diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/__init__.py b/env-llmeval/lib/python3.10/site-packages/mpmath/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..46a7c6f7c0875548f264612b604a9e1574b00a84 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/__init__.py @@ -0,0 +1,468 @@ +__version__ = '1.3.0' + +from .usertools import monitor, timing + +from .ctx_fp import FPContext +from .ctx_mp import MPContext +from .ctx_iv import MPIntervalContext + +fp = FPContext() +mp = MPContext() +iv = MPIntervalContext() + +fp._mp = mp +mp._mp = mp +iv._mp = mp +mp._fp = fp +fp._fp = fp +mp._iv = iv +fp._iv = iv +iv._iv = iv + +# XXX: extremely bad pickle hack +from . import ctx_mp as _ctx_mp +_ctx_mp._mpf_module.mpf = mp.mpf +_ctx_mp._mpf_module.mpc = mp.mpc + +make_mpf = mp.make_mpf +make_mpc = mp.make_mpc + +extraprec = mp.extraprec +extradps = mp.extradps +workprec = mp.workprec +workdps = mp.workdps +autoprec = mp.autoprec +maxcalls = mp.maxcalls +memoize = mp.memoize + +mag = mp.mag + +bernfrac = mp.bernfrac + +qfrom = mp.qfrom +mfrom = mp.mfrom +kfrom = mp.kfrom +taufrom = mp.taufrom +qbarfrom = mp.qbarfrom +ellipfun = mp.ellipfun +jtheta = mp.jtheta +kleinj = mp.kleinj +eta = mp.eta + +qp = mp.qp +qhyper = mp.qhyper +qgamma = mp.qgamma +qfac = mp.qfac + +nint_distance = mp.nint_distance + +plot = mp.plot +cplot = mp.cplot +splot = mp.splot + +odefun = mp.odefun + +jacobian = mp.jacobian +findroot = mp.findroot +multiplicity = mp.multiplicity + +isinf = mp.isinf +isnan = mp.isnan +isnormal = mp.isnormal +isint = mp.isint +isfinite = mp.isfinite +almosteq = mp.almosteq +nan = mp.nan +rand = mp.rand + +absmin = mp.absmin +absmax = mp.absmax + +fraction = mp.fraction + +linspace = mp.linspace +arange = mp.arange + +mpmathify = convert = mp.convert +mpc = mp.mpc + +mpi = iv._mpi + +nstr = mp.nstr +nprint = mp.nprint +chop = mp.chop + +fneg = mp.fneg +fadd = mp.fadd +fsub = mp.fsub +fmul = mp.fmul +fdiv = mp.fdiv +fprod = mp.fprod + +quad = mp.quad +quadgl = mp.quadgl +quadts = mp.quadts +quadosc = mp.quadosc +quadsubdiv = mp.quadsubdiv + +invertlaplace = mp.invertlaplace +invlaptalbot = mp.invlaptalbot +invlapstehfest = mp.invlapstehfest +invlapdehoog = mp.invlapdehoog + +pslq = mp.pslq +identify = mp.identify +findpoly = mp.findpoly + +richardson = mp.richardson +shanks = mp.shanks +levin = mp.levin +cohen_alt = mp.cohen_alt +nsum = mp.nsum +nprod = mp.nprod +difference = mp.difference +diff = mp.diff +diffs = mp.diffs +diffs_prod = mp.diffs_prod +diffs_exp = mp.diffs_exp +diffun = mp.diffun +differint = mp.differint +taylor = mp.taylor +pade = mp.pade +polyval = mp.polyval +polyroots = mp.polyroots +fourier = mp.fourier +fourierval = mp.fourierval +sumem = mp.sumem +sumap = mp.sumap +chebyfit = mp.chebyfit +limit = mp.limit + +matrix = mp.matrix +eye = mp.eye +diag = mp.diag +zeros = mp.zeros +ones = mp.ones +hilbert = mp.hilbert +randmatrix = mp.randmatrix +swap_row = mp.swap_row +extend = mp.extend +norm = mp.norm +mnorm = mp.mnorm + +lu_solve = mp.lu_solve +lu = mp.lu +qr = mp.qr +unitvector = mp.unitvector +inverse = mp.inverse +residual = mp.residual +qr_solve = mp.qr_solve +cholesky = mp.cholesky +cholesky_solve = mp.cholesky_solve +det = mp.det +cond = mp.cond +hessenberg = mp.hessenberg +schur = mp.schur +eig = mp.eig +eig_sort = mp.eig_sort +eigsy = mp.eigsy +eighe = mp.eighe +eigh = mp.eigh +svd_r = mp.svd_r +svd_c = mp.svd_c +svd = mp.svd +gauss_quadrature = mp.gauss_quadrature + +expm = mp.expm +sqrtm = mp.sqrtm +powm = mp.powm +logm = mp.logm +sinm = mp.sinm +cosm = mp.cosm + +mpf = mp.mpf +j = mp.j +exp = mp.exp +expj = mp.expj +expjpi = mp.expjpi +ln = mp.ln +im = mp.im +re = mp.re +inf = mp.inf +ninf = mp.ninf +sign = mp.sign + +eps = mp.eps +pi = mp.pi +ln2 = mp.ln2 +ln10 = mp.ln10 +phi = mp.phi +e = mp.e +euler = mp.euler +catalan = mp.catalan +khinchin = mp.khinchin +glaisher = mp.glaisher +apery = mp.apery +degree = mp.degree +twinprime = mp.twinprime +mertens = mp.mertens + +ldexp = mp.ldexp +frexp = mp.frexp + +fsum = mp.fsum +fdot = mp.fdot + +sqrt = mp.sqrt +cbrt = mp.cbrt +exp = mp.exp +ln = mp.ln +log = mp.log +log10 = mp.log10 +power = mp.power +cos = mp.cos +sin = mp.sin +tan = mp.tan +cosh = mp.cosh +sinh = mp.sinh +tanh = mp.tanh +acos = mp.acos +asin = mp.asin +atan = mp.atan +asinh = mp.asinh +acosh = mp.acosh +atanh = mp.atanh +sec = mp.sec +csc = mp.csc +cot = mp.cot +sech = mp.sech +csch = mp.csch +coth = mp.coth +asec = mp.asec +acsc = mp.acsc +acot = mp.acot +asech = mp.asech +acsch = mp.acsch +acoth = mp.acoth +cospi = mp.cospi +sinpi = mp.sinpi +sinc = mp.sinc +sincpi = mp.sincpi +cos_sin = mp.cos_sin +cospi_sinpi = mp.cospi_sinpi +fabs = mp.fabs +re = mp.re +im = mp.im +conj = mp.conj +floor = mp.floor +ceil = mp.ceil +nint = mp.nint +frac = mp.frac +root = mp.root +nthroot = mp.nthroot +hypot = mp.hypot +fmod = mp.fmod +ldexp = mp.ldexp +frexp = mp.frexp +sign = mp.sign +arg = mp.arg +phase = mp.phase +polar = mp.polar +rect = mp.rect +degrees = mp.degrees +radians = mp.radians +atan2 = mp.atan2 +fib = mp.fib +fibonacci = mp.fibonacci +lambertw = mp.lambertw +zeta = mp.zeta +altzeta = mp.altzeta +gamma = mp.gamma +rgamma = mp.rgamma +factorial = mp.factorial +fac = mp.fac +fac2 = mp.fac2 +beta = mp.beta +betainc = mp.betainc +psi = mp.psi +#psi0 = mp.psi0 +#psi1 = mp.psi1 +#psi2 = mp.psi2 +#psi3 = mp.psi3 +polygamma = mp.polygamma +digamma = mp.digamma +#trigamma = mp.trigamma +#tetragamma = mp.tetragamma +#pentagamma = mp.pentagamma +harmonic = mp.harmonic +bernoulli = mp.bernoulli +bernfrac = mp.bernfrac +stieltjes = mp.stieltjes +hurwitz = mp.hurwitz +dirichlet = mp.dirichlet +bernpoly = mp.bernpoly +eulerpoly = mp.eulerpoly +eulernum = mp.eulernum +polylog = mp.polylog +clsin = mp.clsin +clcos = mp.clcos +gammainc = mp.gammainc +gammaprod = mp.gammaprod +binomial = mp.binomial +rf = mp.rf +ff = mp.ff +hyper = mp.hyper +hyp0f1 = mp.hyp0f1 +hyp1f1 = mp.hyp1f1 +hyp1f2 = mp.hyp1f2 +hyp2f1 = mp.hyp2f1 +hyp2f2 = mp.hyp2f2 +hyp2f0 = mp.hyp2f0 +hyp2f3 = mp.hyp2f3 +hyp3f2 = mp.hyp3f2 +hyperu = mp.hyperu +hypercomb = mp.hypercomb +meijerg = mp.meijerg +appellf1 = mp.appellf1 +appellf2 = mp.appellf2 +appellf3 = mp.appellf3 +appellf4 = mp.appellf4 +hyper2d = mp.hyper2d +bihyper = mp.bihyper +erf = mp.erf +erfc = mp.erfc +erfi = mp.erfi +erfinv = mp.erfinv +npdf = mp.npdf +ncdf = mp.ncdf +expint = mp.expint +e1 = mp.e1 +ei = mp.ei +li = mp.li +ci = mp.ci +si = mp.si +chi = mp.chi +shi = mp.shi +fresnels = mp.fresnels +fresnelc = mp.fresnelc +airyai = mp.airyai +airybi = mp.airybi +airyaizero = mp.airyaizero +airybizero = mp.airybizero +scorergi = mp.scorergi +scorerhi = mp.scorerhi +ellipk = mp.ellipk +ellipe = mp.ellipe +ellipf = mp.ellipf +ellippi = mp.ellippi +elliprc = mp.elliprc +elliprj = mp.elliprj +elliprf = mp.elliprf +elliprd = mp.elliprd +elliprg = mp.elliprg +agm = mp.agm +jacobi = mp.jacobi +chebyt = mp.chebyt +chebyu = mp.chebyu +legendre = mp.legendre +legenp = mp.legenp +legenq = mp.legenq +hermite = mp.hermite +pcfd = mp.pcfd +pcfu = mp.pcfu +pcfv = mp.pcfv +pcfw = mp.pcfw +gegenbauer = mp.gegenbauer +laguerre = mp.laguerre +spherharm = mp.spherharm +besselj = mp.besselj +j0 = mp.j0 +j1 = mp.j1 +besseli = mp.besseli +bessely = mp.bessely +besselk = mp.besselk +besseljzero = mp.besseljzero +besselyzero = mp.besselyzero +hankel1 = mp.hankel1 +hankel2 = mp.hankel2 +struveh = mp.struveh +struvel = mp.struvel +angerj = mp.angerj +webere = mp.webere +lommels1 = mp.lommels1 +lommels2 = mp.lommels2 +whitm = mp.whitm +whitw = mp.whitw +ber = mp.ber +bei = mp.bei +ker = mp.ker +kei = mp.kei +coulombc = mp.coulombc +coulombf = mp.coulombf +coulombg = mp.coulombg +barnesg = mp.barnesg +superfac = mp.superfac +hyperfac = mp.hyperfac +loggamma = mp.loggamma +siegeltheta = mp.siegeltheta +siegelz = mp.siegelz +grampoint = mp.grampoint +zetazero = mp.zetazero +riemannr = mp.riemannr +primepi = mp.primepi +primepi2 = mp.primepi2 +primezeta = mp.primezeta +bell = mp.bell +polyexp = mp.polyexp +expm1 = mp.expm1 +log1p = mp.log1p +powm1 = mp.powm1 +unitroots = mp.unitroots +cyclotomic = mp.cyclotomic +mangoldt = mp.mangoldt +secondzeta = mp.secondzeta +nzeros = mp.nzeros +backlunds = mp.backlunds +lerchphi = mp.lerchphi +stirling1 = mp.stirling1 +stirling2 = mp.stirling2 +squarew = mp.squarew +trianglew = mp.trianglew +sawtoothw = mp.sawtoothw +unit_triangle = mp.unit_triangle +sigmoid = mp.sigmoid + +# be careful when changing this name, don't use test*! +def runtests(): + """ + Run all mpmath tests and print output. + """ + import os.path + from inspect import getsourcefile + from .tests import runtests as tests + testdir = os.path.dirname(os.path.abspath(getsourcefile(tests))) + importdir = os.path.abspath(testdir + '/../..') + tests.testit(importdir, testdir) + +def doctests(filter=[]): + import sys + from timeit import default_timer as clock + for i, arg in enumerate(sys.argv): + if '__init__.py' in arg: + filter = [sn for sn in sys.argv[i+1:] if not sn.startswith("-")] + break + import doctest + globs = globals().copy() + for obj in globs: #sorted(globs.keys()): + if filter: + if not sum([pat in obj for pat in filter]): + continue + sys.stdout.write(str(obj) + " ") + sys.stdout.flush() + t1 = clock() + doctest.run_docstring_examples(globs[obj], {}, verbose=("-v" in sys.argv)) + t2 = clock() + print(round(t2-t1, 3)) + +if __name__ == '__main__': + doctests() diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/__pycache__/ctx_mp_python.cpython-310.pyc b/env-llmeval/lib/python3.10/site-packages/mpmath/__pycache__/ctx_mp_python.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..15c9a316dbfdcf59ab779f42190d2383aa0fe304 Binary files /dev/null and b/env-llmeval/lib/python3.10/site-packages/mpmath/__pycache__/ctx_mp_python.cpython-310.pyc differ diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/__pycache__/function_docs.cpython-310.pyc b/env-llmeval/lib/python3.10/site-packages/mpmath/__pycache__/function_docs.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..350207f3d36a861e913063f8ea7cd9f23b9a369f Binary files /dev/null and b/env-llmeval/lib/python3.10/site-packages/mpmath/__pycache__/function_docs.cpython-310.pyc differ diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_fp.py b/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_fp.py new file mode 100644 index 0000000000000000000000000000000000000000..aa72ea5b03fde4da66b0d8fbf8ffa4012e3f6178 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_fp.py @@ -0,0 +1,253 @@ +from .ctx_base import StandardBaseContext + +import math +import cmath +from . import math2 + +from . import function_docs + +from .libmp import mpf_bernoulli, to_float, int_types +from . import libmp + +class FPContext(StandardBaseContext): + """ + Context for fast low-precision arithmetic (53-bit precision, giving at most + about 15-digit accuracy), using Python's builtin float and complex. + """ + + def __init__(ctx): + StandardBaseContext.__init__(ctx) + + # Override SpecialFunctions implementation + ctx.loggamma = math2.loggamma + ctx._bernoulli_cache = {} + ctx.pretty = False + + ctx._init_aliases() + + _mpq = lambda cls, x: float(x[0])/x[1] + + NoConvergence = libmp.NoConvergence + + def _get_prec(ctx): return 53 + def _set_prec(ctx, p): return + def _get_dps(ctx): return 15 + def _set_dps(ctx, p): return + + _fixed_precision = True + + prec = property(_get_prec, _set_prec) + dps = property(_get_dps, _set_dps) + + zero = 0.0 + one = 1.0 + eps = math2.EPS + inf = math2.INF + ninf = math2.NINF + nan = math2.NAN + j = 1j + + # Called by SpecialFunctions.__init__() + @classmethod + def _wrap_specfun(cls, name, f, wrap): + if wrap: + def f_wrapped(ctx, *args, **kwargs): + convert = ctx.convert + args = [convert(a) for a in args] + return f(ctx, *args, **kwargs) + else: + f_wrapped = f + f_wrapped.__doc__ = function_docs.__dict__.get(name, f.__doc__) + setattr(cls, name, f_wrapped) + + def bernoulli(ctx, n): + cache = ctx._bernoulli_cache + if n in cache: + return cache[n] + cache[n] = to_float(mpf_bernoulli(n, 53, 'n'), strict=True) + return cache[n] + + pi = math2.pi + e = math2.e + euler = math2.euler + sqrt2 = 1.4142135623730950488 + sqrt5 = 2.2360679774997896964 + phi = 1.6180339887498948482 + ln2 = 0.69314718055994530942 + ln10 = 2.302585092994045684 + euler = 0.57721566490153286061 + catalan = 0.91596559417721901505 + khinchin = 2.6854520010653064453 + apery = 1.2020569031595942854 + glaisher = 1.2824271291006226369 + + absmin = absmax = abs + + def is_special(ctx, x): + return x - x != 0.0 + + def isnan(ctx, x): + return x != x + + def isinf(ctx, x): + return abs(x) == math2.INF + + def isnormal(ctx, x): + if x: + return x - x == 0.0 + return False + + def isnpint(ctx, x): + if type(x) is complex: + if x.imag: + return False + x = x.real + return x <= 0.0 and round(x) == x + + mpf = float + mpc = complex + + def convert(ctx, x): + try: + return float(x) + except: + return complex(x) + + power = staticmethod(math2.pow) + sqrt = staticmethod(math2.sqrt) + exp = staticmethod(math2.exp) + ln = log = staticmethod(math2.log) + cos = staticmethod(math2.cos) + sin = staticmethod(math2.sin) + tan = staticmethod(math2.tan) + cos_sin = staticmethod(math2.cos_sin) + acos = staticmethod(math2.acos) + asin = staticmethod(math2.asin) + atan = staticmethod(math2.atan) + cosh = staticmethod(math2.cosh) + sinh = staticmethod(math2.sinh) + tanh = staticmethod(math2.tanh) + gamma = staticmethod(math2.gamma) + rgamma = staticmethod(math2.rgamma) + fac = factorial = staticmethod(math2.factorial) + floor = staticmethod(math2.floor) + ceil = staticmethod(math2.ceil) + cospi = staticmethod(math2.cospi) + sinpi = staticmethod(math2.sinpi) + cbrt = staticmethod(math2.cbrt) + _nthroot = staticmethod(math2.nthroot) + _ei = staticmethod(math2.ei) + _e1 = staticmethod(math2.e1) + _zeta = _zeta_int = staticmethod(math2.zeta) + + # XXX: math2 + def arg(ctx, z): + z = complex(z) + return math.atan2(z.imag, z.real) + + def expj(ctx, x): + return ctx.exp(ctx.j*x) + + def expjpi(ctx, x): + return ctx.exp(ctx.j*ctx.pi*x) + + ldexp = math.ldexp + frexp = math.frexp + + def mag(ctx, z): + if z: + return ctx.frexp(abs(z))[1] + return ctx.ninf + + def isint(ctx, z): + if hasattr(z, "imag"): # float/int don't have .real/.imag in py2.5 + if z.imag: + return False + z = z.real + try: + return z == int(z) + except: + return False + + def nint_distance(ctx, z): + if hasattr(z, "imag"): # float/int don't have .real/.imag in py2.5 + n = round(z.real) + else: + n = round(z) + if n == z: + return n, ctx.ninf + return n, ctx.mag(abs(z-n)) + + def _convert_param(ctx, z): + if type(z) is tuple: + p, q = z + return ctx.mpf(p) / q, 'R' + if hasattr(z, "imag"): # float/int don't have .real/.imag in py2.5 + intz = int(z.real) + else: + intz = int(z) + if z == intz: + return intz, 'Z' + return z, 'R' + + def _is_real_type(ctx, z): + return isinstance(z, float) or isinstance(z, int_types) + + def _is_complex_type(ctx, z): + return isinstance(z, complex) + + def hypsum(ctx, p, q, types, coeffs, z, maxterms=6000, **kwargs): + coeffs = list(coeffs) + num = range(p) + den = range(p,p+q) + tol = ctx.eps + s = t = 1.0 + k = 0 + while 1: + for i in num: t *= (coeffs[i]+k) + for i in den: t /= (coeffs[i]+k) + k += 1; t /= k; t *= z; s += t + if abs(t) < tol: + return s + if k > maxterms: + raise ctx.NoConvergence + + def atan2(ctx, x, y): + return math.atan2(x, y) + + def psi(ctx, m, z): + m = int(m) + if m == 0: + return ctx.digamma(z) + return (-1)**(m+1) * ctx.fac(m) * ctx.zeta(m+1, z) + + digamma = staticmethod(math2.digamma) + + def harmonic(ctx, x): + x = ctx.convert(x) + if x == 0 or x == 1: + return x + return ctx.digamma(x+1) + ctx.euler + + nstr = str + + def to_fixed(ctx, x, prec): + return int(math.ldexp(x, prec)) + + def rand(ctx): + import random + return random.random() + + _erf = staticmethod(math2.erf) + _erfc = staticmethod(math2.erfc) + + def sum_accurately(ctx, terms, check_step=1): + s = ctx.zero + k = 0 + for term in terms(): + s += term + if (not k % check_step) and term: + if abs(term) <= 1e-18*abs(s): + break + k += 1 + return s diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_iv.py b/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_iv.py new file mode 100644 index 0000000000000000000000000000000000000000..c038e00a5677e318d222b63c22d225e3045e1c2b --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_iv.py @@ -0,0 +1,551 @@ +import operator + +from . import libmp + +from .libmp.backend import basestring + +from .libmp import ( + int_types, MPZ_ONE, + prec_to_dps, dps_to_prec, repr_dps, + round_floor, round_ceiling, + fzero, finf, fninf, fnan, + mpf_le, mpf_neg, + from_int, from_float, from_str, from_rational, + mpi_mid, mpi_delta, mpi_str, + mpi_abs, mpi_pos, mpi_neg, mpi_add, mpi_sub, + mpi_mul, mpi_div, mpi_pow_int, mpi_pow, + mpi_from_str, + mpci_pos, mpci_neg, mpci_add, mpci_sub, mpci_mul, mpci_div, mpci_pow, + mpci_abs, mpci_pow, mpci_exp, mpci_log, + ComplexResult, + mpf_hash, mpc_hash) +from .matrices.matrices import _matrix + +mpi_zero = (fzero, fzero) + +from .ctx_base import StandardBaseContext + +new = object.__new__ + +def convert_mpf_(x, prec, rounding): + if hasattr(x, "_mpf_"): return x._mpf_ + if isinstance(x, int_types): return from_int(x, prec, rounding) + if isinstance(x, float): return from_float(x, prec, rounding) + if isinstance(x, basestring): return from_str(x, prec, rounding) + raise NotImplementedError + + +class ivmpf(object): + """ + Interval arithmetic class. Precision is controlled by iv.prec. + """ + + def __new__(cls, x=0): + return cls.ctx.convert(x) + + def cast(self, cls, f_convert): + a, b = self._mpi_ + if a == b: + return cls(f_convert(a)) + raise ValueError + + def __int__(self): + return self.cast(int, libmp.to_int) + + def __float__(self): + return self.cast(float, libmp.to_float) + + def __complex__(self): + return self.cast(complex, libmp.to_float) + + def __hash__(self): + a, b = self._mpi_ + if a == b: + return mpf_hash(a) + else: + return hash(self._mpi_) + + @property + def real(self): return self + + @property + def imag(self): return self.ctx.zero + + def conjugate(self): return self + + @property + def a(self): + a, b = self._mpi_ + return self.ctx.make_mpf((a, a)) + + @property + def b(self): + a, b = self._mpi_ + return self.ctx.make_mpf((b, b)) + + @property + def mid(self): + ctx = self.ctx + v = mpi_mid(self._mpi_, ctx.prec) + return ctx.make_mpf((v, v)) + + @property + def delta(self): + ctx = self.ctx + v = mpi_delta(self._mpi_, ctx.prec) + return ctx.make_mpf((v,v)) + + @property + def _mpci_(self): + return self._mpi_, mpi_zero + + def _compare(*args): + raise TypeError("no ordering relation is defined for intervals") + + __gt__ = _compare + __le__ = _compare + __gt__ = _compare + __ge__ = _compare + + def __contains__(self, t): + t = self.ctx.mpf(t) + return (self.a <= t.a) and (t.b <= self.b) + + def __str__(self): + return mpi_str(self._mpi_, self.ctx.prec) + + def __repr__(self): + if self.ctx.pretty: + return str(self) + a, b = self._mpi_ + n = repr_dps(self.ctx.prec) + a = libmp.to_str(a, n) + b = libmp.to_str(b, n) + return "mpi(%r, %r)" % (a, b) + + def _compare(s, t, cmpfun): + if not hasattr(t, "_mpi_"): + try: + t = s.ctx.convert(t) + except: + return NotImplemented + return cmpfun(s._mpi_, t._mpi_) + + def __eq__(s, t): return s._compare(t, libmp.mpi_eq) + def __ne__(s, t): return s._compare(t, libmp.mpi_ne) + def __lt__(s, t): return s._compare(t, libmp.mpi_lt) + def __le__(s, t): return s._compare(t, libmp.mpi_le) + def __gt__(s, t): return s._compare(t, libmp.mpi_gt) + def __ge__(s, t): return s._compare(t, libmp.mpi_ge) + + def __abs__(self): + return self.ctx.make_mpf(mpi_abs(self._mpi_, self.ctx.prec)) + def __pos__(self): + return self.ctx.make_mpf(mpi_pos(self._mpi_, self.ctx.prec)) + def __neg__(self): + return self.ctx.make_mpf(mpi_neg(self._mpi_, self.ctx.prec)) + + def ae(s, t, rel_eps=None, abs_eps=None): + return s.ctx.almosteq(s, t, rel_eps, abs_eps) + +class ivmpc(object): + + def __new__(cls, re=0, im=0): + re = cls.ctx.convert(re) + im = cls.ctx.convert(im) + y = new(cls) + y._mpci_ = re._mpi_, im._mpi_ + return y + + def __hash__(self): + (a, b), (c,d) = self._mpci_ + if a == b and c == d: + return mpc_hash((a, c)) + else: + return hash(self._mpci_) + + def __repr__(s): + if s.ctx.pretty: + return str(s) + return "iv.mpc(%s, %s)" % (repr(s.real), repr(s.imag)) + + def __str__(s): + return "(%s + %s*j)" % (str(s.real), str(s.imag)) + + @property + def a(self): + (a, b), (c,d) = self._mpci_ + return self.ctx.make_mpf((a, a)) + + @property + def b(self): + (a, b), (c,d) = self._mpci_ + return self.ctx.make_mpf((b, b)) + + @property + def c(self): + (a, b), (c,d) = self._mpci_ + return self.ctx.make_mpf((c, c)) + + @property + def d(self): + (a, b), (c,d) = self._mpci_ + return self.ctx.make_mpf((d, d)) + + @property + def real(s): + return s.ctx.make_mpf(s._mpci_[0]) + + @property + def imag(s): + return s.ctx.make_mpf(s._mpci_[1]) + + def conjugate(s): + a, b = s._mpci_ + return s.ctx.make_mpc((a, mpf_neg(b))) + + def overlap(s, t): + t = s.ctx.convert(t) + real_overlap = (s.a <= t.a <= s.b) or (s.a <= t.b <= s.b) or (t.a <= s.a <= t.b) or (t.a <= s.b <= t.b) + imag_overlap = (s.c <= t.c <= s.d) or (s.c <= t.d <= s.d) or (t.c <= s.c <= t.d) or (t.c <= s.d <= t.d) + return real_overlap and imag_overlap + + def __contains__(s, t): + t = s.ctx.convert(t) + return t.real in s.real and t.imag in s.imag + + def _compare(s, t, ne=False): + if not isinstance(t, s.ctx._types): + try: + t = s.ctx.convert(t) + except: + return NotImplemented + if hasattr(t, '_mpi_'): + tval = t._mpi_, mpi_zero + elif hasattr(t, '_mpci_'): + tval = t._mpci_ + if ne: + return s._mpci_ != tval + return s._mpci_ == tval + + def __eq__(s, t): return s._compare(t) + def __ne__(s, t): return s._compare(t, True) + + def __lt__(s, t): raise TypeError("complex intervals cannot be ordered") + __le__ = __gt__ = __ge__ = __lt__ + + def __neg__(s): return s.ctx.make_mpc(mpci_neg(s._mpci_, s.ctx.prec)) + def __pos__(s): return s.ctx.make_mpc(mpci_pos(s._mpci_, s.ctx.prec)) + def __abs__(s): return s.ctx.make_mpf(mpci_abs(s._mpci_, s.ctx.prec)) + + def ae(s, t, rel_eps=None, abs_eps=None): + return s.ctx.almosteq(s, t, rel_eps, abs_eps) + +def _binary_op(f_real, f_complex): + def g_complex(ctx, sval, tval): + return ctx.make_mpc(f_complex(sval, tval, ctx.prec)) + def g_real(ctx, sval, tval): + try: + return ctx.make_mpf(f_real(sval, tval, ctx.prec)) + except ComplexResult: + sval = (sval, mpi_zero) + tval = (tval, mpi_zero) + return g_complex(ctx, sval, tval) + def lop_real(s, t): + if isinstance(t, _matrix): return NotImplemented + ctx = s.ctx + if not isinstance(t, ctx._types): t = ctx.convert(t) + if hasattr(t, "_mpi_"): return g_real(ctx, s._mpi_, t._mpi_) + if hasattr(t, "_mpci_"): return g_complex(ctx, (s._mpi_, mpi_zero), t._mpci_) + return NotImplemented + def rop_real(s, t): + ctx = s.ctx + if not isinstance(t, ctx._types): t = ctx.convert(t) + if hasattr(t, "_mpi_"): return g_real(ctx, t._mpi_, s._mpi_) + if hasattr(t, "_mpci_"): return g_complex(ctx, t._mpci_, (s._mpi_, mpi_zero)) + return NotImplemented + def lop_complex(s, t): + if isinstance(t, _matrix): return NotImplemented + ctx = s.ctx + if not isinstance(t, s.ctx._types): + try: + t = s.ctx.convert(t) + except (ValueError, TypeError): + return NotImplemented + return g_complex(ctx, s._mpci_, t._mpci_) + def rop_complex(s, t): + ctx = s.ctx + if not isinstance(t, s.ctx._types): + t = s.ctx.convert(t) + return g_complex(ctx, t._mpci_, s._mpci_) + return lop_real, rop_real, lop_complex, rop_complex + +ivmpf.__add__, ivmpf.__radd__, ivmpc.__add__, ivmpc.__radd__ = _binary_op(mpi_add, mpci_add) +ivmpf.__sub__, ivmpf.__rsub__, ivmpc.__sub__, ivmpc.__rsub__ = _binary_op(mpi_sub, mpci_sub) +ivmpf.__mul__, ivmpf.__rmul__, ivmpc.__mul__, ivmpc.__rmul__ = _binary_op(mpi_mul, mpci_mul) +ivmpf.__div__, ivmpf.__rdiv__, ivmpc.__div__, ivmpc.__rdiv__ = _binary_op(mpi_div, mpci_div) +ivmpf.__pow__, ivmpf.__rpow__, ivmpc.__pow__, ivmpc.__rpow__ = _binary_op(mpi_pow, mpci_pow) + +ivmpf.__truediv__ = ivmpf.__div__; ivmpf.__rtruediv__ = ivmpf.__rdiv__ +ivmpc.__truediv__ = ivmpc.__div__; ivmpc.__rtruediv__ = ivmpc.__rdiv__ + +class ivmpf_constant(ivmpf): + def __new__(cls, f): + self = new(cls) + self._f = f + return self + def _get_mpi_(self): + prec = self.ctx._prec[0] + a = self._f(prec, round_floor) + b = self._f(prec, round_ceiling) + return a, b + _mpi_ = property(_get_mpi_) + +class MPIntervalContext(StandardBaseContext): + + def __init__(ctx): + ctx.mpf = type('ivmpf', (ivmpf,), {}) + ctx.mpc = type('ivmpc', (ivmpc,), {}) + ctx._types = (ctx.mpf, ctx.mpc) + ctx._constant = type('ivmpf_constant', (ivmpf_constant,), {}) + ctx._prec = [53] + ctx._set_prec(53) + ctx._constant._ctxdata = ctx.mpf._ctxdata = ctx.mpc._ctxdata = [ctx.mpf, new, ctx._prec] + ctx._constant.ctx = ctx.mpf.ctx = ctx.mpc.ctx = ctx + ctx.pretty = False + StandardBaseContext.__init__(ctx) + ctx._init_builtins() + + def _mpi(ctx, a, b=None): + if b is None: + return ctx.mpf(a) + return ctx.mpf((a,b)) + + def _init_builtins(ctx): + ctx.one = ctx.mpf(1) + ctx.zero = ctx.mpf(0) + ctx.inf = ctx.mpf('inf') + ctx.ninf = -ctx.inf + ctx.nan = ctx.mpf('nan') + ctx.j = ctx.mpc(0,1) + ctx.exp = ctx._wrap_mpi_function(libmp.mpi_exp, libmp.mpci_exp) + ctx.sqrt = ctx._wrap_mpi_function(libmp.mpi_sqrt) + ctx.ln = ctx._wrap_mpi_function(libmp.mpi_log, libmp.mpci_log) + ctx.cos = ctx._wrap_mpi_function(libmp.mpi_cos, libmp.mpci_cos) + ctx.sin = ctx._wrap_mpi_function(libmp.mpi_sin, libmp.mpci_sin) + ctx.tan = ctx._wrap_mpi_function(libmp.mpi_tan) + ctx.gamma = ctx._wrap_mpi_function(libmp.mpi_gamma, libmp.mpci_gamma) + ctx.loggamma = ctx._wrap_mpi_function(libmp.mpi_loggamma, libmp.mpci_loggamma) + ctx.rgamma = ctx._wrap_mpi_function(libmp.mpi_rgamma, libmp.mpci_rgamma) + ctx.factorial = ctx._wrap_mpi_function(libmp.mpi_factorial, libmp.mpci_factorial) + ctx.fac = ctx.factorial + + ctx.eps = ctx._constant(lambda prec, rnd: (0, MPZ_ONE, 1-prec, 1)) + ctx.pi = ctx._constant(libmp.mpf_pi) + ctx.e = ctx._constant(libmp.mpf_e) + ctx.ln2 = ctx._constant(libmp.mpf_ln2) + ctx.ln10 = ctx._constant(libmp.mpf_ln10) + ctx.phi = ctx._constant(libmp.mpf_phi) + ctx.euler = ctx._constant(libmp.mpf_euler) + ctx.catalan = ctx._constant(libmp.mpf_catalan) + ctx.glaisher = ctx._constant(libmp.mpf_glaisher) + ctx.khinchin = ctx._constant(libmp.mpf_khinchin) + ctx.twinprime = ctx._constant(libmp.mpf_twinprime) + + def _wrap_mpi_function(ctx, f_real, f_complex=None): + def g(x, **kwargs): + if kwargs: + prec = kwargs.get('prec', ctx._prec[0]) + else: + prec = ctx._prec[0] + x = ctx.convert(x) + if hasattr(x, "_mpi_"): + return ctx.make_mpf(f_real(x._mpi_, prec)) + if hasattr(x, "_mpci_"): + return ctx.make_mpc(f_complex(x._mpci_, prec)) + raise ValueError + return g + + @classmethod + def _wrap_specfun(cls, name, f, wrap): + if wrap: + def f_wrapped(ctx, *args, **kwargs): + convert = ctx.convert + args = [convert(a) for a in args] + prec = ctx.prec + try: + ctx.prec += 10 + retval = f(ctx, *args, **kwargs) + finally: + ctx.prec = prec + return +retval + else: + f_wrapped = f + setattr(cls, name, f_wrapped) + + def _set_prec(ctx, n): + ctx._prec[0] = max(1, int(n)) + ctx._dps = prec_to_dps(n) + + def _set_dps(ctx, n): + ctx._prec[0] = dps_to_prec(n) + ctx._dps = max(1, int(n)) + + prec = property(lambda ctx: ctx._prec[0], _set_prec) + dps = property(lambda ctx: ctx._dps, _set_dps) + + def make_mpf(ctx, v): + a = new(ctx.mpf) + a._mpi_ = v + return a + + def make_mpc(ctx, v): + a = new(ctx.mpc) + a._mpci_ = v + return a + + def _mpq(ctx, pq): + p, q = pq + a = libmp.from_rational(p, q, ctx.prec, round_floor) + b = libmp.from_rational(p, q, ctx.prec, round_ceiling) + return ctx.make_mpf((a, b)) + + def convert(ctx, x): + if isinstance(x, (ctx.mpf, ctx.mpc)): + return x + if isinstance(x, ctx._constant): + return +x + if isinstance(x, complex) or hasattr(x, "_mpc_"): + re = ctx.convert(x.real) + im = ctx.convert(x.imag) + return ctx.mpc(re,im) + if isinstance(x, basestring): + v = mpi_from_str(x, ctx.prec) + return ctx.make_mpf(v) + if hasattr(x, "_mpi_"): + a, b = x._mpi_ + else: + try: + a, b = x + except (TypeError, ValueError): + a = b = x + if hasattr(a, "_mpi_"): + a = a._mpi_[0] + else: + a = convert_mpf_(a, ctx.prec, round_floor) + if hasattr(b, "_mpi_"): + b = b._mpi_[1] + else: + b = convert_mpf_(b, ctx.prec, round_ceiling) + if a == fnan or b == fnan: + a = fninf + b = finf + assert mpf_le(a, b), "endpoints must be properly ordered" + return ctx.make_mpf((a, b)) + + def nstr(ctx, x, n=5, **kwargs): + x = ctx.convert(x) + if hasattr(x, "_mpi_"): + return libmp.mpi_to_str(x._mpi_, n, **kwargs) + if hasattr(x, "_mpci_"): + re = libmp.mpi_to_str(x._mpci_[0], n, **kwargs) + im = libmp.mpi_to_str(x._mpci_[1], n, **kwargs) + return "(%s + %s*j)" % (re, im) + + def mag(ctx, x): + x = ctx.convert(x) + if isinstance(x, ctx.mpc): + return max(ctx.mag(x.real), ctx.mag(x.imag)) + 1 + a, b = libmp.mpi_abs(x._mpi_) + sign, man, exp, bc = b + if man: + return exp+bc + if b == fzero: + return ctx.ninf + if b == fnan: + return ctx.nan + return ctx.inf + + def isnan(ctx, x): + return False + + def isinf(ctx, x): + return x == ctx.inf + + def isint(ctx, x): + x = ctx.convert(x) + a, b = x._mpi_ + if a == b: + sign, man, exp, bc = a + if man: + return exp >= 0 + return a == fzero + return None + + def ldexp(ctx, x, n): + a, b = ctx.convert(x)._mpi_ + a = libmp.mpf_shift(a, n) + b = libmp.mpf_shift(b, n) + return ctx.make_mpf((a,b)) + + def absmin(ctx, x): + return abs(ctx.convert(x)).a + + def absmax(ctx, x): + return abs(ctx.convert(x)).b + + def atan2(ctx, y, x): + y = ctx.convert(y)._mpi_ + x = ctx.convert(x)._mpi_ + return ctx.make_mpf(libmp.mpi_atan2(y,x,ctx.prec)) + + def _convert_param(ctx, x): + if isinstance(x, libmp.int_types): + return x, 'Z' + if isinstance(x, tuple): + p, q = x + return (ctx.mpf(p) / ctx.mpf(q), 'R') + x = ctx.convert(x) + if isinstance(x, ctx.mpf): + return x, 'R' + if isinstance(x, ctx.mpc): + return x, 'C' + raise ValueError + + def _is_real_type(ctx, z): + return isinstance(z, ctx.mpf) or isinstance(z, int_types) + + def _is_complex_type(ctx, z): + return isinstance(z, ctx.mpc) + + def hypsum(ctx, p, q, types, coeffs, z, maxterms=6000, **kwargs): + coeffs = list(coeffs) + num = range(p) + den = range(p,p+q) + #tol = ctx.eps + s = t = ctx.one + k = 0 + while 1: + for i in num: t *= (coeffs[i]+k) + for i in den: t /= (coeffs[i]+k) + k += 1; t /= k; t *= z; s += t + if t == 0: + return s + #if abs(t) < tol: + # return s + if k > maxterms: + raise ctx.NoConvergence + + +# Register with "numbers" ABC +# We do not subclass, hence we do not use the @abstractmethod checks. While +# this is less invasive it may turn out that we do not actually support +# parts of the expected interfaces. See +# http://docs.python.org/2/library/numbers.html for list of abstract +# methods. +try: + import numbers + numbers.Complex.register(ivmpc) + numbers.Real.register(ivmpf) +except ImportError: + pass diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_mp.py b/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_mp.py new file mode 100644 index 0000000000000000000000000000000000000000..93594dd44474a415c74e4b0beb83bd7012666c9d --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/ctx_mp.py @@ -0,0 +1,1339 @@ +""" +This module defines the mpf, mpc classes, and standard functions for +operating with them. +""" +__docformat__ = 'plaintext' + +import functools + +import re + +from .ctx_base import StandardBaseContext + +from .libmp.backend import basestring, BACKEND + +from . import libmp + +from .libmp import (MPZ, MPZ_ZERO, MPZ_ONE, int_types, repr_dps, + round_floor, round_ceiling, dps_to_prec, round_nearest, prec_to_dps, + ComplexResult, to_pickable, from_pickable, normalize, + from_int, from_float, from_str, to_int, to_float, to_str, + from_rational, from_man_exp, + fone, fzero, finf, fninf, fnan, + mpf_abs, mpf_pos, mpf_neg, mpf_add, mpf_sub, mpf_mul, mpf_mul_int, + mpf_div, mpf_rdiv_int, mpf_pow_int, mpf_mod, + mpf_eq, mpf_cmp, mpf_lt, mpf_gt, mpf_le, mpf_ge, + mpf_hash, mpf_rand, + mpf_sum, + bitcount, to_fixed, + mpc_to_str, + mpc_to_complex, mpc_hash, mpc_pos, mpc_is_nonzero, mpc_neg, mpc_conjugate, + mpc_abs, mpc_add, mpc_add_mpf, mpc_sub, mpc_sub_mpf, mpc_mul, mpc_mul_mpf, + mpc_mul_int, mpc_div, mpc_div_mpf, mpc_pow, mpc_pow_mpf, mpc_pow_int, + mpc_mpf_div, + mpf_pow, + mpf_pi, mpf_degree, mpf_e, mpf_phi, mpf_ln2, mpf_ln10, + mpf_euler, mpf_catalan, mpf_apery, mpf_khinchin, + mpf_glaisher, mpf_twinprime, mpf_mertens, + int_types) + +from . import function_docs +from . import rational + +new = object.__new__ + +get_complex = re.compile(r'^\(?(?P[\+\-]?\d*(\.\d*)?(e[\+\-]?\d+)?)??' + r'(?P[\+\-]?\d*(\.\d*)?(e[\+\-]?\d+)?j)?\)?$') + +if BACKEND == 'sage': + from sage.libs.mpmath.ext_main import Context as BaseMPContext + # pickle hack + import sage.libs.mpmath.ext_main as _mpf_module +else: + from .ctx_mp_python import PythonMPContext as BaseMPContext + from . import ctx_mp_python as _mpf_module + +from .ctx_mp_python import _mpf, _mpc, mpnumeric + +class MPContext(BaseMPContext, StandardBaseContext): + """ + Context for multiprecision arithmetic with a global precision. + """ + + def __init__(ctx): + BaseMPContext.__init__(ctx) + ctx.trap_complex = False + ctx.pretty = False + ctx.types = [ctx.mpf, ctx.mpc, ctx.constant] + ctx._mpq = rational.mpq + ctx.default() + StandardBaseContext.__init__(ctx) + + ctx.mpq = rational.mpq + ctx.init_builtins() + + ctx.hyp_summators = {} + + ctx._init_aliases() + + # XXX: automate + try: + ctx.bernoulli.im_func.func_doc = function_docs.bernoulli + ctx.primepi.im_func.func_doc = function_docs.primepi + ctx.psi.im_func.func_doc = function_docs.psi + ctx.atan2.im_func.func_doc = function_docs.atan2 + except AttributeError: + # python 3 + ctx.bernoulli.__func__.func_doc = function_docs.bernoulli + ctx.primepi.__func__.func_doc = function_docs.primepi + ctx.psi.__func__.func_doc = function_docs.psi + ctx.atan2.__func__.func_doc = function_docs.atan2 + + ctx.digamma.func_doc = function_docs.digamma + ctx.cospi.func_doc = function_docs.cospi + ctx.sinpi.func_doc = function_docs.sinpi + + def init_builtins(ctx): + + mpf = ctx.mpf + mpc = ctx.mpc + + # Exact constants + ctx.one = ctx.make_mpf(fone) + ctx.zero = ctx.make_mpf(fzero) + ctx.j = ctx.make_mpc((fzero,fone)) + ctx.inf = ctx.make_mpf(finf) + ctx.ninf = ctx.make_mpf(fninf) + ctx.nan = ctx.make_mpf(fnan) + + eps = ctx.constant(lambda prec, rnd: (0, MPZ_ONE, 1-prec, 1), + "epsilon of working precision", "eps") + ctx.eps = eps + + # Approximate constants + ctx.pi = ctx.constant(mpf_pi, "pi", "pi") + ctx.ln2 = ctx.constant(mpf_ln2, "ln(2)", "ln2") + ctx.ln10 = ctx.constant(mpf_ln10, "ln(10)", "ln10") + ctx.phi = ctx.constant(mpf_phi, "Golden ratio phi", "phi") + ctx.e = ctx.constant(mpf_e, "e = exp(1)", "e") + ctx.euler = ctx.constant(mpf_euler, "Euler's constant", "euler") + ctx.catalan = ctx.constant(mpf_catalan, "Catalan's constant", "catalan") + ctx.khinchin = ctx.constant(mpf_khinchin, "Khinchin's constant", "khinchin") + ctx.glaisher = ctx.constant(mpf_glaisher, "Glaisher's constant", "glaisher") + ctx.apery = ctx.constant(mpf_apery, "Apery's constant", "apery") + ctx.degree = ctx.constant(mpf_degree, "1 deg = pi / 180", "degree") + ctx.twinprime = ctx.constant(mpf_twinprime, "Twin prime constant", "twinprime") + ctx.mertens = ctx.constant(mpf_mertens, "Mertens' constant", "mertens") + + # Standard functions + ctx.sqrt = ctx._wrap_libmp_function(libmp.mpf_sqrt, libmp.mpc_sqrt) + ctx.cbrt = ctx._wrap_libmp_function(libmp.mpf_cbrt, libmp.mpc_cbrt) + ctx.ln = ctx._wrap_libmp_function(libmp.mpf_log, libmp.mpc_log) + ctx.atan = ctx._wrap_libmp_function(libmp.mpf_atan, libmp.mpc_atan) + ctx.exp = ctx._wrap_libmp_function(libmp.mpf_exp, libmp.mpc_exp) + ctx.expj = ctx._wrap_libmp_function(libmp.mpf_expj, libmp.mpc_expj) + ctx.expjpi = ctx._wrap_libmp_function(libmp.mpf_expjpi, libmp.mpc_expjpi) + ctx.sin = ctx._wrap_libmp_function(libmp.mpf_sin, libmp.mpc_sin) + ctx.cos = ctx._wrap_libmp_function(libmp.mpf_cos, libmp.mpc_cos) + ctx.tan = ctx._wrap_libmp_function(libmp.mpf_tan, libmp.mpc_tan) + ctx.sinh = ctx._wrap_libmp_function(libmp.mpf_sinh, libmp.mpc_sinh) + ctx.cosh = ctx._wrap_libmp_function(libmp.mpf_cosh, libmp.mpc_cosh) + ctx.tanh = ctx._wrap_libmp_function(libmp.mpf_tanh, libmp.mpc_tanh) + ctx.asin = ctx._wrap_libmp_function(libmp.mpf_asin, libmp.mpc_asin) + ctx.acos = ctx._wrap_libmp_function(libmp.mpf_acos, libmp.mpc_acos) + ctx.atan = ctx._wrap_libmp_function(libmp.mpf_atan, libmp.mpc_atan) + ctx.asinh = ctx._wrap_libmp_function(libmp.mpf_asinh, libmp.mpc_asinh) + ctx.acosh = ctx._wrap_libmp_function(libmp.mpf_acosh, libmp.mpc_acosh) + ctx.atanh = ctx._wrap_libmp_function(libmp.mpf_atanh, libmp.mpc_atanh) + ctx.sinpi = ctx._wrap_libmp_function(libmp.mpf_sin_pi, libmp.mpc_sin_pi) + ctx.cospi = ctx._wrap_libmp_function(libmp.mpf_cos_pi, libmp.mpc_cos_pi) + ctx.floor = ctx._wrap_libmp_function(libmp.mpf_floor, libmp.mpc_floor) + ctx.ceil = ctx._wrap_libmp_function(libmp.mpf_ceil, libmp.mpc_ceil) + ctx.nint = ctx._wrap_libmp_function(libmp.mpf_nint, libmp.mpc_nint) + ctx.frac = ctx._wrap_libmp_function(libmp.mpf_frac, libmp.mpc_frac) + ctx.fib = ctx.fibonacci = ctx._wrap_libmp_function(libmp.mpf_fibonacci, libmp.mpc_fibonacci) + + ctx.gamma = ctx._wrap_libmp_function(libmp.mpf_gamma, libmp.mpc_gamma) + ctx.rgamma = ctx._wrap_libmp_function(libmp.mpf_rgamma, libmp.mpc_rgamma) + ctx.loggamma = ctx._wrap_libmp_function(libmp.mpf_loggamma, libmp.mpc_loggamma) + ctx.fac = ctx.factorial = ctx._wrap_libmp_function(libmp.mpf_factorial, libmp.mpc_factorial) + + ctx.digamma = ctx._wrap_libmp_function(libmp.mpf_psi0, libmp.mpc_psi0) + ctx.harmonic = ctx._wrap_libmp_function(libmp.mpf_harmonic, libmp.mpc_harmonic) + ctx.ei = ctx._wrap_libmp_function(libmp.mpf_ei, libmp.mpc_ei) + ctx.e1 = ctx._wrap_libmp_function(libmp.mpf_e1, libmp.mpc_e1) + ctx._ci = ctx._wrap_libmp_function(libmp.mpf_ci, libmp.mpc_ci) + ctx._si = ctx._wrap_libmp_function(libmp.mpf_si, libmp.mpc_si) + ctx.ellipk = ctx._wrap_libmp_function(libmp.mpf_ellipk, libmp.mpc_ellipk) + ctx._ellipe = ctx._wrap_libmp_function(libmp.mpf_ellipe, libmp.mpc_ellipe) + ctx.agm1 = ctx._wrap_libmp_function(libmp.mpf_agm1, libmp.mpc_agm1) + ctx._erf = ctx._wrap_libmp_function(libmp.mpf_erf, None) + ctx._erfc = ctx._wrap_libmp_function(libmp.mpf_erfc, None) + ctx._zeta = ctx._wrap_libmp_function(libmp.mpf_zeta, libmp.mpc_zeta) + ctx._altzeta = ctx._wrap_libmp_function(libmp.mpf_altzeta, libmp.mpc_altzeta) + + # Faster versions + ctx.sqrt = getattr(ctx, "_sage_sqrt", ctx.sqrt) + ctx.exp = getattr(ctx, "_sage_exp", ctx.exp) + ctx.ln = getattr(ctx, "_sage_ln", ctx.ln) + ctx.cos = getattr(ctx, "_sage_cos", ctx.cos) + ctx.sin = getattr(ctx, "_sage_sin", ctx.sin) + + def to_fixed(ctx, x, prec): + return x.to_fixed(prec) + + def hypot(ctx, x, y): + r""" + Computes the Euclidean norm of the vector `(x, y)`, equal + to `\sqrt{x^2 + y^2}`. Both `x` and `y` must be real.""" + x = ctx.convert(x) + y = ctx.convert(y) + return ctx.make_mpf(libmp.mpf_hypot(x._mpf_, y._mpf_, *ctx._prec_rounding)) + + def _gamma_upper_int(ctx, n, z): + n = int(ctx._re(n)) + if n == 0: + return ctx.e1(z) + if not hasattr(z, '_mpf_'): + raise NotImplementedError + prec, rounding = ctx._prec_rounding + real, imag = libmp.mpf_expint(n, z._mpf_, prec, rounding, gamma=True) + if imag is None: + return ctx.make_mpf(real) + else: + return ctx.make_mpc((real, imag)) + + def _expint_int(ctx, n, z): + n = int(n) + if n == 1: + return ctx.e1(z) + if not hasattr(z, '_mpf_'): + raise NotImplementedError + prec, rounding = ctx._prec_rounding + real, imag = libmp.mpf_expint(n, z._mpf_, prec, rounding) + if imag is None: + return ctx.make_mpf(real) + else: + return ctx.make_mpc((real, imag)) + + def _nthroot(ctx, x, n): + if hasattr(x, '_mpf_'): + try: + return ctx.make_mpf(libmp.mpf_nthroot(x._mpf_, n, *ctx._prec_rounding)) + except ComplexResult: + if ctx.trap_complex: + raise + x = (x._mpf_, libmp.fzero) + else: + x = x._mpc_ + return ctx.make_mpc(libmp.mpc_nthroot(x, n, *ctx._prec_rounding)) + + def _besselj(ctx, n, z): + prec, rounding = ctx._prec_rounding + if hasattr(z, '_mpf_'): + return ctx.make_mpf(libmp.mpf_besseljn(n, z._mpf_, prec, rounding)) + elif hasattr(z, '_mpc_'): + return ctx.make_mpc(libmp.mpc_besseljn(n, z._mpc_, prec, rounding)) + + def _agm(ctx, a, b=1): + prec, rounding = ctx._prec_rounding + if hasattr(a, '_mpf_') and hasattr(b, '_mpf_'): + try: + v = libmp.mpf_agm(a._mpf_, b._mpf_, prec, rounding) + return ctx.make_mpf(v) + except ComplexResult: + pass + if hasattr(a, '_mpf_'): a = (a._mpf_, libmp.fzero) + else: a = a._mpc_ + if hasattr(b, '_mpf_'): b = (b._mpf_, libmp.fzero) + else: b = b._mpc_ + return ctx.make_mpc(libmp.mpc_agm(a, b, prec, rounding)) + + def bernoulli(ctx, n): + return ctx.make_mpf(libmp.mpf_bernoulli(int(n), *ctx._prec_rounding)) + + def _zeta_int(ctx, n): + return ctx.make_mpf(libmp.mpf_zeta_int(int(n), *ctx._prec_rounding)) + + def atan2(ctx, y, x): + x = ctx.convert(x) + y = ctx.convert(y) + return ctx.make_mpf(libmp.mpf_atan2(y._mpf_, x._mpf_, *ctx._prec_rounding)) + + def psi(ctx, m, z): + z = ctx.convert(z) + m = int(m) + if ctx._is_real_type(z): + return ctx.make_mpf(libmp.mpf_psi(m, z._mpf_, *ctx._prec_rounding)) + else: + return ctx.make_mpc(libmp.mpc_psi(m, z._mpc_, *ctx._prec_rounding)) + + def cos_sin(ctx, x, **kwargs): + if type(x) not in ctx.types: + x = ctx.convert(x) + prec, rounding = ctx._parse_prec(kwargs) + if hasattr(x, '_mpf_'): + c, s = libmp.mpf_cos_sin(x._mpf_, prec, rounding) + return ctx.make_mpf(c), ctx.make_mpf(s) + elif hasattr(x, '_mpc_'): + c, s = libmp.mpc_cos_sin(x._mpc_, prec, rounding) + return ctx.make_mpc(c), ctx.make_mpc(s) + else: + return ctx.cos(x, **kwargs), ctx.sin(x, **kwargs) + + def cospi_sinpi(ctx, x, **kwargs): + if type(x) not in ctx.types: + x = ctx.convert(x) + prec, rounding = ctx._parse_prec(kwargs) + if hasattr(x, '_mpf_'): + c, s = libmp.mpf_cos_sin_pi(x._mpf_, prec, rounding) + return ctx.make_mpf(c), ctx.make_mpf(s) + elif hasattr(x, '_mpc_'): + c, s = libmp.mpc_cos_sin_pi(x._mpc_, prec, rounding) + return ctx.make_mpc(c), ctx.make_mpc(s) + else: + return ctx.cos(x, **kwargs), ctx.sin(x, **kwargs) + + def clone(ctx): + """ + Create a copy of the context, with the same working precision. + """ + a = ctx.__class__() + a.prec = ctx.prec + return a + + # Several helper methods + # TODO: add more of these, make consistent, write docstrings, ... + + def _is_real_type(ctx, x): + if hasattr(x, '_mpc_') or type(x) is complex: + return False + return True + + def _is_complex_type(ctx, x): + if hasattr(x, '_mpc_') or type(x) is complex: + return True + return False + + def isnan(ctx, x): + """ + Return *True* if *x* is a NaN (not-a-number), or for a complex + number, whether either the real or complex part is NaN; + otherwise return *False*:: + + >>> from mpmath import * + >>> isnan(3.14) + False + >>> isnan(nan) + True + >>> isnan(mpc(3.14,2.72)) + False + >>> isnan(mpc(3.14,nan)) + True + + """ + if hasattr(x, "_mpf_"): + return x._mpf_ == fnan + if hasattr(x, "_mpc_"): + return fnan in x._mpc_ + if isinstance(x, int_types) or isinstance(x, rational.mpq): + return False + x = ctx.convert(x) + if hasattr(x, '_mpf_') or hasattr(x, '_mpc_'): + return ctx.isnan(x) + raise TypeError("isnan() needs a number as input") + + def isfinite(ctx, x): + """ + Return *True* if *x* is a finite number, i.e. neither + an infinity or a NaN. + + >>> from mpmath import * + >>> isfinite(inf) + False + >>> isfinite(-inf) + False + >>> isfinite(3) + True + >>> isfinite(nan) + False + >>> isfinite(3+4j) + True + >>> isfinite(mpc(3,inf)) + False + >>> isfinite(mpc(nan,3)) + False + + """ + if ctx.isinf(x) or ctx.isnan(x): + return False + return True + + def isnpint(ctx, x): + """ + Determine if *x* is a nonpositive integer. + """ + if not x: + return True + if hasattr(x, '_mpf_'): + sign, man, exp, bc = x._mpf_ + return sign and exp >= 0 + if hasattr(x, '_mpc_'): + return not x.imag and ctx.isnpint(x.real) + if type(x) in int_types: + return x <= 0 + if isinstance(x, ctx.mpq): + p, q = x._mpq_ + if not p: + return True + return q == 1 and p <= 0 + return ctx.isnpint(ctx.convert(x)) + + def __str__(ctx): + lines = ["Mpmath settings:", + (" mp.prec = %s" % ctx.prec).ljust(30) + "[default: 53]", + (" mp.dps = %s" % ctx.dps).ljust(30) + "[default: 15]", + (" mp.trap_complex = %s" % ctx.trap_complex).ljust(30) + "[default: False]", + ] + return "\n".join(lines) + + @property + def _repr_digits(ctx): + return repr_dps(ctx._prec) + + @property + def _str_digits(ctx): + return ctx._dps + + def extraprec(ctx, n, normalize_output=False): + """ + The block + + with extraprec(n): + + + increases the precision n bits, executes , and then + restores the precision. + + extraprec(n)(f) returns a decorated version of the function f + that increases the working precision by n bits before execution, + and restores the parent precision afterwards. With + normalize_output=True, it rounds the return value to the parent + precision. + """ + return PrecisionManager(ctx, lambda p: p + n, None, normalize_output) + + def extradps(ctx, n, normalize_output=False): + """ + This function is analogous to extraprec (see documentation) + but changes the decimal precision instead of the number of bits. + """ + return PrecisionManager(ctx, None, lambda d: d + n, normalize_output) + + def workprec(ctx, n, normalize_output=False): + """ + The block + + with workprec(n): + + + sets the precision to n bits, executes , and then restores + the precision. + + workprec(n)(f) returns a decorated version of the function f + that sets the precision to n bits before execution, + and restores the precision afterwards. With normalize_output=True, + it rounds the return value to the parent precision. + """ + return PrecisionManager(ctx, lambda p: n, None, normalize_output) + + def workdps(ctx, n, normalize_output=False): + """ + This function is analogous to workprec (see documentation) + but changes the decimal precision instead of the number of bits. + """ + return PrecisionManager(ctx, None, lambda d: n, normalize_output) + + def autoprec(ctx, f, maxprec=None, catch=(), verbose=False): + r""" + Return a wrapped copy of *f* that repeatedly evaluates *f* + with increasing precision until the result converges to the + full precision used at the point of the call. + + This heuristically protects against rounding errors, at the cost of + roughly a 2x slowdown compared to manually setting the optimal + precision. This method can, however, easily be fooled if the results + from *f* depend "discontinuously" on the precision, for instance + if catastrophic cancellation can occur. Therefore, :func:`~mpmath.autoprec` + should be used judiciously. + + **Examples** + + Many functions are sensitive to perturbations of the input arguments. + If the arguments are decimal numbers, they may have to be converted + to binary at a much higher precision. If the amount of required + extra precision is unknown, :func:`~mpmath.autoprec` is convenient:: + + >>> from mpmath import * + >>> mp.dps = 15 + >>> mp.pretty = True + >>> besselj(5, 125 * 10**28) # Exact input + -8.03284785591801e-17 + >>> besselj(5, '1.25e30') # Bad + 7.12954868316652e-16 + >>> autoprec(besselj)(5, '1.25e30') # Good + -8.03284785591801e-17 + + The following fails to converge because `\sin(\pi) = 0` whereas all + finite-precision approximations of `\pi` give nonzero values:: + + >>> autoprec(sin)(pi) # doctest: +IGNORE_EXCEPTION_DETAIL + Traceback (most recent call last): + ... + NoConvergence: autoprec: prec increased to 2910 without convergence + + As the following example shows, :func:`~mpmath.autoprec` can protect against + cancellation, but is fooled by too severe cancellation:: + + >>> x = 1e-10 + >>> exp(x)-1; expm1(x); autoprec(lambda t: exp(t)-1)(x) + 1.00000008274037e-10 + 1.00000000005e-10 + 1.00000000005e-10 + >>> x = 1e-50 + >>> exp(x)-1; expm1(x); autoprec(lambda t: exp(t)-1)(x) + 0.0 + 1.0e-50 + 0.0 + + With *catch*, an exception or list of exceptions to intercept + may be specified. The raised exception is interpreted + as signaling insufficient precision. This permits, for example, + evaluating a function where a too low precision results in a + division by zero:: + + >>> f = lambda x: 1/(exp(x)-1) + >>> f(1e-30) + Traceback (most recent call last): + ... + ZeroDivisionError + >>> autoprec(f, catch=ZeroDivisionError)(1e-30) + 1.0e+30 + + + """ + def f_autoprec_wrapped(*args, **kwargs): + prec = ctx.prec + if maxprec is None: + maxprec2 = ctx._default_hyper_maxprec(prec) + else: + maxprec2 = maxprec + try: + ctx.prec = prec + 10 + try: + v1 = f(*args, **kwargs) + except catch: + v1 = ctx.nan + prec2 = prec + 20 + while 1: + ctx.prec = prec2 + try: + v2 = f(*args, **kwargs) + except catch: + v2 = ctx.nan + if v1 == v2: + break + err = ctx.mag(v2-v1) - ctx.mag(v2) + if err < (-prec): + break + if verbose: + print("autoprec: target=%s, prec=%s, accuracy=%s" \ + % (prec, prec2, -err)) + v1 = v2 + if prec2 >= maxprec2: + raise ctx.NoConvergence(\ + "autoprec: prec increased to %i without convergence"\ + % prec2) + prec2 += int(prec2*2) + prec2 = min(prec2, maxprec2) + finally: + ctx.prec = prec + return +v2 + return f_autoprec_wrapped + + def nstr(ctx, x, n=6, **kwargs): + """ + Convert an ``mpf`` or ``mpc`` to a decimal string literal with *n* + significant digits. The small default value for *n* is chosen to + make this function useful for printing collections of numbers + (lists, matrices, etc). + + If *x* is a list or tuple, :func:`~mpmath.nstr` is applied recursively + to each element. For unrecognized classes, :func:`~mpmath.nstr` + simply returns ``str(x)``. + + The companion function :func:`~mpmath.nprint` prints the result + instead of returning it. + + The keyword arguments *strip_zeros*, *min_fixed*, *max_fixed* + and *show_zero_exponent* are forwarded to :func:`~mpmath.libmp.to_str`. + + The number will be printed in fixed-point format if the position + of the leading digit is strictly between min_fixed + (default = min(-dps/3,-5)) and max_fixed (default = dps). + + To force fixed-point format always, set min_fixed = -inf, + max_fixed = +inf. To force floating-point format, set + min_fixed >= max_fixed. + + >>> from mpmath import * + >>> nstr([+pi, ldexp(1,-500)]) + '[3.14159, 3.05494e-151]' + >>> nprint([+pi, ldexp(1,-500)]) + [3.14159, 3.05494e-151] + >>> nstr(mpf("5e-10"), 5) + '5.0e-10' + >>> nstr(mpf("5e-10"), 5, strip_zeros=False) + '5.0000e-10' + >>> nstr(mpf("5e-10"), 5, strip_zeros=False, min_fixed=-11) + '0.00000000050000' + >>> nstr(mpf(0), 5, show_zero_exponent=True) + '0.0e+0' + + """ + if isinstance(x, list): + return "[%s]" % (", ".join(ctx.nstr(c, n, **kwargs) for c in x)) + if isinstance(x, tuple): + return "(%s)" % (", ".join(ctx.nstr(c, n, **kwargs) for c in x)) + if hasattr(x, '_mpf_'): + return to_str(x._mpf_, n, **kwargs) + if hasattr(x, '_mpc_'): + return "(" + mpc_to_str(x._mpc_, n, **kwargs) + ")" + if isinstance(x, basestring): + return repr(x) + if isinstance(x, ctx.matrix): + return x.__nstr__(n, **kwargs) + return str(x) + + def _convert_fallback(ctx, x, strings): + if strings and isinstance(x, basestring): + if 'j' in x.lower(): + x = x.lower().replace(' ', '') + match = get_complex.match(x) + re = match.group('re') + if not re: + re = 0 + im = match.group('im').rstrip('j') + return ctx.mpc(ctx.convert(re), ctx.convert(im)) + if hasattr(x, "_mpi_"): + a, b = x._mpi_ + if a == b: + return ctx.make_mpf(a) + else: + raise ValueError("can only create mpf from zero-width interval") + raise TypeError("cannot create mpf from " + repr(x)) + + def mpmathify(ctx, *args, **kwargs): + return ctx.convert(*args, **kwargs) + + def _parse_prec(ctx, kwargs): + if kwargs: + if kwargs.get('exact'): + return 0, 'f' + prec, rounding = ctx._prec_rounding + if 'rounding' in kwargs: + rounding = kwargs['rounding'] + if 'prec' in kwargs: + prec = kwargs['prec'] + if prec == ctx.inf: + return 0, 'f' + else: + prec = int(prec) + elif 'dps' in kwargs: + dps = kwargs['dps'] + if dps == ctx.inf: + return 0, 'f' + prec = dps_to_prec(dps) + return prec, rounding + return ctx._prec_rounding + + _exact_overflow_msg = "the exact result does not fit in memory" + + _hypsum_msg = """hypsum() failed to converge to the requested %i bits of accuracy +using a working precision of %i bits. Try with a higher maxprec, +maxterms, or set zeroprec.""" + + def hypsum(ctx, p, q, flags, coeffs, z, accurate_small=True, **kwargs): + if hasattr(z, "_mpf_"): + key = p, q, flags, 'R' + v = z._mpf_ + elif hasattr(z, "_mpc_"): + key = p, q, flags, 'C' + v = z._mpc_ + if key not in ctx.hyp_summators: + ctx.hyp_summators[key] = libmp.make_hyp_summator(key)[1] + summator = ctx.hyp_summators[key] + prec = ctx.prec + maxprec = kwargs.get('maxprec', ctx._default_hyper_maxprec(prec)) + extraprec = 50 + epsshift = 25 + # Jumps in magnitude occur when parameters are close to negative + # integers. We must ensure that these terms are included in + # the sum and added accurately + magnitude_check = {} + max_total_jump = 0 + for i, c in enumerate(coeffs): + if flags[i] == 'Z': + if i >= p and c <= 0: + ok = False + for ii, cc in enumerate(coeffs[:p]): + # Note: c <= cc or c < cc, depending on convention + if flags[ii] == 'Z' and cc <= 0 and c <= cc: + ok = True + if not ok: + raise ZeroDivisionError("pole in hypergeometric series") + continue + n, d = ctx.nint_distance(c) + n = -int(n) + d = -d + if i >= p and n >= 0 and d > 4: + if n in magnitude_check: + magnitude_check[n] += d + else: + magnitude_check[n] = d + extraprec = max(extraprec, d - prec + 60) + max_total_jump += abs(d) + while 1: + if extraprec > maxprec: + raise ValueError(ctx._hypsum_msg % (prec, prec+extraprec)) + wp = prec + extraprec + if magnitude_check: + mag_dict = dict((n,None) for n in magnitude_check) + else: + mag_dict = {} + zv, have_complex, magnitude = summator(coeffs, v, prec, wp, \ + epsshift, mag_dict, **kwargs) + cancel = -magnitude + jumps_resolved = True + if extraprec < max_total_jump: + for n in mag_dict.values(): + if (n is None) or (n < prec): + jumps_resolved = False + break + accurate = (cancel < extraprec-25-5 or not accurate_small) + if jumps_resolved: + if accurate: + break + # zero? + zeroprec = kwargs.get('zeroprec') + if zeroprec is not None: + if cancel > zeroprec: + if have_complex: + return ctx.mpc(0) + else: + return ctx.zero + + # Some near-singularities were not included, so increase + # precision and repeat until they are + extraprec *= 2 + # Possible workaround for bad roundoff in fixed-point arithmetic + epsshift += 5 + extraprec += 5 + + if type(zv) is tuple: + if have_complex: + return ctx.make_mpc(zv) + else: + return ctx.make_mpf(zv) + else: + return zv + + def ldexp(ctx, x, n): + r""" + Computes `x 2^n` efficiently. No rounding is performed. + The argument `x` must be a real floating-point number (or + possible to convert into one) and `n` must be a Python ``int``. + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> ldexp(1, 10) + mpf('1024.0') + >>> ldexp(1, -3) + mpf('0.125') + + """ + x = ctx.convert(x) + return ctx.make_mpf(libmp.mpf_shift(x._mpf_, n)) + + def frexp(ctx, x): + r""" + Given a real number `x`, returns `(y, n)` with `y \in [0.5, 1)`, + `n` a Python integer, and such that `x = y 2^n`. No rounding is + performed. + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> frexp(7.5) + (mpf('0.9375'), 3) + + """ + x = ctx.convert(x) + y, n = libmp.mpf_frexp(x._mpf_) + return ctx.make_mpf(y), n + + def fneg(ctx, x, **kwargs): + """ + Negates the number *x*, giving a floating-point result, optionally + using a custom precision and rounding mode. + + See the documentation of :func:`~mpmath.fadd` for a detailed description + of how to specify precision and rounding. + + **Examples** + + An mpmath number is returned:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fneg(2.5) + mpf('-2.5') + >>> fneg(-5+2j) + mpc(real='5.0', imag='-2.0') + + Precise control over rounding is possible:: + + >>> x = fadd(2, 1e-100, exact=True) + >>> fneg(x) + mpf('-2.0') + >>> fneg(x, rounding='f') + mpf('-2.0000000000000004') + + Negating with and without roundoff:: + + >>> n = 200000000000000000000001 + >>> print(int(-mpf(n))) + -200000000000000016777216 + >>> print(int(fneg(n))) + -200000000000000016777216 + >>> print(int(fneg(n, prec=log(n,2)+1))) + -200000000000000000000001 + >>> print(int(fneg(n, dps=log(n,10)+1))) + -200000000000000000000001 + >>> print(int(fneg(n, prec=inf))) + -200000000000000000000001 + >>> print(int(fneg(n, dps=inf))) + -200000000000000000000001 + >>> print(int(fneg(n, exact=True))) + -200000000000000000000001 + + """ + prec, rounding = ctx._parse_prec(kwargs) + x = ctx.convert(x) + if hasattr(x, '_mpf_'): + return ctx.make_mpf(mpf_neg(x._mpf_, prec, rounding)) + if hasattr(x, '_mpc_'): + return ctx.make_mpc(mpc_neg(x._mpc_, prec, rounding)) + raise ValueError("Arguments need to be mpf or mpc compatible numbers") + + def fadd(ctx, x, y, **kwargs): + """ + Adds the numbers *x* and *y*, giving a floating-point result, + optionally using a custom precision and rounding mode. + + The default precision is the working precision of the context. + You can specify a custom precision in bits by passing the *prec* keyword + argument, or by providing an equivalent decimal precision with the *dps* + keyword argument. If the precision is set to ``+inf``, or if the flag + *exact=True* is passed, an exact addition with no rounding is performed. + + When the precision is finite, the optional *rounding* keyword argument + specifies the direction of rounding. Valid options are ``'n'`` for + nearest (default), ``'f'`` for floor, ``'c'`` for ceiling, ``'d'`` + for down, ``'u'`` for up. + + **Examples** + + Using :func:`~mpmath.fadd` with precision and rounding control:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fadd(2, 1e-20) + mpf('2.0') + >>> fadd(2, 1e-20, rounding='u') + mpf('2.0000000000000004') + >>> nprint(fadd(2, 1e-20, prec=100), 25) + 2.00000000000000000001 + >>> nprint(fadd(2, 1e-20, dps=15), 25) + 2.0 + >>> nprint(fadd(2, 1e-20, dps=25), 25) + 2.00000000000000000001 + >>> nprint(fadd(2, 1e-20, exact=True), 25) + 2.00000000000000000001 + + Exact addition avoids cancellation errors, enforcing familiar laws + of numbers such as `x+y-x = y`, which don't hold in floating-point + arithmetic with finite precision:: + + >>> x, y = mpf(2), mpf('1e-1000') + >>> print(x + y - x) + 0.0 + >>> print(fadd(x, y, prec=inf) - x) + 1.0e-1000 + >>> print(fadd(x, y, exact=True) - x) + 1.0e-1000 + + Exact addition can be inefficient and may be impossible to perform + with large magnitude differences:: + + >>> fadd(1, '1e-100000000000000000000', prec=inf) + Traceback (most recent call last): + ... + OverflowError: the exact result does not fit in memory + + """ + prec, rounding = ctx._parse_prec(kwargs) + x = ctx.convert(x) + y = ctx.convert(y) + try: + if hasattr(x, '_mpf_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpf(mpf_add(x._mpf_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_add_mpf(y._mpc_, x._mpf_, prec, rounding)) + if hasattr(x, '_mpc_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpc(mpc_add_mpf(x._mpc_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_add(x._mpc_, y._mpc_, prec, rounding)) + except (ValueError, OverflowError): + raise OverflowError(ctx._exact_overflow_msg) + raise ValueError("Arguments need to be mpf or mpc compatible numbers") + + def fsub(ctx, x, y, **kwargs): + """ + Subtracts the numbers *x* and *y*, giving a floating-point result, + optionally using a custom precision and rounding mode. + + See the documentation of :func:`~mpmath.fadd` for a detailed description + of how to specify precision and rounding. + + **Examples** + + Using :func:`~mpmath.fsub` with precision and rounding control:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fsub(2, 1e-20) + mpf('2.0') + >>> fsub(2, 1e-20, rounding='d') + mpf('1.9999999999999998') + >>> nprint(fsub(2, 1e-20, prec=100), 25) + 1.99999999999999999999 + >>> nprint(fsub(2, 1e-20, dps=15), 25) + 2.0 + >>> nprint(fsub(2, 1e-20, dps=25), 25) + 1.99999999999999999999 + >>> nprint(fsub(2, 1e-20, exact=True), 25) + 1.99999999999999999999 + + Exact subtraction avoids cancellation errors, enforcing familiar laws + of numbers such as `x-y+y = x`, which don't hold in floating-point + arithmetic with finite precision:: + + >>> x, y = mpf(2), mpf('1e1000') + >>> print(x - y + y) + 0.0 + >>> print(fsub(x, y, prec=inf) + y) + 2.0 + >>> print(fsub(x, y, exact=True) + y) + 2.0 + + Exact addition can be inefficient and may be impossible to perform + with large magnitude