds.
Returns
=======
partial_velocities : tuple of Vector
The partial velocity vectors corresponding to the provided
generalized speeds.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import dynamicsymbols
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> p = Point('p')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> p.set_vel(N, u1 * N.x + u2 * A.y)
>>> p.partial_velocity(N, u1)
N.x
>>> p.partial_velocity(N, u1, u2)
(N.x, A.y)
"""
partials = [self.vel(frame).diff(speed, frame, var_in_dcm=False) for
speed in gen_speeds]
if len(partials) == 1:
return partials[0]
else:
return tuple(partials)
|
169dc5c747689cd50f8fdae4674df1142f07854cace14be880aebd6a1d92777a | from functools import reduce
from sympy.core.backend import (sympify, diff, sin, cos, Matrix, symbols,
Function, S, Symbol)
from sympy.integrals.integrals import integrate
from sympy.simplify.trigsimp import trigsimp
from .vector import Vector, _check_vector
from .frame import CoordinateSym, _check_frame
from .dyadic import Dyadic
from .printing import vprint, vsprint, vpprint, vlatex, init_vprinting
from sympy.utilities.iterables import iterable
from sympy.utilities.misc import translate
__all__ = ['cross', 'dot', 'express', 'time_derivative', 'outer',
'kinematic_equations', 'get_motion_params', 'partial_velocity',
'dynamicsymbols', 'vprint', 'vsprint', 'vpprint', 'vlatex',
'init_vprinting']
def cross(vec1, vec2):
"""Cross product convenience wrapper for Vector.cross(): \n"""
if not isinstance(vec1, (Vector, Dyadic)):
raise TypeError('Cross product is between two vectors')
return vec1 ^ vec2
cross.__doc__ += Vector.cross.__doc__ # type: ignore
def dot(vec1, vec2):
"""Dot product convenience wrapper for Vector.dot(): \n"""
if not isinstance(vec1, (Vector, Dyadic)):
raise TypeError('Dot product is between two vectors')
return vec1 & vec2
dot.__doc__ += Vector.dot.__doc__ # type: ignore
def express(expr, frame, frame2=None, variables=False):
"""
Global function for 'express' functionality.
Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame.
Refer to the local methods of Vector and Dyadic for details.
If 'variables' is True, then the coordinate variables (CoordinateSym
instances) of other frames present in the vector/scalar field or
dyadic expression are also substituted in terms of the base scalars of
this frame.
Parameters
==========
expr : Vector/Dyadic/scalar(sympyfiable)
The expression to re-express in ReferenceFrame 'frame'
frame: ReferenceFrame
The reference frame to express expr in
frame2 : ReferenceFrame
The other frame required for re-expression(only for Dyadic expr)
variables : boolean
Specifies whether to substitute the coordinate variables present
in expr, in terms of those of frame
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> from sympy.physics.vector import express
>>> express(d, B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
>>> express(B.x, N)
cos(q)*N.x + sin(q)*N.y
>>> express(N[0], B, variables=True)
B_x*cos(q) - B_y*sin(q)
"""
_check_frame(frame)
if expr == 0:
return expr
if isinstance(expr, Vector):
# Given expr is a Vector
if variables:
# If variables attribute is True, substitute the coordinate
# variables in the Vector
frame_list = [x[-1] for x in expr.args]
subs_dict = {}
for f in frame_list:
subs_dict.update(f.variable_map(frame))
expr = expr.subs(subs_dict)
# Re-express in this frame
outvec = Vector([])
for i, v in enumerate(expr.args):
if v[1] != frame:
temp = frame.dcm(v[1]) * v[0]
if Vector.simp:
temp = temp.applyfunc(lambda x:
trigsimp(x, method='fu'))
outvec += Vector([(temp, frame)])
else:
outvec += Vector([v])
return outvec
if isinstance(expr, Dyadic):
if frame2 is None:
frame2 = frame
_check_frame(frame2)
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += express(v[0], frame, variables=variables) * \
(express(v[1], frame, variables=variables) |
express(v[2], frame2, variables=variables))
return ol
else:
if variables:
# Given expr is a scalar field
frame_set = set()
expr = sympify(expr)
# Substitute all the coordinate variables
for x in expr.free_symbols:
if isinstance(x, CoordinateSym) and x.frame != frame:
frame_set.add(x.frame)
subs_dict = {}
for f in frame_set:
subs_dict.update(f.variable_map(frame))
return expr.subs(subs_dict)
return expr
def time_derivative(expr, frame, order=1):
"""
Calculate the time derivative of a vector/scalar field function
or dyadic expression in given frame.
References
==========
https://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames
Parameters
==========
expr : Vector/Dyadic/sympifyable
The expression whose time derivative is to be calculated
frame : ReferenceFrame
The reference frame to calculate the time derivative in
order : integer
The order of the derivative to be calculated
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> from sympy import Symbol
>>> q1 = Symbol('q1')
>>> u1 = dynamicsymbols('u1')
>>> N = ReferenceFrame('N')
>>> A = N.orientnew('A', 'Axis', [q1, N.x])
>>> v = u1 * N.x
>>> A.set_ang_vel(N, 10*A.x)
>>> from sympy.physics.vector import time_derivative
>>> time_derivative(v, N)
u1'*N.x
>>> time_derivative(u1*A[0], N)
N_x*u1'
>>> B = N.orientnew('B', 'Axis', [u1, N.z])
>>> from sympy.physics.vector import outer
>>> d = outer(N.x, N.x)
>>> time_derivative(d, B)
- u1'*(N.y|N.x) - u1'*(N.x|N.y)
"""
t = dynamicsymbols._t
_check_frame(frame)
if order == 0:
return expr
if order % 1 != 0 or order < 0:
raise ValueError("Unsupported value of order entered")
if isinstance(expr, Vector):
outlist = []
for i, v in enumerate(expr.args):
if v[1] == frame:
outlist += [(express(v[0], frame, variables=True).diff(t),
frame)]
else:
outlist += (time_derivative(Vector([v]), v[1]) +
(v[1].ang_vel_in(frame) ^ Vector([v]))).args
outvec = Vector(outlist)
return time_derivative(outvec, frame, order - 1)
if isinstance(expr, Dyadic):
ol = Dyadic(0)
for i, v in enumerate(expr.args):
ol += (v[0].diff(t) * (v[1] | v[2]))
ol += (v[0] * (time_derivative(v[1], frame) | v[2]))
ol += (v[0] * (v[1] | time_derivative(v[2], frame)))
return time_derivative(ol, frame, order - 1)
else:
return diff(express(expr, frame, variables=True), t, order)
def outer(vec1, vec2):
"""Outer product convenience wrapper for Vector.outer():\n"""
if not isinstance(vec1, Vector):
raise TypeError('Outer product is between two Vectors')
return vec1 | vec2
outer.__doc__ += Vector.outer.__doc__ # type: ignore
def kinematic_equations(speeds, coords, rot_type, rot_order=''):
"""Gives equations relating the qdot's to u's for a rotation type.
Supply rotation type and order as in orient. Speeds are assumed to be
body-fixed; if we are defining the orientation of B in A using by rot_type,
the angular velocity of B in A is assumed to be in the form: speed[0]*B.x +
speed[1]*B.y + speed[2]*B.z
Parameters
==========
speeds : list of length 3
The body fixed angular velocity measure numbers.
coords : list of length 3 or 4
The coordinates used to define the orientation of the two frames.
rot_type : str
The type of rotation used to create the equations. Body, Space, or
Quaternion only
rot_order : str or int
If applicable, the order of a series of rotations.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import kinematic_equations, vprint
>>> u1, u2, u3 = dynamicsymbols('u1 u2 u3')
>>> q1, q2, q3 = dynamicsymbols('q1 q2 q3')
>>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'),
... order=None)
[-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3']
"""
# Code below is checking and sanitizing input
approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131',
'212', '232', '313', '323', '1', '2', '3', '')
# make sure XYZ => 123 and rot_type is in lower case
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.lower()
if not isinstance(speeds, (list, tuple)):
raise TypeError('Need to supply speeds in a list')
if len(speeds) != 3:
raise TypeError('Need to supply 3 body-fixed speeds')
if not isinstance(coords, (list, tuple)):
raise TypeError('Need to supply coordinates in a list')
if rot_type in ['body', 'space']:
if rot_order not in approved_orders:
raise ValueError('Not an acceptable rotation order')
if len(coords) != 3:
raise ValueError('Need 3 coordinates for body or space')
# Actual hard-coded kinematic differential equations
w1, w2, w3 = speeds
if w1 == w2 == w3 == 0:
return [S.Zero]*3
q1, q2, q3 = coords
q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords]
s1, s2, s3 = [sin(q1), sin(q2), sin(q3)]
c1, c2, c3 = [cos(q1), cos(q2), cos(q3)]
if rot_type == 'body':
if rot_order == '123':
return [q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 *
c3, q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3]
if rot_order == '231':
return [q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (- w2 * c3 + w3 * s3) * s2 / c2]
if rot_order == '312':
return [q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 *
s3, q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2]
if rot_order == '132':
return [q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 *
c3, q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2]
if rot_order == '213':
return [q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 *
s3, q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3]
if rot_order == '321':
return [q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 *
s3, q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2]
if rot_order == '121':
return [q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 *
s3, q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2]
if rot_order == '131':
return [q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 *
c3, q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2]
if rot_order == '212':
return [q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 *
s3, q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2]
if rot_order == '232':
return [q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 *
c3, q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2]
if rot_order == '313':
return [q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 *
s3, q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3]
if rot_order == '323':
return [q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 *
c3, q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3]
if rot_type == 'space':
if rot_order == '123':
return [q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2]
if rot_order == '231':
return [q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2]
if rot_order == '312':
return [q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2]
if rot_order == '132':
return [q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2, q2d - w2 *
s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2]
if rot_order == '213':
return [q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2]
if rot_order == '321':
return [q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2]
if rot_order == '121':
return [q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2, q2d - w2 *
c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2]
if rot_order == '131':
return [q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2, q2d - w2 *
s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2]
if rot_order == '212':
return [q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2, q2d - w1 *
c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2]
if rot_order == '232':
return [q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2, q2d + w1 *
s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2]
if rot_order == '313':
return [q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3, q2d - w1 *
c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2]
if rot_order == '323':
return [q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3, q2d - w1 *
s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2]
elif rot_type == 'quaternion':
if rot_order != '':
raise ValueError('Cannot have rotation order for quaternion')
if len(coords) != 4:
raise ValueError('Need 4 coordinates for quaternion')
# Actual hard-coded kinematic differential equations
e0, e1, e2, e3 = coords
w = Matrix(speeds + [0])
E = Matrix([[e0, -e3, e2, e1],
[e3, e0, -e1, e2],
[-e2, e1, e0, e3],
[-e1, -e2, -e3, e0]])
edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]])
return list(edots.T - 0.5 * w.T * E.T)
else:
raise ValueError('Not an approved rotation type for this function')
def get_motion_params(frame, **kwargs):
"""
Returns the three motion parameters - (acceleration, velocity, and
position) as vectorial functions of time in the given frame.
If a higher order differential function is provided, the lower order
functions are used as boundary conditions. For example, given the
acceleration, the velocity and position parameters are taken as
boundary conditions.
The values of time at which the boundary conditions are specified
are taken from timevalue1(for position boundary condition) and
timevalue2(for velocity boundary condition).
If any of the boundary conditions are not provided, they are taken
to be zero by default (zero vectors, in case of vectorial inputs). If
the boundary conditions are also functions of time, they are converted
to constants by substituting the time values in the dynamicsymbols._t
time Symbol.
This function can also be used for calculating rotational motion
parameters. Have a look at the Parameters and Examples for more clarity.
Parameters
==========
frame : ReferenceFrame
The frame to express the motion parameters in
acceleration : Vector
Acceleration of the object/frame as a function of time
velocity : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
position : Vector
Velocity as function of time or as boundary condition
of velocity at time = timevalue1
timevalue1 : sympyfiable
Value of time for position boundary condition
timevalue2 : sympyfiable
Value of time for velocity boundary condition
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> from sympy import symbols
>>> R = ReferenceFrame('R')
>>> v1, v2, v3 = dynamicsymbols('v1 v2 v3')
>>> v = v1*R.x + v2*R.y + v3*R.z
>>> get_motion_params(R, position = v)
(v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z)
>>> a, b, c = symbols('a b c')
>>> v = a*R.x + b*R.y + c*R.z
>>> get_motion_params(R, velocity = v)
(0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z)
>>> parameters = get_motion_params(R, acceleration = v)
>>> parameters[1]
a*t*R.x + b*t*R.y + c*t*R.z
>>> parameters[2]
a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z
"""
def _process_vector_differential(vectdiff, condition, variable, ordinate,
frame):
"""
Helper function for get_motion methods. Finds derivative of vectdiff
wrt variable, and its integral using the specified boundary condition
at value of variable = ordinate.
Returns a tuple of - (derivative, function and integral) wrt vectdiff
"""
# Make sure boundary condition is independent of 'variable'
if condition != 0:
condition = express(condition, frame, variables=True)
# Special case of vectdiff == 0
if vectdiff == Vector(0):
return (0, 0, condition)
# Express vectdiff completely in condition's frame to give vectdiff1
vectdiff1 = express(vectdiff, frame)
# Find derivative of vectdiff
vectdiff2 = time_derivative(vectdiff, frame)
# Integrate and use boundary condition
vectdiff0 = Vector(0)
lims = (variable, ordinate, variable)
for dim in frame:
function1 = vectdiff1.dot(dim)
abscissa = dim.dot(condition).subs({variable: ordinate})
# Indefinite integral of 'function1' wrt 'variable', using
# the given initial condition (ordinate, abscissa).
vectdiff0 += (integrate(function1, lims) + abscissa) * dim
# Return tuple
return (vectdiff2, vectdiff, vectdiff0)
_check_frame(frame)
# Decide mode of operation based on user's input
if 'acceleration' in kwargs:
mode = 2
elif 'velocity' in kwargs:
mode = 1
else:
mode = 0
# All the possible parameters in kwargs
# Not all are required for every case
# If not specified, set to default values(may or may not be used in
# calculations)
conditions = ['acceleration', 'velocity', 'position',
'timevalue', 'timevalue1', 'timevalue2']
for i, x in enumerate(conditions):
if x not in kwargs:
if i < 3:
kwargs[x] = Vector(0)
else:
kwargs[x] = S.Zero
elif i < 3:
_check_vector(kwargs[x])
else:
kwargs[x] = sympify(kwargs[x])
if mode == 2:
vel = _process_vector_differential(kwargs['acceleration'],
kwargs['velocity'],
dynamicsymbols._t,
kwargs['timevalue2'], frame)[2]
pos = _process_vector_differential(vel, kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)[2]
return (kwargs['acceleration'], vel, pos)
elif mode == 1:
return _process_vector_differential(kwargs['velocity'],
kwargs['position'],
dynamicsymbols._t,
kwargs['timevalue1'], frame)
else:
vel = time_derivative(kwargs['position'], frame)
acc = time_derivative(vel, frame)
return (acc, vel, kwargs['position'])
def partial_velocity(vel_vecs, gen_speeds, frame):
"""Returns a list of partial velocities with respect to the provided
generalized speeds in the given reference frame for each of the supplied
velocity vectors.
The output is a list of lists. The outer list has a number of elements
equal to the number of supplied velocity vectors. The inner lists are, for
each velocity vector, the partial derivatives of that velocity vector with
respect to the generalized speeds supplied.
Parameters
==========
vel_vecs : iterable
An iterable of velocity vectors (angular or linear).
gen_speeds : iterable
An iterable of generalized speeds.
frame : ReferenceFrame
The reference frame that the partial derivatives are going to be taken
in.
