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"""Efficient functions for generating orthogonal polynomials."""
from sympy.core.symbol import Dummy
from sympy.polys.densearith import (dup_mul, dup_mul_ground,
dup_lshift, dup_sub, dup_add)
from sympy.polys.domains import ZZ, QQ
from sympy.polys.polytools import named_poly
from sympy.utilities import public
def dup_jacobi(n, a, b, K):
"""Low-level implementation of Jacobi polynomials."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [(a+b)/K(2) + K.one, (a-b)/K(2)]
for i in range(2, n+1):
den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2))
f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den)
f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den)
f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den
p0 = dup_mul_ground(m1, f0, K)
p1 = dup_mul_ground(dup_lshift(m1, 1, K), f1, K)
p2 = dup_mul_ground(m2, f2, K)
m2, m1 = m1, dup_sub(dup_add(p0, p1, K), p2, K)
return m1
@public
def jacobi_poly(n, a, b, x=None, polys=False):
r"""Generates the Jacobi polynomial `P_n^{(a,b)}(x)`.
Parameters
==========
n : int
Degree of the polynomial.
a
Lower limit of minimal domain for the list of coefficients.
b
Upper limit of minimal domain for the list of coefficients.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_jacobi, None, "Jacobi polynomial", (x, a, b), polys)
def dup_gegenbauer(n, a, K):
"""Low-level implementation of Gegenbauer polynomials."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [K(2)*a, K.zero]
for i in range(2, n+1):
p1 = dup_mul_ground(dup_lshift(m1, 1, K), K(2)*(a-K.one)/K(i) + K(2), K)
p2 = dup_mul_ground(m2, K(2)*(a-K.one)/K(i) + K.one, K)
m2, m1 = m1, dup_sub(p1, p2, K)
return m1
def gegenbauer_poly(n, a, x=None, polys=False):
r"""Generates the Gegenbauer polynomial `C_n^{(a)}(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
a
Decides minimal domain for the list of coefficients.
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_gegenbauer, None, "Gegenbauer polynomial", (x, a), polys)
def dup_chebyshevt(n, K):
"""Low-level implementation of Chebyshev polynomials of the first kind."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [K.one, K.zero]
for i in range(2, n+1):
m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2), K), m2, K)
return m1
def dup_chebyshevu(n, K):
"""Low-level implementation of Chebyshev polynomials of the second kind."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [K(2), K.zero]
for i in range(2, n+1):
m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2), K), m2, K)
return m1
@public
def chebyshevt_poly(n, x=None, polys=False):
r"""Generates the Chebyshev polynomial of the first kind `T_n(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_chebyshevt, ZZ,
"Chebyshev polynomial of the first kind", (x,), polys)
@public
def chebyshevu_poly(n, x=None, polys=False):
r"""Generates the Chebyshev polynomial of the second kind `U_n(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_chebyshevu, ZZ,
"Chebyshev polynomial of the second kind", (x,), polys)
def dup_hermite(n, K):
"""Low-level implementation of Hermite polynomials."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [K(2), K.zero]
for i in range(2, n+1):
a = dup_lshift(m1, 1, K)
b = dup_mul_ground(m2, K(i-1), K)
m2, m1 = m1, dup_mul_ground(dup_sub(a, b, K), K(2), K)
return m1
def dup_hermite_prob(n, K):
"""Low-level implementation of probabilist's Hermite polynomials."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [K.one, K.zero]
for i in range(2, n+1):
a = dup_lshift(m1, 1, K)
b = dup_mul_ground(m2, K(i-1), K)
m2, m1 = m1, dup_sub(a, b, K)
return m1
@public
def hermite_poly(n, x=None, polys=False):
r"""Generates the Hermite polynomial `H_n(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_hermite, ZZ, "Hermite polynomial", (x,), polys)
@public
def hermite_prob_poly(n, x=None, polys=False):
r"""Generates the probabilist's Hermite polynomial `He_n(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_hermite_prob, ZZ,
"probabilist's Hermite polynomial", (x,), polys)
def dup_legendre(n, K):
"""Low-level implementation of Legendre polynomials."""
if n < 1:
return [K.one]
m2, m1 = [K.one], [K.one, K.zero]
for i in range(2, n+1):
a = dup_mul_ground(dup_lshift(m1, 1, K), K(2*i-1, i), K)
b = dup_mul_ground(m2, K(i-1, i), K)
m2, m1 = m1, dup_sub(a, b, K)
return m1
@public
def legendre_poly(n, x=None, polys=False):
r"""Generates the Legendre polynomial `P_n(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_legendre, QQ, "Legendre polynomial", (x,), polys)
def dup_laguerre(n, alpha, K):
"""Low-level implementation of Laguerre polynomials."""
m2, m1 = [K.zero], [K.one]
for i in range(1, n+1):
a = dup_mul(m1, [-K.one/K(i), (alpha-K.one)/K(i) + K(2)], K)
b = dup_mul_ground(m2, (alpha-K.one)/K(i) + K.one, K)
m2, m1 = m1, dup_sub(a, b, K)
return m1
@public
def laguerre_poly(n, x=None, alpha=0, polys=False):
r"""Generates the Laguerre polynomial `L_n^{(\alpha)}(x)`.
Parameters
==========
n : int
Degree of the polynomial.
x : optional
alpha : optional
Decides minimal domain for the list of coefficients.
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
"""
return named_poly(n, dup_laguerre, None, "Laguerre polynomial", (x, alpha), polys)
def dup_spherical_bessel_fn(n, K):
"""Low-level implementation of fn(n, x)."""
if n < 1:
return [K.one, K.zero]
m2, m1 = [K.one], [K.one, K.zero]
for i in range(2, n+1):
m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2*i-1), K), m2, K)
return dup_lshift(m1, 1, K)
def dup_spherical_bessel_fn_minus(n, K):
"""Low-level implementation of fn(-n, x)."""
m2, m1 = [K.one, K.zero], [K.zero]
for i in range(2, n+1):
m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(3-2*i), K), m2, K)
return m1
def spherical_bessel_fn(n, x=None, polys=False):
"""
Coefficients for the spherical Bessel functions.
These are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Parameters
==========
n : int
Degree of the polynomial.
x : optional
polys : bool, optional
If True, return a Poly, otherwise (default) return an expression.
Examples
========
>>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
>>> from sympy import Symbol
>>> z = Symbol("z")
>>> fn(1, z)
z**(-2)
>>> fn(2, z)
-1/z + 3/z**3
>>> fn(3, z)
-6/z**2 + 15/z**4
>>> fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
if x is None:
x = Dummy("x")
f = dup_spherical_bessel_fn_minus if n < 0 else dup_spherical_bessel_fn
return named_poly(abs(n), f, ZZ, "", (QQ(1)/x,), polys)
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