differences:: + + >>> fsub(1, '1e-100000000000000000000', prec=inf) + Traceback (most recent call last): + ... + OverflowError: the exact result does not fit in memory + + """ + prec, rounding = ctx._parse_prec(kwargs) + x = ctx.convert(x) + y = ctx.convert(y) + try: + if hasattr(x, '_mpf_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpf(mpf_sub(x._mpf_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_sub((x._mpf_, fzero), y._mpc_, prec, rounding)) + if hasattr(x, '_mpc_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpc(mpc_sub_mpf(x._mpc_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_sub(x._mpc_, y._mpc_, prec, rounding)) + except (ValueError, OverflowError): + raise OverflowError(ctx._exact_overflow_msg) + raise ValueError("Arguments need to be mpf or mpc compatible numbers") + + def fmul(ctx, x, y, **kwargs): + """ + Multiplies the numbers *x* and *y*, giving a floating-point result, + optionally using a custom precision and rounding mode. + + See the documentation of :func:`~mpmath.fadd` for a detailed description + of how to specify precision and rounding. + + **Examples** + + The result is an mpmath number:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fmul(2, 5.0) + mpf('10.0') + >>> fmul(0.5j, 0.5) + mpc(real='0.0', imag='0.25') + + Avoiding roundoff:: + + >>> x, y = 10**10+1, 10**15+1 + >>> print(x*y) + 10000000001000010000000001 + >>> print(mpf(x) * mpf(y)) + 1.0000000001e+25 + >>> print(int(mpf(x) * mpf(y))) + 10000000001000011026399232 + >>> print(int(fmul(x, y))) + 10000000001000011026399232 + >>> print(int(fmul(x, y, dps=25))) + 10000000001000010000000001 + >>> print(int(fmul(x, y, exact=True))) + 10000000001000010000000001 + + Exact multiplication with complex numbers can be inefficient and may + be impossible to perform with large magnitude differences between + real and imaginary parts:: + + >>> x = 1+2j + >>> y = mpc(2, '1e-100000000000000000000') + >>> fmul(x, y) + mpc(real='2.0', imag='4.0') + >>> fmul(x, y, rounding='u') + mpc(real='2.0', imag='4.0000000000000009') + >>> fmul(x, y, exact=True) + Traceback (most recent call last): + ... + OverflowError: the exact result does not fit in memory + + """ + prec, rounding = ctx._parse_prec(kwargs) + x = ctx.convert(x) + y = ctx.convert(y) + try: + if hasattr(x, '_mpf_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpf(mpf_mul(x._mpf_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_mul_mpf(y._mpc_, x._mpf_, prec, rounding)) + if hasattr(x, '_mpc_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpc(mpc_mul_mpf(x._mpc_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_mul(x._mpc_, y._mpc_, prec, rounding)) + except (ValueError, OverflowError): + raise OverflowError(ctx._exact_overflow_msg) + raise ValueError("Arguments need to be mpf or mpc compatible numbers") + + def fdiv(ctx, x, y, **kwargs): + """ + Divides the numbers *x* and *y*, giving a floating-point result, + optionally using a custom precision and rounding mode. + + See the documentation of :func:`~mpmath.fadd` for a detailed description + of how to specify precision and rounding. + + **Examples** + + The result is an mpmath number:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fdiv(3, 2) + mpf('1.5') + >>> fdiv(2, 3) + mpf('0.66666666666666663') + >>> fdiv(2+4j, 0.5) + mpc(real='4.0', imag='8.0') + + The rounding direction and precision can be controlled:: + + >>> fdiv(2, 3, dps=3) # Should be accurate to at least 3 digits + mpf('0.6666259765625') + >>> fdiv(2, 3, rounding='d') + mpf('0.66666666666666663') + >>> fdiv(2, 3, prec=60) + mpf('0.66666666666666667') + >>> fdiv(2, 3, rounding='u') + mpf('0.66666666666666674') + + Checking the error of a division by performing it at higher precision:: + + >>> fdiv(2, 3) - fdiv(2, 3, prec=100) + mpf('-3.7007434154172148e-17') + + Unlike :func:`~mpmath.fadd`, :func:`~mpmath.fmul`, etc., exact division is not + allowed since the quotient of two floating-point numbers generally + does not have an exact floating-point representation. (In the + future this might be changed to allow the case where the division + is actually exact.) + + >>> fdiv(2, 3, exact=True) + Traceback (most recent call last): + ... + ValueError: division is not an exact operation + + """ + prec, rounding = ctx._parse_prec(kwargs) + if not prec: + raise ValueError("division is not an exact operation") + x = ctx.convert(x) + y = ctx.convert(y) + if hasattr(x, '_mpf_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpf(mpf_div(x._mpf_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_div((x._mpf_, fzero), y._mpc_, prec, rounding)) + if hasattr(x, '_mpc_'): + if hasattr(y, '_mpf_'): + return ctx.make_mpc(mpc_div_mpf(x._mpc_, y._mpf_, prec, rounding)) + if hasattr(y, '_mpc_'): + return ctx.make_mpc(mpc_div(x._mpc_, y._mpc_, prec, rounding)) + raise ValueError("Arguments need to be mpf or mpc compatible numbers") + + def nint_distance(ctx, x): + r""" + Return `(n,d)` where `n` is the nearest integer to `x` and `d` is + an estimate of `\log_2(|x-n|)`. If `d < 0`, `-d` gives the precision + (measured in bits) lost to cancellation when computing `x-n`. + + >>> from mpmath import * + >>> n, d = nint_distance(5) + >>> print(n); print(d) + 5 + -inf + >>> n, d = nint_distance(mpf(5)) + >>> print(n); print(d) + 5 + -inf + >>> n, d = nint_distance(mpf(5.00000001)) + >>> print(n); print(d) + 5 + -26 + >>> n, d = nint_distance(mpf(4.99999999)) + >>> print(n); print(d) + 5 + -26 + >>> n, d = nint_distance(mpc(5,10)) + >>> print(n); print(d) + 5 + 4 + >>> n, d = nint_distance(mpc(5,0.000001)) + >>> print(n); print(d) + 5 + -19 + + """ + typx = type(x) + if typx in int_types: + return int(x), ctx.ninf + elif typx is rational.mpq: + p, q = x._mpq_ + n, r = divmod(p, q) + if 2*r >= q: + n += 1 + elif not r: + return n, ctx.ninf + # log(p/q-n) = log((p-nq)/q) = log(p-nq) - log(q) + d = bitcount(abs(p-n*q)) - bitcount(q) + return n, d + if hasattr(x, "_mpf_"): + re = x._mpf_ + im_dist = ctx.ninf + elif hasattr(x, "_mpc_"): + re, im = x._mpc_ + isign, iman, iexp, ibc = im + if iman: + im_dist = iexp + ibc + elif im == fzero: + im_dist = ctx.ninf + else: + raise ValueError("requires a finite number") + else: + x = ctx.convert(x) + if hasattr(x, "_mpf_") or hasattr(x, "_mpc_"): + return ctx.nint_distance(x) + else: + raise TypeError("requires an mpf/mpc") + sign, man, exp, bc = re + mag = exp+bc + # |x| < 0.5 + if mag < 0: + n = 0 + re_dist = mag + elif man: + # exact integer + if exp >= 0: + n = man << exp + re_dist = ctx.ninf + # exact half-integer + elif exp == -1: + n = (man>>1)+1 + re_dist = 0 + else: + d = (-exp-1) + t = man >> d + if t & 1: + t += 1 + man = (t<>1 # int(t)>>1 + re_dist = exp+bitcount(man) + if sign: + n = -n + elif re == fzero: + re_dist = ctx.ninf + n = 0 + else: + raise ValueError("requires a finite number") + return n, max(re_dist, im_dist) + + def fprod(ctx, factors): + r""" + Calculates a product containing a finite number of factors (for + infinite products, see :func:`~mpmath.nprod`). The factors will be + converted to mpmath numbers. + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fprod([1, 2, 0.5, 7]) + mpf('7.0') + + """ + orig = ctx.prec + try: + v = ctx.one + for p in factors: + v *= p + finally: + ctx.prec = orig + return +v + + def rand(ctx): + """ + Returns an ``mpf`` with value chosen randomly from `[0, 1)`. + The number of randomly generated bits in the mantissa is equal + to the working precision. + """ + return ctx.make_mpf(mpf_rand(ctx._prec)) + + def fraction(ctx, p, q): + """ + Given Python integers `(p, q)`, returns a lazy ``mpf`` representing + the fraction `p/q`. The value is updated with the precision. + + >>> from mpmath import * + >>> mp.dps = 15 + >>> a = fraction(1,100) + >>> b = mpf(1)/100 + >>> print(a); print(b) + 0.01 + 0.01 + >>> mp.dps = 30 + >>> print(a); print(b) # a will be accurate + 0.01 + 0.0100000000000000002081668171172 + >>> mp.dps = 15 + """ + return ctx.constant(lambda prec, rnd: from_rational(p, q, prec, rnd), + '%s/%s' % (p, q)) + + def absmin(ctx, x): + return abs(ctx.convert(x)) + + def absmax(ctx, x): + return abs(ctx.convert(x)) + + def _as_points(ctx, x): + # XXX: remove this? + if hasattr(x, '_mpi_'): + a, b = x._mpi_ + return [ctx.make_mpf(a), ctx.make_mpf(b)] + return x + + ''' + def _zetasum(ctx, s, a, b): + """ + Computes sum of k^(-s) for k = a, a+1, ..., b with a, b both small + integers. + """ + a = int(a) + b = int(b) + s = ctx.convert(s) + prec, rounding = ctx._prec_rounding + if hasattr(s, '_mpf_'): + v = ctx.make_mpf(libmp.mpf_zetasum(s._mpf_, a, b, prec)) + elif hasattr(s, '_mpc_'): + v = ctx.make_mpc(libmp.mpc_zetasum(s._mpc_, a, b, prec)) + return v + ''' + + def _zetasum_fast(ctx, s, a, n, derivatives=[0], reflect=False): + if not (ctx.isint(a) and hasattr(s, "_mpc_")): + raise NotImplementedError + a = int(a) + prec = ctx._prec + xs, ys = libmp.mpc_zetasum(s._mpc_, a, n, derivatives, reflect, prec) + xs = [ctx.make_mpc(x) for x in xs] + ys = [ctx.make_mpc(y) for y in ys] + return xs, ys + +class PrecisionManager: + def __init__(self, ctx, precfun, dpsfun, normalize_output=False): + self.ctx = ctx + self.precfun = precfun + self.dpsfun = dpsfun + self.normalize_output = normalize_output + def __call__(self, f): + @functools.wraps(f) + def g(*args, **kwargs): + orig = self.ctx.prec + try: + if self.precfun: + self.ctx.prec = self.precfun(self.ctx.prec) + else: + self.ctx.dps = self.dpsfun(self.ctx.dps) + if self.normalize_output: + v = f(*args, **kwargs) + if type(v) is tuple: + return tuple([+a for a in v]) + return +v + else: + return f(*args, **kwargs) + finally: + self.ctx.prec = orig + return g + def __enter__(self): + self.origp = self.ctx.prec + if self.precfun: + self.ctx.prec = self.precfun(self.ctx.prec) + else: + self.ctx.dps = self.dpsfun(self.ctx.dps) + def __exit__(self, exc_type, exc_val, exc_tb): + self.ctx.prec = self.origp + return False + + +if __name__ == '__main__': + import doctest + doctest.testmod() diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/function_docs.py b/env-llmeval/lib/python3.10/site-packages/mpmath/function_docs.py new file mode 100644 index 0000000000000000000000000000000000000000..73c071dc30a25c0ea1366e06a407a20206bd18a2 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/function_docs.py @@ -0,0 +1,10201 @@ +""" +Extended docstrings for functions.py +""" + + +pi = r""" +`\pi`, roughly equal to 3.141592654, represents the area of the unit +circle, the half-period of trigonometric functions, and many other +things in mathematics. + +Mpmath can evaluate `\pi` to arbitrary precision:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +pi + 3.1415926535897932384626433832795028841971693993751 + +This shows digits 99991-100000 of `\pi` (the last digit is actually +a 4 when the decimal expansion is truncated, but here the nearest +rounding is used):: + + >>> mp.dps = 100000 + >>> str(pi)[-10:] + '5549362465' + +**Possible issues** + +:data:`pi` always rounds to the nearest floating-point +number when used. This means that exact mathematical identities +involving `\pi` will generally not be preserved in floating-point +arithmetic. In particular, multiples of :data:`pi` (except for +the trivial case ``0*pi``) are *not* the exact roots of +:func:`~mpmath.sin`, but differ roughly by the current epsilon:: + + >>> mp.dps = 15 + >>> sin(pi) + 1.22464679914735e-16 + +One solution is to use the :func:`~mpmath.sinpi` function instead:: + + >>> sinpi(1) + 0.0 + +See the documentation of trigonometric functions for additional +details. + +**References** + +* [BorweinBorwein]_ + +""" + +degree = r""" +Represents one degree of angle, `1^{\circ} = \pi/180`, or +about 0.01745329. This constant may be evaluated to arbitrary +precision:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +degree + 0.017453292519943295769236907684886127134428718885417 + +The :data:`degree` object is convenient for conversion +to radians:: + + >>> sin(30 * degree) + 0.5 + >>> asin(0.5) / degree + 30.0 +""" + +e = r""" +The transcendental number `e` = 2.718281828... is the base of the +natural logarithm (:func:`~mpmath.ln`) and of the exponential function +(:func:`~mpmath.exp`). + +Mpmath can be evaluate `e` to arbitrary precision:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +e + 2.7182818284590452353602874713526624977572470937 + +This shows digits 99991-100000 of `e` (the last digit is actually +a 5 when the decimal expansion is truncated, but here the nearest +rounding is used):: + + >>> mp.dps = 100000 + >>> str(e)[-10:] + '2100427166' + +**Possible issues** + +:data:`e` always rounds to the nearest floating-point number +when used, and mathematical identities involving `e` may not +hold in floating-point arithmetic. For example, ``ln(e)`` +might not evaluate exactly to 1. + +In particular, don't use ``e**x`` to compute the exponential +function. Use ``exp(x)`` instead; this is both faster and more +accurate. +""" + +phi = r""" +Represents the golden ratio `\phi = (1+\sqrt 5)/2`, +approximately equal to 1.6180339887. To high precision, +its value is:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +phi + 1.6180339887498948482045868343656381177203091798058 + +Formulas for the golden ratio include the following:: + + >>> (1+sqrt(5))/2 + 1.6180339887498948482045868343656381177203091798058 + >>> findroot(lambda x: x**2-x-1, 1) + 1.6180339887498948482045868343656381177203091798058 + >>> limit(lambda n: fib(n+1)/fib(n), inf) + 1.6180339887498948482045868343656381177203091798058 +""" + +euler = r""" +Euler's constant or the Euler-Mascheroni constant `\gamma` += 0.57721566... is a number of central importance to +number theory and special functions. It is defined as the limit + +.. math :: + + \gamma = \lim_{n\to\infty} H_n - \log n + +where `H_n = 1 + \frac{1}{2} + \ldots + \frac{1}{n}` is a harmonic +number (see :func:`~mpmath.harmonic`). + +Evaluation of `\gamma` is supported at arbitrary precision:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +euler + 0.57721566490153286060651209008240243104215933593992 + +We can also compute `\gamma` directly from the definition, +although this is less efficient:: + + >>> limit(lambda n: harmonic(n)-log(n), inf) + 0.57721566490153286060651209008240243104215933593992 + +This shows digits 9991-10000 of `\gamma` (the last digit is actually +a 5 when the decimal expansion is truncated, but here the nearest +rounding is used):: + + >>> mp.dps = 10000 + >>> str(euler)[-10:] + '4679858166' + +Integrals, series, and representations for `\gamma` in terms of +special functions include the following (there are many others):: + + >>> mp.dps = 25 + >>> -quad(lambda x: exp(-x)*log(x), [0,inf]) + 0.5772156649015328606065121 + >>> quad(lambda x,y: (x-1)/(1-x*y)/log(x*y), [0,1], [0,1]) + 0.5772156649015328606065121 + >>> nsum(lambda k: 1/k-log(1+1/k), [1,inf]) + 0.5772156649015328606065121 + >>> nsum(lambda k: (-1)**k*zeta(k)/k, [2,inf]) + 0.5772156649015328606065121 + >>> -diff(gamma, 1) + 0.5772156649015328606065121 + >>> limit(lambda x: 1/x-gamma(x), 0) + 0.5772156649015328606065121 + >>> limit(lambda x: zeta(x)-1/(x-1), 1) + 0.5772156649015328606065121 + >>> (log(2*pi*nprod(lambda n: + ... exp(-2+2/n)*(1+2/n)**n, [1,inf]))-3)/2 + 0.5772156649015328606065121 + +For generalizations of the identities `\gamma = -\Gamma'(1)` +and `\gamma = \lim_{x\to1} \zeta(x)-1/(x-1)`, see +:func:`~mpmath.psi` and :func:`~mpmath.stieltjes` respectively. + +**References** + +* [BorweinBailey]_ + +""" + +catalan = r""" +Catalan's constant `K` = 0.91596559... is given by the infinite +series + +.. math :: + + K = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}. + +Mpmath can evaluate it to arbitrary precision:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +catalan + 0.91596559417721901505460351493238411077414937428167 + +One can also compute `K` directly from the definition, although +this is significantly less efficient:: + + >>> nsum(lambda k: (-1)**k/(2*k+1)**2, [0, inf]) + 0.91596559417721901505460351493238411077414937428167 + +This shows digits 9991-10000 of `K` (the last digit is actually +a 3 when the decimal expansion is truncated, but here the nearest +rounding is used):: + + >>> mp.dps = 10000 + >>> str(catalan)[-10:] + '9537871504' + +Catalan's constant has numerous integral representations:: + + >>> mp.dps = 50 + >>> quad(lambda x: -log(x)/(1+x**2), [0, 1]) + 0.91596559417721901505460351493238411077414937428167 + >>> quad(lambda x: atan(x)/x, [0, 1]) + 0.91596559417721901505460351493238411077414937428167 + >>> quad(lambda x: ellipk(x**2)/2, [0, 1]) + 0.91596559417721901505460351493238411077414937428167 + >>> quad(lambda x,y: 1/(1+(x*y)**2), [0, 1], [0, 1]) + 0.91596559417721901505460351493238411077414937428167 + +As well as series representations:: + + >>> pi*log(sqrt(3)+2)/8 + 3*nsum(lambda n: + ... (fac(n)/(2*n+1))**2/fac(2*n), [0, inf])/8 + 0.91596559417721901505460351493238411077414937428167 + >>> 1-nsum(lambda n: n*zeta(2*n+1)/16**n, [1,inf]) + 0.91596559417721901505460351493238411077414937428167 +""" + +khinchin = r""" +Khinchin's constant `K` = 2.68542... is a number that +appears in the theory of continued fractions. Mpmath can evaluate +it to arbitrary precision:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +khinchin + 2.6854520010653064453097148354817956938203822939945 + +An integral representation is:: + + >>> I = quad(lambda x: log((1-x**2)/sincpi(x))/x/(1+x), [0, 1]) + >>> 2*exp(1/log(2)*I) + 2.6854520010653064453097148354817956938203822939945 + +The computation of ``khinchin`` is based on an efficient +implementation of the following series:: + + >>> f = lambda n: (zeta(2*n)-1)/n*sum((-1)**(k+1)/mpf(k) + ... for k in range(1,2*int(n))) + >>> exp(nsum(f, [1,inf])/log(2)) + 2.6854520010653064453097148354817956938203822939945 +""" + +glaisher = r""" +Glaisher's constant `A`, also known as the Glaisher-Kinkelin +constant, is a number approximately equal to 1.282427129 that +sometimes appears in formulas related to gamma and zeta functions. +It is also related to the Barnes G-function (see :func:`~mpmath.barnesg`). + +The constant is defined as `A = \exp(1/12-\zeta'(-1))` where +`\zeta'(s)` denotes the derivative of the Riemann zeta function +(see :func:`~mpmath.zeta`). + +Mpmath can evaluate Glaisher's constant to arbitrary precision: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +glaisher + 1.282427129100622636875342568869791727767688927325 + +We can verify that the value computed by :data:`glaisher` is +correct using mpmath's facilities for numerical +differentiation and arbitrary evaluation of the zeta function: + + >>> exp(mpf(1)/12 - diff(zeta, -1)) + 1.282427129100622636875342568869791727767688927325 + +Here is an example of an integral that can be evaluated in +terms of Glaisher's constant: + + >>> mp.dps = 15 + >>> quad(lambda x: log(gamma(x)), [1, 1.5]) + -0.0428537406502909 + >>> -0.5 - 7*log(2)/24 + log(pi)/4 + 3*log(glaisher)/2 + -0.042853740650291 + +Mpmath computes Glaisher's constant by applying Euler-Maclaurin +summation to a slowly convergent series. The implementation is +reasonably efficient up to about 10,000 digits. See the source +code for additional details. + +References: +http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html +""" + +apery = r""" +Represents Apery's constant, which is the irrational number +approximately equal to 1.2020569 given by + +.. math :: + + \zeta(3) = \sum_{k=1}^\infty\frac{1}{k^3}. + +The calculation is based on an efficient hypergeometric +series. To 50 decimal places, the value is given by:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +apery + 1.2020569031595942853997381615114499907649862923405 + +Other ways to evaluate Apery's constant using mpmath +include:: + + >>> zeta(3) + 1.2020569031595942853997381615114499907649862923405 + >>> -psi(2,1)/2 + 1.2020569031595942853997381615114499907649862923405 + >>> 8*nsum(lambda k: 1/(2*k+1)**3, [0,inf])/7 + 1.2020569031595942853997381615114499907649862923405 + >>> f = lambda k: 2/k**3/(exp(2*pi*k)-1) + >>> 7*pi**3/180 - nsum(f, [1,inf]) + 1.2020569031595942853997381615114499907649862923405 + +This shows digits 9991-10000 of Apery's constant:: + + >>> mp.dps = 10000 + >>> str(apery)[-10:] + '3189504235' +""" + +mertens = r""" +Represents the Mertens or Meissel-Mertens constant, which is the +prime number analog of Euler's constant: + +.. math :: + + B_1 = \lim_{N\to\infty} + \left(\sum_{p_k \le N} \frac{1}{p_k} - \log \log N \right) + +Here `p_k` denotes the `k`-th prime number. Other names for this +constant include the Hadamard-de la Vallee-Poussin constant or +the prime reciprocal constant. + +The following gives the Mertens constant to 50 digits:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +mertens + 0.2614972128476427837554268386086958590515666482612 + +References: +http://mathworld.wolfram.com/MertensConstant.html +""" + +twinprime = r""" +Represents the twin prime constant, which is the factor `C_2` +featuring in the Hardy-Littlewood conjecture for the growth of the +twin prime counting function, + +.. math :: + + \pi_2(n) \sim 2 C_2 \frac{n}{\log^2 n}. + +It is given by the product over primes + +.. math :: + + C_2 = \prod_{p\ge3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016 + +Computing `C_2` to 50 digits:: + + >>> from mpmath import * + >>> mp.dps = 50; mp.pretty = True + >>> +twinprime + 0.66016181584686957392781211001455577843262336028473 + +References: +http://mathworld.wolfram.com/TwinPrimesConstant.html +""" + +ln = r""" +Computes the natural logarithm of `x`, `\ln x`. +See :func:`~mpmath.log` for additional documentation.""" + +sqrt = r""" +``sqrt(x)`` gives the principal square root of `x`, `\sqrt x`. +For positive real numbers, the principal root is simply the +positive square root. For arbitrary complex numbers, the principal +square root is defined to satisfy `\sqrt x = \exp(\log(x)/2)`. +The function thus has a branch cut along the negative half real axis. + +For all mpmath numbers ``x``, calling ``sqrt(x)`` is equivalent to +performing ``x**0.5``. + +**Examples** + +Basic examples and limits:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> sqrt(10) + 3.16227766016838 + >>> sqrt(100) + 10.0 + >>> sqrt(-4) + (0.0 + 2.0j) + >>> sqrt(1+1j) + (1.09868411346781 + 0.455089860562227j) + >>> sqrt(inf) + +inf + +Square root evaluation is fast at huge precision:: + + >>> mp.dps = 50000 + >>> a = sqrt(3) + >>> str(a)[-10:] + '9329332815' + +:func:`mpmath.iv.sqrt` supports interval arguments:: + + >>> iv.dps = 15; iv.pretty = True + >>> iv.sqrt([16,100]) + [4.0, 10.0] + >>> iv.sqrt(2) + [1.4142135623730949234, 1.4142135623730951455] + >>> iv.sqrt(2) ** 2 + [1.9999999999999995559, 2.0000000000000004441] + +""" + +cbrt = r""" +``cbrt(x)`` computes the cube root of `x`, `x^{1/3}`. This +function is faster and more accurate than raising to a floating-point +fraction:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> 125**(mpf(1)/3) + mpf('4.9999999999999991') + >>> cbrt(125) + mpf('5.0') + +Every nonzero complex number has three cube roots. This function +returns the cube root defined by `\exp(\log(x)/3)` where the +principal branch of the natural logarithm is used. Note that this +does not give a real cube root for negative real numbers:: + + >>> mp.pretty = True + >>> cbrt(-1) + (0.5 + 0.866025403784439j) +""" + +exp = r""" +Computes the exponential function, + +.. math :: + + \exp(x) = e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}. + +For complex numbers, the exponential function also satisfies + +.. math :: + + \exp(x+yi) = e^x (\cos y + i \sin y). + +**Basic examples** + +Some values of the exponential function:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> exp(0) + 1.0 + >>> exp(1) + 2.718281828459045235360287 + >>> exp(-1) + 0.3678794411714423215955238 + >>> exp(inf) + +inf + >>> exp(-inf) + 0.0 + +Arguments can be arbitrarily large:: + + >>> exp(10000) + 8.806818225662921587261496e+4342 + >>> exp(-10000) + 1.135483865314736098540939e-4343 + +Evaluation is supported for interval arguments via +:func:`mpmath.iv.exp`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.exp([-inf,0]) + [0.0, 1.0] + >>> iv.exp([0,1]) + [1.0, 2.71828182845904523536028749558] + +The exponential function can be evaluated efficiently to arbitrary +precision:: + + >>> mp.dps = 10000 + >>> exp(pi) #doctest: +ELLIPSIS + 23.140692632779269005729...8984304016040616 + +**Functional properties** + +Numerical verification of Euler's identity for the complex +exponential function:: + + >>> mp.dps = 15 + >>> exp(j*pi)+1 + (0.0 + 1.22464679914735e-16j) + >>> chop(exp(j*pi)+1) + 0.0 + +This recovers the coefficients (reciprocal factorials) in the +Maclaurin series expansion of exp:: + + >>> nprint(taylor(exp, 0, 5)) + [1.0, 1.0, 0.5, 0.166667, 0.0416667, 0.00833333] + +The exponential function is its own derivative and antiderivative:: + + >>> exp(pi) + 23.1406926327793 + >>> diff(exp, pi) + 23.1406926327793 + >>> quad(exp, [-inf, pi]) + 23.1406926327793 + +The exponential function can be evaluated using various methods, +including direct summation of the series, limits, and solving +the defining differential equation:: + + >>> nsum(lambda k: pi**k/fac(k), [0,inf]) + 23.1406926327793 + >>> limit(lambda k: (1+pi/k)**k, inf) + 23.1406926327793 + >>> odefun(lambda t, x: x, 0, 1)(pi) + 23.1406926327793 +""" + +cosh = r""" +Computes the hyperbolic cosine of `x`, +`\cosh(x) = (e^x + e^{-x})/2`. Values and limits include:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> cosh(0) + 1.0 + >>> cosh(1) + 1.543080634815243778477906 + >>> cosh(-inf), cosh(+inf) + (+inf, +inf) + +The hyperbolic cosine is an even, convex function with +a global minimum at `x = 0`, having a Maclaurin series +that starts:: + + >>> nprint(chop(taylor(cosh, 0, 5))) + [1.0, 0.0, 0.5, 0.0, 0.0416667, 0.0] + +Generalized to complex numbers, the hyperbolic cosine is +equivalent to a cosine with the argument rotated +in the imaginary direction, or `\cosh x = \cos ix`:: + + >>> cosh(2+3j) + (-3.724545504915322565473971 + 0.5118225699873846088344638j) + >>> cos(3-2j) + (-3.724545504915322565473971 + 0.5118225699873846088344638j) +""" + +sinh = r""" +Computes the hyperbolic sine of `x`, +`\sinh(x) = (e^x - e^{-x})/2`. Values and limits include:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> sinh(0) + 0.0 + >>> sinh(1) + 1.175201193643801456882382 + >>> sinh(-inf), sinh(+inf) + (-inf, +inf) + +The hyperbolic sine is an odd function, with a Maclaurin +series that starts:: + + >>> nprint(chop(taylor(sinh, 0, 5))) + [0.0, 1.0, 0.0, 0.166667, 0.0, 0.00833333] + +Generalized to complex numbers, the hyperbolic sine is +essentially a sine with a rotation `i` applied to +the argument; more precisely, `\sinh x = -i \sin ix`:: + + >>> sinh(2+3j) + (-3.590564589985779952012565 + 0.5309210862485198052670401j) + >>> j*sin(3-2j) + (-3.590564589985779952012565 + 0.5309210862485198052670401j) +""" + +tanh = r""" +Computes the hyperbolic tangent of `x`, +`\tanh(x) = \sinh(x)/\cosh(x)`. Values and limits include:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> tanh(0) + 0.0 + >>> tanh(1) + 0.7615941559557648881194583 + >>> tanh(-inf), tanh(inf) + (-1.0, 1.0) + +The hyperbolic tangent is an odd, sigmoidal function, similar +to the inverse tangent and error function. Its Maclaurin +series is:: + + >>> nprint(chop(taylor(tanh, 0, 5))) + [0.0, 1.0, 0.0, -0.333333, 0.0, 0.133333] + +Generalized to complex numbers, the hyperbolic tangent is +essentially a tangent with a rotation `i` applied to +the argument; more precisely, `\tanh x = -i \tan ix`:: + + >>> tanh(2+3j) + (0.9653858790221331242784803 - 0.009884375038322493720314034j) + >>> j*tan(3-2j) + (0.9653858790221331242784803 - 0.009884375038322493720314034j) +""" + +cos = r""" +Computes the cosine of `x`, `\cos(x)`. + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> cos(pi/3) + 0.5 + >>> cos(100000001) + -0.9802850113244713353133243 + >>> cos(2+3j) + (-4.189625690968807230132555 - 9.109227893755336597979197j) + >>> cos(inf) + nan + >>> nprint(chop(taylor(cos, 0, 6))) + [1.0, 0.0, -0.5, 0.0, 0.0416667, 0.0, -0.00138889] + +Intervals are supported via :func:`mpmath.iv.cos`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.cos([0,1]) + [0.540302305868139717400936602301, 1.0] + >>> iv.cos([0,2]) + [-0.41614683654714238699756823214, 1.0] +""" + +sin = r""" +Computes the sine of `x`, `\sin(x)`. + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> sin(pi/3) + 0.8660254037844386467637232 + >>> sin(100000001) + 0.1975887055794968911438743 + >>> sin(2+3j) + (9.1544991469114295734673 - 4.168906959966564350754813j) + >>> sin(inf) + nan + >>> nprint(chop(taylor(sin, 0, 6))) + [0.0, 1.0, 0.0, -0.166667, 0.0, 0.00833333, 0.0] + +Intervals are supported via :func:`mpmath.iv.sin`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.sin([0,1]) + [0.0, 0.841470984807896506652502331201] + >>> iv.sin([0,2]) + [0.0, 1.0] +""" + +tan = r""" +Computes the tangent of `x`, `\tan(x) = \frac{\sin(x)}{\cos(x)}`. +The tangent function is singular at `x = (n+1/2)\pi`, but +``tan(x)`` always returns a finite result since `(n+1/2)\pi` +cannot be represented exactly using floating-point arithmetic. + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> tan(pi/3) + 1.732050807568877293527446 + >>> tan(100000001) + -0.2015625081449864533091058 + >>> tan(2+3j) + (-0.003764025641504248292751221 + 1.003238627353609801446359j) + >>> tan(inf) + nan + >>> nprint(chop(taylor(tan, 0, 6))) + [0.0, 1.0, 0.0, 0.333333, 0.0, 0.133333, 0.0] + +Intervals are supported via :func:`mpmath.iv.tan`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.tan([0,1]) + [0.0, 1.55740772465490223050697482944] + >>> iv.tan([0,2]) # Interval includes a singularity + [-inf, +inf] +""" + +sec = r""" +Computes the secant of `x`, `\mathrm{sec}(x) = \frac{1}{\cos(x)}`. +The secant function is singular at `x = (n+1/2)\pi`, but +``sec(x)`` always returns a finite result since `(n+1/2)\pi` +cannot be represented exactly using floating-point arithmetic. + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> sec(pi/3) + 2.0 + >>> sec(10000001) + -1.184723164360392819100265 + >>> sec(2+3j) + (-0.04167496441114427004834991 + 0.0906111371962375965296612j) + >>> sec(inf) + nan + >>> nprint(chop(taylor(sec, 0, 6))) + [1.0, 0.0, 0.5, 0.0, 0.208333, 0.0, 0.0847222] + +Intervals are supported via :func:`mpmath.iv.sec`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.sec([0,1]) + [1.0, 1.85081571768092561791175326276] + >>> iv.sec([0,2]) # Interval includes a singularity + [-inf, +inf] +""" + +csc = r""" +Computes the cosecant of `x`, `\mathrm{csc}(x) = \frac{1}{\sin(x)}`. +This cosecant function is singular at `x = n \pi`, but with the +exception of the point `x = 0`, ``csc(x)`` returns a finite result +since `n \pi` cannot be represented exactly using floating-point +arithmetic. + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> csc(pi/3) + 1.154700538379251529018298 + >>> csc(10000001) + -1.864910497503629858938891 + >>> csc(2+3j) + (0.09047320975320743980579048 + 0.04120098628857412646300981j) + >>> csc(inf) + nan + +Intervals are supported via :func:`mpmath.iv.csc`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.csc([0,1]) # Interval includes a singularity + [1.18839510577812121626159943988, +inf] + >>> iv.csc([0,2]) + [1.0, +inf] +""" + +cot = r""" +Computes the cotangent of `x`, +`\mathrm{cot}(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}`. +This cotangent function is singular at `x = n \pi`, but with the +exception of the point `x = 0`, ``cot(x)`` returns a finite result +since `n \pi` cannot be represented exactly using floating-point +arithmetic. + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> cot(pi/3) + 0.5773502691896257645091488 + >>> cot(10000001) + 1.574131876209625656003562 + >>> cot(2+3j) + (-0.003739710376336956660117409 - 0.9967577965693583104609688j) + >>> cot(inf) + nan + +Intervals are supported via :func:`mpmath.iv.cot`:: + + >>> iv.dps = 25; iv.pretty = True + >>> iv.cot([0,1]) # Interval includes a singularity + [0.642092615934330703006419974862, +inf] + >>> iv.cot([1,2]) + [-inf, +inf] +""" + +acos = r""" +Computes the inverse cosine or arccosine of `x`, `\cos^{-1}(x)`. +Since `-1 \le \cos(x) \le 1` for real `x`, the inverse +cosine is real-valued only for `-1 \le x \le 1`. On this interval, +:func:`~mpmath.acos` is defined to be a monotonically decreasing +function assuming values between `+\pi` and `0`. + +Basic values are:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> acos(-1) + 3.141592653589793238462643 + >>> acos(0) + 1.570796326794896619231322 + >>> acos(1) + 0.0 + >>> nprint(chop(taylor(acos, 0, 6))) + [1.5708, -1.0, 0.0, -0.166667, 0.0, -0.075, 0.0] + +:func:`~mpmath.acos` is defined so as to be a proper inverse function of +`\cos(\theta)` for `0 \le \theta < \pi`. +We have `\cos(\cos^{-1}(x)) = x` for all `x`, but +`\cos^{-1}(\cos(x)) = x` only for `0 \le \Re[x] < \pi`:: + + >>> for x in [1, 10, -1, 2+3j, 10+3j]: + ... print("%s %s" % (cos(acos(x)), acos(cos(x)))) + ... + 1.0 1.0 + (10.0 + 0.0j) 2.566370614359172953850574 + -1.0 1.0 + (2.0 + 3.0j) (2.0 + 3.0j) + (10.0 + 3.0j) (2.566370614359172953850574 - 3.0j) + +The inverse cosine has two branch points: `x = \pm 1`. :func:`~mpmath.acos` +places the branch cuts along the line segments `(-\infty, -1)` and +`(+1, +\infty)`. In general, + +.. math :: + + \cos^{-1}(x) = \frac{\pi}{2} + i \log\left(ix + \sqrt{1-x^2} \right) + +where the principal-branch log and square root are implied. +""" + +asin = r""" +Computes the inverse sine or arcsine of `x`, `\sin^{-1}(x)`. +Since `-1 \le \sin(x) \le 1` for real `x`, the inverse +sine is real-valued only for `-1 \le x \le 1`. +On this interval, it is defined to be a monotonically increasing +function assuming values between `-\pi/2` and `\pi/2`. + +Basic values are:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> asin(-1) + -1.570796326794896619231322 + >>> asin(0) + 0.0 + >>> asin(1) + 1.570796326794896619231322 + >>> nprint(chop(taylor(asin, 0, 6))) + [0.0, 1.0, 0.0, 0.166667, 0.0, 0.075, 0.0] + +:func:`~mpmath.asin` is defined so as to be a proper inverse function of +`\sin(\theta)` for `-\pi/2 < \theta < \pi/2`. +We have `\sin(\sin^{-1}(x)) = x` for all `x`, but +`\sin^{-1}(\sin(x)) = x` only for `-\pi/2 < \Re[x] < \pi/2`:: + + >>> for x in [1, 10, -1, 1+3j, -2+3j]: + ... print("%s %s" % (chop(sin(asin(x))), asin(sin(x)))) + ... + 1.0 1.0 + 10.0 -0.5752220392306202846120698 + -1.0 -1.0 + (1.0 + 3.0j) (1.0 + 3.0j) + (-2.0 + 3.0j) (-1.141592653589793238462643 - 3.0j) + +The inverse sine has two branch points: `x = \pm 1`. :func:`~mpmath.asin` +places the branch cuts along the line segments `(-\infty, -1)` and +`(+1, +\infty)`. In general, + +.. math :: + + \sin^{-1}(x) = -i \log\left(ix + \sqrt{1-x^2} \right) + +where the principal-branch log and square root are implied. +""" + +atan = r""" +Computes the inverse tangent or arctangent of `x`, `\tan^{-1}(x)`. +This is a real-valued function for all real `x`, with range +`(-\pi/2, \pi/2)`. + +Basic values are:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> atan(-inf) + -1.570796326794896619231322 + >>> atan(-1) + -0.7853981633974483096156609 + >>> atan(0) + 0.0 + >>> atan(1) + 0.7853981633974483096156609 + >>> atan(inf) + 1.570796326794896619231322 + >>> nprint(chop(taylor(atan, 0, 6))) + [0.0, 1.0, 0.0, -0.333333, 0.0, 0.2, 0.0] + +The inverse tangent is often used to compute angles. However, +the atan2 function is often better for this as it preserves sign +(see :func:`~mpmath.atan2`). + +:func:`~mpmath.atan` is defined so as to be a proper inverse function of +`\tan(\theta)` for `-\pi/2 < \theta < \pi/2`. +We have `\tan(\tan^{-1}(x)) = x` for all `x`, but +`\tan^{-1}(\tan(x)) = x` only for `-\pi/2 < \Re[x] < \pi/2`:: + + >>> mp.dps = 25 + >>> for x in [1, 10, -1, 1+3j, -2+3j]: + ... print("%s %s" % (tan(atan(x)), atan(tan(x)))) + ... + 1.0 1.0 + 10.0 0.5752220392306202846120698 + -1.0 -1.0 + (1.0 + 3.0j) (1.000000000000000000000001 + 3.0j) + (-2.0 + 3.0j) (1.141592653589793238462644 + 3.0j) + +The inverse tangent has two branch points: `x = \pm i`. :func:`~mpmath.atan` +places the branch cuts along the line segments `(-i \infty, -i)` and +`(+i, +i \infty)`. In general, + +.. math :: + + \tan^{-1}(x) = \frac{i}{2}\left(\log(1-ix)-\log(1+ix)\right) + +where the principal-branch log is implied. +""" + +acot = r"""Computes the inverse cotangent of `x`, +`\mathrm{cot}^{-1}(x) = \tan^{-1}(1/x)`.""" + +asec = r"""Computes the inverse secant of `x`, +`\mathrm{sec}^{-1}(x) = \cos^{-1}(1/x)`.""" + +acsc = r"""Computes the inverse cosecant of `x`, +`\mathrm{csc}^{-1}(x) = \sin^{-1}(1/x)`.""" + +coth = r"""Computes the hyperbolic cotangent of `x`, +`\mathrm{coth}(x) = \frac{\cosh(x)}{\sinh(x)}`. +""" + +sech = r"""Computes the hyperbolic secant of `x`, +`\mathrm{sech}(x) = \frac{1}{\cosh(x)}`. +""" + +csch = r"""Computes the hyperbolic cosecant of `x`, +`\mathrm{csch}(x) = \frac{1}{\sinh(x)}`. +""" + +acosh = r"""Computes the inverse hyperbolic cosine of `x`, +`\mathrm{cosh}^{-1}(x) = \log(x+\sqrt{x+1}\sqrt{x-1})`. +""" + +asinh = r"""Computes the inverse hyperbolic sine of `x`, +`\mathrm{sinh}^{-1}(x) = \log(x+\sqrt{1+x^2})`. +""" + +atanh = r"""Computes the inverse hyperbolic tangent of `x`, +`\mathrm{tanh}^{-1}(x) = \frac{1}{2}\left(\log(1+x)-\log(1-x)\right)`. +""" + +acoth = r"""Computes the inverse hyperbolic cotangent of `x`, +`\mathrm{coth}^{-1}(x) = \tanh^{-1}(1/x)`.""" + +asech = r"""Computes the inverse hyperbolic secant of `x`, +`\mathrm{sech}^{-1}(x) = \cosh^{-1}(1/x)`.""" + +acsch = r"""Computes the inverse hyperbolic cosecant of `x`, +`\mathrm{csch}^{-1}(x) = \sinh^{-1}(1/x)`.""" + + + +sinpi = r""" +Computes `\sin(\pi x)`, more accurately than the expression +``sin(pi*x)``:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> sinpi(10**10), sin(pi*(10**10)) + (0.0, -2.23936276195592e-6) + >>> sinpi(10**10+0.5), sin(pi*(10**10+0.5)) + (1.0, 0.999999999998721) +""" + +cospi = r""" +Computes `\cos(\pi x)`, more accurately than the expression +``cos(pi*x)``:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> cospi(10**10), cos(pi*(10**10)) + (1.0, 0.999999999997493) + >>> cospi(10**10+0.5), cos(pi*(10**10+0.5)) + (0.0, 1.59960492420134e-6) +""" + +sinc = r""" +``sinc(x)`` computes the unnormalized sinc function, defined as + +.. math :: + + \mathrm{sinc}(x) = \begin{cases} + \sin(x)/x, & \mbox{if } x \ne 0 \\ + 1, & \mbox{if } x = 0. + \end{cases} + +See :func:`~mpmath.sincpi` for the normalized sinc function. + +Simple values and limits include:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> sinc(0) + 1.0 + >>> sinc(1) + 0.841470984807897 + >>> sinc(inf) + 0.0 + +The integral of the sinc function is the sine integral Si:: + + >>> quad(sinc, [0, 1]) + 0.946083070367183 + >>> si(1) + 0.946083070367183 +""" + +sincpi = r""" +``sincpi(x)`` computes the normalized sinc function, defined as + +.. math :: + + \mathrm{sinc}_{\pi}(x) = \begin{cases} + \sin(\pi x)/(\pi x), & \mbox{if } x \ne 0 \\ + 1, & \mbox{if } x = 0. + \end{cases} + +Equivalently, we have +`\mathrm{sinc}_{\pi}(x) = \mathrm{sinc}(\pi x)`. + +The normalization entails that the function integrates +to unity over the entire real line:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> quadosc(sincpi, [-inf, inf], period=2.0) + 1.0 + +Like, :func:`~mpmath.sinpi`, :func:`~mpmath.sincpi` is evaluated accurately +at its roots:: + + >>> sincpi(10) + 0.0 +""" + +expj = r""" +Convenience function for computing `e^{ix}`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> expj(0) + (1.0 + 0.0j) + >>> expj(-1) + (0.5403023058681397174009366 - 0.8414709848078965066525023j) + >>> expj(j) + (0.3678794411714423215955238 + 0.0j) + >>> expj(1+j) + (0.1987661103464129406288032 + 0.3095598756531121984439128j) +""" + +expjpi = r""" +Convenience function for computing `e^{i \pi x}`. +Evaluation is accurate near zeros (see also :func:`~mpmath.cospi`, +:func:`~mpmath.sinpi`):: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> expjpi(0) + (1.0 + 0.0j) + >>> expjpi(1) + (-1.0 + 0.0j) + >>> expjpi(0.5) + (0.0 + 1.0j) + >>> expjpi(-1) + (-1.0 + 0.0j) + >>> expjpi(j) + (0.04321391826377224977441774 + 0.0j) + >>> expjpi(1+j) + (-0.04321391826377224977441774 + 0.0j) +""" + +floor = r""" +Computes the floor of `x`, `\lfloor x \rfloor`, defined as +the largest integer less than or equal to `x`:: + + >>> from mpmath import * + >>> mp.pretty = False + >>> floor(3.5) + mpf('3.0') + +.. note :: + + :func:`~mpmath.floor`, :func:`~mpmath.ceil` and :func:`~mpmath.nint` return a + floating-point number, not a Python ``int``. If `\lfloor x \rfloor` is + too large to be represented exactly at the present working precision, + the result will be rounded, not necessarily in the direction + implied by the mathematical definition of the function. + +To avoid rounding, use *prec=0*:: + + >>> mp.dps = 15 + >>> print(int(floor(10**30+1))) + 1000000000000000019884624838656 + >>> print(int(floor(10**30+1, prec=0))) + 1000000000000000000000000000001 + +The floor function is defined for complex numbers and +acts on the real and imaginary parts separately:: + + >>> floor(3.25+4.75j) + mpc(real='3.0', imag='4.0') +""" + +ceil = r""" +Computes the ceiling of `x`, `\lceil x \rceil`, defined as +the smallest integer greater than or equal to `x`:: + + >>> from mpmath import * + >>> mp.pretty = False + >>> ceil(3.5) + mpf('4.0') + +The ceiling function is defined for complex numbers and +acts on the real and imaginary parts separately:: + + >>> ceil(3.25+4.75j) + mpc(real='4.0', imag='5.0') + +See notes about rounding for :func:`~mpmath.floor`. +""" + +nint = r""" +Evaluates the nearest integer function, `\mathrm{nint}(x)`. +This gives the nearest integer to `x`; on a tie, it +gives the nearest even integer:: + + >>> from mpmath import * + >>> mp.pretty = False + >>> nint(3.2) + mpf('3.0') + >>> nint(3.8) + mpf('4.0') + >>> nint(3.5) + mpf('4.0') + >>> nint(4.5) + mpf('4.0') + +The nearest integer function is defined for complex numbers and +acts on the real and imaginary parts separately:: + + >>> nint(3.25+4.75j) + mpc(real='3.0', imag='5.0') + +See notes about rounding for :func:`~mpmath.floor`. +""" + +frac = r""" +Gives the fractional part of `x`, defined as +`\mathrm{frac}(x) = x - \lfloor x \rfloor` (see :func:`~mpmath.floor`). +In effect, this computes `x` modulo 1, or `x+n` where +`n \in \mathbb{Z}` is such that `x+n \in [0,1)`:: + + >>> from mpmath import * + >>> mp.pretty = False + >>> frac(1.25) + mpf('0.25') + >>> frac(3) + mpf('0.0') + >>> frac(-1.25) + mpf('0.75') + +For a complex number, the fractional part function applies to +the real and imaginary parts separately:: + + >>> frac(2.25+3.75j) + mpc(real='0.25', imag='0.75') + +Plotted, the fractional part function gives a sawtooth +wave. The Fourier series coefficients have a simple +form:: + + >>> mp.dps = 15 + >>> nprint(fourier(lambda x: frac(x)-0.5, [0,1], 4)) + ([0.0, 0.0, 0.0, 0.0, 0.0], [0.0, -0.31831, -0.159155, -0.106103, -0.0795775]) + >>> nprint([-1/(pi*k) for k in range(1,5)]) + [-0.31831, -0.159155, -0.106103, -0.0795775] + +.. note:: + + The fractional part is sometimes defined as a symmetric + function, i.e. returning `-\mathrm{frac}(-x)` if `x < 0`. + This convention is used, for instance, by Mathematica's + ``FractionalPart``. + +""" + +sign = r""" +Returns the sign of `x`, defined as `\mathrm{sign}(x) = x / |x|` +(with the special case `\mathrm{sign}(0) = 0`):: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> sign(10) + mpf('1.0') + >>> sign(-10) + mpf('-1.0') + >>> sign(0) + mpf('0.0') + +Note that the sign function is also defined for complex numbers, +for which it gives the projection onto the unit circle:: + + >>> mp.dps = 15; mp.pretty = True + >>> sign(1+j) + (0.707106781186547 + 0.707106781186547j) + +""" + +arg = r""" +Computes the complex argument (phase) of `x`, defined as the +signed angle between the positive real axis and `x` in the +complex plane:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> arg(3) + 0.0 + >>> arg(3+3j) + 0.785398163397448 + >>> arg(3j) + 1.5707963267949 + >>> arg(-3) + 3.14159265358979 + >>> arg(-3j) + -1.5707963267949 + +The angle is defined to satisfy `-\pi < \arg(x) \le \pi` and +with the sign convention that a nonnegative imaginary part +results in a nonnegative argument. + +The value returned by :func:`~mpmath.arg` is an ``mpf`` instance. +""" + +fabs = r""" +Returns the absolute value of `x`, `|x|`. Unlike :func:`abs`, +:func:`~mpmath.fabs` converts non-mpmath numbers (such as ``int``) +into mpmath numbers:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> fabs(3) + mpf('3.0') + >>> fabs(-3) + mpf('3.0') + >>> fabs(3+4j) + mpf('5.0') +""" + +re = r""" +Returns the real part of `x`, `\Re(x)`. :func:`~mpmath.re` +converts a non-mpmath number to an mpmath number:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> re(3) + mpf('3.0') + >>> re(-1+4j) + mpf('-1.0') +""" + +im = r""" +Returns the imaginary part of `x`, `\Im(x)`. :func:`~mpmath.im` +converts a non-mpmath number to an mpmath number:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> im(3) + mpf('0.0') + >>> im(-1+4j) + mpf('4.0') +""" + +conj = r""" +Returns the complex conjugate of `x`, `\overline{x}`. Unlike +``x.conjugate()``, :func:`~mpmath.im` converts `x` to a mpmath number:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> conj(3) + mpf('3.0') + >>> conj(-1+4j) + mpc(real='-1.0', imag='-4.0') +""" + +polar = r""" +Returns the polar representation of the complex number `z` +as a pair `(r, \phi)` such that `z = r e^{i \phi}`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> polar(-2) + (2.0, 3.14159265358979) + >>> polar(3-4j) + (5.0, -0.927295218001612) +""" + +rect = r""" +Returns the complex number represented by polar +coordinates `(r, \phi)`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> chop(rect(2, pi)) + -2.0 + >>> rect(sqrt(2), -pi/4) + (1.0 - 1.0j) +""" + +expm1 = r""" +Computes `e^x - 1`, accurately for small `x`. + +Unlike the expression ``exp(x) - 1``, ``expm1(x)`` does not suffer from +potentially catastrophic cancellation:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> exp(1e-10)-1; print(expm1(1e-10)) + 1.00000008274037e-10 + 1.00000000005e-10 + >>> exp(1e-20)-1; print(expm1(1e-20)) + 0.0 + 1.0e-20 + >>> 1/(exp(1e-20)-1) + Traceback (most recent call last): + ... + ZeroDivisionError + >>> 1/expm1(1e-20) + 1.0e+20 + +Evaluation works for extremely tiny values:: + + >>> expm1(0) + 0.0 + >>> expm1('1e-10000000') + 1.0e-10000000 + +""" + +log1p = r""" +Computes `\log(1+x)`, accurately for small `x`. + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> log(1+1e-10); print(mp.log1p(1e-10)) + 1.00000008269037e-10 + 9.9999999995e-11 + >>> mp.log1p(1e-100j) + (5.0e-201 + 1.0e-100j) + >>> mp.log1p(0) + 0.0 + +""" + + +powm1 = r""" +Computes `x^y - 1`, accurately when `x^y` is very close to 1. + +This avoids potentially catastrophic cancellation:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> power(0.99999995, 1e-10) - 1 + 0.0 + >>> powm1(0.99999995, 1e-10) + -5.00000012791934e-18 + +Powers exactly equal to 1, and only those powers, yield 0 exactly:: + + >>> powm1(-j, 4) + (0.0 + 0.0j) + >>> powm1(3, 0) + 0.0 + >>> powm1(fadd(-1, 1e-100, exact=True), 4) + -4.0e-100 + +Evaluation works for extremely tiny `y`:: + + >>> powm1(2, '1e-100000') + 6.93147180559945e-100001 + >>> powm1(j, '1e-1000') + (-1.23370055013617e-2000 + 1.5707963267949e-1000j) + +""" + +root = r""" +``root(z, n, k=0)`` computes an `n`-th root of `z`, i.e. returns a number +`r` that (up to possible approximation error) satisfies `r^n = z`. +(``nthroot`` is available as an alias for ``root``.) + +Every complex number `z \ne 0` has `n` distinct `n`-th roots, which are +equidistant points on a circle with radius `|z|^{1/n}`, centered around the +origin. A specific root may be selected using the optional index +`k`. The roots are indexed counterclockwise, starting with `k = 0` for the root +closest to the positive real half-axis. + +The `k = 0` root is the so-called principal `n`-th root, often denoted by +`\sqrt[n]{z}` or `z^{1/n}`, and also given by `\exp(\log(z) / n)`. If `z` is +a positive real number, the principal root is just the unique positive +`n`-th root of `z`. Under some circumstances, non-principal real roots exist: +for positive real `z`, `n` even, there is a negative root given by `k = n/2`; +for negative real `z`, `n` odd, there is a negative root given by `k = (n-1)/2`. + +To obtain all roots with a simple expression, use +``[root(z,n,k) for k in range(n)]``. + +An important special case, ``root(1, n, k)`` returns the `k`-th `n`-th root of +unity, `\zeta_k = e^{2 \pi i k / n}`. Alternatively, :func:`~mpmath.unitroots` +provides a slightly more convenient way to obtain the roots of unity, +including the option to compute only the primitive roots of unity. + +Both `k` and `n` should be integers; `k` outside of ``range(n)`` will be +reduced modulo `n`. If `n` is negative, `x^{-1/n} = 1/x^{1/n}` (or +the equivalent reciprocal for a non-principal root with `k \ne 0`) is computed. + +:func:`~mpmath.root` is implemented to use Newton's method for small +`n`. At high precision, this makes `x^{1/n}` not much more +expensive than the regular exponentiation, `x^n`. For very large +`n`, :func:`~mpmath.nthroot` falls back to use the exponential function. + +**Examples** + +:func:`~mpmath.nthroot`/:func:`~mpmath.root` is faster and more accurate than raising to a +floating-point fraction:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = False + >>> 16807 ** (mpf(1)/5) + mpf('7.0000000000000009') + >>> root(16807, 5) + mpf('7.0') + >>> nthroot(16807, 5) # Alias + mpf('7.0') + +A high-precision root:: + + >>> mp.dps = 50; mp.pretty = True + >>> nthroot(10, 5) + 1.584893192461113485202101373391507013269442133825 + >>> nthroot(10, 5) ** 5 + 10.0 + +Computing principal and non-principal square and cube roots:: + + >>> mp.dps = 15 + >>> root(10, 2) + 3.16227766016838 + >>> root(10, 2, 1) + -3.16227766016838 + >>> root(-10, 3) + (1.07721734501594 + 1.86579517236206j) + >>> root(-10, 3, 1) + -2.15443469003188 + >>> root(-10, 3, 2) + (1.07721734501594 - 1.86579517236206j) + +All the 7th roots of a complex number:: + + >>> for r in [root(3+4j, 7, k) for k in range(7)]: + ... print("%s %s" % (r, r**7)) + ... + (1.24747270589553 + 0.166227124177353j) (3.0 + 4.0j) + (0.647824911301003 + 1.07895435170559j) (3.0 + 4.0j) + (-0.439648254723098 + 1.17920694574172j) (3.0 + 4.0j) + (-1.19605731775069 + 0.391492658196305j) (3.0 + 4.0j) + (-1.05181082538903 - 0.691023585965793j) (3.0 + 4.0j) + (-0.115529328478668 - 1.25318497558335j) (3.0 + 4.0j) + (0.907748109144957 - 0.871672518271819j) (3.0 + 4.0j) + +Cube roots of unity:: + + >>> for k in range(3): print(root(1, 3, k)) + ... + 1.0 + (-0.5 + 0.866025403784439j) + (-0.5 - 0.866025403784439j) + +Some exact high order roots:: + + >>> root(75**210, 105) + 5625.0 + >>> root(1, 128, 96) + (0.0 - 1.0j) + >>> root(4**128, 128, 96) + (0.0 - 4.0j) + +""" + +unitroots = r""" +``unitroots(n)`` returns `\zeta_0, \zeta_1, \ldots, \zeta_{n-1}`, +all the distinct `n`-th roots of unity, as a list. If the option +*primitive=True* is passed, only the primitive roots are returned. + +Every `n`-th root of unity satisfies `(\zeta_k)^n = 1`. There are `n` distinct +roots for each `n` (`\zeta_k` and `\zeta_j` are the same when +`k = j \pmod n`), which form a regular polygon with vertices on the unit +circle. They are ordered counterclockwise with increasing `k`, starting +with `\zeta_0 = 1`. + +**Examples** + +The roots of unity up to `n = 4`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> nprint(unitroots(1)) + [1.0] + >>> nprint(unitroots(2)) + [1.0, -1.0] + >>> nprint(unitroots(3)) + [1.0, (-0.5 + 0.866025j), (-0.5 - 0.866025j)] + >>> nprint(unitroots(4)) + [1.0, (0.0 + 1.0j), -1.0, (0.0 - 1.0j)] + +Roots of unity form a geometric series that sums to 0:: + + >>> mp.dps = 50 + >>> chop(fsum(unitroots(25))) + 0.0 + +Primitive roots up to `n = 4`:: + + >>> mp.dps = 15 + >>> nprint(unitroots(1, primitive=True)) + [1.0] + >>> nprint(unitroots(2, primitive=True)) + [-1.0] + >>> nprint(unitroots(3, primitive=True)) + [(-0.5 + 0.866025j), (-0.5 - 0.866025j)] + >>> nprint(unitroots(4, primitive=True)) + [(0.0 + 1.0j), (0.0 - 1.0j)] + +There are only four primitive 12th roots:: + + >>> nprint(unitroots(12, primitive=True)) + [(0.866025 + 0.5j), (-0.866025 + 0.5j), (-0.866025 - 0.5j), (0.866025 - 0.5j)] + +The `n`-th roots of unity form a group, the cyclic group of order `n`. +Any primitive root `r` is a generator for this group, meaning that +`r^0, r^1, \ldots, r^{n-1}` gives the whole set of unit roots (in +some permuted order):: + + >>> for r in unitroots(6): print(r) + ... + 1.0 + (0.5 + 0.866025403784439j) + (-0.5 + 0.866025403784439j) + -1.0 + (-0.5 - 0.866025403784439j) + (0.5 - 0.866025403784439j) + >>> r = unitroots(6, primitive=True)[1] + >>> for k in range(6): print(chop(r**k)) + ... + 1.0 + (0.5 - 0.866025403784439j) + (-0.5 - 0.866025403784439j) + -1.0 + (-0.5 + 0.866025403784438j) + (0.5 + 0.866025403784438j) + +The number of primitive roots equals the Euler totient function `\phi(n)`:: + + >>> [len(unitroots(n, primitive=True)) for n in range(1,20)] + [1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18] + +""" + + +log = r""" +Computes the base-`b` logarithm of `x`, `\log_b(x)`. If `b` is +unspecified, :func:`~mpmath.log` computes the natural (base `e`) logarithm +and is equivalent to :func:`~mpmath.ln`. In general, the base `b` logarithm +is defined in terms of the natural logarithm as +`\log_b(x) = \ln(x)/\ln(b)`. + +By convention, we take `\log(0) = -\infty`. + +The natural logarithm is real if `x > 0` and complex if `x < 0` or if +`x` is complex. The principal branch of the complex logarithm is +used, meaning that `\Im(\ln(x)) = -\pi < \arg(x) \le \pi`. + +**Examples** + +Some basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> log(1) + 0.0 + >>> log(2) + 0.693147180559945 + >>> log(1000,10) + 3.0 + >>> log(4, 16) + 0.5 + >>> log(j) + (0.0 + 1.5707963267949j) + >>> log(-1) + (0.0 + 3.14159265358979j) + >>> log(0) + -inf + >>> log(inf) + +inf + +The natural logarithm is the antiderivative of `1/x`:: + + >>> quad(lambda x: 1/x, [1, 5]) + 1.6094379124341 + >>> log(5) + 1.6094379124341 + >>> diff(log, 10) + 0.1 + +The Taylor series expansion of the natural logarithm around +`x = 1` has coefficients `(-1)^{n+1}/n`:: + + >>> nprint(taylor(log, 1, 7)) + [0.0, 1.0, -0.5, 0.333333, -0.25, 0.2, -0.166667, 0.142857] + +:func:`~mpmath.log` supports arbitrary precision evaluation:: + + >>> mp.dps = 50 + >>> log(pi) + 1.1447298858494001741434273513530587116472948129153 + >>> log(pi, pi**3) + 0.33333333333333333333333333333333333333333333333333 + >>> mp.dps = 25 + >>> log(3+4j) + (1.609437912434100374600759 + 0.9272952180016122324285125j) +""" + +log10 = r""" +Computes the base-10 logarithm of `x`, `\log_{10}(x)`. ``log10(x)`` +is equivalent to ``log(x, 10)``. +""" + +fmod = r""" +Converts `x` and `y` to mpmath numbers and returns `x \mod y`. +For mpmath numbers, this is equivalent to ``x % y``. + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> fmod(100, pi) + 2.61062773871641 + +You can use :func:`~mpmath.fmod` to compute fractional parts of numbers:: + + >>> fmod(10.25, 1) + 0.25 + +""" + +radians = r""" +Converts the degree angle `x` to radians:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> radians(60) + 1.0471975511966 +""" + +degrees = r""" +Converts the radian angle `x` to a degree angle:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> degrees(pi/3) + 60.0 +""" + +atan2 = r""" +Computes the two-argument arctangent, `\mathrm{atan2}(y, x)`, +giving the signed angle between the positive `x`-axis and the +point `(x, y)` in the 2D plane. This function is defined for +real `x` and `y` only. + +The two-argument arctangent essentially computes +`\mathrm{atan}(y/x)`, but accounts for the signs of both +`x` and `y` to give the angle for the correct quadrant. The +following examples illustrate the difference:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> atan2(1,1), atan(1/1.) + (0.785398163397448, 0.785398163397448) + >>> atan2(1,-1), atan(1/-1.) + (2.35619449019234, -0.785398163397448) + >>> atan2(-1,1), atan(-1/1.) + (-0.785398163397448, -0.785398163397448) + >>> atan2(-1,-1), atan(-1/-1.) + (-2.35619449019234, 0.785398163397448) + +The angle convention is the same as that used for the complex +argument; see :func:`~mpmath.arg`. +""" + +fibonacci = r""" +``fibonacci(n)`` computes the `n`-th Fibonacci number, `F(n)`. The +Fibonacci numbers are defined by the recurrence `F(n) = F(n-1) + F(n-2)` +with the initial values `F(0) = 0`, `F(1) = 1`. :func:`~mpmath.fibonacci` +extends this definition to arbitrary real and complex arguments +using the formula + +.. math :: + + F(z) = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5} + +where `\phi` is the golden ratio. :func:`~mpmath.fibonacci` also uses this +continuous formula to compute `F(n)` for extremely large `n`, where +calculating the exact integer would be wasteful. + +For convenience, :func:`~mpmath.fib` is available as an alias for +:func:`~mpmath.fibonacci`. + +**Basic examples** + +Some small Fibonacci numbers are:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for i in range(10): + ... print(fibonacci(i)) + ... + 0.0 + 1.0 + 1.0 + 2.0 + 3.0 + 5.0 + 8.0 + 13.0 + 21.0 + 34.0 + >>> fibonacci(50) + 12586269025.0 + +The recurrence for `F(n)` extends backwards to negative `n`:: + + >>> for i in range(10): + ... print(fibonacci(-i)) + ... + 0.0 + 1.0 + -1.0 + 2.0 + -3.0 + 5.0 + -8.0 + 13.0 + -21.0 + 34.0 + +Large Fibonacci numbers will be computed approximately unless +the precision is set high enough:: + + >>> fib(200) + 2.8057117299251e+41 + >>> mp.dps = 45 + >>> fib(200) + 280571172992510140037611932413038677189525.0 + +:func:`~mpmath.fibonacci` can compute approximate Fibonacci numbers +of stupendous size:: + + >>> mp.dps = 15 + >>> fibonacci(10**25) + 3.49052338550226e+2089876402499787337692720 + +**Real and complex arguments** + +The extended Fibonacci function is an analytic function. The +property `F(z) = F(z-1) + F(z-2)` holds for arbitrary `z`:: + + >>> mp.dps = 15 + >>> fib(pi) + 2.1170270579161 + >>> fib(pi-1) + fib(pi-2) + 2.1170270579161 + >>> fib(3+4j) + (-5248.51130728372 - 14195.962288353j) + >>> fib(2+4j) + fib(1+4j) + (-5248.51130728372 - 14195.962288353j) + +The Fibonacci function has infinitely many roots on the +negative half-real axis. The first root is at 0, the second is +close to -0.18, and then there are infinitely many roots that +asymptotically approach `-n+1/2`:: + + >>> findroot(fib, -0.2) + -0.183802359692956 + >>> findroot(fib, -2) + -1.57077646820395 + >>> findroot(fib, -17) + -16.4999999596115 + >>> findroot(fib, -24) + -23.5000000000479 + +**Mathematical relationships** + +For large `n`, `F(n+1)/F(n)` approaches the golden ratio:: + + >>> mp.dps = 50 + >>> fibonacci(101)/fibonacci(100) + 1.6180339887498948482045868343656381177203127439638 + >>> +phi + 1.6180339887498948482045868343656381177203091798058 + +The sum of reciprocal Fibonacci numbers converges to an irrational +number for which no closed form expression is known:: + + >>> mp.dps = 15 + >>> nsum(lambda n: 1/fib(n), [1, inf]) + 3.35988566624318 + +Amazingly, however, the sum of odd-index reciprocal Fibonacci +numbers can be expressed in terms of a Jacobi theta function:: + + >>> nsum(lambda n: 1/fib(2*n+1), [0, inf]) + 1.82451515740692 + >>> sqrt(5)*jtheta(2,0,(3-sqrt(5))/2)**2/4 + 1.82451515740692 + +Some related sums can be done in closed form:: + + >>> nsum(lambda k: 1/(1+fib(2*k+1)), [0, inf]) + 1.11803398874989 + >>> phi - 0.5 + 1.11803398874989 + >>> f = lambda k:(-1)**(k+1) / sum(fib(n)**2 for n in range(1,int(k+1))) + >>> nsum(f, [1, inf]) + 0.618033988749895 + >>> phi-1 + 0.618033988749895 + +**References** + +1. http://mathworld.wolfram.com/FibonacciNumber.html +""" + +altzeta = r""" +Gives the Dirichlet eta function, `\eta(s)`, also known as the +alternating zeta function. This function is defined in analogy +with the Riemann zeta function as providing the sum of the +alternating series + +.. math :: + + \eta(s) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k^s} + = 1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+\ldots + +The eta function, unlike the Riemann zeta function, is an entire +function, having a finite value for all complex `s`. The special case +`\eta(1) = \log(2)` gives the value of the alternating harmonic series. + +The alternating zeta function may expressed using the Riemann zeta function +as `\eta(s) = (1 - 2^{1-s}) \zeta(s)`. It can also be expressed +in terms of the Hurwitz zeta function, for example using +:func:`~mpmath.dirichlet` (see documentation for that function). + +**Examples** + +Some special values are:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> altzeta(1) + 0.693147180559945 + >>> altzeta(0) + 0.5 + >>> altzeta(-1) + 0.25 + >>> altzeta(-2) + 0.0 + +An example of a sum that can be computed more accurately and +efficiently via :func:`~mpmath.altzeta` than via numerical summation:: + + >>> sum(-(-1)**n / mpf(n)**2.5 for n in range(1, 100)) + 0.867204951503984 + >>> altzeta(2.5) + 0.867199889012184 + +At positive even integers, the Dirichlet eta function +evaluates to a rational multiple of a power of `\pi`:: + + >>> altzeta(2) + 0.822467033424113 + >>> pi**2/12 + 0.822467033424113 + +Like the Riemann zeta function, `\eta(s)`, approaches 1 +as `s` approaches positive infinity, although it does +so from below rather than from above:: + + >>> altzeta(30) + 0.999999999068682 + >>> altzeta(inf) + 1.0 + >>> mp.pretty = False + >>> altzeta(1000, rounding='d') + mpf('0.99999999999999989') + >>> altzeta(1000, rounding='u') + mpf('1.0') + +**References** + +1. http://mathworld.wolfram.com/DirichletEtaFunction.html + +2. http://en.wikipedia.org/wiki/Dirichlet_eta_function +""" + +factorial = r""" +Computes the factorial, `x!