Examples
========
>>> from sympy.physics.vector import Point, ReferenceFrame
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy.physics.vector import partial_velocity
>>> u = dynamicsymbols('u')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, u * N.x)
>>> vel_vecs = [P.vel(N)]
>>> gen_speeds = [u]
>>> partial_velocity(vel_vecs, gen_speeds, N)
[[N.x]]
"""
if not iterable(vel_vecs):
raise TypeError('Velocity vectors must be contained in an iterable.')
if not iterable(gen_speeds):
raise TypeError('Generalized speeds must be contained in an iterable')
vec_partials = []
for vec in vel_vecs:
partials = []
for speed in gen_speeds:
partials.append(vec.diff(speed, frame, var_in_dcm=False))
vec_partials.append(partials)
return vec_partials
def dynamicsymbols(names, level=0, **assumptions):
"""Uses symbols and Function for functions of time.
Creates a SymPy UndefinedFunction, which is then initialized as a function
of a variable, the default being Symbol('t').
Parameters
==========
names : str
Names of the dynamic symbols you want to create; works the same way as
inputs to symbols
level : int
Level of differentiation of the returned function; d/dt once of t,
twice of t, etc.
assumptions :
- real(bool) : This is used to set the dynamicsymbol as real,
by default is False.
- positive(bool) : This is used to set the dynamicsymbol as positive,
by default is False.
- commutative(bool) : This is used to set the commutative property of
a dynamicsymbol, by default is True.
- integer(bool) : This is used to set the dynamicsymbol as integer,
by default is False.
Examples
========
>>> from sympy.physics.vector import dynamicsymbols
>>> from sympy import diff, Symbol
>>> q1 = dynamicsymbols('q1')
>>> q1
q1(t)
>>> q2 = dynamicsymbols('q2', real=True)
>>> q2.is_real
True
>>> q3 = dynamicsymbols('q3', positive=True)
>>> q3.is_positive
True
>>> q4, q5 = dynamicsymbols('q4,q5', commutative=False)
>>> bool(q4*q5 != q5*q4)
True
>>> q6 = dynamicsymbols('q6', integer=True)
>>> q6.is_integer
True
>>> diff(q1, Symbol('t'))
Derivative(q1(t), t)
"""
esses = symbols(names, cls=Function, **assumptions)
t = dynamicsymbols._t
if iterable(esses):
esses = [reduce(diff, [t] * level, e(t)) for e in esses]
return esses
else:
return reduce(diff, [t] * level, esses(t))
dynamicsymbols._t = Symbol('t') # type: ignore
dynamicsymbols._str = '\'' # type: ignore
|
33b99cd296b8d4264743a55c726d7ee1547e366b48a91ee39a915c50680578d5 | from sympy.core.function import diff
from sympy.core.singleton import S
from sympy.integrals.integrals import integrate
from sympy.physics.vector import Vector, express
from sympy.physics.vector.frame import _check_frame
from sympy.physics.vector.vector import _check_vector
__all__ = ['curl', 'divergence', 'gradient', 'is_conservative',
'is_solenoidal', 'scalar_potential',
'scalar_potential_difference']
def curl(vect, frame):
"""
Returns the curl of a vector field computed wrt the coordinate
symbols of the given frame.
Parameters
==========
vect : Vector
The vector operand
frame : ReferenceFrame
The reference frame to calculate the curl in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import curl
>>> R = ReferenceFrame('R')
>>> v1 = R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z
>>> curl(v1, R)
0
>>> v2 = R[0]*R[1]*R[2]*R.x
>>> curl(v2, R)
R_x*R_y*R.y - R_x*R_z*R.z
"""
_check_vector(vect)
if vect == 0:
return Vector(0)
vect = express(vect, frame, variables=True)
# A mechanical approach to avoid looping overheads
vectx = vect.dot(frame.x)
vecty = vect.dot(frame.y)
vectz = vect.dot(frame.z)
outvec = Vector(0)
outvec += (diff(vectz, frame[1]) - diff(vecty, frame[2])) * frame.x
outvec += (diff(vectx, frame[2]) - diff(vectz, frame[0])) * frame.y
outvec += (diff(vecty, frame[0]) - diff(vectx, frame[1])) * frame.z
return outvec
def divergence(vect, frame):
"""
Returns the divergence of a vector field computed wrt the coordinate
symbols of the given frame.
Parameters
==========
vect : Vector
The vector operand
frame : ReferenceFrame
The reference frame to calculate the divergence in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import divergence
>>> R = ReferenceFrame('R')
>>> v1 = R[0]*R[1]*R[2] * (R.x+R.y+R.z)
>>> divergence(v1, R)
R_x*R_y + R_x*R_z + R_y*R_z
>>> v2 = 2*R[1]*R[2]*R.y
>>> divergence(v2, R)
2*R_z
"""
_check_vector(vect)
if vect == 0:
return S.Zero
vect = express(vect, frame, variables=True)
vectx = vect.dot(frame.x)
vecty = vect.dot(frame.y)
vectz = vect.dot(frame.z)
out = S.Zero
out += diff(vectx, frame[0])
out += diff(vecty, frame[1])
out += diff(vectz, frame[2])
return out
def gradient(scalar, frame):
"""
Returns the vector gradient of a scalar field computed wrt the
coordinate symbols of the given frame.
Parameters
==========
scalar : sympifiable
The scalar field to take the gradient of
frame : ReferenceFrame
The frame to calculate the gradient in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import gradient
>>> R = ReferenceFrame('R')
>>> s1 = R[0]*R[1]*R[2]
>>> gradient(s1, R)
R_y*R_z*R.x + R_x*R_z*R.y + R_x*R_y*R.z
>>> s2 = 5*R[0]**2*R[2]
>>> gradient(s2, R)
10*R_x*R_z*R.x + 5*R_x**2*R.z
"""
_check_frame(frame)
outvec = Vector(0)
scalar = express(scalar, frame, variables=True)
for i, x in enumerate(frame):
outvec += diff(scalar, frame[i]) * x
return outvec
def is_conservative(field):
"""
Checks if a field is conservative.
Parameters
==========
field : Vector
The field to check for conservative property
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import is_conservative
>>> R = ReferenceFrame('R')
>>> is_conservative(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z)
True
>>> is_conservative(R[2] * R.y)
False
"""
# Field is conservative irrespective of frame
# Take the first frame in the result of the separate method of Vector
if field == Vector(0):
return True
frame = list(field.separate())[0]
return curl(field, frame).simplify() == Vector(0)
def is_solenoidal(field):
"""
Checks if a field is solenoidal.
Parameters
==========
field : Vector
The field to check for solenoidal property
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import is_solenoidal
>>> R = ReferenceFrame('R')
>>> is_solenoidal(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z)
True
>>> is_solenoidal(R[1] * R.y)
False
"""
# Field is solenoidal irrespective of frame
# Take the first frame in the result of the separate method in Vector
if field == Vector(0):
return True
frame = list(field.separate())[0]
return divergence(field, frame).simplify() is S.Zero
def scalar_potential(field, frame):
"""
Returns the scalar potential function of a field in a given frame
(without the added integration constant).
Parameters
==========
field : Vector
The vector field whose scalar potential function is to be
calculated
frame : ReferenceFrame
The frame to do the calculation in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy.physics.vector import scalar_potential, gradient
>>> R = ReferenceFrame('R')
>>> scalar_potential(R.z, R) == R[2]
True
>>> scalar_field = 2*R[0]**2*R[1]*R[2]
>>> grad_field = gradient(scalar_field, R)
>>> scalar_potential(grad_field, R)
2*R_x**2*R_y*R_z
"""
# Check whether field is conservative
if not is_conservative(field):
raise ValueError("Field is not conservative")
if field == Vector(0):
return S.Zero
# Express the field exntirely in frame
# Substitute coordinate variables also
_check_frame(frame)
field = express(field, frame, variables=True)
# Make a list of dimensions of the frame
dimensions = [x for x in frame]
# Calculate scalar potential function
temp_function = integrate(field.dot(dimensions[0]), frame[0])
for i, dim in enumerate(dimensions[1:]):
partial_diff = diff(temp_function, frame[i + 1])
partial_diff = field.dot(dim) - partial_diff
temp_function += integrate(partial_diff, frame[i + 1])
return temp_function
def scalar_potential_difference(field, frame, point1, point2, origin):
"""
Returns the scalar potential difference between two points in a
certain frame, wrt a given field.
If a scalar field is provided, its values at the two points are
considered. If a conservative vector field is provided, the values
of its scalar potential function at the two points are used.
Returns (potential at position 2) - (potential at position 1)
Parameters
==========
field : Vector/sympyfiable
The field to calculate wrt
frame : ReferenceFrame
The frame to do the calculations in
point1 : Point
The initial Point in given frame
position2 : Point
The second Point in the given frame
origin : Point
The Point to use as reference point for position vector
calculation
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, Point
>>> from sympy.physics.vector import scalar_potential_difference
>>> R = ReferenceFrame('R')
>>> O = Point('O')
>>> P = O.locatenew('P', R[0]*R.x + R[1]*R.y + R[2]*R.z)
>>> vectfield = 4*R[0]*R[1]*R.x + 2*R[0]**2*R.y
>>> scalar_potential_difference(vectfield, R, O, P, O)
2*R_x**2*R_y
>>> Q = O.locatenew('O', 3*R.x + R.y + 2*R.z)
>>> scalar_potential_difference(vectfield, R, P, Q, O)
-2*R_x**2*R_y + 18
"""
_check_frame(frame)
if isinstance(field, Vector):
# Get the scalar potential function
scalar_fn = scalar_potential(field, frame)
else:
# Field is a scalar
scalar_fn = field
# Express positions in required frame
position1 = express(point1.pos_from(origin), frame, variables=True)
position2 = express(point2.pos_from(origin), frame, variables=True)
# Get the two positions as substitution dicts for coordinate variables
subs_dict1 = {}
subs_dict2 = {}
for i, x in enumerate(frame):
subs_dict1[frame[i]] = x.dot(position1)
subs_dict2[frame[i]] = x.dot(position2)
return scalar_fn.subs(subs_dict2) - scalar_fn.subs(subs_dict1)
|
e7be8a6ba90dae721cfaf5753c1cc94ea5d880b23c36b4586c68eadb4b104c69 | from sympy.core.backend import (diff, expand, sin, cos, sympify, eye, symbols,
ImmutableMatrix as Matrix, MatrixBase)
from sympy.core.symbol import (Dummy, Symbol)
from sympy.simplify.trigsimp import trigsimp
from sympy.solvers.solvers import solve
from sympy.physics.vector.vector import Vector, _check_vector
from sympy.utilities.misc import translate
from warnings import warn
__all__ = ['CoordinateSym', 'ReferenceFrame']
class CoordinateSym(Symbol):
"""
A coordinate symbol/base scalar associated wrt a Reference Frame.
Ideally, users should not instantiate this class. Instances of
this class must only be accessed through the corresponding frame
as 'frame[index]'.
CoordinateSyms having the same frame and index parameters are equal
(even though they may be instantiated separately).
Parameters
==========
name : string
The display name of the CoordinateSym
frame : ReferenceFrame
The reference frame this base scalar belongs to
index : 0, 1 or 2
The index of the dimension denoted by this coordinate variable
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, CoordinateSym
>>> A = ReferenceFrame('A')
>>> A[1]
A_y
>>> type(A[0])
<class 'sympy.physics.vector.frame.CoordinateSym'>
>>> a_y = CoordinateSym('a_y', A, 1)
>>> a_y == A[1]
True
"""
def __new__(cls, name, frame, index):
# We can't use the cached Symbol.__new__ because this class depends on
# frame and index, which are not passed to Symbol.__xnew__.
assumptions = {}
super()._sanitize(assumptions, cls)
obj = super().__xnew__(cls, name, **assumptions)
_check_frame(frame)
if index not in range(0, 3):
raise ValueError("Invalid index specified")
obj._id = (frame, index)
return obj
@property
def frame(self):
return self._id[0]
def __eq__(self, other):
# Check if the other object is a CoordinateSym of the same frame and
# same index
if isinstance(other, CoordinateSym):
if other._id == self._id:
return True
return False
def __ne__(self, other):
return not self == other
def __hash__(self):
return tuple((self._id[0].__hash__(), self._id[1])).__hash__()
class ReferenceFrame:
"""A reference frame in classical mechanics.
ReferenceFrame is a class used to represent a reference frame in classical
mechanics. It has a standard basis of three unit vectors in the frame's
x, y, and z directions.
It also can have a rotation relative to a parent frame; this rotation is
defined by a direction cosine matrix relating this frame's basis vectors to
the parent frame's basis vectors. It can also have an angular velocity
vector, defined in another frame.
"""
_count = 0
def __init__(self, name, indices=None, latexs=None, variables=None):
"""ReferenceFrame initialization method.
A ReferenceFrame has a set of orthonormal basis vectors, along with
orientations relative to other ReferenceFrames and angular velocities
relative to other ReferenceFrames.
Parameters
==========
indices : tuple of str
Enables the reference frame's basis unit vectors to be accessed by
Python's square bracket indexing notation using the provided three
indice strings and alters the printing of the unit vectors to
reflect this choice.
latexs : tuple of str
Alters the LaTeX printing of the reference frame's basis unit
vectors to the provided three valid LaTeX strings.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, vlatex
>>> N = ReferenceFrame('N')
>>> N.x
N.x
>>> O = ReferenceFrame('O', indices=('1', '2', '3'))
>>> O.x
O['1']
>>> O['1']
O['1']
>>> P = ReferenceFrame('P', latexs=('A1', 'A2', 'A3'))
>>> vlatex(P.x)
'A1'
``symbols()`` can be used to create multiple Reference Frames in one
step, for example:
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import symbols
>>> A, B, C = symbols('A B C', cls=ReferenceFrame)
>>> D, E = symbols('D E', cls=ReferenceFrame, indices=('1', '2', '3'))
>>> A[0]
A_x
>>> D.x
D['1']
>>> E.y
E['2']
>>> type(A) == type(D)
True
"""
if not isinstance(name, str):
raise TypeError('Need to supply a valid name')
# The if statements below are for custom printing of basis-vectors for
# each frame.
# First case, when custom indices are supplied
if indices is not None:
if not isinstance(indices, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(indices) != 3:
raise ValueError('Supply 3 indices')
for i in indices:
if not isinstance(i, str):
raise TypeError('Indices must be strings')
self.str_vecs = [(name + '[\'' + indices[0] + '\']'),
(name + '[\'' + indices[1] + '\']'),
(name + '[\'' + indices[2] + '\']')]
self.pretty_vecs = [(name.lower() + "_" + indices[0]),
(name.lower() + "_" + indices[1]),
(name.lower() + "_" + indices[2])]
self.latex_vecs = [(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[0])),
(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[1])),
(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(),
indices[2]))]
self.indices = indices
# Second case, when no custom indices are supplied
else:
self.str_vecs = [(name + '.x'), (name + '.y'), (name + '.z')]
self.pretty_vecs = [name.lower() + "_x",
name.lower() + "_y",
name.lower() + "_z"]
self.latex_vecs = [(r"\mathbf{\hat{%s}_x}" % name.lower()),
(r"\mathbf{\hat{%s}_y}" % name.lower()),
(r"\mathbf{\hat{%s}_z}" % name.lower())]
self.indices = ['x', 'y', 'z']
# Different step, for custom latex basis vectors
if latexs is not None:
if not isinstance(latexs, (tuple, list)):
raise TypeError('Supply the indices as a list')
if len(latexs) != 3:
raise ValueError('Supply 3 indices')
for i in latexs:
if not isinstance(i, str):
raise TypeError('Latex entries must be strings')
self.latex_vecs = latexs
self.name = name
self._var_dict = {}
# The _dcm_dict dictionary will only store the dcms of adjacent
# parent-child relationships. The _dcm_cache dictionary will store
# calculated dcm along with all content of _dcm_dict for faster
# retrieval of dcms.
self._dcm_dict = {}
self._dcm_cache = {}
self._ang_vel_dict = {}
self._ang_acc_dict = {}
self._dlist = [self._dcm_dict, self._ang_vel_dict, self._ang_acc_dict]
self._cur = 0
self._x = Vector([(Matrix([1, 0, 0]), self)])
self._y = Vector([(Matrix([0, 1, 0]), self)])
self._z = Vector([(Matrix([0, 0, 1]), self)])
# Associate coordinate symbols wrt this frame
if variables is not None:
if not isinstance(variables, (tuple, list)):
raise TypeError('Supply the variable names as a list/tuple')
if len(variables) != 3:
raise ValueError('Supply 3 variable names')
for i in variables:
if not isinstance(i, str):
raise TypeError('Variable names must be strings')
else:
variables = [name + '_x', name + '_y', name + '_z']
self.varlist = (CoordinateSym(variables[0], self, 0),
CoordinateSym(variables[1], self, 1),
CoordinateSym(variables[2], self, 2))
ReferenceFrame._count += 1
self.index = ReferenceFrame._count
def __getitem__(self, ind):
"""
Returns basis vector for the provided index, if the index is a string.