`. For integers `n \ge 0`, we have +`n! = 1 \cdot 2 \cdots (n-1) \cdot n` and more generally the factorial +is defined for real or complex `x` by `x! = \Gamma(x+1)`. + +**Examples** + +Basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for k in range(6): + ... print("%s %s" % (k, fac(k))) + ... + 0 1.0 + 1 1.0 + 2 2.0 + 3 6.0 + 4 24.0 + 5 120.0 + >>> fac(inf) + +inf + >>> fac(0.5), sqrt(pi)/2 + (0.886226925452758, 0.886226925452758) + +For large positive `x`, `x!` can be approximated by +Stirling's formula:: + + >>> x = 10**10 + >>> fac(x) + 2.32579620567308e+95657055186 + >>> sqrt(2*pi*x)*(x/e)**x + 2.32579597597705e+95657055186 + +:func:`~mpmath.fac` supports evaluation for astronomically large values:: + + >>> fac(10**30) + 6.22311232304258e+29565705518096748172348871081098 + +Reciprocal factorials appear in the Taylor series of the +exponential function (among many other contexts):: + + >>> nsum(lambda k: 1/fac(k), [0, inf]), exp(1) + (2.71828182845905, 2.71828182845905) + >>> nsum(lambda k: pi**k/fac(k), [0, inf]), exp(pi) + (23.1406926327793, 23.1406926327793) + +""" + +gamma = r""" +Computes the gamma function, `\Gamma(x)`. The gamma function is a +shifted version of the ordinary factorial, satisfying +`\Gamma(n) = (n-1)!` for integers `n > 0`. More generally, it +is defined by + +.. math :: + + \Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t}\, dt + +for any real or complex `x` with `\Re(x) > 0` and for `\Re(x) < 0` +by analytic continuation. + +**Examples** + +Basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for k in range(1, 6): + ... print("%s %s" % (k, gamma(k))) + ... + 1 1.0 + 2 1.0 + 3 2.0 + 4 6.0 + 5 24.0 + >>> gamma(inf) + +inf + >>> gamma(0) + Traceback (most recent call last): + ... + ValueError: gamma function pole + +The gamma function of a half-integer is a rational multiple of +`\sqrt{\pi}`:: + + >>> gamma(0.5), sqrt(pi) + (1.77245385090552, 1.77245385090552) + >>> gamma(1.5), sqrt(pi)/2 + (0.886226925452758, 0.886226925452758) + +We can check the integral definition:: + + >>> gamma(3.5) + 3.32335097044784 + >>> quad(lambda t: t**2.5*exp(-t), [0,inf]) + 3.32335097044784 + +:func:`~mpmath.gamma` supports arbitrary-precision evaluation and +complex arguments:: + + >>> mp.dps = 50 + >>> gamma(sqrt(3)) + 0.91510229697308632046045539308226554038315280564184 + >>> mp.dps = 25 + >>> gamma(2j) + (0.009902440080927490985955066 - 0.07595200133501806872408048j) + +Arguments can also be large. Note that the gamma function grows +very quickly:: + + >>> mp.dps = 15 + >>> gamma(10**20) + 1.9328495143101e+1956570551809674817225 + +**References** + +* [Spouge]_ + +""" + +psi = r""" +Gives the polygamma function of order `m` of `z`, `\psi^{(m)}(z)`. +Special cases are known as the *digamma function* (`\psi^{(0)}(z)`), +the *trigamma function* (`\psi^{(1)}(z)`), etc. The polygamma +functions are defined as the logarithmic derivatives of the gamma +function: + +.. math :: + + \psi^{(m)}(z) = \left(\frac{d}{dz}\right)^{m+1} \log \Gamma(z) + +In particular, `\psi^{(0)}(z) = \Gamma'(z)/\Gamma(z)`. In the +present implementation of :func:`~mpmath.psi`, the order `m` must be a +nonnegative integer, while the argument `z` may be an arbitrary +complex number (with exception for the polygamma function's poles +at `z = 0, -1, -2, \ldots`). + +**Examples** + +For various rational arguments, the polygamma function reduces to +a combination of standard mathematical constants:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> psi(0, 1), -euler + (-0.5772156649015328606065121, -0.5772156649015328606065121) + >>> psi(1, '1/4'), pi**2+8*catalan + (17.19732915450711073927132, 17.19732915450711073927132) + >>> psi(2, '1/2'), -14*apery + (-16.82879664423431999559633, -16.82879664423431999559633) + +The polygamma functions are derivatives of each other:: + + >>> diff(lambda x: psi(3, x), pi), psi(4, pi) + (-0.1105749312578862734526952, -0.1105749312578862734526952) + >>> quad(lambda x: psi(4, x), [2, 3]), psi(3,3)-psi(3,2) + (-0.375, -0.375) + +The digamma function diverges logarithmically as `z \to \infty`, +while higher orders tend to zero:: + + >>> psi(0,inf), psi(1,inf), psi(2,inf) + (+inf, 0.0, 0.0) + +Evaluation for a complex argument:: + + >>> psi(2, -1-2j) + (0.03902435405364952654838445 + 0.1574325240413029954685366j) + +Evaluation is supported for large orders `m` and/or large +arguments `z`:: + + >>> psi(3, 10**100) + 2.0e-300 + >>> psi(250, 10**30+10**20*j) + (-1.293142504363642687204865e-7010 + 3.232856260909107391513108e-7018j) + +**Application to infinite series** + +Any infinite series where the summand is a rational function of +the index `k` can be evaluated in closed form in terms of polygamma +functions of the roots and poles of the summand:: + + >>> a = sqrt(2) + >>> b = sqrt(3) + >>> nsum(lambda k: 1/((k+a)**2*(k+b)), [0, inf]) + 0.4049668927517857061917531 + >>> (psi(0,a)-psi(0,b)-a*psi(1,a)+b*psi(1,a))/(a-b)**2 + 0.4049668927517857061917531 + +This follows from the series representation (`m > 0`) + +.. math :: + + \psi^{(m)}(z) = (-1)^{m+1} m! \sum_{k=0}^{\infty} + \frac{1}{(z+k)^{m+1}}. + +Since the roots of a polynomial may be complex, it is sometimes +necessary to use the complex polygamma function to evaluate +an entirely real-valued sum:: + + >>> nsum(lambda k: 1/(k**2-2*k+3), [0, inf]) + 1.694361433907061256154665 + >>> nprint(polyroots([1,-2,3])) + [(1.0 - 1.41421j), (1.0 + 1.41421j)] + >>> r1 = 1-sqrt(2)*j + >>> r2 = r1.conjugate() + >>> (psi(0,-r2)-psi(0,-r1))/(r1-r2) + (1.694361433907061256154665 + 0.0j) + +""" + +digamma = r""" +Shortcut for ``psi(0,z)``. +""" + +harmonic = r""" +If `n` is an integer, ``harmonic(n)`` gives a floating-point +approximation of the `n`-th harmonic number `H(n)`, defined as + +.. math :: + + H(n) = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} + +The first few harmonic numbers are:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(8): + ... print("%s %s" % (n, harmonic(n))) + ... + 0 0.0 + 1 1.0 + 2 1.5 + 3 1.83333333333333 + 4 2.08333333333333 + 5 2.28333333333333 + 6 2.45 + 7 2.59285714285714 + +The infinite harmonic series `1 + 1/2 + 1/3 + \ldots` diverges:: + + >>> harmonic(inf) + +inf + +:func:`~mpmath.harmonic` is evaluated using the digamma function rather +than by summing the harmonic series term by term. It can therefore +be computed quickly for arbitrarily large `n`, and even for +nonintegral arguments:: + + >>> harmonic(10**100) + 230.835724964306 + >>> harmonic(0.5) + 0.613705638880109 + >>> harmonic(3+4j) + (2.24757548223494 + 0.850502209186044j) + +:func:`~mpmath.harmonic` supports arbitrary precision evaluation:: + + >>> mp.dps = 50 + >>> harmonic(11) + 3.0198773448773448773448773448773448773448773448773 + >>> harmonic(pi) + 1.8727388590273302654363491032336134987519132374152 + +The harmonic series diverges, but at a glacial pace. It is possible +to calculate the exact number of terms required before the sum +exceeds a given amount, say 100:: + + >>> mp.dps = 50 + >>> v = 10**findroot(lambda x: harmonic(10**x) - 100, 10) + >>> v + 15092688622113788323693563264538101449859496.864101 + >>> v = int(ceil(v)) + >>> print(v) + 15092688622113788323693563264538101449859497 + >>> harmonic(v-1) + 99.999999999999999999999999999999999999999999942747 + >>> harmonic(v) + 100.000000000000000000000000000000000000000000009 + +""" + +bernoulli = r""" +Computes the nth Bernoulli number, `B_n`, for any integer `n \ge 0`. + +The Bernoulli numbers are rational numbers, but this function +returns a floating-point approximation. To obtain an exact +fraction, use :func:`~mpmath.bernfrac` instead. + +**Examples** + +Numerical values of the first few Bernoulli numbers:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(15): + ... print("%s %s" % (n, bernoulli(n))) + ... + 0 1.0 + 1 -0.5 + 2 0.166666666666667 + 3 0.0 + 4 -0.0333333333333333 + 5 0.0 + 6 0.0238095238095238 + 7 0.0 + 8 -0.0333333333333333 + 9 0.0 + 10 0.0757575757575758 + 11 0.0 + 12 -0.253113553113553 + 13 0.0 + 14 1.16666666666667 + +Bernoulli numbers can be approximated with arbitrary precision:: + + >>> mp.dps = 50 + >>> bernoulli(100) + -2.8382249570693706959264156336481764738284680928013e+78 + +Arbitrarily large `n` are supported:: + + >>> mp.dps = 15 + >>> bernoulli(10**20 + 2) + 3.09136296657021e+1876752564973863312327 + +The Bernoulli numbers are related to the Riemann zeta function +at integer arguments:: + + >>> -bernoulli(8) * (2*pi)**8 / (2*fac(8)) + 1.00407735619794 + >>> zeta(8) + 1.00407735619794 + +**Algorithm** + +For small `n` (`n < 3000`) :func:`~mpmath.bernoulli` uses a recurrence +formula due to Ramanujan. All results in this range are cached, +so sequential computation of small Bernoulli numbers is +guaranteed to be fast. + +For larger `n`, `B_n` is evaluated in terms of the Riemann zeta +function. +""" + +stieltjes = r""" +For a nonnegative integer `n`, ``stieltjes(n)`` computes the +`n`-th Stieltjes constant `\gamma_n`, defined as the +`n`-th coefficient in the Laurent series expansion of the +Riemann zeta function around the pole at `s = 1`. That is, +we have: + +.. math :: + + \zeta(s) = \frac{1}{s-1} \sum_{n=0}^{\infty} + \frac{(-1)^n}{n!} \gamma_n (s-1)^n + +More generally, ``stieltjes(n, a)`` gives the corresponding +coefficient `\gamma_n(a)` for the Hurwitz zeta function +`\zeta(s,a)` (with `\gamma_n = \gamma_n(1)`). + +**Examples** + +The zeroth Stieltjes constant is just Euler's constant `\gamma`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> stieltjes(0) + 0.577215664901533 + +Some more values are:: + + >>> stieltjes(1) + -0.0728158454836767 + >>> stieltjes(10) + 0.000205332814909065 + >>> stieltjes(30) + 0.00355772885557316 + >>> stieltjes(1000) + -1.57095384420474e+486 + >>> stieltjes(2000) + 2.680424678918e+1109 + >>> stieltjes(1, 2.5) + -0.23747539175716 + +An alternative way to compute `\gamma_1`:: + + >>> diff(extradps(15)(lambda x: 1/(x-1) - zeta(x)), 1) + -0.0728158454836767 + +:func:`~mpmath.stieltjes` supports arbitrary precision evaluation:: + + >>> mp.dps = 50 + >>> stieltjes(2) + -0.0096903631928723184845303860352125293590658061013408 + +**Algorithm** + +:func:`~mpmath.stieltjes` numerically evaluates the integral in +the following representation due to Ainsworth, Howell and +Coffey [1], [2]: + +.. math :: + + \gamma_n(a) = \frac{\log^n a}{2a} - \frac{\log^{n+1}(a)}{n+1} + + \frac{2}{a} \Re \int_0^{\infty} + \frac{(x/a-i)\log^n(a-ix)}{(1+x^2/a^2)(e^{2\pi x}-1)} dx. + +For some reference values with `a = 1`, see e.g. [4]. + +**References** + +1. O. R. Ainsworth & L. W. Howell, "An integral representation of + the generalized Euler-Mascheroni constants", NASA Technical + Paper 2456 (1985), + http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19850014994_1985014994.pdf + +2. M. W. Coffey, "The Stieltjes constants, their relation to the + `\eta_j` coefficients, and representation of the Hurwitz + zeta function", arXiv:0706.0343v1 http://arxiv.org/abs/0706.0343 + +3. http://mathworld.wolfram.com/StieltjesConstants.html + +4. http://pi.lacim.uqam.ca/piDATA/stieltjesgamma.txt + +""" + +gammaprod = r""" +Given iterables `a` and `b`, ``gammaprod(a, b)`` computes the +product / quotient of gamma functions: + +.. math :: + + \frac{\Gamma(a_0) \Gamma(a_1) \cdots \Gamma(a_p)} + {\Gamma(b_0) \Gamma(b_1) \cdots \Gamma(b_q)} + +Unlike direct calls to :func:`~mpmath.gamma`, :func:`~mpmath.gammaprod` considers +the entire product as a limit and evaluates this limit properly if +any of the numerator or denominator arguments are nonpositive +integers such that poles of the gamma function are encountered. +That is, :func:`~mpmath.gammaprod` evaluates + +.. math :: + + \lim_{\epsilon \to 0} + \frac{\Gamma(a_0+\epsilon) \Gamma(a_1+\epsilon) \cdots + \Gamma(a_p+\epsilon)} + {\Gamma(b_0+\epsilon) \Gamma(b_1+\epsilon) \cdots + \Gamma(b_q+\epsilon)} + +In particular: + +* If there are equally many poles in the numerator and the + denominator, the limit is a rational number times the remaining, + regular part of the product. + +* If there are more poles in the numerator, :func:`~mpmath.gammaprod` + returns ``+inf``. + +* If there are more poles in the denominator, :func:`~mpmath.gammaprod` + returns 0. + +**Examples** + +The reciprocal gamma function `1/\Gamma(x)` evaluated at `x = 0`:: + + >>> from mpmath import * + >>> mp.dps = 15 + >>> gammaprod([], [0]) + 0.0 + +A limit:: + + >>> gammaprod([-4], [-3]) + -0.25 + >>> limit(lambda x: gamma(x-1)/gamma(x), -3, direction=1) + -0.25 + >>> limit(lambda x: gamma(x-1)/gamma(x), -3, direction=-1) + -0.25 + +""" + +beta = r""" +Computes the beta function, +`B(x,y) = \Gamma(x) \Gamma(y) / \Gamma(x+y)`. +The beta function is also commonly defined by the integral +representation + +.. math :: + + B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt + +**Examples** + +For integer and half-integer arguments where all three gamma +functions are finite, the beta function becomes either rational +number or a rational multiple of `\pi`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> beta(5, 2) + 0.0333333333333333 + >>> beta(1.5, 2) + 0.266666666666667 + >>> 16*beta(2.5, 1.5) + 3.14159265358979 + +Where appropriate, :func:`~mpmath.beta` evaluates limits. A pole +of the beta function is taken to result in ``+inf``:: + + >>> beta(-0.5, 0.5) + 0.0 + >>> beta(-3, 3) + -0.333333333333333 + >>> beta(-2, 3) + +inf + >>> beta(inf, 1) + 0.0 + >>> beta(inf, 0) + nan + +:func:`~mpmath.beta` supports complex numbers and arbitrary precision +evaluation:: + + >>> beta(1, 2+j) + (0.4 - 0.2j) + >>> mp.dps = 25 + >>> beta(j,0.5) + (1.079424249270925780135675 - 1.410032405664160838288752j) + >>> mp.dps = 50 + >>> beta(pi, e) + 0.037890298781212201348153837138927165984170287886464 + +Various integrals can be computed by means of the +beta function:: + + >>> mp.dps = 15 + >>> quad(lambda t: t**2.5*(1-t)**2, [0, 1]) + 0.0230880230880231 + >>> beta(3.5, 3) + 0.0230880230880231 + >>> quad(lambda t: sin(t)**4 * sqrt(cos(t)), [0, pi/2]) + 0.319504062596158 + >>> beta(2.5, 0.75)/2 + 0.319504062596158 + +""" + +betainc = r""" +``betainc(a, b, x1=0, x2=1, regularized=False)`` gives the generalized +incomplete beta function, + +.. math :: + + I_{x_1}^{x_2}(a,b) = \int_{x_1}^{x_2} t^{a-1} (1-t)^{b-1} dt. + +When `x_1 = 0, x_2 = 1`, this reduces to the ordinary (complete) +beta function `B(a,b)`; see :func:`~mpmath.beta`. + +With the keyword argument ``regularized=True``, :func:`~mpmath.betainc` +computes the regularized incomplete beta function +`I_{x_1}^{x_2}(a,b) / B(a,b)`. This is the cumulative distribution of the +beta distribution with parameters `a`, `b`. + +.. note : + + Implementations of the incomplete beta function in some other + software uses a different argument order. For example, Mathematica uses the + reversed argument order ``Beta[x1,x2,a,b]``. For the equivalent of SciPy's + three-argument incomplete beta integral (implicitly with `x1 = 0`), use + ``betainc(a,b,0,x2,regularized=True)``. + +**Examples** + +Verifying that :func:`~mpmath.betainc` computes the integral in the +definition:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> x,y,a,b = 3, 4, 0, 6 + >>> betainc(x, y, a, b) + -4010.4 + >>> quad(lambda t: t**(x-1) * (1-t)**(y-1), [a, b]) + -4010.4 + +The arguments may be arbitrary complex numbers:: + + >>> betainc(0.75, 1-4j, 0, 2+3j) + (0.2241657956955709603655887 + 0.3619619242700451992411724j) + +With regularization:: + + >>> betainc(1, 2, 0, 0.25, regularized=True) + 0.4375 + >>> betainc(pi, e, 0, 1, regularized=True) # Complete + 1.0 + +The beta integral satisfies some simple argument transformation +symmetries:: + + >>> mp.dps = 15 + >>> betainc(2,3,4,5), -betainc(2,3,5,4), betainc(3,2,1-5,1-4) + (56.0833333333333, 56.0833333333333, 56.0833333333333) + +The beta integral can often be evaluated analytically. For integer and +rational arguments, the incomplete beta function typically reduces to a +simple algebraic-logarithmic expression:: + + >>> mp.dps = 25 + >>> identify(chop(betainc(0, 0, 3, 4))) + '-(log((9/8)))' + >>> identify(betainc(2, 3, 4, 5)) + '(673/12)' + >>> identify(betainc(1.5, 1, 1, 2)) + '((-12+sqrt(1152))/18)' + +""" + +binomial = r""" +Computes the binomial coefficient + +.. math :: + + {n \choose k} = \frac{n!}{k!(n-k)!}. + +The binomial coefficient gives the number of ways that `k` items +can be chosen from a set of `n` items. More generally, the binomial +coefficient is a well-defined function of arbitrary real or +complex `n` and `k`, via the gamma function. + +**Examples** + +Generate Pascal's triangle:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(5): + ... nprint([binomial(n,k) for k in range(n+1)]) + ... + [1.0] + [1.0, 1.0] + [1.0, 2.0, 1.0] + [1.0, 3.0, 3.0, 1.0] + [1.0, 4.0, 6.0, 4.0, 1.0] + +There is 1 way to select 0 items from the empty set, and 0 ways to +select 1 item from the empty set:: + + >>> binomial(0, 0) + 1.0 + >>> binomial(0, 1) + 0.0 + +:func:`~mpmath.binomial` supports large arguments:: + + >>> binomial(10**20, 10**20-5) + 8.33333333333333e+97 + >>> binomial(10**20, 10**10) + 2.60784095465201e+104342944813 + +Nonintegral binomial coefficients find use in series +expansions:: + + >>> nprint(taylor(lambda x: (1+x)**0.25, 0, 4)) + [1.0, 0.25, -0.09375, 0.0546875, -0.0375977] + >>> nprint([binomial(0.25, k) for k in range(5)]) + [1.0, 0.25, -0.09375, 0.0546875, -0.0375977] + +An integral representation:: + + >>> n, k = 5, 3 + >>> f = lambda t: exp(-j*k*t)*(1+exp(j*t))**n + >>> chop(quad(f, [-pi,pi])/(2*pi)) + 10.0 + >>> binomial(n,k) + 10.0 + +""" + +rf = r""" +Computes the rising factorial or Pochhammer symbol, + +.. math :: + + x^{(n)} = x (x+1) \cdots (x+n-1) = \frac{\Gamma(x+n)}{\Gamma(x)} + +where the rightmost expression is valid for nonintegral `n`. + +**Examples** + +For integral `n`, the rising factorial is a polynomial:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(5): + ... nprint(taylor(lambda x: rf(x,n), 0, n)) + ... + [1.0] + [0.0, 1.0] + [0.0, 1.0, 1.0] + [0.0, 2.0, 3.0, 1.0] + [0.0, 6.0, 11.0, 6.0, 1.0] + +Evaluation is supported for arbitrary arguments:: + + >>> rf(2+3j, 5.5) + (-7202.03920483347 - 3777.58810701527j) +""" + +ff = r""" +Computes the falling factorial, + +.. math :: + + (x)_n = x (x-1) \cdots (x-n+1) = \frac{\Gamma(x+1)}{\Gamma(x-n+1)} + +where the rightmost expression is valid for nonintegral `n`. + +**Examples** + +For integral `n`, the falling factorial is a polynomial:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(5): + ... nprint(taylor(lambda x: ff(x,n), 0, n)) + ... + [1.0] + [0.0, 1.0] + [0.0, -1.0, 1.0] + [0.0, 2.0, -3.0, 1.0] + [0.0, -6.0, 11.0, -6.0, 1.0] + +Evaluation is supported for arbitrary arguments:: + + >>> ff(2+3j, 5.5) + (-720.41085888203 + 316.101124983878j) +""" + +fac2 = r""" +Computes the double factorial `x!!`, defined for integers +`x > 0` by + +.. math :: + + x!! = \begin{cases} + 1 \cdot 3 \cdots (x-2) \cdot x & x \;\mathrm{odd} \\ + 2 \cdot 4 \cdots (x-2) \cdot x & x \;\mathrm{even} + \end{cases} + +and more generally by [1] + +.. math :: + + x!! = 2^{x/2} \left(\frac{\pi}{2}\right)^{(\cos(\pi x)-1)/4} + \Gamma\left(\frac{x}{2}+1\right). + +**Examples** + +The integer sequence of double factorials begins:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> nprint([fac2(n) for n in range(10)]) + [1.0, 1.0, 2.0, 3.0, 8.0, 15.0, 48.0, 105.0, 384.0, 945.0] + +For large `x`, double factorials follow a Stirling-like asymptotic +approximation:: + + >>> x = mpf(10000) + >>> fac2(x) + 5.97272691416282e+17830 + >>> sqrt(pi)*x**((x+1)/2)*exp(-x/2) + 5.97262736954392e+17830 + +The recurrence formula `x!! = x (x-2)!!` can be reversed to +define the double factorial of negative odd integers (but +not negative even integers):: + + >>> fac2(-1), fac2(-3), fac2(-5), fac2(-7) + (1.0, -1.0, 0.333333333333333, -0.0666666666666667) + >>> fac2(-2) + Traceback (most recent call last): + ... + ValueError: gamma function pole + +With the exception of the poles at negative even integers, +:func:`~mpmath.fac2` supports evaluation for arbitrary complex arguments. +The recurrence formula is valid generally:: + + >>> fac2(pi+2j) + (-1.3697207890154e-12 + 3.93665300979176e-12j) + >>> (pi+2j)*fac2(pi-2+2j) + (-1.3697207890154e-12 + 3.93665300979176e-12j) + +Double factorials should not be confused with nested factorials, +which are immensely larger:: + + >>> fac(fac(20)) + 5.13805976125208e+43675043585825292774 + >>> fac2(20) + 3715891200.0 + +Double factorials appear, among other things, in series expansions +of Gaussian functions and the error function. Infinite series +include:: + + >>> nsum(lambda k: 1/fac2(k), [0, inf]) + 3.05940740534258 + >>> sqrt(e)*(1+sqrt(pi/2)*erf(sqrt(2)/2)) + 3.05940740534258 + >>> nsum(lambda k: 2**k/fac2(2*k-1), [1, inf]) + 4.06015693855741 + >>> e * erf(1) * sqrt(pi) + 4.06015693855741 + +A beautiful Ramanujan sum:: + + >>> nsum(lambda k: (-1)**k*(fac2(2*k-1)/fac2(2*k))**3, [0,inf]) + 0.90917279454693 + >>> (gamma('9/8')/gamma('5/4')/gamma('7/8'))**2 + 0.90917279454693 + +**References** + +1. http://functions.wolfram.com/GammaBetaErf/Factorial2/27/01/0002/ + +2. http://mathworld.wolfram.com/DoubleFactorial.html + +""" + +hyper = r""" +Evaluates the generalized hypergeometric function + +.. math :: + + \,_pF_q(a_1,\ldots,a_p; b_1,\ldots,b_q; z) = + \sum_{n=0}^\infty \frac{(a_1)_n (a_2)_n \ldots (a_p)_n} + {(b_1)_n(b_2)_n\ldots(b_q)_n} \frac{z^n}{n!} + +where `(x)_n` denotes the rising factorial (see :func:`~mpmath.rf`). + +The parameters lists ``a_s`` and ``b_s`` may contain integers, +real numbers, complex numbers, as well as exact fractions given in +the form of tuples `(p, q)`. :func:`~mpmath.hyper` is optimized to handle +integers and fractions more efficiently than arbitrary +floating-point parameters (since rational parameters are by +far the most common). + +**Examples** + +Verifying that :func:`~mpmath.hyper` gives the sum in the definition, by +comparison with :func:`~mpmath.nsum`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> a,b,c,d = 2,3,4,5 + >>> x = 0.25 + >>> hyper([a,b],[c,d],x) + 1.078903941164934876086237 + >>> fn = lambda n: rf(a,n)*rf(b,n)/rf(c,n)/rf(d,n)*x**n/fac(n) + >>> nsum(fn, [0, inf]) + 1.078903941164934876086237 + +The parameters can be any combination of integers, fractions, +floats and complex numbers:: + + >>> a, b, c, d, e = 1, (-1,2), pi, 3+4j, (2,3) + >>> x = 0.2j + >>> hyper([a,b],[c,d,e],x) + (0.9923571616434024810831887 - 0.005753848733883879742993122j) + >>> b, e = -0.5, mpf(2)/3 + >>> fn = lambda n: rf(a,n)*rf(b,n)/rf(c,n)/rf(d,n)/rf(e,n)*x**n/fac(n) + >>> nsum(fn, [0, inf]) + (0.9923571616434024810831887 - 0.005753848733883879742993122j) + +The `\,_0F_0` and `\,_1F_0` series are just elementary functions:: + + >>> a, z = sqrt(2), +pi + >>> hyper([],[],z) + 23.14069263277926900572909 + >>> exp(z) + 23.14069263277926900572909 + >>> hyper([a],[],z) + (-0.09069132879922920160334114 + 0.3283224323946162083579656j) + >>> (1-z)**(-a) + (-0.09069132879922920160334114 + 0.3283224323946162083579656j) + +If any `a_k` coefficient is a nonpositive integer, the series terminates +into a finite polynomial:: + + >>> hyper([1,1,1,-3],[2,5],1) + 0.7904761904761904761904762 + >>> identify(_) + '(83/105)' + +If any `b_k` is a nonpositive integer, the function is undefined (unless the +series terminates before the division by zero occurs):: + + >>> hyper([1,1,1,-3],[-2,5],1) + Traceback (most recent call last): + ... + ZeroDivisionError: pole in hypergeometric series + >>> hyper([1,1,1,-1],[-2,5],1) + 1.1 + +Except for polynomial cases, the radius of convergence `R` of the hypergeometric +series is either `R = \infty` (if `p \le q`), `R = 1` (if `p = q+1`), or +`R = 0` (if `p > q+1`). + +The analytic continuations of the functions with `p = q+1`, i.e. `\,_2F_1`, +`\,_3F_2`, `\,_4F_3`, etc, are all implemented and therefore these functions +can be evaluated for `|z| \ge 1`. The shortcuts :func:`~mpmath.hyp2f1`, :func:`~mpmath.hyp3f2` +are available to handle the most common cases (see their documentation), +but functions of higher degree are also supported via :func:`~mpmath.hyper`:: + + >>> hyper([1,2,3,4], [5,6,7], 1) # 4F3 at finite-valued branch point + 1.141783505526870731311423 + >>> hyper([4,5,6,7], [1,2,3], 1) # 4F3 at pole + +inf + >>> hyper([1,2,3,4,5], [6,7,8,9], 10) # 5F4 + (1.543998916527972259717257 - 0.5876309929580408028816365j) + >>> hyper([1,2,3,4,5,6], [7,8,9,10,11], 1j) # 6F5 + (0.9996565821853579063502466 + 0.0129721075905630604445669j) + +Near `z = 1` with noninteger parameters:: + + >>> hyper(['1/3',1,'3/2',2], ['1/5','11/6','41/8'], 1) + 2.219433352235586121250027 + >>> hyper(['1/3',1,'3/2',2], ['1/5','11/6','5/4'], 1) + +inf + >>> eps1 = extradps(6)(lambda: 1 - mpf('1e-6'))() + >>> hyper(['1/3',1,'3/2',2], ['1/5','11/6','5/4'], eps1) + 2923978034.412973409330956 + +Please note that, as currently implemented, evaluation of `\,_pF_{p-1}` +with `p \ge 3` may be slow or inaccurate when `|z-1|` is small, +for some parameter values. + +Evaluation may be aborted if convergence appears to be too slow. +The optional ``maxterms`` (limiting the number of series terms) and ``maxprec`` +(limiting the internal precision) keyword arguments can be used +to control evaluation:: + + >>> hyper([1,2,3], [4,5,6], 10000) # doctest: +IGNORE_EXCEPTION_DETAIL + Traceback (most recent call last): + ... + NoConvergence: Hypergeometric series converges too slowly. Try increasing maxterms. + >>> hyper([1,2,3], [4,5,6], 10000, maxterms=10**6) + 7.622806053177969474396918e+4310 + +Additional options include ``force_series`` (which forces direct use of +a hypergeometric series even if another evaluation method might work better) +and ``asymp_tol`` which controls the target tolerance for using +asymptotic series. + +When `p > q+1`, ``hyper`` computes the (iterated) Borel sum of the divergent +series. For `\,_2F_0` the Borel sum has an analytic solution and can be +computed efficiently (see :func:`~mpmath.hyp2f0`). For higher degrees, the functions +is evaluated first by attempting to sum it directly as an asymptotic +series (this only works for tiny `|z|`), and then by evaluating the Borel +regularized sum using numerical integration. Except for +special parameter combinations, this can be extremely slow. + + >>> hyper([1,1], [], 0.5) # regularization of 2F0 + (1.340965419580146562086448 + 0.8503366631752726568782447j) + >>> hyper([1,1,1,1], [1], 0.5) # regularization of 4F1 + (1.108287213689475145830699 + 0.5327107430640678181200491j) + +With the following magnitude of argument, the asymptotic series for `\,_3F_1` +gives only a few digits. Using Borel summation, ``hyper`` can produce +a value with full accuracy:: + + >>> mp.dps = 15 + >>> hyper([2,0.5,4], [5.25], '0.08', force_series=True) # doctest: +IGNORE_EXCEPTION_DETAIL + Traceback (most recent call last): + ... + NoConvergence: Hypergeometric series converges too slowly. Try increasing maxterms. + >>> hyper([2,0.5,4], [5.25], '0.08', asymp_tol=1e-4) + 1.0725535790737 + >>> hyper([2,0.5,4], [5.25], '0.08') + (1.07269542893559 + 5.54668863216891e-5j) + >>> hyper([2,0.5,4], [5.25], '-0.08', asymp_tol=1e-4) + 0.946344925484879 + >>> hyper([2,0.5,4], [5.25], '-0.08') + 0.946312503737771 + >>> mp.dps = 25 + >>> hyper([2,0.5,4], [5.25], '-0.08') + 0.9463125037377662296700858 + +Note that with the positive `z` value, there is a complex part in the +correct result, which falls below the tolerance of the asymptotic series. + +By default, a parameter that appears in both ``a_s`` and ``b_s`` will be removed +unless it is a nonpositive integer. This generally speeds up evaluation +by producing a hypergeometric function of lower order. +This optimization can be disabled by passing ``eliminate=False``. + + >>> hyper([1,2,3], [4,5,3], 10000) + 1.268943190440206905892212e+4321 + >>> hyper([1,2,3], [4,5,3], 10000, eliminate=False) # doctest: +IGNORE_EXCEPTION_DETAIL + Traceback (most recent call last): + ... + NoConvergence: Hypergeometric series converges too slowly. Try increasing maxterms. + >>> hyper([1,2,3], [4,5,3], 10000, eliminate=False, maxterms=10**6) + 1.268943190440206905892212e+4321 + +If a nonpositive integer `-n` appears in both ``a_s`` and ``b_s``, this parameter +cannot be unambiguously removed since it creates a term 0 / 0. +In this case the hypergeometric series is understood to terminate before +the division by zero occurs. This convention is consistent with Mathematica. +An alternative convention of eliminating the parameters can be toggled +with ``eliminate_all=True``: + + >>> hyper([2,-1], [-1], 3) + 7.0 + >>> hyper([2,-1], [-1], 3, eliminate_all=True) + 0.25 + >>> hyper([2], [], 3) + 0.25 + +""" + +hypercomb = r""" +Computes a weighted combination of hypergeometric functions + +.. math :: + + \sum_{r=1}^N \left[ \prod_{k=1}^{l_r} {w_{r,k}}^{c_{r,k}} + \frac{\prod_{k=1}^{m_r} \Gamma(\alpha_{r,k})}{\prod_{k=1}^{n_r} + \Gamma(\beta_{r,k})} + \,_{p_r}F_{q_r}(a_{r,1},\ldots,a_{r,p}; b_{r,1}, + \ldots, b_{r,q}; z_r)\right]. + +Typically the parameters are linear combinations of a small set of base +parameters; :func:`~mpmath.hypercomb` permits computing a correct value in +the case that some of the `\alpha`, `\beta`, `b` turn out to be +nonpositive integers, or if division by zero occurs for some `w^c`, +assuming that there are opposing singularities that cancel out. +The limit is computed by evaluating the function with the base +parameters perturbed, at a higher working precision. + +The first argument should be a function that takes the perturbable +base parameters ``params`` as input and returns `N` tuples +``(w, c, alpha, beta, a, b, z)``, where the coefficients ``w``, ``c``, +gamma factors ``alpha``, ``beta``, and hypergeometric coefficients +``a``, ``b`` each should be lists of numbers, and ``z`` should be a single +number. + +**Examples** + +The following evaluates + +.. math :: + + (a-1) \frac{\Gamma(a-3)}{\Gamma(a-4)} \,_1F_1(a,a-1,z) = e^z(a-4)(a+z-1) + +with `a=1, z=3`. There is a zero factor, two gamma function poles, and +the 1F1 function is singular; all singularities cancel out to give a finite +value:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> hypercomb(lambda a: [([a-1],[1],[a-3],[a-4],[a],[a-1],3)], [1]) + -180.769832308689 + >>> -9*exp(3) + -180.769832308689 + +""" + +hyp0f1 = r""" +Gives the hypergeometric function `\,_0F_1`, sometimes known as the +confluent limit function, defined as + +.. math :: + + \,_0F_1(a,z) = \sum_{k=0}^{\infty} \frac{1}{(a)_k} \frac{z^k}{k!}. + +This function satisfies the differential equation `z f''(z) + a f'(z) = f(z)`, +and is related to the Bessel function of the first kind (see :func:`~mpmath.besselj`). + +``hyp0f1(a,z)`` is equivalent to ``hyper([],[a],z)``; see documentation for +:func:`~mpmath.hyper` for more information. + +**Examples** + +Evaluation for arbitrary arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hyp0f1(2, 0.25) + 1.130318207984970054415392 + >>> hyp0f1((1,2), 1234567) + 6.27287187546220705604627e+964 + >>> hyp0f1(3+4j, 1000000j) + (3.905169561300910030267132e+606 + 3.807708544441684513934213e+606j) + +Evaluation is supported for arbitrarily large values of `z`, +using asymptotic expansions:: + + >>> hyp0f1(1, 10**50) + 2.131705322874965310390701e+8685889638065036553022565 + >>> hyp0f1(1, -10**50) + 1.115945364792025420300208e-13 + +Verifying the differential equation:: + + >>> a = 2.5 + >>> f = lambda z: hyp0f1(a,z) + >>> for z in [0, 10, 3+4j]: + ... chop(z*diff(f,z,2) + a*diff(f,z) - f(z)) + ... + 0.0 + 0.0 + 0.0 + +""" + +hyp1f1 = r""" +Gives the confluent hypergeometric function of the first kind, + +.. math :: + + \,_1F_1(a,b,z) = \sum_{k=0}^{\infty} \frac{(a)_k}{(b)_k} \frac{z^k}{k!}, + +also known as Kummer's function and sometimes denoted by `M(a,b,z)`. This +function gives one solution to the confluent (Kummer's) differential equation + +.. math :: + + z f''(z) + (b-z) f'(z) - af(z) = 0. + +A second solution is given by the `U` function; see :func:`~mpmath.hyperu`. +Solutions are also given in an alternate form by the Whittaker +functions (:func:`~mpmath.whitm`, :func:`~mpmath.whitw`). + +``hyp1f1(a,b,z)`` is equivalent +to ``hyper([a],[b],z)``; see documentation for :func:`~mpmath.hyper` for more +information. + +**Examples** + +Evaluation for real and complex values of the argument `z`, with +fixed parameters `a = 2, b = -1/3`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hyp1f1(2, (-1,3), 3.25) + -2815.956856924817275640248 + >>> hyp1f1(2, (-1,3), -3.25) + -1.145036502407444445553107 + >>> hyp1f1(2, (-1,3), 1000) + -8.021799872770764149793693e+441 + >>> hyp1f1(2, (-1,3), -1000) + 0.000003131987633006813594535331 + >>> hyp1f1(2, (-1,3), 100+100j) + (-3.189190365227034385898282e+48 - 1.106169926814270418999315e+49j) + +Parameters may be complex:: + + >>> hyp1f1(2+3j, -1+j, 10j) + (261.8977905181045142673351 + 160.8930312845682213562172j) + +Arbitrarily large values of `z` are supported:: + + >>> hyp1f1(3, 4, 10**20) + 3.890569218254486878220752e+43429448190325182745 + >>> hyp1f1(3, 4, -10**20) + 6.0e-60 + >>> hyp1f1(3, 4, 10**20*j) + (-1.935753855797342532571597e-20 - 2.291911213325184901239155e-20j) + +Verifying the differential equation:: + + >>> a, b = 1.5, 2 + >>> f = lambda z: hyp1f1(a,b,z) + >>> for z in [0, -10, 3, 3+4j]: + ... chop(z*diff(f,z,2) + (b-z)*diff(f,z) - a*f(z)) + ... + 0.0 + 0.0 + 0.0 + 0.0 + +An integral representation:: + + >>> a, b = 1.5, 3 + >>> z = 1.5 + >>> hyp1f1(a,b,z) + 2.269381460919952778587441 + >>> g = lambda t: exp(z*t)*t**(a-1)*(1-t)**(b-a-1) + >>> gammaprod([b],[a,b-a])*quad(g, [0,1]) + 2.269381460919952778587441 + + +""" + +hyp1f2 = r""" +Gives the hypergeometric function `\,_1F_2(a_1,a_2;b_1,b_2; z)`. +The call ``hyp1f2(a1,b1,b2,z)`` is equivalent to +``hyper([a1],[b1,b2],z)``. + +Evaluation works for complex and arbitrarily large arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> a, b, c = 1.5, (-1,3), 2.25 + >>> hyp1f2(a, b, c, 10**20) + -1.159388148811981535941434e+8685889639 + >>> hyp1f2(a, b, c, -10**20) + -12.60262607892655945795907 + >>> hyp1f2(a, b, c, 10**20*j) + (4.237220401382240876065501e+6141851464 - 2.950930337531768015892987e+6141851464j) + >>> hyp1f2(2+3j, -2j, 0.5j, 10-20j) + (135881.9905586966432662004 - 86681.95885418079535738828j) + +""" + +hyp2f2 = r""" +Gives the hypergeometric function `\,_2F_2(a_1,a_2;b_1,b_2; z)`. +The call ``hyp2f2(a1,a2,b1,b2,z)`` is equivalent to +``hyper([a1,a2],[b1,b2],z)``. + +Evaluation works for complex and arbitrarily large arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> a, b, c, d = 1.5, (-1,3), 2.25, 4 + >>> hyp2f2(a, b, c, d, 10**20) + -5.275758229007902299823821e+43429448190325182663 + >>> hyp2f2(a, b, c, d, -10**20) + 2561445.079983207701073448 + >>> hyp2f2(a, b, c, d, 10**20*j) + (2218276.509664121194836667 - 1280722.539991603850462856j) + >>> hyp2f2(2+3j, -2j, 0.5j, 4j, 10-20j) + (80500.68321405666957342788 - 20346.82752982813540993502j) + +""" + +hyp2f3 = r""" +Gives the hypergeometric function `\,_2F_3(a_1,a_2;b_1,b_2,b_3; z)`. +The call ``hyp2f3(a1,a2,b1,b2,b3,z)`` is equivalent to +``hyper([a1,a2],[b1,b2,b3],z)``. + +Evaluation works for arbitrarily large arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> a1,a2,b1,b2,b3 = 1.5, (-1,3), 2.25, 4, (1,5) + >>> hyp2f3(a1,a2,b1,b2,b3,10**20) + -4.169178177065714963568963e+8685889590 + >>> hyp2f3(a1,a2,b1,b2,b3,-10**20) + 7064472.587757755088178629 + >>> hyp2f3(a1,a2,b1,b2,b3,10**20*j) + (-5.163368465314934589818543e+6141851415 + 1.783578125755972803440364e+6141851416j) + >>> hyp2f3(2+3j, -2j, 0.5j, 4j, -1-j, 10-20j) + (-2280.938956687033150740228 + 13620.97336609573659199632j) + >>> hyp2f3(2+3j, -2j, 0.5j, 4j, -1-j, 10000000-20000000j) + (4.849835186175096516193e+3504 - 3.365981529122220091353633e+3504j) + +""" + +hyp2f1 = r""" +Gives the Gauss hypergeometric function `\,_2F_1` (often simply referred to as +*the* hypergeometric function), defined for `|z| < 1` as + +.. math :: + + \,_2F_1(a,b,c,z) = \sum_{k=0}^{\infty} + \frac{(a)_k (b)_k}{(c)_k} \frac{z^k}{k!}. + +and for `|z| \ge 1` by analytic continuation, with a branch cut on `(1, \infty)` +when necessary. + +Special cases of this function include many of the orthogonal polynomials as +well as the incomplete beta function and other functions. Properties of the +Gauss hypergeometric function are documented comprehensively in many references, +for example Abramowitz & Stegun, section 15. + +The implementation supports the analytic continuation as well as evaluation +close to the unit circle where `|z| \approx 1`. The syntax ``hyp2f1(a,b,c,z)`` +is equivalent to ``hyper([a,b],[c],z)``. + +**Examples** + +Evaluation with `z` inside, outside and on the unit circle, for +fixed parameters:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hyp2f1(2, (1,2), 4, 0.75) + 1.303703703703703703703704 + >>> hyp2f1(2, (1,2), 4, -1.75) + 0.7431290566046919177853916 + >>> hyp2f1(2, (1,2), 4, 1.75) + (1.418075801749271137026239 - 1.114976146679907015775102j) + >>> hyp2f1(2, (1,2), 4, 1) + 1.6 + >>> hyp2f1(2, (1,2), 4, -1) + 0.8235498012182875315037882 + >>> hyp2f1(2, (1,2), 4, j) + (0.9144026291433065674259078 + 0.2050415770437884900574923j) + >>> hyp2f1(2, (1,2), 4, 2+j) + (0.9274013540258103029011549 + 0.7455257875808100868984496j) + >>> hyp2f1(2, (1,2), 4, 0.25j) + (0.9931169055799728251931672 + 0.06154836525312066938147793j) + +Evaluation with complex parameter values:: + + >>> hyp2f1(1+j, 0.75, 10j, 1+5j) + (0.8834833319713479923389638 + 0.7053886880648105068343509j) + +Evaluation with `z = 1`:: + + >>> hyp2f1(-2.5, 3.5, 1.5, 1) + 0.0 + >>> hyp2f1(-2.5, 3, 4, 1) + 0.06926406926406926406926407 + >>> hyp2f1(2, 3, 4, 1) + +inf + +Evaluation for huge arguments:: + + >>> hyp2f1((-1,3), 1.75, 4, '1e100') + (7.883714220959876246415651e+32 + 1.365499358305579597618785e+33j) + >>> hyp2f1((-1,3), 1.75, 4, '1e1000000') + (7.883714220959876246415651e+333332 + 1.365499358305579597618785e+333333j) + >>> hyp2f1((-1,3), 1.75, 4, '1e1000000j') + (1.365499358305579597618785e+333333 - 7.883714220959876246415651e+333332j) + +An integral representation:: + + >>> a,b,c,z = -0.5, 1, 2.5, 0.25 + >>> g = lambda t: t**(b-1) * (1-t)**(c-b-1) * (1-t*z)**(-a) + >>> gammaprod([c],[b,c-b]) * quad(g, [0,1]) + 0.9480458814362824478852618 + >>> hyp2f1(a,b,c,z) + 0.9480458814362824478852618 + +Verifying the hypergeometric differential equation:: + + >>> f = lambda z: hyp2f1(a,b,c,z) + >>> chop(z*(1-z)*diff(f,z,2) + (c-(a+b+1)*z)*diff(f,z) - a*b*f(z)) + 0.0 + +""" + +hyp3f2 = r""" +Gives the generalized hypergeometric function `\,_3F_2`, defined for `|z| < 1` +as + +.. math :: + + \,_3F_2(a_1,a_2,a_3,b_1,b_2,z) = \sum_{k=0}^{\infty} + \frac{(a_1)_k (a_2)_k (a_3)_k}{(b_1)_k (b_2)_k} \frac{z^k}{k!}. + +and for `|z| \ge 1` by analytic continuation. The analytic structure of this +function is similar to that of `\,_2F_1`, generally with a singularity at +`z = 1` and a branch cut on `(1, \infty)`. + +Evaluation is supported inside, on, and outside +the circle of convergence `|z| = 1`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hyp3f2(1,2,3,4,5,0.25) + 1.083533123380934241548707 + >>> hyp3f2(1,2+2j,3,4,5,-10+10j) + (0.1574651066006004632914361 - 0.03194209021885226400892963j) + >>> hyp3f2(1,2,3,4,5,-10) + 0.3071141169208772603266489 + >>> hyp3f2(1,2,3,4,5,10) + (-0.4857045320523947050581423 - 0.5988311440454888436888028j) + >>> hyp3f2(0.25,1,1,2,1.5,1) + 1.157370995096772047567631 + >>> (8-pi-2*ln2)/3 + 1.157370995096772047567631 + >>> hyp3f2(1+j,0.5j,2,1,-2j,-1) + (1.74518490615029486475959 + 0.1454701525056682297614029j) + >>> hyp3f2(1+j,0.5j,2,1,-2j,sqrt(j)) + (0.9829816481834277511138055 - 0.4059040020276937085081127j) + >>> hyp3f2(-3,2,1,-5,4,1) + 1.41 + >>> hyp3f2(-3,2,1,-5,4,2) + 2.12 + +Evaluation very close to the unit circle:: + + >>> hyp3f2(1,2,3,4,5,'1.0001') + (1.564877796743282766872279 - 3.76821518787438186031973e-11j) + >>> hyp3f2(1,2,3,4,5,'1+0.0001j') + (1.564747153061671573212831 + 0.0001305757570366084557648482j) + >>> hyp3f2(1,2,3,4,5,'0.9999') + 1.564616644881686134983664 + >>> hyp3f2(1,2,3,4,5,'-0.9999') + 0.7823896253461678060196207 + +.. note :: + + Evaluation for `|z-1|` small can currently be inaccurate or slow + for some parameter combinations. + +For various parameter combinations, `\,_3F_2` admits representation in terms +of hypergeometric functions of lower degree, or in terms of +simpler functions:: + + >>> for a, b, z in [(1,2,-1), (2,0.5,1)]: + ... hyp2f1(a,b,a+b+0.5,z)**2 + ... hyp3f2(2*a,a+b,2*b,a+b+0.5,2*a+2*b,z) + ... + 0.4246104461966439006086308 + 0.4246104461966439006086308 + 7.111111111111111111111111 + 7.111111111111111111111111 + + >>> z = 2+3j + >>> hyp3f2(0.5,1,1.5,2,2,z) + (0.7621440939243342419729144 + 0.4249117735058037649915723j) + >>> 4*(pi-2*ellipe(z))/(pi*z) + (0.7621440939243342419729144 + 0.4249117735058037649915723j) + +""" + +hyperu = r""" +Gives the Tricomi confluent hypergeometric function `U`, also known as +the Kummer or confluent hypergeometric function of the second kind. This +function gives a second linearly independent solution to the confluent +hypergeometric differential equation (the first is provided by `\,_1F_1` -- +see :func:`~mpmath.hyp1f1`). + +**Examples** + +Evaluation for arbitrary complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hyperu(2,3,4) + 0.0625 + >>> hyperu(0.25, 5, 1000) + 0.1779949416140579573763523 + >>> hyperu(0.25, 5, -1000) + (0.1256256609322773150118907 - 0.1256256609322773150118907j) + +The `U` function may be singular at `z = 0`:: + + >>> hyperu(1.5, 2, 0) + +inf + >>> hyperu(1.5, -2, 0) + 0.1719434921288400112603671 + +Verifying the differential equation:: + + >>> a, b = 1.5, 2 + >>> f = lambda z: hyperu(a,b,z) + >>> for z in [-10, 3, 3+4j]: + ... chop(z*diff(f,z,2) + (b-z)*diff(f,z) - a*f(z)) + ... + 0.0 + 0.0 + 0.0 + +An integral representation:: + + >>> a,b,z = 2, 3.5, 4.25 + >>> hyperu(a,b,z) + 0.06674960718150520648014567 + >>> quad(lambda t: exp(-z*t)*t**(a-1)*(1+t)**(b-a-1),[0,inf]) / gamma(a) + 0.06674960718150520648014567 + + +[1] http://people.math.sfu.ca/~cbm/aands/page_504.htm +""" + +hyp2f0 = r""" +Gives the hypergeometric function `\,_2F_0`, defined formally by the +series + +.. math :: + + \,_2F_0(a,b;;z) = \sum_{n=0}^{\infty} (a)_n (b)_n \frac{z^n}{n!}. + +This series usually does not converge. For small enough `z`, it can be viewed +as an asymptotic series that may be summed directly with an appropriate +truncation. When this is not the case, :func:`~mpmath.hyp2f0` gives a regularized sum, +or equivalently, it uses a representation in terms of the +hypergeometric U function [1]. The series also converges when either `a` or `b` +is a nonpositive integer, as it then terminates into a polynomial +after `-a` or `-b` terms. + +**Examples** + +Evaluation is supported for arbitrary complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hyp2f0((2,3), 1.25, -100) + 0.07095851870980052763312791 + >>> hyp2f0((2,3), 1.25, 100) + (-0.03254379032170590665041131 + 0.07269254613282301012735797j) + >>> hyp2f0(-0.75, 1-j, 4j) + (-0.3579987031082732264862155 - 3.052951783922142735255881j) + +Even with real arguments, the regularized value of 2F0 is often complex-valued, +but the imaginary part decreases exponentially as `z \to 0`. In the following +example, the first call uses complex evaluation while the second has a small +enough `z` to evaluate using the direct series and thus the returned value +is strictly real (this should be taken to indicate that the imaginary +part is less than ``eps``):: + + >>> mp.dps = 15 + >>> hyp2f0(1.5, 0.5, 0.05) + (1.04166637647907 + 8.34584913683906e-8j) + >>> hyp2f0(1.5, 0.5, 0.0005) + 1.00037535207621 + +The imaginary part can be retrieved by increasing the working precision:: + + >>> mp.dps = 80 + >>> nprint(hyp2f0(1.5, 0.5, 0.009).imag) + 1.23828e-46 + +In the polynomial case (the series terminating), 2F0 can evaluate exactly:: + + >>> mp.dps = 15 + >>> hyp2f0(-6,-6,2) + 291793.0 + >>> identify(hyp2f0(-2,1,0.25)) + '(5/8)' + +The coefficients of the polynomials can be recovered using Taylor expansion:: + + >>> nprint(taylor(lambda x: hyp2f0(-3,0.5,x), 0, 10)) + [1.0, -1.5, 2.25, -1.875, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] + >>> nprint(taylor(lambda x: hyp2f0(-4,0.5,x), 0, 10)) + [1.0, -2.0, 4.5, -7.5, 6.5625, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] + + +[1] http://people.math.sfu.ca/~cbm/aands/page_504.htm +""" + + +gammainc = r""" +``gammainc(z, a=0, b=inf)`` computes the (generalized) incomplete +gamma function with integration limits `[a, b]`: + +.. math :: + + \Gamma(z,a,b) = \int_a^b t^{z-1} e^{-t} \, dt + +The generalized incomplete gamma function reduces to the +following special cases when one or both endpoints are fixed: + +* `\Gamma(z,0,\infty)` is the standard ("complete") + gamma function, `\Gamma(z)` (available directly + as the mpmath function :func:`~mpmath.gamma`) +* `\Gamma(z,a,\infty)` is the "upper" incomplete gamma + function, `\Gamma(z,a)` +* `\Gamma(z,0,b)` is the "lower" incomplete gamma + function, `\gamma(z,b)`. + +Of course, we have +`\Gamma(z,0,x) + \Gamma(z,x,\infty) = \Gamma(z)` +for all `z` and `x`. + +Note however that some authors reverse the order of the +arguments when defining the lower and upper incomplete +gamma function, so one should be careful to get the correct +definition. + +If also given the keyword argument ``regularized=True``, +:func:`~mpmath.gammainc` computes the "regularized" incomplete gamma +function + +.. math :: + + P(z,a,b) = \frac{\Gamma(z,a,b)}{\Gamma(z)}. + +**Examples** + +We can compare with numerical quadrature to verify that +:func:`~mpmath.gammainc` computes the integral in the definition:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> gammainc(2+3j, 4, 10) + (0.00977212668627705160602312 - 0.0770637306312989892451977j) + >>> quad(lambda t: t**(2+3j-1) * exp(-t), [4, 10]) + (0.00977212668627705160602312 - 0.0770637306312989892451977j) + +Argument symmetries follow directly from the integral definition:: + + >>> gammainc(3, 4, 5) + gammainc(3, 5, 4) + 0.0 + >>> gammainc(3,0,2) + gammainc(3,2,4); gammainc(3,0,4) + 1.523793388892911312363331 + 1.523793388892911312363331 + >>> findroot(lambda z: gammainc(2,z,3), 1) + 3.0 + +Evaluation for arbitrarily large arguments:: + + >>> gammainc(10, 100) + 4.083660630910611272288592e-26 + >>> gammainc(10, 10000000000000000) + 5.290402449901174752972486e-4342944819032375 + >>> gammainc(3+4j, 1000000+1000000j) + (-1.257913707524362408877881e-434284 + 2.556691003883483531962095e-434284j) + +Evaluation of a generalized incomplete gamma function automatically chooses +the representation that gives a more accurate result, depending on which +parameter is larger:: + + >>> gammainc(10000000, 3) - gammainc(10000000, 2) # Bad + 0.0 + >>> gammainc(10000000, 2, 3) # Good + 1.755146243738946045873491e+4771204 + >>> gammainc(2, 0, 100000001) - gammainc(2, 0, 100000000) # Bad + 0.0 + >>> gammainc(2, 100000000, 100000001) # Good + 4.078258353474186729184421e-43429441 + +The incomplete gamma functions satisfy simple recurrence +relations:: + + >>> mp.dps = 25 + >>> z, a = mpf(3.5), mpf(2) + >>> gammainc(z+1, a); z*gammainc(z,a) + a**z*exp(-a) + 10.60130296933533459267329 + 10.60130296933533459267329 + >>> gammainc(z+1,0,a); z*gammainc(z,0,a) - a**z*exp(-a) + 1.030425427232114336470932 + 1.030425427232114336470932 + +Evaluation at integers and poles:: + + >>> gammainc(-3, -4, -5) + (-0.2214577048967798566234192 + 0.0j) + >>> gammainc(-3, 0, 5) + +inf + +If `z` is an integer, the recurrence reduces the incomplete gamma +function to `P(a) \exp(-a) + Q(b) \exp(-b)` where `P` and +`Q` are polynomials:: + + >>> gammainc(1, 2); exp(-2) + 0.1353352832366126918939995 + 0.1353352832366126918939995 + >>> mp.dps = 50 + >>> identify(gammainc(6, 1, 2), ['exp(-1)', 'exp(-2)']) + '(326*exp(-1) + (-872)*exp(-2))' + +The incomplete gamma functions reduce to functions such as +the exponential integral Ei and the error function for special +arguments:: + + >>> mp.dps = 25 + >>> gammainc(0, 4); -ei(-4) + 0.00377935240984890647887486 + 0.00377935240984890647887486 + >>> gammainc(0.5, 0, 2); sqrt(pi)*erf(sqrt(2)) + 1.691806732945198336509541 + 1.691806732945198336509541 + +""" + +erf = r""" +Computes the error function, `\mathrm{erf}(x)`. The error +function is the normalized antiderivative of the Gaussian function +`\exp(-t^2)`. More precisely, + +.. math:: + + \mathrm{erf}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(-t^2) \,dt + +**Basic examples** + +Simple values and limits include:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> erf(0) + 0.0 + >>> erf(1) + 0.842700792949715 + >>> erf(-1) + -0.842700792949715 + >>> erf(inf) + 1.0 + >>> erf(-inf) + -1.0 + +For large real `x`, `\mathrm{erf}(x)` approaches 1 very +rapidly:: + + >>> erf(3) + 0.999977909503001 + >>> erf(5) + 0.999999999998463 + +The error function is an odd function:: + + >>> nprint(chop(taylor(erf, 0, 5))) + [0.0, 1.12838, 0.0, -0.376126, 0.0, 0.112838] + +:func:`~mpmath.erf` implements arbitrary-precision evaluation and +supports complex numbers:: + + >>> mp.dps = 50 + >>> erf(0.5) + 0.52049987781304653768274665389196452873645157575796 + >>> mp.dps = 25 + >>> erf(1+j) + (1.316151281697947644880271 + 0.1904534692378346862841089j) + +Evaluation is supported for large arguments:: + + >>> mp.dps = 25 + >>> erf('1e1000') + 1.0 + >>> erf('-1e1000') + -1.0 + >>> erf('1e-1000') + 1.128379167095512573896159e-1000 + >>> erf('1e7j') + (0.0 + 8.593897639029319267398803e+43429448190317j) + >>> erf('1e7+1e7j') + (0.9999999858172446172631323 + 3.728805278735270407053139e-8j) + +**Related functions** + +See also :func:`~mpmath.erfc`, which is more accurate for large `x`, +and :func:`~mpmath.erfi` which gives the antiderivative of +`\exp(t^2)`. + +The Fresnel integrals :func:`~mpmath.fresnels` and :func:`~mpmath.fresnelc` +are also related to the error function. +""" + +erfc = r""" +Computes the complementary error function, +`\mathrm{erfc}(x) = 1-\mathrm{erf}(x)`. +This function avoids cancellation that occurs when naively +computing the complementary error function as ``1-erf(x)``:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> 1 - erf(10) + 0.0 + >>> erfc(10) + 2.08848758376254e-45 + +:func:`~mpmath.erfc` works accurately even for ludicrously large +arguments:: + + >>> erfc(10**10) + 4.3504398860243e-43429448190325182776 + +Complex arguments are supported:: + + >>> erfc(500+50j) + (1.19739830969552e-107492 + 1.46072418957528e-107491j) + +""" + + +erfi = r""" +Computes the imaginary error function, `\mathrm{erfi}(x)`. +The imaginary error function is defined in analogy with the +error function, but with a positive sign in the integrand: + +.. math :: + + \mathrm{erfi}(x) = \frac{2}{\sqrt \pi} \int_0^x \exp(t^2) \,dt + +Whereas the error function rapidly converges to 1 as `x` grows, +the imaginary error function rapidly diverges to infinity. +The functions are related as +`\mathrm{erfi}(x) = -i\,\mathrm{erf}(ix)` for all complex +numbers `x`. + +**Examples** + +Basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> erfi(0) + 0.0 + >>> erfi(1) + 1.65042575879754 + >>> erfi(-1) + -1.65042575879754 + >>> erfi(inf) + +inf + >>> erfi(-inf) + -inf + +Note the symmetry between erf and erfi:: + + >>> erfi(3j) + (0.0 + 0.999977909503001j) + >>> erf(3) + 0.999977909503001 + >>> erf(1+2j) + (-0.536643565778565 - 5.04914370344703j) + >>> erfi(2+1j) + (-5.04914370344703 - 0.536643565778565j) + +Large arguments are supported:: + + >>> erfi(1000) + 1.71130938718796e+434291 + >>> erfi(10**10) + 7.3167287567024e+43429448190325182754 + >>> erfi(-10**10) + -7.3167287567024e+43429448190325182754 + >>> erfi(1000-500j) + (2.49895233563961e+325717 + 2.6846779342253e+325717j) + >>> erfi(100000j) + (0.0 + 1.0j) + >>> erfi(-100000j) + (0.0 - 1.0j) + + +""" + +erfinv = r""" +Computes the inverse error function, satisfying + +.. math :: + + \mathrm{erf}(\mathrm{erfinv}(x)) = + \mathrm{erfinv}(\mathrm{erf}(x)) = x. + +This function is defined only for `-1 \le x \le 1`. + +**Examples** + +Special values include:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> erfinv(0) + 0.0 + >>> erfinv(1) + +inf + >>> erfinv(-1) + -inf + +The domain is limited to the standard interval:: + + >>> erfinv(2) + Traceback (most recent call last): + ... + ValueError: erfinv(x) is defined only for -1 <= x <= 1 + +It is simple to check that :func:`~mpmath.erfinv` computes inverse values of +:func:`~mpmath.erf` as promised:: + + >>> erf(erfinv(0.75)) + 0.75 + >>> erf(erfinv(-0.995)) + -0.995 + +:func:`~mpmath.erfinv` supports arbitrary-precision evaluation:: + + >>> mp.dps = 50 + >>> x = erf(2) + >>> x + 0.99532226501895273416206925636725292861089179704006 + >>> erfinv(x) + 2.0 + +A definite integral involving the inverse error function:: + + >>> mp.dps = 15 + >>> quad(erfinv, [0, 1]) + 0.564189583547756 + >>> 1/sqrt(pi) + 0.564189583547756 + +The inverse error function can be used to generate random numbers +with a Gaussian distribution (although this is a relatively +inefficient algorithm):: + + >>> nprint([erfinv(2*rand()-1) for n in range(6)]) # doctest: +SKIP + [-0.586747, 1.10233, -0.376796, 0.926037, -0.708142, -0.732012] + +""" + +npdf = r""" +``npdf(x, mu=0, sigma=1)`` evaluates the probability density +function of a normal distribution with mean value `\mu` +and variance `\sigma^2`. + +Elementary properties of the probability distribution can +be verified using numerical integration:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> quad(npdf, [-inf, inf]) + 1.0 + >>> quad(lambda x: npdf(x, 3), [3, inf]) + 0.5 + >>> quad(lambda x: npdf(x, 3, 2), [3, inf]) + 0.5 + +See also :func:`~mpmath.ncdf`, which gives the cumulative +distribution. +""" + +ncdf = r""" +``ncdf(x, mu=0, sigma=1)`` evaluates the cumulative distribution +function of a normal distribution with mean value `\mu` +and variance `\sigma^2`. + +See also :func:`~mpmath.npdf`, which gives the probability density. + +Elementary properties include:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> ncdf(pi, mu=pi) + 0.5 + >>> ncdf(-inf) + 0.0 + >>> ncdf(+inf) + 1.0 + +The cumulative distribution is the integral of the density +function having identical mu and sigma:: + + >>> mp.dps = 15 + >>> diff(ncdf, 2) + 0.053990966513188 + >>> npdf(2) + 0.053990966513188 + >>> diff(lambda x: ncdf(x, 1, 0.5), 0) + 0.107981933026376 + >>> npdf(0, 1, 0.5) + 0.107981933026376 +""" + +expint = r""" +:func:`~mpmath.expint(n,z)` gives the generalized exponential integral +or En-function, + +.. math :: + + \mathrm{E}_n(z) = \int_1^{\infty} \frac{e^{-zt}}{t^n} dt, + +where `n` and `z` may both be complex numbers. The case with `n = 1` is +also given by :func:`~mpmath.e1`. + +**Examples** + +Evaluation at real and complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> expint(1, 6.25) + 0.0002704758872637179088496194 + >>> expint(-3, 2+3j) + (0.00299658467335472929656159 + 0.06100816202125885450319632j) + >>> expint(2+3j, 4-5j) + (0.001803529474663565056945248 - 0.002235061547756185403349091j) + +At negative integer values of `n`, `E_n(z)` reduces to a +rational-exponential function:: + + >>> f = lambda n, z: fac(n)*sum(z**k/fac(k-1) for k in range(1,n+2))/\ + ... exp(z)/z**(n+2) + >>> n = 3 + >>> z = 1/pi + >>> expint(-n,z) + 584.2604820613019908668219 + >>> f(n,z) + 584.2604820613019908668219 + >>> n = 5 + >>> expint(-n,z) + 115366.5762594725451811138 + >>> f(n,z) + 115366.5762594725451811138 +""" + +e1 = r""" +Computes the exponential integral `\mathrm{E}_1(z)`, given by + +.. math :: + + \mathrm{E}_1(z) = \int_z^{\infty} \frac{e^{-t}}{t} dt. + +This is equivalent to :func:`~mpmath.expint` with `n = 1`. + +**Examples** + +Two ways to evaluate this function:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> e1(6.25) + 0.0002704758872637179088496194 + >>> expint(1,6.25) + 0.0002704758872637179088496194 + +The E1-function is essentially the same as the Ei-function (:func:`~mpmath.ei`) +with negated argument, except for an imaginary branch cut term:: + + >>> e1(2.5) + 0.02491491787026973549562801 + >>> -ei(-2.5) + 0.02491491787026973549562801 + >>> e1(-2.5) + (-7.073765894578600711923552 - 3.141592653589793238462643j) + >>> -ei(2.5) + -7.073765894578600711923552 + +""" + +ei = r""" +Computes the exponential integral or Ei-function, `\mathrm{Ei}(x)`. +The exponential integral is defined as + +.. math :: + + \mathrm{Ei}(x) = \int_{-\infty\,}^x \frac{e^t}{t} \, dt. + +When the integration range includes `t = 0`, the exponential +integral is interpreted as providing the Cauchy principal value. + +For real `x`, the Ei-function behaves roughly like +`\mathrm{Ei}(x) \approx \exp(x) + \log(|x|)`. + +The Ei-function is related to the more general family of exponential +integral functions denoted by `E_n`, which are available as :func:`~mpmath.expint`. + +**Basic examples** + +Some basic values and limits are:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> ei(0) + -inf + >>> ei(1) + 1.89511781635594 + >>> ei(inf) + +inf + >>> ei(-inf) + 0.0 + +For `x < 0`, the defining integral can be evaluated +numerically as a reference:: + + >>> ei(-4) + -0.00377935240984891 + >>> quad(lambda t: exp(t)/t, [-inf, -4]) + -0.00377935240984891 + +:func:`~mpmath.ei` supports complex arguments and arbitrary +precision evaluation:: + + >>> mp.dps = 50 + >>> ei(pi) + 10.928374389331410348638445906907535171566338835056 + >>> mp.dps = 25 + >>> ei(3+4j) + (-4.154091651642689822535359 + 4.294418620024357476985535j) + +**Related functions** + +The exponential integral is closely related to the logarithmic +integral. See :func:`~mpmath.li` for additional information. + +The exponential integral is related to the hyperbolic +and trigonometric integrals (see :func:`~mpmath.chi`, :func:`~mpmath.shi`, +:func:`~mpmath.ci`, :func:`~mpmath.si`) similarly to how the ordinary +exponential function is related to the hyperbolic and +trigonometric functions:: + + >>> mp.dps = 15 + >>> ei(3) + 9.93383257062542 + >>> chi(3) + shi(3) + 9.93383257062542 + >>> chop(ci(3j) - j*si(3j) - pi*j/2) + 9.93383257062542 + +Beware that logarithmic corrections, as in the last example +above, are required to obtain the correct branch in general. +For details, see [1]. + +The exponential integral is also a special case of the +hypergeometric function `\,_2F_2`:: + + >>> z = 0.6 + >>> z*hyper([1,1],[2,2],z) + (ln(z)-ln(1/z))/2 + euler + 0.769881289937359 + >>> ei(z) + 0.769881289937359 + +**References** + +1. Relations between Ei and other functions: + http://functions.wolfram.com/GammaBetaErf/ExpIntegralEi/27/01/ + +2. Abramowitz & Stegun, section 5: + http://people.math.sfu.ca/~cbm/aands/page_228.htm + +3. Asymptotic expansion for Ei: + http://mathworld.wolfram.com/En-Function.html +""" + +li = r""" +Computes the logarithmic integral or li-function +`\mathrm{li}(x)`, defined by + +.. math :: + + \mathrm{li}(x) = \int_0^x \frac{1}{\log t} \, dt + +The logarithmic integral has a singularity at `x = 1`. + +Alternatively, ``li(x, offset=True)`` computes the offset +logarithmic integral (used in number theory) + +.. math :: + + \mathrm{Li}(x) = \int_2^x \frac{1}{\log t} \, dt. + +These two functions are related via the simple identity +`\mathrm{Li}(x) = \mathrm{li}(x) - \mathrm{li}(2)`. + +The logarithmic integral should also not be confused with +the polylogarithm (also denoted by Li), which is implemented +as :func:`~mpmath.polylog`. + +**Examples** + +Some basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 30; mp.pretty = True + >>> li(0) + 0.0 + >>> li(1) + -inf + >>> li(1) + -inf + >>> li(2) + 1.04516378011749278484458888919 + >>> findroot(li, 2) + 1.45136923488338105028396848589 + >>> li(inf) + +inf + >>> li(2, offset=True) + 0.0 + >>> li(1, offset=True) + -inf + >>> li(0, offset=True) + -1.04516378011749278484458888919 + >>> li(10, offset=True) + 5.12043572466980515267839286347 + +The logarithmic integral can be evaluated for arbitrary +complex arguments:: + + >>> mp.dps = 20 + >>> li(3+4j) + (3.1343755504645775265 + 2.6769247817778742392j) + +The logarithmic integral is related to the exponential integral:: + + >>> ei(log(3)) + 2.1635885946671919729 + >>> li(3) + 2.1635885946671919729 + +The logarithmic integral grows like `O(x/\log(x))`:: + + >>> mp.dps = 15 + >>> x = 10**100 + >>> x/log(x) + 4.34294481903252e+97 + >>> li(x) + 4.3619719871407e+97 + +The prime number theorem states that the number of primes less +than `x` is asymptotic to `\mathrm{Li}(x)` (equivalently +`\mathrm{li}(x)`). For example, it is known that there are +exactly 1,925,320,391,606,803,968,923 prime numbers less than +`10^{23}` [1]. The logarithmic integral provides a very +accurate estimate:: + + >>> li(10**23, offset=True) + 1.92532039161405e+21 + +A definite integral is:: + + >>> quad(li, [0, 1]) + -0.693147180559945 + >>> -ln(2) + -0.693147180559945 + +**References** + +1. http://mathworld.wolfram.com/PrimeCountingFunction.html + +2. http://mathworld.wolfram.com/LogarithmicIntegral.html + +""" + +ci = r""" +Computes the cosine integral, + +.. math :: + + \mathrm{Ci}(x) = -\int_x^{\infty} \frac{\cos t}{t}\,dt + = \gamma + \log x + \int_0^x \frac{\cos t - 1}{t}\,dt + +**Examples** + +Some values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> ci(0) + -inf + >>> ci(1) + 0.3374039229009681346626462 + >>> ci(pi) + 0.07366791204642548599010096 + >>> ci(inf) + 0.0 + >>> ci(-inf) + (0.0 + 3.141592653589793238462643j) + >>> ci(2+3j) + (1.408292501520849518759125 - 2.983617742029605093121118j) + +The cosine integral behaves roughly like the sinc function +(see :func:`~mpmath.sinc`) for large real `x`:: + + >>> ci(10**10) + -4.875060251748226537857298e-11 + >>> sinc(10**10) + -4.875060250875106915277943e-11 + >>> chop(limit(ci, inf)) + 0.0 + +It has infinitely many roots on the positive real axis:: + + >>> findroot(ci, 1) + 0.6165054856207162337971104 + >>> findroot(ci, 2) + 3.384180422551186426397851 + +Evaluation is supported for `z` anywhere in the complex plane:: + + >>> ci(10**6*(1+j)) + (4.449410587611035724984376e+434287 + 9.75744874290013526417059e+434287j) + +We can evaluate the defining integral as a reference:: + + >>> mp.dps = 15 + >>> -quadosc(lambda t: cos(t)/t, [5, inf], omega=1) + -0.190029749656644 + >>> ci(5) + -0.190029749656644 + +Some infinite series can be evaluated using the +cosine integral:: + + >>> nsum(lambda k: (-1)**k/(fac(2*k)*(2*k)), [1,inf]) + -0.239811742000565 + >>> ci(1) - euler + -0.239811742000565 + +""" + +si = r""" +Computes the sine integral, + +.. math :: + + \mathrm{Si}(x) = \int_0^x \frac{\sin t}{t}\,dt. + +The sine integral is thus the antiderivative of the sinc +function (see :func:`~mpmath.sinc`). + +**Examples** + +Some values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> si(0) + 0.0 + >>> si(1) + 0.9460830703671830149413533 + >>> si(-1) + -0.9460830703671830149413533 + >>> si(pi) + 1.851937051982466170361053 + >>> si(inf) + 1.570796326794896619231322 + >>> si(-inf) + -1.570796326794896619231322 + >>> si(2+3j) + (4.547513889562289219853204 + 1.399196580646054789459839j) + +The sine integral approaches `\pi/2` for large real `x`:: + + >>> si(10**10) + 1.570796326707584656968511 + >>> pi/2 + 1.570796326794896619231322 + +Evaluation is supported for `z` anywhere in the complex plane:: + + >>> si(10**6*(1+j)) + (-9.75744874290013526417059e+434287 + 4.449410587611035724984376e+434287j) + +We can evaluate the defining integral as a reference:: + + >>> mp.dps = 15 + >>> quad(sinc, [0, 5]) + 1.54993124494467 + >>> si(5) + 1.54993124494467 + +Some infinite series can be evaluated using the +sine integral:: + + >>> nsum(lambda k: (-1)**k/(fac(2*k+1)*(2*k+1)), [0,inf]) + 0.946083070367183 + >>> si(1) + 0.946083070367183 + +""" + +chi = r""" +Computes the hyperbolic cosine integral, defined +in analogy with the cosine integral (see :func:`~mpmath.ci`) as + +.. math :: + + \mathrm{Chi}(x) = -\int_x^{\infty} \frac{\cosh t}{t}\,dt + = \gamma + \log x + \int_0^x \frac{\cosh t - 1}{t}\,dt + +Some values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> chi(0) + -inf + >>> chi(1) + 0.8378669409802082408946786 + >>> chi(inf) + +inf + >>> findroot(chi, 0.5) + 0.5238225713898644064509583 + >>> chi(2+3j) + (-0.1683628683277204662429321 + 2.625115880451325002151688j) + +Evaluation is supported for `z` anywhere in the complex plane:: + + >>> chi(10**6*(1+j)) + (4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j) + +""" + +shi = r""" +Computes the hyperbolic sine integral, defined +in analogy with the sine integral (see :func:`~mpmath.si`) as + +.. math :: + + \mathrm{Shi}(x) = \int_0^x \frac{\sinh t}{t}\,dt. + +Some values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> shi(0) + 0.0 + >>> shi(1) + 1.057250875375728514571842 + >>> shi(-1) + -1.057250875375728514571842 + >>> shi(inf) + +inf + >>> shi(2+3j) + (-0.1931890762719198291678095 + 2.645432555362369624818525j) + +Evaluation is supported for `z` anywhere in the complex plane:: + + >>> shi(10**6*(1+j)) + (4.449410587611035724984376e+434287 - 9.75744874290013526417059e+434287j) + +""" + +fresnels = r""" +Computes the Fresnel sine integral + +.. math :: + + S(x) = \int_0^x \sin\left(\frac{\pi t^2}{2}\right) \,dt + +Note that some sources define this function +without the normalization factor `\pi/2`. + +**Examples** + +Some basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> fresnels(0) + 0.0 + >>> fresnels(inf) + 0.5 + >>> fresnels(-inf) + -0.5 + >>> fresnels(1) + 0.4382591473903547660767567 + >>> fresnels(1+2j) + (36.72546488399143842838788 + 15.58775110440458732748279j) + +Comparing with the definition:: + + >>> fresnels(3) + 0.4963129989673750360976123 + >>> quad(lambda t: sin(pi*t**2/2), [0,3]) + 0.4963129989673750360976123 +""" + +fresnelc = r""" +Computes the Fresnel cosine integral + +.. math :: + + C(x) = \int_0^x \cos\left(\frac{\pi t^2}{2}\right) \,dt + +Note that some sources define this function +without the normalization factor `\pi/2`. + +**Examples** + +Some basic values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> fresnelc(0) + 0.0 + >>> fresnelc(inf) + 0.5 + >>> fresnelc(-inf) + -0.5 + >>> fresnelc(1) + 0.7798934003768228294742064 + >>> fresnelc(1+2j) + (16.08787137412548041729489 - 36.22568799288165021578758j) + +Comparing with the definition:: + + >>> fresnelc(3) + 0.6057207892976856295561611 + >>> quad(lambda t: cos(pi*t**2/2), [0,3]) + 0.6057207892976856295561611 +""" + +airyai = r""" +Computes the Airy function `\operatorname{Ai}(z)`, which is +the solution of the Airy differential equation `f''(z) - z f(z) = 0` +with initial conditions + +.. math :: + + \operatorname{Ai}(0) = + \frac{1}{3^{2/3}\Gamma\left(\frac{2}{3}\right)} + + \operatorname{Ai}'(0) = + -\frac{1}{3^{1/3}\Gamma\left(\frac{1}{3}\right)}. + +Other common ways of defining the Ai-function include +integrals such as + +.. math :: + + \operatorname{Ai}(x) = \frac{1}{\pi} + \int_0^{\infty} \cos\left(\frac{1}{3}t^3+xt\right) dt + \qquad x \in \mathbb{R} + + \operatorname{Ai}(z) = \frac{\sqrt{3}}{2\pi} + \int_0^{\infty} + \exp\left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right) dt. + +The Ai-function is an entire function with a turning point, +behaving roughly like a slowly decaying sine wave for `z < 0` and +like a rapidly decreasing exponential for `z > 0`. +A second solution of the Airy differential equation +is given by `\operatorname{Bi}(z)` (see :func:`~mpmath.airybi`). + +Optionally, with *derivative=alpha*, :func:`airyai` can compute the +`\alpha`-th order fractional derivative with respect to `z`. +For `\alpha = n = 1,2,3,\ldots` this gives the derivative +`\operatorname{Ai}^{(n)}(z)`, and for `\alpha = -n = -1,-2,-3,\ldots` +this gives the `n`-fold iterated integral + +.. math :: + + f_0(z) = \operatorname{Ai}(z) + + f_n(z) = \int_0^z f_{n-1}(t) dt. + +The Ai-function has infinitely many zeros, all located along the +negative half of the real axis. They can be computed with +:func:`~mpmath.airyaizero`. + +**Plots** + +.. literalinclude :: /plots/ai.py +.. image :: /plots/ai.png +.. literalinclude :: /plots/ai_c.py +.. image :: /plots/ai_c.png + +**Basic examples** + +Limits and values include:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> airyai(0); 1/(power(3,'2/3')*gamma('2/3')) + 0.3550280538878172392600632 + 0.3550280538878172392600632 + >>> airyai(1) + 0.1352924163128814155241474 + >>> airyai(-1) + 0.5355608832923521187995166 + >>> airyai(inf); airyai(-inf) + 0.0 + 0.0 + +Evaluation is supported for large magnitudes of the argument:: + + >>> airyai(-100) + 0.1767533932395528780908311 + >>> airyai(100) + 2.634482152088184489550553e-291 + >>> airyai(50+50j) + (-5.31790195707456404099817e-68 - 1.163588003770709748720107e-67j) + >>> airyai(-50+50j) + (1.041242537363167632587245e+158 + 3.347525544923600321838281e+157j) + +Huge arguments are also fine:: + + >>> airyai(10**10) + 1.162235978298741779953693e-289529654602171 + >>> airyai(-10**10) + 0.0001736206448152818510510181 + >>> w = airyai(10**10*(1+j)) + >>> w.real + 5.711508683721355528322567e-186339621747698 + >>> w.imag + 1.867245506962312577848166e-186339621747697 + +The first root of the Ai-function is:: + + >>> findroot(airyai, -2) + -2.338107410459767038489197 + >>> airyaizero(1) + -2.338107410459767038489197 + +**Properties and relations** + +Verifying the Airy differential equation:: + + >>> for z in [-3.4, 0, 2.5, 1+2j]: + ... chop(airyai(z,2) - z*airyai(z)) + ... + 0.0 + 0.0 + 0.0 + 0.0 + +The first few terms of the Taylor series expansion around `z = 0` +(every third term is zero):: + + >>> nprint(taylor(airyai, 0, 5)) + [0.355028, -0.258819, 0.0, 0.0591713, -0.0215683, 0.0] + +The Airy functions satisfy the Wronskian relation +`\operatorname{Ai}(z) \operatorname{Bi}'(z) - +\operatorname{Ai}'(z) \operatorname{Bi}(z) = 1/\pi`:: + + >>> z = -0.5 + >>> airyai(z)*airybi(z,1) - airyai(z,1)*airybi(z) + 0.3183098861837906715377675 + >>> 1/pi + 0.3183098861837906715377675 + +The Airy functions can be expressed in terms of Bessel +functions of order `\pm 1/3`. For `\Re[z] \le 0`, we have:: + + >>> z = -3 + >>> airyai(z) + -0.3788142936776580743472439 + >>> y = 2*power(-z,'3/2')/3 + >>> (sqrt(-z) * (besselj('1/3',y) + besselj('-1/3',y)))/3 + -0.3788142936776580743472439 + +**Derivatives and integrals** + +Derivatives of the Ai-function (directly and using :func:`~mpmath.diff`):: + + >>> airyai(-3,1); diff(airyai,-3) + 0.3145837692165988136507873 + 0.3145837692165988136507873 + >>> airyai(-3,2); diff(airyai,-3,2) + 1.136442881032974223041732 + 1.136442881032974223041732 + >>> airyai(1000,1); diff(airyai,1000) + -2.943133917910336090459748e-9156 + -2.943133917910336090459748e-9156 + +Several derivatives at `z = 0`:: + + >>> airyai(0,0); airyai(0,1); airyai(0,2) + 0.3550280538878172392600632 + -0.2588194037928067984051836 + 0.0 + >>> airyai(0,3); airyai(0,4); airyai(0,5) + 0.3550280538878172392600632 + -0.5176388075856135968103671 + 0.0 + >>> airyai(0,15); airyai(0,16); airyai(0,17) + 1292.30211615165475090663 + -3188.655054727379756351861 + 0.0 + +The integral of the Ai-function:: + + >>> airyai(3,-1); quad(airyai, [0,3]) + 0.3299203760070217725002701 + 0.3299203760070217725002701 + >>> airyai(-10,-1); quad(airyai, [0,-10]) + -0.765698403134212917425148 + -0.765698403134212917425148 + +Integrals of high or fractional order:: + + >>> airyai(-2,0.5); differint(airyai,-2,0.5,0) + (0.0 + 0.2453596101351438273844725j) + (0.0 + 0.2453596101351438273844725j) + >>> airyai(-2,-4); differint(airyai,-2,-4,0) + 0.2939176441636809580339365 + 0.2939176441636809580339365 + >>> airyai(0,-1); airyai(0,-2); airyai(0,-3) + 0.0 + 0.0 + 0.0 + +Integrals of the Ai-function can be evaluated at limit points:: + + >>> airyai(-1000000,-1); airyai(-inf,-1) + -0.6666843728311539978751512 + -0.6666666666666666666666667 + >>> airyai(10,-1); airyai(+inf,-1) + 0.3333333332991690159427932 + 0.3333333333333333333333333 + >>> airyai(+inf,-2); airyai(+inf,-3) + +inf + +inf + >>> airyai(-1000000,-2); airyai(-inf,-2) + 666666.4078472650651209742 + +inf + >>> airyai(-1000000,-3); airyai(-inf,-3) + -333333074513.7520264995733 + -inf + +**References** + +1. [DLMF]_ Chapter 9: Airy and Related Functions +2. [WolframFunctions]_ section: Bessel-Type Functions + +""" + +airybi = r""" +Computes the Airy function `\operatorname{Bi}(z)`, which is +the solution of the Airy differential equation `f''(z) - z f(z) = 0` +with initial conditions + +.. math :: + + \operatorname{Bi}(0) = + \frac{1}{3^{1/6}\Gamma\left(\frac{2}{3}\right)} + + \operatorname{Bi}'(0) = + \frac{3^{1/6}}{\Gamma\left(\frac{1}{3}\right)}. + +Like the Ai-function (see :func:`~mpmath.airyai`), the Bi-function +is oscillatory for `z < 0`, but it grows rather than decreases +for `z > 0`. + +Optionally, as for :func:`~mpmath.airyai`, derivatives, integrals +and fractional derivatives can be computed with the *derivative* +parameter. + +The Bi-function has infinitely many zeros along the negative +half-axis, as well as complex zeros, which can all be computed +with :func:`~mpmath.airybizero`. + +**Plots** + +.. literalinclude :: /plots/bi.py +.. image :: /plots/bi.png +.. literalinclude :: /plots/bi_c.py +.. image :: /plots/bi_c.png + +**Basic examples** + +Limits and values include:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> airybi(0); 1/(power(3,'1/6')*gamma('2/3')) + 0.6149266274460007351509224 + 0.6149266274460007351509224 + >>> airybi(1) + 1.207423594952871259436379 + >>> airybi(-1) + 0.10399738949694461188869 + >>> airybi(inf); airybi(-inf) + +inf + 0.0 + +Evaluation is supported for large magnitudes of the argument:: + + >>> airybi(-100) + 0.02427388768016013160566747 + >>> airybi(100) + 6.041223996670201399005265e+288 + >>> airybi(50+50j) + (-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j) + >>> airybi(-50+50j) + (-3.347525544923600321838281e+157 + 1.041242537363167632587245e+158j) + +Huge arguments:: + + >>> airybi(10**10) + 1.369385787943539818688433e+289529654602165 + >>> airybi(-10**10) + 0.001775656141692932747610973 + >>> w = airybi(10**10*(1+j)) + >>> w.real + -6.559955931096196875845858e+186339621747689 + >>> w.imag + -6.822462726981357180929024e+186339621747690 + +The first real root of the Bi-function is:: + + >>> findroot(airybi, -1); airybizero(1) + -1.17371322270912792491998 + -1.17371322270912792491998 + +**Properties and relations** + +Verifying the Airy differential equation:: + + >>> for z in [-3.4, 0, 2.5, 1+2j]: + ... chop(airybi(z,2) - z*airybi(z)) + ... + 0.0 + 0.0 + 0.0 + 0.0 + +The first few terms of the Taylor series expansion around `z = 0` +(every third term is zero):: + + >>> nprint(taylor(airybi, 0, 5)) + [0.614927, 0.448288, 0.0, 0.102488, 0.0373574, 0.0] + +The Airy functions can be expressed in terms of Bessel +functions of order `\pm 1/3`. For `\Re[z] \le 0`, we have:: + + >>> z = -3 + >>> airybi(z) + -0.1982896263749265432206449 + >>> p = 2*power(-z,'3/2')/3 + >>> sqrt(-mpf(z)/3)*(besselj('-1/3',p) - besselj('1/3',p)) + -0.1982896263749265432206449 + +**Derivatives and integrals** + +Derivatives of the Bi-function (directly and using :func:`~mpmath.diff`):: + + >>> airybi(-3,1); diff(airybi,-3) + -0.675611222685258537668032 + -0.675611222685258537668032 + >>> airybi(-3,2); diff(airybi,-3,2) + 0.5948688791247796296619346 + 0.5948688791247796296619346 + >>> airybi(1000,1); diff(airybi,1000) + 1.710055114624614989262335e+9156 + 1.710055114624614989262335e+9156 + +Several derivatives at `z = 0`:: + + >>> airybi(0,0); airybi(0,1); airybi(0,2) + 0.6149266274460007351509224 + 0.4482883573538263579148237 + 0.0 + >>> airybi(0,3); airybi(0,4); airybi(0,5) + 0.6149266274460007351509224 + 0.8965767147076527158296474 + 0.0 + >>> airybi(0,15); airybi(0,16); airybi(0,17) + 2238.332923903442675949357 + 5522.912562599140729510628 + 0.0 + +The integral of the Bi-function:: + + >>> airybi(3,-1); quad(airybi, [0,3]) + 10.06200303130620056316655 + 10.06200303130620056316655 + >>> airybi(-10,-1); quad(airybi, [0,-10]) + -0.01504042480614002045135483 + -0.01504042480614002045135483 + +Integrals of high or fractional order:: + + >>> airybi(-2,0.5); differint(airybi, -2, 0.5, 0) + (0.0 + 0.5019859055341699223453257j) + (0.0 + 0.5019859055341699223453257j) + >>> airybi(-2,-4); differint(airybi,-2,-4,0) + 0.2809314599922447252139092 + 0.2809314599922447252139092 + >>> airybi(0,-1); airybi(0,-2); airybi(0,-3) + 0.0 + 0.0 + 0.0 + +Integrals of the Bi-function can be evaluated at limit points:: + + >>> airybi(-1000000,-1); airybi(-inf,-1) + 0.000002191261128063434047966873 + 0.0 + >>> airybi(10,-1); airybi(+inf,-1) + 147809803.1074067161675853 + +inf + >>> airybi(+inf,-2); airybi(+inf,-3) + +inf + +inf + >>> airybi(-1000000,-2); airybi(-inf,-2) + 0.4482883750599908479851085 + 0.4482883573538263579148237 + >>> gamma('2/3')*power(3,'2/3')/(2*pi) + 0.4482883573538263579148237 + >>> airybi(-100000,-3); airybi(-inf,-3) + -44828.52827206932872493133 + -inf + >>> airybi(-100000,-4); airybi(-inf,-4) + 2241411040.437759489540248 + +inf + +""" + +airyaizero = r""" +Gives the `k`-th zero of the Airy Ai-function, +i.e. the `k`-th number `a_k` ordered by magnitude for which +`\operatorname{Ai}(a_k) = 0`. + +Optionally, with *derivative=1*, the corresponding +zero `a'_k` of the derivative function, i.e. +`\operatorname{Ai}'(a'_k) = 0`, is computed. + +**Examples** + +Some values of `a_k`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> airyaizero(1) + -2.338107410459767038489197 + >>> airyaizero(2) + -4.087949444130970616636989 + >>> airyaizero(3) + -5.520559828095551059129856 + >>> airyaizero(1000) + -281.0315196125215528353364 + +Some values of `a'_k`:: + + >>> airyaizero(1,1) + -1.018792971647471089017325 + >>> airyaizero(2,1) + -3.248197582179836537875424 + >>> airyaizero(3,1) + -4.820099211178735639400616 + >>> airyaizero(1000,1) + -280.9378080358935070607097 + +Verification:: + + >>> chop(airyai(airyaizero(1))) + 0.0 + >>> chop(airyai(airyaizero(1,1),1)) + 0.0 + +""" + +airybizero = r""" +With *complex=False*, gives the `k`-th real zero of the Airy Bi-function, +i.e. the `k`-th number `b_k` ordered by magnitude for which +`\operatorname{Bi}(b_k) = 0`. + +With *complex=True*, gives the `k`-th complex zero in the upper +half plane `\beta_k`. Also the conjugate `\overline{\beta_k}` +is a zero. + +Optionally, with *derivative=1*, the corresponding +zero `b'_k` or `\beta'_k` of the derivative function, i.e. +`\operatorname{Bi}'(b'_k) = 0` or `\operatorname{Bi}'(\beta'_k) = 0`, +is computed. + +**Examples** + +Some values of `b_k`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> airybizero(1) + -1.17371322270912792491998 + >>> airybizero(2) + -3.271093302836352715680228 + >>> airybizero(3) + -4.830737841662015932667709 + >>> airybizero(1000) + -280.9378112034152401578834 + +Some values of `b_k`:: + + >>> airybizero(1,1) + -2.294439682614123246622459 + >>> airybizero(2,1) + -4.073155089071828215552369 + >>> airybizero(3,1) + -5.512395729663599496259593 + >>> airybizero(1000,1) + -281.0315164471118527161362 + +Some values of `\beta_k`:: + + >>> airybizero(1,complex=True) + (0.9775448867316206859469927 + 2.141290706038744575749139j) + >>> airybizero(2,complex=True) + (1.896775013895336346627217 + 3.627291764358919410440499j) + >>> airybizero(3,complex=True) + (2.633157739354946595708019 + 4.855468179979844983174628j) + >>> airybizero(1000,complex=True) + (140.4978560578493018899793 + 243.3907724215792121244867j) + +Some values of `\beta'_k`:: + + >>> airybizero(1,1,complex=True) + (0.2149470745374305676088329 + 1.100600143302797880647194j) + >>> airybizero(2,1,complex=True) + (1.458168309223507392028211 + 2.912249367458445419235083j) + >>> airybizero(3,1,complex=True) + (2.273760763013482299792362 + 4.254528549217097862167015j) + >>> airybizero(1000,1,complex=True) + (140.4509972835270559730423 + 243.3096175398562811896208j) + +Verification:: + + >>> chop(airybi(airybizero(1))) + 0.0 + >>> chop(airybi(airybizero(1,1),1)) + 0.0 + >>> u = airybizero(1,complex=True) + >>> chop(airybi(u)) + 0.0 + >>> chop(airybi(conj(u))) + 0.0 + +The complex zeros (in the upper and lower half-planes respectively) +asymptotically approach the rays `z = R \exp(\pm i \pi /3)`:: + + >>> arg(airybizero(1,complex=True)) + 1.142532510286334022305364 + >>> arg(airybizero(1000,complex=True)) + 1.047271114786212061583917 + >>> arg(airybizero(1000000,complex=True)) + 1.047197624741816183341355 + >>> pi/3 + 1.047197551196597746154214 + +""" + + +ellipk = r""" +Evaluates the complete elliptic integral of the first kind, +`K(m)`, defined by + +.. math :: + + K(m) = \int_0^{\pi/2} \frac{dt}{\sqrt{1-m \sin^2 t}} \, = \, + \frac{\pi}{2} \,_2F_1\left(\frac{1}{2}, \frac{1}{2}, 1, m\right). + +Note that the argument is the parameter `m = k^2`, +not the modulus `k` which is sometimes used. + +**Plots** + +.. literalinclude :: /plots/ellipk.py +.. image :: /plots/ellipk.png + +**Examples** + +Values and limits include:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> ellipk(0) + 1.570796326794896619231322 + >>> ellipk(inf) + (0.0 + 0.0j) + >>> ellipk(-inf) + 0.0 + >>> ellipk(1) + +inf + >>> ellipk(-1) + 1.31102877714605990523242 + >>> ellipk(2) + (1.31102877714605990523242 - 1.31102877714605990523242j) + +Verifying the defining integral and hypergeometric +representation:: + + >>> ellipk(0.5) + 1.85407467730137191843385 + >>> quad(lambda t: (1-0.5*sin(t)**2)**-0.5, [0, pi/2]) + 1.85407467730137191843385 + >>> pi/2*hyp2f1(0.5,0.5,1,0.5) + 1.85407467730137191843385 + +Evaluation is supported for arbitrary complex `m`:: + + >>> ellipk(3+4j) + (0.9111955638049650086562171 + 0.6313342832413452438845091j) + +A definite integral:: + + >>> quad(ellipk, [0, 1]) + 2.0 +""" + +agm = r""" +``agm(a, b)`` computes the arithmetic-geometric mean of `a` and +`b`, defined as the limit of the following iteration: + +.. math :: + + a_0 = a + + b_0 = b + + a_{n+1} = \frac{a_n+b_n}{2} + + b_{n+1} = \sqrt{a_n b_n} + +This function can be called with a single argument, computing +`\mathrm{agm}(a,1) = \mathrm{agm}(1,a)`. + +**Examples** + +It is a well-known theorem that the geometric mean of +two distinct positive numbers is less than the arithmetic +mean. It follows that the arithmetic-geometric mean lies +between the two means:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> a = mpf(3) + >>> b = mpf(4) + >>> sqrt(a*b) + 3.46410161513775 + >>> agm(a,b) + 3.48202767635957 + >>> (a+b)/2 + 3.5 + +The arithmetic-geometric mean is scale-invariant:: + + >>> agm(10*e, 10*pi) + 29.261085515723 + >>> 10*agm(e, pi) + 29.261085515723 + +As an order-of-magnitude estimate, `\mathrm{agm}(1,x) \approx x` +for large `x`:: + + >>> agm(10**10) + 643448704.760133 + >>> agm(10**50) + 1.34814309345871e+48 + +For tiny `x`, `\mathrm{agm}(1,x) \approx -\pi/(2 \log(x/4))`:: + + >>> agm('0.01') + 0.262166887202249 + >>> -pi/2/log('0.0025') + 0.262172347753122 + +The arithmetic-geometric mean can also be computed for complex +numbers:: + + >>> agm(3, 2+j) + (2.51055133276184 + 0.547394054060638j) + +The AGM iteration converges very quickly (each step doubles +the number of correct digits), so :func:`~mpmath.agm` supports efficient +high-precision evaluation:: + + >>> mp.dps = 10000 + >>> a = agm(1,2) + >>> str(a)[-10:] + '1679581912' + +**Mathematical relations** + +The arithmetic-geometric mean may be used to evaluate the +following two parametric definite integrals: + +.. math :: + + I_1 = \int_0^{\infty} + \frac{1}{\sqrt{(x^2+a^2)(x^2+b^2)}} \,dx + + I_2 = \int_0^{\pi/2} + \frac{1}{\sqrt{a^2 \cos^2(x) + b^2 \sin^2(x)}} \,dx + +We have:: + + >>> mp.dps = 15 + >>> a = 3 + >>> b = 4 + >>> f1 = lambda x: ((x**2+a**2)*(x**2+b**2))**-0.5 + >>> f2 = lambda x: ((a*cos(x))**2 + (b*sin(x))**2)**-0.5 + >>> quad(f1, [0, inf]) + 0.451115405388492 + >>> quad(f2, [0, pi/2]) + 0.451115405388492 + >>> pi/(2*agm(a,b)) + 0.451115405388492 + +A formula for `\Gamma(1/4)`:: + + >>> gamma(0.25) + 3.62560990822191 + >>> sqrt(2*sqrt(2*pi**3)/agm(1,sqrt(2))) + 3.62560990822191 + +**Possible issues** + +The branch cut chosen for complex `a` and `b` is somewhat +arbitrary. + +""" + +gegenbauer = r""" +Evaluates the Gegenbauer polynomial, or ultraspherical polynomial, + +.. math :: + + C_n^{(a)}(z) = {n+2a-1 \choose n} \,_2F_1\left(-n, n+2a; + a+\frac{1}{2}; \frac{1}{2}(1-z)\right). + +When `n` is a nonnegative integer, this formula gives a polynomial +in `z` of degree `n`, but all parameters are permitted to be +complex numbers. With `a = 1/2`, the Gegenbauer polynomial +reduces to a Legendre polynomial. + +**Examples** + +Evaluation for arbitrary arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> gegenbauer(3, 0.5, -10) + -2485.0 + >>> gegenbauer(1000, 10, 100) + 3.012757178975667428359374e+2322 + >>> gegenbauer(2+3j, -0.75, -1000j) + (-5038991.358609026523401901 + 9414549.285447104177860806j) + +Evaluation at negative integer orders:: + + >>> gegenbauer(-4, 2, 1.75) + -1.0 + >>> gegenbauer(-4, 3, 1.75) + 0.0 + >>> gegenbauer(-4, 2j, 1.75) + 0.0 + >>> gegenbauer(-7, 0.5, 3) + 8989.0 + +The Gegenbauer polynomials solve the differential equation:: + + >>> n, a = 4.5, 1+2j + >>> f = lambda z: gegenbauer(n, a, z) + >>> for z in [0, 0.75, -0.5j]: + ... chop((1-z**2)*diff(f,z,2) - (2*a+1)*z*diff(f,z) + n*(n+2*a)*f(z)) + ... + 0.0 + 0.0 + 0.0 + +The Gegenbauer polynomials have generating function +`(1-2zt+t^2)^{-a}`:: + + >>> a, z = 2.5, 1 + >>> taylor(lambda t: (1-2*z*t+t**2)**(-a), 0, 3) + [1.0, 5.0, 15.0, 35.0] + >>> [gegenbauer(n,a,z) for n in range(4)] + [1.0, 5.0, 15.0, 35.0] + +The Gegenbauer polynomials are orthogonal on `[-1, 1]` with respect +to the weight `(1-z^2)^{a-\frac{1}{2}}`:: + + >>> a, n, m = 2.5, 4, 5 + >>> Cn = lambda z: gegenbauer(n, a, z, zeroprec=1000) + >>> Cm = lambda z: gegenbauer(m, a, z, zeroprec=1000) + >>> chop(quad(lambda z: Cn(z)*Cm(z)*(1-z**2)*(a-0.5), [-1, 1])) + 0.0 +""" + +laguerre = r""" +Gives the generalized (associated) Laguerre polynomial, defined by + +.. math :: + + L_n^a(z) = \frac{\Gamma(n+b+1)}{\Gamma(b+1) \Gamma(n+1)} + \,_1F_1(-n, a+1, z). + +With `a = 0` and `n` a nonnegative integer, this reduces to an ordinary +Laguerre polynomial, the sequence of which begins +`L_0(z) = 1, L_1(z) = 1-z, L_2(z) = z^2-2z+1, \ldots`. + +The Laguerre polynomials are orthogonal with respect to the weight +`z^a e^{-z}` on `[0, \infty)`. + +**Plots** + +.. literalinclude :: /plots/laguerre.py +.. image :: /plots/laguerre.png + +**Examples** + +Evaluation for arbitrary arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> laguerre(5, 0, 0.25) + 0.03726399739583333333333333 + >>> laguerre(1+j, 0.5, 2+3j) + (4.474921610704496808379097 - 11.02058050372068958069241j) + >>> laguerre(2, 0, 10000) + 49980001.0 + >>> laguerre(2.5, 0, 10000) + -9.327764910194842158583189e+4328 + +The first few Laguerre polynomials, normalized to have integer +coefficients:: + + >>> for n in range(7): + ... chop(taylor(lambda z: fac(n)*laguerre(n, 0, z), 0, n)) + ... + [1.0] + [1.0, -1.0] + [2.0, -4.0, 1.0] + [6.0, -18.0, 9.0, -1.0] + [24.0, -96.0, 72.0, -16.0, 1.0] + [120.0, -600.0, 600.0, -200.0, 25.0, -1.0] + [720.0, -4320.0, 5400.0, -2400.0, 450.0, -36.0, 1.0] + +Verifying orthogonality:: + + >>> Lm = lambda t: laguerre(m,a,t) + >>> Ln = lambda t: laguerre(n,a,t) + >>> a, n, m = 2.5, 2, 3 + >>> chop(quad(lambda t: exp(-t)*t**a*Lm(t)*Ln(t), [0,inf])) + 0.0 + + +""" + +hermite = r""" +Evaluates the Hermite polynomial `H_n(z)`, which may be defined using +the recurrence + +.. math :: + + H_0(z) = 1 + + H_1(z) = 2z + + H_{n+1} = 2z H_n(z) - 2n H_{n-1}(z). + +The Hermite polynomials are orthogonal on `(-\infty, \infty)` with +respect to the weight `e^{-z^2}`. More generally, allowing arbitrary complex +values of `n`, the Hermite function `H_n(z)` is defined as + +.. math :: + + H_n(z) = (2z)^n \,_2F_0\left(-\frac{n}{2}, \frac{1-n}{2}, + -\frac{1}{z^2}\right) + +for `\Re{z} > 0`, or generally + +.. math :: + + H_n(z) = 2^n \sqrt{\pi} \left( + \frac{1}{\Gamma\left(\frac{1-n}{2}\right)} + \,_1F_1\left(-\frac{n}{2}, \frac{1}{2}, z^2\right) - + \frac{2z}{\Gamma\left(-\frac{n}{2}\right)} + \,_1F_1\left(\frac{1-n}{2}, \frac{3}{2}, z^2\right) + \right). + +**Plots** + +.. literalinclude :: /plots/hermite.py +.. image :: /plots/hermite.png + +**Examples** + +Evaluation for arbitrary arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hermite(0, 10) + 1.0 + >>> hermite(1, 10); hermite(2, 10) + 20.0 + 398.0 + >>> hermite(10000, 2) + 4.950440066552087387515653e+19334 + >>> hermite(3, -10**8) + -7999999999999998800000000.0 + >>> hermite(-3, -10**8) + 1.675159751729877682920301e+4342944819032534 + >>> hermite(2+3j, -1+2j) + (-0.07652130602993513389421901 - 0.1084662449961914580276007j) + +Coefficients of the first few Hermite polynomials are:: + + >>> for n in range(7): + ... chop(taylor(lambda z: hermite(n, z), 0, n)) + ... + [1.0] + [0.0, 2.0] + [-2.0, 0.0, 4.0] + [0.0, -12.0, 0.0, 8.0] + [12.0, 0.0, -48.0, 0.0, 16.0] + [0.0, 120.0, 0.0, -160.0, 0.0, 32.0] + [-120.0, 0.0, 720.0, 0.0, -480.0, 0.0, 64.0] + +Values at `z = 0`:: + + >>> for n in range(-5, 9): + ... hermite(n, 0) + ... + 0.02769459142039868792653387 + 0.08333333333333333333333333 + 0.2215567313631895034122709 + 0.5 + 0.8862269254527580136490837 + 1.0 + 0.0 + -2.0 + 0.0 + 12.0 + 0.0 + -120.0 + 0.0 + 1680.0 + +Hermite functions satisfy the differential equation:: + + >>> n = 4 + >>> f = lambda z: hermite(n, z) + >>> z = 1.5 + >>> chop(diff(f,z,2) - 2*z*diff(f,z) + 2*n*f(z)) + 0.0 + +Verifying orthogonality:: + + >>> chop(quad(lambda t: hermite(2,t)*hermite(4,t)*exp(-t**2), [-inf,inf])) + 0.0 + +""" + +jacobi = r""" +``jacobi(n, a, b, x)`` evaluates the Jacobi polynomial +`P_n^{(a,b)}(x)`. The Jacobi polynomials are a special +case of the hypergeometric function `\,_2F_1` given by: + +.. math :: + + P_n^{(a,b)}(x) = {n+a \choose n} + \,_2F_1\left(-n,1+a+b+n,a+1,\frac{1-x}{2}\right). + +Note that this definition generalizes to nonintegral values +of `n`. When `n` is an integer, the hypergeometric series +terminates after a finite number of terms, giving +a polynomial in `x`. + +**Evaluation of Jacobi polynomials** + +A special evaluation is `P_n^{(a,b)}(1) = {n+a \choose n}`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> jacobi(4, 0.5, 0.25, 1) + 2.4609375 + >>> binomial(4+0.5, 4) + 2.4609375 + +A Jacobi polynomial of degree `n` is equal to its +Taylor polynomial of degree `n`. The explicit +coefficients of Jacobi polynomials can therefore +be recovered easily using :func:`~mpmath.taylor`:: + + >>> for n in range(5): + ... nprint(taylor(lambda x: jacobi(n,1,2,x), 0, n)) + ... + [1.0] + [-0.5, 2.5] + [-0.75, -1.5, 5.25] + [0.5, -3.5, -3.5, 10.5] + [0.625, 2.5, -11.25, -7.5, 20.625] + +For nonintegral `n`, the Jacobi "polynomial" is no longer +a polynomial:: + + >>> nprint(taylor(lambda x: jacobi(0.5,1,2,x), 0, 4)) + [0.309983, 1.84119, -1.26933, 1.26699, -1.34808] + +**Orthogonality** + +The Jacobi polynomials are orthogonal on the interval +`[-1, 1]` with respect to the weight function +`w(x) = (1-x)^a (1+x)^b`. That is, +`w(x) P_n^{(a,b)}(x) P_m^{(a,b)}(x)` integrates to +zero if `m \ne n` and to a nonzero number if `m = n`. + +The orthogonality is easy to verify using numerical +quadrature:: + + >>> P = jacobi + >>> f = lambda x: (1-x)**a * (1+x)**b * P(m,a,b,x) * P(n,a,b,x) + >>> a = 2 + >>> b = 3 + >>> m, n = 3, 4 + >>> chop(quad(f, [-1, 1]), 1) + 0.0 + >>> m, n = 4, 4 + >>> quad(f, [-1, 1]) + 1.9047619047619 + +**Differential equation** + +The Jacobi polynomials are solutions of the differential +equation + +.. math :: + + (1-x^2) y'' + (b-a-(a+b+2)x) y' + n (n+a+b+1) y = 0. + +We can verify that :func:`~mpmath.jacobi` approximately satisfies +this equation:: + + >>> from mpmath import * + >>> mp.dps = 15 + >>> a = 2.5 + >>> b = 4 + >>> n = 3 + >>> y = lambda x: jacobi(n,a,b,x) + >>> x = pi + >>> A0 = n*(n+a+b+1)*y(x) + >>> A1 = (b-a-(a+b+2)*x)*diff(y,x) + >>> A2 = (1-x**2)*diff(y,x,2) + >>> nprint(A2 + A1 + A0, 1) + 4.0e-12 + +The difference of order `10^{-12}` is as close to zero as +it could be at 15-digit working precision, since the terms +are large:: + + >>> A0, A1, A2 + (26560.2328981879, -21503.7641037294, -5056.46879445852) + +""" + +legendre = r""" +``legendre(n, x)`` evaluates the Legendre polynomial `P_n(x)`. +The Legendre polynomials are given by the formula + +.. math :: + + P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n. + +Alternatively, they can be computed recursively using + +.. math :: + + P_0(x) = 1 + + P_1(x) = x + + (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x). + +A third definition is in terms of the hypergeometric function +`\,_2F_1`, whereby they can be generalized to arbitrary `n`: + +.. math :: + + P_n(x) = \,_2F_1\left(-n, n+1, 1, \frac{1-x}{2}\right) + +**Plots** + +.. literalinclude :: /plots/legendre.py +.. image :: /plots/legendre.png + +**Basic evaluation** + +The Legendre polynomials assume fixed values at the points +`x = -1` and `x = 1`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> nprint([legendre(n, 1) for n in range(6)]) + [1.0, 1.0, 1.0, 1.0, 1.0, 1.0] + >>> nprint([legendre(n, -1) for n in range(6)]) + [1.0, -1.0, 1.0, -1.0, 1.0, -1.0] + +The coefficients of Legendre polynomials can be recovered +using degree-`n` Taylor expansion:: + + >>> for n in range(5): + ... nprint(chop(taylor(lambda x: legendre(n, x), 0, n))) + ... + [1.0] + [0.0, 1.0] + [-0.5, 0.0, 1.5] + [0.0, -1.5, 0.0, 2.5] + [0.375, 0.0, -3.75, 0.0, 4.375] + +The roots of Legendre polynomials are located symmetrically +on the interval `[-1, 1]`:: + + >>> for n in range(5): + ... nprint(polyroots(taylor(lambda x: legendre(n, x), 0, n)[::-1])) + ... + [] + [0.0] + [-0.57735, 0.57735] + [-0.774597, 0.0, 0.774597] + [-0.861136, -0.339981, 0.339981, 0.861136] + +An example of an evaluation for arbitrary `n`:: + + >>> legendre(0.75, 2+4j) + (1.94952805264875 + 2.1071073099422j) + +**Orthogonality** + +The Legendre polynomials are orthogonal on `[-1, 1]` with respect +to the trivial weight `w(x) = 1`. That is, `P_m(x) P_n(x)` +integrates to zero if `m \ne n` and to `2/(2n+1)` if `m = n`:: + + >>> m, n = 3, 4 + >>> quad(lambda x: legendre(m,x)*legendre(n,x), [-1, 1]) + 0.0 + >>> m, n = 4, 4 + >>> quad(lambda x: legendre(m,x)*legendre(n,x), [-1, 1]) + 0.222222222222222 + +**Differential equation** + +The Legendre polynomials satisfy the differential equation + +.. math :: + + ((1-x^2) y')' + n(n+1) y' = 0. + +We can verify this numerically:: + + >>> n = 3.6 + >>> x = 0.73 + >>> P = legendre + >>> A = diff(lambda t: (1-t**2)*diff(lambda u: P(n,u), t), x) + >>> B = n*(n+1)*P(n,x) + >>> nprint(A+B,1) + 9.0e-16 + +""" + + +legenp = r""" +Calculates the (associated) Legendre function of the first kind of +degree *n* and order *m*, `P_n^m(z)`. Taking `m = 0` gives the ordinary +Legendre function of the first kind, `P_n(z)`. The parameters may be +complex numbers. + +In terms of the Gauss hypergeometric function, the (associated) Legendre +function is defined as + +.. math :: + + P_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(1+z)^{m/2}}{(1-z)^{m/2}} + \,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right). + +With *type=3* instead of *type=2*, the alternative +definition + +.. math :: + + \hat{P}_n^m(z) = \frac{1}{\Gamma(1-m)} \frac{(z+1)^{m/2}}{(z-1)^{m/2}} + \,_2F_1\left(-n, n+1, 1-m, \frac{1-z}{2}\right). + +is used. These functions correspond respectively to ``LegendreP[n,m,2,z]`` +and ``LegendreP[n,m,3,z]`` in Mathematica. + +The general solution of the (associated) Legendre differential equation + +.. math :: + + (1-z^2) f''(z) - 2zf'(z) + \left(n(n+1)-\frac{m^2}{1-z^2}\right)f(z) = 0 + +is given by `C_1 P_n^m(z) + C_2 Q_n^m(z)` for arbitrary constants +`C_1`, `C_2`, where `Q_n^m(z)` is a Legendre function of the +second kind as implemented by :func:`~mpmath.legenq`. + +**Examples** + +Evaluation for arbitrary parameters and arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> legenp(2, 0, 10); legendre(2, 10) + 149.5 + 149.5 + >>> legenp(-2, 0.5, 2.5) + (1.972260393822275434196053 - 1.972260393822275434196053j) + >>> legenp(2+3j, 1-j, -0.5+4j) + (-3.335677248386698208736542 - 5.663270217461022307645625j) + >>> chop(legenp(3, 2, -1.5, type=2)) + 28.125 + >>> chop(legenp(3, 2, -1.5, type=3)) + -28.125 + +Verifying the associated Legendre differential equation:: + + >>> n, m = 2, -0.5 + >>> C1, C2 = 1, -3 + >>> f = lambda z: C1*legenp(n,m,z) + C2*legenq(n,m,z) + >>> deq = lambda z: (1-z**2)*diff(f,z,2) - 2*z*diff(f,z) + \ + ... (n*(n+1)-m**2/(1-z**2))*f(z) + >>> for z in [0, 2, -1.5, 0.5+2j]: + ... chop(deq(mpmathify(z))) + ... + 0.0 + 0.0 + 0.0 + 0.0 +""" + +legenq = r""" +Calculates the (associated) Legendre function of the second kind of +degree *n* and order *m*, `Q_n^m(z)`. Taking `m = 0` gives the ordinary +Legendre function of the second kind, `Q_n(z)`. The parameters may be +complex numbers. + +The Legendre functions of the second kind give a second set of +solutions to the (associated) Legendre differential equation. +(See :func:`~mpmath.legenp`.) +Unlike the Legendre functions of the first kind, they are not +polynomials of `z` for integer `n`, `m` but rational or logarithmic +functions with poles at `z = \pm 1`. + +There are various ways to define Legendre functions of +the second kind, giving rise to different complex structure. +A version can be selected using the *type* keyword argument. +The *type=2* and *type=3* functions are given respectively by + +.. math :: + + Q_n^m(z) = \frac{\pi}{2 \sin(\pi m)} + \left( \cos(\pi m) P_n^m(z) - + \frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} P_n^{-m}(z)\right) + + \hat{Q}_n^m(z) = \frac{\pi}{2 \sin(\pi m)} e^{\pi i m} + \left( \hat{P}_n^m(z) - + \frac{\Gamma(1+m+n)}{\Gamma(1-m+n)} \hat{P}_n^{-m}(z)\right) + +where `P` and `\hat{P}` are the *type=2* and *type=3* Legendre functions +of the first kind. The formulas above should be understood as limits +when `m` is an integer. + +These functions correspond to ``LegendreQ[n,m,2,z]`` (or ``LegendreQ[n,m,z]``) +and ``LegendreQ[n,m,3,z]`` in Mathematica. The *type=3* function +is essentially the same as the function defined in +Abramowitz & Stegun (eq. 8.1.3) but with `(z+1)^{m/2}(z-1)^{m/2}` instead +of `(z^2-1)^{m/2}`, giving slightly different branches. + +**Examples** + +Evaluation for arbitrary parameters and arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> legenq(2, 0, 0.5) + -0.8186632680417568557122028 + >>> legenq(-1.5, -2, 2.5) + (0.6655964618250228714288277 + 0.3937692045497259717762649j) + >>> legenq(2-j, 3+4j, -6+5j) + (-10001.95256487468541686564 - 6011.691337610097577791134j) + +Different versions of the function:: + + >>> legenq(2, 1, 0.5) + 0.7298060598018049369381857 + >>> legenq(2, 1, 1.5) + (-7.902916572420817192300921 + 0.1998650072605976600724502j) + >>> legenq(2, 1, 0.5, type=3) + (2.040524284763495081918338 - 0.7298060598018049369381857j) + >>> chop(legenq(2, 1, 1.5, type=3)) + -0.1998650072605976600724502 + +""" + +chebyt = r""" +``chebyt(n, x)`` evaluates the Chebyshev polynomial of the first +kind `T_n(x)`, defined by the identity + +.. math :: + + T_n(\cos x) = \cos(n x). + +The Chebyshev polynomials of the first kind are a special +case of the Jacobi polynomials, and by extension of the +hypergeometric function `\,_2F_1`. They can thus also be +evaluated for nonintegral `n`. + +**Plots** + +.. literalinclude :: /plots/chebyt.py +.. image :: /plots/chebyt.png + +**Basic evaluation** + +The coefficients of the `n`-th polynomial can be recovered +using using degree-`n` Taylor expansion:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(5): + ... nprint(chop(taylor(lambda x: chebyt(n, x), 0, n))) + ... + [1.0] + [0.0, 1.0] + [-1.0, 0.0, 2.0] + [0.0, -3.0, 0.0, 4.0] + [1.0, 0.0, -8.0, 0.0, 8.0] + +**Orthogonality** + +The Chebyshev polynomials of the first kind are orthogonal +on the interval `[-1, 1]` with respect to the weight +function `w(x) = 1/\sqrt{1-x^2}`:: + + >>> f = lambda x: chebyt(m,x)*chebyt(n,x)/sqrt(1-x**2) + >>> m, n = 3, 4 + >>> nprint(quad(f, [-1, 1]),1) + 0.0 + >>> m, n = 4, 4 + >>> quad(f, [-1, 1]) + 1.57079632596448 + +""" + +chebyu = r""" +``chebyu(n, x)`` evaluates the Chebyshev polynomial of the second +kind `U_n(x)`, defined by the identity + +.. math :: + + U_n(\cos x) = \frac{\sin((n+1)x)}{\sin(x)}. + +The Chebyshev polynomials of the second kind are a special +case of the Jacobi polynomials, and by extension of the +hypergeometric function `\,_2F_1`. They can thus also be +evaluated for nonintegral `n`. + +**Plots** + +.. literalinclude :: /plots/chebyu.py +.. image :: /plots/chebyu.png + +**Basic evaluation** + +The coefficients of the `n`-th polynomial can be recovered +using using degree-`n` Taylor expansion:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(5): + ... nprint(chop(taylor(lambda x: chebyu(n, x), 0, n))) + ... + [1.0] + [0.0, 2.0] + [-1.0, 0.0, 4.0] + [0.0, -4.0, 0.0, 8.0] + [1.0, 0.0, -12.0, 0.0, 16.0] + +**Orthogonality** + +The Chebyshev polynomials of the second kind are orthogonal +on the interval `[-1, 1]` with respect to the weight +function `w(x) = \sqrt{1-x^2}`:: + + >>> f = lambda x: chebyu(m,x)*chebyu(n,x)*sqrt(1-x**2) + >>> m, n = 3, 4 + >>> quad(f, [-1, 1]) + 0.0 + >>> m, n = 4, 4 + >>> quad(f, [-1, 1]) + 1.5707963267949 +""" + +besselj = r""" +``besselj(n, x, derivative=0)`` gives the Bessel function of the first kind +`J_n(x)`. Bessel functions of the first kind are defined as +solutions of the differential equation + +.. math :: + + x^2 y'' + x y' + (x^2 - n^2) y = 0 + +which appears, among other things, when solving the radial +part of Laplace's equation in cylindrical coordinates. This +equation has two solutions for given `n`, where the +`J_n`-function is the solution that is nonsingular at `x = 0`. +For positive integer `n`, `J_n(x)` behaves roughly like a sine +(odd `n`) or cosine (even `n`) multiplied by a magnitude factor +that decays slowly as `x \to \pm\infty`. + +Generally, `J_n` is a special case of the hypergeometric +function `\,_0F_1`: + +.. math :: + + J_n(x) = \frac{x^n}{2^n \Gamma(n+1)} + \,_0F_1\left(n+1,-\frac{x^2}{4}\right) + +With *derivative* = `m \ne 0`, the `m`-th derivative + +.. math :: + + \frac{d^m}{dx^m} J_n(x) + +is computed. + +**Plots** + +.. literalinclude :: /plots/besselj.py +.. image :: /plots/besselj.png +.. literalinclude :: /plots/besselj_c.py +.. image :: /plots/besselj_c.png + +**Examples** + +Evaluation is supported for arbitrary arguments, and at +arbitrary precision:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> besselj(2, 1000) + -0.024777229528606 + >>> besselj(4, 0.75) + 0.000801070086542314 + >>> besselj(2, 1000j) + (-2.48071721019185e+432 + 6.41567059811949e-437j) + >>> mp.dps = 25 + >>> besselj(0.75j, 3+4j) + (-2.778118364828153309919653 - 1.5863603889018621585533j) + >>> mp.dps = 50 + >>> besselj(1, pi) + 0.28461534317975275734531059968613140570981118184947 + +Arguments may be large:: + + >>> mp.dps = 25 + >>> besselj(0, 10000) + -0.007096160353388801477265164 + >>> besselj(0, 10**10) + 0.000002175591750246891726859055 + >>> besselj(2, 10**100) + 7.337048736538615712436929e-51 + >>> besselj(2, 10**5*j) + (-3.540725411970948860173735e+43426 + 4.4949812409615803110051e-43433j) + +The Bessel functions of the first kind satisfy simple +symmetries around `x = 0`:: + + >>> mp.dps = 15 + >>> nprint([besselj(n,0) for n in range(5)]) + [1.0, 0.0, 0.0, 0.0, 0.0] + >>> nprint([besselj(n,pi) for n in range(5)]) + [-0.304242, 0.284615, 0.485434, 0.333458, 0.151425] + >>> nprint([besselj(n,-pi) for n in range(5)]) + [-0.304242, -0.284615, 0.485434, -0.333458, 0.151425] + +Roots of Bessel functions are often used:: + + >>> nprint([findroot(j0, k) for k in [2, 5, 8, 11, 14]]) + [2.40483, 5.52008, 8.65373, 11.7915, 14.9309] + >>> nprint([findroot(j1, k) for k in [3, 7, 10, 13, 16]]) + [3.83171, 7.01559, 10.1735, 13.3237, 16.4706] + +The roots are not periodic, but the distance between successive +roots asymptotically approaches `2 \pi`. Bessel functions of +the first kind have the following normalization:: + + >>> quadosc(j0, [0, inf], period=2*pi) + 1.0 + >>> quadosc(j1, [0, inf], period=2*pi) + 1.0 + +For `n = 1/2` or `n = -1/2`, the Bessel function reduces to a +trigonometric function:: + + >>> x = 10 + >>> besselj(0.5, x), sqrt(2/(pi*x))*sin(x) + (-0.13726373575505, -0.13726373575505) + >>> besselj(-0.5, x), sqrt(2/(pi*x))*cos(x) + (-0.211708866331398, -0.211708866331398) + +Derivatives of any order can be computed (negative orders +correspond to integration):: + + >>> mp.dps = 25 + >>> besselj(0, 7.5, 1) + -0.1352484275797055051822405 + >>> diff(lambda x: besselj(0,x), 7.5) + -0.1352484275797055051822405 + >>> besselj(0, 7.5, 10) + -0.1377811164763244890135677 + >>> diff(lambda x: besselj(0,x), 7.5, 10) + -0.1377811164763244890135677 + >>> besselj(0,7.5,-1) - besselj(0,3.5,-1) + -0.1241343240399987693521378 + >>> quad(j0, [3.5, 7.5]) + -0.1241343240399987693521378 + +Differentiation with a noninteger order gives the fractional derivative +in the sense of the Riemann-Liouville differintegral, as computed by +:func:`~mpmath.differint`:: + + >>> mp.dps = 15 + >>> besselj(1, 3.5, 0.75) + -0.385977722939384 + >>> differint(lambda x: besselj(1, x), 3.5, 0.75) + -0.385977722939384 + +""" + +besseli = r""" +``besseli(n, x, derivative=0)`` gives the modified Bessel function of the +first kind, + +.. math :: + + I_n(x) = i^{-n} J_n(ix). + +With *derivative* = `m \ne 0`, the `m`-th derivative + +.. math :: + + \frac{d^m}{dx^m} I_n(x) + +is computed. + +**Plots** + +.. literalinclude :: /plots/besseli.py +.. image :: /plots/besseli.png +.. literalinclude :: /plots/besseli_c.py +.. image :: /plots/besseli_c.png + +**Examples** + +Some values of `I_n(x)`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> besseli(0,0) + 1.0 + >>> besseli(1,0) + 0.0 + >>> besseli(0,1) + 1.266065877752008335598245 + >>> besseli(3.5, 2+3j) + (-0.2904369752642538144289025 - 0.4469098397654815837307006j) + +Arguments may be large:: + + >>> besseli(2, 1000) + 2.480717210191852440616782e+432 + >>> besseli(2, 10**10) + 4.299602851624027900335391e+4342944813 + >>> besseli(2, 6000+10000j) + (-2.114650753239580827144204e+2603 + 4.385040221241629041351886e+2602j) + +For integers `n`, the following integral representation holds:: + + >>> mp.dps = 15 + >>> n = 3 + >>> x = 2.3 + >>> quad(lambda t: exp(x*cos(t))*cos(n*t), [0,pi])/pi + 0.349223221159309 + >>> besseli(n,x) + 0.349223221159309 + +Derivatives and antiderivatives of any order can be computed:: + + >>> mp.dps = 25 + >>> besseli(2, 7.5, 1) + 195.8229038931399062565883 + >>> diff(lambda x: besseli(2,x), 7.5) + 195.8229038931399062565883 + >>> besseli(2, 7.5, 10) + 153.3296508971734525525176 + >>> diff(lambda x: besseli(2,x), 7.5, 10) + 153.3296508971734525525176 + >>> besseli(2,7.5,-1) - besseli(2,3.5,-1) + 202.5043900051930141956876 + >>> quad(lambda x: besseli(2,x), [3.5, 7.5]) + 202.5043900051930141956876 + +""" + +bessely = r""" +``bessely(n, x, derivative=0)`` gives the Bessel function of the second kind, + +.. math :: + + Y_n(x) = \frac{J_n(x) \cos(\pi n) - J_{-n}(x)}{\sin(\pi n)}. + +For `n` an integer, this formula should be understood as a +limit. With *derivative* = `m \ne 0`, the `m`-th derivative + +.. math :: + + \frac{d^m}{dx^m} Y_n(x) + +is computed. + +**Plots** + +.. literalinclude :: /plots/bessely.py +.. image :: /plots/bessely.png +.. literalinclude :: /plots/bessely_c.py +.. image :: /plots/bessely_c.png + +**Examples** + +Some values of `Y_n(x)`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> bessely(0,0), bessely(1,0), bessely(2,0) + (-inf, -inf, -inf) + >>> bessely(1, pi) + 0.3588729167767189594679827 + >>> bessely(0.5, 3+4j) + (9.242861436961450520325216 - 3.085042824915332562522402j) + +Arguments may be large:: + + >>> bessely(0, 10000) + 0.00364780555898660588668872 + >>> bessely(2.5, 10**50) + -4.8952500412050989295774e-26 + >>> bessely(2.5, -10**50) + (0.0 + 4.8952500412050989295774e-26j) + +Derivatives and antiderivatives of any order can be computed:: + + >>> bessely(2, 3.5, 1) + 0.3842618820422660066089231 + >>> diff(lambda x: bessely(2, x), 3.5) + 0.3842618820422660066089231 + >>> bessely(0.5, 3.5, 1) + -0.2066598304156764337900417 + >>> diff(lambda x: bessely(0.5, x), 3.5) + -0.2066598304156764337900417 + >>> diff(lambda x: bessely(2, x), 0.5, 10) + -208173867409.5547350101511 + >>> bessely(2, 0.5, 10) + -208173867409.5547350101511 + >>> bessely(2, 100.5, 100) + 0.02668487547301372334849043 + >>> quad(lambda x: bessely(2,x), [1,3]) + -1.377046859093181969213262 + >>> bessely(2,3,-1) - bessely(2,1,-1) + -1.377046859093181969213262 + +""" + +besselk = r""" +``besselk(n, x)`` gives the modified Bessel function of the +second kind, + +.. math :: + + K_n(x) = \frac{\pi}{2} \frac{I_{-n}(x)-I_{n}(x)}{\sin(\pi n)} + +For `n` an integer, this formula should be understood as a +limit. + +**Plots** + +.. literalinclude :: /plots/besselk.py +.. image :: /plots/besselk.png +.. literalinclude :: /plots/besselk_c.py +.. image :: /plots/besselk_c.png + +**Examples** + +Evaluation is supported for arbitrary complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> besselk(0,1) + 0.4210244382407083333356274 + >>> besselk(0, -1) + (0.4210244382407083333356274 - 3.97746326050642263725661j) + >>> besselk(3.5, 2+3j) + (-0.02090732889633760668464128 + 0.2464022641351420167819697j) + >>> besselk(2+3j, 0.5) + (0.9615816021726349402626083 + 0.1918250181801757416908224j) + +Arguments may be large:: + + >>> besselk(0, 100) + 4.656628229175902018939005e-45 + >>> besselk(1, 10**6) + 4.131967049321725588398296e-434298 + >>> besselk(1, 10**6*j) + (0.001140348428252385844876706 - 0.0005200017201681152909000961j) + >>> besselk(4.5, fmul(10**50, j, exact=True)) + (1.561034538142413947789221e-26 + 1.243554598118700063281496e-25j) + +The point `x = 0` is a singularity (logarithmic if `n = 0`):: + + >>> besselk(0,0) + +inf + >>> besselk(1,0) + +inf + >>> for n in range(-4, 5): + ... print(besselk(n, '1e-1000')) + ... + 4.8e+4001 + 8.0e+3000 + 2.0e+2000 + 1.0e+1000 + 2302.701024509704096466802 + 1.0e+1000 + 2.0e+2000 + 8.0e+3000 + 4.8e+4001 + +""" + +hankel1 = r""" +``hankel1(n,x)`` computes the Hankel function of the first kind, +which is the complex combination of Bessel functions given by + +.. math :: + + H_n^{(1)}(x) = J_n(x) + i Y_n(x). + +**Plots** + +.. literalinclude :: /plots/hankel1.py +.. image :: /plots/hankel1.png +.. literalinclude :: /plots/hankel1_c.py +.. image :: /plots/hankel1_c.png + +**Examples** + +The Hankel function is generally complex-valued:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hankel1(2, pi) + (0.4854339326315091097054957 - 0.0999007139290278787734903j) + >>> hankel1(3.5, pi) + (0.2340002029630507922628888 - 0.6419643823412927142424049j) +""" + +hankel2 = r""" +``hankel2(n,x)`` computes the Hankel function of the second kind, +which is the complex combination of Bessel functions given by + +.. math :: + + H_n^{(2)}(x) = J_n(x) - i Y_n(x). + +**Plots** + +.. literalinclude :: /plots/hankel2.py +.. image :: /plots/hankel2.png +.. literalinclude :: /plots/hankel2_c.py +.. image :: /plots/hankel2_c.png + +**Examples** + +The Hankel function is generally complex-valued:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> hankel2(2, pi) + (0.4854339326315091097054957 + 0.0999007139290278787734903j) + >>> hankel2(3.5, pi) + (0.2340002029630507922628888 + 0.6419643823412927142424049j) +""" + +lambertw = r""" +The Lambert W function `W(z)` is defined as the inverse function +of `w \exp(w)`. In other words, the value of `W(z)` is such that +`z = W(z) \exp(W(z))` for any complex number `z`. + +The Lambert W function is a multivalued function with infinitely +many branches `W_k(z)`, indexed by `k \in \mathbb{Z}`. Each branch +gives a different solution `w` of the equation `z = w \exp(w)`. +All branches are supported by :func:`~mpmath.lambertw`: + +* ``lambertw(z)`` gives the principal solution (branch 0) + +* ``lambertw(z, k)`` gives the solution on branch `k` + +The Lambert W function has two partially real branches: the +principal branch (`k = 0`) is real for real `z > -1/e`, and the +`k = -1` branch is real for `-1/e < z < 0`. All branches except +`k = 0` have a logarithmic singularity at `z = 0`. + +The definition, implementation and choice of branches +is based on [Corless]_. + +**Plots** + +.. literalinclude :: /plots/lambertw.py +.. image :: /plots/lambertw.png +.. literalinclude :: /plots/lambertw_c.py +.. image :: /plots/lambertw_c.png + +**Basic examples** + +The Lambert W function is the inverse of `w \exp(w)`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> w = lambertw(1) + >>> w + 0.5671432904097838729999687 + >>> w*exp(w) + 1.0 + +Any branch gives a valid inverse:: + + >>> w = lambertw(1, k=3) + >>> w + (-2.853581755409037807206819 + 17.11353553941214591260783j) + >>> w = lambertw(1, k=25) + >>> w + (-5.047020464221569709378686 + 155.4763860949415867162066j) + >>> chop(w*exp(w)) + 1.0 + +**Applications to equation-solving** + +The Lambert W function may be used to solve various kinds of +equations, such as finding the value of the infinite power +tower `z^{z^{z^{\ldots}}}`:: + + >>> def tower(z, n): + ... if n == 0: + ... return z + ... return z ** tower(z, n-1) + ... + >>> tower(mpf(0.5), 100) + 0.6411857445049859844862005 + >>> -lambertw(-log(0.5))/log(0.5) + 0.6411857445049859844862005 + +**Properties** + +The Lambert W function grows roughly like the natural logarithm +for large arguments:: + + >>> lambertw(1000); log(1000) + 5.249602852401596227126056 + 6.907755278982137052053974 + >>> lambertw(10**100); log(10**100) + 224.8431064451185015393731 + 230.2585092994045684017991 + +The principal branch of the Lambert W function has a rational +Taylor series expansion around `z = 0`:: + + >>> nprint(taylor(lambertw, 0, 6), 10) + [0.0, 1.0, -1.0, 1.5, -2.666666667, 5.208333333, -10.8] + +Some special values and limits are:: + + >>> lambertw(0) + 0.0 + >>> lambertw(1) + 0.5671432904097838729999687 + >>> lambertw(e) + 1.0 + >>> lambertw(inf) + +inf + >>> lambertw(0, k=-1) + -inf + >>> lambertw(0, k=3) + -inf + >>> lambertw(inf, k=2) + (+inf + 12.56637061435917295385057j) + >>> lambertw(inf, k=3) + (+inf + 18.84955592153875943077586j) + >>> lambertw(-inf, k=3) + (+inf + 21.9911485751285526692385j) + +The `k = 0` and `k = -1` branches join at `z = -1/e` where +`W(z) = -1` for both branches. Since `-1/e` can only be represented +approximately with binary floating-point numbers, evaluating the +Lambert W function at this point only gives `-1` approximately:: + + >>> lambertw(-1/e, 0) + -0.9999999999998371330228251 + >>> lambertw(-1/e, -1) + -1.000000000000162866977175 + +If `-1/e` happens to round in the negative direction, there might be +a small imaginary part:: + + >>> mp.dps = 15 + >>> lambertw(-1/e) + (-1.0 + 8.22007971483662e-9j) + >>> lambertw(-1/e+eps) + -0.999999966242188 + +**References** + +1. [Corless]_ +""" + +barnesg = r""" +Evaluates the Barnes G-function, which generalizes the +superfactorial (:func:`~mpmath.superfac`) and by extension also the +hyperfactorial (:func:`~mpmath.hyperfac`) to the complex numbers +in an analogous way to how the gamma function generalizes +the ordinary factorial. + +The Barnes G-function may be defined in terms of a Weierstrass +product: + +.. math :: + + G(z+1) = (2\pi)^{z/2} e^{-[z(z+1)+\gamma z^2]/2} + \prod_{n=1}^\infty + \left[\left(1+\frac{z}{n}\right)^ne^{-z+z^2/(2n)}\right] + +For positive integers `n`, we have have relation to superfactorials +`G(n) = \mathrm{sf}(n-2) = 0! \cdot 1! \cdots (n-2)!`. + +**Examples** + +Some elementary values and limits of the Barnes G-function:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> barnesg(1), barnesg(2), barnesg(3) + (1.0, 1.0, 1.0) + >>> barnesg(4) + 2.0 + >>> barnesg(5) + 12.0 + >>> barnesg(6) + 288.0 + >>> barnesg(7) + 34560.0 + >>> barnesg(8) + 24883200.0 + >>> barnesg(inf) + +inf + >>> barnesg(0), barnesg(-1), barnesg(-2) + (0.0, 0.0, 0.0) + +Closed-form values are known for some rational arguments:: + + >>> barnesg('1/2') + 0.603244281209446 + >>> sqrt(exp(0.25+log(2)/12)/sqrt(pi)/glaisher**3) + 0.603244281209446 + >>> barnesg('1/4') + 0.29375596533861 + >>> nthroot(exp('3/8')/exp(catalan/pi)/ + ... gamma(0.25)**3/sqrt(glaisher)**9, 4) + 0.29375596533861 + +The Barnes G-function satisfies the functional equation +`G(z+1) = \Gamma(z) G(z)`:: + + >>> z = pi + >>> barnesg(z+1) + 2.39292119327948 + >>> gamma(z)*barnesg(z) + 2.39292119327948 + +The asymptotic growth rate of the Barnes G-function is related to +the Glaisher-Kinkelin constant:: + + >>> limit(lambda n: barnesg(n+1)/(n**(n**2/2-mpf(1)/12)* + ... (2*pi)**(n/2)*exp(-3*n**2/4)), inf) + 0.847536694177301 + >>> exp('1/12')/glaisher + 0.847536694177301 + +The Barnes G-function can be differentiated in closed form:: + + >>> z = 3 + >>> diff(barnesg, z) + 0.264507203401607 + >>> barnesg(z)*((z-1)*psi(0,z)-z+(log(2*pi)+1)/2) + 0.264507203401607 + +Evaluation is supported for arbitrary arguments and at arbitrary +precision:: + + >>> barnesg(6.5) + 2548.7457695685 + >>> barnesg(-pi) + 0.00535976768353037 + >>> barnesg(3+4j) + (-0.000676375932234244 - 4.42236140124728e-5j) + >>> mp.dps = 50 + >>> barnesg(1/sqrt(2)) + 0.81305501090451340843586085064413533788206204124732 + >>> q = barnesg(10j) + >>> q.real + 0.000000000021852360840356557241543036724799812371995850552234 + >>> q.imag + -0.00000000000070035335320062304849020654215545839053210041457588 + >>> mp.dps = 15 + >>> barnesg(100) + 3.10361006263698e+6626 + >>> barnesg(-101) + 0.0 + >>> barnesg(-10.5) + 5.94463017605008e+25 + >>> barnesg(-10000.5) + -6.14322868174828e+167480422 + >>> barnesg(1000j) + (5.21133054865546e-1173597 + 4.27461836811016e-1173597j) + >>> barnesg(-1000+1000j) + (2.43114569750291e+1026623 + 2.24851410674842e+1026623j) + + +**References** + +1. Whittaker & Watson, *A Course of Modern Analysis*, + Cambridge University Press, 4th edition (1927), p.264 +2. http://en.wikipedia.org/wiki/Barnes_G-function +3. http://mathworld.wolfram.com/BarnesG-Function.html + +""" + +superfac = r""" +Computes the superfactorial, defined as the product of +consecutive factorials + +.. math :: + + \mathrm{sf}(n) = \prod_{k=1}^n k! + +For general complex `z`, `\mathrm{sf}(z)` is defined +in terms of the Barnes G-function (see :func:`~mpmath.barnesg`). + +**Examples** + +The first few superfactorials are (OEIS A000178):: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(10): + ... print("%s %s" % (n, superfac(n))) + ... + 0 1.0 + 1 1.0 + 2 2.0 + 3 12.0 + 4 288.0 + 5 34560.0 + 6 24883200.0 + 7 125411328000.0 + 8 5.05658474496e+15 + 9 1.83493347225108e+21 + +Superfactorials grow very rapidly:: + + >>> superfac(1000) + 3.24570818422368e+1177245 + >>> superfac(10**10) + 2.61398543581249e+467427913956904067453 + +Evaluation is supported for arbitrary arguments:: + + >>> mp.dps = 25 + >>> superfac(pi) + 17.20051550121297985285333 + >>> superfac(2+3j) + (-0.005915485633199789627466468 + 0.008156449464604044948738263j) + >>> diff(superfac, 1) + 0.2645072034016070205673056 + +**References** + +1. http://oeis.org/A000178 + +""" + + +hyperfac = r""" +Computes the hyperfactorial, defined for integers as the product + +.. math :: + + H(n) = \prod_{k=1}^n k^k. + + +The hyperfactorial satisfies the recurrence formula `H(z) = z^z H(z-1)`. +It can be defined more generally in terms of the Barnes G-function (see +:func:`~mpmath.barnesg`) and the gamma function by the formula + +.. math :: + + H(z) = \frac{\Gamma(z+1)^z}{G(z)}. + +The extension to complex numbers can also be done via +the integral representation + +.. math :: + + H(z) = (2\pi)^{-z/2} \exp \left[ + {z+1 \choose 2} + \int_0^z \log(t!)