If the index is a number, returns the coordinate variable correspon-
-ding to that index.
"""
if not isinstance(ind, str):
if ind < 3:
return self.varlist[ind]
else:
raise ValueError("Invalid index provided")
if self.indices[0] == ind:
return self.x
if self.indices[1] == ind:
return self.y
if self.indices[2] == ind:
return self.z
else:
raise ValueError('Not a defined index')
def __iter__(self):
return iter([self.x, self.y, self.z])
def __str__(self):
"""Returns the name of the frame. """
return self.name
__repr__ = __str__
def _dict_list(self, other, num):
"""Returns an inclusive list of reference frames that connect this
reference frame to the provided reference frame.
Parameters
==========
other : ReferenceFrame
The other reference frame to look for a connecting relationship to.
num : integer
``0``, ``1``, and ``2`` will look for orientation, angular
velocity, and angular acceleration relationships between the two
frames, respectively.
Returns
=======
list
Inclusive list of reference frames that connect this reference
frame to the other reference frame.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> A = ReferenceFrame('A')
>>> B = ReferenceFrame('B')
>>> C = ReferenceFrame('C')
>>> D = ReferenceFrame('D')
>>> B.orient_axis(A, A.x, 1.0)
>>> C.orient_axis(B, B.x, 1.0)
>>> D.orient_axis(C, C.x, 1.0)
>>> D._dict_list(A, 0)
[D, C, B, A]
Raises
======
ValueError
When no path is found between the two reference frames or ``num``
is an incorrect value.
"""
connect_type = {0: 'orientation',
1: 'angular velocity',
2: 'angular acceleration'}
if num not in connect_type.keys():
raise ValueError('Valid values for num are 0, 1, or 2.')
possible_connecting_paths = [[self]]
oldlist = [[]]
while possible_connecting_paths != oldlist:
oldlist = possible_connecting_paths[:] # make a copy
for frame_list in possible_connecting_paths:
frames_adjacent_to_last = frame_list[-1]._dlist[num].keys()
for adjacent_frame in frames_adjacent_to_last:
if adjacent_frame not in frame_list:
connecting_path = frame_list + [adjacent_frame]
if connecting_path not in possible_connecting_paths:
possible_connecting_paths.append(connecting_path)
for connecting_path in oldlist:
if connecting_path[-1] != other:
possible_connecting_paths.remove(connecting_path)
possible_connecting_paths.sort(key=len)
if len(possible_connecting_paths) != 0:
return possible_connecting_paths[0] # selects the shortest path
msg = 'No connecting {} path found between {} and {}.'
raise ValueError(msg.format(connect_type[num], self.name, other.name))
def _w_diff_dcm(self, otherframe):
"""Angular velocity from time differentiating the DCM. """
from sympy.physics.vector.functions import dynamicsymbols
dcm2diff = otherframe.dcm(self)
diffed = dcm2diff.diff(dynamicsymbols._t)
angvelmat = diffed * dcm2diff.T
w1 = trigsimp(expand(angvelmat[7]), recursive=True)
w2 = trigsimp(expand(angvelmat[2]), recursive=True)
w3 = trigsimp(expand(angvelmat[3]), recursive=True)
return Vector([(Matrix([w1, w2, w3]), otherframe)])
def variable_map(self, otherframe):
"""
Returns a dictionary which expresses the coordinate variables
of this frame in terms of the variables of otherframe.
If Vector.simp is True, returns a simplified version of the mapped
values. Else, returns them without simplification.
Simplification of the expressions may take time.
Parameters
==========
otherframe : ReferenceFrame
The other frame to map the variables to
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> A = ReferenceFrame('A')
>>> q = dynamicsymbols('q')
>>> B = A.orientnew('B', 'Axis', [q, A.z])
>>> A.variable_map(B)
{A_x: B_x*cos(q(t)) - B_y*sin(q(t)), A_y: B_x*sin(q(t)) + B_y*cos(q(t)), A_z: B_z}
"""
_check_frame(otherframe)
if (otherframe, Vector.simp) in self._var_dict:
return self._var_dict[(otherframe, Vector.simp)]
else:
vars_matrix = self.dcm(otherframe) * Matrix(otherframe.varlist)
mapping = {}
for i, x in enumerate(self):
if Vector.simp:
mapping[self.varlist[i]] = trigsimp(vars_matrix[i],
method='fu')
else:
mapping[self.varlist[i]] = vars_matrix[i]
self._var_dict[(otherframe, Vector.simp)] = mapping
return mapping
def ang_acc_in(self, otherframe):
"""Returns the angular acceleration Vector of the ReferenceFrame.
Effectively returns the Vector:
``N_alpha_B``
which represent the angular acceleration of B in N, where B is self,
and N is otherframe.
Parameters
==========
otherframe : ReferenceFrame
The ReferenceFrame which the angular acceleration is returned in.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_acc(N, V)
>>> A.ang_acc_in(N)
10*N.x
"""
_check_frame(otherframe)
if otherframe in self._ang_acc_dict:
return self._ang_acc_dict[otherframe]
else:
return self.ang_vel_in(otherframe).dt(otherframe)
def ang_vel_in(self, otherframe):
"""Returns the angular velocity Vector of the ReferenceFrame.
Effectively returns the Vector:
^N omega ^B
which represent the angular velocity of B in N, where B is self, and
N is otherframe.
Parameters
==========
otherframe : ReferenceFrame
The ReferenceFrame which the angular velocity is returned in.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_vel(N, V)
>>> A.ang_vel_in(N)
10*N.x
"""
_check_frame(otherframe)
flist = self._dict_list(otherframe, 1)
outvec = Vector(0)
for i in range(len(flist) - 1):
outvec += flist[i]._ang_vel_dict[flist[i + 1]]
return outvec
def dcm(self, otherframe):
r"""Returns the direction cosine matrix of this reference frame
relative to the provided reference frame.
The returned matrix can be used to express the orthogonal unit vectors
of this frame in terms of the orthogonal unit vectors of
``otherframe``.
Parameters
==========
otherframe : ReferenceFrame
The reference frame which the direction cosine matrix of this frame
is formed relative to.
Examples
========
The following example rotates the reference frame A relative to N by a
simple rotation and then calculates the direction cosine matrix of N
relative to A.
>>> from sympy import symbols, sin, cos
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> A.orient_axis(N, q1, N.x)
>>> N.dcm(A)
Matrix([
[1, 0, 0],
[0, cos(q1), -sin(q1)],
[0, sin(q1), cos(q1)]])
The second row of the above direction cosine matrix represents the
``N.y`` unit vector in N expressed in A. Like so:
>>> Ny = 0*A.x + cos(q1)*A.y - sin(q1)*A.z
Thus, expressing ``N.y`` in A should return the same result:
>>> N.y.express(A)
cos(q1)*A.y - sin(q1)*A.z
Notes
=====
It is important to know what form of the direction cosine matrix is
returned. If ``B.dcm(A)`` is called, it means the "direction cosine
matrix of B rotated relative to A". This is the matrix
:math:`{}^B\mathbf{C}^A` shown in the following relationship:
.. math::
\begin{bmatrix}
\hat{\mathbf{b}}_1 \\
\hat{\mathbf{b}}_2 \\
\hat{\mathbf{b}}_3
\end{bmatrix}
=
{}^B\mathbf{C}^A
\begin{bmatrix}
\hat{\mathbf{a}}_1 \\
\hat{\mathbf{a}}_2 \\
\hat{\mathbf{a}}_3
\end{bmatrix}.
:math:`{}^B\mathbf{C}^A` is the matrix that expresses the B unit
vectors in terms of the A unit vectors.
"""
_check_frame(otherframe)
# Check if the dcm wrt that frame has already been calculated
if otherframe in self._dcm_cache:
return self._dcm_cache[otherframe]
flist = self._dict_list(otherframe, 0)
outdcm = eye(3)
for i in range(len(flist) - 1):
outdcm = outdcm * flist[i]._dcm_dict[flist[i + 1]]
# After calculation, store the dcm in dcm cache for faster future
# retrieval
self._dcm_cache[otherframe] = outdcm
otherframe._dcm_cache[self] = outdcm.T
return outdcm
def _dcm(self, parent, parent_orient):
# If parent.oreint(self) is already defined,then
# update the _dcm_dict of parent while over write
# all content of self._dcm_dict and self._dcm_cache
# with new dcm relation.
# Else update _dcm_cache and _dcm_dict of both
# self and parent.
frames = self._dcm_cache.keys()
dcm_dict_del = []
dcm_cache_del = []
if parent in frames:
for frame in frames:
if frame in self._dcm_dict:
dcm_dict_del += [frame]
dcm_cache_del += [frame]
# Reset the _dcm_cache of this frame, and remove it from the
# _dcm_caches of the frames it is linked to. Also remove it from
# the _dcm_dict of its parent
for frame in dcm_dict_del:
del frame._dcm_dict[self]
for frame in dcm_cache_del:
del frame._dcm_cache[self]
# Reset the _dcm_dict
self._dcm_dict = self._dlist[0] = {}
# Reset the _dcm_cache
self._dcm_cache = {}
else:
# Check for loops and raise warning accordingly.
visited = []
queue = list(frames)
cont = True # Flag to control queue loop.
while queue and cont:
node = queue.pop(0)
if node not in visited:
visited.append(node)
neighbors = node._dcm_dict.keys()
for neighbor in neighbors:
if neighbor == parent:
warn('Loops are defined among the orientation of '
'frames. This is likely not desired and may '
'cause errors in your calculations.')
cont = False
break
queue.append(neighbor)
# Add the dcm relationship to _dcm_dict
self._dcm_dict.update({parent: parent_orient.T})
parent._dcm_dict.update({self: parent_orient})
# Update the dcm cache
self._dcm_cache.update({parent: parent_orient.T})
parent._dcm_cache.update({self: parent_orient})
def orient_axis(self, parent, axis, angle):
"""Sets the orientation of this reference frame with respect to a
parent reference frame by rotating through an angle about an axis fixed
in the parent reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
axis : Vector
Vector fixed in the parent frame about about which this frame is
rotated. It need not be a unit vector and the rotation follows the
right hand rule.
angle : sympifiable
Angle in radians by which it the frame is to be rotated.
Warns
======
UserWarning
If the orientation creates a kinematic loop.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B.orient_axis(N, N.x, q1)
The ``orient_axis()`` method generates a direction cosine matrix and
its transpose which defines the orientation of B relative to N and vice
versa. Once orient is called, ``dcm()`` outputs the appropriate
direction cosine matrix:
>>> B.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
>>> N.dcm(B)
Matrix([
[1, 0, 0],
[0, cos(q1), -sin(q1)],
[0, sin(q1), cos(q1)]])
The following two lines show that the sense of the rotation can be
defined by negating the vector direction or the angle. Both lines
produce the same result.
>>> B.orient_axis(N, -N.x, q1)
>>> B.orient_axis(N, N.x, -q1)
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
if not isinstance(axis, Vector) and isinstance(angle, Vector):
axis, angle = angle, axis
axis = _check_vector(axis)
amount = sympify(angle)
theta = amount
parent_orient_axis = []
if not axis.dt(parent) == 0:
raise ValueError('Axis cannot be time-varying.')
unit_axis = axis.express(parent).normalize()
unit_col = unit_axis.args[0][0]
parent_orient_axis = (
(eye(3) - unit_col * unit_col.T) * cos(theta) +
Matrix([[0, -unit_col[2], unit_col[1]],
[unit_col[2], 0, -unit_col[0]],
[-unit_col[1], unit_col[0], 0]]) *
sin(theta) + unit_col * unit_col.T)
self._dcm(parent, parent_orient_axis)
thetad = (amount).diff(dynamicsymbols._t)
wvec = thetad*axis.express(parent).normalize()
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orient_explicit(self, parent, dcm):
"""Sets the orientation of this reference frame relative to a parent
reference frame by explicitly setting the direction cosine matrix.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
dcm : Matrix, shape(3, 3)
Direction cosine matrix that specifies the relative rotation
between the two reference frames.
Warns
======
UserWarning
If the orientation creates a kinematic loop.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols, Matrix, sin, cos
>>> from sympy.physics.vector import ReferenceFrame
>>> q1 = symbols('q1')
>>> A = ReferenceFrame('A')
>>> B = ReferenceFrame('B')
>>> N = ReferenceFrame('N')
A simple rotation of ``A`` relative to ``N`` about ``N.x`` is defined
by the following direction cosine matrix:
>>> dcm = Matrix([[1, 0, 0],
... [0, cos(q1), -sin(q1)],
... [0, sin(q1), cos(q1)]])
>>> A.orient_explicit(N, dcm)
>>> A.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
This is equivalent to using ``orient_axis()``:
>>> B.orient_axis(N, N.x, q1)
>>> B.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
**Note carefully that** ``N.dcm(B)`` **(the transpose) would be passed
into** ``orient_explicit()`` **for** ``A.dcm(N)`` **to match**
``B.dcm(N)``:
>>> A.orient_explicit(N, N.dcm(B))
>>> A.dcm(N)
Matrix([
[1, 0, 0],
[0, cos(q1), sin(q1)],
[0, -sin(q1), cos(q1)]])
"""
_check_frame(parent)
# amounts must be a Matrix type object
# (e.g. sympy.matrices.dense.MutableDenseMatrix).
if not isinstance(dcm, MatrixBase):
raise TypeError("Amounts must be a SymPy Matrix type object.")
parent_orient_dcm = []
parent_orient_dcm = dcm
self._dcm(parent, parent_orient_dcm)
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def _rot(self, axis, angle):
"""DCM for simple axis 1,2,or 3 rotations."""
if axis == 1:
return Matrix([[1, 0, 0],
[0, cos(angle), -sin(angle)],
[0, sin(angle), cos(angle)]])
elif axis == 2:
return Matrix([[cos(angle), 0, sin(angle)],
[0, 1, 0],
[-sin(angle), 0, cos(angle)]])
elif axis == 3:
return Matrix([[cos(angle), -sin(angle), 0],
[sin(angle), cos(angle), 0],
[0, 0, 1]])
def orient_body_fixed(self, parent, angles, rotation_order):
"""Rotates this reference frame relative to the parent reference frame
by right hand rotating through three successive body fixed simple axis
rotations. Each subsequent axis of rotation is about the "body fixed"
unit vectors of a new intermediate reference frame. This type of
rotation is also referred to rotating through the `Euler and Tait-Bryan
Angles`_.