\,dt + \right]. + +**Examples** + +The rapidly-growing sequence of hyperfactorials begins +(OEIS A002109):: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(10): + ... print("%s %s" % (n, hyperfac(n))) + ... + 0 1.0 + 1 1.0 + 2 4.0 + 3 108.0 + 4 27648.0 + 5 86400000.0 + 6 4031078400000.0 + 7 3.3197663987712e+18 + 8 5.56964379417266e+25 + 9 2.15779412229419e+34 + +Some even larger hyperfactorials are:: + + >>> hyperfac(1000) + 5.46458120882585e+1392926 + >>> hyperfac(10**10) + 4.60408207642219e+489142638002418704309 + +The hyperfactorial can be evaluated for arbitrary arguments:: + + >>> hyperfac(0.5) + 0.880449235173423 + >>> diff(hyperfac, 1) + 0.581061466795327 + >>> hyperfac(pi) + 205.211134637462 + >>> hyperfac(-10+1j) + (3.01144471378225e+46 - 2.45285242480185e+46j) + +The recurrence property of the hyperfactorial holds +generally:: + + >>> z = 3-4*j + >>> hyperfac(z) + (-4.49795891462086e-7 - 6.33262283196162e-7j) + >>> z**z * hyperfac(z-1) + (-4.49795891462086e-7 - 6.33262283196162e-7j) + >>> z = mpf(-0.6) + >>> chop(z**z * hyperfac(z-1)) + 1.28170142849352 + >>> hyperfac(z) + 1.28170142849352 + +The hyperfactorial may also be computed using the integral +definition:: + + >>> z = 2.5 + >>> hyperfac(z) + 15.9842119922237 + >>> (2*pi)**(-z/2)*exp(binomial(z+1,2) + + ... quad(lambda t: loggamma(t+1), [0, z])) + 15.9842119922237 + +:func:`~mpmath.hyperfac` supports arbitrary-precision evaluation:: + + >>> mp.dps = 50 + >>> hyperfac(10) + 215779412229418562091680268288000000000000000.0 + >>> hyperfac(1/sqrt(2)) + 0.89404818005227001975423476035729076375705084390942 + +**References** + +1. http://oeis.org/A002109 +2. http://mathworld.wolfram.com/Hyperfactorial.html + +""" + +rgamma = r""" +Computes the reciprocal of the gamma function, `1/\Gamma(z)`. This +function evaluates to zero at the poles +of the gamma function, `z = 0, -1, -2, \ldots`. + +**Examples** + +Basic examples:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> rgamma(1) + 1.0 + >>> rgamma(4) + 0.1666666666666666666666667 + >>> rgamma(0); rgamma(-1) + 0.0 + 0.0 + >>> rgamma(1000) + 2.485168143266784862783596e-2565 + >>> rgamma(inf) + 0.0 + +A definite integral that can be evaluated in terms of elementary +integrals:: + + >>> quad(rgamma, [0,inf]) + 2.807770242028519365221501 + >>> e + quad(lambda t: exp(-t)/(pi**2+log(t)**2), [0,inf]) + 2.807770242028519365221501 +""" + +loggamma = r""" +Computes the principal branch of the log-gamma function, +`\ln \Gamma(z)`. Unlike `\ln(\Gamma(z))`, which has infinitely many +complex branch cuts, the principal log-gamma function only has a single +branch cut along the negative half-axis. The principal branch +continuously matches the asymptotic Stirling expansion + +.. math :: + + \ln \Gamma(z) \sim \frac{\ln(2 \pi)}{2} + + \left(z-\frac{1}{2}\right) \ln(z) - z + O(z^{-1}). + +The real parts of both functions agree, but their imaginary +parts generally differ by `2 n \pi` for some `n \in \mathbb{Z}`. +They coincide for `z \in \mathbb{R}, z > 0`. + +Computationally, it is advantageous to use :func:`~mpmath.loggamma` +instead of :func:`~mpmath.gamma` for extremely large arguments. + +**Examples** + +Comparing with `\ln(\Gamma(z))`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> loggamma('13.2'); log(gamma('13.2')) + 20.49400419456603678498394 + 20.49400419456603678498394 + >>> loggamma(3+4j) + (-1.756626784603784110530604 + 4.742664438034657928194889j) + >>> log(gamma(3+4j)) + (-1.756626784603784110530604 - 1.540520869144928548730397j) + >>> log(gamma(3+4j)) + 2*pi*j + (-1.756626784603784110530604 + 4.742664438034657928194889j) + +Note the imaginary parts for negative arguments:: + + >>> loggamma(-0.5); loggamma(-1.5); loggamma(-2.5) + (1.265512123484645396488946 - 3.141592653589793238462643j) + (0.8600470153764810145109327 - 6.283185307179586476925287j) + (-0.05624371649767405067259453 - 9.42477796076937971538793j) + +Some special values:: + + >>> loggamma(1); loggamma(2) + 0.0 + 0.0 + >>> loggamma(3); +ln2 + 0.6931471805599453094172321 + 0.6931471805599453094172321 + >>> loggamma(3.5); log(15*sqrt(pi)/8) + 1.200973602347074224816022 + 1.200973602347074224816022 + >>> loggamma(inf) + +inf + +Huge arguments are permitted:: + + >>> loggamma('1e30') + 6.807755278982137052053974e+31 + >>> loggamma('1e300') + 6.897755278982137052053974e+302 + >>> loggamma('1e3000') + 6.906755278982137052053974e+3003 + >>> loggamma('1e100000000000000000000') + 2.302585092994045684007991e+100000000000000000020 + >>> loggamma('1e30j') + (-1.570796326794896619231322e+30 + 6.807755278982137052053974e+31j) + >>> loggamma('1e300j') + (-1.570796326794896619231322e+300 + 6.897755278982137052053974e+302j) + >>> loggamma('1e3000j') + (-1.570796326794896619231322e+3000 + 6.906755278982137052053974e+3003j) + +The log-gamma function can be integrated analytically +on any interval of unit length:: + + >>> z = 0 + >>> quad(loggamma, [z,z+1]); log(2*pi)/2 + 0.9189385332046727417803297 + 0.9189385332046727417803297 + >>> z = 3+4j + >>> quad(loggamma, [z,z+1]); (log(z)-1)*z + log(2*pi)/2 + (-0.9619286014994750641314421 + 5.219637303741238195688575j) + (-0.9619286014994750641314421 + 5.219637303741238195688575j) + +The derivatives of the log-gamma function are given by the +polygamma function (:func:`~mpmath.psi`):: + + >>> diff(loggamma, -4+3j); psi(0, -4+3j) + (1.688493531222971393607153 + 2.554898911356806978892748j) + (1.688493531222971393607153 + 2.554898911356806978892748j) + >>> diff(loggamma, -4+3j, 2); psi(1, -4+3j) + (-0.1539414829219882371561038 - 0.1020485197430267719746479j) + (-0.1539414829219882371561038 - 0.1020485197430267719746479j) + +The log-gamma function satisfies an additive form of the +recurrence relation for the ordinary gamma function:: + + >>> z = 2+3j + >>> loggamma(z); loggamma(z+1) - log(z) + (-2.092851753092733349564189 + 2.302396543466867626153708j) + (-2.092851753092733349564189 + 2.302396543466867626153708j) + +""" + +siegeltheta = r""" +Computes the Riemann-Siegel theta function, + +.. math :: + + \theta(t) = \frac{ + \log\Gamma\left(\frac{1+2it}{4}\right) - + \log\Gamma\left(\frac{1-2it}{4}\right) + }{2i} - \frac{\log \pi}{2} t. + +The Riemann-Siegel theta function is important in +providing the phase factor for the Z-function +(see :func:`~mpmath.siegelz`). Evaluation is supported for real and +complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> siegeltheta(0) + 0.0 + >>> siegeltheta(inf) + +inf + >>> siegeltheta(-inf) + -inf + >>> siegeltheta(1) + -1.767547952812290388302216 + >>> siegeltheta(10+0.25j) + (-3.068638039426838572528867 + 0.05804937947429712998395177j) + +Arbitrary derivatives may be computed with derivative = k + + >>> siegeltheta(1234, derivative=2) + 0.0004051864079114053109473741 + >>> diff(siegeltheta, 1234, n=2) + 0.0004051864079114053109473741 + + +The Riemann-Siegel theta function has odd symmetry around `t = 0`, +two local extreme points and three real roots including 0 (located +symmetrically):: + + >>> nprint(chop(taylor(siegeltheta, 0, 5))) + [0.0, -2.68609, 0.0, 2.69433, 0.0, -6.40218] + >>> findroot(diffun(siegeltheta), 7) + 6.28983598883690277966509 + >>> findroot(siegeltheta, 20) + 17.84559954041086081682634 + +For large `t`, there is a famous asymptotic formula +for `\theta(t)`, to first order given by:: + + >>> t = mpf(10**6) + >>> siegeltheta(t) + 5488816.353078403444882823 + >>> -t*log(2*pi/t)/2-t/2 + 5488816.745777464310273645 +""" + +grampoint = r""" +Gives the `n`-th Gram point `g_n`, defined as the solution +to the equation `\theta(g_n) = \pi n` where `\theta(t)` +is the Riemann-Siegel theta function (:func:`~mpmath.siegeltheta`). + +The first few Gram points are:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> grampoint(0) + 17.84559954041086081682634 + >>> grampoint(1) + 23.17028270124630927899664 + >>> grampoint(2) + 27.67018221781633796093849 + >>> grampoint(3) + 31.71797995476405317955149 + +Checking the definition:: + + >>> siegeltheta(grampoint(3)) + 9.42477796076937971538793 + >>> 3*pi + 9.42477796076937971538793 + +A large Gram point:: + + >>> grampoint(10**10) + 3293531632.728335454561153 + +Gram points are useful when studying the Z-function +(:func:`~mpmath.siegelz`). See the documentation of that function +for additional examples. + +:func:`~mpmath.grampoint` can solve the defining equation for +nonintegral `n`. There is a fixed point where `g(x) = x`:: + + >>> findroot(lambda x: grampoint(x) - x, 10000) + 9146.698193171459265866198 + +**References** + +1. http://mathworld.wolfram.com/GramPoint.html + +""" + +siegelz = r""" +Computes the Z-function, also known as the Riemann-Siegel Z function, + +.. math :: + + Z(t) = e^{i \theta(t)} \zeta(1/2+it) + +where `\zeta(s)` is the Riemann zeta function (:func:`~mpmath.zeta`) +and where `\theta(t)` denotes the Riemann-Siegel theta function +(see :func:`~mpmath.siegeltheta`). + +Evaluation is supported for real and complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> siegelz(1) + -0.7363054628673177346778998 + >>> siegelz(3+4j) + (-0.1852895764366314976003936 - 0.2773099198055652246992479j) + +The first four derivatives are supported, using the +optional *derivative* keyword argument:: + + >>> siegelz(1234567, derivative=3) + 56.89689348495089294249178 + >>> diff(siegelz, 1234567, n=3) + 56.89689348495089294249178 + + +The Z-function has a Maclaurin expansion:: + + >>> nprint(chop(taylor(siegelz, 0, 4))) + [-1.46035, 0.0, 2.73588, 0.0, -8.39357] + +The Z-function `Z(t)` is equal to `\pm |\zeta(s)|` on the +critical line `s = 1/2+it` (i.e. for real arguments `t` +to `Z`). Its zeros coincide with those of the Riemann zeta +function:: + + >>> findroot(siegelz, 14) + 14.13472514173469379045725 + >>> findroot(siegelz, 20) + 21.02203963877155499262848 + >>> findroot(zeta, 0.5+14j) + (0.5 + 14.13472514173469379045725j) + >>> findroot(zeta, 0.5+20j) + (0.5 + 21.02203963877155499262848j) + +Since the Z-function is real-valued on the critical line +(and unlike `|\zeta(s)|` analytic), it is useful for +investigating the zeros of the Riemann zeta function. +For example, one can use a root-finding algorithm based +on sign changes:: + + >>> findroot(siegelz, [100, 200], solver='bisect') + 176.4414342977104188888926 + +To locate roots, Gram points `g_n` which can be computed +by :func:`~mpmath.grampoint` are useful. If `(-1)^n Z(g_n)` is +positive for two consecutive `n`, then `Z(t)` must have +a zero between those points:: + + >>> g10 = grampoint(10) + >>> g11 = grampoint(11) + >>> (-1)**10 * siegelz(g10) > 0 + True + >>> (-1)**11 * siegelz(g11) > 0 + True + >>> findroot(siegelz, [g10, g11], solver='bisect') + 56.44624769706339480436776 + >>> g10, g11 + (54.67523744685325626632663, 57.54516517954725443703014) + +""" + +riemannr = r""" +Evaluates the Riemann R function, a smooth approximation of the +prime counting function `\pi(x)` (see :func:`~mpmath.primepi`). The Riemann +R function gives a fast numerical approximation useful e.g. to +roughly estimate the number of primes in a given interval. + +The Riemann R function is computed using the rapidly convergent Gram +series, + +.. math :: + + R(x) = 1 + \sum_{k=1}^{\infty} + \frac{\log^k x}{k k! \zeta(k+1)}. + +From the Gram series, one sees that the Riemann R function is a +well-defined analytic function (except for a branch cut along +the negative real half-axis); it can be evaluated for arbitrary +real or complex arguments. + +The Riemann R function gives a very accurate approximation +of the prime counting function. For example, it is wrong by at +most 2 for `x < 1000`, and for `x = 10^9` differs from the exact +value of `\pi(x)` by 79, or less than two parts in a million. +It is about 10 times more accurate than the logarithmic integral +estimate (see :func:`~mpmath.li`), which however is even faster to evaluate. +It is orders of magnitude more accurate than the extremely +fast `x/\log x` estimate. + +**Examples** + +For small arguments, the Riemann R function almost exactly +gives the prime counting function if rounded to the nearest +integer:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> primepi(50), riemannr(50) + (15, 14.9757023241462) + >>> max(abs(primepi(n)-int(round(riemannr(n)))) for n in range(100)) + 1 + >>> max(abs(primepi(n)-int(round(riemannr(n)))) for n in range(300)) + 2 + +The Riemann R function can be evaluated for arguments far too large +for exact determination of `\pi(x)` to be computationally +feasible with any presently known algorithm:: + + >>> riemannr(10**30) + 1.46923988977204e+28 + >>> riemannr(10**100) + 4.3619719871407e+97 + >>> riemannr(10**1000) + 4.3448325764012e+996 + +A comparison of the Riemann R function and logarithmic integral estimates +for `\pi(x)` using exact values of `\pi(10^n)` up to `n = 9`. +The fractional error is shown in parentheses:: + + >>> exact = [4,25,168,1229,9592,78498,664579,5761455,50847534] + >>> for n, p in enumerate(exact): + ... n += 1 + ... r, l = riemannr(10**n), li(10**n) + ... rerr, lerr = nstr((r-p)/p,3), nstr((l-p)/p,3) + ... print("%i %i %s(%s) %s(%s)" % (n, p, r, rerr, l, lerr)) + ... + 1 4 4.56458314100509(0.141) 6.1655995047873(0.541) + 2 25 25.6616332669242(0.0265) 30.1261415840796(0.205) + 3 168 168.359446281167(0.00214) 177.609657990152(0.0572) + 4 1229 1226.93121834343(-0.00168) 1246.13721589939(0.0139) + 5 9592 9587.43173884197(-0.000476) 9629.8090010508(0.00394) + 6 78498 78527.3994291277(0.000375) 78627.5491594622(0.00165) + 7 664579 664667.447564748(0.000133) 664918.405048569(0.000511) + 8 5761455 5761551.86732017(1.68e-5) 5762209.37544803(0.000131) + 9 50847534 50847455.4277214(-1.55e-6) 50849234.9570018(3.35e-5) + +The derivative of the Riemann R function gives the approximate +probability for a number of magnitude `x` to be prime:: + + >>> diff(riemannr, 1000) + 0.141903028110784 + >>> mpf(primepi(1050) - primepi(950)) / 100 + 0.15 + +Evaluation is supported for arbitrary arguments and at arbitrary +precision:: + + >>> mp.dps = 30 + >>> riemannr(7.5) + 3.72934743264966261918857135136 + >>> riemannr(-4+2j) + (-0.551002208155486427591793957644 + 2.16966398138119450043195899746j) + +""" + +primepi = r""" +Evaluates the prime counting function, `\pi(x)`, which gives +the number of primes less than or equal to `x`. The argument +`x` may be fractional. + +The prime counting function is very expensive to evaluate +precisely for large `x`, and the present implementation is +not optimized in any way. For numerical approximation of the +prime counting function, it is better to use :func:`~mpmath.primepi2` +or :func:`~mpmath.riemannr`. + +Some values of the prime counting function:: + + >>> from mpmath import * + >>> [primepi(k) for k in range(20)] + [0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8] + >>> primepi(3.5) + 2 + >>> primepi(100000) + 9592 + +""" + +primepi2 = r""" +Returns an interval (as an ``mpi`` instance) providing bounds +for the value of the prime counting function `\pi(x)`. For small +`x`, :func:`~mpmath.primepi2` returns an exact interval based on +the output of :func:`~mpmath.primepi`. For `x > 2656`, a loose interval +based on Schoenfeld's inequality + +.. math :: + + |\pi(x) - \mathrm{li}(x)| < \frac{\sqrt x \log x}{8 \pi} + +is returned. This estimate is rigorous assuming the truth of +the Riemann hypothesis, and can be computed very quickly. + +**Examples** + +Exact values of the prime counting function for small `x`:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> iv.dps = 15; iv.pretty = True + >>> primepi2(10) + [4.0, 4.0] + >>> primepi2(100) + [25.0, 25.0] + >>> primepi2(1000) + [168.0, 168.0] + +Loose intervals are generated for moderately large `x`: + + >>> primepi2(10000), primepi(10000) + ([1209.0, 1283.0], 1229) + >>> primepi2(50000), primepi(50000) + ([5070.0, 5263.0], 5133) + +As `x` increases, the absolute error gets worse while the relative +error improves. The exact value of `\pi(10^{23})` is +1925320391606803968923, and :func:`~mpmath.primepi2` gives 9 significant +digits:: + + >>> p = primepi2(10**23) + >>> p + [1.9253203909477020467e+21, 1.925320392280406229e+21] + >>> mpf(p.delta) / mpf(p.a) + 6.9219865355293e-10 + +A more precise, nonrigorous estimate for `\pi(x)` can be +obtained using the Riemann R function (:func:`~mpmath.riemannr`). +For large enough `x`, the value returned by :func:`~mpmath.primepi2` +essentially amounts to a small perturbation of the value returned by +:func:`~mpmath.riemannr`:: + + >>> primepi2(10**100) + [4.3619719871407024816e+97, 4.3619719871407032404e+97] + >>> riemannr(10**100) + 4.3619719871407e+97 +""" + +primezeta = r""" +Computes the prime zeta function, which is defined +in analogy with the Riemann zeta function (:func:`~mpmath.zeta`) +as + +.. math :: + + P(s) = \sum_p \frac{1}{p^s} + +where the sum is taken over all prime numbers `p`. Although +this sum only converges for `\mathrm{Re}(s) > 1`, the +function is defined by analytic continuation in the +half-plane `\mathrm{Re}(s) > 0`. + +**Examples** + +Arbitrary-precision evaluation for real and complex arguments is +supported:: + + >>> from mpmath import * + >>> mp.dps = 30; mp.pretty = True + >>> primezeta(2) + 0.452247420041065498506543364832 + >>> primezeta(pi) + 0.15483752698840284272036497397 + >>> mp.dps = 50 + >>> primezeta(3) + 0.17476263929944353642311331466570670097541212192615 + >>> mp.dps = 20 + >>> primezeta(3+4j) + (-0.12085382601645763295 - 0.013370403397787023602j) + +The prime zeta function has a logarithmic pole at `s = 1`, +with residue equal to the difference of the Mertens and +Euler constants:: + + >>> primezeta(1) + +inf + >>> extradps(25)(lambda x: primezeta(1+x)+log(x))(+eps) + -0.31571845205389007685 + >>> mertens-euler + -0.31571845205389007685 + +The analytic continuation to `0 < \mathrm{Re}(s) \le 1` +is implemented. In this strip the function exhibits +very complex behavior; on the unit interval, it has poles at +`1/n` for every squarefree integer `n`:: + + >>> primezeta(0.5) # Pole at s = 1/2 + (-inf + 3.1415926535897932385j) + >>> primezeta(0.25) + (-1.0416106801757269036 + 0.52359877559829887308j) + >>> primezeta(0.5+10j) + (0.54892423556409790529 + 0.45626803423487934264j) + +Although evaluation works in principle for any `\mathrm{Re}(s) > 0`, +it should be noted that the evaluation time increases exponentially +as `s` approaches the imaginary axis. + +For large `\mathrm{Re}(s)`, `P(s)` is asymptotic to `2^{-s}`:: + + >>> primezeta(inf) + 0.0 + >>> primezeta(10), mpf(2)**-10 + (0.00099360357443698021786, 0.0009765625) + >>> primezeta(1000) + 9.3326361850321887899e-302 + >>> primezeta(1000+1000j) + (-3.8565440833654995949e-302 - 8.4985390447553234305e-302j) + +**References** + +Carl-Erik Froberg, "On the prime zeta function", +BIT 8 (1968), pp. 187-202. + +""" + +bernpoly = r""" +Evaluates the Bernoulli polynomial `B_n(z)`. + +The first few Bernoulli polynomials are:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(6): + ... nprint(chop(taylor(lambda x: bernpoly(n,x), 0, n))) + ... + [1.0] + [-0.5, 1.0] + [0.166667, -1.0, 1.0] + [0.0, 0.5, -1.5, 1.0] + [-0.0333333, 0.0, 1.0, -2.0, 1.0] + [0.0, -0.166667, 0.0, 1.66667, -2.5, 1.0] + +At `z = 0`, the Bernoulli polynomial evaluates to a +Bernoulli number (see :func:`~mpmath.bernoulli`):: + + >>> bernpoly(12, 0), bernoulli(12) + (-0.253113553113553, -0.253113553113553) + >>> bernpoly(13, 0), bernoulli(13) + (0.0, 0.0) + +Evaluation is accurate for large `n` and small `z`:: + + >>> mp.dps = 25 + >>> bernpoly(100, 0.5) + 2.838224957069370695926416e+78 + >>> bernpoly(1000, 10.5) + 5.318704469415522036482914e+1769 + +""" + +polylog = r""" +Computes the polylogarithm, defined by the sum + +.. math :: + + \mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^s}. + +This series is convergent only for `|z| < 1`, so elsewhere +the analytic continuation is implied. + +The polylogarithm should not be confused with the logarithmic +integral (also denoted by Li or li), which is implemented +as :func:`~mpmath.li`. + +**Examples** + +The polylogarithm satisfies a huge number of functional identities. +A sample of polylogarithm evaluations is shown below:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> polylog(1,0.5), log(2) + (0.693147180559945, 0.693147180559945) + >>> polylog(2,0.5), (pi**2-6*log(2)**2)/12 + (0.582240526465012, 0.582240526465012) + >>> polylog(2,-phi), -log(phi)**2-pi**2/10 + (-1.21852526068613, -1.21852526068613) + >>> polylog(3,0.5), 7*zeta(3)/8-pi**2*log(2)/12+log(2)**3/6 + (0.53721319360804, 0.53721319360804) + +:func:`~mpmath.polylog` can evaluate the analytic continuation of the +polylogarithm when `s` is an integer:: + + >>> polylog(2, 10) + (0.536301287357863 - 7.23378441241546j) + >>> polylog(2, -10) + -4.1982778868581 + >>> polylog(2, 10j) + (-3.05968879432873 + 3.71678149306807j) + >>> polylog(-2, 10) + -0.150891632373114 + >>> polylog(-2, -10) + 0.067618332081142 + >>> polylog(-2, 10j) + (0.0384353698579347 + 0.0912451798066779j) + +Some more examples, with arguments on the unit circle (note that +the series definition cannot be used for computation here):: + + >>> polylog(2,j) + (-0.205616758356028 + 0.915965594177219j) + >>> j*catalan-pi**2/48 + (-0.205616758356028 + 0.915965594177219j) + >>> polylog(3,exp(2*pi*j/3)) + (-0.534247512515375 + 0.765587078525922j) + >>> -4*zeta(3)/9 + 2*j*pi**3/81 + (-0.534247512515375 + 0.765587078525921j) + +Polylogarithms of different order are related by integration +and differentiation:: + + >>> s, z = 3, 0.5 + >>> polylog(s+1, z) + 0.517479061673899 + >>> quad(lambda t: polylog(s,t)/t, [0, z]) + 0.517479061673899 + >>> z*diff(lambda t: polylog(s+2,t), z) + 0.517479061673899 + +Taylor series expansions around `z = 0` are:: + + >>> for n in range(-3, 4): + ... nprint(taylor(lambda x: polylog(n,x), 0, 5)) + ... + [0.0, 1.0, 8.0, 27.0, 64.0, 125.0] + [0.0, 1.0, 4.0, 9.0, 16.0, 25.0] + [0.0, 1.0, 2.0, 3.0, 4.0, 5.0] + [0.0, 1.0, 1.0, 1.0, 1.0, 1.0] + [0.0, 1.0, 0.5, 0.333333, 0.25, 0.2] + [0.0, 1.0, 0.25, 0.111111, 0.0625, 0.04] + [0.0, 1.0, 0.125, 0.037037, 0.015625, 0.008] + +The series defining the polylogarithm is simultaneously +a Taylor series and an L-series. For certain values of `z`, the +polylogarithm reduces to a pure zeta function:: + + >>> polylog(pi, 1), zeta(pi) + (1.17624173838258, 1.17624173838258) + >>> polylog(pi, -1), -altzeta(pi) + (-0.909670702980385, -0.909670702980385) + +Evaluation for arbitrary, nonintegral `s` is supported +for `z` within the unit circle: + + >>> polylog(3+4j, 0.25) + (0.24258605789446 - 0.00222938275488344j) + >>> nsum(lambda k: 0.25**k / k**(3+4j), [1,inf]) + (0.24258605789446 - 0.00222938275488344j) + +It is also supported outside of the unit circle:: + + >>> polylog(1+j, 20+40j) + (-7.1421172179728 - 3.92726697721369j) + >>> polylog(1+j, 200+400j) + (-5.41934747194626 - 9.94037752563927j) + +**References** + +1. Richard Crandall, "Note on fast polylogarithm computation" + http://www.reed.edu/physics/faculty/crandall/papers/Polylog.pdf +2. http://en.wikipedia.org/wiki/Polylogarithm +3. http://mathworld.wolfram.com/Polylogarithm.html + +""" + +bell = r""" +For `n` a nonnegative integer, ``bell(n,x)`` evaluates the Bell +polynomial `B_n(x)`, the first few of which are + +.. math :: + + B_0(x) = 1 + + B_1(x) = x + + B_2(x) = x^2+x + + B_3(x) = x^3+3x^2+x + +If `x = 1` or :func:`~mpmath.bell` is called with only one argument, it +gives the `n`-th Bell number `B_n`, which is the number of +partitions of a set with `n` elements. By setting the precision to +at least `\log_{10} B_n` digits, :func:`~mpmath.bell` provides fast +calculation of exact Bell numbers. + +In general, :func:`~mpmath.bell` computes + +.. math :: + + B_n(x) = e^{-x} \left(\mathrm{sinc}(\pi n) + E_n(x)\right) + +where `E_n(x)` is the generalized exponential function implemented +by :func:`~mpmath.polyexp`. This is an extension of Dobinski's formula [1], +where the modification is the sinc term ensuring that `B_n(x)` is +continuous in `n`; :func:`~mpmath.bell` can thus be evaluated, +differentiated, etc for arbitrary complex arguments. + +**Examples** + +Simple evaluations:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> bell(0, 2.5) + 1.0 + >>> bell(1, 2.5) + 2.5 + >>> bell(2, 2.5) + 8.75 + +Evaluation for arbitrary complex arguments:: + + >>> bell(5.75+1j, 2-3j) + (-10767.71345136587098445143 - 15449.55065599872579097221j) + +The first few Bell polynomials:: + + >>> for k in range(7): + ... nprint(taylor(lambda x: bell(k,x), 0, k)) + ... + [1.0] + [0.0, 1.0] + [0.0, 1.0, 1.0] + [0.0, 1.0, 3.0, 1.0] + [0.0, 1.0, 7.0, 6.0, 1.0] + [0.0, 1.0, 15.0, 25.0, 10.0, 1.0] + [0.0, 1.0, 31.0, 90.0, 65.0, 15.0, 1.0] + +The first few Bell numbers and complementary Bell numbers:: + + >>> [int(bell(k)) for k in range(10)] + [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147] + >>> [int(bell(k,-1)) for k in range(10)] + [1, -1, 0, 1, 1, -2, -9, -9, 50, 267] + +Large Bell numbers:: + + >>> mp.dps = 50 + >>> bell(50) + 185724268771078270438257767181908917499221852770.0 + >>> bell(50,-1) + -29113173035759403920216141265491160286912.0 + +Some even larger values:: + + >>> mp.dps = 25 + >>> bell(1000,-1) + -1.237132026969293954162816e+1869 + >>> bell(1000) + 2.989901335682408421480422e+1927 + >>> bell(1000,2) + 6.591553486811969380442171e+1987 + >>> bell(1000,100.5) + 9.101014101401543575679639e+2529 + +A determinant identity satisfied by Bell numbers:: + + >>> mp.dps = 15 + >>> N = 8 + >>> det([[bell(k+j) for j in range(N)] for k in range(N)]) + 125411328000.0 + >>> superfac(N-1) + 125411328000.0 + +**References** + +1. http://mathworld.wolfram.com/DobinskisFormula.html + +""" + +polyexp = r""" +Evaluates the polyexponential function, defined for arbitrary +complex `s`, `z` by the series + +.. math :: + + E_s(z) = \sum_{k=1}^{\infty} \frac{k^s}{k!} z^k. + +`E_s(z)` is constructed from the exponential function analogously +to how the polylogarithm is constructed from the ordinary +logarithm; as a function of `s` (with `z` fixed), `E_s` is an L-series +It is an entire function of both `s` and `z`. + +The polyexponential function provides a generalization of the +Bell polynomials `B_n(x)` (see :func:`~mpmath.bell`) to noninteger orders `n`. +In terms of the Bell polynomials, + +.. math :: + + E_s(z) = e^z B_s(z) - \mathrm{sinc}(\pi s). + +Note that `B_n(x)` and `e^{-x} E_n(x)` are identical if `n` +is a nonzero integer, but not otherwise. In particular, they differ +at `n = 0`. + +**Examples** + +Evaluating a series:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> nsum(lambda k: sqrt(k)/fac(k), [1,inf]) + 2.101755547733791780315904 + >>> polyexp(0.5,1) + 2.101755547733791780315904 + +Evaluation for arbitrary arguments:: + + >>> polyexp(-3-4j, 2.5+2j) + (2.351660261190434618268706 + 1.202966666673054671364215j) + +Evaluation is accurate for tiny function values:: + + >>> polyexp(4, -100) + 3.499471750566824369520223e-36 + +If `n` is a nonpositive integer, `E_n` reduces to a special +instance of the hypergeometric function `\,_pF_q`:: + + >>> n = 3 + >>> x = pi + >>> polyexp(-n,x) + 4.042192318847986561771779 + >>> x*hyper([1]*(n+1), [2]*(n+1), x) + 4.042192318847986561771779 + +""" + +cyclotomic = r""" +Evaluates the cyclotomic polynomial `\Phi_n(x)`, defined by + +.. math :: + + \Phi_n(x) = \prod_{\zeta} (x - \zeta) + +where `\zeta` ranges over all primitive `n`-th roots of unity +(see :func:`~mpmath.unitroots`). An equivalent representation, used +for computation, is + +.. math :: + + \Phi_n(x) = \prod_{d\mid n}(x^d-1)^{\mu(n/d)} = \Phi_n(x) + +where `\mu(m)` denotes the Moebius function. The cyclotomic +polynomials are integer polynomials, the first of which can be +written explicitly as + +.. math :: + + \Phi_0(x) = 1 + + \Phi_1(x) = x - 1 + + \Phi_2(x) = x + 1 + + \Phi_3(x) = x^3 + x^2 + 1 + + \Phi_4(x) = x^2 + 1 + + \Phi_5(x) = x^4 + x^3 + x^2 + x + 1 + + \Phi_6(x) = x^2 - x + 1 + +**Examples** + +The coefficients of low-order cyclotomic polynomials can be recovered +using Taylor expansion:: + + >>> from mpmath import * + >>> mp.dps = 15; mp.pretty = True + >>> for n in range(9): + ... p = chop(taylor(lambda x: cyclotomic(n,x), 0, 10)) + ... print("%s %s" % (n, nstr(p[:10+1-p[::-1].index(1)]))) + ... + 0 [1.0] + 1 [-1.0, 1.0] + 2 [1.0, 1.0] + 3 [1.0, 1.0, 1.0] + 4 [1.0, 0.0, 1.0] + 5 [1.0, 1.0, 1.0, 1.0, 1.0] + 6 [1.0, -1.0, 1.0] + 7 [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0] + 8 [1.0, 0.0, 0.0, 0.0, 1.0] + +The definition as a product over primitive roots may be checked +by computing the product explicitly (for a real argument, this +method will generally introduce numerical noise in the imaginary +part):: + + >>> mp.dps = 25 + >>> z = 3+4j + >>> cyclotomic(10, z) + (-419.0 - 360.0j) + >>> fprod(z-r for r in unitroots(10, primitive=True)) + (-419.0 - 360.0j) + >>> z = 3 + >>> cyclotomic(10, z) + 61.0 + >>> fprod(z-r for r in unitroots(10, primitive=True)) + (61.0 - 3.146045605088568607055454e-25j) + +Up to permutation, the roots of a given cyclotomic polynomial +can be checked to agree with the list of primitive roots:: + + >>> p = taylor(lambda x: cyclotomic(6,x), 0, 6)[:3] + >>> for r in polyroots(p[::-1]): + ... print(r) + ... + (0.5 - 0.8660254037844386467637232j) + (0.5 + 0.8660254037844386467637232j) + >>> + >>> for r in unitroots(6, primitive=True): + ... print(r) + ... + (0.5 + 0.8660254037844386467637232j) + (0.5 - 0.8660254037844386467637232j) + +""" + +meijerg = r""" +Evaluates the Meijer G-function, defined as + +.. math :: + + G^{m,n}_{p,q} \left( \left. \begin{matrix} + a_1, \dots, a_n ; a_{n+1} \dots a_p \\ + b_1, \dots, b_m ; b_{m+1} \dots b_q + \end{matrix}\; \right| \; z ; r \right) = + \frac{1}{2 \pi i} \int_L + \frac{\prod_{j=1}^m \Gamma(b_j+s) \prod_{j=1}^n\Gamma(1-a_j-s)} + {\prod_{j=n+1}^{p}\Gamma(a_j+s) \prod_{j=m+1}^q \Gamma(1-b_j-s)} + z^{-s/r} ds + +for an appropriate choice of the contour `L` (see references). + +There are `p` elements `a_j`. +The argument *a_s* should be a pair of lists, the first containing the +`n` elements `a_1, \ldots, a_n` and the second containing +the `p-n` elements `a_{n+1}, \ldots a_p`. + +There are `q` elements `b_j`. +The argument *b_s* should be a pair of lists, the first containing the +`m` elements `b_1, \ldots, b_m` and the second containing +the `q-m` elements `b_{m+1}, \ldots b_q`. + +The implicit tuple `(m, n, p, q)` constitutes the order or degree of the +Meijer G-function, and is determined by the lengths of the coefficient +vectors. Confusingly, the indices in this tuple appear in a different order +from the coefficients, but this notation is standard. The many examples +given below should hopefully clear up any potential confusion. + +**Algorithm** + +The Meijer G-function is evaluated as a combination of hypergeometric series. +There are two versions of the function, which can be selected with +the optional *series* argument. + +*series=1* uses a sum of `m` `\,_pF_{q-1}` functions of `z` + +*series=2* uses a sum of `n` `\,_qF_{p-1}` functions of `1/z` + +The default series is chosen based on the degree and `|z|` in order +to be consistent with Mathematica's. This definition of the Meijer G-function +has a discontinuity at `|z| = 1` for some orders, which can +be avoided by explicitly specifying a series. + +Keyword arguments are forwarded to :func:`~mpmath.hypercomb`. + +**Examples** + +Many standard functions are special cases of the Meijer G-function +(possibly rescaled and/or with branch cut corrections). We define +some test parameters:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> a = mpf(0.75) + >>> b = mpf(1.5) + >>> z = mpf(2.25) + +The exponential function: +`e^z = G^{1,0}_{0,1} \left( \left. \begin{matrix} - \\ 0 \end{matrix} \; +\right| \; -z \right)` + + >>> meijerg([[],[]], [[0],[]], -z) + 9.487735836358525720550369 + >>> exp(z) + 9.487735836358525720550369 + +The natural logarithm: +`\log(1+z) = G^{1,2}_{2,2} \left( \left. \begin{matrix} 1, 1 \\ 1, 0 +\end{matrix} \; \right| \; -z \right)` + + >>> meijerg([[1,1],[]], [[1],[0]], z) + 1.178654996341646117219023 + >>> log(1+z) + 1.178654996341646117219023 + +A rational function: +`\frac{z}{z+1} = G^{1,2}_{2,2} \left( \left. \begin{matrix} 1, 1 \\ 1, 1 +\end{matrix} \; \right| \; z \right)` + + >>> meijerg([[1,1],[]], [[1],[1]], z) + 0.6923076923076923076923077 + >>> z/(z+1) + 0.6923076923076923076923077 + +The sine and cosine functions: + +`\frac{1}{\sqrt \pi} \sin(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. \begin{matrix} +- \\ \frac{1}{2}, 0 \end{matrix} \; \right| \; z \right)` + +`\frac{1}{\sqrt \pi} \cos(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. \begin{matrix} +- \\ 0, \frac{1}{2} \end{matrix} \; \right| \; z \right)` + + >>> meijerg([[],[]], [[0.5],[0]], (z/2)**2) + 0.4389807929218676682296453 + >>> sin(z)/sqrt(pi) + 0.4389807929218676682296453 + >>> meijerg([[],[]], [[0],[0.5]], (z/2)**2) + -0.3544090145996275423331762 + >>> cos(z)/sqrt(pi) + -0.3544090145996275423331762 + +Bessel functions: + +`J_a(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. +\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2} +\end{matrix} \; \right| \; z \right)` + +`Y_a(2 \sqrt z) = G^{2,0}_{1,3} \left( \left. +\begin{matrix} \frac{-a-1}{2} \\ \frac{a}{2}, -\frac{a}{2}, \frac{-a-1}{2} +\end{matrix} \; \right| \; z \right)` + +`(-z)^{a/2} z^{-a/2} I_a(2 \sqrt z) = G^{1,0}_{0,2} \left( \left. +\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2} +\end{matrix} \; \right| \; -z \right)` + +`2 K_a(2 \sqrt z) = G^{2,0}_{0,2} \left( \left. +\begin{matrix} - \\ \frac{a}{2}, -\frac{a}{2} +\end{matrix} \; \right| \; z \right)` + +As the example with the Bessel *I* function shows, a branch +factor is required for some arguments when inverting the square root. + + >>> meijerg([[],[]], [[a/2],[-a/2]], (z/2)**2) + 0.5059425789597154858527264 + >>> besselj(a,z) + 0.5059425789597154858527264 + >>> meijerg([[],[(-a-1)/2]], [[a/2,-a/2],[(-a-1)/2]], (z/2)**2) + 0.1853868950066556941442559 + >>> bessely(a, z) + 0.1853868950066556941442559 + >>> meijerg([[],[]], [[a/2],[-a/2]], -(z/2)**2) + (0.8685913322427653875717476 + 2.096964974460199200551738j) + >>> (-z)**(a/2) / z**(a/2) * besseli(a, z) + (0.8685913322427653875717476 + 2.096964974460199200551738j) + >>> 0.5*meijerg([[],[]], [[a/2,-a/2],[]], (z/2)**2) + 0.09334163695597828403796071 + >>> besselk(a,z) + 0.09334163695597828403796071 + +Error functions: + +`\sqrt{\pi} z^{2(a-1)} \mathrm{erfc}(z) = G^{2,0}_{1,2} \left( \left. +\begin{matrix} a \\ a-1, a-\frac{1}{2} +\end{matrix} \; \right| \; z, \frac{1}{2} \right)` + + >>> meijerg([[],[a]], [[a-1,a-0.5],[]], z, 0.5) + 0.00172839843123091957468712 + >>> sqrt(pi) * z**(2*a-2) * erfc(z) + 0.00172839843123091957468712 + +A Meijer G-function of higher degree, (1,1,2,3): + + >>> meijerg([[a],[b]], [[a],[b,a-1]], z) + 1.55984467443050210115617 + >>> sin((b-a)*pi)/pi*(exp(z)-1)*z**(a-1) + 1.55984467443050210115617 + +A Meijer G-function of still higher degree, (4,1,2,4), that can +be expanded as a messy combination of exponential integrals: + + >>> meijerg([[a],[2*b-a]], [[b,a,b-0.5,-1-a+2*b],[]], z) + 0.3323667133658557271898061 + >>> chop(4**(a-b+1)*sqrt(pi)*gamma(2*b-2*a)*z**a*\ + ... expint(2*b-2*a, -2*sqrt(-z))*expint(2*b-2*a, 2*sqrt(-z))) + 0.3323667133658557271898061 + +In the following case, different series give different values:: + + >>> chop(meijerg([[1],[0.25]],[[3],[0.5]],-2)) + -0.06417628097442437076207337 + >>> meijerg([[1],[0.25]],[[3],[0.5]],-2,series=1) + 0.1428699426155117511873047 + >>> chop(meijerg([[1],[0.25]],[[3],[0.5]],-2,series=2)) + -0.06417628097442437076207337 + +**References** + +1. http://en.wikipedia.org/wiki/Meijer_G-function + +2. http://mathworld.wolfram.com/MeijerG-Function.html + +3. http://functions.wolfram.com/HypergeometricFunctions/MeijerG/ + +4. http://functions.wolfram.com/HypergeometricFunctions/MeijerG1/ + +""" + +clsin = r""" +Computes the Clausen sine function, defined formally by the series + +.. math :: + + \mathrm{Cl}_s(z) = \sum_{k=1}^{\infty} \frac{\sin(kz)}{k^s}. + +The special case `\mathrm{Cl}_2(z)` (i.e. ``clsin(2,z)``) is the classical +"Clausen function". More generally, the Clausen function is defined for +complex `s` and `z`, even when the series does not converge. The +Clausen function is related to the polylogarithm (:func:`~mpmath.polylog`) as + +.. math :: + + \mathrm{Cl}_s(z) = \frac{1}{2i}\left(\mathrm{Li}_s\left(e^{iz}\right) - + \mathrm{Li}_s\left(e^{-iz}\right)\right) + + = \mathrm{Im}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}), + +and this representation can be taken to provide the analytic continuation of the +series. The complementary function :func:`~mpmath.clcos` gives the corresponding +cosine sum. + +**Examples** + +Evaluation for arbitrarily chosen `s` and `z`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> s, z = 3, 4 + >>> clsin(s, z); nsum(lambda k: sin(z*k)/k**s, [1,inf]) + -0.6533010136329338746275795 + -0.6533010136329338746275795 + +Using `z + \pi` instead of `z` gives an alternating series:: + + >>> clsin(s, z+pi) + 0.8860032351260589402871624 + >>> nsum(lambda k: (-1)**k*sin(z*k)/k**s, [1,inf]) + 0.8860032351260589402871624 + +With `s = 1`, the sum can be expressed in closed form +using elementary functions:: + + >>> z = 1 + sqrt(3) + >>> clsin(1, z) + 0.2047709230104579724675985 + >>> chop((log(1-exp(-j*z)) - log(1-exp(j*z)))/(2*j)) + 0.2047709230104579724675985 + >>> nsum(lambda k: sin(k*z)/k, [1,inf]) + 0.2047709230104579724675985 + +The classical Clausen function `\mathrm{Cl}_2(\theta)` gives the +value of the integral `\int_0^{\theta} -\ln(2\sin(x/2)) dx` for +`0 < \theta < 2 \pi`:: + + >>> cl2 = lambda t: clsin(2, t) + >>> cl2(3.5) + -0.2465045302347694216534255 + >>> -quad(lambda x: ln(2*sin(0.5*x)), [0, 3.5]) + -0.2465045302347694216534255 + +This function is symmetric about `\theta = \pi` with zeros and extreme +points:: + + >>> cl2(0); cl2(pi/3); chop(cl2(pi)); cl2(5*pi/3); chop(cl2(2*pi)) + 0.0 + 1.014941606409653625021203 + 0.0 + -1.014941606409653625021203 + 0.0 + +Catalan's constant is a special value:: + + >>> cl2(pi/2) + 0.9159655941772190150546035 + >>> +catalan + 0.9159655941772190150546035 + +The Clausen sine function can be expressed in closed form when +`s` is an odd integer (becoming zero when `s` < 0):: + + >>> z = 1 + sqrt(2) + >>> clsin(1, z); (pi-z)/2 + 0.3636895456083490948304773 + 0.3636895456083490948304773 + >>> clsin(3, z); pi**2/6*z - pi*z**2/4 + z**3/12 + 0.5661751584451144991707161 + 0.5661751584451144991707161 + >>> clsin(-1, z) + 0.0 + >>> clsin(-3, z) + 0.0 + +It can also be expressed in closed form for even integer `s \le 0`, +providing a finite sum for series such as +`\sin(z) + \sin(2z) + \sin(3z) + \ldots`:: + + >>> z = 1 + sqrt(2) + >>> clsin(0, z) + 0.1903105029507513881275865 + >>> cot(z/2)/2 + 0.1903105029507513881275865 + >>> clsin(-2, z) + -0.1089406163841548817581392 + >>> -cot(z/2)*csc(z/2)**2/4 + -0.1089406163841548817581392 + +Call with ``pi=True`` to multiply `z` by `\pi` exactly:: + + >>> clsin(3, 3*pi) + -8.892316224968072424732898e-26 + >>> clsin(3, 3, pi=True) + 0.0 + +Evaluation for complex `s`, `z` in a nonconvergent case:: + + >>> s, z = -1-j, 1+2j + >>> clsin(s, z) + (-0.593079480117379002516034 + 0.9038644233367868273362446j) + >>> extraprec(20)(nsum)(lambda k: sin(k*z)/k**s, [1,inf]) + (-0.593079480117379002516034 + 0.9038644233367868273362446j) + +""" + +clcos = r""" +Computes the Clausen cosine function, defined formally by the series + +.. math :: + + \mathrm{\widetilde{Cl}}_s(z) = \sum_{k=1}^{\infty} \frac{\cos(kz)}{k^s}. + +This function is complementary to the Clausen sine function +:func:`~mpmath.clsin`. In terms of the polylogarithm, + +.. math :: + + \mathrm{\widetilde{Cl}}_s(z) = + \frac{1}{2}\left(\mathrm{Li}_s\left(e^{iz}\right) + + \mathrm{Li}_s\left(e^{-iz}\right)\right) + + = \mathrm{Re}\left[\mathrm{Li}_s(e^{iz})\right] \quad (s, z \in \mathbb{R}). + +**Examples** + +Evaluation for arbitrarily chosen `s` and `z`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> s, z = 3, 4 + >>> clcos(s, z); nsum(lambda k: cos(z*k)/k**s, [1,inf]) + -0.6518926267198991308332759 + -0.6518926267198991308332759 + +Using `z + \pi` instead of `z` gives an alternating series:: + + >>> s, z = 3, 0.5 + >>> clcos(s, z+pi) + -0.8155530586502260817855618 + >>> nsum(lambda k: (-1)**k*cos(z*k)/k**s, [1,inf]) + -0.8155530586502260817855618 + +With `s = 1`, the sum can be expressed in closed form +using elementary functions:: + + >>> z = 1 + sqrt(3) + >>> clcos(1, z) + -0.6720334373369714849797918 + >>> chop(-0.5*(log(1-exp(j*z))+log(1-exp(-j*z)))) + -0.6720334373369714849797918 + >>> -log(abs(2*sin(0.5*z))) # Equivalent to above when z is real + -0.6720334373369714849797918 + >>> nsum(lambda k: cos(k*z)/k, [1,inf]) + -0.6720334373369714849797918 + +It can also be expressed in closed form when `s` is an even integer. +For example, + + >>> clcos(2,z) + -0.7805359025135583118863007 + >>> pi**2/6 - pi*z/2 + z**2/4 + -0.7805359025135583118863007 + +The case `s = 0` gives the renormalized sum of +`\cos(z) + \cos(2z) + \cos(3z) + \ldots` (which happens to be the same for +any value of `z`):: + + >>> clcos(0, z) + -0.5 + >>> nsum(lambda k: cos(k*z), [1,inf]) + -0.5 + +Also the sums + +.. math :: + + \cos(z) + 2\cos(2z) + 3\cos(3z) + \ldots + +and + +.. math :: + + \cos(z) + 2^n \cos(2z) + 3^n \cos(3z) + \ldots + +for higher integer powers `n = -s` can be done in closed form. They are zero +when `n` is positive and even (`s` negative and even):: + + >>> clcos(-1, z); 1/(2*cos(z)-2) + -0.2607829375240542480694126 + -0.2607829375240542480694126 + >>> clcos(-3, z); (2+cos(z))*csc(z/2)**4/8 + 0.1472635054979944390848006 + 0.1472635054979944390848006 + >>> clcos(-2, z); clcos(-4, z); clcos(-6, z) + 0.0 + 0.0 + 0.0 + +With `z = \pi`, the series reduces to that of the Riemann zeta function +(more generally, if `z = p \pi/q`, it is a finite sum over Hurwitz zeta +function values):: + + >>> clcos(2.5, 0); zeta(2.5) + 1.34148725725091717975677 + 1.34148725725091717975677 + >>> clcos(2.5, pi); -altzeta(2.5) + -0.8671998890121841381913472 + -0.8671998890121841381913472 + +Call with ``pi=True`` to multiply `z` by `\pi` exactly:: + + >>> clcos(-3, 2*pi) + 2.997921055881167659267063e+102 + >>> clcos(-3, 2, pi=True) + 0.008333333333333333333333333 + +Evaluation for complex `s`, `z` in a nonconvergent case:: + + >>> s, z = -1-j, 1+2j + >>> clcos(s, z) + (0.9407430121562251476136807 + 0.715826296033590204557054j) + >>> extraprec(20)(nsum)(lambda k: cos(k*z)/k**s, [1,inf]) + (0.9407430121562251476136807 + 0.715826296033590204557054j) + +""" + +whitm = r""" +Evaluates the Whittaker function `M(k,m,z)`, which gives a solution +to the Whittaker differential equation + +.. math :: + + \frac{d^2f}{dz^2} + \left(-\frac{1}{4}+\frac{k}{z}+ + \frac{(\frac{1}{4}-m^2)}{z^2}\right) f = 0. + +A second solution is given by :func:`~mpmath.whitw`. + +The Whittaker functions are defined in Abramowitz & Stegun, section 13.1. +They are alternate forms of the confluent hypergeometric functions +`\,_1F_1` and `U`: + +.. math :: + + M(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} + \,_1F_1(\tfrac{1}{2}+m-k, 1+2m, z) + + W(k,m,z) = e^{-\frac{1}{2}z} z^{\frac{1}{2}+m} + U(\tfrac{1}{2}+m-k, 1+2m, z). + +**Examples** + +Evaluation for arbitrary real and complex arguments is supported:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> whitm(1, 1, 1) + 0.7302596799460411820509668 + >>> whitm(1, 1, -1) + (0.0 - 1.417977827655098025684246j) + >>> whitm(j, j/2, 2+3j) + (3.245477713363581112736478 - 0.822879187542699127327782j) + >>> whitm(2, 3, 100000) + 4.303985255686378497193063e+21707 + +Evaluation at zero:: + + >>> whitm(1,-1,0); whitm(1,-0.5,0); whitm(1,0,0) + +inf + nan + 0.0 + +We can verify that :func:`~mpmath.whitm` numerically satisfies the +differential equation for arbitrarily chosen values:: + + >>> k = mpf(0.25) + >>> m = mpf(1.5) + >>> f = lambda z: whitm(k,m,z) + >>> for z in [-1, 2.5, 3, 1+2j]: + ... chop(diff(f,z,2) + (-0.25 + k/z + (0.25-m**2)/z**2)*f(z)) + ... + 0.0 + 0.0 + 0.0 + 0.0 + +An integral involving both :func:`~mpmath.whitm` and :func:`~mpmath.whitw`, +verifying evaluation along the real axis:: + + >>> quad(lambda x: exp(-x)*whitm(3,2,x)*whitw(1,-2,x), [0,inf]) + 3.438869842576800225207341 + >>> 128/(21*sqrt(pi)) + 3.438869842576800225207341 + +""" + +whitw = r""" +Evaluates the Whittaker function `W(k,m,z)`, which gives a second +solution to the Whittaker differential equation. (See :func:`~mpmath.whitm`.) + +**Examples** + +Evaluation for arbitrary real and complex arguments is supported:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> whitw(1, 1, 1) + 1.19532063107581155661012 + >>> whitw(1, 1, -1) + (-0.9424875979222187313924639 - 0.2607738054097702293308689j) + >>> whitw(j, j/2, 2+3j) + (0.1782899315111033879430369 - 0.01609578360403649340169406j) + >>> whitw(2, 3, 100000) + 1.887705114889527446891274e-21705 + >>> whitw(-1, -1, 100) + 1.905250692824046162462058e-24 + +Evaluation at zero:: + + >>> for m in [-1, -0.5, 0, 0.5, 1]: + ... whitw(1, m, 0) + ... + +inf + nan + 0.0 + nan + +inf + +We can verify that :func:`~mpmath.whitw` numerically satisfies the +differential equation for arbitrarily chosen values:: + + >>> k = mpf(0.25) + >>> m = mpf(1.5) + >>> f = lambda z: whitw(k,m,z) + >>> for z in [-1, 2.5, 3, 1+2j]: + ... chop(diff(f,z,2) + (-0.25 + k/z + (0.25-m**2)/z**2)*f(z)) + ... + 0.0 + 0.0 + 0.0 + 0.0 + +""" + +ber = r""" +Computes the Kelvin function ber, which for real arguments gives the real part +of the Bessel J function of a rotated argument + +.. math :: + + J_n\left(x e^{3\pi i/4}\right) = \mathrm{ber}_n(x) + i \mathrm{bei}_n(x). + +The imaginary part is given by :func:`~mpmath.bei`. + +**Plots** + +.. literalinclude :: /plots/ber.py +.. image :: /plots/ber.png + +**Examples** + +Verifying the defining relation:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> n, x = 2, 3.5 + >>> ber(n,x) + 1.442338852571888752631129 + >>> bei(n,x) + -0.948359035324558320217678 + >>> besselj(n, x*root(1,8,3)) + (1.442338852571888752631129 - 0.948359035324558320217678j) + +The ber and bei functions are also defined by analytic continuation +for complex arguments:: + + >>> ber(1+j, 2+3j) + (4.675445984756614424069563 - 15.84901771719130765656316j) + >>> bei(1+j, 2+3j) + (15.83886679193707699364398 + 4.684053288183046528703611j) + +""" + +bei = r""" +Computes the Kelvin function bei, which for real arguments gives the +imaginary part of the Bessel J function of a rotated argument. +See :func:`~mpmath.ber`. +""" + +ker = r""" +Computes the Kelvin function ker, which for real arguments gives the real part +of the (rescaled) Bessel K function of a rotated argument + +.. math :: + + e^{-\pi i/2} K_n\left(x e^{3\pi i/4}\right) = \mathrm{ker}_n(x) + i \mathrm{kei}_n(x). + +The imaginary part is given by :func:`~mpmath.kei`. + +**Plots** + +.. literalinclude :: /plots/ker.py +.. image :: /plots/ker.png + +**Examples** + +Verifying the defining relation:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> n, x = 2, 4.5 + >>> ker(n,x) + 0.02542895201906369640249801 + >>> kei(n,x) + -0.02074960467222823237055351 + >>> exp(-n*pi*j/2) * besselk(n, x*root(1,8,1)) + (0.02542895201906369640249801 - 0.02074960467222823237055351j) + +The ker and kei functions are also defined by analytic continuation +for complex arguments:: + + >>> ker(1+j, 3+4j) + (1.586084268115490421090533 - 2.939717517906339193598719j) + >>> kei(1+j, 3+4j) + (-2.940403256319453402690132 - 1.585621643835618941044855j) + +""" + +kei = r""" +Computes the Kelvin function kei, which for real arguments gives the +imaginary part of the (rescaled) Bessel K function of a rotated argument. +See :func:`~mpmath.ker`. +""" + +struveh = r""" +Gives the Struve function + +.. math :: + + \,\mathbf{H}_n(z) = + \sum_{k=0}^\infty \frac{(-1)^k}{\Gamma(k+\frac{3}{2}) + \Gamma(k+n+\frac{3}{2})} {\left({\frac{z}{2}}\right)}^{2k+n+1} + +which is a solution to the Struve differential equation + +.. math :: + + z^2 f''(z) + z f'(z) + (z^2-n^2) f(z) = \frac{2 z^{n+1}}{\pi (2n-1)!!}. + +**Examples** + +Evaluation for arbitrary real and complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> struveh(0, 3.5) + 0.3608207733778295024977797 + >>> struveh(-1, 10) + -0.255212719726956768034732 + >>> struveh(1, -100.5) + 0.5819566816797362287502246 + >>> struveh(2.5, 10000000000000) + 3153915652525200060.308937 + >>> struveh(2.5, -10000000000000) + (0.0 - 3153915652525200060.308937j) + >>> struveh(1+j, 1000000+4000000j) + (-3.066421087689197632388731e+1737173 - 1.596619701076529803290973e+1737173j) + +A Struve function of half-integer order is elementary; for example: + + >>> z = 3 + >>> struveh(0.5, 3) + 0.9167076867564138178671595 + >>> sqrt(2/(pi*z))*(1-cos(z)) + 0.9167076867564138178671595 + +Numerically verifying the differential equation:: + + >>> z = mpf(4.5) + >>> n = 3 + >>> f = lambda z: struveh(n,z) + >>> lhs = z**2*diff(f,z,2) + z*diff(f,z) + (z**2-n**2)*f(z) + >>> rhs = 2*z**(n+1)/fac2(2*n-1)/pi + >>> lhs + 17.40359302709875496632744 + >>> rhs + 17.40359302709875496632744 + +""" + +struvel = r""" +Gives the modified Struve function + +.. math :: + + \,\mathbf{L}_n(z) = -i e^{-n\pi i/2} \mathbf{H}_n(i z) + +which solves to the modified Struve differential equation + +.. math :: + + z^2 f''(z) + z f'(z) - (z^2+n^2) f(z) = \frac{2 z^{n+1}}{\pi (2n-1)!!}. + +**Examples** + +Evaluation for arbitrary real and complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> struvel(0, 3.5) + 7.180846515103737996249972 + >>> struvel(-1, 10) + 2670.994904980850550721511 + >>> struvel(1, -100.5) + 1.757089288053346261497686e+42 + >>> struvel(2.5, 10000000000000) + 4.160893281017115450519948e+4342944819025 + >>> struvel(2.5, -10000000000000) + (0.0 - 4.160893281017115450519948e+4342944819025j) + >>> struvel(1+j, 700j) + (-0.1721150049480079451246076 + 0.1240770953126831093464055j) + >>> struvel(1+j, 1000000+4000000j) + (-2.973341637511505389128708e+434290 - 5.164633059729968297147448e+434290j) + +Numerically verifying the differential equation:: + + >>> z = mpf(3.5) + >>> n = 3 + >>> f = lambda z: struvel(n,z) + >>> lhs = z**2*diff(f,z,2) + z*diff(f,z) - (z**2+n**2)*f(z) + >>> rhs = 2*z**(n+1)/fac2(2*n-1)/pi + >>> lhs + 6.368850306060678353018165 + >>> rhs + 6.368850306060678353018165 +""" + +appellf1 = r""" +Gives the Appell F1 hypergeometric function of two variables, + +.. math :: + + F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}} + \frac{x^m y^n}{m! n!}. + +This series is only generally convergent when `|x| < 1` and `|y| < 1`, +although :func:`~mpmath.appellf1` can evaluate an analytic continuation +with respecto to either variable, and sometimes both. + +**Examples** + +Evaluation is supported for real and complex parameters:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> appellf1(1,0,0.5,1,0.5,0.25) + 1.154700538379251529018298 + >>> appellf1(1,1+j,0.5,1,0.5,0.5j) + (1.138403860350148085179415 + 1.510544741058517621110615j) + +For some integer parameters, the F1 series reduces to a polynomial:: + + >>> appellf1(2,-4,-3,1,2,5) + -816.0 + >>> appellf1(-5,1,2,1,4,5) + -20528.0 + +The analytic continuation with respect to either `x` or `y`, +and sometimes with respect to both, can be evaluated:: + + >>> appellf1(2,3,4,5,100,0.5) + (0.0006231042714165329279738662 + 0.0000005769149277148425774499857j) + >>> appellf1('1.1', '0.3', '0.2+2j', '0.4', '0.2', 1.5+3j) + (-0.1782604566893954897128702 + 0.002472407104546216117161499j) + >>> appellf1(1,2,3,4,10,12) + -0.07122993830066776374929313 + +For certain arguments, F1 reduces to an ordinary hypergeometric function:: + + >>> appellf1(1,2,3,5,0.5,0.25) + 1.547902270302684019335555 + >>> 4*hyp2f1(1,2,5,'1/3')/3 + 1.547902270302684019335555 + >>> appellf1(1,2,3,4,0,1.5) + (-1.717202506168937502740238 - 2.792526803190927323077905j) + >>> hyp2f1(1,3,4,1.5) + (-1.717202506168937502740238 - 2.792526803190927323077905j) + +The F1 function satisfies a system of partial differential equations:: + + >>> a,b1,b2,c,x,y = map(mpf, [1,0.5,0.25,1.125,0.25,-0.25]) + >>> F = lambda x,y: appellf1(a,b1,b2,c,x,y) + >>> chop(x*(1-x)*diff(F,(x,y),(2,0)) + + ... y*(1-x)*diff(F,(x,y),(1,1)) + + ... (c-(a+b1+1)*x)*diff(F,(x,y),(1,0)) - + ... b1*y*diff(F,(x,y),(0,1)) - + ... a*b1*F(x,y)) + 0.0 + >>> + >>> chop(y*(1-y)*diff(F,(x,y),(0,2)) + + ... x*(1-y)*diff(F,(x,y),(1,1)) + + ... (c-(a+b2+1)*y)*diff(F,(x,y),(0,1)) - + ... b2*x*diff(F,(x,y),(1,0)) - + ... a*b2*F(x,y)) + 0.0 + +The Appell F1 function allows for closed-form evaluation of various +integrals, such as any integral of the form +`\int x^r (x+a)^p (x+b)^q dx`:: + + >>> def integral(a,b,p,q,r,x1,x2): + ... a,b,p,q,r,x1,x2 = map(mpmathify, [a,b,p,q,r,x1,x2]) + ... f = lambda x: x**r * (x+a)**p * (x+b)**q + ... def F(x): + ... v = x**(r+1)/(r+1) * (a+x)**p * (b+x)**q + ... v *= (1+x/a)**(-p) + ... v *= (1+x/b)**(-q) + ... v *= appellf1(r+1,-p,-q,2+r,-x/a,-x/b) + ... return v + ... print("Num. quad: %s" % quad(f, [x1,x2])) + ... print("Appell F1: %s" % (F(x2)-F(x1))) + ... + >>> integral('1/5','4/3','-2','3','1/2',0,1) + Num. quad: 9.073335358785776206576981 + Appell F1: 9.073335358785776206576981 + >>> integral('3/2','4/3','-2','3','1/2',0,1) + Num. quad: 1.092829171999626454344678 + Appell F1: 1.092829171999626454344678 + >>> integral('3/2','4/3','-2','3','1/2',12,25) + Num. quad: 1106.323225040235116498927 + Appell F1: 1106.323225040235116498927 + +Also incomplete elliptic integrals fall into this category [1]:: + + >>> def E(z, m): + ... if (pi/2).ae(z): + ... return ellipe(m) + ... return 2*round(re(z)/pi)*ellipe(m) + mpf(-1)**round(re(z)/pi)*\ + ... sin(z)*appellf1(0.5,0.5,-0.5,1.5,sin(z)**2,m*sin(z)**2) + ... + >>> z, m = 1, 0.5 + >>> E(z,m); quad(lambda t: sqrt(1-m*sin(t)**2), [0,pi/4,3*pi/4,z]) + 0.9273298836244400669659042 + 0.9273298836244400669659042 + >>> z, m = 3, 2 + >>> E(z,m); quad(lambda t: sqrt(1-m*sin(t)**2), [0,pi/4,3*pi/4,z]) + (1.057495752337234229715836 + 1.198140234735592207439922j) + (1.057495752337234229715836 + 1.198140234735592207439922j) + +**References** + +1. [WolframFunctions]_ http://functions.wolfram.com/EllipticIntegrals/EllipticE2/26/01/ +2. [SrivastavaKarlsson]_ +3. [CabralRosetti]_ +4. [Vidunas]_ +5. [Slater]_ + +""" + +angerj = r""" +Gives the Anger function + +.. math :: + + \mathbf{J}_{\nu}(z) = \frac{1}{\pi} + \int_0^{\pi} \cos(\nu t - z \sin t) dt + +which is an entire function of both the parameter `\nu` and +the argument `z`. It solves the inhomogeneous Bessel differential +equation + +.. math :: + + f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z) + = \frac{(z-\nu)}{\pi z^2} \sin(\pi \nu). + +**Examples** + +Evaluation for real and complex parameter and argument:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> angerj(2,3) + 0.4860912605858910769078311 + >>> angerj(-3+4j, 2+5j) + (-5033.358320403384472395612 + 585.8011892476145118551756j) + >>> angerj(3.25, 1e6j) + (4.630743639715893346570743e+434290 - 1.117960409887505906848456e+434291j) + >>> angerj(-1.5, 1e6) + 0.0002795719747073879393087011 + +The Anger function coincides with the Bessel J-function when `\nu` +is an integer:: + + >>> angerj(1,3); besselj(1,3) + 0.3390589585259364589255146 + 0.3390589585259364589255146 + >>> angerj(1.5,3); besselj(1.5,3) + 0.4088969848691080859328847 + 0.4777182150870917715515015 + +Verifying the differential equation:: + + >>> v,z = mpf(2.25), 0.75 + >>> f = lambda z: angerj(v,z) + >>> diff(f,z,2) + diff(f,z)/z + (1-(v/z)**2)*f(z) + -0.6002108774380707130367995 + >>> (z-v)/(pi*z**2) * sinpi(v) + -0.6002108774380707130367995 + +Verifying the integral representation:: + + >>> angerj(v,z) + 0.1145380759919333180900501 + >>> quad(lambda t: cos(v*t-z*sin(t))/pi, [0,pi]) + 0.1145380759919333180900501 + +**References** + +1. [DLMF]_ section 11.10: Anger-Weber Functions +""" + +webere = r""" +Gives the Weber function + +.. math :: + + \mathbf{E}_{\nu}(z) = \frac{1}{\pi} + \int_0^{\pi} \sin(\nu t - z \sin t) dt + +which is an entire function of both the parameter `\nu` and +the argument `z`. It solves the inhomogeneous Bessel differential +equation + +.. math :: + + f''(z) + \frac{1}{z}f'(z) + \left(1-\frac{\nu^2}{z^2}\right) f(z) + = -\frac{1}{\pi z^2} (z+\nu+(z-\nu)\cos(\pi \nu)). + +**Examples** + +Evaluation for real and complex parameter and argument:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> webere(2,3) + -0.1057668973099018425662646 + >>> webere(-3+4j, 2+5j) + (-585.8081418209852019290498 - 5033.314488899926921597203j) + >>> webere(3.25, 1e6j) + (-1.117960409887505906848456e+434291 - 4.630743639715893346570743e+434290j) + >>> webere(3.25, 1e6) + -0.00002812518265894315604914453 + +Up to addition of a rational function of `z`, the Weber function coincides +with the Struve H-function when `\nu` is an integer:: + + >>> webere(1,3); 2/pi-struveh(1,3) + -0.3834897968188690177372881 + -0.3834897968188690177372881 + >>> webere(5,3); 26/(35*pi)-struveh(5,3) + 0.2009680659308154011878075 + 0.2009680659308154011878075 + +Verifying the differential equation:: + + >>> v,z = mpf(2.25), 0.75 + >>> f = lambda z: webere(v,z) + >>> diff(f,z,2) + diff(f,z)/z + (1-(v/z)**2)*f(z) + -1.097441848875479535164627 + >>> -(z+v+(z-v)*cospi(v))/(pi*z**2) + -1.097441848875479535164627 + +Verifying the integral representation:: + + >>> webere(v,z) + 0.1486507351534283744485421 + >>> quad(lambda t: sin(v*t-z*sin(t))/pi, [0,pi]) + 0.1486507351534283744485421 + +**References** + +1. [DLMF]_ section 11.10: Anger-Weber Functions +""" + +lommels1 = r""" +Gives the Lommel function `s_{\mu,\nu}` or `s^{(1)}_{\mu,\nu}` + +.. math :: + + s_{\mu,\nu}(z) = \frac{z^{\mu+1}}{(\mu-\nu+1)(\mu+\nu+1)} + \,_1F_2\left(1; \frac{\mu-\nu+3}{2}, \frac{\mu+\nu+3}{2}; + -\frac{z^2}{4} \right) + +which solves the inhomogeneous Bessel equation + +.. math :: + + z^2 f''(z) + z f'(z) + (z^2-\nu^2) f(z) = z^{\mu+1}. + +A second solution is given by :func:`~mpmath.lommels2`. + +**Plots** + +.. literalinclude :: /plots/lommels1.py +.. image :: /plots/lommels1.png + +**Examples** + +An integral representation:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> u,v,z = 0.25, 0.125, mpf(0.75) + >>> lommels1(u,v,z) + 0.4276243877565150372999126 + >>> (bessely(v,z)*quad(lambda t: t**u*besselj(v,t), [0,z]) - \ + ... besselj(v,z)*quad(lambda t: t**u*bessely(v,t), [0,z]))*(pi/2) + 0.4276243877565150372999126 + +A special value:: + + >>> lommels1(v,v,z) + 0.5461221367746048054932553 + >>> gamma(v+0.5)*sqrt(pi)*power(2,v-1)*struveh(v,z) + 0.5461221367746048054932553 + +Verifying the differential equation:: + + >>> f = lambda z: lommels1(u,v,z) + >>> z**2*diff(f,z,2) + z*diff(f,z) + (z**2-v**2)*f(z) + 0.6979536443265746992059141 + >>> z**(u+1) + 0.6979536443265746992059141 + +**References** + +1. [GradshteynRyzhik]_ +2. [Weisstein]_ http://mathworld.wolfram.com/LommelFunction.html +""" + +lommels2 = r""" +Gives the second Lommel function `S_{\mu,\nu}` or `s^{(2)}_{\mu,\nu}` + +.. math :: + + S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} + \Gamma\left(\tfrac{1}{2}(\mu-\nu+1)\right) + \Gamma\left(\tfrac{1}{2}(\mu+\nu+1)\right) \times + + \left[\sin(\tfrac{1}{2}(\mu-\nu)\pi) J_{\nu}(z) - + \cos(\tfrac{1}{2}(\mu-\nu)\pi) Y_{\nu}(z) + \right] + +which solves the same differential equation as +:func:`~mpmath.lommels1`. + +**Plots** + +.. literalinclude :: /plots/lommels2.py +.. image :: /plots/lommels2.png + +**Examples** + +For large `|z|`, `S_{\mu,\nu} \sim z^{\mu-1}`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> lommels2(10,2,30000) + 1.968299831601008419949804e+40 + >>> power(30000,9) + 1.9683e+40 + +A special value:: + + >>> u,v,z = 0.5, 0.125, mpf(0.75) + >>> lommels2(v,v,z) + 0.9589683199624672099969765 + >>> (struveh(v,z)-bessely(v,z))*power(2,v-1)*sqrt(pi)*gamma(v+0.5) + 0.9589683199624672099969765 + +Verifying the differential equation:: + + >>> f = lambda z: lommels2(u,v,z) + >>> z**2*diff(f,z,2) + z*diff(f,z) + (z**2-v**2)*f(z) + 0.6495190528383289850727924 + >>> z**(u+1) + 0.6495190528383289850727924 + +**References** + +1. [GradshteynRyzhik]_ +2. [Weisstein]_ http://mathworld.wolfram.com/LommelFunction.html +""" + +appellf2 = r""" +Gives the Appell F2 hypergeometric function of two variables + +.. math :: + + F_2(a,b_1,b_2,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c_1)_m (c_2)_n} + \frac{x^m y^n}{m! n!}. + +The series is generally absolutely convergent for `|x| + |y| < 1`. + +**Examples** + +Evaluation for real and complex arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> appellf2(1,2,3,4,5,0.25,0.125) + 1.257417193533135344785602 + >>> appellf2(1,-3,-4,2,3,2,3) + -42.8 + >>> appellf2(0.5,0.25,-0.25,2,3,0.25j,0.25) + (0.9880539519421899867041719 + 0.01497616165031102661476978j) + >>> chop(appellf2(1,1+j,1-j,3j,-3j,0.25,0.25)) + 1.201311219287411337955192 + >>> appellf2(1,1,1,4,6,0.125,16) + (-0.09455532250274744282125152 - 0.7647282253046207836769297j) + +A transformation formula:: + + >>> a,b1,b2,c1,c2,x,y = map(mpf, [1,2,0.5,0.25,1.625,-0.125,0.125]) + >>> appellf2(a,b1,b2,c1,c2,x,y) + 0.2299211717841180783309688 + >>> (1-x)**(-a)*appellf2(a,c1-b1,b2,c1,c2,x/(x-1),y/(1-x)) + 0.2299211717841180783309688 + +A system of partial differential equations satisfied by F2:: + + >>> a,b1,b2,c1,c2,x,y = map(mpf, [1,0.5,0.25,1.125,1.5,0.0625,-0.0625]) + >>> F = lambda x,y: appellf2(a,b1,b2,c1,c2,x,y) + >>> chop(x*(1-x)*diff(F,(x,y),(2,0)) - + ... x*y*diff(F,(x,y),(1,1)) + + ... (c1-(a+b1+1)*x)*diff(F,(x,y),(1,0)) - + ... b1*y*diff(F,(x,y),(0,1)) - + ... a*b1*F(x,y)) + 0.0 + >>> chop(y*(1-y)*diff(F,(x,y),(0,2)) - + ... x*y*diff(F,(x,y),(1,1)) + + ... (c2-(a+b2+1)*y)*diff(F,(x,y),(0,1)) - + ... b2*x*diff(F,(x,y),(1,0)) - + ... a*b2*F(x,y)) + 0.0 + +**References** + +See references for :func:`~mpmath.appellf1`. +""" + +appellf3 = r""" +Gives the Appell F3 hypergeometric function of two variables + +.. math :: + + F_3(a_1,a_2,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \frac{(a_1)_m (a_2)_n (b_1)_m (b_2)_n}{(c)_{m+n}} + \frac{x^m y^n}{m! n!}. + +The series is generally absolutely convergent for `|x| < 1, |y| < 1`. + +**Examples** + +Evaluation for various parameters and variables:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> appellf3(1,2,3,4,5,0.5,0.25) + 2.221557778107438938158705 + >>> appellf3(1,2,3,4,5,6,0); hyp2f1(1,3,5,6) + (-0.5189554589089861284537389 - 0.1454441043328607980769742j) + (-0.5189554589089861284537389 - 0.1454441043328607980769742j) + >>> appellf3(1,-2,-3,1,1,4,6) + -17.4 + >>> appellf3(1,2,-3,1,1,4,6) + (17.7876136773677356641825 + 19.54768762233649126154534j) + >>> appellf3(1,2,-3,1,1,6,4) + (85.02054175067929402953645 + 148.4402528821177305173599j) + >>> chop(appellf3(1+j,2,1-j,2,3,0.25,0.25)) + 1.719992169545200286696007 + +Many transformations and evaluations for special combinations +of the parameters are possible, e.g.: + + >>> a,b,c,x,y = map(mpf, [0.5,0.25,0.125,0.125,-0.125]) + >>> appellf3(a,c-a,b,c-b,c,x,y) + 1.093432340896087107444363 + >>> (1-y)**(a+b-c)*hyp2f1(a,b,c,x+y-x*y) + 1.093432340896087107444363 + >>> x**2*appellf3(1,1,1,1,3,x,-x) + 0.01568646277445385390945083 + >>> polylog(2,x**2) + 0.01568646277445385390945083 + >>> a1,a2,b1,b2,c,x = map(mpf, [0.5,0.25,0.125,0.5,4.25,0.125]) + >>> appellf3(a1,a2,b1,b2,c,x,1) + 1.03947361709111140096947 + >>> gammaprod([c,c-a2-b2],[c-a2,c-b2])*hyp3f2(a1,b1,c-a2-b2,c-a2,c-b2,x) + 1.03947361709111140096947 + +The Appell F3 function satisfies a pair of partial +differential equations:: + + >>> a1,a2,b1,b2,c,x,y = map(mpf, [0.5,0.25,0.125,0.5,0.625,0.0625,-0.0625]) + >>> F = lambda x,y: appellf3(a1,a2,b1,b2,c,x,y) + >>> chop(x*(1-x)*diff(F,(x,y),(2,0)) + + ... y*diff(F,(x,y),(1,1)) + + ... (c-(a1+b1+1)*x)*diff(F,(x,y),(1,0)) - + ... a1*b1*F(x,y)) + 0.0 + >>> chop(y*(1-y)*diff(F,(x,y),(0,2)) + + ... x*diff(F,(x,y),(1,1)) + + ... (c-(a2+b2+1)*y)*diff(F,(x,y),(0,1)) - + ... a2*b2*F(x,y)) + 0.0 + +**References** + +See references for :func:`~mpmath.appellf1`. +""" + +appellf4 = r""" +Gives the Appell F4 hypergeometric function of two variables + +.. math :: + + F_4(a,b,c_1,c_2,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} + \frac{(a)_{m+n} (b)_{m+n}}{(c_1)_m (c_2)_n} + \frac{x^m y^n}{m! n!}. + +The series is generally absolutely convergent for +`\sqrt{|x|} + \sqrt{|y|} < 1`. + +**Examples** + +Evaluation for various parameters and arguments:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> appellf4(1,1,2,2,0.25,0.125) + 1.286182069079718313546608 + >>> appellf4(-2,-3,4,5,4,5) + 34.8 + >>> appellf4(5,4,2,3,0.25j,-0.125j) + (-0.2585967215437846642163352 + 2.436102233553582711818743j) + +Reduction to `\,_2F_1` in a special case:: + + >>> a,b,c,x,y = map(mpf, [0.5,0.25,0.125,0.125,-0.125]) + >>> appellf4(a,b,c,a+b-c+1,x*(1-y),y*(1-x)) + 1.129143488466850868248364 + >>> hyp2f1(a,b,c,x)*hyp2f1(a,b,a+b-c+1,y) + 1.129143488466850868248364 + +A system of partial differential equations satisfied by F4:: + + >>> a,b,c1,c2,x,y = map(mpf, [1,0.5,0.25,1.125,0.0625,-0.0625]) + >>> F = lambda x,y: appellf4(a,b,c1,c2,x,y) + >>> chop(x*(1-x)*diff(F,(x,y),(2,0)) - + ... y**2*diff(F,(x,y),(0,2)) - + ... 2*x*y*diff(F,(x,y),(1,1)) + + ... (c1-(a+b+1)*x)*diff(F,(x,y),(1,0)) - + ... ((a+b+1)*y)*diff(F,(x,y),(0,1)) - + ... a*b*F(x,y)) + 0.0 + >>> chop(y*(1-y)*diff(F,(x,y),(0,2)) - + ... x**2*diff(F,(x,y),(2,0)) - + ... 2*x*y*diff(F,(x,y),(1,1)) + + ... (c2-(a+b+1)*y)*diff(F,(x,y),(0,1)) - + ... ((a+b+1)*x)*diff(F,(x,y),(1,0)) - + ... a*b*F(x,y)) + 0.0 + +**References** + +See references for :func:`~mpmath.appellf1`. +""" + +zeta = r""" +Computes the Riemann zeta function + +.. math :: + + \zeta(s) = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\ldots + +or, with `a \ne 1`, the more general Hurwitz zeta function + +.. math :: + + \zeta(s,a) = \sum_{k=0}^\infty \frac{1}{(a+k)^s}. + +Optionally, ``zeta(s, a, n)`` computes the `n`-th derivative with +respect to `s`, + +.. math :: + + \zeta^{(n)}(s,a) = (-1)^n \sum_{k=0}^\infty \frac{\log^n(a+k)}{(a+k)^s}. + +Although these series only converge for `\Re(s) > 1`, the Riemann and Hurwitz +zeta functions are defined through analytic continuation for arbitrary +complex `s \ne 1` (`s = 1` is a pole). + +The implementation uses three algorithms: the Borwein algorithm for +the Riemann zeta function when `s` is close to the real line; +the Riemann-Siegel formula for the Riemann zeta function when `s` is +large imaginary, and Euler-Maclaurin summation in all other cases. +The reflection formula for `\Re(s) < 0` is implemented in some cases. +The algorithm can be chosen with ``method = 'borwein'``, +``method='riemann-siegel'`` or ``method = 'euler-maclaurin'``. + +The parameter `a` is usually a rational number `a = p/q`, and may be specified +as such by passing an integer tuple `(p, q)`. Evaluation is supported for +arbitrary complex `a`, but may be slow and/or inaccurate when `\Re(s) < 0` for +nonrational `a` or when computing derivatives. + +**Examples** + +Some values of the Riemann zeta function:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> zeta(2); pi**2 / 6 + 1.644934066848226436472415 + 1.644934066848226436472415 + >>> zeta(0) + -0.5 + >>> zeta(-1) + -0.08333333333333333333333333 + >>> zeta(-2) + 0.0 + +For large positive `s`, `\zeta(s)` rapidly approaches 1:: + + >>> zeta(50) + 1.000000000000000888178421 + >>> zeta(100) + 1.0 + >>> zeta(inf) + 1.0 + >>> 1-sum((zeta(k)-1)/k for k in range(2,85)); +euler + 0.5772156649015328606065121 + 0.5772156649015328606065121 + >>> nsum(lambda k: zeta(k)-1, [2, inf]) + 1.0 + +Evaluation is supported for complex `s` and `a`: + + >>> zeta(-3+4j) + (-0.03373057338827757067584698 + 0.2774499251557093745297677j) + >>> zeta(2+3j, -1+j) + (389.6841230140842816370741 + 295.2674610150305334025962j) + +The Riemann zeta function has so-called nontrivial zeros on +the critical line `s = 1/2 + it`:: + + >>> findroot(zeta, 0.5+14j); zetazero(1) + (0.5 + 14.13472514173469379045725j) + (0.5 + 14.13472514173469379045725j) + >>> findroot(zeta, 0.5+21j); zetazero(2) + (0.5 + 21.02203963877155499262848j) + (0.5 + 21.02203963877155499262848j) + >>> findroot(zeta, 0.5+25j); zetazero(3) + (0.5 + 25.01085758014568876321379j) + (0.5 + 25.01085758014568876321379j) + >>> chop(zeta(zetazero(10))) + 0.0 + +Evaluation on and near the critical line is supported for large +heights `t` by means of the Riemann-Siegel formula (currently +for `a = 1`, `n \le 4`):: + + >>> zeta(0.5+100000j) + (1.073032014857753132114076 + 5.780848544363503984261041j) + >>> zeta(0.75+1000000j) + (0.9535316058375145020351559 + 0.9525945894834273060175651j) + >>> zeta(0.5+10000000j) + (11.45804061057709254500227 - 8.643437226836021723818215j) + >>> zeta(0.5+100000000j, derivative=1) + (51.12433106710194942681869 + 43.87221167872304520599418j) + >>> zeta(0.5+100000000j, derivative=2) + (-444.2760822795430400549229 - 896.3789978119185981665403j) + >>> zeta(0.5+100000000j, derivative=3) + (3230.72682687670422215339 + 14374.36950073615897616781j) + >>> zeta(0.5+100000000j, derivative=4) + (-11967.35573095046402130602 - 218945.7817789262839266148j) + >>> zeta(1+10000000j) # off the line + (2.859846483332530337008882 + 0.491808047480981808903986j) + >>> zeta(1+10000000j, derivative=1) + (-4.333835494679647915673205 - 0.08405337962602933636096103j) + >>> zeta(1+10000000j, derivative=4) + (453.2764822702057701894278 - 581.963625832768189140995j) + +For investigation of the zeta function zeros, the Riemann-Siegel +Z-function is often more convenient than working with the Riemann +zeta function directly (see :func:`~mpmath.siegelz`). + +Some values of the Hurwitz zeta function:: + + >>> zeta(2, 3); -5./4 + pi**2/6 + 0.3949340668482264364724152 + 0.3949340668482264364724152 + >>> zeta(2, (3,4)); pi**2 - 8*catalan + 2.541879647671606498397663 + 2.541879647671606498397663 + +For positive integer values of `s`, the Hurwitz zeta function is +equivalent to a polygamma function (except for a normalizing factor):: + + >>> zeta(4, (1,5)); psi(3, '1/5')/6 + 625.5408324774542966919938 + 625.5408324774542966919938 + +Evaluation of derivatives:: + + >>> zeta(0, 3+4j, 1); loggamma(3+4j) - ln(2*pi)/2 + (-2.675565317808456852310934 + 4.742664438034657928194889j) + (-2.675565317808456852310934 + 4.742664438034657928194889j) + >>> zeta(2, 1, 20) + 2432902008176640000.000242 + >>> zeta(3+4j, 5.5+2j, 4) + (-0.140075548947797130681075 - 0.3109263360275413251313634j) + >>> zeta(0.5+100000j, 1, 4) + (-10407.16081931495861539236 + 13777.78669862804508537384j) + >>> zeta(-100+0.5j, (1,3), derivative=4) + (4.007180821099823942702249e+79 + 4.916117957092593868321778e+78j) + +Generating a Taylor series at `s = 2` using derivatives:: + + >>> for k in range(11): print("%s * (s-2)^%i" % (zeta(2,1,k)/fac(k), k)) + ... + 1.644934066848226436472415 * (s-2)^0 + -0.9375482543158437537025741 * (s-2)^1 + 0.9946401171494505117104293 * (s-2)^2 + -1.000024300473840810940657 * (s-2)^3 + 1.000061933072352565457512 * (s-2)^4 + -1.000006869443931806408941 * (s-2)^5 + 1.000000173233769531820592 * (s-2)^6 + -0.9999999569989868493432399 * (s-2)^7 + 0.9999999937218844508684206 * (s-2)^8 + -0.9999999996355013916608284 * (s-2)^9 + 1.000000000004610645020747 * (s-2)^10 + +Evaluation at zero and for negative integer `s`:: + + >>> zeta(0, 10) + -9.5 + >>> zeta(-2, (2,3)); mpf(1)/81 + 0.01234567901234567901234568 + 0.01234567901234567901234568 + >>> zeta(-3+4j, (5,4)) + (0.2899236037682695182085988 + 0.06561206166091757973112783j) + >>> zeta(-3.25, 1/pi) + -0.0005117269627574430494396877 + >>> zeta(-3.5, pi, 1) + 11.156360390440003294709 + >>> zeta(-100.5, (8,3)) + -4.68162300487989766727122e+77 + >>> zeta(-10.5, (-8,3)) + (-0.01521913704446246609237979 + 29907.72510874248161608216j) + >>> zeta(-1000.5, (-8,3)) + (1.031911949062334538202567e+1770 + 1.519555750556794218804724e+426j) + >>> zeta(-1+j, 3+4j) + (-16.32988355630802510888631 - 22.17706465801374033261383j) + >>> zeta(-1+j, 3+4j, 2) + (32.48985276392056641594055 - 51.11604466157397267043655j) + >>> diff(lambda s: zeta(s, 3+4j), -1+j, 2) + (32.48985276392056641594055 - 51.11604466157397267043655j) + +**References** + +1. http://mathworld.wolfram.com/RiemannZetaFunction.html + +2. http://mathworld.wolfram.com/HurwitzZetaFunction.html + +3. [BorweinZeta]_ + +""" + +dirichlet = r""" +Evaluates the Dirichlet L-function + +.. math :: + + L(s,\chi) = \sum_{k=1}^\infty \frac{\chi(k)}{k^s}. + +where `\chi` is a periodic sequence of length `q` which should be supplied +in the form of a list `[\chi(0), \chi(1), \ldots, \chi(q-1)]`. +Strictly, `\chi` should be a Dirichlet character, but any periodic +sequence will work. + +For example, ``dirichlet(s, [1])`` gives the ordinary +Riemann zeta function and ``dirichlet(s, [-1,1])`` gives +the alternating zeta function (Dirichlet eta function). + +Also the derivative with respect to `s` (currently only a first +derivative) can be evaluated. + +**Examples** + +The ordinary Riemann zeta function:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> dirichlet(3, [1]); zeta(3) + 1.202056903159594285399738 + 1.202056903159594285399738 + >>> dirichlet(1, [1]) + +inf + +The alternating zeta function:: + + >>> dirichlet(1, [-1,1]); ln(2) + 0.6931471805599453094172321 + 0.6931471805599453094172321 + +The following defines the Dirichlet beta function +`\beta(s) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^s}` and verifies +several values of this function:: + + >>> B = lambda s, d=0: dirichlet(s, [0, 1, 0, -1], d) + >>> B(0); 1./2 + 0.5 + 0.5 + >>> B(1); pi/4 + 0.7853981633974483096156609 + 0.7853981633974483096156609 + >>> B(2); +catalan + 0.9159655941772190150546035 + 0.9159655941772190150546035 + >>> B(2,1); diff(B, 2) + 0.08158073611659279510291217 + 0.08158073611659279510291217 + >>> B(-1,1); 2*catalan/pi + 0.5831218080616375602767689 + 0.5831218080616375602767689 + >>> B(0,1); log(gamma(0.25)**2/(2*pi*sqrt(2))) + 0.3915943927068367764719453 + 0.3915943927068367764719454 + >>> B(1,1); 0.25*pi*(euler+2*ln2+3*ln(pi)-4*ln(gamma(0.25))) + 0.1929013167969124293631898 + 0.1929013167969124293631898 + +A custom L-series of period 3:: + + >>> dirichlet(2, [2,0,1]) + 0.7059715047839078092146831 + >>> 2*nsum(lambda k: (3*k)**-2, [1,inf]) + \ + ... nsum(lambda k: (3*k+2)**-2, [0,inf]) + 0.7059715047839078092146831 + +""" + +coulombf = r""" +Calculates the regular Coulomb wave function + +.. math :: + + F_l(\eta,z) = C_l(\eta) z^{l+1} e^{-iz} \,_1F_1(l+1-i\eta, 2l+2, 2iz) + +where the normalization constant `C_l(\eta)` is as calculated by +:func:`~mpmath.coulombc`. This function solves the differential equation + +.. math :: + + f''(z) + \left(1-\frac{2\eta}{z}-\frac{l(l+1)}{z^2}\right) f(z) = 0. + +A second linearly independent solution is given by the irregular +Coulomb wave function `G_l(\eta,z)` (see :func:`~mpmath.coulombg`) +and thus the general solution is +`f(z) = C_1 F_l(\eta,z) + C_2 G_l(\eta,z)` for arbitrary +constants `C_1`, `C_2`. +Physically, the Coulomb wave functions give the radial solution +to the Schrodinger equation for a point particle in a `1/z` potential; `z` is +then the radius and `l`, `\eta` are quantum numbers. + +The Coulomb wave functions with real parameters are defined +in Abramowitz & Stegun, section 14. However, all parameters are permitted +to be complex in this implementation (see references). + +**Plots** + +.. literalinclude :: /plots/coulombf.py +.. image :: /plots/coulombf.png +.. literalinclude :: /plots/coulombf_c.py +.. image :: /plots/coulombf_c.png + +**Examples** + +Evaluation is supported for arbitrary magnitudes of `z`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> coulombf(2, 1.5, 3.5) + 0.4080998961088761187426445 + >>> coulombf(-2, 1.5, 3.5) + 0.7103040849492536747533465 + >>> coulombf(2, 1.5, '1e-10') + 4.143324917492256448770769e-33 + >>> coulombf(2, 1.5, 1000) + 0.4482623140325567050716179 + >>> coulombf(2, 1.5, 10**10) + -0.066804196437694360046619 + +Verifying the differential equation:: + + >>> l, eta, z = 2, 3, mpf(2.75) + >>> A, B = 1, 2 + >>> f = lambda z: A*coulombf(l,eta,z) + B*coulombg(l,eta,z) + >>> chop(diff(f,z,2) + (1-2*eta/z - l*(l+1)/z**2)*f(z)) + 0.0 + +A Wronskian relation satisfied by the Coulomb wave functions:: + + >>> l = 2 + >>> eta = 1.5 + >>> F = lambda z: coulombf(l,eta,z) + >>> G = lambda z: coulombg(l,eta,z) + >>> for z in [3.5, -1, 2+3j]: + ... chop(diff(F,z)*G(z) - F(z)*diff(G,z)) + ... + 1.0 + 1.0 + 1.0 + +Another Wronskian relation:: + + >>> F = coulombf + >>> G = coulombg + >>> for z in [3.5, -1, 2+3j]: + ... chop(F(l-1,eta,z)*G(l,eta,z)-F(l,eta,z)*G(l-1,eta,z) - l/sqrt(l**2+eta**2)) + ... + 0.0 + 0.0 + 0.0 + +An integral identity connecting the regular and irregular wave functions:: + + >>> l, eta, z = 4+j, 2-j, 5+2j + >>> coulombf(l,eta,z) + j*coulombg(l,eta,z) + (0.7997977752284033239714479 + 0.9294486669502295512503127j) + >>> g = lambda t: exp(-t)*t**(l-j*eta)*(t+2*j*z)**(l+j*eta) + >>> j*exp(-j*z)*z**(-l)/fac(2*l+1)/coulombc(l,eta)*quad(g, [0,inf]) + (0.7997977752284033239714479 + 0.9294486669502295512503127j) + +Some test case with complex parameters, taken from Michel [2]:: + + >>> mp.dps = 15 + >>> coulombf(1+0.1j, 50+50j, 100.156) + (-1.02107292320897e+15 - 2.83675545731519e+15j) + >>> coulombg(1+0.1j, 50+50j, 100.156) + (2.83675545731519e+15 - 1.02107292320897e+15j) + >>> coulombf(1e-5j, 10+1e-5j, 0.1+1e-6j) + (4.30566371247811e-14 - 9.03347835361657e-19j) + >>> coulombg(1e-5j, 10+1e-5j, 0.1+1e-6j) + (778709182061.134 + 18418936.2660553j) + +The following reproduces a table in Abramowitz & Stegun, at twice +the precision:: + + >>> mp.dps = 10 + >>> eta = 2; z = 5 + >>> for l in [5, 4, 3, 2, 1, 0]: + ... print("%s %s %s" % (l, coulombf(l,eta,z), + ... diff(lambda z: coulombf(l,eta,z), z))) + ... + 5 0.09079533488 0.1042553261 + 4 0.2148205331 0.2029591779 + 3 0.4313159311 0.320534053 + 2 0.7212774133 0.3952408216 + 1 0.9935056752 0.3708676452 + 0 1.143337392 0.2937960375 + +**References** + +1. I.J. Thompson & A.R. Barnett, "Coulomb and Bessel Functions of Complex + Arguments and Order", J. Comp. Phys., vol 64, no. 2, June 1986. + +2. N. Michel, "Precise Coulomb wave functions for a wide range of + complex `l`, `\eta` and `z`", http://arxiv.org/abs/physics/0702051v1 + +""" + +coulombg = r""" +Calculates the irregular Coulomb wave function + +.. math :: + + G_l(\eta,z) = \frac{F_l(\eta,z) \cos(\chi) - F_{-l-1}(\eta,z)}{\sin(\chi)} + +where `\chi = \sigma_l - \sigma_{-l-1} - (l+1/2) \pi` +and `\sigma_l(\eta) = (\ln \Gamma(1+l+i\eta)-\ln \Gamma(1+l-i\eta))/(2i)`. + +See :func:`~mpmath.coulombf` for additional information. + +**Plots** + +.. literalinclude :: /plots/coulombg.py +.. image :: /plots/coulombg.png +.. literalinclude :: /plots/coulombg_c.py +.. image :: /plots/coulombg_c.png + +**Examples** + +Evaluation is supported for arbitrary magnitudes of `z`:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> coulombg(-2, 1.5, 3.5) + 1.380011900612186346255524 + >>> coulombg(2, 1.5, 3.5) + 1.919153700722748795245926 + >>> coulombg(-2, 1.5, '1e-10') + 201126715824.7329115106793 + >>> coulombg(-2, 1.5, 1000) + 0.1802071520691149410425512 + >>> coulombg(-2, 1.5, 10**10) + 0.652103020061678070929794 + +The following reproduces a table in Abramowitz & Stegun, +at twice the precision:: + + >>> mp.dps = 10 + >>> eta = 2; z = 5 + >>> for l in [1, 2, 3, 4, 5]: + ... print("%s %s %s" % (l, coulombg(l,eta,z), + ... -diff(lambda z: coulombg(l,eta,z), z))) + ... + 1 1.08148276 0.6028279961 + 2 1.496877075 0.5661803178 + 3 2.048694714 0.7959909551 + 4 3.09408669 1.731802374 + 5 5.629840456 4.549343289 + +Evaluation close to the singularity at `z = 0`:: + + >>> mp.dps = 15 + >>> coulombg(0,10,1) + 3088184933.67358 + >>> coulombg(0,10,'1e-10') + 5554866000719.8 + >>> coulombg(0,10,'1e-100') + 5554866221524.1 + +Evaluation with a half-integer value for `l`:: + + >>> coulombg(1.5, 1, 10) + 0.852320038297334 +""" + +coulombc = r""" +Gives the normalizing Gamow constant for Coulomb wave functions, + +.. math :: + + C_l(\eta) = 2^l \exp\left(-\pi \eta/2 + [\ln \Gamma(1+l+i\eta) + + \ln \Gamma(1+l-i\eta)]/2 - \ln \Gamma(2l+2)\right), + +where the log gamma function with continuous imaginary part +away from the negative half axis (see :func:`~mpmath.loggamma`) is implied. + +This function is used internally for the calculation of +Coulomb wave functions, and automatically cached to make multiple +evaluations with fixed `l`, `\eta` fast. +""" + +ellipfun = r""" +Computes any of the Jacobi elliptic functions, defined +in terms of Jacobi theta functions as + +.. math :: + + \mathrm{sn}(u,m) = \frac{\vartheta_3(0,q)}{\vartheta_2(0,q)} + \frac{\vartheta_1(t,q)}{\vartheta_4(t,q)} + + \mathrm{cn}(u,m) = \frac{\vartheta_4(0,q)}{\vartheta_2(0,q)} + \frac{\vartheta_2(t,q)}{\vartheta_4(t,q)} + + \mathrm{dn}(u,m) = \frac{\vartheta_4(0,q)}{\vartheta_3(0,q)} + \frac{\vartheta_3(t,q)}{\vartheta_4(t,q)}, + +or more generally computes a ratio of two such functions. Here +`t = u/\vartheta_3(0,q)^2`, and `q = q(m)` denotes the nome (see +:func:`~mpmath.nome`). Optionally, you can specify the nome directly +instead of `m` by passing ``q=``, or you can directly +specify the elliptic parameter `k` with ``k=``. + +The first argument should be a two-character string specifying the +function using any combination of ``'s'``, ``'c'``, ``'d'``, ``'n'``. These +letters respectively denote the basic functions +`\mathrm{sn}(u,m)`, `\mathrm{cn}(u,m)`, `\mathrm{dn}(u,m)`, and `1`. +The identifier specifies the ratio of two such functions. +For example, ``'ns'`` identifies the function + +.. math :: + + \mathrm{ns}(u,m) = \frac{1}{\mathrm{sn}(u,m)} + +and ``'cd'`` identifies the function + +.. math :: + + \mathrm{cd}(u,m) = \frac{\mathrm{cn}(u,m)}{\mathrm{dn}(u,m)}. + +If called with only the first argument, a function object +evaluating the chosen function for given arguments is returned. + +**Examples** + +Basic evaluation:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> ellipfun('cd', 3.5, 0.5) + -0.9891101840595543931308394 + >>> ellipfun('cd', 3.5, q=0.25) + 0.07111979240214668158441418 + +The sn-function is doubly periodic in the complex plane with periods +`4 K(m)` and `2 i K(1-m)` (see :func:`~mpmath.ellipk`):: + + >>> sn = ellipfun('sn') + >>> sn(2, 0.25) + 0.9628981775982774425751399 + >>> sn(2+4*ellipk(0.25), 0.25) + 0.9628981775982774425751399 + >>> chop(sn(2+2*j*ellipk(1-0.25), 0.25)) + 0.9628981775982774425751399 + +The cn-function is doubly periodic with periods `4 K(m)` and `2 K(m) + 2 i K(1-m)`:: + + >>> cn = ellipfun('cn') + >>> cn(2, 0.25) + -0.2698649654510865792581416 + >>> cn(2+4*ellipk(0.25), 0.25) + -0.2698649654510865792581416 + >>> chop(cn(2+2*ellipk(0.25)+2*j*ellipk(1-0.25), 0.25)) + -0.2698649654510865792581416 + +The dn-function is doubly periodic with periods `2 K(m)` and `4 i K(1-m)`:: + + >>> dn = ellipfun('dn') + >>> dn(2, 0.25) + 0.8764740583123262286931578 + >>> dn(2+2*ellipk(0.25), 0.25) + 0.8764740583123262286931578 + >>> chop(dn(2+4*j*ellipk(1-0.25), 0.25)) + 0.8764740583123262286931578 + +""" + + +jtheta = r""" +Computes the Jacobi theta function `\vartheta_n(z, q)`, where +`n = 1, 2, 3, 4`, defined by the infinite series: + +.. math :: + + \vartheta_1(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty} + (-1)^n q^{n^2+n\,} \sin((2n+1)z) + + \vartheta_2(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty} + q^{n^{2\,} + n} \cos((2n+1)z) + + \vartheta_3(z,q) = 1 + 2 \sum_{n=1}^{\infty} + q^{n^2\,} \cos(2 n z) + + \vartheta_4(z,q) = 1 + 2 \sum_{n=1}^{\infty} + (-q)^{n^2\,} \cos(2 n z) + +The theta functions are functions of two variables: + +* `z` is the *argument*, an arbitrary real or complex number + +* `q` is the *nome*, which must be a real or complex number + in the unit disk (i.e. `|q| < 1`). For `|q| \ll 1`, the + series converge very quickly, so the Jacobi theta functions + can efficiently be evaluated to high precision. + +The compact notations `\vartheta_n(q) = \vartheta_n(0,q)` +and `\vartheta_n = \vartheta_n(0,q)` are also frequently +encountered. Finally, Jacobi theta functions are frequently +considered as functions of the half-period ratio `\tau` +and then usually denoted by `\vartheta_n(z|\tau)`. + +Optionally, ``jtheta(n, z, q, derivative=d)`` with `d > 0` computes +a `d`-th derivative with respect to `z`. + +**Examples and basic properties** + +Considered as functions of `z`, the Jacobi theta functions may be +viewed as generalizations of the ordinary trigonometric functions +cos and sin. They are periodic functions:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> jtheta(1, 0.25, '0.2') + 0.2945120798627300045053104 + >>> jtheta(1, 0.25 + 2*pi, '0.2') + 0.2945120798627300045053104 + +Indeed, the series defining the theta functions are essentially +trigonometric Fourier series. The coefficients can be retrieved +using :func:`~mpmath.fourier`:: + + >>> mp.dps = 10 + >>> nprint(fourier(lambda x: jtheta(2, x, 0.5), [-pi, pi], 4)) + ([0.0, 1.68179, 0.0, 0.420448, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0]) + +The Jacobi theta functions are also so-called quasiperiodic +functions of `z` and `\tau`, meaning that for fixed `\tau`, +`\vartheta_n(z, q)` and `\vartheta_n(z+\pi \tau, q)` are the same +except for an exponential factor:: + + >>> mp.dps = 25 + >>> tau = 3*j/10 + >>> q = exp(pi*j*tau) + >>> z = 10 + >>> jtheta(4, z+tau*pi, q) + (-0.682420280786034687520568 + 1.526683999721399103332021j) + >>> -exp(-2*j*z)/q * jtheta(4, z, q) + (-0.682420280786034687520568 + 1.526683999721399103332021j) + +The Jacobi theta functions satisfy a huge number of other +functional equations, such as the following identity (valid for +any `q`):: + + >>> q = mpf(3)/10 + >>> jtheta(3,0,q)**4 + 6.823744089352763305137427 + >>> jtheta(2,0,q)**4 + jtheta(4,0,q)**4 + 6.823744089352763305137427 + +Extensive listings of identities satisfied by the Jacobi theta +functions can be found in standard reference works. + +The Jacobi theta functions are related to the gamma function +for special arguments:: + + >>> jtheta(3, 0, exp(-pi)) + 1.086434811213308014575316 + >>> pi**(1/4.) / gamma(3/4.) + 1.086434811213308014575316 + +:func:`~mpmath.jtheta` supports arbitrary precision evaluation and complex +arguments:: + + >>> mp.dps = 50 + >>> jtheta(4, sqrt(2), 0.5) + 2.0549510717571539127004115835148878097035750653737 + >>> mp.dps = 25 + >>> jtheta(4, 1+2j, (1+j)/5) + (7.180331760146805926356634 - 1.634292858119162417301683j) + +Evaluation of derivatives:: + + >>> mp.dps = 25 + >>> jtheta(1, 7, 0.25, 1); diff(lambda z: jtheta(1, z, 0.25), 7) + 1.209857192844475388637236 + 1.209857192844475388637236 + >>> jtheta(1, 7, 0.25, 2); diff(lambda z: jtheta(1, z, 0.25), 7, 2) + -0.2598718791650217206533052 + -0.2598718791650217206533052 + >>> jtheta(2, 7, 0.25, 1); diff(lambda z: jtheta(2, z, 0.25), 7) + -1.150231437070259644461474 + -1.150231437070259644461474 + >>> jtheta(2, 7, 0.25, 2); diff(lambda z: jtheta(2, z, 0.25), 7, 2) + -0.6226636990043777445898114 + -0.6226636990043777445898114 + >>> jtheta(3, 7, 0.25, 1); diff(lambda z: jtheta(3, z, 0.25), 7) + -0.9990312046096634316587882 + -0.9990312046096634316587882 + >>> jtheta(3, 7, 0.25, 2); diff(lambda z: jtheta(3, z, 0.25), 7, 2) + -0.1530388693066334936151174 + -0.1530388693066334936151174 + >>> jtheta(4, 7, 0.25, 1); diff(lambda z: jtheta(4, z, 0.25), 7) + 0.9820995967262793943571139 + 0.9820995967262793943571139 + >>> jtheta(4, 7, 0.25, 2); diff(lambda z: jtheta(4, z, 0.25), 7, 2) + 0.3936902850291437081667755 + 0.3936902850291437081667755 + +**Possible issues** + +For `|q| \ge 1` or `\Im(\tau) \le 0`, :func:`~mpmath.jtheta` raises +``ValueError``. This exception is also raised for `|q|` extremely +close to 1 (or equivalently `\tau` very close to 0), since the +series would converge too slowly:: + + >>> jtheta(1, 10, 0.99999999 * exp(0.5*j)) + Traceback (most recent call last): + ... + ValueError: abs(q) > THETA_Q_LIM = 1.000000 + +""" + +eulernum = r""" +Gives the `n`-th Euler number, defined as the `n`-th derivative of +`\mathrm{sech}(t) = 1/\cosh(t)` evaluated at `t = 0`. Equivalently, the +Euler numbers give the coefficients of the Taylor series + +.. math :: + + \mathrm{sech}(t) = \sum_{n=0}^{\infty} \frac{E_n}{n!} t^n. + +The Euler numbers are closely related to Bernoulli numbers +and Bernoulli polynomials. They can also be evaluated in terms of +Euler polynomials (see :func:`~mpmath.eulerpoly`) as `E_n = 2^n E_n(1/2)`. + +**Examples** + +Computing the first few Euler numbers and verifying that they +agree with the Taylor series:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> [eulernum(n) for n in range(11)] + [1.0, 0.0, -1.0, 0.0, 5.0, 0.0, -61.0, 0.0, 1385.0, 0.0, -50521.0] + >>> chop(diffs(sech, 0, 10)) + [1.0, 0.0, -1.0, 0.0, 5.0, 0.0, -61.0, 0.0, 1385.0, 0.0, -50521.0] + +Euler numbers grow very rapidly. :func:`~mpmath.eulernum` efficiently +computes numerical approximations for large indices:: + + >>> eulernum(50) + -6.053285248188621896314384e+54 + >>> eulernum(1000) + 3.887561841253070615257336e+2371 + >>> eulernum(10**20) + 4.346791453661149089338186e+1936958564106659551331 + +Comparing with an asymptotic formula for the Euler numbers:: + + >>> n = 10**5 + >>> (-1)**(n//2) * 8 * sqrt(n/(2*pi)) * (2*n/(pi*e))**n + 3.69919063017432362805663e+436961 + >>> eulernum(n) + 3.699193712834466537941283e+436961 + +Pass ``exact=True`` to obtain exact values of Euler numbers as integers:: + + >>> print(eulernum(50, exact=True)) + -6053285248188621896314383785111649088103498225146815121 + >>> print(eulernum(200, exact=True) % 10**10) + 1925859625 + >>> eulernum(1001, exact=True) + 0 +""" + +eulerpoly = r""" +Evaluates the Euler polynomial `E_n(z)`, defined by the generating function +representation + +.. math :: + + \frac{2e^{zt}}{e^t+1} = \sum_{n=0}^\infty E_n(z) \frac{t^n}{n!}. + +The Euler polynomials may also be represented in terms of +Bernoulli polynomials (see :func:`~mpmath.bernpoly`) using various formulas, for +example + +.. math :: + + E_n(z) = \frac{2}{n+1} \left( + B_n(z)-2^{n+1}B_n\left(\frac{z}{2}\right) + \right). + +Special values include the Euler numbers `E_n = 2^n E_n(1/2)` (see +:func:`~mpmath.eulernum`). + +**Examples** + +Computing the coefficients of the first few Euler polynomials:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> for n in range(6): + ... chop(taylor(lambda z: eulerpoly(n,z), 0, n)) + ... + [1.0] + [-0.5, 1.0] + [0.0, -1.0, 1.0] + [0.25, 0.0, -1.5, 1.0] + [0.0, 1.0, 0.0, -2.0, 1.0] + [-0.5, 0.0, 2.5, 0.0, -2.5, 1.0] + +Evaluation for arbitrary `z`:: + + >>> eulerpoly(2,3) + 6.0 + >>> eulerpoly(5,4) + 423.5 + >>> eulerpoly(35, 11111111112) + 3.994957561486776072734601e+351 + >>> eulerpoly(4, 10+20j) + (-47990.0 - 235980.0j) + >>> eulerpoly(2, '-3.5e-5') + 0.000035001225 + >>> eulerpoly(3, 0.5) + 0.0 + >>> eulerpoly(55, -10**80) + -1.0e+4400 + >>> eulerpoly(5, -inf) + -inf + >>> eulerpoly(6, -inf) + +inf + +Computing Euler numbers:: + + >>> 2**26 * eulerpoly(26,0.5) + -4087072509293123892361.0 + >>> eulernum(26) + -4087072509293123892361.0 + +Evaluation is accurate for large `n` and small `z`:: + + >>> eulerpoly(100, 0.5) + 2.29047999988194114177943e+108 + >>> eulerpoly(1000, 10.5) + 3.628120031122876847764566e+2070 + >>> eulerpoly(10000, 10.5) + 1.149364285543783412210773e+30688 +""" + +spherharm = r""" +Evaluates the spherical harmonic `Y_l^m(\theta,\phi)`, + +.. math :: + + Y_l^m(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}} + P_l^m(\cos \theta) e^{i m \phi} + +where `P_l^m` is an associated Legendre function (see :func:`~mpmath.legenp`). + +Here `\theta \in [0, \pi]` denotes the polar coordinate (ranging +from the north pole to the south pole) and `\phi \in [0, 2 \pi]` denotes the +azimuthal coordinate on a sphere. Care should be used since many different +conventions for spherical coordinate variables are used. + +Usually spherical harmonics are considered for `l \in \mathbb{N}`, +`m \in \mathbb{Z}`, `|m| \le l`. More generally, `l,m,\theta,\phi` +are permitted to be complex numbers. + +.. note :: + + :func:`~mpmath.spherharm` returns a complex number, even if the value is + purely real. + +**Plots** + +.. literalinclude :: /plots/spherharm40.py + +`Y_{4,0}`: + +.. image :: /plots/spherharm40.png + +`Y_{4,1}`: + +.. image :: /plots/spherharm41.png + +`Y_{4,2}`: + +.. image :: /plots/spherharm42.png + +`Y_{4,3}`: + +.. image :: /plots/spherharm43.png + +`Y_{4,4}`: + +.. image :: /plots/spherharm44.png + +**Examples** + +Some low-order spherical harmonics with reference values:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> theta = pi/4 + >>> phi = pi/3 + >>> spherharm(0,0,theta,phi); 0.5*sqrt(1/pi)*expj(0) + (0.2820947917738781434740397 + 0.0j) + (0.2820947917738781434740397 + 0.0j) + >>> spherharm(1,-1,theta,phi); 0.5*sqrt(3/(2*pi))*expj(-phi)*sin(theta) + (0.1221506279757299803965962 - 0.2115710938304086076055298j) + (0.1221506279757299803965962 - 0.2115710938304086076055298j) + >>> spherharm(1,0,theta,phi); 0.5*sqrt(3/pi)*cos(theta)*expj(0) + (0.3454941494713354792652446 + 0.0j) + (0.3454941494713354792652446 + 0.0j) + >>> spherharm(1,1,theta,phi); -0.5*sqrt(3/(2*pi))*expj(phi)*sin(theta) + (-0.1221506279757299803965962 - 0.2115710938304086076055298j) + (-0.1221506279757299803965962 - 0.2115710938304086076055298j) + +With the normalization convention used, the spherical harmonics are orthonormal +on the unit sphere:: + + >>> sphere = [0,pi], [0,2*pi] + >>> dS = lambda t,p: fp.sin(t) # differential element + >>> Y1 = lambda t,p: fp.spherharm(l1,m1,t,p) + >>> Y2 = lambda t,p: fp.conj(fp.spherharm(l2,m2,t,p)) + >>> l1 = l2 = 3; m1 = m2 = 2 + >>> fp.chop(fp.quad(lambda t,p: Y1(t,p)*Y2(t,p)*dS(t,p), *sphere)) + 1.0000000000000007 + >>> m2 = 1 # m1 != m2 + >>> print(fp.chop(fp.quad(lambda t,p: Y1(t,p)*Y2(t,p)*dS(t,p), *sphere))) + 0.0 + +Evaluation is accurate for large orders:: + + >>> spherharm(1000,750,0.5,0.25) + (3.776445785304252879026585e-102 - 5.82441278771834794493484e-102j) + +Evaluation works with complex parameter values:: + + >>> spherharm(1+j, 2j, 2+3j, -0.5j) + (64.44922331113759992154992 + 1981.693919841408089681743j) +""" + +scorergi = r""" +Evaluates the Scorer function + +.. math :: + + \operatorname{Gi}(z) = + \operatorname{Ai}(z) \int_0^z \operatorname{Bi}(t) dt + + \operatorname{Bi}(z) \int_z^{\infty} \operatorname{Ai}(t) dt + +which gives a particular solution to the inhomogeneous Airy +differential equation `f''(z) - z f(z) = 1/\pi`. Another +particular solution is given by the Scorer Hi-function +(:func:`~mpmath.scorerhi`). The two functions are related as +`\operatorname{Gi}(z) + \operatorname{Hi}(z) = \operatorname{Bi}(z)`. + +**Plots** + +.. literalinclude :: /plots/gi.py +.. image :: /plots/gi.png +.. literalinclude :: /plots/gi_c.py +.. image :: /plots/gi_c.png + +**Examples** + +Some values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> scorergi(0); 1/(power(3,'7/6')*gamma('2/3')) + 0.2049755424820002450503075 + 0.2049755424820002450503075 + >>> diff(scorergi, 0); 1/(power(3,'5/6')*gamma('1/3')) + 0.1494294524512754526382746 + 0.1494294524512754526382746 + >>> scorergi(+inf); scorergi(-inf) + 0.0 + 0.0 + >>> scorergi(1) + 0.2352184398104379375986902 + >>> scorergi(-1) + -0.1166722172960152826494198 + +Evaluation for large arguments:: + + >>> scorergi(10) + 0.03189600510067958798062034 + >>> scorergi(100) + 0.003183105228162961476590531 + >>> scorergi(1000000) + 0.0000003183098861837906721743873 + >>> 1/(pi*1000000) + 0.0000003183098861837906715377675 + >>> scorergi(-1000) + -0.08358288400262780392338014 + >>> scorergi(-100000) + 0.02886866118619660226809581 + >>> scorergi(50+10j) + (0.0061214102799778578790984 - 0.001224335676457532180747917j) + >>> scorergi(-50-10j) + (5.236047850352252236372551e+29 - 3.08254224233701381482228e+29j) + >>> scorergi(100000j) + (-8.806659285336231052679025e+6474077 + 8.684731303500835514850962e+6474077j) + +Verifying the connection between Gi and Hi:: + + >>> z = 0.25 + >>> scorergi(z) + scorerhi(z) + 0.7287469039362150078694543 + >>> airybi(z) + 0.7287469039362150078694543 + +Verifying the differential equation:: + + >>> for z in [-3.4, 0, 2.5, 1+2j]: + ... chop(diff(scorergi,z,2) - z*scorergi(z)) + ... + -0.3183098861837906715377675 + -0.3183098861837906715377675 + -0.3183098861837906715377675 + -0.3183098861837906715377675 + +Verifying the integral representation:: + + >>> z = 0.5 + >>> scorergi(z) + 0.2447210432765581976910539 + >>> Ai,Bi = airyai,airybi + >>> Bi(z)*(Ai(inf,-1)-Ai(z,-1)) + Ai(z)*(Bi(z,-1)-Bi(0,-1)) + 0.2447210432765581976910539 + +**References** + +1. [DLMF]_ section 9.12: Scorer Functions + +""" + +scorerhi = r""" +Evaluates the second Scorer function + +.. math :: + + \operatorname{Hi}(z) = + \operatorname{Bi}(z) \int_{-\infty}^z \operatorname{Ai}(t) dt - + \operatorname{Ai}(z) \int_{-\infty}^z \operatorname{Bi}(t) dt + +which gives a particular solution to the inhomogeneous Airy +differential equation `f''(z) - z f(z) = 1/\pi`. See also +:func:`~mpmath.scorergi`. + +**Plots** + +.. literalinclude :: /plots/hi.py +.. image :: /plots/hi.png +.. literalinclude :: /plots/hi_c.py +.. image :: /plots/hi_c.png + +**Examples** + +Some values and limits:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> scorerhi(0); 2/(power(3,'7/6')*gamma('2/3')) + 0.4099510849640004901006149 + 0.4099510849640004901006149 + >>> diff(scorerhi,0); 2/(power(3,'5/6')*gamma('1/3')) + 0.2988589049025509052765491 + 0.2988589049025509052765491 + >>> scorerhi(+inf); scorerhi(-inf) + +inf + 0.0 + >>> scorerhi(1) + 0.9722051551424333218376886 + >>> scorerhi(-1) + 0.2206696067929598945381098 + +Evaluation for large arguments:: + + >>> scorerhi(10) + 455641153.5163291358991077 + >>> scorerhi(100) + 6.041223996670201399005265e+288 + >>> scorerhi(1000000) + 7.138269638197858094311122e+289529652 + >>> scorerhi(-10) + 0.0317685352825022727415011 + >>> scorerhi(-100) + 0.003183092495767499864680483 + >>> scorerhi(100j) + (-6.366197716545672122983857e-9 + 0.003183098861710582761688475j) + >>> scorerhi(50+50j) + (-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j) + >>> scorerhi(-1000-1000j) + (0.0001591549432510502796565538 - 0.000159154943091895334973109j) + +Verifying the differential equation:: + + >>> for z in [-3.4, 0, 2, 1+2j]: + ... chop(diff(scorerhi,z,2) - z*scorerhi(z)) + ... + 0.3183098861837906715377675 + 0.3183098861837906715377675 + 0.3183098861837906715377675 + 0.3183098861837906715377675 + +Verifying the integral representation:: + + >>> z = 0.5 + >>> scorerhi(z) + 0.6095559998265972956089949 + >>> Ai,Bi = airyai,airybi + >>> Bi(z)*(Ai(z,-1)-Ai(-inf,-1)) - Ai(z)*(Bi(z,-1)-Bi(-inf,-1)) + 0.6095559998265972956089949 + +""" + + +stirling1 = r""" +Gives the Stirling number of the first kind `s(n,k)`, defined by + +.. math :: + + x(x-1)(x-2)\cdots(x-n+1) = \sum_{k=0}^n s(n,k) x^k. + +The value is computed using an integer recurrence. The implementation +is not optimized for approximating large values quickly. + +**Examples** + +Comparing with the generating function:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> taylor(lambda x: ff(x, 5), 0, 5) + [0.0, 24.0, -50.0, 35.0, -10.0, 1.0] + >>> [stirling1(5, k) for k in range(6)] + [0.0, 24.0, -50.0, 35.0, -10.0, 1.0] + +Recurrence relation:: + + >>> n, k = 5, 3 + >>> stirling1(n+1,k) + n*stirling1(n,k) - stirling1(n,k-1) + 0.0 + +The matrices of Stirling numbers of first and second kind are inverses +of each other:: + + >>> A = matrix(5, 5); B = matrix(5, 5) + >>> for n in range(5): + ... for k in range(5): + ... A[n,k] = stirling1(n,k) + ... B[n,k] = stirling2(n,k) + ... + >>> A * B + [1.0 0.0 0.0 0.0 0.0] + [0.0 1.0 0.0 0.0 0.0] + [0.0 0.0 1.0 0.0 0.0] + [0.0 0.0 0.0 1.0 0.0] + [0.0 0.0 0.0 0.0 1.0] + +Pass ``exact=True`` to obtain exact values of Stirling numbers as integers:: + + >>> stirling1(42, 5) + -2.864498971768501633736628e+50 + >>> print(stirling1(42, 5, exact=True)) + -286449897176850163373662803014001546235808317440000 + +""" + +stirling2 = r""" +Gives the Stirling number of the second kind `S(n,k)`, defined by + +.. math :: + + x^n = \sum_{k=0}^n S(n,k) x(x-1)(x-2)\cdots(x-k+1) + +The value is computed using integer arithmetic to evaluate a power sum. +The implementation is not optimized for approximating large values quickly. + +**Examples** + +Comparing with the generating function:: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> taylor(lambda x: sum(stirling2(5,k) * ff(x,k) for k in range(6)), 0, 5) + [0.0, 0.0, 0.0, 0.0, 0.0, 1.0] + +Recurrence relation:: + + >>> n, k = 5, 3 + >>> stirling2(n+1,k) - k*stirling2(n,k) - stirling2(n,k-1) + 0.0 + +Pass ``exact=True`` to obtain exact values of Stirling numbers as integers:: + + >>> stirling2(52, 10) + 2.641822121003543906807485e+45 + >>> print(stirling2(52, 10, exact=True)) + 2641822121003543906807485307053638921722527655 + + +""" + +squarew = r""" +Computes the square wave function using the definition: + +.. math:: + x(t) = A(-1)^{\left\lfloor{2t / P}\right\rfloor} + +where `P` is the period of the wave and `A` is the amplitude. + +**Examples** + +Square wave with period = 2, amplitude = 1 :: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> squarew(0,1,2) + 1.0 + >>> squarew(0.5,1,2) + 1.0 + >>> squarew(1,1,2) + -1.0 + >>> squarew(1.5,1,2) + -1.0 + >>> squarew(2,1,2) + 1.0 +""" + +trianglew = r""" +Computes the triangle wave function using the definition: + +.. math:: + x(t) = 2A\left(\frac{1}{2}-\left|1-2 \operatorname{frac}\left(\frac{x}{P}+\frac{1}{4}\right)\right|\right) + +where :math:`\operatorname{frac}\left(\frac{t}{T}\right) = \frac{t}{T}-\left\lfloor{\frac{t}{T}}\right\rfloor` +, `P` is the period of the wave, and `A` is the amplitude. + +**Examples** + +Triangle wave with period = 2, amplitude = 1 :: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> trianglew(0,1,2) + 0.0 + >>> trianglew(0.25,1,2) + 0.5 + >>> trianglew(0.5,1,2) + 1.0 + >>> trianglew(1,1,2) + 0.0 + >>> trianglew(1.5,1,2) + -1.0 + >>> trianglew(2,1,2) + 0.0 +""" + +sawtoothw = r""" +Computes the sawtooth wave function using the definition: + +.. math:: + x(t) = A\operatorname{frac}\left(\frac{t}{T}\right) + +where :math:`\operatorname{frac}\left(\frac{t}{T}\right) = \frac{t}{T}-\left\lfloor{\frac{t}{T}}\right\rfloor`, +`P` is the period of the wave, and `A` is the amplitude. + +**Examples** + +Sawtooth wave with period = 2, amplitude = 1 :: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> sawtoothw(0,1,2) + 0.0 + >>> sawtoothw(0.5,1,2) + 0.25 + >>> sawtoothw(1,1,2) + 0.5 + >>> sawtoothw(1.5,1,2) + 0.75 + >>> sawtoothw(2,1,2) + 0.0 +""" + +unit_triangle = r""" +Computes the unit triangle using the definition: + +.. math:: + x(t) = A(-\left| t \right| + 1) + +where `A` is the amplitude. + +**Examples** + +Unit triangle with amplitude = 1 :: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> unit_triangle(-1,1) + 0.0 + >>> unit_triangle(-0.5,1) + 0.5 + >>> unit_triangle(0,1) + 1.0 + >>> unit_triangle(0.5,1) + 0.5 + >>> unit_triangle(1,1) + 0.0 +""" + +sigmoid = r""" +Computes the sigmoid function using the definition: + +.. math:: + x(t) = \frac{A}{1 + e^{-t}} + +where `A` is the amplitude. + +**Examples** + +Sigmoid function with amplitude = 1 :: + + >>> from mpmath import * + >>> mp.dps = 25; mp.pretty = True + >>> sigmoid(-1,1) + 0.2689414213699951207488408 + >>> sigmoid(-0.5,1) + 0.3775406687981454353610994 + >>> sigmoid(0,1) + 0.5 + >>> sigmoid(0.5,1) + 0.6224593312018545646389006 + >>> sigmoid(1,1) + 0.7310585786300048792511592 + +""" diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/math2.py b/env-llmeval/lib/python3.10/site-packages/mpmath/math2.py new file mode 100644 index 0000000000000000000000000000000000000000..302e25f509c18b2c76a2b62611f2765db84ab13e --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/math2.py @@ -0,0 +1,672 @@ +""" +This module complements the math and cmath builtin modules by providing +fast machine precision versions of some additional functions (gamma, ...) +and wrapping math/cmath functions so that they can be called with either +real or complex arguments. +""" + +import operator +import math +import cmath + +# Irrational (?) constants +pi = 3.1415926535897932385 +e = 2.7182818284590452354 +sqrt2 = 1.4142135623730950488 +sqrt5 = 2.2360679774997896964 +phi = 1.6180339887498948482 +ln2 = 0.69314718055994530942 +ln10 = 2.302585092994045684 +euler = 0.57721566490153286061 +catalan = 0.91596559417721901505 +khinchin = 2.6854520010653064453 +apery = 1.2020569031595942854 + +logpi = 1.1447298858494001741 + +def _mathfun_real(f_real, f_complex): + def f(x, **kwargs): + if type(x) is float: + return f_real(x) + if type(x) is complex: + return f_complex(x) + try: + x = float(x) + return f_real(x) + except (TypeError, ValueError): + x = complex(x) + return f_complex(x) + f.__name__ = f_real.__name__ + return f + +def _mathfun(f_real, f_complex): + def f(x, **kwargs): + if type(x) is complex: + return f_complex(x) + try: + return f_real(float(x)) + except (TypeError, ValueError): + return f_complex(complex(x)) + f.__name__ = f_real.__name__ + return f + +def _mathfun_n(f_real, f_complex): + def f(*args, **kwargs): + try: + return f_real(*(float(x) for x in args)) + except (TypeError, ValueError): + return f_complex(*(complex(x) for x in args)) + f.__name__ = f_real.__name__ + return f + +# Workaround for non-raising log and sqrt in Python 2.5 and 2.4 +# on Unix system +try: + math.log(-2.0) + def math_log(x): + if x <= 0.0: + raise ValueError("math domain error") + return math.log(x) + def math_sqrt(x): + if x < 0.0: + raise ValueError("math domain error") + return math.sqrt(x) +except (ValueError, TypeError): + math_log = math.log + math_sqrt = math.sqrt + +pow = _mathfun_n(operator.pow, lambda x, y: complex(x)**y) +log = _mathfun_n(math_log, cmath.log) +sqrt = _mathfun(math_sqrt, cmath.sqrt) +exp = _mathfun_real(math.exp, cmath.exp) + +cos = _mathfun_real(math.cos, cmath.cos) +sin = _mathfun_real(math.sin, cmath.sin) +tan = _mathfun_real(math.tan, cmath.tan) + +acos = _mathfun(math.acos, cmath.acos) +asin = _mathfun(math.asin, cmath.asin) +atan = _mathfun_real(math.atan, cmath.atan) + +cosh = _mathfun_real(math.cosh, cmath.cosh) +sinh = _mathfun_real(math.sinh, cmath.sinh) +tanh = _mathfun_real(math.tanh, cmath.tanh) + +floor = _mathfun_real(math.floor, + lambda z: complex(math.floor(z.real), math.floor(z.imag))) +ceil = _mathfun_real(math.ceil, + lambda z: complex(math.ceil(z.real), math.ceil(z.imag))) + + +cos_sin = _mathfun_real(lambda x: (math.cos(x), math.sin(x)), + lambda z: (cmath.cos(z), cmath.sin(z))) + +cbrt = _mathfun(lambda x: x**(1./3), lambda z: z**(1./3)) + +def nthroot(x, n): + r = 1./n + try: + return float(x) ** r + except (ValueError, TypeError): + return complex(x) ** r + +def _sinpi_real(x): + if x < 0: + return -_sinpi_real(-x) + n, r = divmod(x, 0.5) + r *= pi + n %= 4 + if n == 0: return math.sin(r) + if n == 1: return math.cos(r) + if n == 2: return -math.sin(r) + if n == 3: return -math.cos(r) + +def _cospi_real(x): + if x < 0: + x = -x + n, r = divmod(x, 0.5) + r *= pi + n %= 4 + if n == 0: return math.cos(r) + if n == 1: return -math.sin(r) + if n == 2: return -math.cos(r) + if n == 3: return math.sin(r) + +def _sinpi_complex(z): + if z.real < 0: + return -_sinpi_complex(-z) + n, r = divmod(z.real, 0.5) + z = pi*complex(r, z.imag) + n %= 4 + if n == 0: return cmath.sin(z) + if n == 1: return cmath.cos(z) + if n == 2: return -cmath.sin(z) + if n == 3: return -cmath.cos(z) + +def _cospi_complex(z): + if z.real < 0: + z = -z + n, r = divmod(z.real, 0.5) + z = pi*complex(r, z.imag) + n %= 4 + if n == 0: return cmath.cos(z) + if n == 1: return -cmath.sin(z) + if n == 2: return -cmath.cos(z) + if n == 3: return cmath.sin(z) + +cospi = _mathfun_real(_cospi_real, _cospi_complex) +sinpi = _mathfun_real(_sinpi_real, _sinpi_complex) + +def tanpi(x): + try: + return sinpi(x) / cospi(x) + except OverflowError: + if complex(x).imag > 10: + return 1j + if complex(x).imag < 10: + return -1j + raise + +def cotpi(x): + try: + return cospi(x) / sinpi(x) + except OverflowError: + if complex(x).imag > 10: + return -1j + if complex(x).imag < 10: + return 1j + raise + +INF = 1e300*1e300 +NINF = -INF +NAN = INF-INF +EPS = 2.2204460492503131e-16 + +_exact_gamma = (INF, 1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, + 362880.0, 3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0, + 1307674368000.0, 20922789888000.0, 355687428096000.0, 6402373705728000.0, + 121645100408832000.0, 2432902008176640000.0) + +_max_exact_gamma = len(_exact_gamma)-1 + +# Lanczos coefficients used by the GNU Scientific Library +_lanczos_g = 7 +_lanczos_p = (0.99999999999980993, 676.5203681218851, -1259.1392167224028, + 771.32342877765313, -176.61502916214059, 12.507343278686905, + -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7) + +def _gamma_real(x): + _intx = int(x) + if _intx == x: + if _intx <= 0: + #return (-1)**_intx * INF + raise ZeroDivisionError("gamma function pole") + if _intx <= _max_exact_gamma: + return _exact_gamma[_intx] + if x < 0.5: + # TODO: sinpi + return pi / (_sinpi_real(x)*_gamma_real(1-x)) + else: + x -= 1.0 + r = _lanczos_p[0] + for i in range(1, _lanczos_g+2): + r += _lanczos_p[i]/(x+i) + t = x + _lanczos_g + 0.5 + return 2.506628274631000502417 * t**(x+0.5) * math.exp(-t) * r + +def _gamma_complex(x): + if not x.imag: + return complex(_gamma_real(x.real)) + if x.real < 0.5: + # TODO: sinpi + return pi / (_sinpi_complex(x)*_gamma_complex(1-x)) + else: + x -= 1.0 + r = _lanczos_p[0] + for i in range(1, _lanczos_g+2): + r += _lanczos_p[i]/(x+i) + t = x + _lanczos_g + 0.5 + return 2.506628274631000502417 * t**(x+0.5) * cmath.exp(-t) * r + +gamma = _mathfun_real(_gamma_real, _gamma_complex) + +def rgamma(x): + try: + return 1./gamma(x) + except ZeroDivisionError: + return x*0.0 + +def factorial(x): + return gamma(x+1.0) + +def arg(x): + if type(x) is float: + return math.atan2(0.0,x) + return math.atan2(x.imag,x.real) + +# XXX: broken for negatives +def loggamma(x): + if type(x) not in (float, complex): + try: + x = float(x) + except (ValueError, TypeError): + x = complex(x) + try: + xreal = x.real + ximag = x.imag + except AttributeError: # py2.5 + xreal = x + ximag = 0.0 + # Reflection formula + # http://functions.wolfram.com/GammaBetaErf/LogGamma/16/01/01/0003/ + if xreal < 0.0: + if abs(x) < 0.5: + v = log(gamma(x)) + if ximag == 0: + v = v.conjugate() + return v + z = 1-x + try: + re = z.real + im = z.imag + except AttributeError: # py2.5 + re = z + im = 0.0 + refloor = floor(re) + if im == 0.0: + imsign = 0 + elif im < 0.0: + imsign = -1 + else: + imsign = 1 + return (-pi*1j)*abs(refloor)*(1-abs(imsign)) + logpi - \ + log(sinpi(z-refloor)) - loggamma(z) + 1j*pi*refloor*imsign + if x == 1.0 or x == 2.0: + return x*0 + p = 0. + while abs(x) < 11: + p -= log(x) + x += 1.0 + s = 0.918938533204672742 + (x-0.5)*log(x) - x + r = 1./x + r2 = r*r + s += 0.083333333333333333333*r; r *= r2 + s += -0.0027777777777777777778*r; r *= r2 + s += 0.00079365079365079365079*r; r *= r2 + s += -0.0005952380952380952381*r; r *= r2 + s += 0.00084175084175084175084*r; r *= r2 + s += -0.0019175269175269175269*r; r *= r2 + s += 0.0064102564102564102564*r; r *= r2 + s += -0.02955065359477124183*r + return s + p + +_psi_coeff = [ +0.083333333333333333333, +-0.0083333333333333333333, +0.003968253968253968254, +-0.0041666666666666666667, +0.0075757575757575757576, +-0.021092796092796092796, +0.083333333333333333333, +-0.44325980392156862745, +3.0539543302701197438, +-26.456212121212121212] + +def _digamma_real(x): + _intx = int(x) + if _intx == x: + if _intx <= 0: + raise ZeroDivisionError("polygamma pole") + if x < 0.5: + x = 1.0-x + s = pi*cotpi(x) + else: + s = 0.0 + while x < 10.0: + s -= 1.0/x + x += 1.0 + x2 = x**-2 + t = x2 + for c in _psi_coeff: + s -= c*t + if t < 1e-20: + break + t *= x2 + return s + math_log(x) - 0.5/x + +def _digamma_complex(x): + if not x.imag: + return complex(_digamma_real(x.real)) + if x.real < 0.5: + x = 1.0-x + s = pi*cotpi(x) + else: + s = 0.0 + while abs(x) < 10.0: + s -= 1.0/x + x += 1.0 + x2 = x**-2 + t = x2 + for c in _psi_coeff: + s -= c*t + if abs(t) < 1e-20: + break + t *= x2 + return s + cmath.log(x) - 0.5/x + +digamma = _mathfun_real(_digamma_real, _digamma_complex) + +# TODO: could implement complex erf and erfc here. Need +# to find an accurate method (avoiding cancellation) +# for approx. 1 < abs(x) < 9. + +_erfc_coeff_P = [ + 1.0000000161203922312, + 2.1275306946297962644, + 2.2280433377390253297, + 1.4695509105618423961, + 0.66275911699770787537, + 0.20924776504163751585, + 0.045459713768411264339, + 0.0063065951710717791934, + 0.00044560259661560421715][::-1] + +_erfc_coeff_Q = [ + 1.0000000000000000000, + 3.2559100272784894318, + 4.9019435608903239131, + 4.4971472894498014205, + 2.7845640601891186528, + 1.2146026030046904138, + 0.37647108453729465912, + 0.080970149639040548613, + 0.011178148899483545902, + 0.00078981003831980423513][::-1] + +def _polyval(coeffs, x): + p = coeffs[0] + for c in coeffs[1:]: + p = c + x*p + return p + +def _erf_taylor(x): + # Taylor series assuming 0 <= x <= 1 + x2 = x*x + s = t = x + n = 1 + while abs(t) > 1e-17: + t *= x2/n + s -= t/(n+n+1) + n += 1 + t *= x2/n + s += t/(n+n+1) + n += 1 + return 1.1283791670955125739*s + +def _erfc_mid(x): + # Rational approximation assuming 0 <= x <= 9 + return exp(-x*x)*_polyval(_erfc_coeff_P,x)/_polyval(_erfc_coeff_Q,x) + +def _erfc_asymp(x): + # Asymptotic expansion assuming x >= 9 + x2 = x*x + v = exp(-x2)/x*0.56418958354775628695 + r = t = 0.5 / x2 + s = 1.0 + for n in range(1,22,4): + s -= t + t *= r * (n+2) + s += t + t *= r * (n+4) + if abs(t) < 1e-17: + break + return s * v + +def erf(x): + """ + erf of a real number. + """ + x = float(x) + if x != x: + return x + if x < 0.0: + return -erf(-x) + if x >= 1.0: + if x >= 6.0: + return 1.0 + return 1.0 - _erfc_mid(x) + return _erf_taylor(x) + +def erfc(x): + """ + erfc of a real number. + """ + x = float(x) + if x != x: + return x + if x < 0.0: + if x < -6.0: + return 2.0 + return 2.0-erfc(-x) + if x > 9.0: + return _erfc_asymp(x) + if x >= 1.0: + return _erfc_mid(x) + return 1.0 - _erf_taylor(x) + +gauss42 = [\ +(0.99839961899006235, 0.0041059986046490839), +(-0.99839961899006235, 0.0041059986046490839), +(0.9915772883408609, 0.009536220301748501), +(-0.9915772883408609,0.009536220301748501), +(0.97934250806374812, 0.014922443697357493), +(-0.97934250806374812, 0.014922443697357493), +(0.96175936533820439,0.020227869569052644), +(-0.96175936533820439, 0.020227869569052644), +(0.93892355735498811, 0.025422959526113047), +(-0.93892355735498811,0.025422959526113047), +(0.91095972490412735, 0.030479240699603467), +(-0.91095972490412735, 0.030479240699603467), +(0.87802056981217269,0.03536907109759211), +(-0.87802056981217269, 0.03536907109759211), +(0.8402859832618168, 0.040065735180692258), +(-0.8402859832618168,0.040065735180692258), +(0.7979620532554873, 0.044543577771965874), +(-0.7979620532554873, 0.044543577771965874), +(0.75127993568948048,0.048778140792803244), +(-0.75127993568948048, 0.048778140792803244), +(0.70049459055617114, 0.052746295699174064), +(-0.70049459055617114,0.052746295699174064), +(0.64588338886924779, 0.056426369358018376), +(-0.64588338886924779, 0.056426369358018376), +(0.58774459748510932, 0.059798262227586649), +(-0.58774459748510932, 0.059798262227586649), +(0.5263957499311922, 0.062843558045002565), +(-0.5263957499311922, 0.062843558045002565), +(0.46217191207042191, 0.065545624364908975), +(-0.46217191207042191, 0.065545624364908975), +(0.39542385204297503, 0.067889703376521934), +(-0.39542385204297503, 0.067889703376521934), +(0.32651612446541151, 0.069862992492594159), +(-0.32651612446541151, 0.069862992492594159), +(0.25582507934287907, 0.071454714265170971), +(-0.25582507934287907, 0.071454714265170971), +(0.18373680656485453, 0.072656175243804091), +(-0.18373680656485453, 0.072656175243804091), +(0.11064502720851986, 0.073460813453467527), +(-0.11064502720851986, 0.073460813453467527), +(0.036948943165351772, 0.073864234232172879), +(-0.036948943165351772, 0.073864234232172879)] + +EI_ASYMP_CONVERGENCE_RADIUS = 40.0 + +def ei_asymp(z, _e1=False): + r = 1./z + s = t = 1.0 + k = 1 + while 1: + t *= k*r + s += t + if abs(t) < 1e-16: + break + k += 1 + v = s*exp(z)/z + if _e1: + if type(z) is complex: + zreal = z.real + zimag = z.imag + else: + zreal = z + zimag = 0.0 + if zimag == 0.0 and zreal > 0.0: + v += pi*1j + else: + if type(z) is complex: + if z.imag > 0: + v += pi*1j + if z.imag < 0: + v -= pi*1j + return v + +def ei_taylor(z, _e1=False): + s = t = z + k = 2 + while 1: + t = t*z/k + term = t/k + if abs(term) < 1e-17: + break + s += term + k += 1 + s += euler + if _e1: + s += log(-z) + else: + if type(z) is float or z.imag == 0.0: + s += math_log(abs(z)) + else: + s += cmath.log(z) + return s + +def ei(z, _e1=False): + typez = type(z) + if typez not in (float, complex): + try: + z = float(z) + typez = float + except (TypeError, ValueError): + z = complex(z) + typez = complex + if not z: + return -INF + absz = abs(z) + if absz > EI_ASYMP_CONVERGENCE_RADIUS: + return ei_asymp(z, _e1) + elif absz <= 2.0 or (typez is float and z > 0.0): + return ei_taylor(z, _e1) + # Integrate, starting from whichever is smaller of a Taylor + # series value or an asymptotic series value + if typez is complex and z.real > 0.0: + zref = z / absz + ref = ei_taylor(zref, _e1) + else: + zref = EI_ASYMP_CONVERGENCE_RADIUS * z / absz + ref = ei_asymp(zref, _e1) + C = (zref-z)*0.5 + D = (zref+z)*0.5 + s = 0.0 + if type(z) is complex: + _exp = cmath.exp + else: + _exp = math.exp + for x,w in gauss42: + t = C*x+D + s += w*_exp(t)/t + ref -= C*s + return ref + +def e1(z): + # hack to get consistent signs if the imaginary part if 0 + # and signed + typez = type(z) + if type(z) not in (float, complex): + try: + z = float(z) + typez = float + except (TypeError, ValueError): + z = complex(z) + typez = complex + if typez is complex and not z.imag: + z = complex(z.real, 0.0) + # end hack + return -ei(-z, _e1=True) + +_zeta_int = [\ +-0.5, +0.0, +1.6449340668482264365,1.2020569031595942854,1.0823232337111381915, +1.0369277551433699263,1.0173430619844491397,1.0083492773819228268, +1.0040773561979443394,1.0020083928260822144,1.0009945751278180853, +1.0004941886041194646,1.0002460865533080483,1.0001227133475784891, +1.0000612481350587048,1.0000305882363070205,1.0000152822594086519, +1.0000076371976378998,1.0000038172932649998,1.0000019082127165539, +1.0000009539620338728,1.0000004769329867878,1.0000002384505027277, +1.0000001192199259653,1.0000000596081890513,1.0000000298035035147, +1.0000000149015548284] + +_zeta_P = [-3.50000000087575873, -0.701274355654678147, +-0.0672313458590012612, -0.00398731457954257841, +-0.000160948723019303141, -4.67633010038383371e-6, +-1.02078104417700585e-7, -1.68030037095896287e-9, +-1.85231868742346722e-11][::-1] + +_zeta_Q = [1.00000000000000000, -0.936552848762465319, +-0.0588835413263763741, -0.00441498861482948666, +-0.000143416758067432622, -5.10691659585090782e-6, +-9.58813053268913799e-8, -1.72963791443181972e-9, +-1.83527919681474132e-11][::-1] + +_zeta_1 = [3.03768838606128127e-10, -1.21924525236601262e-8, +2.01201845887608893e-7, -1.53917240683468381e-6, +-5.09890411005967954e-7, 0.000122464707271619326, +-0.000905721539353130232, -0.00239315326074843037, +0.084239750013159168, 0.418938517907442414, 0.500000001921884009] + +_zeta_0 = [-3.46092485016748794e-10, -6.42610089468292485e-9, +1.76409071536679773e-7, -1.47141263991560698e-6, -6.38880222546167613e-7, +0.000122641099800668209, -0.000905894913516772796, -0.00239303348507992713, +0.0842396947501199816, 0.418938533204660256, 0.500000000000000052] + +def zeta(s): + """ + Riemann zeta function, real argument + """ + if not isinstance(s, (float, int)): + try: + s = float(s) + except (ValueError, TypeError): + try: + s = complex(s) + if not s.imag: + return complex(zeta(s.real)) + except (ValueError, TypeError): + pass + raise NotImplementedError + if s == 1: + raise ValueError("zeta(1) pole") + if s >= 27: + return 1.0 + 2.0**(-s) + 3.0**(-s) + n = int(s) + if n == s: + if n >= 0: + return _zeta_int[n] + if not (n % 2): + return 0.0 + if s <= 0.0: + return 2.**s*pi**(s-1)*_sinpi_real(0.5*s)*_gamma_real(1-s)*zeta(1-s) + if s <= 2.0: + if s <= 1.0: + return _polyval(_zeta_0,s)/(s-1) + return _polyval(_zeta_1,s)/(s-1) + z = _polyval(_zeta_P,s) / _polyval(_zeta_Q,s) + return 1.0 + 2.0**(-s) + 3.0**(-s) + 4.0**(-s)*z diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/matrices/__init__.py b/env-llmeval/lib/python3.10/site-packages/mpmath/matrices/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..293697b9fcf8bd82d58ac4ff45acd73fadac82f9 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/matrices/__init__.py @@ -0,0 +1,2 @@ +from . import eigen # to set methods +from . import eigen_symmetric # to set methods diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/matrices/eigen_symmetric.py b/env-llmeval/lib/python3.10/site-packages/mpmath/matrices/eigen_symmetric.py new file mode 100644 index 0000000000000000000000000000000000000000..c82c0bb061d22c37a89f82a0b9bdab3e9ba7ddde --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/matrices/eigen_symmetric.py @@ -0,0 +1,1807 @@ +#!