.. _Euler and Tait-Bryan Angles: https://en.wikipedia.org/wiki/Euler_angles
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
angles : 3-tuple of sympifiable
Three angles in radians used for the successive rotations.
rotation_order : 3 character string or 3 digit integer
Order of the rotations about each intermediate reference frames'
unit vectors. The Euler rotation about the X, Z', X'' axes can be
specified by the strings ``'XZX'``, ``'131'``, or the integer
``131``. There are 12 unique valid rotation orders (6 Euler and 6
Tait-Bryan): zxz, xyx, yzy, zyz, xzx, yxy, xyz, yzx, zxy, xzy, zyx,
and yxz.
Warns
======
UserWarning
If the orientation creates a kinematic loop.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1, q2, q3 = symbols('q1, q2, q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B1 = ReferenceFrame('B1')
>>> B2 = ReferenceFrame('B2')
>>> B3 = ReferenceFrame('B3')
For example, a classic Euler Angle rotation can be done by:
>>> B.orient_body_fixed(N, (q1, q2, q3), 'XYX')
>>> B.dcm(N)
Matrix([
[ cos(q2), sin(q1)*sin(q2), -sin(q2)*cos(q1)],
[sin(q2)*sin(q3), -sin(q1)*sin(q3)*cos(q2) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q3)*cos(q1)*cos(q2)],
[sin(q2)*cos(q3), -sin(q1)*cos(q2)*cos(q3) - sin(q3)*cos(q1), -sin(q1)*sin(q3) + cos(q1)*cos(q2)*cos(q3)]])
This rotates reference frame B relative to reference frame N through
``q1`` about ``N.x``, then rotates B again through ``q2`` about
``B.y``, and finally through ``q3`` about ``B.x``. It is equivalent to
three successive ``orient_axis()`` calls:
>>> B1.orient_axis(N, N.x, q1)
>>> B2.orient_axis(B1, B1.y, q2)
>>> B3.orient_axis(B2, B2.x, q3)
>>> B3.dcm(N)
Matrix([
[ cos(q2), sin(q1)*sin(q2), -sin(q2)*cos(q1)],
[sin(q2)*sin(q3), -sin(q1)*sin(q3)*cos(q2) + cos(q1)*cos(q3), sin(q1)*cos(q3) + sin(q3)*cos(q1)*cos(q2)],
[sin(q2)*cos(q3), -sin(q1)*cos(q2)*cos(q3) - sin(q3)*cos(q1), -sin(q1)*sin(q3) + cos(q1)*cos(q2)*cos(q3)]])
Acceptable rotation orders are of length 3, expressed in as a string
``'XYZ'`` or ``'123'`` or integer ``123``. Rotations about an axis
twice in a row are prohibited.
>>> B.orient_body_fixed(N, (q1, q2, 0), 'ZXZ')
>>> B.orient_body_fixed(N, (q1, q2, 0), '121')
>>> B.orient_body_fixed(N, (q1, q2, q3), 123)
"""
_check_frame(parent)
amounts = list(angles)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123
rot_order = translate(str(rotation_order), 'XYZxyz', '123123')
if rot_order not in approved_orders:
raise TypeError('The rotation order is not a valid order.')
parent_orient_body = []
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Body orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient_body = (self._rot(a1, amounts[0]) *
self._rot(a2, amounts[1]) *
self._rot(a3, amounts[2]))
self._dcm(parent, parent_orient_body)
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
'body', rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
# NOTE : SymPy 1.7 removed the call to simplify() that occured
# inside the solve() function, so this restores the pre-1.7
# behavior. See:
# https://github.com/sympy/sympy/issues/23140
# and
# https://github.com/sympy/sympy/issues/23130
wvec = wvec.simplify()
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orient_space_fixed(self, parent, angles, rotation_order):
"""Rotates this reference frame relative to the parent reference frame
by right hand rotating through three successive space fixed simple axis
rotations. Each subsequent axis of rotation is about the "space fixed"
unit vectors of the parent reference frame.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
angles : 3-tuple of sympifiable
Three angles in radians used for the successive rotations.
rotation_order : 3 character string or 3 digit integer
Order of the rotations about the parent reference frame's unit
vectors. The order can be specified by the strings ``'XZX'``,
``'131'``, or the integer ``131``. There are 12 unique valid
rotation orders.
Warns
======
UserWarning
If the orientation creates a kinematic loop.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q1, q2, q3 = symbols('q1, q2, q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
>>> B1 = ReferenceFrame('B1')
>>> B2 = ReferenceFrame('B2')
>>> B3 = ReferenceFrame('B3')
>>> B.orient_space_fixed(N, (q1, q2, q3), '312')
>>> B.dcm(N)
Matrix([
[ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1)],
[-sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3)],
[ sin(q3)*cos(q2), -sin(q2), cos(q2)*cos(q3)]])
is equivalent to:
>>> B1.orient_axis(N, N.z, q1)
>>> B2.orient_axis(B1, N.x, q2)
>>> B3.orient_axis(B2, N.y, q3)
>>> B3.dcm(N).simplify()
Matrix([
[ sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1)],
[-sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3)],
[ sin(q3)*cos(q2), -sin(q2), cos(q2)*cos(q3)]])
It is worth noting that space-fixed and body-fixed rotations are
related by the order of the rotations, i.e. the reverse order of body
fixed will give space fixed and vice versa.
>>> B.orient_space_fixed(N, (q1, q2, q3), '231')
>>> B.dcm(N)
Matrix([
[cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[ -sin(q2), cos(q2)*cos(q3), sin(q3)*cos(q2)],
[sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3)]])
>>> B.orient_body_fixed(N, (q3, q2, q1), '132')
>>> B.dcm(N)
Matrix([
[cos(q1)*cos(q2), sin(q1)*sin(q3) + sin(q2)*cos(q1)*cos(q3), -sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)],
[ -sin(q2), cos(q2)*cos(q3), sin(q3)*cos(q2)],
[sin(q1)*cos(q2), sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3)]])
"""
_check_frame(parent)
amounts = list(angles)
for i, v in enumerate(amounts):
if not isinstance(v, Vector):
amounts[i] = sympify(v)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
# make sure XYZ => 123
rot_order = translate(str(rotation_order), 'XYZxyz', '123123')
if rot_order not in approved_orders:
raise TypeError('The supplied order is not an approved type')
parent_orient_space = []
if not (len(amounts) == 3 & len(rot_order) == 3):
raise TypeError('Space orientation takes 3 values & 3 orders')
a1 = int(rot_order[0])
a2 = int(rot_order[1])
a3 = int(rot_order[2])
parent_orient_space = (self._rot(a3, amounts[2]) *
self._rot(a2, amounts[1]) *
self._rot(a1, amounts[0]))
self._dcm(parent, parent_orient_space)
try:
from sympy.polys.polyerrors import CoercionFailed
from sympy.physics.vector.functions import kinematic_equations
q1, q2, q3 = amounts
u1, u2, u3 = symbols('u1, u2, u3', cls=Dummy)
templist = kinematic_equations([u1, u2, u3], [q1, q2, q3],
'space', rot_order)
templist = [expand(i) for i in templist]
td = solve(templist, [u1, u2, u3])
u1 = expand(td[u1])
u2 = expand(td[u2])
u3 = expand(td[u3])
wvec = u1 * self.x + u2 * self.y + u3 * self.z
except (CoercionFailed, AssertionError):
wvec = self._w_diff_dcm(parent)
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orient_quaternion(self, parent, numbers):
"""Sets the orientation of this reference frame relative to a parent
reference frame via an orientation quaternion. An orientation
quaternion is defined as a finite rotation a unit vector, ``(lambda_x,
lambda_y, lambda_z)``, by an angle ``theta``. The orientation
quaternion is described by four parameters:
- ``q0 = cos(theta/2)``
- ``q1 = lambda_x*sin(theta/2)``
- ``q2 = lambda_y*sin(theta/2)``
- ``q3 = lambda_z*sin(theta/2)``
See `Quaternions and Spatial Rotation
<https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation>`_ on
Wikipedia for more information.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
numbers : 4-tuple of sympifiable
The four quaternion scalar numbers as defined above: ``q0``,
``q1``, ``q2``, ``q3``.
Warns
======
UserWarning
If the orientation creates a kinematic loop.
Examples
========
Setup variables for the examples:
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
>>> B = ReferenceFrame('B')
Set the orientation:
>>> B.orient_quaternion(N, (q0, q1, q2, q3))
>>> B.dcm(N)
Matrix([
[q0**2 + q1**2 - q2**2 - q3**2, 2*q0*q3 + 2*q1*q2, -2*q0*q2 + 2*q1*q3],
[ -2*q0*q3 + 2*q1*q2, q0**2 - q1**2 + q2**2 - q3**2, 2*q0*q1 + 2*q2*q3],
[ 2*q0*q2 + 2*q1*q3, -2*q0*q1 + 2*q2*q3, q0**2 - q1**2 - q2**2 + q3**2]])
"""
from sympy.physics.vector.functions import dynamicsymbols
_check_frame(parent)
numbers = list(numbers)
for i, v in enumerate(numbers):
if not isinstance(v, Vector):
numbers[i] = sympify(v)
parent_orient_quaternion = []
if not (isinstance(numbers, (list, tuple)) & (len(numbers) == 4)):
raise TypeError('Amounts are a list or tuple of length 4')
q0, q1, q2, q3 = numbers
parent_orient_quaternion = (
Matrix([[q0**2 + q1**2 - q2**2 - q3**2,
2 * (q1 * q2 - q0 * q3),
2 * (q0 * q2 + q1 * q3)],
[2 * (q1 * q2 + q0 * q3),
q0**2 - q1**2 + q2**2 - q3**2,
2 * (q2 * q3 - q0 * q1)],
[2 * (q1 * q3 - q0 * q2),
2 * (q0 * q1 + q2 * q3),
q0**2 - q1**2 - q2**2 + q3**2]]))
self._dcm(parent, parent_orient_quaternion)
t = dynamicsymbols._t
q0, q1, q2, q3 = numbers
q0d = diff(q0, t)
q1d = diff(q1, t)
q2d = diff(q2, t)
q3d = diff(q3, t)
w1 = 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1)
w2 = 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2)
w3 = 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3)
wvec = Vector([(Matrix([w1, w2, w3]), self)])
self._ang_vel_dict.update({parent: wvec})
parent._ang_vel_dict.update({self: -wvec})
self._var_dict = {}
def orient(self, parent, rot_type, amounts, rot_order=''):
"""Sets the orientation of this reference frame relative to another
(parent) reference frame.
.. note:: It is now recommended to use the ``.orient_axis,
.orient_body_fixed, .orient_space_fixed, .orient_quaternion``
methods for the different rotation types.
Parameters
==========
parent : ReferenceFrame
Reference frame that this reference frame will be rotated relative
to.
rot_type : str
The method used to generate the direction cosine matrix. Supported
methods are:
- ``'Axis'``: simple rotations about a single common axis
- ``'DCM'``: for setting the direction cosine matrix directly
- ``'Body'``: three successive rotations about new intermediate
axes, also called "Euler and Tait-Bryan angles"
- ``'Space'``: three successive rotations about the parent
frames' unit vectors
- ``'Quaternion'``: rotations defined by four parameters which
result in a singularity free direction cosine matrix
amounts :
Expressions defining the rotation angles or direction cosine
matrix. These must match the ``rot_type``. See examples below for
details. The input types are:
- ``'Axis'``: 2-tuple (expr/sym/func, Vector)
- ``'DCM'``: Matrix, shape(3,3)
- ``'Body'``: 3-tuple of expressions, symbols, or functions
- ``'Space'``: 3-tuple of expressions, symbols, or functions
- ``'Quaternion'``: 4-tuple of expressions, symbols, or
functions
rot_order : str or int, optional
If applicable, the order of the successive of rotations. The string
``'123'`` and integer ``123`` are equivalent, for example. Required
for ``'Body'`` and ``'Space'``.
Warns
======
UserWarning
If the orientation creates a kinematic loop.
"""
_check_frame(parent)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.upper()
if rot_order not in approved_orders:
raise TypeError('The supplied order is not an approved type')
if rot_type == 'AXIS':
self.orient_axis(parent, amounts[1], amounts[0])
elif rot_type == 'DCM':
self.orient_explicit(parent, amounts)
elif rot_type == 'BODY':
self.orient_body_fixed(parent, amounts, rot_order)
elif rot_type == 'SPACE':
self.orient_space_fixed(parent, amounts, rot_order)
elif rot_type == 'QUATERNION':
self.orient_quaternion(parent, amounts)
else:
raise NotImplementedError('That is not an implemented rotation')
def orientnew(self, newname, rot_type, amounts, rot_order='',
variables=None, indices=None, latexs=None):
r"""Returns a new reference frame oriented with respect to this
reference frame.
See ``ReferenceFrame.orient()`` for detailed examples of how to orient
reference frames.
Parameters
==========
newname : str
Name for the new reference frame.
rot_type : str
The method used to generate the direction cosine matrix. Supported
methods are:
- ``'Axis'``: simple rotations about a single common axis
- ``'DCM'``: for setting the direction cosine matrix directly
- ``'Body'``: three successive rotations about new intermediate
axes, also called "Euler and Tait-Bryan angles"
- ``'Space'``: three successive rotations about the parent
frames' unit vectors
- ``'Quaternion'``: rotations defined by four parameters which
result in a singularity free direction cosine matrix
amounts :
Expressions defining the rotation angles or direction cosine
matrix. These must match the ``rot_type``. See examples below for
details. The input types are:
- ``'Axis'``: 2-tuple (expr/sym/func, Vector)
- ``'DCM'``: Matrix, shape(3,3)
- ``'Body'``: 3-tuple of expressions, symbols, or functions
- ``'Space'``: 3-tuple of expressions, symbols, or functions
- ``'Quaternion'``: 4-tuple of expressions, symbols, or
functions
rot_order : str or int, optional
If applicable, the order of the successive of rotations. The string
``'123'`` and integer ``123`` are equivalent, for example. Required
for ``'Body'`` and ``'Space'``.
indices : tuple of str
Enables the reference frame's basis unit vectors to be accessed by
Python's square bracket indexing notation using the provided three
indice strings and alters the printing of the unit vectors to
reflect this choice.
latexs : tuple of str
Alters the LaTeX printing of the reference frame's basis unit
vectors to the provided three valid LaTeX strings.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame, vlatex
>>> q0, q1, q2, q3 = symbols('q0 q1 q2 q3')
>>> N = ReferenceFrame('N')
Create a new reference frame A rotated relative to N through a simple
rotation.
>>> A = N.orientnew('A', 'Axis', (q0, N.x))
Create a new reference frame B rotated relative to N through body-fixed
rotations.
>>> B = N.orientnew('B', 'Body', (q1, q2, q3), '123')
Create a new reference frame C rotated relative to N through a simple
rotation with unique indices and LaTeX printing.
>>> C = N.orientnew('C', 'Axis', (q0, N.x), indices=('1', '2', '3'),
... latexs=(r'\hat{\mathbf{c}}_1',r'\hat{\mathbf{c}}_2',
... r'\hat{\mathbf{c}}_3'))
>>> C['1']
C['1']
>>> print(vlatex(C['1']))
\hat{\mathbf{c}}_1
"""
newframe = self.__class__(newname, variables=variables,
indices=indices, latexs=latexs)
approved_orders = ('123', '231', '312', '132', '213', '321', '121',
'131', '212', '232', '313', '323', '')
rot_order = translate(str(rot_order), 'XYZxyz', '123123')
rot_type = rot_type.upper()
if rot_order not in approved_orders:
raise TypeError('The supplied order is not an approved type')
if rot_type == 'AXIS':
newframe.orient_axis(self, amounts[1], amounts[0])
elif rot_type == 'DCM':
newframe.orient_explicit(self, amounts)
elif rot_type == 'BODY':
newframe.orient_body_fixed(self, amounts, rot_order)
elif rot_type == 'SPACE':
newframe.orient_space_fixed(self, amounts, rot_order)
elif rot_type == 'QUATERNION':
newframe.orient_quaternion(self, amounts)
else:
raise NotImplementedError('That is not an implemented rotation')
return newframe
def set_ang_acc(self, otherframe, value):
"""Define the angular acceleration Vector in a ReferenceFrame.
Defines the angular acceleration of this ReferenceFrame, in another.
Angular acceleration can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.
Parameters
==========
otherframe : ReferenceFrame
A ReferenceFrame to define the angular acceleration in
value : Vector
The Vector representing angular acceleration
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_acc(N, V)
>>> A.ang_acc_in(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(otherframe)
self._ang_acc_dict.update({otherframe: value})
otherframe._ang_acc_dict.update({self: -value})
def set_ang_vel(self, otherframe, value):
"""Define the angular velocity vector in a ReferenceFrame.
Defines the angular velocity of this ReferenceFrame, in another.
Angular velocity can be defined with respect to multiple different
ReferenceFrames. Care must be taken to not create loops which are
inconsistent.
Parameters
==========
otherframe : ReferenceFrame
A ReferenceFrame to define the angular velocity in
value : Vector
The Vector representing angular velocity
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> V = 10 * N.x
>>> A.set_ang_vel(N, V)
>>> A.ang_vel_in(N)
10*N.x
"""
if value == 0:
value = Vector(0)
value = _check_vector(value)
_check_frame(otherframe)
self._ang_vel_dict.update({otherframe: value})
otherframe._ang_vel_dict.update({self: -value})
@property
def x(self):
"""The basis Vector for the ReferenceFrame, in the x direction. """
return self._x
@property
def y(self):
"""The basis Vector for the ReferenceFrame, in the y direction. """
return self._y
@property
def z(self):
"""The basis Vector for the ReferenceFrame, in the z direction. """
return self._z
def partial_velocity(self, frame, *gen_speeds):
"""Returns the partial angular velocities of this frame in the given
frame with respect to one or more provided generalized speeds.
Parameters
==========
frame : ReferenceFrame
The frame with which the angular velocity is defined in.
gen_speeds : functions of time
The generalized speeds.
Returns
=======
partial_velocities : tuple of Vector
The partial angular velocity vectors corresponding to the provided
generalized speeds.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols
>>> N = ReferenceFrame('N')
>>> A = ReferenceFrame('A')
>>> u1, u2 = dynamicsymbols('u1, u2')
>>> A.set_ang_vel(N, u1 * A.x + u2 * N.y)
>>> A.partial_velocity(N, u1)
A.x
>>> A.partial_velocity(N, u1, u2)
(A.x, N.y)
"""
partials = [self.ang_vel_in(frame).diff(speed, frame, var_in_dcm=False)
for speed in gen_speeds]
if len(partials) == 1:
return partials[0]
else:
return tuple(partials)
def _check_frame(other):
from .vector import VectorTypeError
if not isinstance(other, ReferenceFrame):
raise VectorTypeError(other, ReferenceFrame('A'))
|
2ccc6c59aa35dfa28524fdc48907455141af8ff1fd5e1b28b90f0896b55bd784 | from sympy.core.backend import sympify, Add, ImmutableMatrix as Matrix
from sympy.core.evalf import EvalfMixin
from sympy.printing.defaults import Printable
from mpmath.libmp.libmpf import prec_to_dps
__all__ = ['Dyadic']
class Dyadic(Printable, EvalfMixin):
"""A Dyadic object.
See:
https://en.wikipedia.org/wiki/Dyadic_tensor
Kane, T., Levinson, D. Dynamics Theory and Applications. 1985 McGraw-Hill
A more powerful way to represent a rigid body's inertia. While it is more
complex, by choosing Dyadic components to be in body fixed basis vectors,
the resulting matrix is equivalent to the inertia tensor.
"""
is_number = False
def __init__(self, inlist):
"""
Just like Vector's init, you should not call this unless creating a
zero dyadic.
zd = Dyadic(0)
Stores a Dyadic as a list of lists; the inner list has the measure
number and the two unit vectors; the outerlist holds each unique
unit vector pair.
"""
self.args = []
if inlist == 0:
inlist = []
while len(inlist) != 0:
added = 0
for i, v in enumerate(self.args):
if ((str(inlist[0][1]) == str(self.args[i][1])) and
(str(inlist[0][2]) == str(self.args[i][2]))):
self.args[i] = (self.args[i][0] + inlist[0][0],
inlist[0][1], inlist[0][2])
inlist.remove(inlist[0])
added = 1
break
if added != 1:
self.args.append(inlist[0])
inlist.remove(inlist[0])
i = 0
# This code is to remove empty parts from the list
while i < len(self.args):
if ((self.args[i][0] == 0) | (self.args[i][1] == 0) |
(self.args[i][2] == 0)):
self.args.remove(self.args[i])
i -= 1
i += 1
@property
def func(self):
"""Returns the class Dyadic. """
return Dyadic
def __add__(self, other):
"""The add operator for Dyadic. """
other = _check_dyadic(other)
return Dyadic(self.args + other.args)
def __and__(self, other):
"""The inner product operator for a Dyadic and a Dyadic or Vector.
Parameters
==========
other : Dyadic or Vector
The other Dyadic or Vector to take the inner product with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> D1 = outer(N.x, N.y)
>>> D2 = outer(N.y, N.y)
>>> D1.dot(D2)
(N.x|N.y)
>>> D1.dot(N.y)
N.x
"""
from sympy.physics.vector.vector import Vector, _check_vector
if isinstance(other, Dyadic):
other = _check_dyadic(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
for i2, v2 in enumerate(other.args):
ol += v[0] * v2[0] * (v[2] & v2[1]) * (v[1] | v2[2])
else:
other = _check_vector(other)
ol = Vector(0)
for i, v in enumerate(self.args):
ol += v[0] * v[1] * (v[2] & other)
return ol
def __truediv__(self, other):
"""Divides the Dyadic by a sympifyable expression. """
return self.__mul__(1 / other)
def __eq__(self, other):
"""Tests for equality.
Is currently weak; needs stronger comparison testing
"""
if other == 0:
other = Dyadic(0)
other = _check_dyadic(other)
if (self.args == []) and (other.args == []):
return True
elif (self.args == []) or (other.args == []):
return False
return set(self.args) == set(other.args)
def __mul__(self, other):
"""Multiplies the Dyadic by a sympifyable expression.
Parameters
==========
other : Sympafiable
The scalar to multiply this Dyadic with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> 5 * d
5*(N.x|N.x)
"""
newlist = [v for v in self.args]
for i, v in enumerate(newlist):
newlist[i] = (sympify(other) * newlist[i][0], newlist[i][1],
newlist[i][2])
return Dyadic(newlist)
def __ne__(self, other):
return not self == other
def __neg__(self):
return self * -1
def _latex(self, printer):
ar = self.args # just to shorten things
if len(ar) == 0:
return str(0)
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(' + ' + printer._print(ar[i][1]) + r"\otimes " +
printer._print(ar[i][2]))
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(' - ' +
printer._print(ar[i][1]) +
r"\otimes " +
printer._print(ar[i][2]))
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = printer._print(ar[i][0])
if isinstance(ar[i][0], Add):
arg_str = '(%s)' % arg_str
if arg_str.startswith('-'):
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + printer._print(ar[i][1]) +
r"\otimes " + printer._print(ar[i][2]))
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def _pretty(self, printer):
e = self
class Fake:
baseline = 0
def render(self, *args, **kwargs):
ar = e.args # just to shorten things
mpp = printer
if len(ar) == 0:
return str(0)
bar = "\N{CIRCLED TIMES}" if printer._use_unicode else "|"
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.extend([" + ",
mpp.doprint(ar[i][1]),
bar,
mpp.doprint(ar[i][2])])
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.extend([" - ",
mpp.doprint(ar[i][1]),
bar,
mpp.doprint(ar[i][2])])
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
if isinstance(ar[i][0], Add):
arg_str = mpp._print(
ar[i][0]).parens()[0]
else:
arg_str = mpp.doprint(ar[i][0])
if arg_str.startswith("-"):
arg_str = arg_str[1:]
str_start = " - "
else:
str_start = " + "
ol.extend([str_start, arg_str, " ",
mpp.doprint(ar[i][1]),
bar,
mpp.doprint(ar[i][2])])
outstr = "".join(ol)
if outstr.startswith(" + "):
outstr = outstr[3:]
elif outstr.startswith(" "):
outstr = outstr[1:]
return outstr
return Fake()
def __rand__(self, other):
"""The inner product operator for a Vector or Dyadic, and a Dyadic
This is for: Vector dot Dyadic
Parameters
==========
other : Vector
The vector we are dotting with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, dot, outer
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> dot(N.x, d)
N.x
"""
from sympy.physics.vector.vector import Vector, _check_vector
other = _check_vector(other)
ol = Vector(0)
for i, v in enumerate(self.args):
ol += v[0] * v[2] * (v[1] & other)
return ol
def __rsub__(self, other):
return (-1 * self) + other
def __rxor__(self, other):
"""For a cross product in the form: Vector x Dyadic
Parameters
==========
other : Vector
The Vector that we are crossing this Dyadic with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, cross
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> cross(N.y, d)
- (N.z|N.x)
"""
from sympy.physics.vector.vector import _check_vector
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
ol += v[0] * ((other ^ v[1]) | v[2])
return ol
def _sympystr(self, printer):
"""Printing method. """
ar = self.args # just to shorten things
if len(ar) == 0:
return printer._print(0)
ol = [] # output list, to be concatenated to a string
for i, v in enumerate(ar):
# if the coef of the dyadic is 1, we skip the 1
if ar[i][0] == 1:
ol.append(' + (' + printer._print(ar[i][1]) + '|' +
printer._print(ar[i][2]) + ')')
# if the coef of the dyadic is -1, we skip the 1
elif ar[i][0] == -1:
ol.append(' - (' + printer._print(ar[i][1]) + '|' +
printer._print(ar[i][2]) + ')')
# If the coefficient of the dyadic is not 1 or -1,
# we might wrap it in parentheses, for readability.
elif ar[i][0] != 0:
arg_str = printer._print(ar[i][0])
if isinstance(ar[i][0], Add):
arg_str = "(%s)" % arg_str
if arg_str[0] == '-':
arg_str = arg_str[1:]
str_start = ' - '
else:
str_start = ' + '
ol.append(str_start + arg_str + '*(' +
printer._print(ar[i][1]) +
'|' + printer._print(ar[i][2]) + ')')
outstr = ''.join(ol)
if outstr.startswith(' + '):
outstr = outstr[3:]
elif outstr.startswith(' '):
outstr = outstr[1:]
return outstr
def __sub__(self, other):
"""The subtraction operator. """
return self.__add__(other * -1)
def __xor__(self, other):
"""For a cross product in the form: Dyadic x Vector.
Parameters
==========
other : Vector
The Vector that we are crossing this Dyadic with
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, cross
>>> N = ReferenceFrame('N')
>>> d = outer(N.x, N.x)
>>> cross(d, N.y)
(N.x|N.z)
"""
from sympy.physics.vector.vector import _check_vector
other = _check_vector(other)
ol = Dyadic(0)
for i, v in enumerate(self.args):
ol += v[0] * (v[1] | (v[2] ^ other))
return ol
__radd__ = __add__
__rmul__ = __mul__
def express(self, frame1, frame2=None):
"""Expresses this Dyadic in alternate frame(s)
The first frame is the list side expression, the second frame is the
right side; if Dyadic is in form A.x|B.y, you can express it in two
different frames. If no second frame is given, the Dyadic is
expressed in only one frame.
Calls the global express function
Parameters
==========
frame1 : ReferenceFrame
The frame to express the left side of the Dyadic in
frame2 : ReferenceFrame
If provided, the frame to express the right side of the Dyadic in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.express(B, N)
cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x)
"""
from sympy.physics.vector.functions import express
return express(self, frame1, frame2)
def to_matrix(self, reference_frame, second_reference_frame=None):
"""Returns the matrix form of the dyadic with respect to one or two
reference frames.
Parameters
----------
reference_frame : ReferenceFrame
The reference frame that the rows and columns of the matrix
correspond to. If a second reference frame is provided, this
only corresponds to the rows of the matrix.
second_reference_frame : ReferenceFrame, optional, default=None
The reference frame that the columns of the matrix correspond
to.
Returns
-------
matrix : ImmutableMatrix, shape(3,3)
The matrix that gives the 2D tensor form.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.vector import ReferenceFrame, Vector
>>> Vector.simp = True
>>> from sympy.physics.mechanics import inertia
>>> Ixx, Iyy, Izz, Ixy, Iyz, Ixz = symbols('Ixx, Iyy, Izz, Ixy, Iyz, Ixz')
>>> N = ReferenceFrame('N')
>>> inertia_dyadic = inertia(N, Ixx, Iyy, Izz, Ixy, Iyz, Ixz)
>>> inertia_dyadic.to_matrix(N)
Matrix([
[Ixx, Ixy, Ixz],
[Ixy, Iyy, Iyz],
[Ixz, Iyz, Izz]])
>>> beta = symbols('beta')
>>> A = N.orientnew('A', 'Axis', (beta, N.x))
>>> inertia_dyadic.to_matrix(A)
Matrix([
[ Ixx, Ixy*cos(beta) + Ixz*sin(beta), -Ixy*sin(beta) + Ixz*cos(beta)],
[ Ixy*cos(beta) + Ixz*sin(beta), Iyy*cos(2*beta)/2 + Iyy/2 + Iyz*sin(2*beta) - Izz*cos(2*beta)/2 + Izz/2, -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2],
[-Ixy*sin(beta) + Ixz*cos(beta), -Iyy*sin(2*beta)/2 + Iyz*cos(2*beta) + Izz*sin(2*beta)/2, -Iyy*cos(2*beta)/2 + Iyy/2 - Iyz*sin(2*beta) + Izz*cos(2*beta)/2 + Izz/2]])
"""
if second_reference_frame is None:
second_reference_frame = reference_frame
return Matrix([i.dot(self).dot(j) for i in reference_frame for j in
second_reference_frame]).reshape(3, 3)
def doit(self, **hints):
"""Calls .doit() on each term in the Dyadic"""
return sum([Dyadic([(v[0].doit(**hints), v[1], v[2])])
for v in self.args], Dyadic(0))
def dt(self, frame):
"""Take the time derivative of this Dyadic in a frame.
This function calls the global time_derivative method
Parameters
==========
frame : ReferenceFrame
The frame to take the time derivative in
Examples
========
>>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> N = ReferenceFrame('N')
>>> q = dynamicsymbols('q')
>>> B = N.orientnew('B', 'Axis', [q, N.z])
>>> d = outer(N.x, N.x)
>>> d.dt(B)
- q'*(N.y|N.x) - q'*(N.x|N.y)
"""
from sympy.physics.vector.functions import time_derivative
return time_derivative(self, frame)
def simplify(self):
"""Returns a simplified Dyadic."""
out = Dyadic(0)
for v in self.args:
out += Dyadic([(v[0].simplify(), v[1], v[2])])
return out
def subs(self, *args, **kwargs):
"""Substitution on the Dyadic.