/usr/bin/python +# -*- coding: utf-8 -*- + +################################################################################################## +# module for the symmetric eigenvalue problem +# Copyright 2013 Timo Hartmann (thartmann15 at gmail.com) +# +# todo: +# - implement balancing +# +################################################################################################## + +""" +The symmetric eigenvalue problem. +--------------------------------- + +This file contains routines for the symmetric eigenvalue problem. + +high level routines: + + eigsy : real symmetric (ordinary) eigenvalue problem + eighe : complex hermitian (ordinary) eigenvalue problem + eigh : unified interface for eigsy and eighe + svd_r : singular value decomposition for real matrices + svd_c : singular value decomposition for complex matrices + svd : unified interface for svd_r and svd_c + + +low level routines: + + r_sy_tridiag : reduction of real symmetric matrix to real symmetric tridiagonal matrix + c_he_tridiag_0 : reduction of complex hermitian matrix to real symmetric tridiagonal matrix + c_he_tridiag_1 : auxiliary routine to c_he_tridiag_0 + c_he_tridiag_2 : auxiliary routine to c_he_tridiag_0 + tridiag_eigen : solves the real symmetric tridiagonal matrix eigenvalue problem + svd_r_raw : raw singular value decomposition for real matrices + svd_c_raw : raw singular value decomposition for complex matrices +""" + +from ..libmp.backend import xrange +from .eigen import defun + + +def r_sy_tridiag(ctx, A, D, E, calc_ev = True): + """ + This routine transforms a real symmetric matrix A to a real symmetric + tridiagonal matrix T using an orthogonal similarity transformation: + Q' * A * Q = T (here ' denotes the matrix transpose). + The orthogonal matrix Q is build up from Householder reflectors. + + parameters: + A (input/output) On input, A contains the real symmetric matrix of + dimension (n,n). On output, if calc_ev is true, A contains the + orthogonal matrix Q, otherwise A is destroyed. + + D (output) real array of length n, contains the diagonal elements + of the tridiagonal matrix + + E (output) real array of length n, contains the offdiagonal elements + of the tridiagonal matrix in E[0:(n-1)] where is the dimension of + the matrix A. E[n-1] is undefined. + + calc_ev (input) If calc_ev is true, this routine explicitly calculates the + orthogonal matrix Q which is then returned in A. If calc_ev is + false, Q is not explicitly calculated resulting in a shorter run time. + + This routine is a python translation of the fortran routine tred2.f in the + software library EISPACK (see netlib.org) which itself is based on the algol + procedure tred2 described in: + - Num. Math. 11, p.181-195 (1968) by Martin, Reinsch and Wilkonson + - Handbook for auto. comp., Vol II, Linear Algebra, p.212-226 (1971) + + For a good introduction to Householder reflections, see also + Stoer, Bulirsch - Introduction to Numerical Analysis. + """ + + # note : the vector v of the i-th houshoulder reflector is stored in a[(i+1):,i] + # whereas v/ is stored in a[i,(i+1):] + + n = A.rows + for i in xrange(n - 1, 0, -1): + # scale the vector + + scale = 0 + for k in xrange(0, i): + scale += abs(A[k,i]) + + scale_inv = 0 + if scale != 0: + scale_inv = 1/scale + + # sadly there are floating point numbers not equal to zero whose reciprocal is infinity + + if i == 1 or scale == 0 or ctx.isinf(scale_inv): + E[i] = A[i-1,i] # nothing to do + D[i] = 0 + continue + + # calculate parameters for housholder transformation + + H = 0 + for k in xrange(0, i): + A[k,i] *= scale_inv + H += A[k,i] * A[k,i] + + F = A[i-1,i] + G = ctx.sqrt(H) + if F > 0: + G = -G + E[i] = scale * G + H -= F * G + A[i-1,i] = F - G + F = 0 + + # apply housholder transformation + + for j in xrange(0, i): + if calc_ev: + A[i,j] = A[j,i] / H + + G = 0 # calculate A*U + for k in xrange(0, j + 1): + G += A[k,j] * A[k,i] + for k in xrange(j + 1, i): + G += A[j,k] * A[k,i] + + E[j] = G / H # calculate P + F += E[j] * A[j,i] + + HH = F / (2 * H) + + for j in xrange(0, i): # calculate reduced A + F = A[j,i] + G = E[j] - HH * F # calculate Q + E[j] = G + + for k in xrange(0, j + 1): + A[k,j] -= F * E[k] + G * A[k,i] + + D[i] = H + + for i in xrange(1, n): # better for compatibility + E[i-1] = E[i] + E[n-1] = 0 + + if calc_ev: + D[0] = 0 + for i in xrange(0, n): + if D[i] != 0: + for j in xrange(0, i): # accumulate transformation matrices + G = 0 + for k in xrange(0, i): + G += A[i,k] * A[k,j] + for k in xrange(0, i): + A[k,j] -= G * A[k,i] + + D[i] = A[i,i] + A[i,i] = 1 + + for j in xrange(0, i): + A[j,i] = A[i,j] = 0 + else: + for i in xrange(0, n): + D[i] = A[i,i] + + + + + +def c_he_tridiag_0(ctx, A, D, E, T): + """ + This routine transforms a complex hermitian matrix A to a real symmetric + tridiagonal matrix T using an unitary similarity transformation: + Q' * A * Q = T (here ' denotes the hermitian matrix transpose, + i.e. transposition und conjugation). + The unitary matrix Q is build up from Householder reflectors and + an unitary diagonal matrix. + + parameters: + A (input/output) On input, A contains the complex hermitian matrix + of dimension (n,n). On output, A contains the unitary matrix Q + in compressed form. + + D (output) real array of length n, contains the diagonal elements + of the tridiagonal matrix. + + E (output) real array of length n, contains the offdiagonal elements + of the tridiagonal matrix in E[0:(n-1)] where is the dimension of + the matrix A. E[n-1] is undefined. + + T (output) complex array of length n, contains a unitary diagonal + matrix. + + This routine is a python translation (in slightly modified form) of the fortran + routine htridi.f in the software library EISPACK (see netlib.org) which itself + is a complex version of the algol procedure tred1 described in: + - Num. Math. 11, p.181-195 (1968) by Martin, Reinsch and Wilkonson + - Handbook for auto. comp., Vol II, Linear Algebra, p.212-226 (1971) + + For a good introduction to Householder reflections, see also + Stoer, Bulirsch - Introduction to Numerical Analysis. + """ + + n = A.rows + T[n-1] = 1 + for i in xrange(n - 1, 0, -1): + + # scale the vector + + scale = 0 + for k in xrange(0, i): + scale += abs(ctx.re(A[k,i])) + abs(ctx.im(A[k,i])) + + scale_inv = 0 + if scale != 0: + scale_inv = 1 / scale + + # sadly there are floating point numbers not equal to zero whose reciprocal is infinity + + if scale == 0 or ctx.isinf(scale_inv): + E[i] = 0 + D[i] = 0 + T[i-1] = 1 + continue + + if i == 1: + F = A[i-1,i] + f = abs(F) + E[i] = f + D[i] = 0 + if f != 0: + T[i-1] = T[i] * F / f + else: + T[i-1] = T[i] + continue + + # calculate parameters for housholder transformation + + H = 0 + for k in xrange(0, i): + A[k,i] *= scale_inv + rr = ctx.re(A[k,i]) + ii = ctx.im(A[k,i]) + H += rr * rr + ii * ii + + F = A[i-1,i] + f = abs(F) + G = ctx.sqrt(H) + H += G * f + E[i] = scale * G + if f != 0: + F = F / f + TZ = - T[i] * F # T[i-1]=-T[i]*F, but we need T[i-1] as temporary storage + G *= F + else: + TZ = -T[i] # T[i-1]=-T[i] + A[i-1,i] += G + F = 0 + + # apply housholder transformation + + for j in xrange(0, i): + A[i,j] = A[j,i] / H + + G = 0 # calculate A*U + for k in xrange(0, j + 1): + G += ctx.conj(A[k,j]) * A[k,i] + for k in xrange(j + 1, i): + G += A[j,k] * A[k,i] + + T[j] = G / H # calculate P + F += ctx.conj(T[j]) * A[j,i] + + HH = F / (2 * H) + + for j in xrange(0, i): # calculate reduced A + F = A[j,i] + G = T[j] - HH * F # calculate Q + T[j] = G + + for k in xrange(0, j + 1): + A[k,j] -= ctx.conj(F) * T[k] + ctx.conj(G) * A[k,i] + # as we use the lower left part for storage + # we have to use the transpose of the normal formula + + T[i-1] = TZ + D[i] = H + + for i in xrange(1, n): # better for compatibility + E[i-1] = E[i] + E[n-1] = 0 + + D[0] = 0 + for i in xrange(0, n): + zw = D[i] + D[i] = ctx.re(A[i,i]) + A[i,i] = zw + + + + + + + +def c_he_tridiag_1(ctx, A, T): + """ + This routine forms the unitary matrix Q described in c_he_tridiag_0. + + parameters: + A (input/output) On input, A is the same matrix as delivered by + c_he_tridiag_0. On output, A is set to Q. + + T (input) On input, T is the same array as delivered by c_he_tridiag_0. + + """ + + n = A.rows + + for i in xrange(0, n): + if A[i,i] != 0: + for j in xrange(0, i): + G = 0 + for k in xrange(0, i): + G += ctx.conj(A[i,k]) * A[k,j] + for k in xrange(0, i): + A[k,j] -= G * A[k,i] + + A[i,i] = 1 + + for j in xrange(0, i): + A[j,i] = A[i,j] = 0 + + for i in xrange(0, n): + for k in xrange(0, n): + A[i,k] *= T[k] + + + + +def c_he_tridiag_2(ctx, A, T, B): + """ + This routine applied the unitary matrix Q described in c_he_tridiag_0 + onto the the matrix B, i.e. it forms Q*B. + + parameters: + A (input) On input, A is the same matrix as delivered by c_he_tridiag_0. + + T (input) On input, T is the same array as delivered by c_he_tridiag_0. + + B (input/output) On input, B is a complex matrix. On output B is replaced + by Q*B. + + This routine is a python translation of the fortran routine htribk.f in the + software library EISPACK (see netlib.org). See c_he_tridiag_0 for more + references. + """ + + n = A.rows + + for i in xrange(0, n): + for k in xrange(0, n): + B[k,i] *= T[k] + + for i in xrange(0, n): + if A[i,i] != 0: + for j in xrange(0, n): + G = 0 + for k in xrange(0, i): + G += ctx.conj(A[i,k]) * B[k,j] + for k in xrange(0, i): + B[k,j] -= G * A[k,i] + + + + + +def tridiag_eigen(ctx, d, e, z = False): + """ + This subroutine find the eigenvalues and the first components of the + eigenvectors of a real symmetric tridiagonal matrix using the implicit + QL method. + + parameters: + + d (input/output) real array of length n. on input, d contains the diagonal + elements of the input matrix. on output, d contains the eigenvalues in + ascending order. + + e (input) real array of length n. on input, e contains the offdiagonal + elements of the input matrix in e[0:(n-1)]. On output, e has been + destroyed. + + z (input/output) If z is equal to False, no eigenvectors will be computed. + Otherwise on input z should have the format z[0:m,0:n] (i.e. a real or + complex matrix of dimension (m,n) ). On output this matrix will be + multiplied by the matrix of the eigenvectors (i.e. the columns of this + matrix are the eigenvectors): z --> z*EV + That means if z[i,j]={1 if j==j; 0 otherwise} on input, then on output + z will contain the first m components of the eigenvectors. That means + if m is equal to n, the i-th eigenvector will be z[:,i]. + + This routine is a python translation (in slightly modified form) of the + fortran routine imtql2.f in the software library EISPACK (see netlib.org) + which itself is based on the algol procudure imtql2 desribed in: + - num. math. 12, p. 377-383(1968) by matrin and wilkinson + - modified in num. math. 15, p. 450(1970) by dubrulle + - handbook for auto. comp., vol. II-linear algebra, p. 241-248 (1971) + See also the routine gaussq.f in netlog.org or acm algorithm 726. + """ + + n = len(d) + e[n-1] = 0 + iterlim = 2 * ctx.dps + + for l in xrange(n): + j = 0 + while 1: + m = l + while 1: + # look for a small subdiagonal element + if m + 1 == n: + break + if abs(e[m]) <= ctx.eps * (abs(d[m]) + abs(d[m + 1])): + break + m = m + 1 + if m == l: + break + + if j >= iterlim: + raise RuntimeError("tridiag_eigen: no convergence to an eigenvalue after %d iterations" % iterlim) + + j += 1 + + # form shift + + p = d[l] + g = (d[l + 1] - p) / (2 * e[l]) + r = ctx.hypot(g, 1) + + if g < 0: + s = g - r + else: + s = g + r + + g = d[m] - p + e[l] / s + + s, c, p = 1, 1, 0 + + for i in xrange(m - 1, l - 1, -1): + f = s * e[i] + b = c * e[i] + if abs(f) > abs(g): # this here is a slight improvement also used in gaussq.f or acm algorithm 726. + c = g / f + r = ctx.hypot(c, 1) + e[i + 1] = f * r + s = 1 / r + c = c * s + else: + s = f / g + r = ctx.hypot(s, 1) + e[i + 1] = g * r + c = 1 / r + s = s * c + g = d[i + 1] - p + r = (d[i] - g) * s + 2 * c * b + p = s * r + d[i + 1] = g + p + g = c * r - b + + if not isinstance(z, bool): + # calculate eigenvectors + for w in xrange(z.rows): + f = z[w,i+1] + z[w,i+1] = s * z[w,i] + c * f + z[w,i ] = c * z[w,i] - s * f + + d[l] = d[l] - p + e[l] = g + e[m] = 0 + + for ii in xrange(1, n): + # sort eigenvalues and eigenvectors (bubble-sort) + i = ii - 1 + k = i + p = d[i] + for j in xrange(ii, n): + if d[j] >= p: + continue + k = j + p = d[k] + if k == i: + continue + d[k] = d[i] + d[i] = p + + if not isinstance(z, bool): + for w in xrange(z.rows): + p = z[w,i] + z[w,i] = z[w,k] + z[w,k] = p + +######################################################################################## + +@defun +def eigsy(ctx, A, eigvals_only = False, overwrite_a = False): + """ + This routine solves the (ordinary) eigenvalue problem for a real symmetric + square matrix A. Given A, an orthogonal matrix Q is calculated which + diagonalizes A: + + Q' A Q = diag(E) and Q Q' = Q' Q = 1 + + Here diag(E) is a diagonal matrix whose diagonal is E. + ' denotes the transpose. + + The columns of Q are the eigenvectors of A and E contains the eigenvalues: + + A Q[:,i] = E[i] Q[:,i] + + + input: + + A: real matrix of format (n,n) which is symmetric + (i.e. A=A' or A[i,j]=A[j,i]) + + eigvals_only: if true, calculates only the eigenvalues E. + if false, calculates both eigenvectors and eigenvalues. + + overwrite_a: if true, allows modification of A which may improve + performance. if false, A is not modified. + + output: + + E: vector of format (n). contains the eigenvalues of A in ascending order. + + Q: orthogonal matrix of format (n,n). contains the eigenvectors + of A as columns. + + return value: + + E if eigvals_only is true + (E, Q) if eigvals_only is false + + example: + >>> from mpmath import mp + >>> A = mp.matrix([[3, 2], [2, 0]]) + >>> E = mp.eigsy(A, eigvals_only = True) + >>> print(E) + [-1.0] + [ 4.0] + + >>> A = mp.matrix([[1, 2], [2, 3]]) + >>> E, Q = mp.eigsy(A) + >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) + [0.0] + [0.0] + + see also: eighe, eigh, eig + """ + + if not overwrite_a: + A = A.copy() + + d = ctx.zeros(A.rows, 1) + e = ctx.zeros(A.rows, 1) + + if eigvals_only: + r_sy_tridiag(ctx, A, d, e, calc_ev = False) + tridiag_eigen(ctx, d, e, False) + return d + else: + r_sy_tridiag(ctx, A, d, e, calc_ev = True) + tridiag_eigen(ctx, d, e, A) + return (d, A) + + +@defun +def eighe(ctx, A, eigvals_only = False, overwrite_a = False): + """ + This routine solves the (ordinary) eigenvalue problem for a complex + hermitian square matrix A. Given A, an unitary matrix Q is calculated which + diagonalizes A: + + Q' A Q = diag(E) and Q Q' = Q' Q = 1 + + Here diag(E) a is diagonal matrix whose diagonal is E. + ' denotes the hermitian transpose (i.e. ordinary transposition and + complex conjugation). + + The columns of Q are the eigenvectors of A and E contains the eigenvalues: + + A Q[:,i] = E[i] Q[:,i] + + + input: + + A: complex matrix of format (n,n) which is hermitian + (i.e. A=A' or A[i,j]=conj(A[j,i])) + + eigvals_only: if true, calculates only the eigenvalues E. + if false, calculates both eigenvectors and eigenvalues. + + overwrite_a: if true, allows modification of A which may improve + performance. if false, A is not modified. + + output: + + E: vector of format (n). contains the eigenvalues of A in ascending order. + + Q: unitary matrix of format (n,n). contains the eigenvectors + of A as columns. + + return value: + + E if eigvals_only is true + (E, Q) if eigvals_only is false + + example: + >>> from mpmath import mp + >>> A = mp.matrix([[1, -3 - 1j], [-3 + 1j, -2]]) + >>> E = mp.eighe(A, eigvals_only = True) + >>> print(E) + [-4.0] + [ 3.0] + + >>> A = mp.matrix([[1, 2 + 5j], [2 - 5j, 3]]) + >>> E, Q = mp.eighe(A) + >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) + [0.0] + [0.0] + + see also: eigsy, eigh, eig + """ + + if not overwrite_a: + A = A.copy() + + d = ctx.zeros(A.rows, 1) + e = ctx.zeros(A.rows, 1) + t = ctx.zeros(A.rows, 1) + + if eigvals_only: + c_he_tridiag_0(ctx, A, d, e, t) + tridiag_eigen(ctx, d, e, False) + return d + else: + c_he_tridiag_0(ctx, A, d, e, t) + B = ctx.eye(A.rows) + tridiag_eigen(ctx, d, e, B) + c_he_tridiag_2(ctx, A, t, B) + return (d, B) + +@defun +def eigh(ctx, A, eigvals_only = False, overwrite_a = False): + """ + "eigh" is a unified interface for "eigsy" and "eighe". Depending on + whether A is real or complex the appropriate function is called. + + This routine solves the (ordinary) eigenvalue problem for a real symmetric + or complex hermitian square matrix A. Given A, an orthogonal (A real) or + unitary (A complex) matrix Q is calculated which diagonalizes A: + + Q' A Q = diag(E) and Q Q' = Q' Q = 1 + + Here diag(E) a is diagonal matrix whose diagonal is E. + ' denotes the hermitian transpose (i.e. ordinary transposition and + complex conjugation). + + The columns of Q are the eigenvectors of A and E contains the eigenvalues: + + A Q[:,i] = E[i] Q[:,i] + + input: + + A: a real or complex square matrix of format (n,n) which is symmetric + (i.e. A[i,j]=A[j,i]) or hermitian (i.e. A[i,j]=conj(A[j,i])). + + eigvals_only: if true, calculates only the eigenvalues E. + if false, calculates both eigenvectors and eigenvalues. + + overwrite_a: if true, allows modification of A which may improve + performance. if false, A is not modified. + + output: + + E: vector of format (n). contains the eigenvalues of A in ascending order. + + Q: an orthogonal or unitary matrix of format (n,n). contains the + eigenvectors of A as columns. + + return value: + + E if eigvals_only is true + (E, Q) if eigvals_only is false + + example: + >>> from mpmath import mp + >>> A = mp.matrix([[3, 2], [2, 0]]) + >>> E = mp.eigh(A, eigvals_only = True) + >>> print(E) + [-1.0] + [ 4.0] + + >>> A = mp.matrix([[1, 2], [2, 3]]) + >>> E, Q = mp.eigh(A) + >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) + [0.0] + [0.0] + + >>> A = mp.matrix([[1, 2 + 5j], [2 - 5j, 3]]) + >>> E, Q = mp.eigh(A) + >>> print(mp.chop(A * Q[:,0] - E[0] * Q[:,0])) + [0.0] + [0.0] + + see also: eigsy, eighe, eig + """ + + iscomplex = any(type(x) is ctx.mpc for x in A) + + if iscomplex: + return ctx.eighe(A, eigvals_only = eigvals_only, overwrite_a = overwrite_a) + else: + return ctx.eigsy(A, eigvals_only = eigvals_only, overwrite_a = overwrite_a) + + +@defun +def gauss_quadrature(ctx, n, qtype = "legendre", alpha = 0, beta = 0): + """ + This routine calulates gaussian quadrature rules for different + families of orthogonal polynomials. Let (a, b) be an interval, + W(x) a positive weight function and n a positive integer. + Then the purpose of this routine is to calculate pairs (x_k, w_k) + for k=0, 1, 2, ... (n-1) which give + + int(W(x) * F(x), x = a..b) = sum(w_k * F(x_k),k = 0..(n-1)) + + exact for all polynomials F(x) of degree (strictly) less than 2*n. For all + integrable functions F(x) the sum is a (more or less) good approximation to + the integral. The x_k are called nodes (which are the zeros of the + related orthogonal polynomials) and the w_k are called the weights. + + parameters + n (input) The degree of the quadrature rule, i.e. its number of + nodes. + + qtype (input) The family of orthogonal polynmomials for which to + compute the quadrature rule. See the list below. + + alpha (input) real number, used as parameter for some orthogonal + polynomials + + beta (input) real number, used as parameter for some orthogonal + polynomials. + + return value + + (X, W) a pair of two real arrays where x_k = X[k] and w_k = W[k]. + + + orthogonal polynomials: + + qtype polynomial + ----- ---------- + + "legendre" Legendre polynomials, W(x)=1 on the interval (-1, +1) + "legendre01" shifted Legendre polynomials, W(x)=1 on the interval (0, +1) + "hermite" Hermite polynomials, W(x)=exp(-x*x) on (-infinity,+infinity) + "laguerre" Laguerre polynomials, W(x)=exp(-x) on (0,+infinity) + "glaguerre" generalized Laguerre polynomials, W(x)=exp(-x)*x**alpha + on (0, +infinity) + "chebyshev1" Chebyshev polynomials of the first kind, W(x)=1/sqrt(1-x*x) + on (-1, +1) + "chebyshev2" Chebyshev polynomials of the second kind, W(x)=sqrt(1-x*x) + on (-1, +1) + "jacobi" Jacobi polynomials, W(x)=(1-x)**alpha * (1+x)**beta on (-1, +1) + with alpha>-1 and beta>-1 + + examples: + >>> from mpmath import mp + >>> f = lambda x: x**8 + 2 * x**6 - 3 * x**4 + 5 * x**2 - 7 + >>> X, W = mp.gauss_quadrature(5, "hermite") + >>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)]) + >>> B = mp.sqrt(mp.pi) * 57 / 16 + >>> C = mp.quad(lambda x: mp.exp(- x * x) * f(x), [-mp.inf, +mp.inf]) + >>> mp.nprint((mp.chop(A-B, tol = 1e-10), mp.chop(A-C, tol = 1e-10))) + (0.0, 0.0) + + >>> f = lambda x: x**5 - 2 * x**4 + 3 * x**3 - 5 * x**2 + 7 * x - 11 + >>> X, W = mp.gauss_quadrature(3, "laguerre") + >>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)]) + >>> B = 76 + >>> C = mp.quad(lambda x: mp.exp(-x) * f(x), [0, +mp.inf]) + >>> mp.nprint(mp.chop(A-B, tol = 1e-10), mp.chop(A-C, tol = 1e-10)) + .0 + + # orthogonality of the chebyshev polynomials: + >>> f = lambda x: mp.chebyt(3, x) * mp.chebyt(2, x) + >>> X, W = mp.gauss_quadrature(3, "chebyshev1") + >>> A = mp.fdot([(f(x), w) for x, w in zip(X, W)]) + >>> print(mp.chop(A, tol = 1e-10)) + 0.0 + + references: + - golub and welsch, "calculations of gaussian quadrature rules", mathematics of + computation 23, p. 221-230 (1969) + - golub, "some modified matrix eigenvalue problems", siam review 15, p. 318-334 (1973) + - stroud and secrest, "gaussian quadrature formulas", prentice-hall (1966) + + See also the routine gaussq.f in netlog.org or ACM Transactions on + Mathematical Software algorithm 726. + """ + + d = ctx.zeros(n, 1) + e = ctx.zeros(n, 1) + z = ctx.zeros(1, n) + + z[0,0] = 1 + + if qtype == "legendre": + # legendre on the range -1 +1 , abramowitz, table 25.4, p.916 + w = 2 + for i in xrange(n): + j = i + 1 + e[i] = ctx.sqrt(j * j / (4 * j * j - ctx.mpf(1))) + elif qtype == "legendre01": + # legendre shifted to 0 1 , abramowitz, table 25.8, p.921 + w = 1 + for i in xrange(n): + d[i] = 1 / ctx.mpf(2) + j = i + 1 + e[i] = ctx.sqrt(j * j / (16 * j * j - ctx.mpf(4))) + elif qtype == "hermite": + # hermite on the range -inf +inf , abramowitz, table 25.10,p.924 + w = ctx.sqrt(ctx.pi) + for i in xrange(n): + j = i + 1 + e[i] = ctx.sqrt(j / ctx.mpf(2)) + elif qtype == "laguerre": + # laguerre on the range 0 +inf , abramowitz, table 25.9, p. 923 + w = 1 + for i in xrange(n): + j = i + 1 + d[i] = 2 * j - 1 + e[i] = j + elif qtype=="chebyshev1": + # chebyshev polynimials of the first kind + w = ctx.pi + for i in xrange(n): + e[i] = 1 / ctx.mpf(2) + e[0] = ctx.sqrt(1 / ctx.mpf(2)) + elif qtype == "chebyshev2": + # chebyshev polynimials of the second kind + w = ctx.pi / 2 + for i in xrange(n): + e[i] = 1 / ctx.mpf(2) + elif qtype == "glaguerre": + # generalized laguerre on the range 0 +inf + w = ctx.gamma(1 + alpha) + for i in xrange(n): + j = i + 1 + d[i] = 2 * j - 1 + alpha + e[i] = ctx.sqrt(j * (j + alpha)) + elif qtype == "jacobi": + # jacobi polynomials + alpha = ctx.mpf(alpha) + beta = ctx.mpf(beta) + ab = alpha + beta + abi = ab + 2 + w = (2**(ab+1)) * ctx.gamma(alpha + 1) * ctx.gamma(beta + 1) / ctx.gamma(abi) + d[0] = (beta - alpha) / abi + e[0] = ctx.sqrt(4 * (1 + alpha) * (1 + beta) / ((abi + 1) * (abi * abi))) + a2b2 = beta * beta - alpha * alpha + for i in xrange(1, n): + j = i + 1 + abi = 2 * j + ab + d[i] = a2b2 / ((abi - 2) * abi) + e[i] = ctx.sqrt(4 * j * (j + alpha) * (j + beta) * (j + ab) / ((abi * abi - 1) * abi * abi)) + elif isinstance(qtype, str): + raise ValueError("unknown quadrature rule \"%s\"" % qtype) + elif not isinstance(qtype, str): + w = qtype(d, e) + else: + assert 0 + + tridiag_eigen(ctx, d, e, z) + + for i in xrange(len(z)): + z[i] *= z[i] + + z = z.transpose() + return (d, w * z) + +################################################################################################## +################################################################################################## +################################################################################################## + +def svd_r_raw(ctx, A, V = False, calc_u = False): + """ + This routine computes the singular value decomposition of a matrix A. + Given A, two orthogonal matrices U and V are calculated such that + + A = U S V + + where S is a suitable shaped matrix whose off-diagonal elements are zero. + The diagonal elements of S are the singular values of A, i.e. the + squareroots of the eigenvalues of A' A or A A'. Here ' denotes the transpose. + Householder bidiagonalization and a variant of the QR algorithm is used. + + overview of the matrices : + + A : m*n A gets replaced by U + U : m*n U replaces A. If n>m then only the first m*m block of U is + non-zero. column-orthogonal: U' U = B + here B is a n*n matrix whose first min(m,n) diagonal + elements are 1 and all other elements are zero. + S : n*n diagonal matrix, only the diagonal elements are stored in + the array S. only the first min(m,n) diagonal elements are non-zero. + V : n*n orthogonal: V V' = V' V = 1 + + parameters: + A (input/output) On input, A contains a real matrix of shape m*n. + On output, if calc_u is true A contains the column-orthogonal + matrix U; otherwise A is simply used as workspace and thus destroyed. + + V (input/output) if false, the matrix V is not calculated. otherwise + V must be a matrix of shape n*n. + + calc_u (input) If true, the matrix U is calculated and replaces A. + if false, U is not calculated and A is simply destroyed + + return value: + S an array of length n containing the singular values of A sorted by + decreasing magnitude. only the first min(m,n) elements are non-zero. + + This routine is a python translation of the fortran routine svd.f in the + software library EISPACK (see netlib.org) which itself is based on the + algol procedure svd described in: + - num. math. 14, 403-420(1970) by golub and reinsch. + - wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971). + + """ + + m, n = A.rows, A.cols + + S = ctx.zeros(n, 1) + + # work is a temporary array of size n + work = ctx.zeros(n, 1) + + g = scale = anorm = 0 + maxits = 3 * ctx.dps + + for i in xrange(n): # householder reduction to bidiagonal form + work[i] = scale*g + g = s = scale = 0 + if i < m: + for k in xrange(i, m): + scale += ctx.fabs(A[k,i]) + if scale != 0: + for k in xrange(i, m): + A[k,i] /= scale + s += A[k,i] * A[k,i] + f = A[i,i] + g = -ctx.sqrt(s) + if f < 0: + g = -g + h = f * g - s + A[i,i] = f - g + for j in xrange(i+1, n): + s = 0 + for k in xrange(i, m): + s += A[k,i] * A[k,j] + f = s / h + for k in xrange(i, m): + A[k,j] += f * A[k,i] + for k in xrange(i,m): + A[k,i] *= scale + + S[i] = scale * g + g = s = scale = 0 + + if i < m and i != n - 1: + for k in xrange(i+1, n): + scale += ctx.fabs(A[i,k]) + if scale: + for k in xrange(i+1, n): + A[i,k] /= scale + s += A[i,k] * A[i,k] + f = A[i,i+1] + g = -ctx.sqrt(s) + if f < 0: + g = -g + h = f * g - s + A[i,i+1] = f - g + + for k in xrange(i+1, n): + work[k] = A[i,k] / h + + for j in xrange(i+1, m): + s = 0 + for k in xrange(i+1, n): + s += A[j,k] * A[i,k] + for k in xrange(i+1, n): + A[j,k] += s * work[k] + + for k in xrange(i+1, n): + A[i,k] *= scale + + anorm = max(anorm, ctx.fabs(S[i]) + ctx.fabs(work[i])) + + if not isinstance(V, bool): + for i in xrange(n-2, -1, -1): # accumulation of right hand transformations + V[i+1,i+1] = 1 + + if work[i+1] != 0: + for j in xrange(i+1, n): + V[i,j] = (A[i,j] / A[i,i+1]) / work[i+1] + for j in xrange(i+1, n): + s = 0 + for k in xrange(i+1, n): + s += A[i,k] * V[j,k] + for k in xrange(i+1, n): + V[j,k] += s * V[i,k] + + for j in xrange(i+1, n): + V[j,i] = V[i,j] = 0 + + V[0,0] = 1 + + if m= maxits: + raise RuntimeError("svd: no convergence to an eigenvalue after %d iterations" % its) + + x = S[l] # shift from bottom 2 by 2 minor + nm = k-1 + y = S[nm] + g = work[nm] + h = work[k] + f = ((y - z) * (y + z) + (g - h) * (g + h))/(2 * h * y) + g = ctx.hypot(f, 1) + if f >= 0: f = ((x - z) * (x + z) + h * ((y / (f + g)) - h)) / x + else: f = ((x - z) * (x + z) + h * ((y / (f - g)) - h)) / x + + c = s = 1 # next qt transformation + + for j in xrange(l, nm + 1): + g = work[j+1] + y = S[j+1] + h = s * g + g = c * g + z = ctx.hypot(f, h) + work[j] = z + c = f / z + s = h / z + f = x * c + g * s + g = g * c - x * s + h = y * s + y *= c + if not isinstance(V, bool): + for jj in xrange(n): + x = V[j ,jj] + z = V[j+1,jj] + V[j ,jj]= x * c + z * s + V[j+1 ,jj]= z * c - x * s + z = ctx.hypot(f, h) + S[j] = z + if z != 0: # rotation can be arbitray if z=0 + z = 1 / z + c = f * z + s = h * z + f = c * g + s * y + x = c * y - s * g + + if calc_u: + for jj in xrange(m): + y = A[jj,j ] + z = A[jj,j+1] + A[jj,j ] = y * c + z * s + A[jj,j+1 ] = z * c - y * s + + work[l] = 0 + work[k] = f + S[k] = x + + ########################## + + # Sort singular values into decreasing order (bubble-sort) + + for i in xrange(n): + imax = i + s = ctx.fabs(S[i]) # s is the current maximal element + + for j in xrange(i + 1, n): + c = ctx.fabs(S[j]) + if c > s: + s = c + imax = j + + if imax != i: + # swap singular values + + z = S[i] + S[i] = S[imax] + S[imax] = z + + if calc_u: + for j in xrange(m): + z = A[j,i] + A[j,i] = A[j,imax] + A[j,imax] = z + + if not isinstance(V, bool): + for j in xrange(n): + z = V[i,j] + V[i,j] = V[imax,j] + V[imax,j] = z + + return S + +####################### + +def svd_c_raw(ctx, A, V = False, calc_u = False): + """ + This routine computes the singular value decomposition of a matrix A. + Given A, two unitary matrices U and V are calculated such that + + A = U S V + + where S is a suitable shaped matrix whose off-diagonal elements are zero. + The diagonal elements of S are the singular values of A, i.e. the + squareroots of the eigenvalues of A' A or A A'. Here ' denotes the hermitian + transpose (i.e. transposition and conjugation). Householder bidiagonalization + and a variant of the QR algorithm is used. + + overview of the matrices : + + A : m*n A gets replaced by U + U : m*n U replaces A. If n>m then only the first m*m block of U is + non-zero. column-unitary: U' U = B + here B is a n*n matrix whose first min(m,n) diagonal + elements are 1 and all other elements are zero. + S : n*n diagonal matrix, only the diagonal elements are stored in + the array S. only the first min(m,n) diagonal elements are non-zero. + V : n*n unitary: V V' = V' V = 1 + + parameters: + A (input/output) On input, A contains a complex matrix of shape m*n. + On output, if calc_u is true A contains the column-unitary + matrix U; otherwise A is simply used as workspace and thus destroyed. + + V (input/output) if false, the matrix V is not calculated. otherwise + V must be a matrix of shape n*n. + + calc_u (input) If true, the matrix U is calculated and replaces A. + if false, U is not calculated and A is simply destroyed + + return value: + S an array of length n containing the singular values of A sorted by + decreasing magnitude. only the first min(m,n) elements are non-zero. + + This routine is a python translation of the fortran routine svd.f in the + software library EISPACK (see netlib.org) which itself is based on the + algol procedure svd described in: + - num. math. 14, 403-420(1970) by golub and reinsch. + - wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971). + + """ + + m, n = A.rows, A.cols + + S = ctx.zeros(n, 1) + + # work is a temporary array of size n + work = ctx.zeros(n, 1) + lbeta = ctx.zeros(n, 1) + rbeta = ctx.zeros(n, 1) + dwork = ctx.zeros(n, 1) + + g = scale = anorm = 0 + maxits = 3 * ctx.dps + + for i in xrange(n): # householder reduction to bidiagonal form + dwork[i] = scale * g # dwork are the side-diagonal elements + g = s = scale = 0 + if i < m: + for k in xrange(i, m): + scale += ctx.fabs(ctx.re(A[k,i])) + ctx.fabs(ctx.im(A[k,i])) + if scale != 0: + for k in xrange(i, m): + A[k,i] /= scale + ar = ctx.re(A[k,i]) + ai = ctx.im(A[k,i]) + s += ar * ar + ai * ai + f = A[i,i] + g = -ctx.sqrt(s) + if ctx.re(f) < 0: + beta = -g - ctx.conj(f) + g = -g + else: + beta = -g + ctx.conj(f) + beta /= ctx.conj(beta) + beta += 1 + h = 2 * (ctx.re(f) * g - s) + A[i,i] = f - g + beta /= h + lbeta[i] = (beta / scale) / scale + for j in xrange(i+1, n): + s = 0 + for k in xrange(i, m): + s += ctx.conj(A[k,i]) * A[k,j] + f = beta * s + for k in xrange(i, m): + A[k,j] += f * A[k,i] + for k in xrange(i, m): + A[k,i] *= scale + + S[i] = scale * g # S are the diagonal elements + g = s = scale = 0 + + if i < m and i != n - 1: + for k in xrange(i+1, n): + scale += ctx.fabs(ctx.re(A[i,k])) + ctx.fabs(ctx.im(A[i,k])) + if scale: + for k in xrange(i+1, n): + A[i,k] /= scale + ar = ctx.re(A[i,k]) + ai = ctx.im(A[i,k]) + s += ar * ar + ai * ai + f = A[i,i+1] + g = -ctx.sqrt(s) + if ctx.re(f) < 0: + beta = -g - ctx.conj(f) + g = -g + else: + beta = -g + ctx.conj(f) + + beta /= ctx.conj(beta) + beta += 1 + + h = 2 * (ctx.re(f) * g - s) + A[i,i+1] = f - g + + beta /= h + rbeta[i] = (beta / scale) / scale + + for k in xrange(i+1, n): + work[k] = A[i, k] + + for j in xrange(i+1, m): + s = 0 + for k in xrange(i+1, n): + s += ctx.conj(A[i,k]) * A[j,k] + f = s * beta + for k in xrange(i+1,n): + A[j,k] += f * work[k] + + for k in xrange(i+1, n): + A[i,k] *= scale + + anorm = max(anorm,ctx.fabs(S[i]) + ctx.fabs(dwork[i])) + + if not isinstance(V, bool): + for i in xrange(n-2, -1, -1): # accumulation of right hand transformations + V[i+1,i+1] = 1 + + if dwork[i+1] != 0: + f = ctx.conj(rbeta[i]) + for j in xrange(i+1, n): + V[i,j] = A[i,j] * f + for j in xrange(i+1, n): + s = 0 + for k in xrange(i+1, n): + s += ctx.conj(A[i,k]) * V[j,k] + for k in xrange(i+1, n): + V[j,k] += s * V[i,k] + + for j in xrange(i+1,n): + V[j,i] = V[i,j] = 0 + + V[0,0] = 1 + + if m < n : minnm = m + else : minnm = n + + if calc_u: + for i in xrange(minnm-1, -1, -1): # accumulation of left hand transformations + g = S[i] + for j in xrange(i+1, n): + A[i,j] = 0 + if g != 0: + g = 1 / g + for j in xrange(i+1, n): + s = 0 + for k in xrange(i+1, m): + s += ctx.conj(A[k,i]) * A[k,j] + f = s * ctx.conj(lbeta[i]) + for k in xrange(i, m): + A[k,j] += f * A[k,i] + for j in xrange(i, m): + A[j,i] *= g + else: + for j in xrange(i, m): + A[j,i] = 0 + A[i,i] += 1 + + for k in xrange(n-1, -1, -1): + # diagonalization of the bidiagonal form: + # loop over singular values, and over allowed itations + + its = 0 + while 1: + its += 1 + flag = True + + for l in xrange(k, -1, -1): + nm = l - 1 + + if ctx.fabs(dwork[l]) + anorm == anorm: + flag = False + break + + if ctx.fabs(S[nm]) + anorm == anorm: + break + + if flag: + c = 0 + s = 1 + for i in xrange(l, k+1): + f = s * dwork[i] + dwork[i] *= c + if ctx.fabs(f) + anorm == anorm: + break + g = S[i] + h = ctx.hypot(f, g) + S[i] = h + h = 1 / h + c = g * h + s = -f * h + + if calc_u: + for j in xrange(m): + y = A[j,nm] + z = A[j,i] + A[j,nm]= y * c + z * s + A[j,i] = z * c - y * s + + z = S[k] + + if l == k: # convergence + if z < 0: # singular value is made nonnegative + S[k] = -z + if not isinstance(V, bool): + for j in xrange(n): + V[k,j] = -V[k,j] + break + + if its >= maxits: + raise RuntimeError("svd: no convergence to an eigenvalue after %d iterations" % its) + + x = S[l] # shift from bottom 2 by 2 minor + nm = k-1 + y = S[nm] + g = dwork[nm] + h = dwork[k] + f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2 * h * y) + g = ctx.hypot(f, 1) + if f >=0: f = (( x - z) *( x + z) + h *((y / (f + g)) - h)) / x + else: f = (( x - z) *( x + z) + h *((y / (f - g)) - h)) / x + + c = s = 1 # next qt transformation + + for j in xrange(l, nm + 1): + g = dwork[j+1] + y = S[j+1] + h = s * g + g = c * g + z = ctx.hypot(f, h) + dwork[j] = z + c = f / z + s = h / z + f = x * c + g * s + g = g * c - x * s + h = y * s + y *= c + if not isinstance(V, bool): + for jj in xrange(n): + x = V[j ,jj] + z = V[j+1,jj] + V[j ,jj]= x * c + z * s + V[j+1,jj ]= z * c - x * s + z = ctx.hypot(f, h) + S[j] = z + if z != 0: # rotation can be arbitray if z=0 + z = 1 / z + c = f * z + s = h * z + f = c * g + s * y + x = c * y - s * g + if calc_u: + for jj in xrange(m): + y = A[jj,j ] + z = A[jj,j+1] + A[jj,j ]= y * c + z * s + A[jj,j+1 ]= z * c - y * s + + dwork[l] = 0 + dwork[k] = f + S[k] = x + + ########################## + + # Sort singular values into decreasing order (bubble-sort) + + for i in xrange(n): + imax = i + s = ctx.fabs(S[i]) # s is the current maximal element + + for j in xrange(i + 1, n): + c = ctx.fabs(S[j]) + if c > s: + s = c + imax = j + + if imax != i: + # swap singular values + + z = S[i] + S[i] = S[imax] + S[imax] = z + + if calc_u: + for j in xrange(m): + z = A[j,i] + A[j,i] = A[j,imax] + A[j,imax] = z + + if not isinstance(V, bool): + for j in xrange(n): + z = V[i,j] + V[i,j] = V[imax,j] + V[imax,j] = z + + return S + +################################################################################################## + +@defun +def svd_r(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False): + """ + This routine computes the singular value decomposition of a matrix A. + Given A, two orthogonal matrices U and V are calculated such that + + A = U S V and U' U = 1 and V V' = 1 + + where S is a suitable shaped matrix whose off-diagonal elements are zero. + Here ' denotes the transpose. The diagonal elements of S are the singular + values of A, i.e. the squareroots of the eigenvalues of A' A or A A'. + + input: + A : a real matrix of shape (m, n) + full_matrices : if true, U and V are of shape (m, m) and (n, n). + if false, U and V are of shape (m, min(m, n)) and (min(m, n), n). + compute_uv : if true, U and V are calculated. if false, only S is calculated. + overwrite_a : if true, allows modification of A which may improve + performance. if false, A is not modified. + + output: + U : an orthogonal matrix: U' U = 1. if full_matrices is true, U is of + shape (m, m). ortherwise it is of shape (m, min(m, n)). + + S : an array of length min(m, n) containing the singular values of A sorted by + decreasing magnitude. + + V : an orthogonal matrix: V V' = 1. if full_matrices is true, V is of + shape (n, n). ortherwise it is of shape (min(m, n), n). + + return value: + + S if compute_uv is false + (U, S, V) if compute_uv is true + + overview of the matrices: + + full_matrices true: + A : m*n + U : m*m U' U = 1 + S as matrix : m*n + V : n*n V V' = 1 + + full_matrices false: + A : m*n + U : m*min(n,m) U' U = 1 + S as matrix : min(m,n)*min(m,n) + V : min(m,n)*n V V' = 1 + + examples: + + >>> from mpmath import mp + >>> A = mp.matrix([[2, -2, -1], [3, 4, -2], [-2, -2, 0]]) + >>> S = mp.svd_r(A, compute_uv = False) + >>> print(S) + [6.0] + [3.0] + [1.0] + + >>> U, S, V = mp.svd_r(A) + >>> print(mp.chop(A - U * mp.diag(S) * V)) + [0.0 0.0 0.0] + [0.0 0.0 0.0] + [0.0 0.0 0.0] + + + see also: svd, svd_c + """ + + m, n = A.rows, A.cols + + if not compute_uv: + if not overwrite_a: + A = A.copy() + S = svd_r_raw(ctx, A, V = False, calc_u = False) + S = S[:min(m,n)] + return S + + if full_matrices and n < m: + V = ctx.zeros(m, m) + A0 = ctx.zeros(m, m) + A0[:,:n] = A + S = svd_r_raw(ctx, A0, V, calc_u = True) + + S = S[:n] + V = V[:n,:n] + + return (A0, S, V) + else: + if not overwrite_a: + A = A.copy() + V = ctx.zeros(n, n) + S = svd_r_raw(ctx, A, V, calc_u = True) + + if n > m: + if full_matrices == False: + V = V[:m,:] + + S = S[:m] + A = A[:,:m] + + return (A, S, V) + +############################## + +@defun +def svd_c(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False): + """ + This routine computes the singular value decomposition of a matrix A. + Given A, two unitary matrices U and V are calculated such that + + A = U S V and U' U = 1 and V V' = 1 + + where S is a suitable shaped matrix whose off-diagonal elements are zero. + Here ' denotes the hermitian transpose (i.e. transposition and complex + conjugation). The diagonal elements of S are the singular values of A, + i.e. the squareroots of the eigenvalues of A' A or A A'. + + input: + A : a complex matrix of shape (m, n) + full_matrices : if true, U and V are of shape (m, m) and (n, n). + if false, U and V are of shape (m, min(m, n)) and (min(m, n), n). + compute_uv : if true, U and V are calculated. if false, only S is calculated. + overwrite_a : if true, allows modification of A which may improve + performance. if false, A is not modified. + + output: + U : an unitary matrix: U' U = 1. if full_matrices is true, U is of + shape (m, m). ortherwise it is of shape (m, min(m, n)). + + S : an array of length min(m, n) containing the singular values of A sorted by + decreasing magnitude. + + V : an unitary matrix: V V' = 1. if full_matrices is true, V is of + shape (n, n). ortherwise it is of shape (min(m, n), n). + + return value: + + S if compute_uv is false + (U, S, V) if compute_uv is true + + overview of the matrices: + + full_matrices true: + A : m*n + U : m*m U' U = 1 + S as matrix : m*n + V : n*n V V' = 1 + + full_matrices false: + A : m*n + U : m*min(n,m) U' U = 1 + S as matrix : min(m,n)*min(m,n) + V : min(m,n)*n V V' = 1 + + example: + >>> from mpmath import mp + >>> A = mp.matrix([[-2j, -1-3j, -2+2j], [2-2j, -1-3j, 1], [-3+1j,-2j,0]]) + >>> S = mp.svd_c(A, compute_uv = False) + >>> print(mp.chop(S - mp.matrix([mp.sqrt(34), mp.sqrt(15), mp.sqrt(6)]))) + [0.0] + [0.0] + [0.0] + + >>> U, S, V = mp.svd_c(A) + >>> print(mp.chop(A - U * mp.diag(S) * V)) + [0.0 0.0 0.0] + [0.0 0.0 0.0] + [0.0 0.0 0.0] + + see also: svd, svd_r + """ + + m, n = A.rows, A.cols + + if not compute_uv: + if not overwrite_a: + A = A.copy() + S = svd_c_raw(ctx, A, V = False, calc_u = False) + S = S[:min(m,n)] + return S + + if full_matrices and n < m: + V = ctx.zeros(m, m) + A0 = ctx.zeros(m, m) + A0[:,:n] = A + S = svd_c_raw(ctx, A0, V, calc_u = True) + + S = S[:n] + V = V[:n,:n] + + return (A0, S, V) + else: + if not overwrite_a: + A = A.copy() + V = ctx.zeros(n, n) + S = svd_c_raw(ctx, A, V, calc_u = True) + + if n > m: + if full_matrices == False: + V = V[:m,:] + + S = S[:m] + A = A[:,:m] + + return (A, S, V) + +@defun +def svd(ctx, A, full_matrices = False, compute_uv = True, overwrite_a = False): + """ + "svd" is a unified interface for "svd_r" and "svd_c". Depending on + whether A is real or complex the appropriate function is called. + + This routine computes the singular value decomposition of a matrix A. + Given A, two orthogonal (A real) or unitary (A complex) matrices U and V + are calculated such that + + A = U S V and U' U = 1 and V V' = 1 + + where S is a suitable shaped matrix whose off-diagonal elements are zero. + Here ' denotes the hermitian transpose (i.e. transposition and complex + conjugation). The diagonal elements of S are the singular values of A, + i.e. the squareroots of the eigenvalues of A' A or A A'. + + input: + A : a real or complex matrix of shape (m, n) + full_matrices : if true, U and V are of shape (m, m) and (n, n). + if false, U and V are of shape (m, min(m, n)) and (min(m, n), n). + compute_uv : if true, U and V are calculated. if false, only S is calculated. + overwrite_a : if true, allows modification of A which may improve + performance. if false, A is not modified. + + output: + U : an orthogonal or unitary matrix: U' U = 1. if full_matrices is true, U is of + shape (m, m). ortherwise it is of shape (m, min(m, n)). + + S : an array of length min(m, n) containing the singular values of A sorted by + decreasing magnitude. + + V : an orthogonal or unitary matrix: V V' = 1. if full_matrices is true, V is of + shape (n, n). ortherwise it is of shape (min(m, n), n). + + return value: + + S if compute_uv is false + (U, S, V) if compute_uv is true + + overview of the matrices: + + full_matrices true: + A : m*n + U : m*m U' U = 1 + S as matrix : m*n + V : n*n V V' = 1 + + full_matrices false: + A : m*n + U : m*min(n,m) U' U = 1 + S as matrix : min(m,n)*min(m,n) + V : min(m,n)*n V V' = 1 + + examples: + + >>> from mpmath import mp + >>> A = mp.matrix([[2, -2, -1], [3, 4, -2], [-2, -2, 0]]) + >>> S = mp.svd(A, compute_uv = False) + >>> print(S) + [6.0] + [3.0] + [1.0] + + >>> U, S, V = mp.svd(A) + >>> print(mp.chop(A - U * mp.diag(S) * V)) + [0.0 0.0 0.0] + [0.0 0.0 0.0] + [0.0 0.0 0.0] + + see also: svd_r, svd_c + """ + + iscomplex = any(type(x) is ctx.mpc for x in A) + + if iscomplex: + return ctx.svd_c(A, full_matrices = full_matrices, compute_uv = compute_uv, overwrite_a = overwrite_a) + else: + return ctx.svd_r(A, full_matrices = full_matrices, compute_uv = compute_uv, overwrite_a = overwrite_a) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/rational.py b/env-llmeval/lib/python3.10/site-packages/mpmath/rational.py new file mode 100644 index 0000000000000000000000000000000000000000..58745205319ac3548ad5feb49371d2d154b2d3c8 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/rational.py @@ -0,0 +1,240 @@ +import operator +import sys +from .libmp import int_types, mpf_hash, bitcount, from_man_exp, HASH_MODULUS + +new = object.__new__ + +def create_reduced(p, q, _cache={}): + key = p, q + if key in _cache: + return _cache[key] + x, y = p, q + while y: + x, y = y, x % y + if x != 1: + p //= x + q //= x + v = new(mpq) + v._mpq_ = p, q + # Speedup integers, half-integers and other small fractions + if q <= 4 and abs(key[0]) < 100: + _cache[key] = v + return v + +class mpq(object): + """ + Exact rational type, currently only intended for internal use. + """ + + __slots__ = ["_mpq_"] + + def __new__(cls, p, q=1): + if type(p) is tuple: + p, q = p + elif hasattr(p, '_mpq_'): + p, q = p._mpq_ + return create_reduced(p, q) + + def __repr__(s): + return "mpq(%s,%s)" % s._mpq_ + + def __str__(s): + return "(%s/%s)" % s._mpq_ + + def __int__(s): + a, b = s._mpq_ + return a // b + + def __nonzero__(s): + return bool(s._mpq_[0]) + + __bool__ = __nonzero__ + + def __hash__(s): + a, b = s._mpq_ + if sys.version_info >= (3, 2): + inverse = pow(b, HASH_MODULUS-2, HASH_MODULUS) + if not inverse: + h = sys.hash_info.inf + else: + h = (abs(a) * inverse) % HASH_MODULUS + if a < 0: h = -h + if h == -1: h = -2 + return h + else: + if b == 1: + return hash(a) + # Power of two: mpf compatible hash + if not (b & (b-1)): + return mpf_hash(from_man_exp(a, 1-bitcount(b))) + return hash((a,b)) + + def __eq__(s, t): + ttype = type(t) + if ttype is mpq: + return s._mpq_ == t._mpq_ + if ttype in int_types: + a, b = s._mpq_ + if b != 1: + return False + return a == t + return NotImplemented + + def __ne__(s, t): + ttype = type(t) + if ttype is mpq: + return s._mpq_ != t._mpq_ + if ttype in int_types: + a, b = s._mpq_ + if b != 1: + return True + return a != t + return NotImplemented + + def _cmp(s, t, op): + ttype = type(t) + if ttype in int_types: + a, b = s._mpq_ + return op(a, t*b) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return op(a*d, b*c) + return NotImplementedError + + def __lt__(s, t): return s._cmp(t, operator.lt) + def __le__(s, t): return s._cmp(t, operator.le) + def __gt__(s, t): return s._cmp(t, operator.gt) + def __ge__(s, t): return s._cmp(t, operator.ge) + + def __abs__(s): + a, b = s._mpq_ + if a >= 0: + return s + v = new(mpq) + v._mpq_ = -a, b + return v + + def __neg__(s): + a, b = s._mpq_ + v = new(mpq) + v._mpq_ = -a, b + return v + + def __pos__(s): + return s + + def __add__(s, t): + ttype = type(t) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return create_reduced(a*d+b*c, b*d) + if ttype in int_types: + a, b = s._mpq_ + v = new(mpq) + v._mpq_ = a+b*t, b + return v + return NotImplemented + + __radd__ = __add__ + + def __sub__(s, t): + ttype = type(t) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return create_reduced(a*d-b*c, b*d) + if ttype in int_types: + a, b = s._mpq_ + v = new(mpq) + v._mpq_ = a-b*t, b + return v + return NotImplemented + + def __rsub__(s, t): + ttype = type(t) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return create_reduced(b*c-a*d, b*d) + if ttype in int_types: + a, b = s._mpq_ + v = new(mpq) + v._mpq_ = b*t-a, b + return v + return NotImplemented + + def __mul__(s, t): + ttype = type(t) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return create_reduced(a*c, b*d) + if ttype in int_types: + a, b = s._mpq_ + return create_reduced(a*t, b) + return NotImplemented + + __rmul__ = __mul__ + + def __div__(s, t): + ttype = type(t) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return create_reduced(a*d, b*c) + if ttype in int_types: + a, b = s._mpq_ + return create_reduced(a, b*t) + return NotImplemented + + def __rdiv__(s, t): + ttype = type(t) + if ttype is mpq: + a, b = s._mpq_ + c, d = t._mpq_ + return create_reduced(b*c, a*d) + if ttype in int_types: + a, b = s._mpq_ + return create_reduced(b*t, a) + return NotImplemented + + def __pow__(s, t): + ttype = type(t) + if ttype in int_types: + a, b = s._mpq_ + if t: + if t < 0: + a, b, t = b, a, -t + v = new(mpq) + v._mpq_ = a**t, b**t + return v + raise ZeroDivisionError + return NotImplemented + + +mpq_1 = mpq((1,1)) +mpq_0 = mpq((0,1)) +mpq_1_2 = mpq((1,2)) +mpq_3_2 = mpq((3,2)) +mpq_1_4 = mpq((1,4)) +mpq_1_16 = mpq((1,16)) +mpq_3_16 = mpq((3,16)) +mpq_5_2 = mpq((5,2)) +mpq_3_4 = mpq((3,4)) +mpq_7_4 = mpq((7,4)) +mpq_5_4 = mpq((5,4)) + + +# Register with "numbers" ABC +# We do not subclass, hence we do not use the @abstractmethod checks. While +# this is less invasive it may turn out that we do not actually support +# parts of the expected interfaces. See +# http://docs.python.org/2/library/numbers.html for list of abstract +# methods. +try: + import numbers + numbers.Rational.register(mpq) +except ImportError: + pass diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/__init__.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/__pycache__/__init__.cpython-310.pyc b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..68cb9e3a22280fd6096570f02471e742c9cd08ea Binary files /dev/null and b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/__pycache__/__init__.cpython-310.pyc differ diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/__pycache__/extratest_zeta.cpython-310.pyc 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b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/extratest_gamma.py @@ -0,0 +1,215 @@ +from mpmath import * +from mpmath.libmp import ifac + +import sys +if "-dps" in sys.argv: + maxdps = int(sys.argv[sys.argv.index("-dps")+1]) +else: + maxdps = 1000 + +raise_ = "-raise" in sys.argv + +errcount = 0 + +def check(name, func, z, y): + global errcount + try: + x = func(z) + except: + errcount += 1 + if raise_: + raise + print() + print(name) + print("EXCEPTION") + import traceback + traceback.print_tb(sys.exc_info()[2]) + print() + return + xre = x.real + xim = x.imag + yre = y.real + yim = y.imag + tol = eps*8 + err = 0 + if abs(xre-yre) > abs(yre)*tol: + err = 1 + print() + print("Error! %s (re = %s, wanted %s, err=%s)" % (name, nstr(xre,10), nstr(yre,10), nstr(abs(xre-yre)))) + errcount += 1 + if raise_: + raise SystemExit + if abs(xim-yim) > abs(yim)*tol: + err = 1 + print() + print("Error! %s (im = %s, wanted %s, err=%s)" % (name, nstr(xim,10), nstr(yim,10), nstr(abs(xim-yim)))) + errcount += 1 + if raise_: + raise SystemExit + if not err: + sys.stdout.write("%s ok; " % name) + +def testcase(case): + z, result = case + print("Testing z =", z) + mp.dps = 1010 + z = eval(z) + mp.dps = maxdps + 50 + if result is None: + gamma_val = gamma(z) + loggamma_val = loggamma(z) + factorial_val = factorial(z) + rgamma_val = rgamma(z) + else: + loggamma_val = eval(result) + gamma_val = exp(loggamma_val) + factorial_val = z * gamma_val + rgamma_val = 1/gamma_val + for dps in [5, 10, 15, 25, 40, 60, 90, 120, 250, 600, 1000, 1800, 3600]: + if dps > maxdps: + break + mp.dps = dps + print("dps = %s" % dps) + check("gamma", gamma, z, gamma_val) + check("rgamma", rgamma, z, rgamma_val) + check("loggamma", loggamma, z, loggamma_val) + check("factorial", factorial, z, factorial_val) + print() + mp.dps = 15 + +testcases = [] + +# Basic values +for n in list(range(1,200)) + list(range(201,2000,17)): + testcases.append(["%s" % n, None]) +for n in range(-200,200): + testcases.append(["%s+0.5" % n, None]) + testcases.append(["%s+0.37" % n, None]) + +testcases += [\ +["(0.1+1j)", None], +["(-0.1+1j)", None], +["(0.1-1j)", None], +["(-0.1-1j)", None], +["10j", None], +["-10j", None], +["100j", None], +["10000j", None], +["-10000000j", None], +["(10**100)*j", None], +["125+(10**100)*j", None], +["-125+(10**100)*j", None], +["(10**10)*(1+j)", None], +["(10**10)*(-1+j)", None], +["(10**100)*(1+j)", None], +["(10**100)*(-1+j)", None], +["(1.5-1j)", None], +["(6+4j)", None], +["(4+1j)", None], +["(3.5+2j)", None], +["(1.5-1j)", None], +["(-6-4j)", None], +["(-2-3j)", None], +["(-2.5-2j)", None], +["(4+1j)", None], +["(3+3j)", None], +["(2-2j)", None], +["1", "0"], +["2", "0"], +["3", "log(2)"], +["4", "log(6)"], +["5", "log(24)"], +["0.5", "log(pi)/2"], +["1.5", "log(sqrt(pi)/2)"], +["2.5", "log(3*sqrt(pi)/4)"], +["mpf('0.37')", None], +["0.25", "log(sqrt(2*sqrt(2*pi**3)/agm(1,sqrt(2))))"], +["-0.4", None], +["mpf('-1.9')", None], +["mpf('12.8')", None], +["mpf('33.7')", None], +["mpf('95.2')", None], +["mpf('160.3')", None], +["mpf('2057.8')", None], +["25", "log(ifac(24))"], +["80", "log(ifac(79))"], +["500", "log(ifac(500-1))"], +["8000", "log(ifac(8000-1))"], +["8000.5", None], +["mpf('8000.1')", None], +["mpf('1.37e10')", None], +["mpf('1.37e10')*(1+j)", None], +["mpf('1.37e10')*(-1+j)", None], +["mpf('1.37e10')*(-1-j)", None], +["mpf('1.37e10')*(-1+j)", None], +["mpf('1.37e100')", None], +["mpf('1.37e100')*(1+j)", None], +["mpf('1.37e100')*(-1+j)", None], +["mpf('1.37e100')*(-1-j)", None], +["mpf('1.37e100')*(-1+j)", None], +["3+4j", +"mpc('" +"-1.7566267846037841105306041816232757851567066070613445016197619371316057169" +"4723618263960834804618463052988607348289672535780644470689771115236512106002" +"5970873471563240537307638968509556191696167970488390423963867031934333890838" +"8009531786948197210025029725361069435208930363494971027388382086721660805397" +"9163230643216054580167976201709951509519218635460317367338612500626714783631" +"7498317478048447525674016344322545858832610325861086336204591943822302971823" +"5161814175530618223688296232894588415495615809337292518431903058265147109853" +"1710568942184987827643886816200452860853873815413367529829631430146227470517" +"6579967222200868632179482214312673161276976117132204633283806161971389519137" +"1243359764435612951384238091232760634271570950240717650166551484551654327989" +"9360285030081716934130446150245110557038117075172576825490035434069388648124" +"6678152254554001586736120762641422590778766100376515737713938521275749049949" +"1284143906816424244705094759339932733567910991920631339597278805393743140853" +"391550313363278558195609260225928','" +"4.74266443803465792819488940755002274088830335171164611359052405215840070271" +"5906813009373171139767051863542508136875688550817670379002790304870822775498" +"2809996675877564504192565392367259119610438951593128982646945990372179860613" +"4294436498090428077839141927485901735557543641049637962003652638924845391650" +"9546290137755550107224907606529385248390667634297183361902055842228798984200" +"9591180450211798341715874477629099687609819466457990642030707080894518168924" +"6805549314043258530272479246115112769957368212585759640878745385160943755234" +"9398036774908108204370323896757543121853650025529763655312360354244898913463" +"7115955702828838923393113618205074162812089732064414530813087483533203244056" +"0546577484241423134079056537777170351934430586103623577814746004431994179990" +"5318522939077992613855205801498201930221975721246498720895122345420698451980" +"0051215797310305885845964334761831751370672996984756815410977750799748813563" +"8784405288158432214886648743541773208808731479748217023665577802702269468013" +"673719173759245720489020315779001')"], +] + +for z in [4, 14, 34, 64]: + testcases.append(["(2+j)*%s/3" % z, None]) + testcases.append(["(-2+j)*%s/3" % z, None]) + testcases.append(["(1+2*j)*%s/3" % z, None]) + testcases.append(["(2-j)*%s/3" % z, None]) + testcases.append(["(20+j)*%s/3" % z, None]) + testcases.append(["(-20+j)*%s/3" % z, None]) + testcases.append(["(1+20*j)*%s/3" % z, None]) + testcases.append(["(20-j)*%s/3" % z, None]) + testcases.append(["(200+j)*%s/3" % z, None]) + testcases.append(["(-200+j)*%s/3" % z, None]) + testcases.append(["(1+200*j)*%s/3" % z, None]) + testcases.append(["(200-j)*%s/3" % z, None]) + +# Poles +for n in [0,1,2,3,4,25,-1,-2,-3,-4,-20,-21,-50,-51,-200,-201,-20000,-20001]: + for t in ['1e-5', '1e-20', '1e-100', '1e-10000']: + testcases.append(["fadd(%s,'%s',exact=True)" % (n, t), None]) + testcases.append(["fsub(%s,'%s',exact=True)" % (n, t), None]) + testcases.append(["fadd(%s,'%sj',exact=True)" % (n, t), None]) + testcases.append(["fsub(%s,'%sj',exact=True)" % (n, t), None]) + +if __name__ == "__main__": + from timeit import default_timer as clock + tot_time = 0.0 + for case in testcases: + t1 = clock() + testcase(case) + t2 = clock() + print("Test time:", t2-t1) + print() + tot_time += (t2-t1) + print("Total time:", tot_time) + print("Errors:", errcount) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/extratest_zeta.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/extratest_zeta.py new file mode 100644 index 0000000000000000000000000000000000000000..582b3d9cbd956b9cdf94309e0e718371fe716101 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/extratest_zeta.py @@ -0,0 +1,30 @@ +from mpmath import zetazero +from timeit import default_timer as clock + +def test_zetazero(): + cases = [\ + (399999999, 156762524.6750591511), + (241389216, 97490234.2276711795), + (526196239, 202950727.691229534), + (542964976, 209039046.578535272), + (1048449112, 388858885.231056486), + (1048449113, 388858885.384337406), + (1048449114, 388858886.002285122), + (1048449115, 388858886.00239369), + (1048449116, 388858886.690745053) + ] + for n, v in cases: + print(n, v) + t1 = clock() + ok = zetazero(n).ae(complex(0.5,v)) + t2 = clock() + print("ok =", ok, ("(time = %s)" % round(t2-t1,3))) + print("Now computing two huge zeros (this may take hours)") + print("Computing zetazero(8637740722917)") + ok = zetazero(8637740722917).ae(complex(0.5,2124447368584.39296466152)) + print("ok =", ok) + ok = zetazero(8637740722918).ae(complex(0.5,2124447368584.39298170604)) + print("ok =", ok) + +if __name__ == "__main__": + test_zetazero() diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/runtests.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/runtests.py new file mode 100644 index 0000000000000000000000000000000000000000..70fde272fdc0e05e3d8951edddca380bd36139ab --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/runtests.py @@ -0,0 +1,161 @@ +#!/usr/bin/env python + +""" +python runtests.py -py + Use py.test to run tests (more useful for debugging) + +python runtests.py -coverage + Generate test coverage report. Statistics are written to /tmp + +python runtests.py -profile + Generate profile stats (this is much slower) + +python runtests.py -nogmpy + Run tests without using GMPY even if it exists + +python runtests.py -strict + Enforce extra tests in normalize() + +python runtests.py -local + Insert '../..' at the beginning of sys.path to use local mpmath + +python runtests.py -skip ... + Skip tests from the listed modules + +Additional arguments are used to filter the tests to run. Only files that have +one of the arguments in their name are executed. + +""" + +import sys, os, traceback + +profile = False +if "-profile" in sys.argv: + sys.argv.remove('-profile') + profile = True + +coverage = False +if "-coverage" in sys.argv: + sys.argv.remove('-coverage') + coverage = True + +if "-nogmpy" in sys.argv: + sys.argv.remove('-nogmpy') + os.environ['MPMATH_NOGMPY'] = 'Y' + +if "-strict" in sys.argv: + sys.argv.remove('-strict') + os.environ['MPMATH_STRICT'] = 'Y' + +if "-local" in sys.argv: + sys.argv.remove('-local') + importdir = os.path.abspath(os.path.join(os.path.dirname(sys.argv[0]), + '../..')) +else: + importdir = '' + +# TODO: add a flag for this +testdir = '' + +def testit(importdir='', testdir='', exit_on_fail=False): + """Run all tests in testdir while importing from importdir.""" + if importdir: + sys.path.insert(1, importdir) + if testdir: + sys.path.insert(1, testdir) + import os.path + import mpmath + print("mpmath imported from %s" % os.path.dirname(mpmath.__file__)) + print("mpmath backend: %s" % mpmath.libmp.backend.BACKEND) + print("mpmath mp class: %s" % repr(mpmath.mp)) + print("mpmath version: %s" % mpmath.__version__) + print("Python version: %s" % sys.version) + print("") + if "-py" in sys.argv: + sys.argv.remove('-py') + import py + py.test.cmdline.main() + else: + import glob + from timeit import default_timer as clock + modules = [] + args = sys.argv[1:] + excluded = [] + if '-skip' in args: + excluded = args[args.index('-skip')+1:] + args = args[:args.index('-skip')] + # search for tests in directory of this file if not otherwise specified + if not testdir: + pattern = os.path.dirname(sys.argv[0]) + else: + pattern = testdir + if pattern: + pattern += '/' + pattern += 'test*.py' + # look for tests (respecting specified filter) + for f in glob.glob(pattern): + name = os.path.splitext(os.path.basename(f))[0] + # If run as a script, only run tests given as args, if any are given + if args and __name__ == "__main__": + ok = False + for arg in args: + if arg in name: + ok = True + break + if not ok: + continue + elif name in excluded: + continue + module = __import__(name) + priority = module.__dict__.get('priority', 100) + if priority == 666: + modules = [[priority, name, module]] + break + modules.append([priority, name, module]) + # execute tests + modules.sort() + tstart = clock() + for priority, name, module in modules: + print(name) + for f in sorted(module.__dict__.keys()): + if f.startswith('test_'): + if coverage and ('numpy' in f): + continue + sys.stdout.write(" " + f[5:].ljust(25) + " ") + t1 = clock() + try: + module.