Examples
========
>>> from sympy.physics.vector import ReferenceFrame
>>> from sympy import Symbol
>>> N = ReferenceFrame('N')
>>> s = Symbol('s')
>>> a = s*(N.x|N.x)
>>> a.subs({s: 2})
2*(N.x|N.x)
"""
return sum([Dyadic([(v[0].subs(*args, **kwargs), v[1], v[2])])
for v in self.args], Dyadic(0))
def applyfunc(self, f):
"""Apply a function to each component of a Dyadic."""
if not callable(f):
raise TypeError("`f` must be callable.")
out = Dyadic(0)
for a, b, c in self.args:
out += f(a) * (b | c)
return out
dot = __and__
cross = __xor__
def _eval_evalf(self, prec):
if not self.args:
return self
new_args = []
dps = prec_to_dps(prec)
for inlist in self.args:
new_inlist = list(inlist)
new_inlist[0] = inlist[0].evalf(n=dps)
new_args.append(tuple(new_inlist))
return Dyadic(new_args)
def xreplace(self, rule):
"""
Replace occurrences of objects within the measure numbers of the
Dyadic.
Parameters
==========
rule : dict-like
Expresses a replacement rule.
Returns
=======
Dyadic
Result of the replacement.
Examples
========
>>> from sympy import symbols, pi
>>> from sympy.physics.vector import ReferenceFrame, outer
>>> N = ReferenceFrame('N')
>>> D = outer(N.x, N.x)
>>> x, y, z = symbols('x y z')
>>> ((1 + x*y) * D).xreplace({x: pi})
(pi*y + 1)*(N.x|N.x)
>>> ((1 + x*y) * D).xreplace({x: pi, y: 2})
(1 + 2*pi)*(N.x|N.x)
Replacements occur only if an entire node in the expression tree is
matched:
>>> ((x*y + z) * D).xreplace({x*y: pi})
(z + pi)*(N.x|N.x)
>>> ((x*y*z) * D).xreplace({x*y: pi})
x*y*z*(N.x|N.x)
"""
new_args = []
for inlist in self.args:
new_inlist = list(inlist)
new_inlist[0] = new_inlist[0].xreplace(rule)
new_args.append(tuple(new_inlist))
return Dyadic(new_args)
def _check_dyadic(other):
if not isinstance(other, Dyadic):
raise TypeError('A Dyadic must be supplied')
return other
|
c4ca185323f2876c9cc8d8f98289b2ff43e434a2876bd5d1944dfd9995106aa2 | """
Gaussian optics.
The module implements:
- Ray transfer matrices for geometrical and gaussian optics.
See RayTransferMatrix, GeometricRay and BeamParameter
- Conjugation relations for geometrical and gaussian optics.
See geometric_conj*, gauss_conj and conjugate_gauss_beams
The conventions for the distances are as follows:
focal distance
positive for convergent lenses
object distance
positive for real objects
image distance
positive for real images
"""
__all__ = [
'RayTransferMatrix',
'FreeSpace',
'FlatRefraction',
'CurvedRefraction',
'FlatMirror',
'CurvedMirror',
'ThinLens',
'GeometricRay',
'BeamParameter',
'waist2rayleigh',
'rayleigh2waist',
'geometric_conj_ab',
'geometric_conj_af',
'geometric_conj_bf',
'gaussian_conj',
'conjugate_gauss_beams',
]
from sympy.core.expr import Expr
from sympy.core.numbers import (I, pi)
from sympy.core.sympify import sympify
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import atan2
from sympy.matrices.dense import Matrix, MutableDenseMatrix
from sympy.polys.rationaltools import together
from sympy.utilities.misc import filldedent
###
# A, B, C, D matrices
###
class RayTransferMatrix(MutableDenseMatrix):
"""
Base class for a Ray Transfer Matrix.
It should be used if there is not already a more specific subclass mentioned
in See Also.
Parameters
==========
parameters :
A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D]))
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix, ThinLens
>>> from sympy import Symbol, Matrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat
Matrix([
[1, 2],
[3, 4]])
>>> RayTransferMatrix(Matrix([[1, 2], [3, 4]]))
Matrix([
[1, 2],
[3, 4]])
>>> mat.A
1
>>> f = Symbol('f')
>>> lens = ThinLens(f)
>>> lens
Matrix([
[ 1, 0],
[-1/f, 1]])
>>> lens.C
-1/f
See Also
========
GeometricRay, BeamParameter,
FreeSpace, FlatRefraction, CurvedRefraction,
FlatMirror, CurvedMirror, ThinLens
References
==========
.. [1] https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis
"""
def __new__(cls, *args):
if len(args) == 4:
temp = ((args[0], args[1]), (args[2], args[3]))
elif len(args) == 1 \
and isinstance(args[0], Matrix) \
and args[0].shape == (2, 2):
temp = args[0]
else:
raise ValueError(filldedent('''
Expecting 2x2 Matrix or the 4 elements of
the Matrix but got %s''' % str(args)))
return Matrix.__new__(cls, temp)
def __mul__(self, other):
if isinstance(other, RayTransferMatrix):
return RayTransferMatrix(Matrix.__mul__(self, other))
elif isinstance(other, GeometricRay):
return GeometricRay(Matrix.__mul__(self, other))
elif isinstance(other, BeamParameter):
temp = self*Matrix(((other.q,), (1,)))
q = (temp[0]/temp[1]).expand(complex=True)
return BeamParameter(other.wavelen,
together(re(q)),
z_r=together(im(q)))
else:
return Matrix.__mul__(self, other)
@property
def A(self):
"""
The A parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.A
1
"""
return self[0, 0]
@property
def B(self):
"""
The B parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.B
2
"""
return self[0, 1]
@property
def C(self):
"""
The C parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.C
3
"""
return self[1, 0]
@property
def D(self):
"""
The D parameter of the Matrix.
Examples
========
>>> from sympy.physics.optics import RayTransferMatrix
>>> mat = RayTransferMatrix(1, 2, 3, 4)
>>> mat.D
4
"""
return self[1, 1]
class FreeSpace(RayTransferMatrix):
"""
Ray Transfer Matrix for free space.
Parameters
==========
distance
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import FreeSpace
>>> from sympy import symbols
>>> d = symbols('d')
>>> FreeSpace(d)
Matrix([
[1, d],
[0, 1]])
"""
def __new__(cls, d):
return RayTransferMatrix.__new__(cls, 1, d, 0, 1)
class FlatRefraction(RayTransferMatrix):
"""
Ray Transfer Matrix for refraction.
Parameters
==========
n1 :
Refractive index of one medium.
n2 :
Refractive index of other medium.
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import FlatRefraction
>>> from sympy import symbols
>>> n1, n2 = symbols('n1 n2')
>>> FlatRefraction(n1, n2)
Matrix([
[1, 0],
[0, n1/n2]])
"""
def __new__(cls, n1, n2):
n1, n2 = map(sympify, (n1, n2))
return RayTransferMatrix.__new__(cls, 1, 0, 0, n1/n2)
class CurvedRefraction(RayTransferMatrix):
"""
Ray Transfer Matrix for refraction on curved interface.
Parameters
==========
R :
Radius of curvature (positive for concave).
n1 :
Refractive index of one medium.
n2 :
Refractive index of other medium.
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import CurvedRefraction
>>> from sympy import symbols
>>> R, n1, n2 = symbols('R n1 n2')
>>> CurvedRefraction(R, n1, n2)
Matrix([
[ 1, 0],
[(n1 - n2)/(R*n2), n1/n2]])
"""
def __new__(cls, R, n1, n2):
R, n1, n2 = map(sympify, (R, n1, n2))
return RayTransferMatrix.__new__(cls, 1, 0, (n1 - n2)/R/n2, n1/n2)
class FlatMirror(RayTransferMatrix):
"""
Ray Transfer Matrix for reflection.
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import FlatMirror
>>> FlatMirror()
Matrix([
[1, 0],
[0, 1]])
"""
def __new__(cls):
return RayTransferMatrix.__new__(cls, 1, 0, 0, 1)
class CurvedMirror(RayTransferMatrix):
"""
Ray Transfer Matrix for reflection from curved surface.
Parameters
==========
R : radius of curvature (positive for concave)
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import CurvedMirror
>>> from sympy import symbols
>>> R = symbols('R')
>>> CurvedMirror(R)
Matrix([
[ 1, 0],
[-2/R, 1]])
"""
def __new__(cls, R):
R = sympify(R)
return RayTransferMatrix.__new__(cls, 1, 0, -2/R, 1)
class ThinLens(RayTransferMatrix):
"""
Ray Transfer Matrix for a thin lens.
Parameters
==========
f :
The focal distance.
See Also
========
RayTransferMatrix
Examples
========
>>> from sympy.physics.optics import ThinLens
>>> from sympy import symbols
>>> f = symbols('f')
>>> ThinLens(f)
Matrix([
[ 1, 0],
[-1/f, 1]])
"""
def __new__(cls, f):
f = sympify(f)
return RayTransferMatrix.__new__(cls, 1, 0, -1/f, 1)
###
# Representation for geometric ray
###
class GeometricRay(MutableDenseMatrix):
"""
Representation for a geometric ray in the Ray Transfer Matrix formalism.
Parameters
==========
h : height, and
angle : angle, or
matrix : a 2x1 matrix (Matrix(2, 1, [height, angle]))
Examples
========
>>> from sympy.physics.optics import GeometricRay, FreeSpace
>>> from sympy import symbols, Matrix
>>> d, h, angle = symbols('d, h, angle')
>>> GeometricRay(h, angle)
Matrix([
[ h],
[angle]])
>>> FreeSpace(d)*GeometricRay(h, angle)
Matrix([
[angle*d + h],
[ angle]])
>>> GeometricRay( Matrix( ((h,), (angle,)) ) )
Matrix([
[ h],
[angle]])
See Also
========
RayTransferMatrix
"""
def __new__(cls, *args):
if len(args) == 1 and isinstance(args[0], Matrix) \
and args[0].shape == (2, 1):
temp = args[0]
elif len(args) == 2:
temp = ((args[0],), (args[1],))
else:
raise ValueError(filldedent('''
Expecting 2x1 Matrix or the 2 elements of
the Matrix but got %s''' % str(args)))
return Matrix.__new__(cls, temp)
@property
def height(self):
"""
The distance from the optical axis.
Examples
========
>>> from sympy.physics.optics import GeometricRay
>>> from sympy import symbols
>>> h, angle = symbols('h, angle')
>>> gRay = GeometricRay(h, angle)
>>> gRay.height
h
"""
return self[0]
@property
def angle(self):
"""
The angle with the optical axis.
Examples
========
>>> from sympy.physics.optics import GeometricRay
>>> from sympy import symbols
>>> h, angle = symbols('h, angle')
>>> gRay = GeometricRay(h, angle)
>>> gRay.angle
angle
"""
return self[1]
###
# Representation for gauss beam
###
class BeamParameter(Expr):
"""
Representation for a gaussian ray in the Ray Transfer Matrix formalism.
Parameters
==========
wavelen : the wavelength,
z : the distance to waist, and
w : the waist, or
z_r : the rayleigh range.
n : the refractive index of medium.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.q
1 + 1.88679245283019*I*pi
>>> p.q.n()
1.0 + 5.92753330865999*I
>>> p.w_0.n()
0.00100000000000000
>>> p.z_r.n()
5.92753330865999
>>> from sympy.physics.optics import FreeSpace
>>> fs = FreeSpace(10)
>>> p1 = fs*p
>>> p.w.n()
0.00101413072159615
>>> p1.w.n()
0.00210803120913829
See Also
========
RayTransferMatrix
References
==========
.. [1] https://en.wikipedia.org/wiki/Complex_beam_parameter
.. [2] https://en.wikipedia.org/wiki/Gaussian_beam
"""
#TODO A class Complex may be implemented. The BeamParameter may
# subclass it. See:
# https://groups.google.com/d/topic/sympy/7XkU07NRBEs/discussion
def __new__(cls, wavelen, z, z_r=None, w=None, n=1):
wavelen = sympify(wavelen)
z = sympify(z)
n = sympify(n)
if z_r is not None and w is None:
z_r = sympify(z_r)
elif w is not None and z_r is None:
z_r = waist2rayleigh(sympify(w), wavelen, n)
elif z_r is None and w is None:
raise ValueError('Must specify one of w and z_r.')
return Expr.__new__(cls, wavelen, z, z_r, n)
@property
def wavelen(self):
return self.args[0]
@property
def z(self):
return self.args[1]
@property
def z_r(self):
return self.args[2]
@property
def n(self):
return self.args[3]
@property
def q(self):
"""
The complex parameter representing the beam.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.q
1 + 1.88679245283019*I*pi
"""
return self.z + I*self.z_r
@property
def radius(self):
"""
The radius of curvature of the phase front.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.radius
1 + 3.55998576005696*pi**2
"""
return self.z*(1 + (self.z_r/self.z)**2)
@property
def w(self):
"""
The radius of the beam w(z), at any position z along the beam.
The beam radius at `1/e^2` intensity (axial value).
See Also
========
w_0 :
The minimal radius of beam.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.w
0.001*sqrt(0.2809/pi**2 + 1)
"""
return self.w_0*sqrt(1 + (self.z/self.z_r)**2)
@property
def w_0(self):
"""
The minimal radius of beam at `1/e^2` intensity (peak value).
See Also
========
w : the beam radius at `1/e^2` intensity (axial value).
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.w_0
0.00100000000000000
"""
return sqrt(self.z_r/(pi*self.n)*self.wavelen)
@property
def divergence(self):
"""
Half of the total angular spread.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.divergence
0.00053/pi
"""
return self.wavelen/pi/self.w_0
@property
def gouy(self):
"""
The Gouy phase.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.gouy
atan(0.53/pi)
"""
return atan2(self.z, self.z_r)
@property
def waist_approximation_limit(self):
"""
The minimal waist for which the gauss beam approximation is valid.
Explanation
===========
The gauss beam is a solution to the paraxial equation. For curvatures
that are too great it is not a valid approximation.
Examples
========
>>> from sympy.physics.optics import BeamParameter
>>> p = BeamParameter(530e-9, 1, w=1e-3)
>>> p.waist_approximation_limit
1.06e-6/pi
"""
return 2*self.wavelen/pi
###
# Utilities
###
def waist2rayleigh(w, wavelen, n=1):
"""
Calculate the rayleigh range from the waist of a gaussian beam.
See Also
========
rayleigh2waist, BeamParameter
Examples
========
>>> from sympy.physics.optics import waist2rayleigh
>>> from sympy import symbols
>>> w, wavelen = symbols('w wavelen')
>>> waist2rayleigh(w, wavelen)
pi*w**2/wavelen
"""
w, wavelen = map(sympify, (w, wavelen))
return w**2*n*pi/wavelen
def rayleigh2waist(z_r, wavelen):
"""Calculate the waist from the rayleigh range of a gaussian beam.
See Also
========
waist2rayleigh, BeamParameter
Examples
========
>>> from sympy.physics.optics import rayleigh2waist
>>> from sympy import symbols
>>> z_r, wavelen = symbols('z_r wavelen')
>>> rayleigh2waist(z_r, wavelen)
sqrt(wavelen*z_r)/sqrt(pi)
"""
z_r, wavelen = map(sympify, (z_r, wavelen))
return sqrt(z_r/pi*wavelen)
def geometric_conj_ab(a, b):
"""
Conjugation relation for geometrical beams under paraxial conditions.
Explanation
===========
Takes the distances to the optical element and returns the needed
focal distance.
See Also
========
geometric_conj_af, geometric_conj_bf
Examples
========
>>> from sympy.physics.optics import geometric_conj_ab
>>> from sympy import symbols
>>> a, b = symbols('a b')
>>> geometric_conj_ab(a, b)
a*b/(a + b)
"""
a, b = map(sympify, (a, b))
if a.is_infinite or b.is_infinite:
return a if b.is_infinite else b
else:
return a*b/(a + b)
def geometric_conj_af(a, f):
"""
Conjugation relation for geometrical beams under paraxial conditions.
Explanation
===========
Takes the object distance (for geometric_conj_af) or the image distance
(for geometric_conj_bf) to the optical element and the focal distance.
Then it returns the other distance needed for conjugation.
See Also
========
geometric_conj_ab
Examples
========
>>> from sympy.physics.optics.gaussopt import geometric_conj_af, geometric_conj_bf
>>> from sympy import symbols
>>> a, b, f = symbols('a b f')
>>> geometric_conj_af(a, f)
a*f/(a - f)
>>> geometric_conj_bf(b, f)
b*f/(b - f)
"""
a, f = map(sympify, (a, f))
return -geometric_conj_ab(a, -f)
geometric_conj_bf = geometric_conj_af
def gaussian_conj(s_in, z_r_in, f):
"""
Conjugation relation for gaussian beams.
Parameters
==========
s_in :
The distance to optical element from the waist.
z_r_in :
The rayleigh range of the incident beam.
f :
The focal length of the optical element.
Returns
=======
a tuple containing (s_out, z_r_out, m)
s_out :
The distance between the new waist and the optical element.
z_r_out :
The rayleigh range of the emergent beam.
m :
The ration between the new and the old waists.
Examples
========
>>> from sympy.physics.optics import gaussian_conj
>>> from sympy import symbols
>>> s_in, z_r_in, f = symbols('s_in z_r_in f')
>>> gaussian_conj(s_in, z_r_in, f)[0]
1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f)
>>> gaussian_conj(s_in, z_r_in, f)[1]
z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2)
>>> gaussian_conj(s_in, z_r_in, f)[2]
1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2)
"""
s_in, z_r_in, f = map(sympify, (s_in, z_r_in, f))
s_out = 1 / ( -1/(s_in + z_r_in**2/(s_in - f)) + 1/f )
m = 1/sqrt((1 - (s_in/f)**2) + (z_r_in/f)**2)
z_r_out = z_r_in / ((1 - (s_in/f)**2) + (z_r_in/f)**2)
return (s_out, z_r_out, m)
def conjugate_gauss_beams(wavelen, waist_in, waist_out, **kwargs):
"""
Find the optical setup conjugating the object/image waists.
Parameters
==========
wavelen :
The wavelength of the beam.
waist_in and waist_out :
The waists to be conjugated.
f :
The focal distance of the element used in the conjugation.
Returns
=======
a tuple containing (s_in, s_out, f)
s_in :
The distance before the optical element.
s_out :
The distance after the optical element.
f :
The focal distance of the optical element.
Examples
========
>>> from sympy.physics.optics import conjugate_gauss_beams
>>> from sympy import symbols, factor
>>> l, w_i, w_o, f = symbols('l w_i w_o f')
>>> conjugate_gauss_beams(l, w_i, w_o, f=f)[0]
f*(1 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)))
>>> factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1])
f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 -
pi**2*w_i**4/(f**2*l**2)))/w_i**2
>>> conjugate_gauss_beams(l, w_i, w_o, f=f)[2]
f
"""
#TODO add the other possible arguments
wavelen, waist_in, waist_out = map(sympify, (wavelen, waist_in, waist_out))
m = waist_out / waist_in
z = waist2rayleigh(waist_in, wavelen)
if len(kwargs) != 1:
raise ValueError("The function expects only one named argument")
elif 'dist' in kwargs:
raise NotImplementedError(filldedent('''
Currently only focal length is supported as a parameter'''))
elif 'f' in kwargs:
f = sympify(kwargs['f'])
s_in = f * (1 - sqrt(1/m**2 - z**2/f**2))
s_out = gaussian_conj(s_in, z, f)[0]
elif 's_in' in kwargs:
raise NotImplementedError(filldedent('''
Currently only focal length is supported as a parameter'''))
else:
raise ValueError(filldedent('''
The functions expects the focal length as a named argument'''))
return (s_in, s_out, f)
#TODO
#def plot_beam():
# """Plot the beam radius as it propagates in space."""
# pass
#TODO
#def plot_beam_conjugation():
# """
# Plot the intersection of two beams.
#
# Represents the conjugation relation.
#
# See Also
# ========
#
# conjugate_gauss_beams
# """
# pass
|
761b2a1fe983e81615b96851b75a42b31815ea5c25f42fa3cd4720148a3b6e59 | from sympy.core.numbers import I
from sympy.core.symbol import Dummy
from sympy.functions.elementary.complexes import (Abs, arg)
from sympy.functions.elementary.exponential import log
from sympy.abc import s, p, a
from sympy.external import import_module
from sympy.physics.control.control_plots import \
(pole_zero_numerical_data, pole_zero_plot, step_response_numerical_data,
step_response_plot, impulse_response_numerical_data,
impulse_response_plot, ramp_response_numerical_data,
ramp_response_plot, bode_magnitude_numerical_data,
bode_phase_numerical_data, bode_plot)
from sympy.physics.control.lti import (TransferFunction,
Series, Parallel, TransferFunctionMatrix)
from sympy.testing.pytest import raises, skip
matplotlib = import_module(
'matplotlib', import_kwargs={'fromlist': ['pyplot']},
catch=(RuntimeError,))
numpy = import_module('numpy')
tf1 = TransferFunction(1, p**2 + 0.5*p + 2, p)
tf2 = TransferFunction(p, 6*p**2 + 3*p + 1, p)
tf3 = TransferFunction(p, p**3 - 1, p)
tf4 = TransferFunction(10, p**3, p)
tf5 = TransferFunction(5, s**2 + 2*s + 10, s)
tf6 = TransferFunction(1, 1, s)
tf7 = TransferFunction(4*s*3 + 9*s**2 + 0.1*s + 11, 8*s**6 + 9*s**4 + 11, s)
tf8 = TransferFunction(5, s**2 + (2+I)*s + 10, s)
ser1 = Series(tf4, TransferFunction(1, p - 5, p))
ser2 = Series(tf3, TransferFunction(p, p + 2, p))
par1 = Parallel(tf1, tf2)
par2 = Parallel(tf1, tf2, tf3)
def _to_tuple(a, b):
return tuple(a), tuple(b)
def _trim_tuple(a, b):
a, b = _to_tuple(a, b)
return tuple(a[0: 2] + a[len(a)//2 : len(a)//2 + 1] + a[-2:]), \
tuple(b[0: 2] + b[len(b)//2 : len(b)//2 + 1] + b[-2:])
def y_coordinate_equality(plot_data_func, evalf_func, system):
"""Checks whether the y-coordinate value of the plotted
data point is equal to the value of the function at a
particular x."""
x, y = plot_data_func(system)
x, y = _trim_tuple(x, y)
y_exp = tuple(evalf_func(system, x_i) for x_i in x)
return all(Abs(y_exp_i - y_i) < 1e-8 for y_exp_i, y_i in zip(y_exp, y))
def test_errors():
if not matplotlib:
skip("Matplotlib not the default backend")
# Invalid `system` check
tfm = TransferFunctionMatrix([[tf6, tf5], [tf5, tf6]])
expr = 1/(s**2 - 1)
raises(NotImplementedError, lambda: pole_zero_plot(tfm))
raises(NotImplementedError, lambda: pole_zero_numerical_data(expr))
raises(NotImplementedError, lambda: impulse_response_plot(expr))
raises(NotImplementedError, lambda: impulse_response_numerical_data(tfm))
raises(NotImplementedError, lambda: step_response_plot(tfm))
raises(NotImplementedError, lambda: step_response_numerical_data(expr))
raises(NotImplementedError, lambda: ramp_response_plot(expr))
raises(NotImplementedError, lambda: ramp_response_numerical_data(tfm))
raises(NotImplementedError, lambda: bode_plot(tfm))
# More than 1 variables
tf_a = TransferFunction(a, s + 1, s)
raises(ValueError, lambda: pole_zero_plot(tf_a))
raises(ValueError, lambda: pole_zero_numerical_data(tf_a))
raises(ValueError, lambda: impulse_response_plot(tf_a))
raises(ValueError, lambda: impulse_response_numerical_data(tf_a))
raises(ValueError, lambda: step_response_plot(tf_a))
raises(ValueError, lambda: step_response_numerical_data(tf_a))
raises(ValueError, lambda: ramp_response_plot(tf_a))
raises(ValueError, lambda: ramp_response_numerical_data(tf_a))
raises(ValueError, lambda: bode_plot(tf_a))
# lower_limit > 0 for response plots
raises(ValueError, lambda: impulse_response_plot(tf1, lower_limit=-1))
raises(ValueError, lambda: step_response_plot(tf1, lower_limit=-0.1))
raises(ValueError, lambda: ramp_response_plot(tf1, lower_limit=-4/3))
# slope in ramp_response_plot() is negative
raises(ValueError, lambda: ramp_response_plot(tf1, slope=-0.1))
# incorrect frequency or phase unit
raises(ValueError, lambda: bode_plot(tf1,freq_unit = 'hz'))
raises(ValueError, lambda: bode_plot(tf1,phase_unit = 'degree'))
def test_pole_zero():
if not matplotlib:
skip("Matplotlib not the default backend")
assert _to_tuple(*pole_zero_numerical_data(tf1)) == \
((), ((-0.24999999999999994+1.3919410907075054j), (-0.24999999999999994-1.3919410907075054j)))
assert _to_tuple(*pole_zero_numerical_data(tf2)) == \
((0.0,), ((-0.25+0.3227486121839514j), (-0.25-0.3227486121839514j)))
assert _to_tuple(*pole_zero_numerical_data(tf3)) == \
((0.0,), ((-0.5000000000000004+0.8660254037844395j),
(-0.5000000000000004-0.8660254037844395j), (0.9999999999999998+0j)))
assert _to_tuple(*pole_zero_numerical_data(tf7)) == \
(((-0.6722222222222222+0.8776898690157247j), (-0.6722222222222222-0.8776898690157247j)),
((2.220446049250313e-16+1.2797182176061541j), (2.220446049250313e-16-1.2797182176061541j),
(-0.7657146670186428+0.5744385024099056j), (-0.7657146670186428-0.5744385024099056j),
(0.7657146670186427+0.5744385024099052j), (0.7657146670186427-0.5744385024099052j)))
assert _to_tuple(*pole_zero_numerical_data(ser1)) == \
((), (5.0, 0.0, 0.0, 0.0))
assert _to_tuple(*pole_zero_numerical_data(par1)) == \
((-5.645751311064592, -0.5000000000000008, -0.3542486889354093),
((-0.24999999999999986+1.3919410907075052j),
(-0.24999999999999986-1.3919410907075052j), (-0.2499999999999998+0.32274861218395134j),
(-0.2499999999999998-0.32274861218395134j)))
assert _to_tuple(*pole_zero_numerical_data(tf8)) == \
((),
((-1.1641600331447917-3.545808351896439j),
(-0.8358399668552097+2.5458083518964383j)))
def test_bode():
if not matplotlib:
skip("Matplotlib not the default backend")
def bode_phase_evalf(system, point):
expr = system.to_expr()
_w = Dummy("w", real=True)
w_expr = expr.subs({system.var: I*_w})
return arg(w_expr).subs({_w: point}).evalf()
def bode_mag_evalf(system, point):
expr = system.to_expr()
_w = Dummy("w", real=True)
w_expr = expr.subs({system.var: I*_w})
return 20*log(Abs(w_expr), 10).subs({_w: point}).evalf()
def test_bode_data(sys):
return y_coordinate_equality(bode_magnitude_numerical_data, bode_mag_evalf, sys) \
and y_coordinate_equality(bode_phase_numerical_data, bode_phase_evalf, sys)
assert test_bode_data(tf1)
assert test_bode_data(tf2)
assert test_bode_data(tf3)
assert test_bode_data(tf4)
assert test_bode_data(tf5)
def check_point_accuracy(a, b):
return all(Abs(a_i - b_i) < 1e-12 for \
a_i, b_i in zip(a, b))
def test_impulse_response():
if not matplotlib:
skip("Matplotlib not the default backend")
def impulse_res_tester(sys, expected_value):
x, y = _to_tuple(*impulse_response_numerical_data(sys,
adaptive=False, nb_of_points=10))
x_check = check_point_accuracy(x, expected_value[0])
y_check = check_point_accuracy(y, expected_value[1])
return x_check and y_check
exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(0.0, 0.544019738507865, 0.01993849743234938, -0.31140243360893216, -0.022852779906491996, 0.1778306498155759,
0.01962941084328499, -0.1013115194573652, -0.014975541213105696, 0.0575789724730714))
exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.1666666675, 0.08389223412935855,
0.02338051973475047, -0.014966807776379383, -0.034645954223054234, -0.040560075735512804,
-0.037658628907103885, -0.030149507719590022, -0.021162090730736834, -0.012721292737437523))
exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (4.369893391586999e-09, 1.1750333000630964,
3.2922404058312473, 9.432290008148343, 28.37098083007151, 86.18577464367974, 261.90356653762115,
795.6538758627842, 2416.9920942096983, 7342.159505206647))
exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 6.17283950617284, 24.69135802469136,
55.555555555555564, 98.76543209876544, 154.320987654321, 222.22222222222226, 302.46913580246917,
395.0617283950618, 500.0))
exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, -0.10455606138085417,
0.06757671513476461, -0.03234567568833768, 0.013582514927757873, -0.005273419510705473,
0.0019364083003354075, -0.000680070134067832, 0.00022969845960406913, -7.476094359583917e-05))
exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(-6.016699583000218e-09, 0.35039802056107394, 3.3728423827689884, 12.119846079276684,
25.86101014293389, 29.352480635282088, -30.49475907497664, -273.8717189554019, -863.2381702029659,
-1747.0262164682233))
exp7 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335,
4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779,
8.88888888888889, 10.0), (0.0, 18.934638095560974, 5346.93244680907, 1384609.8718249386,
358161126.65801865, 92645770015.70108, 23964739753087.42, 6198974342083139.0, 1.603492601616059e+18,
4.147764422869658e+20))
assert impulse_res_tester(tf1, exp1)
assert impulse_res_tester(tf2, exp2)
assert impulse_res_tester(tf3, exp3)
assert impulse_res_tester(tf4, exp4)
assert impulse_res_tester(tf5, exp5)
assert impulse_res_tester(tf7, exp6)
assert impulse_res_tester(ser1, exp7)
def test_step_response():
if not matplotlib:
skip("Matplotlib not the default backend")
def step_res_tester(sys, expected_value):
x, y = _to_tuple(*step_response_numerical_data(sys,
adaptive=False, nb_of_points=10))
x_check = check_point_accuracy(x, expected_value[0])
y_check = check_point_accuracy(y, expected_value[1])
return x_check and y_check
exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(-1.9193285738516863e-08, 0.42283495488246126, 0.7840485977945262, 0.5546841805655717,
0.33903033806932087, 0.4627251747410237, 0.5909907598988051, 0.5247213989553071,
0.4486997874319281, 0.4839358435839171))
exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(0.0, 0.13728409095645816, 0.19474559355325086, 0.1974909129243011, 0.16841657696573073,
0.12559777736159378, 0.08153828016664713, 0.04360471317348958, 0.015072994568868221,
-0.003636420058445484))
exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(0.0, 0.6314542141914303, 2.9356520038101035, 9.37731009663807, 28.452300356688376,
86.25721933273988, 261.9236645044672, 795.6435410577224, 2416.9786984578764, 7342.154119725917))
exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(0.0, 2.286236899862826, 18.28989519890261, 61.72839629629631, 146.31916159122088, 285.7796124828532,
493.8271703703705, 784.1792566529494, 1170.553292729767, 1666.6667))
exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(-3.999999997894577e-09, 0.6720357068882895, 0.4429938256137113, 0.5182010838004518,
0.4944139147159695, 0.5016379853883338, 0.4995466896527733, 0.5001154784851325,
0.49997448824584123, 0.5000039745919259))
exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(-1.5433688493882158e-09, 0.3428705539937336, 1.1253619102202777, 3.1849962651016517,
9.47532757182671, 28.727231099148135, 87.29426924860557, 265.2138681048606, 805.6636260007757,
2447.387582370878))
assert step_res_tester(tf1, exp1)
assert step_res_tester(tf2, exp2)
assert step_res_tester(tf3, exp3)
assert step_res_tester(tf4, exp4)
assert step_res_tester(tf5, exp5)
assert step_res_tester(ser2, exp6)
def test_ramp_response():
if not matplotlib:
skip("Matplotlib not the default backend")
def ramp_res_tester(sys, num_points, expected_value, slope=1):
x, y = _to_tuple(*ramp_response_numerical_data(sys,
slope=slope, adaptive=False, nb_of_points=num_points))
x_check = check_point_accuracy(x, expected_value[0])
y_check = check_point_accuracy(y, expected_value[1])
return x_check and y_check
exp1 = ((0.0, 2.0, 4.0, 6.0, 8.0, 10.0), (0.0, 0.7324667795033895, 1.9909720978650398,
2.7956587704217783, 3.9224897567931514, 4.85022655284895))
exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445,
5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0),
(2.4360213402019326e-08, 0.10175320182493253, 0.33057612497658406, 0.5967937263298935,
0.8431511866718248, 1.0398805391471613, 1.1776043125035738, 1.2600994825747305, 1.2981042689274653,
1.304684417610106))
exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-3.9329040468771836e-08,
0.34686634635794555, 2.9998828170537903, 12.33303690737476, 40.993913948137795, 127.84145222317912,
391.41713691996, 1192.0006858708389, 3623.9808672503405, 11011.728034546572))
exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.9051973784484078, 30.483158055174524,
154.32098765432104, 487.7305288827924, 1190.7483615302544, 2469.1358024691367, 4574.3789056546275,
7803.688462124678, 12500.0))
exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 3.8844361856975635, 9.141792069209865,
14.096349157657231, 19.09783068994694, 24.10179770390321, 29.09907319114121, 34.10040420185154,
39.09983919254265, 44.10006013058409))
exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555,
6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.1111111111111112, 2.2222222222222223,
3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0))
assert ramp_res_tester(tf1, 6, exp1)
assert ramp_res_tester(tf2, 10, exp2, 1.2)
assert ramp_res_tester(tf3, 10, exp3, 1.5)
assert ramp_res_tester(tf4, 10, exp4, 3)
assert ramp_res_tester(tf5, 10, exp5, 9)
assert ramp_res_tester(tf6, 10, exp6)
|
1b4298a159831972d33bdf241a664103934542db1109f504582e8b7b11150f18 | from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import log
from sympy.external import import_module
from sympy.physics.quantum.density import Density, entropy, fidelity
from sympy.physics.quantum.state import Ket, TimeDepKet
from sympy.physics.quantum.qubit import Qubit
from sympy.physics.quantum.represent import represent
from sympy.physics.quantum.dagger import Dagger
from sympy.physics.quantum.cartesian import XKet, PxKet, PxOp, XOp
from sympy.physics.quantum.spin import JzKet
from sympy.physics.quantum.operator import OuterProduct
from sympy.physics.quantum.trace import Tr
from sympy.functions import sqrt
from sympy.testing.pytest import raises
from sympy.physics.quantum.matrixutils import scipy_sparse_matrix
from sympy.physics.quantum.tensorproduct import TensorProduct
def test_eval_args():
# check instance created
assert isinstance(Density([Ket(0), 0.5], [Ket(1), 0.5]), Density)
assert isinstance(Density([Qubit('00'), 1/sqrt(2)],
[Qubit('11'), 1/sqrt(2)]), Density)
#test if Qubit object type preserved
d = Density([Qubit('00'), 1/sqrt(2)], [Qubit('11'), 1/sqrt(2)])
for (state, prob) in d.args:
assert isinstance(state, Qubit)
# check for value error, when prob is not provided
raises(ValueError, lambda: Density([Ket(0)], [Ket(1)]))
def test_doit():
x, y = symbols('x y')
A, B, C, D, E, F = symbols('A B C D E F', commutative=False)
d = Density([XKet(), 0.5], [PxKet(), 0.5])
assert (0.5*(PxKet()*Dagger(PxKet())) +
0.5*(XKet()*Dagger(XKet()))) == d.doit()
# check for kets with expr in them
d_with_sym = Density([XKet(x*y), 0.5], [PxKet(x*y), 0.5])
assert (0.5*(PxKet(x*y)*Dagger(PxKet(x*y))) +
0.5*(XKet(x*y)*Dagger(XKet(x*y)))) == d_with_sym.doit()
d = Density([(A + B)*C, 1.0])
assert d.doit() == (1.0*A*C*Dagger(C)*Dagger(A) +
1.0*A*C*Dagger(C)*Dagger(B) +
1.0*B*C*Dagger(C)*Dagger(A) +
1.0*B*C*Dagger(C)*Dagger(B))
# With TensorProducts as args
# Density with simple tensor products as args
t = TensorProduct(A, B, C)
d = Density([t, 1.0])
assert d.doit() == \
1.0 * TensorProduct(A*Dagger(A), B*Dagger(B), C*Dagger(C))
# Density with multiple Tensorproducts as states
t2 = TensorProduct(A, B)
t3 = TensorProduct(C, D)
d = Density([t2, 0.5], [t3, 0.5])
assert d.doit() == (0.5 * TensorProduct(A*Dagger(A), B*Dagger(B)) +
0.5 * TensorProduct(C*Dagger(C), D*Dagger(D)))
#Density with mixed states
d = Density([t2 + t3, 1.0])
assert d.doit() == (1.0 * TensorProduct(A*Dagger(A), B*Dagger(B)) +
1.0 * TensorProduct(A*Dagger(C), B*Dagger(D)) +
1.0 * TensorProduct(C*Dagger(A), D*Dagger(B)) +
1.0 * TensorProduct(C*Dagger(C), D*Dagger(D)))
#Density operators with spin states
tp1 = TensorProduct(JzKet(1, 1), JzKet(1, -1))
d = Density([tp1, 1])
# full trace
t = Tr(d)
assert t.doit() == 1
#Partial trace on density operators with spin states
t = Tr(d, [0])
assert t.doit() == JzKet(1, -1) * Dagger(JzKet(1, -1))
t = Tr(d, [1])
assert t.doit() == JzKet(1, 1) * Dagger(JzKet(1, 1))
# with another spin state
tp2 = TensorProduct(JzKet(S.Half, S.Half), JzKet(S.Half, Rational(-1, 2)))
d = Density([tp2, 1])
#full trace
t = Tr(d)
assert t.doit() == 1
#Partial trace on density operators with spin states
t = Tr(d, [0])
assert t.doit() == JzKet(S.Half, Rational(-1, 2)) * Dagger(JzKet(S.Half, Rational(-1, 2)))
t = Tr(d, [1])
assert t.doit() == JzKet(S.Half, S.Half) * Dagger(JzKet(S.Half, S.Half))
def test_apply_op():
d = Density([Ket(0), 0.5], [Ket(1), 0.5])
assert d.apply_op(XOp()) == Density([XOp()*Ket(0), 0.5],
[XOp()*Ket(1), 0.5])
def test_represent():
x, y = symbols('x y')
d = Density([XKet(), 0.5], [PxKet(), 0.5])
assert (represent(0.5*(PxKet()*Dagger(PxKet()))) +
represent(0.5*(XKet()*Dagger(XKet())))) == represent(d)
# check for kets with expr in them
d_with_sym = Density([XKet(x*y), 0.5], [PxKet(x*y), 0.5])
assert (represent(0.5*(PxKet(x*y)*Dagger(PxKet(x*y)))) +
represent(0.5*(XKet(x*y)*Dagger(XKet(x*y))))) == \
represent(d_with_sym)
# check when given explicit basis
assert (represent(0.5*(XKet()*Dagger(XKet())), basis=PxOp()) +
represent(0.5*(PxKet()*Dagger(PxKet())), basis=PxOp())) == \
represent(d, basis=PxOp())
def test_states():
d = Density([Ket(0), 0.5], [Ket(1), 0.5])
states = d.states()
assert states[0] == Ket(0) and states[1] == Ket(1)
def test_probs():
d = Density([Ket(0), .75], [Ket(1), 0.25])
probs = d.probs()
assert probs[0] == 0.75 and probs[1] == 0.25
#probs can be symbols
x, y = symbols('x y')
d = Density([Ket(0), x], [Ket(1), y])
probs = d.probs()
assert probs[0] == x and probs[1] == y
def test_get_state():
x, y = symbols('x y')
d = Density([Ket(0), x], [Ket(1), y])
states = (d.get_state(0), d.get_state(1))
assert states[0] == Ket(0) and states[1] == Ket(1)
def test_get_prob():
x, y = symbols('x y')
d = Density([Ket(0), x], [Ket(1), y])
probs = (d.get_prob(0), d.get_prob(1))
assert probs[0] == x and probs[1] == y
def test_entropy():
up = JzKet(S.Half, S.Half)
down = JzKet(S.Half, Rational(-1, 2))
d = Density((up, S.Half), (down, S.Half))
# test for density object
ent = entropy(d)
assert entropy(d) == log(2)/2
assert d.entropy() == log(2)/2
np = import_module('numpy', min_module_version='1.4.0')
if np:
#do this test only if 'numpy' is available on test machine
np_mat = represent(d, format='numpy')
ent = entropy(np_mat)
assert isinstance(np_mat, np.ndarray)
assert ent.real == 0.69314718055994529
assert ent.imag == 0
scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
if scipy and np:
#do this test only if numpy and scipy are available
mat = represent(d, format="scipy.sparse")
assert isinstance(mat, scipy_sparse_matrix)
assert ent.real == 0.69314718055994529
assert ent.imag == 0
def test_eval_trace():
up = JzKet(S.Half, S.Half)
down = JzKet(S.Half, Rational(-1, 2))
d = Density((up, 0.5), (down, 0.5))
t = Tr(d)
assert t.doit() == 1
#test dummy time dependent states
class TestTimeDepKet(TimeDepKet):
def _eval_trace(self, bra, **options):
return 1
x, t = symbols('x t')
k1 = TestTimeDepKet(0, 0.5)
k2 = TestTimeDepKet(0, 1)
d = Density([k1, 0.5], [k2, 0.5])
assert d.doit() == (0.5 * OuterProduct(k1, k1.dual) +
0.5 * OuterProduct(k2, k2.dual))
t = Tr(d)
assert t.doit() == 1
def test_fidelity():
#test with kets
up = JzKet(S.Half, S.Half)
down = JzKet(S.Half, Rational(-1, 2))
updown = (S.One/sqrt(2))*up + (S.One/sqrt(2))*down
#check with matrices
up_dm = represent(up * Dagger(up))
down_dm = represent(down * Dagger(down))
updown_dm = represent(updown * Dagger(updown))
assert abs(fidelity(up_dm, up_dm) - 1) < 1e-3
assert fidelity(up_dm, down_dm) < 1e-3
assert abs(fidelity(up_dm, updown_dm) - (S.One/sqrt(2))) < 1e-3
assert abs(fidelity(updown_dm, down_dm) - (S.One/sqrt(2))) < 1e-3
#check with density
up_dm = Density([up, 1.0])
down_dm = Density([down, 1.0])
updown_dm = Density([updown, 1.0])
assert abs(fidelity(up_dm, up_dm) - 1) < 1e-3
assert abs(fidelity(up_dm, down_dm)) < 1e-3
assert abs(fidelity(up_dm, updown_dm) - (S.One/sqrt(2))) < 1e-3
assert abs(fidelity(updown_dm, down_dm) - (S.One/sqrt(2))) < 1e-3
#check mixed states with density
updown2 = sqrt(3)/2*up + S.Half*down
d1 = Density([updown, 0.25], [updown2, 0.75])
d2 = Density([updown, 0.75], [updown2, 0.25])
assert abs(fidelity(d1, d2) - 0.991) < 1e-3
assert abs(fidelity(d2, d1) - fidelity(d1, d2)) < 1e-3
#using qubits/density(pure states)
state1 = Qubit('0')
state2 = Qubit('1')
state3 = S.One/sqrt(2)*state1 + S.One/sqrt(2)*state2
state4 = sqrt(Rational(2, 3))*state1 + S.One/sqrt(3)*state2
state1_dm = Density([state1, 1])
state2_dm = Density([state2, 1])
state3_dm = Density([state3, 1])
assert fidelity(state1_dm, state1_dm) == 1
assert fidelity(state1_dm, state2_dm) == 0
assert abs(fidelity(state1_dm, state3_dm) - 1/sqrt(2)) < 1e-3
assert abs(fidelity(state3_dm, state2_dm) - 1/sqrt(2)) < 1e-3
#using qubits/density(mixed states)
d1 = Density([state3, 0.70], [state4, 0.30])
d2 = Density([state3, 0.20], [state4, 0.80])
assert abs(fidelity(d1, d1) - 1) < 1e-3
assert abs(fidelity(d1, d2) - 0.996) < 1e-3
assert abs(fidelity(d1, d2) - fidelity(d2, d1)) < 1e-3
#TODO: test for invalid arguments
# non-square matrix
mat1 = [[0, 0],
[0, 0],
[0, 0]]
mat2 = [[0, 0],
[0, 0]]
raises(ValueError, lambda: fidelity(mat1, mat2))
# unequal dimensions
mat1 = [[0, 0],
[0, 0]]
mat2 = [[0, 0, 0],
[0, 0, 0],
[0, 0, 0]]
raises(ValueError, lambda: fidelity(mat1, mat2))
# unsupported data-type
x, y = 1, 2 # random values that is not a matrix
raises(ValueError, lambda: fidelity(x, y))
|
3d1d7d82853ad115cbdda537854d82277572ae519e3fac7fe77d6723d9aabe6f | from sympy.core.expr import Expr
from sympy.core.mul import Mul
from sympy.core.numbers import (I, Integer)
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import conjugate
from sympy.matrices.dense import Matrix
from sympy.physics.quantum.dagger import adjoint, Dagger
from sympy.external import import_module
from sympy.testing.pytest import skip
from sympy.physics.quantum.operator import Operator, IdentityOperator
def test_scalars():
x = symbols('x', complex=True)
assert Dagger(x) == conjugate(x)
assert Dagger(I*x) == -I*conjugate(x)
i = symbols('i', real=True)
assert Dagger(i) == i
p = symbols('p')
assert isinstance(Dagger(p), adjoint)
i = Integer(3)
assert Dagger(i) == i
A = symbols('A', commutative=False)
assert Dagger(A).is_commutative is False
def test_matrix():
x = symbols('x')
m = Matrix([[I, x*I], [2, 4]])
assert Dagger(m) == m.H
def test_dagger_mul():
O = Operator('O')
I = IdentityOperator()
assert Dagger(O)*O == Dagger(O)*O
assert Dagger(O)*O*I == Mul(Dagger(O), O)*I
assert Dagger(O)*Dagger(O) == Dagger(O)**2
assert Dagger(O)*Dagger(I) == Dagger(O)
class Foo(Expr):
def _eval_adjoint(self):
return I
def test_eval_adjoint():
f = Foo()
d = Dagger(f)
assert d == I
np = import_module('numpy')
def test_numpy_dagger():
if not np:
skip("numpy not installed.")
a = np.array([[1.0, 2.0j], [-1.0j, 2.0]])
adag = a.copy().transpose().conjugate()
assert (Dagger(a) == adag).all()
scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
def test_scipy_sparse_dagger():
if not np:
skip("numpy not installed.")
if not scipy:
skip("scipy not installed.")
else:
sparse = scipy.sparse
a = sparse.csr_matrix([[1.0 + 0.0j, 2.0j], [-1.0j, 2.0 + 0.0j]])
adag = a.copy().transpose().conjugate()
assert np.linalg.norm((Dagger(a) - adag).todense()) == 0.0
|
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