__dict__[f]() + except: + etype, evalue, trb = sys.exc_info() + if etype in (KeyboardInterrupt, SystemExit): + raise + print("") + print("TEST FAILED!") + print("") + traceback.print_exc() + if exit_on_fail: + return + t2 = clock() + print("ok " + " " + ("%.7f" % (t2-t1)) + " s") + tend = clock() + print("") + print("finished tests in " + ("%.2f" % (tend-tstart)) + " seconds") + # clean sys.path + if importdir: + sys.path.remove(importdir) + if testdir: + sys.path.remove(testdir) + +if __name__ == '__main__': + if profile: + import cProfile + cProfile.run("testit('%s', '%s')" % (importdir, testdir), sort=1) + elif coverage: + import trace + tracer = trace.Trace(ignoredirs=[sys.prefix, sys.exec_prefix], + trace=0, count=1) + tracer.run('testit(importdir, testdir)') + r = tracer.results() + r.write_results(show_missing=True, summary=True, coverdir="/tmp") + else: + testit(importdir, testdir) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_basic_ops.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_basic_ops.py new file mode 100644 index 0000000000000000000000000000000000000000..f577c7fa9f9734876b6767f6cc21144df305d82f --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_basic_ops.py @@ -0,0 +1,451 @@ +import mpmath +from mpmath import * +from mpmath.libmp import * +import random +import sys + +try: + long = long +except NameError: + long = int + +def test_type_compare(): + assert mpf(2) == mpc(2,0) + assert mpf(0) == mpc(0) + assert mpf(2) != mpc(2, 0.00001) + assert mpf(2) == 2.0 + assert mpf(2) != 3.0 + assert mpf(2) == 2 + assert mpf(2) != '2.0' + assert mpc(2) != '2.0' + +def test_add(): + assert mpf(2.5) + mpf(3) == 5.5 + assert mpf(2.5) + 3 == 5.5 + assert mpf(2.5) + 3.0 == 5.5 + assert 3 + mpf(2.5) == 5.5 + assert 3.0 + mpf(2.5) == 5.5 + assert (3+0j) + mpf(2.5) == 5.5 + assert mpc(2.5) + mpf(3) == 5.5 + assert mpc(2.5) + 3 == 5.5 + assert mpc(2.5) + 3.0 == 5.5 + assert mpc(2.5) + (3+0j) == 5.5 + assert 3 + mpc(2.5) == 5.5 + assert 3.0 + mpc(2.5) == 5.5 + assert (3+0j) + mpc(2.5) == 5.5 + +def test_sub(): + assert mpf(2.5) - mpf(3) == -0.5 + assert mpf(2.5) - 3 == -0.5 + assert mpf(2.5) - 3.0 == -0.5 + assert 3 - mpf(2.5) == 0.5 + assert 3.0 - mpf(2.5) == 0.5 + assert (3+0j) - mpf(2.5) == 0.5 + assert mpc(2.5) - mpf(3) == -0.5 + assert mpc(2.5) - 3 == -0.5 + assert mpc(2.5) - 3.0 == -0.5 + assert mpc(2.5) - (3+0j) == -0.5 + assert 3 - mpc(2.5) == 0.5 + assert 3.0 - mpc(2.5) == 0.5 + assert (3+0j) - mpc(2.5) == 0.5 + +def test_mul(): + assert mpf(2.5) * mpf(3) == 7.5 + assert mpf(2.5) * 3 == 7.5 + assert mpf(2.5) * 3.0 == 7.5 + assert 3 * mpf(2.5) == 7.5 + assert 3.0 * mpf(2.5) == 7.5 + assert (3+0j) * mpf(2.5) == 7.5 + assert mpc(2.5) * mpf(3) == 7.5 + assert mpc(2.5) * 3 == 7.5 + assert mpc(2.5) * 3.0 == 7.5 + assert mpc(2.5) * (3+0j) == 7.5 + assert 3 * mpc(2.5) == 7.5 + assert 3.0 * mpc(2.5) == 7.5 + assert (3+0j) * mpc(2.5) == 7.5 + +def test_div(): + assert mpf(6) / mpf(3) == 2.0 + assert mpf(6) / 3 == 2.0 + assert mpf(6) / 3.0 == 2.0 + assert 6 / mpf(3) == 2.0 + assert 6.0 / mpf(3) == 2.0 + assert (6+0j) / mpf(3.0) == 2.0 + assert mpc(6) / mpf(3) == 2.0 + assert mpc(6) / 3 == 2.0 + assert mpc(6) / 3.0 == 2.0 + assert mpc(6) / (3+0j) == 2.0 + assert 6 / mpc(3) == 2.0 + assert 6.0 / mpc(3) == 2.0 + assert (6+0j) / mpc(3) == 2.0 + +def test_pow(): + assert mpf(6) ** mpf(3) == 216.0 + assert mpf(6) ** 3 == 216.0 + assert mpf(6) ** 3.0 == 216.0 + assert 6 ** mpf(3) == 216.0 + assert 6.0 ** mpf(3) == 216.0 + assert (6+0j) ** mpf(3.0) == 216.0 + assert mpc(6) ** mpf(3) == 216.0 + assert mpc(6) ** 3 == 216.0 + assert mpc(6) ** 3.0 == 216.0 + assert mpc(6) ** (3+0j) == 216.0 + assert 6 ** mpc(3) == 216.0 + assert 6.0 ** mpc(3) == 216.0 + assert (6+0j) ** mpc(3) == 216.0 + +def test_mixed_misc(): + assert 1 + mpf(3) == mpf(3) + 1 == 4 + assert 1 - mpf(3) == -(mpf(3) - 1) == -2 + assert 3 * mpf(2) == mpf(2) * 3 == 6 + assert 6 / mpf(2) == mpf(6) / 2 == 3 + assert 1.0 + mpf(3) == mpf(3) + 1.0 == 4 + assert 1.0 - mpf(3) == -(mpf(3) - 1.0) == -2 + assert 3.0 * mpf(2) == mpf(2) * 3.0 == 6 + assert 6.0 / mpf(2) == mpf(6) / 2.0 == 3 + +def test_add_misc(): + mp.dps = 15 + assert mpf(4) + mpf(-70) == -66 + assert mpf(1) + mpf(1.1)/80 == 1 + 1.1/80 + assert mpf((1, 10000000000)) + mpf(3) == mpf((1, 10000000000)) + assert mpf(3) + mpf((1, 10000000000)) == mpf((1, 10000000000)) + assert mpf((1, -10000000000)) + mpf(3) == mpf(3) + assert mpf(3) + mpf((1, -10000000000)) == mpf(3) + assert mpf(1) + 1e-15 != 1 + assert mpf(1) + 1e-20 == 1 + assert mpf(1.07e-22) + 0 == mpf(1.07e-22) + assert mpf(0) + mpf(1.07e-22) == mpf(1.07e-22) + +def test_complex_misc(): + # many more tests needed + assert 1 + mpc(2) == 3 + assert not mpc(2).ae(2 + 1e-13) + assert mpc(2+1e-15j).ae(2) + +def test_complex_zeros(): + for a in [0,2]: + for b in [0,3]: + for c in [0,4]: + for d in [0,5]: + assert mpc(a,b)*mpc(c,d) == complex(a,b)*complex(c,d) + +def test_hash(): + for i in range(-256, 256): + assert hash(mpf(i)) == hash(i) + assert hash(mpf(0.5)) == hash(0.5) + assert hash(mpc(2,3)) == hash(2+3j) + # Check that this doesn't fail + assert hash(inf) + # Check that overflow doesn't assign equal hashes to large numbers + assert hash(mpf('1e1000')) != hash('1e10000') + assert hash(mpc(100,'1e1000')) != hash(mpc(200,'1e1000')) + from mpmath.rational import mpq + assert hash(mp.mpq(1,3)) + assert hash(mp.mpq(0,1)) == 0 + assert hash(mp.mpq(-1,1)) == hash(-1) + assert hash(mp.mpq(1,1)) == hash(1) + assert hash(mp.mpq(5,1)) == hash(5) + assert hash(mp.mpq(1,2)) == hash(0.5) + if sys.version_info >= (3, 2): + assert hash(mpf(1)*2**2000) == hash(2**2000) + assert hash(mpf(1)/2**2000) == hash(mpq(1,2**2000)) + +# Advanced rounding test +def test_add_rounding(): + mp.dps = 15 + a = from_float(1e-50) + assert mpf_sub(mpf_add(fone, a, 53, round_up), fone, 53, round_up) == from_float(2.2204460492503131e-16) + assert mpf_sub(fone, a, 53, round_up) == fone + assert mpf_sub(fone, mpf_sub(fone, a, 53, round_down), 53, round_down) == from_float(1.1102230246251565e-16) + assert mpf_add(fone, a, 53, round_down) == fone + +def test_almost_equal(): + assert mpf(1.2).ae(mpf(1.20000001), 1e-7) + assert not mpf(1.2).ae(mpf(1.20000001), 1e-9) + assert not mpf(-0.7818314824680298).ae(mpf(-0.774695868667929)) + +def test_arithmetic_functions(): + import operator + ops = [(operator.add, fadd), (operator.sub, fsub), (operator.mul, fmul), + (operator.truediv, fdiv)] + a = mpf(0.27) + b = mpf(1.13) + c = mpc(0.51+2.16j) + d = mpc(1.08-0.99j) + for x in [a,b,c,d]: + for y in [a,b,c,d]: + for op, fop in ops: + if fop is not fdiv: + mp.prec = 200 + z0 = op(x,y) + mp.prec = 60 + z1 = op(x,y) + mp.prec = 53 + z2 = op(x,y) + assert fop(x, y, prec=60) == z1 + assert fop(x, y) == z2 + if fop is not fdiv: + assert fop(x, y, prec=inf) == z0 + assert fop(x, y, dps=inf) == z0 + assert fop(x, y, exact=True) == z0 + assert fneg(fneg(z1, exact=True), prec=inf) == z1 + assert fneg(z1) == -(+z1) + mp.dps = 15 + +def test_exact_integer_arithmetic(): + # XXX: re-fix this so that all operations are tested with all rounding modes + random.seed(0) + for prec in [6, 10, 25, 40, 100, 250, 725]: + for rounding in ['d', 'u', 'f', 'c', 'n']: + mp.dps = prec + M = 10**(prec-2) + M2 = 10**(prec//2-2) + for i in range(10): + a = random.randint(-M, M) + b = random.randint(-M, M) + assert mpf(a, rounding=rounding) == a + assert int(mpf(a, rounding=rounding)) == a + assert int(mpf(str(a), rounding=rounding)) == a + assert mpf(a) + mpf(b) == a + b + assert mpf(a) - mpf(b) == a - b + assert -mpf(a) == -a + a = random.randint(-M2, M2) + b = random.randint(-M2, M2) + assert mpf(a) * mpf(b) == a*b + assert mpf_mul(from_int(a), from_int(b), mp.prec, rounding) == from_int(a*b) + mp.dps = 15 + +def test_odd_int_bug(): + assert to_int(from_int(3), round_nearest) == 3 + +def test_str_1000_digits(): + mp.dps = 1001 + # last digit may be wrong + assert str(mpf(2)**0.5)[-10:-1] == '9518488472'[:9] + assert str(pi)[-10:-1] == '2164201989'[:9] + mp.dps = 15 + +def test_str_10000_digits(): + mp.dps = 10001 + # last digit may be wrong + assert str(mpf(2)**0.5)[-10:-1] == '5873258351'[:9] + assert str(pi)[-10:-1] == '5256375678'[:9] + mp.dps = 15 + +def test_monitor(): + f = lambda x: x**2 + a = [] + b = [] + g = monitor(f, a.append, b.append) + assert g(3) == 9 + assert g(4) == 16 + assert a[0] == ((3,), {}) + assert b[0] == 9 + +def test_nint_distance(): + assert nint_distance(mpf(-3)) == (-3, -inf) + assert nint_distance(mpc(-3)) == (-3, -inf) + assert nint_distance(mpf(-3.1)) == (-3, -3) + assert nint_distance(mpf(-3.01)) == (-3, -6) + assert nint_distance(mpf(-3.001)) == (-3, -9) + assert nint_distance(mpf(-3.0001)) == (-3, -13) + assert nint_distance(mpf(-2.9)) == (-3, -3) + assert nint_distance(mpf(-2.99)) == (-3, -6) + assert nint_distance(mpf(-2.999)) == (-3, -9) + assert nint_distance(mpf(-2.9999)) == (-3, -13) + assert nint_distance(mpc(-3+0.1j)) == (-3, -3) + assert nint_distance(mpc(-3+0.01j)) == (-3, -6) + assert nint_distance(mpc(-3.1+0.1j)) == (-3, -3) + assert nint_distance(mpc(-3.01+0.01j)) == (-3, -6) + assert nint_distance(mpc(-3.001+0.001j)) == (-3, -9) + assert nint_distance(mpf(0)) == (0, -inf) + assert nint_distance(mpf(0.01)) == (0, -6) + assert nint_distance(mpf('1e-100')) == (0, -332) + +def test_floor_ceil_nint_frac(): + mp.dps = 15 + for n in range(-10,10): + assert floor(n) == n + assert floor(n+0.5) == n + assert ceil(n) == n + assert ceil(n+0.5) == n+1 + assert nint(n) == n + # nint rounds to even + if n % 2 == 1: + assert nint(n+0.5) == n+1 + else: + assert nint(n+0.5) == n + assert floor(inf) == inf + assert floor(ninf) == ninf + assert isnan(floor(nan)) + assert ceil(inf) == inf + assert ceil(ninf) == ninf + assert isnan(ceil(nan)) + assert nint(inf) == inf + assert nint(ninf) == ninf + assert isnan(nint(nan)) + assert floor(0.1) == 0 + assert floor(0.9) == 0 + assert floor(-0.1) == -1 + assert floor(-0.9) == -1 + assert floor(10000000000.1) == 10000000000 + assert floor(10000000000.9) == 10000000000 + assert floor(-10000000000.1) == -10000000000-1 + assert floor(-10000000000.9) == -10000000000-1 + assert floor(1e-100) == 0 + assert floor(-1e-100) == -1 + assert floor(1e100) == 1e100 + assert floor(-1e100) == -1e100 + assert ceil(0.1) == 1 + assert ceil(0.9) == 1 + assert ceil(-0.1) == 0 + assert ceil(-0.9) == 0 + assert ceil(10000000000.1) == 10000000000+1 + assert ceil(10000000000.9) == 10000000000+1 + assert ceil(-10000000000.1) == -10000000000 + assert ceil(-10000000000.9) == -10000000000 + assert ceil(1e-100) == 1 + assert ceil(-1e-100) == 0 + assert ceil(1e100) == 1e100 + assert ceil(-1e100) == -1e100 + assert nint(0.1) == 0 + assert nint(0.9) == 1 + assert nint(-0.1) == 0 + assert nint(-0.9) == -1 + assert nint(10000000000.1) == 10000000000 + assert nint(10000000000.9) == 10000000000+1 + assert nint(-10000000000.1) == -10000000000 + assert nint(-10000000000.9) == -10000000000-1 + assert nint(1e-100) == 0 + assert nint(-1e-100) == 0 + assert nint(1e100) == 1e100 + assert nint(-1e100) == -1e100 + assert floor(3.2+4.6j) == 3+4j + assert ceil(3.2+4.6j) == 4+5j + assert nint(3.2+4.6j) == 3+5j + for n in range(-10,10): + assert frac(n) == 0 + assert frac(0.25) == 0.25 + assert frac(1.25) == 0.25 + assert frac(2.25) == 0.25 + assert frac(-0.25) == 0.75 + assert frac(-1.25) == 0.75 + assert frac(-2.25) == 0.75 + assert frac('1e100000000000000') == 0 + u = mpf('1e-100000000000000') + assert frac(u) == u + assert frac(-u) == 1 # rounding! + u = mpf('1e-400') + assert frac(-u, prec=0) == fsub(1, u, exact=True) + assert frac(3.25+4.75j) == 0.25+0.75j + +def test_isnan_etc(): + from mpmath.rational import mpq + assert isnan(nan) == True + assert isnan(3) == False + assert isnan(mpf(3)) == False + assert isnan(inf) == False + assert isnan(mpc(2,nan)) == True + assert isnan(mpc(2,nan)) == True + assert isnan(mpc(nan,nan)) == True + assert isnan(mpc(2,2)) == False + assert isnan(mpc(nan,inf)) == True + assert isnan(mpc(inf,inf)) == False + assert isnan(mpq((3,2))) == False + assert isnan(mpq((0,1))) == False + assert isinf(inf) == True + assert isinf(-inf) == True + assert isinf(3) == False + assert isinf(nan) == False + assert isinf(3+4j) == False + assert isinf(mpc(inf)) == True + assert isinf(mpc(3,inf)) == True + assert isinf(mpc(inf,3)) == True + assert isinf(mpc(inf,inf)) == True + assert isinf(mpc(nan,inf)) == True + assert isinf(mpc(inf,nan)) == True + assert isinf(mpc(nan,nan)) == False + assert isinf(mpq((3,2))) == False + assert isinf(mpq((0,1))) == False + assert isnormal(3) == True + assert isnormal(3.5) == True + assert isnormal(mpf(3.5)) == True + assert isnormal(0) == False + assert isnormal(mpf(0)) == False + assert isnormal(0.0) == False + assert isnormal(inf) == False + assert isnormal(-inf) == False + assert isnormal(nan) == False + assert isnormal(float(inf)) == False + assert isnormal(mpc(0,0)) == False + assert isnormal(mpc(3,0)) == True + assert isnormal(mpc(0,3)) == True + assert isnormal(mpc(3,3)) == True + assert isnormal(mpc(0,nan)) == False + assert isnormal(mpc(0,inf)) == False + assert isnormal(mpc(3,nan)) == False + assert isnormal(mpc(3,inf)) == False + assert isnormal(mpc(3,-inf)) == False + assert isnormal(mpc(nan,0)) == False + assert isnormal(mpc(inf,0)) == False + assert isnormal(mpc(nan,3)) == False + assert isnormal(mpc(inf,3)) == False + assert isnormal(mpc(inf,nan)) == False + assert isnormal(mpc(nan,inf)) == False + assert isnormal(mpc(nan,nan)) == False + assert isnormal(mpc(inf,inf)) == False + assert isnormal(mpq((3,2))) == True + assert isnormal(mpq((0,1))) == False + assert isint(3) == True + assert isint(0) == True + assert isint(long(3)) == True + assert isint(long(0)) == True + assert isint(mpf(3)) == True + assert isint(mpf(0)) == True + assert isint(mpf(-3)) == True + assert isint(mpf(3.2)) == False + assert isint(3.2) == False + assert isint(nan) == False + assert isint(inf) == False + assert isint(-inf) == False + assert isint(mpc(0)) == True + assert isint(mpc(3)) == True + assert isint(mpc(3.2)) == False + assert isint(mpc(3,inf)) == False + assert isint(mpc(inf)) == False + assert isint(mpc(3,2)) == False + assert isint(mpc(0,2)) == False + assert isint(mpc(3,2),gaussian=True) == True + assert isint(mpc(3,0),gaussian=True) == True + assert isint(mpc(0,3),gaussian=True) == True + assert isint(3+4j) == False + assert isint(3+4j, gaussian=True) == True + assert isint(3+0j) == True + assert isint(mpq((3,2))) == False + assert isint(mpq((3,9))) == False + assert isint(mpq((9,3))) == True + assert isint(mpq((0,4))) == True + assert isint(mpq((1,1))) == True + assert isint(mpq((-1,1))) == True + assert mp.isnpint(0) == True + assert mp.isnpint(1) == False + assert mp.isnpint(-1) == True + assert mp.isnpint(-1.1) == False + assert mp.isnpint(-1.0) == True + assert mp.isnpint(mp.mpq(1,2)) == False + assert mp.isnpint(mp.mpq(-1,2)) == False + assert mp.isnpint(mp.mpq(-3,1)) == True + assert mp.isnpint(mp.mpq(0,1)) == True + assert mp.isnpint(mp.mpq(1,1)) == False + assert mp.isnpint(0+0j) == True + assert mp.isnpint(-1+0j) == True + assert mp.isnpint(-1.1+0j) == False + assert mp.isnpint(-1+0.1j) == False + assert mp.isnpint(0+0.1j) == False + + +def test_issue_438(): + assert mpf(finf) == mpf('inf') + assert mpf(fninf) == mpf('-inf') + assert mpf(fnan)._mpf_ == mpf('nan')._mpf_ diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_bitwise.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_bitwise.py new file mode 100644 index 0000000000000000000000000000000000000000..4f61b69fc8819cf275abaedd98847c58c3b5924a --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_bitwise.py @@ -0,0 +1,188 @@ +""" +Test bit-level integer and mpf operations +""" + +from mpmath import * +from mpmath.libmp import * + +def test_bitcount(): + assert bitcount(0) == 0 + assert bitcount(1) == 1 + assert bitcount(7) == 3 + assert bitcount(8) == 4 + assert bitcount(2**100) == 101 + assert bitcount(2**100-1) == 100 + +def test_trailing(): + assert trailing(0) == 0 + assert trailing(1) == 0 + assert trailing(2) == 1 + assert trailing(7) == 0 + assert trailing(8) == 3 + assert trailing(2**100) == 100 + assert trailing(2**100-1) == 0 + +def test_round_down(): + assert from_man_exp(0, -4, 4, round_down)[:3] == (0, 0, 0) + assert from_man_exp(0xf0, -4, 4, round_down)[:3] == (0, 15, 0) + assert from_man_exp(0xf1, -4, 4, round_down)[:3] == (0, 15, 0) + assert from_man_exp(0xff, -4, 4, round_down)[:3] == (0, 15, 0) + assert from_man_exp(-0xf0, -4, 4, round_down)[:3] == (1, 15, 0) + assert from_man_exp(-0xf1, -4, 4, round_down)[:3] == (1, 15, 0) + assert from_man_exp(-0xff, -4, 4, round_down)[:3] == (1, 15, 0) + +def test_round_up(): + assert from_man_exp(0, -4, 4, round_up)[:3] == (0, 0, 0) + assert from_man_exp(0xf0, -4, 4, round_up)[:3] == (0, 15, 0) + assert from_man_exp(0xf1, -4, 4, round_up)[:3] == (0, 1, 4) + assert from_man_exp(0xff, -4, 4, round_up)[:3] == (0, 1, 4) + assert from_man_exp(-0xf0, -4, 4, round_up)[:3] == (1, 15, 0) + assert from_man_exp(-0xf1, -4, 4, round_up)[:3] == (1, 1, 4) + assert from_man_exp(-0xff, -4, 4, round_up)[:3] == (1, 1, 4) + +def test_round_floor(): + assert from_man_exp(0, -4, 4, round_floor)[:3] == (0, 0, 0) + assert from_man_exp(0xf0, -4, 4, round_floor)[:3] == (0, 15, 0) + assert from_man_exp(0xf1, -4, 4, round_floor)[:3] == (0, 15, 0) + assert from_man_exp(0xff, -4, 4, round_floor)[:3] == (0, 15, 0) + assert from_man_exp(-0xf0, -4, 4, round_floor)[:3] == (1, 15, 0) + assert from_man_exp(-0xf1, -4, 4, round_floor)[:3] == (1, 1, 4) + assert from_man_exp(-0xff, -4, 4, round_floor)[:3] == (1, 1, 4) + +def test_round_ceiling(): + assert from_man_exp(0, -4, 4, round_ceiling)[:3] == (0, 0, 0) + assert from_man_exp(0xf0, -4, 4, round_ceiling)[:3] == (0, 15, 0) + assert from_man_exp(0xf1, -4, 4, round_ceiling)[:3] == (0, 1, 4) + assert from_man_exp(0xff, -4, 4, round_ceiling)[:3] == (0, 1, 4) + assert from_man_exp(-0xf0, -4, 4, round_ceiling)[:3] == (1, 15, 0) + assert from_man_exp(-0xf1, -4, 4, round_ceiling)[:3] == (1, 15, 0) + assert from_man_exp(-0xff, -4, 4, round_ceiling)[:3] == (1, 15, 0) + +def test_round_nearest(): + assert from_man_exp(0, -4, 4, round_nearest)[:3] == (0, 0, 0) + assert from_man_exp(0xf0, -4, 4, round_nearest)[:3] == (0, 15, 0) + assert from_man_exp(0xf7, -4, 4, round_nearest)[:3] == (0, 15, 0) + assert from_man_exp(0xf8, -4, 4, round_nearest)[:3] == (0, 1, 4) # 1111.1000 -> 10000.0 + assert from_man_exp(0xf9, -4, 4, round_nearest)[:3] == (0, 1, 4) # 1111.1001 -> 10000.0 + assert from_man_exp(0xe8, -4, 4, round_nearest)[:3] == (0, 7, 1) # 1110.1000 -> 1110.0 + assert from_man_exp(0xe9, -4, 4, round_nearest)[:3] == (0, 15, 0) # 1110.1001 -> 1111.0 + assert from_man_exp(-0xf0, -4, 4, round_nearest)[:3] == (1, 15, 0) + assert from_man_exp(-0xf7, -4, 4, round_nearest)[:3] == (1, 15, 0) + assert from_man_exp(-0xf8, -4, 4, round_nearest)[:3] == (1, 1, 4) + assert from_man_exp(-0xf9, -4, 4, round_nearest)[:3] == (1, 1, 4) + assert from_man_exp(-0xe8, -4, 4, round_nearest)[:3] == (1, 7, 1) + assert from_man_exp(-0xe9, -4, 4, round_nearest)[:3] == (1, 15, 0) + +def test_rounding_bugs(): + # 1 less than power-of-two cases + assert from_man_exp(72057594037927935, -56, 53, round_up) == (0, 1, 0, 1) + assert from_man_exp(73786976294838205979, -65, 53, round_nearest) == (0, 1, 1, 1) + assert from_man_exp(31, 0, 4, round_up) == (0, 1, 5, 1) + assert from_man_exp(-31, 0, 4, round_floor) == (1, 1, 5, 1) + assert from_man_exp(255, 0, 7, round_up) == (0, 1, 8, 1) + assert from_man_exp(-255, 0, 7, round_floor) == (1, 1, 8, 1) + +def test_rounding_issue_200(): + a = from_man_exp(9867,-100) + b = from_man_exp(9867,-200) + c = from_man_exp(-1,0) + z = (1, 1023, -10, 10) + assert mpf_add(a, c, 10, 'd') == z + assert mpf_add(b, c, 10, 'd') == z + assert mpf_add(c, a, 10, 'd') == z + assert mpf_add(c, b, 10, 'd') == z + +def test_perturb(): + a = fone + b = from_float(0.99999999999999989) + c = from_float(1.0000000000000002) + assert mpf_perturb(a, 0, 53, round_nearest) == a + assert mpf_perturb(a, 1, 53, round_nearest) == a + assert mpf_perturb(a, 0, 53, round_up) == c + assert mpf_perturb(a, 0, 53, round_ceiling) == c + assert mpf_perturb(a, 0, 53, round_down) == a + assert mpf_perturb(a, 0, 53, round_floor) == a + assert mpf_perturb(a, 1, 53, round_up) == a + assert mpf_perturb(a, 1, 53, round_ceiling) == a + assert mpf_perturb(a, 1, 53, round_down) == b + assert mpf_perturb(a, 1, 53, round_floor) == b + a = mpf_neg(a) + b = mpf_neg(b) + c = mpf_neg(c) + assert mpf_perturb(a, 0, 53, round_nearest) == a + assert mpf_perturb(a, 1, 53, round_nearest) == a + assert mpf_perturb(a, 0, 53, round_up) == a + assert mpf_perturb(a, 0, 53, round_floor) == a + assert mpf_perturb(a, 0, 53, round_down) == b + assert mpf_perturb(a, 0, 53, round_ceiling) == b + assert mpf_perturb(a, 1, 53, round_up) == c + assert mpf_perturb(a, 1, 53, round_floor) == c + assert mpf_perturb(a, 1, 53, round_down) == a + assert mpf_perturb(a, 1, 53, round_ceiling) == a + +def test_add_exact(): + ff = from_float + assert mpf_add(ff(3.0), ff(2.5)) == ff(5.5) + assert mpf_add(ff(3.0), ff(-2.5)) == ff(0.5) + assert mpf_add(ff(-3.0), ff(2.5)) == ff(-0.5) + assert mpf_add(ff(-3.0), ff(-2.5)) == ff(-5.5) + assert mpf_sub(mpf_add(fone, ff(1e-100)), fone) == ff(1e-100) + assert mpf_sub(mpf_add(ff(1e-100), fone), fone) == ff(1e-100) + assert mpf_sub(mpf_add(fone, ff(-1e-100)), fone) == ff(-1e-100) + assert mpf_sub(mpf_add(ff(-1e-100), fone), fone) == ff(-1e-100) + assert mpf_add(fone, fzero) == fone + assert mpf_add(fzero, fone) == fone + assert mpf_add(fzero, fzero) == fzero + +def test_long_exponent_shifts(): + mp.dps = 15 + # Check for possible bugs due to exponent arithmetic overflow + # in a C implementation + x = mpf(1) + for p in [32, 64]: + a = ldexp(1,2**(p-1)) + b = ldexp(1,2**p) + c = ldexp(1,2**(p+1)) + d = ldexp(1,-2**(p-1)) + e = ldexp(1,-2**p) + f = ldexp(1,-2**(p+1)) + assert (x+a) == a + assert (x+b) == b + assert (x+c) == c + assert (x+d) == x + assert (x+e) == x + assert (x+f) == x + assert (a+x) == a + assert (b+x) == b + assert (c+x) == c + assert (d+x) == x + assert (e+x) == x + assert (f+x) == x + assert (x-a) == -a + assert (x-b) == -b + assert (x-c) == -c + assert (x-d) == x + assert (x-e) == x + assert (x-f) == x + assert (a-x) == a + assert (b-x) == b + assert (c-x) == c + assert (d-x) == -x + assert (e-x) == -x + assert (f-x) == -x + +def test_float_rounding(): + mp.prec = 64 + for x in [mpf(1), mpf(1)+eps, mpf(1)-eps, -mpf(1)+eps, -mpf(1)-eps]: + fa = float(x) + fb = float(fadd(x,0,prec=53,rounding='n')) + assert fa == fb + z = mpc(x,x) + ca = complex(z) + cb = complex(fadd(z,0,prec=53,rounding='n')) + assert ca == cb + for rnd in ['n', 'd', 'u', 'f', 'c']: + fa = to_float(x._mpf_, rnd=rnd) + fb = to_float(fadd(x,0,prec=53,rounding=rnd)._mpf_, rnd=rnd) + assert fa == fb + mp.prec = 53 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_calculus.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_calculus.py new file mode 100644 index 0000000000000000000000000000000000000000..f0a59773d672f0db20bb5072773472a5a3cc1d1f --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_calculus.py @@ -0,0 +1,216 @@ +import pytest +from mpmath import * + +def test_approximation(): + mp.dps = 15 + f = lambda x: cos(2-2*x)/x + p, err = chebyfit(f, [2, 4], 8, error=True) + assert err < 1e-5 + for i in range(10): + x = 2 + i/5. + assert abs(polyval(p, x) - f(x)) < err + +def test_limits(): + mp.dps = 15 + assert limit(lambda x: (x-sin(x))/x**3, 0).ae(mpf(1)/6) + assert limit(lambda n: (1+1/n)**n, inf).ae(e) + +def test_polyval(): + assert polyval([], 3) == 0 + assert polyval([0], 3) == 0 + assert polyval([5], 3) == 5 + # 4x^3 - 2x + 5 + p = [4, 0, -2, 5] + assert polyval(p,4) == 253 + assert polyval(p,4,derivative=True) == (253, 190) + +def test_polyroots(): + p = polyroots([1,-4]) + assert p[0].ae(4) + p, q = polyroots([1,2,3]) + assert p.ae(-1 - sqrt(2)*j) + assert q.ae(-1 + sqrt(2)*j) + #this is not a real test, it only tests a specific case + assert polyroots([1]) == [] + pytest.raises(ValueError, lambda: polyroots([0])) + +def test_polyroots_legendre(): + n = 64 + coeffs = [11975573020964041433067793888190275875, 0, + -190100434726484311252477736051902332000, 0, + 1437919688271127330313741595496589239248, 0, + -6897338342113537600691931230430793911840, 0, + 23556405536185284408974715545252277554280, 0, + -60969520211303089058522793175947071316960, 0, + 124284021969194758465450309166353645376880, 0, + -204721258548015217049921875719981284186016, 0, + 277415422258095841688223780704620656114900, 0, + -313237834141273382807123548182995095192800, 0, + 297432255354328395601259515935229287637200, 0, + -239057700565161140389797367947941296605600, 0, + 163356095386193445933028201431093219347160, 0, + -95158890516229191805647495979277603503200, 0, + 47310254620162038075933656063247634556400, 0, + -20071017111583894941305187420771723751200, 0, + 7255051932731034189479516844750603752850, 0, + -2228176940331017311443863996901733412640, 0, + 579006552594977616773047095969088431600, 0, + -126584428502545713788439446082310831200, 0, + 23112325428835593809686977515028663000, 0, + -3491517141958743235617737161547844000, 0, + 431305058712550634988073414073557200, 0, + -42927166660756742088912492757452000, 0, + 3378527005707706553294038781836500, 0, + -205277590220215081719131470288800, 0, + 9330799555464321896324157740400, 0, + -304114948474392713657972548576, 0, + 6695289961520387531608984680, 0, + -91048139350447232095702560, 0, + 659769125727878493447120, 0, + -1905929106580294155360, 0, + 916312070471295267] + + with mp.workdps(3): + with pytest.raises(mp.NoConvergence): + polyroots(coeffs, maxsteps=5, cleanup=True, error=False, + extraprec=n*10) + + roots = polyroots(coeffs, maxsteps=50, cleanup=True, error=False, + extraprec=n*10) + roots = [str(r) for r in roots] + assert roots == \ + ['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961', + '-0.946', '-0.93', '-0.911', '-0.889', '-0.866', '-0.841', + '-0.813', '-0.784', '-0.753', '-0.72', '-0.685', '-0.649', + '-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402', + '-0.357', '-0.311', '-0.265', '-0.217', '-0.17', '-0.121', + '-0.073', '-0.0243', '0.0243', '0.073', '0.121', '0.17', '0.217', + '0.265', '0.311', '0.357', '0.402', '0.446', '0.489', '0.531', + '0.572', '0.611', '0.649', '0.685', '0.72', '0.753', '0.784', + '0.813', '0.841', '0.866', '0.889', '0.911', '0.93', '0.946', + '0.961', '0.973', '0.983', '0.991', '0.996', '0.999'] + +def test_polyroots_legendre_init(): + extra_prec = 100 + coeffs = [11975573020964041433067793888190275875, 0, + -190100434726484311252477736051902332000, 0, + 1437919688271127330313741595496589239248, 0, + -6897338342113537600691931230430793911840, 0, + 23556405536185284408974715545252277554280, 0, + -60969520211303089058522793175947071316960, 0, + 124284021969194758465450309166353645376880, 0, + -204721258548015217049921875719981284186016, 0, + 277415422258095841688223780704620656114900, 0, + -313237834141273382807123548182995095192800, 0, + 297432255354328395601259515935229287637200, 0, + -239057700565161140389797367947941296605600, 0, + 163356095386193445933028201431093219347160, 0, + -95158890516229191805647495979277603503200, 0, + 47310254620162038075933656063247634556400, 0, + -20071017111583894941305187420771723751200, 0, + 7255051932731034189479516844750603752850, 0, + -2228176940331017311443863996901733412640, 0, + 579006552594977616773047095969088431600, 0, + -126584428502545713788439446082310831200, 0, + 23112325428835593809686977515028663000, 0, + -3491517141958743235617737161547844000, 0, + 431305058712550634988073414073557200, 0, + -42927166660756742088912492757452000, 0, + 3378527005707706553294038781836500, 0, + -205277590220215081719131470288800, 0, + 9330799555464321896324157740400, 0, + -304114948474392713657972548576, 0, + 6695289961520387531608984680, 0, + -91048139350447232095702560, 0, + 659769125727878493447120, 0, + -1905929106580294155360, 0, + 916312070471295267] + + roots_init = matrix(['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', + '-0.961', '-0.946', '-0.93', '-0.911', '-0.889', + '-0.866', '-0.841', '-0.813', '-0.784', '-0.753', + '-0.72', '-0.685', '-0.649', '-0.611', '-0.572', + '-0.531', '-0.489', '-0.446', '-0.402', '-0.357', + '-0.311', '-0.265', '-0.217', '-0.17', '-0.121', + '-0.073', '-0.0243', '0.0243', '0.073', '0.121', + '0.17', '0.217', '0.265', ' 0.311', '0.357', + '0.402', '0.446', '0.489', '0.531', '0.572', + '0.611', '0.649', '0.685', '0.72', '0.753', + '0.784', '0.813', '0.841', '0.866', '0.889', + '0.911', '0.93', '0.946', '0.961', '0.973', + '0.983', '0.991', '0.996', '0.999', '1.0']) + with mp.workdps(2*mp.dps): + roots_exact = polyroots(coeffs, maxsteps=50, cleanup=True, error=False, + extraprec=2*extra_prec) + with pytest.raises(mp.NoConvergence): + polyroots(coeffs, maxsteps=5, cleanup=True, error=False, + extraprec=extra_prec) + roots,err = polyroots(coeffs, maxsteps=5, cleanup=True, error=True, + extraprec=extra_prec,roots_init=roots_init) + assert max(matrix(roots_exact)-matrix(roots).apply(abs)) < err + roots1,err1 = polyroots(coeffs, maxsteps=25, cleanup=True, error=True, + extraprec=extra_prec,roots_init=roots_init[:60]) + assert max(matrix(roots_exact)-matrix(roots1).apply(abs)) < err1 + +def test_pade(): + one = mpf(1) + mp.dps = 20 + N = 10 + a = [one] + k = 1 + for i in range(1, N+1): + k *= i + a.append(one/k) + p, q = pade(a, N//2, N//2) + for x in arange(0, 1, 0.1): + r = polyval(p[::-1], x)/polyval(q[::-1], x) + assert(r.ae(exp(x), 1.0e-10)) + mp.dps = 15 + +def test_fourier(): + mp.dps = 15 + c, s = fourier(lambda x: x+1, [-1, 2], 2) + #plot([lambda x: x+1, lambda x: fourierval((c, s), [-1, 2], x)], [-1, 2]) + assert c[0].ae(1.5) + assert c[1].ae(-3*sqrt(3)/(2*pi)) + assert c[2].ae(3*sqrt(3)/(4*pi)) + assert s[0] == 0 + assert s[1].ae(3/(2*pi)) + assert s[2].ae(3/(4*pi)) + assert fourierval((c, s), [-1, 2], 1).ae(1.9134966715663442) + +def test_differint(): + mp.dps = 15 + assert differint(lambda t: t, 2, -0.5).ae(8*sqrt(2/pi)/3) + +def test_invlap(): + mp.dps = 15 + t = 0.01 + fp = lambda p: 1/(p+1)**2 + ft = lambda t: t*exp(-t) + ftt = ft(t) + assert invertlaplace(fp,t,method='talbot').ae(ftt) + assert invertlaplace(fp,t,method='stehfest').ae(ftt) + assert invertlaplace(fp,t,method='dehoog').ae(ftt) + assert invertlaplace(fp,t,method='cohen').ae(ftt) + t = 1.0 + ftt = ft(t) + assert invertlaplace(fp,t,method='talbot').ae(ftt) + assert invertlaplace(fp,t,method='stehfest').ae(ftt) + assert invertlaplace(fp,t,method='dehoog').ae(ftt) + assert invertlaplace(fp,t,method='cohen').ae(ftt) + + t = 0.01 + fp = lambda p: log(p)/p + ft = lambda t: -euler-log(t) + ftt = ft(t) + assert invertlaplace(fp,t,method='talbot').ae(ftt) + assert invertlaplace(fp,t,method='stehfest').ae(ftt) + assert invertlaplace(fp,t,method='dehoog').ae(ftt) + assert invertlaplace(fp,t,method='cohen').ae(ftt) + t = 1.0 + ftt = ft(t) + assert invertlaplace(fp,t,method='talbot').ae(ftt) + assert invertlaplace(fp,t,method='stehfest').ae(ftt) + assert invertlaplace(fp,t,method='dehoog').ae(ftt) + assert invertlaplace(fp,t,method='cohen').ae(ftt) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_compatibility.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_compatibility.py new file mode 100644 index 0000000000000000000000000000000000000000..f26d6044b521306b6d1eaeadc5c7839be226dc54 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_compatibility.py @@ -0,0 +1,77 @@ +from mpmath import * +from random import seed, randint, random +import math + +# Test compatibility with Python floats, which are +# IEEE doubles (53-bit) + +N = 5000 +seed(1) + +# Choosing exponents between roughly -140, 140 ensures that +# the Python floats don't overflow or underflow +xs = [(random()-1) * 10**randint(-140, 140) for x in range(N)] +ys = [(random()-1) * 10**randint(-140, 140) for x in range(N)] + +# include some equal values +ys[int(N*0.8):] = xs[int(N*0.8):] + +# Detect whether Python is compiled to use 80-bit floating-point +# instructions, in which case the double compatibility test breaks +uses_x87 = -4.1974624032366689e+117 / -8.4657370748010221e-47 \ + == 4.9581771393902231e+163 + +def test_double_compatibility(): + mp.prec = 53 + for x, y in zip(xs, ys): + mpx = mpf(x) + mpy = mpf(y) + assert mpf(x) == x + assert (mpx < mpy) == (x < y) + assert (mpx > mpy) == (x > y) + assert (mpx == mpy) == (x == y) + assert (mpx != mpy) == (x != y) + assert (mpx <= mpy) == (x <= y) + assert (mpx >= mpy) == (x >= y) + assert mpx == mpx + if uses_x87: + mp.prec = 64 + a = mpx + mpy + b = mpx * mpy + c = mpx / mpy + d = mpx % mpy + mp.prec = 53 + assert +a == x + y + assert +b == x * y + assert +c == x / y + assert +d == x % y + else: + assert mpx + mpy == x + y + assert mpx * mpy == x * y + assert mpx / mpy == x / y + assert mpx % mpy == x % y + assert abs(mpx) == abs(x) + assert mpf(repr(x)) == x + assert ceil(mpx) == math.ceil(x) + assert floor(mpx) == math.floor(x) + +def test_sqrt(): + # this fails quite often. it appers to be float + # that rounds the wrong way, not mpf + fail = 0 + mp.prec = 53 + for x in xs: + x = abs(x) + mp.prec = 100 + mp_high = mpf(x)**0.5 + mp.prec = 53 + mp_low = mpf(x)**0.5 + fp = x**0.5 + assert abs(mp_low-mp_high) <= abs(fp-mp_high) + fail += mp_low != fp + assert fail < N/10 + +def test_bugs(): + # particular bugs + assert mpf(4.4408920985006262E-16) < mpf(1.7763568394002505E-15) + assert mpf(-4.4408920985006262E-16) > mpf(-1.7763568394002505E-15) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_convert.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_convert.py new file mode 100644 index 0000000000000000000000000000000000000000..cb1db5b55c89e980e08fc3fa43cc9715ad68cac9 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_convert.py @@ -0,0 +1,233 @@ +import random +from mpmath import * +from mpmath.libmp import * + + +def test_basic_string(): + """ + Test basic string conversion + """ + mp.dps = 15 + assert mpf('3') == mpf('3.0') == mpf('0003.') == mpf('0.03e2') == mpf(3.0) + assert mpf('30') == mpf('30.0') == mpf('00030.') == mpf(30.0) + for i in range(10): + for j in range(10): + assert mpf('%ie%i' % (i,j)) == i * 10**j + assert str(mpf('25000.0')) == '25000.0' + assert str(mpf('2500.0')) == '2500.0' + assert str(mpf('250.0')) == '250.0' + assert str(mpf('25.0')) == '25.0' + assert str(mpf('2.5')) == '2.5' + assert str(mpf('0.25')) == '0.25' + assert str(mpf('0.025')) == '0.025' + assert str(mpf('0.0025')) == '0.0025' + assert str(mpf('0.00025')) == '0.00025' + assert str(mpf('0.000025')) == '2.5e-5' + assert str(mpf(0)) == '0.0' + assert str(mpf('2.5e1000000000000000000000')) == '2.5e+1000000000000000000000' + assert str(mpf('2.6e-1000000000000000000000')) == '2.6e-1000000000000000000000' + assert str(mpf(1.23402834e-15)) == '1.23402834e-15' + assert str(mpf(-1.23402834e-15)) == '-1.23402834e-15' + assert str(mpf(-1.2344e-15)) == '-1.2344e-15' + assert repr(mpf(-1.2344e-15)) == "mpf('-1.2343999999999999e-15')" + assert str(mpf("2163048125L")) == '2163048125.0' + assert str(mpf("-2163048125l")) == '-2163048125.0' + assert str(mpf("-2163048125L/1088391168")) == '-1.98738118113799' + assert str(mpf("2163048125/1088391168l")) == '1.98738118113799' + +def test_pretty(): + mp.pretty = True + assert repr(mpf(2.5)) == '2.5' + assert repr(mpc(2.5,3.5)) == '(2.5 + 3.5j)' + mp.pretty = False + iv.pretty = True + assert repr(mpi(2.5,3.5)) == '[2.5, 3.5]' + iv.pretty = False + +def test_str_whitespace(): + assert mpf('1.26 ') == 1.26 + +def test_unicode(): + mp.dps = 15 + try: + unicode = unicode + except NameError: + unicode = str + assert mpf(unicode('2.76')) == 2.76 + assert mpf(unicode('inf')) == inf + +def test_str_format(): + assert to_str(from_float(0.1),15,strip_zeros=False) == '0.100000000000000' + assert to_str(from_float(0.0),15,show_zero_exponent=True) == '0.0e+0' + assert to_str(from_float(0.0),0,show_zero_exponent=True) == '.0e+0' + assert to_str(from_float(0.0),0,show_zero_exponent=False) == '.0' + assert to_str(from_float(0.0),1,show_zero_exponent=True) == '0.0e+0' + assert to_str(from_float(0.0),1,show_zero_exponent=False) == '0.0' + assert to_str(from_float(1.23),3,show_zero_exponent=True) == '1.23e+0' + assert to_str(from_float(1.23456789000000e-2),15,strip_zeros=False,min_fixed=0,max_fixed=0) == '1.23456789000000e-2' + assert to_str(from_float(1.23456789000000e+2),15,strip_zeros=False,min_fixed=0,max_fixed=0) == '1.23456789000000e+2' + assert to_str(from_float(2.1287e14), 15, max_fixed=1000) == '212870000000000.0' + assert to_str(from_float(2.1287e15), 15, max_fixed=1000) == '2128700000000000.0' + assert to_str(from_float(2.1287e16), 15, max_fixed=1000) == '21287000000000000.0' + assert to_str(from_float(2.1287e30), 15, max_fixed=1000) == '2128700000000000000000000000000.0' + +def test_tight_string_conversion(): + mp.dps = 15 + # In an old version, '0.5' wasn't recognized as representing + # an exact binary number and was erroneously rounded up or down + assert from_str('0.5', 10, round_floor) == fhalf + assert from_str('0.5', 10, round_ceiling) == fhalf + +def test_eval_repr_invariant(): + """Test that eval(repr(x)) == x""" + random.seed(123) + for dps in [10, 15, 20, 50, 100]: + mp.dps = dps + for i in range(1000): + a = mpf(random.random())**0.5 * 10**random.randint(-100, 100) + assert eval(repr(a)) == a + mp.dps = 15 + +def test_str_bugs(): + mp.dps = 15 + # Decimal rounding used to give the wrong exponent in some cases + assert str(mpf('1e600')) == '1.0e+600' + assert str(mpf('1e10000')) == '1.0e+10000' + +def test_str_prec0(): + assert to_str(from_float(1.234), 0) == '.0e+0' + assert to_str(from_float(1e-15), 0) == '.0e-15' + assert to_str(from_float(1e+15), 0) == '.0e+15' + assert to_str(from_float(-1e-15), 0) == '-.0e-15' + assert to_str(from_float(-1e+15), 0) == '-.0e+15' + +def test_convert_rational(): + mp.dps = 15 + assert from_rational(30, 5, 53, round_nearest) == (0, 3, 1, 2) + assert from_rational(-7, 4, 53, round_nearest) == (1, 7, -2, 3) + assert to_rational((0, 1, -1, 1)) == (1, 2) + +def test_custom_class(): + class mympf: + @property + def _mpf_(self): + return mpf(3.5)._mpf_ + class mympc: + @property + def _mpc_(self): + return mpf(3.5)._mpf_, mpf(2.5)._mpf_ + assert mpf(2) + mympf() == 5.5 + assert mympf() + mpf(2) == 5.5 + assert mpf(mympf()) == 3.5 + assert mympc() + mpc(2) == mpc(5.5, 2.5) + assert mpc(2) + mympc() == mpc(5.5, 2.5) + assert mpc(mympc()) == (3.5+2.5j) + +def test_conversion_methods(): + class SomethingRandom: + pass + class SomethingReal: + def _mpmath_(self, prec, rounding): + return mp.make_mpf(from_str('1.3', prec, rounding)) + class SomethingComplex: + def _mpmath_(self, prec, rounding): + return mp.make_mpc((from_str('1.3', prec, rounding), \ + from_str('1.7', prec, rounding))) + x = mpf(3) + z = mpc(3) + a = SomethingRandom() + y = SomethingReal() + w = SomethingComplex() + for d in [15, 45]: + mp.dps = d + assert (x+y).ae(mpf('4.3')) + assert (y+x).ae(mpf('4.3')) + assert (x+w).ae(mpc('4.3', '1.7')) + assert (w+x).ae(mpc('4.3', '1.7')) + assert (z+y).ae(mpc('4.3')) + assert (y+z).ae(mpc('4.3')) + assert (z+w).ae(mpc('4.3', '1.7')) + assert (w+z).ae(mpc('4.3', '1.7')) + x-y; y-x; x-w; w-x; z-y; y-z; z-w; w-z + x*y; y*x; x*w; w*x; z*y; y*z; z*w; w*z + x/y; y/x; x/w; w/x; z/y; y/z; z/w; w/z + x**y; y**x; x**w; w**x; z**y; y**z; z**w; w**z + x==y; y==x; x==w; w==x; z==y; y==z; z==w; w==z + mp.dps = 15 + assert x.__add__(a) is NotImplemented + assert x.__radd__(a) is NotImplemented + assert x.__lt__(a) is NotImplemented + assert x.__gt__(a) is NotImplemented + assert x.__le__(a) is NotImplemented + assert x.__ge__(a) is NotImplemented + assert x.__eq__(a) is NotImplemented + assert x.__ne__(a) is NotImplemented + # implementation detail + if hasattr(x, "__cmp__"): + assert x.__cmp__(a) is NotImplemented + assert x.__sub__(a) is NotImplemented + assert x.__rsub__(a) is NotImplemented + assert x.__mul__(a) is NotImplemented + assert x.__rmul__(a) is NotImplemented + assert x.__div__(a) is NotImplemented + assert x.__rdiv__(a) is NotImplemented + assert x.__mod__(a) is NotImplemented + assert x.__rmod__(a) is NotImplemented + assert x.__pow__(a) is NotImplemented + assert x.__rpow__(a) is NotImplemented + assert z.__add__(a) is NotImplemented + assert z.__radd__(a) is NotImplemented + assert z.__eq__(a) is NotImplemented + assert z.__ne__(a) is NotImplemented + assert z.__sub__(a) is NotImplemented + assert z.__rsub__(a) is NotImplemented + assert z.__mul__(a) is NotImplemented + assert z.__rmul__(a) is NotImplemented + assert z.__div__(a) is NotImplemented + assert z.__rdiv__(a) is NotImplemented + assert z.__pow__(a) is NotImplemented + assert z.__rpow__(a) is NotImplemented + +def test_mpmathify(): + assert mpmathify('1/2') == 0.5 + assert mpmathify('(1.0+1.0j)') == mpc(1, 1) + assert mpmathify('(1.2e-10 - 3.4e5j)') == mpc('1.2e-10', '-3.4e5') + assert mpmathify('1j') == mpc(1j) + +def test_issue548(): + try: + # This expression is invalid, but may trigger the ReDOS vulnerability + # in the regular expression for parsing complex numbers. + mpmathify('(' + '1' * 5000 + '!j') + except: + return + # The expression is invalid and should raise an exception. + assert False + +def test_compatibility(): + try: + import numpy as np + from fractions import Fraction + from decimal import Decimal + import decimal + except ImportError: + return + # numpy types + for nptype in np.core.numerictypes.typeDict.values(): + if issubclass(nptype, np.complexfloating): + x = nptype(complex(0.5, -0.5)) + elif issubclass(nptype, np.floating): + x = nptype(0.5) + elif issubclass(nptype, np.integer): + x = nptype(2) + # Handle the weird types + try: diff = np.abs(type(np.sqrt(x))(sqrt(x)) - np.sqrt(x)) + except: continue + assert diff < 2.0**-53 + #Fraction and Decimal + oldprec = mp.prec + mp.prec = 1000 + decimal.getcontext().prec = mp.dps + assert sqrt(Fraction(2, 3)).ae(sqrt(mpf('2/3'))) + assert sqrt(Decimal(2)/Decimal(3)).ae(sqrt(mpf('2/3'))) + mp.prec = oldprec diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_diff.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_diff.py new file mode 100644 index 0000000000000000000000000000000000000000..f5711609da38862eb4fd62c88d35f1704c9425a4 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_diff.py @@ -0,0 +1,61 @@ +from mpmath import * + +def test_diff(): + mp.dps = 15 + assert diff(log, 2.0, n=0).ae(log(2)) + assert diff(cos, 1.0).ae(-sin(1)) + assert diff(abs, 0.0) == 0 + assert diff(abs, 0.0, direction=1) == 1 + assert diff(abs, 0.0, direction=-1) == -1 + assert diff(exp, 1.0).ae(e) + assert diff(exp, 1.0, n=5).ae(e) + assert diff(exp, 2.0, n=5, direction=3*j).ae(e**2) + assert diff(lambda x: x**2, 3.0, method='quad').ae(6) + assert diff(lambda x: 3+x**5, 3.0, n=2, method='quad').ae(540) + assert diff(lambda x: 3+x**5, 3.0, n=2, method='step').ae(540) + assert diffun(sin)(2).ae(cos(2)) + assert diffun(sin, n=2)(2).ae(-sin(2)) + +def test_diffs(): + mp.dps = 15 + assert [chop(d) for d in diffs(sin, 0, 1)] == [0, 1] + assert [chop(d) for d in diffs(sin, 0, 1, method='quad')] == [0, 1] + assert [chop(d) for d in diffs(sin, 0, 2)] == [0, 1, 0] + assert [chop(d) for d in diffs(sin, 0, 2, method='quad')] == [0, 1, 0] + +def test_taylor(): + mp.dps = 15 + # Easy to test since the coefficients are exact in floating-point + assert taylor(sqrt, 1, 4) == [1, 0.5, -0.125, 0.0625, -0.0390625] + +def test_diff_partial(): + mp.dps = 15 + x,y,z = xyz = 2,3,7 + f = lambda x,y,z: 3*x**2 * (y+2)**3 * z**5 + assert diff(f, xyz, (0,0,0)).ae(25210500) + assert diff(f, xyz, (0,0,1)).ae(18007500) + assert diff(f, xyz, (0,0,2)).ae(10290000) + assert diff(f, xyz, (0,1,0)).ae(15126300) + assert diff(f, xyz, (0,1,1)).ae(10804500) + assert diff(f, xyz, (0,1,2)).ae(6174000) + assert diff(f, xyz, (0,2,0)).ae(6050520) + assert diff(f, xyz, (0,2,1)).ae(4321800) + assert diff(f, xyz, (0,2,2)).ae(2469600) + assert diff(f, xyz, (1,0,0)).ae(25210500) + assert diff(f, xyz, (1,0,1)).ae(18007500) + assert diff(f, xyz, (1,0,2)).ae(10290000) + assert diff(f, xyz, (1,1,0)).ae(15126300) + assert diff(f, xyz, (1,1,1)).ae(10804500) + assert diff(f, xyz, (1,1,2)).ae(6174000) + assert diff(f, xyz, (1,2,0)).ae(6050520) + assert diff(f, xyz, (1,2,1)).ae(4321800) + assert diff(f, xyz, (1,2,2)).ae(2469600) + assert diff(f, xyz, (2,0,0)).ae(12605250) + assert diff(f, xyz, (2,0,1)).ae(9003750) + assert diff(f, xyz, (2,0,2)).ae(5145000) + assert diff(f, xyz, (2,1,0)).ae(7563150) + assert diff(f, xyz, (2,1,1)).ae(5402250) + assert diff(f, xyz, (2,1,2)).ae(3087000) + assert diff(f, xyz, (2,2,0)).ae(3025260) + assert diff(f, xyz, (2,2,1)).ae(2160900) + assert diff(f, xyz, (2,2,2)).ae(1234800) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_division.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_division.py new file mode 100644 index 0000000000000000000000000000000000000000..c704cadeb953793ac0a887aa09c4278cf68a2824 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_division.py @@ -0,0 +1,143 @@ +from mpmath.libmp import * +from mpmath import mpf, mp + +from random import randint, choice, seed + +all_modes = [round_floor, round_ceiling, round_down, round_up, round_nearest] + +fb = from_bstr +fi = from_int +ff = from_float + + +def test_div_1_3(): + a = fi(1) + b = fi(3) + c = fi(-1) + + # floor rounds down, ceiling rounds up + assert mpf_div(a, b, 7, round_floor) == fb('0.01010101') + assert mpf_div(a, b, 7, round_ceiling) == fb('0.01010110') + assert mpf_div(a, b, 7, round_down) == fb('0.01010101') + assert mpf_div(a, b, 7, round_up) == fb('0.01010110') + assert mpf_div(a, b, 7, round_nearest) == fb('0.01010101') + + # floor rounds up, ceiling rounds down + assert mpf_div(c, b, 7, round_floor) == fb('-0.01010110') + assert mpf_div(c, b, 7, round_ceiling) == fb('-0.01010101') + assert mpf_div(c, b, 7, round_down) == fb('-0.01010101') + assert mpf_div(c, b, 7, round_up) == fb('-0.01010110') + assert mpf_div(c, b, 7, round_nearest) == fb('-0.01010101') + +def test_mpf_divi_1_3(): + a = 1 + b = fi(3) + c = -1 + assert mpf_rdiv_int(a, b, 7, round_floor) == fb('0.01010101') + assert mpf_rdiv_int(a, b, 7, round_ceiling) == fb('0.01010110') + assert mpf_rdiv_int(a, b, 7, round_down) == fb('0.01010101') + assert mpf_rdiv_int(a, b, 7, round_up) == fb('0.01010110') + assert mpf_rdiv_int(a, b, 7, round_nearest) == fb('0.01010101') + assert mpf_rdiv_int(c, b, 7, round_floor) == fb('-0.01010110') + assert mpf_rdiv_int(c, b, 7, round_ceiling) == fb('-0.01010101') + assert mpf_rdiv_int(c, b, 7, round_down) == fb('-0.01010101') + assert mpf_rdiv_int(c, b, 7, round_up) == fb('-0.01010110') + assert mpf_rdiv_int(c, b, 7, round_nearest) == fb('-0.01010101') + + +def test_div_300(): + + q = fi(1000000) + a = fi(300499999) # a/q is a little less than a half-integer + b = fi(300500000) # b/q exactly a half-integer + c = fi(300500001) # c/q is a little more than a half-integer + + # Check nearest integer rounding (prec=9 as 2**8 < 300 < 2**9) + + assert mpf_div(a, q, 9, round_down) == fi(300) + assert mpf_div(b, q, 9, round_down) == fi(300) + assert mpf_div(c, q, 9, round_down) == fi(300) + assert mpf_div(a, q, 9, round_up) == fi(301) + assert mpf_div(b, q, 9, round_up) == fi(301) + assert mpf_div(c, q, 9, round_up) == fi(301) + + # Nearest even integer is down + assert mpf_div(a, q, 9, round_nearest) == fi(300) + assert mpf_div(b, q, 9, round_nearest) == fi(300) + assert mpf_div(c, q, 9, round_nearest) == fi(301) + + # Nearest even integer is up + a = fi(301499999) + b = fi(301500000) + c = fi(301500001) + assert mpf_div(a, q, 9, round_nearest) == fi(301) + assert mpf_div(b, q, 9, round_nearest) == fi(302) + assert mpf_div(c, q, 9, round_nearest) == fi(302) + + +def test_tight_integer_division(): + # Test that integer division at tightest possible precision is exact + N = 100 + seed(1) + for i in range(N): + a = choice([1, -1]) * randint(1, 1< 1: + print("original matrix (hessenberg):\n", A) + + n = A.rows + + Q, H = mp.hessenberg(A) + + if verbose > 1: + print("Q:\n",Q) + print("H:\n",H) + + B = Q * H * Q.transpose_conj() + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + err0 = 0 + for x in xrange(n): + for y in xrange(n): + err0 += abs(A[y,x] - B[y,x]) + err0 /= n * n + + err1 = 0 + for x in xrange(n): + for y in xrange(x + 2, n): + err1 += abs(H[y,x]) + + if verbose > 0: + print("difference (H):", err0, err1) + + if verbose > 1: + print("B:\n", B) + + assert err0 < eps + assert err1 == 0 + + +def run_schur(A, verbose = 0): + if verbose > 1: + print("original matrix (schur):\n", A) + + n = A.rows + + Q, R = mp.schur(A) + + if verbose > 1: + print("Q:\n", Q) + print("R:\n", R) + + B = Q * R * Q.transpose_conj() + C = Q * Q.transpose_conj() + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + err0 = 0 + for x in xrange(n): + for y in xrange(n): + err0 += abs(A[y,x] - B[y,x]) + err0 /= n * n + + err1 = 0 + for x in xrange(n): + for y in xrange(n): + if x == y: + C[y,x] -= 1 + err1 += abs(C[y,x]) + err1 /= n * n + + err2 = 0 + for x in xrange(n): + for y in xrange(x + 1, n): + err2 += abs(R[y,x]) + + if verbose > 0: + print("difference (S):", err0, err1, err2) + + if verbose > 1: + print("B:\n", B) + + assert err0 < eps + assert err1 < eps + assert err2 == 0 + +def run_eig(A, verbose = 0): + if verbose > 1: + print("original matrix (eig):\n", A) + + n = A.rows + + E, EL, ER = mp.eig(A, left = True, right = True) + + if verbose > 1: + print("E:\n", E) + print("EL:\n", EL) + print("ER:\n", ER) + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + err0 = 0 + for i in xrange(n): + B = A * ER[:,i] - E[i] * ER[:,i] + err0 = max(err0, mp.mnorm(B)) + + B = EL[i,:] * A - EL[i,:] * E[i] + err0 = max(err0, mp.mnorm(B)) + + err0 /= n * n + + if verbose > 0: + print("difference (E):", err0) + + assert err0 < eps + +##################### + +def test_eig_dyn(): + v = 0 + for i in xrange(5): + n = 1 + int(mp.rand() * 5) + if mp.rand() > 0.5: + # real + A = 2 * mp.randmatrix(n, n) - 1 + if mp.rand() > 0.5: + A *= 10 + for x in xrange(n): + for y in xrange(n): + A[x,y] = int(A[x,y]) + else: + A = (2 * mp.randmatrix(n, n) - 1) + 1j * (2 * mp.randmatrix(n, n) - 1) + if mp.rand() > 0.5: + A *= 10 + for x in xrange(n): + for y in xrange(n): + A[x,y] = int(mp.re(A[x,y])) + 1j * int(mp.im(A[x,y])) + + run_hessenberg(A, verbose = v) + run_schur(A, verbose = v) + run_eig(A, verbose = v) + +def test_eig(): + v = 0 + AS = [] + + A = mp.matrix([[2, 1, 0], # jordan block of size 3 + [0, 2, 1], + [0, 0, 2]]) + AS.append(A) + AS.append(A.transpose()) + + A = mp.matrix([[2, 0, 0], # jordan block of size 2 + [0, 2, 1], + [0, 0, 2]]) + AS.append(A) + AS.append(A.transpose()) + + A = mp.matrix([[2, 0, 1], # jordan block of size 2 + [0, 2, 0], + [0, 0, 2]]) + AS.append(A) + AS.append(A.transpose()) + + A= mp.matrix([[0, 0, 1], # cyclic + [1, 0, 0], + [0, 1, 0]]) + AS.append(A) + AS.append(A.transpose()) + + for A in AS: + run_hessenberg(A, verbose = v) + run_schur(A, verbose = v) + run_eig(A, verbose = v) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_eigen_symmetric.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_eigen_symmetric.py new file mode 100644 index 0000000000000000000000000000000000000000..aab3d8ea3142aada6e14ad6d3ea25a7e8293554d --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_eigen_symmetric.py @@ -0,0 +1,357 @@ +#!/usr/bin/python +# -*- coding: utf-8 -*- + +from mpmath import mp +from mpmath import libmp + +xrange = libmp.backend.xrange + +def run_eigsy(A, verbose = False): + if verbose: + print("original matrix:\n", str(A)) + + D, Q = mp.eigsy(A) + B = Q * mp.diag(D) * Q.transpose() + C = A - B + E = Q * Q.transpose() - mp.eye(A.rows) + + if verbose: + print("eigenvalues:\n", D) + print("eigenvectors:\n", Q) + + NC = mp.mnorm(C) + NE = mp.mnorm(E) + + if verbose: + print("difference:", NC, "\n", C, "\n") + print("difference:", NE, "\n", E, "\n") + + eps = mp.exp( 0.8 * mp.log(mp.eps)) + + assert NC < eps + assert NE < eps + + return NC + +def run_eighe(A, verbose = False): + if verbose: + print("original matrix:\n", str(A)) + + D, Q = mp.eighe(A) + B = Q * mp.diag(D) * Q.transpose_conj() + C = A - B + E = Q * Q.transpose_conj() - mp.eye(A.rows) + + if verbose: + print("eigenvalues:\n", D) + print("eigenvectors:\n", Q) + + NC = mp.mnorm(C) + NE = mp.mnorm(E) + + if verbose: + print("difference:", NC, "\n", C, "\n") + print("difference:", NE, "\n", E, "\n") + + eps = mp.exp( 0.8 * mp.log(mp.eps)) + + assert NC < eps + assert NE < eps + + return NC + +def run_svd_r(A, full_matrices = False, verbose = True): + + m, n = A.rows, A.cols + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + if verbose: + print("original matrix:\n", str(A)) + print("full", full_matrices) + + U, S0, V = mp.svd_r(A, full_matrices = full_matrices) + + S = mp.zeros(U.cols, V.rows) + for j in xrange(min(m, n)): + S[j,j] = S0[j] + + if verbose: + print("U:\n", str(U)) + print("S:\n", str(S0)) + print("V:\n", str(V)) + + C = U * S * V - A + err = mp.mnorm(C) + if verbose: + print("C\n", str(C), "\n", err) + assert err < eps + + D = V * V.transpose() - mp.eye(V.rows) + err = mp.mnorm(D) + if verbose: + print("D:\n", str(D), "\n", err) + assert err < eps + + E = U.transpose() * U - mp.eye(U.cols) + err = mp.mnorm(E) + if verbose: + print("E:\n", str(E), "\n", err) + assert err < eps + +def run_svd_c(A, full_matrices = False, verbose = True): + + m, n = A.rows, A.cols + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + if verbose: + print("original matrix:\n", str(A)) + print("full", full_matrices) + + U, S0, V = mp.svd_c(A, full_matrices = full_matrices) + + S = mp.zeros(U.cols, V.rows) + for j in xrange(min(m, n)): + S[j,j] = S0[j] + + if verbose: + print("U:\n", str(U)) + print("S:\n", str(S0)) + print("V:\n", str(V)) + + C = U * S * V - A + err = mp.mnorm(C) + if verbose: + print("C\n", str(C), "\n", err) + assert err < eps + + D = V * V.transpose_conj() - mp.eye(V.rows) + err = mp.mnorm(D) + if verbose: + print("D:\n", str(D), "\n", err) + assert err < eps + + E = U.transpose_conj() * U - mp.eye(U.cols) + err = mp.mnorm(E) + if verbose: + print("E:\n", str(E), "\n", err) + assert err < eps + +def run_gauss(qtype, a, b): + eps = 1e-5 + + d, e = mp.gauss_quadrature(len(a), qtype) + d -= mp.matrix(a) + e -= mp.matrix(b) + + assert mp.mnorm(d) < eps + assert mp.mnorm(e) < eps + +def irandmatrix(n, range = 10): + """ + random matrix with integer entries + """ + A = mp.matrix(n, n) + for i in xrange(n): + for j in xrange(n): + A[i,j]=int( (2 * mp.rand() - 1) * range) + return A + +####################### + +def test_eighe_fixed_matrix(): + A = mp.matrix([[2, 3], [3, 5]]) + run_eigsy(A) + run_eighe(A) + + A = mp.matrix([[7, -11], [-11, 13]]) + run_eigsy(A) + run_eighe(A) + + A = mp.matrix([[2, 11, 7], [11, 3, 13], [7, 13, 5]]) + run_eigsy(A) + run_eighe(A) + + A = mp.matrix([[2, 0, 7], [0, 3, 1], [7, 1, 5]]) + run_eigsy(A) + run_eighe(A) + + # + + A = mp.matrix([[2, 3+7j], [3-7j, 5]]) + run_eighe(A) + + A = mp.matrix([[2, -11j, 0], [+11j, 3, 29j], [0, -29j, 5]]) + run_eighe(A) + + A = mp.matrix([[2, 11 + 17j, 7 + 19j], [11 - 17j, 3, -13 + 23j], [7 - 19j, -13 - 23j, 5]]) + run_eighe(A) + +def test_eigsy_randmatrix(): + N = 5 + + for a in xrange(10): + A = 2 * mp.randmatrix(N, N) - 1 + + for i in xrange(0, N): + for j in xrange(i + 1, N): + A[j,i] = A[i,j] + + run_eigsy(A) + +def test_eighe_randmatrix(): + N = 5 + + for a in xrange(10): + A = (2 * mp.randmatrix(N, N) - 1) + 1j * (2 * mp.randmatrix(N, N) - 1) + + for i in xrange(0, N): + A[i,i] = mp.re(A[i,i]) + for j in xrange(i + 1, N): + A[j,i] = mp.conj(A[i,j]) + + run_eighe(A) + +def test_eigsy_irandmatrix(): + N = 4 + R = 4 + + for a in xrange(10): + A=irandmatrix(N, R) + + for i in xrange(0, N): + for j in xrange(i + 1, N): + A[j,i] = A[i,j] + + run_eigsy(A) + +def test_eighe_irandmatrix(): + N = 4 + R = 4 + + for a in xrange(10): + A=irandmatrix(N, R) + 1j * irandmatrix(N, R) + + for i in xrange(0, N): + A[i,i] = mp.re(A[i,i]) + for j in xrange(i + 1, N): + A[j,i] = mp.conj(A[i,j]) + + run_eighe(A) + +def test_svd_r_rand(): + for i in xrange(5): + full = mp.rand() > 0.5 + m = 1 + int(mp.rand() * 10) + n = 1 + int(mp.rand() * 10) + A = 2 * mp.randmatrix(m, n) - 1 + if mp.rand() > 0.5: + A *= 10 + for x in xrange(m): + for y in xrange(n): + A[x,y]=int(A[x,y]) + + run_svd_r(A, full_matrices = full, verbose = False) + +def test_svd_c_rand(): + for i in xrange(5): + full = mp.rand() > 0.5 + m = 1 + int(mp.rand() * 10) + n = 1 + int(mp.rand() * 10) + A = (2 * mp.randmatrix(m, n) - 1) + 1j * (2 * mp.randmatrix(m, n) - 1) + if mp.rand() > 0.5: + A *= 10 + for x in xrange(m): + for y in xrange(n): + A[x,y]=int(mp.re(A[x,y])) + 1j * int(mp.im(A[x,y])) + + run_svd_c(A, full_matrices=full, verbose=False) + +def test_svd_test_case(): + # a test case from Golub and Reinsch + # (see wilkinson/reinsch: handbook for auto. comp., vol ii-linear algebra, 134-151(1971).) + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + a = [[22, 10, 2, 3, 7], + [14, 7, 10, 0, 8], + [-1, 13, -1, -11, 3], + [-3, -2, 13, -2, 4], + [ 9, 8, 1, -2, 4], + [ 9, 1, -7, 5, -1], + [ 2, -6, 6, 5, 1], + [ 4, 5, 0, -2, 2]] + + a = mp.matrix(a) + b = mp.matrix([mp.sqrt(1248), 20, mp.sqrt(384), 0, 0]) + + S = mp.svd_r(a, compute_uv = False) + S -= b + assert mp.mnorm(S) < eps + + S = mp.svd_c(a, compute_uv = False) + S -= b + assert mp.mnorm(S) < eps + + +def test_gauss_quadrature_static(): + a = [-0.57735027, 0.57735027] + b = [ 1, 1] + run_gauss("legendre", a , b) + + a = [ -0.906179846, -0.538469310, 0, 0.538469310, 0.906179846] + b = [ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689] + run_gauss("legendre", a , b) + + a = [ 0.06943184, 0.33000948, 0.66999052, 0.93056816] + b = [ 0.17392742, 0.32607258, 0.32607258, 0.17392742] + run_gauss("legendre01", a , b) + + a = [-0.70710678, 0.70710678] + b = [ 0.88622693, 0.88622693] + run_gauss("hermite", a , b) + + a = [ -2.02018287, -0.958572465, 0, 0.958572465, 2.02018287] + b = [ 0.01995324, 0.39361932, 0.94530872, 0.39361932, 0.01995324] + run_gauss("hermite", a , b) + + a = [ 0.41577456, 2.29428036, 6.28994508] + b = [ 0.71109301, 0.27851773, 0.01038926] + run_gauss("laguerre", a , b) + +def test_gauss_quadrature_dynamic(verbose = False): + n = 5 + + A = mp.randmatrix(2 * n, 1) + + def F(x): + r = 0 + for i in xrange(len(A) - 1, -1, -1): + r = r * x + A[i] + return r + + def run(qtype, FW, R, alpha = 0, beta = 0): + X, W = mp.gauss_quadrature(n, qtype, alpha = alpha, beta = beta) + + a = 0 + for i in xrange(len(X)): + a += W[i] * F(X[i]) + + b = mp.quad(lambda x: FW(x) * F(x), R) + + c = mp.fabs(a - b) + + if verbose: + print(qtype, c, a, b) + + assert c < 1e-5 + + run("legendre", lambda x: 1, [-1, 1]) + run("legendre01", lambda x: 1, [0, 1]) + run("hermite", lambda x: mp.exp(-x*x), [-mp.inf, mp.inf]) + run("laguerre", lambda x: mp.exp(-x), [0, mp.inf]) + run("glaguerre", lambda x: mp.sqrt(x)*mp.exp(-x), [0, mp.inf], alpha = 1 / mp.mpf(2)) + run("chebyshev1", lambda x: 1/mp.sqrt(1-x*x), [-1, 1]) + run("chebyshev2", lambda x: mp.sqrt(1-x*x), [-1, 1]) + run("jacobi", lambda x: (1-x)**(1/mp.mpf(3)) * (1+x)**(1/mp.mpf(5)), [-1, 1], alpha = 1 / mp.mpf(3), beta = 1 / mp.mpf(5) ) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_elliptic.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_elliptic.py new file mode 100644 index 0000000000000000000000000000000000000000..4dddc2df34b8d2fa7f2028b3501e5b7f140d8912 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_elliptic.py @@ -0,0 +1,670 @@ +""" +Limited tests of the elliptic functions module. A full suite of +extensive testing can be found in elliptic_torture_tests.py + +Author of the first version: M.T. Taschuk + +References: + +[1] Abramowitz & Stegun. 'Handbook of Mathematical Functions, 9th Ed.', + (Dover duplicate of 1972 edition) +[2] Whittaker 'A Course of Modern Analysis, 4th Ed.', 1946, + Cambridge University Press + +""" + +import mpmath +import random +import pytest + +from mpmath import * + +def mpc_ae(a, b, eps=eps): + res = True + res = res and a.real.ae(b.real, eps) + res = res and a.imag.ae(b.imag, eps) + return res + +zero = mpf(0) +one = mpf(1) + +jsn = ellipfun('sn') +jcn = ellipfun('cn') +jdn = ellipfun('dn') + +calculate_nome = lambda k: qfrom(k=k) + +def test_ellipfun(): + mp.dps = 15 + assert ellipfun('ss', 0, 0) == 1 + assert ellipfun('cc', 0, 0) == 1 + assert ellipfun('dd', 0, 0) == 1 + assert ellipfun('nn', 0, 0) == 1 + assert ellipfun('sn', 0.25, 0).ae(sin(0.25)) + assert ellipfun('cn', 0.25, 0).ae(cos(0.25)) + assert ellipfun('dn', 0.25, 0).ae(1) + assert ellipfun('ns', 0.25, 0).ae(csc(0.25)) + assert ellipfun('nc', 0.25, 0).ae(sec(0.25)) + assert ellipfun('nd', 0.25, 0).ae(1) + assert ellipfun('sc', 0.25, 0).ae(tan(0.25)) + assert ellipfun('sd', 0.25, 0).ae(sin(0.25)) + assert ellipfun('cd', 0.25, 0).ae(cos(0.25)) + assert ellipfun('cs', 0.25, 0).ae(cot(0.25)) + assert ellipfun('dc', 0.25, 0).ae(sec(0.25)) + assert ellipfun('ds', 0.25, 0).ae(csc(0.25)) + assert ellipfun('sn', 0.25, 1).ae(tanh(0.25)) + assert ellipfun('cn', 0.25, 1).ae(sech(0.25)) + assert ellipfun('dn', 0.25, 1).ae(sech(0.25)) + assert ellipfun('ns', 0.25, 1).ae(coth(0.25)) + assert ellipfun('nc', 0.25, 1).ae(cosh(0.25)) + assert ellipfun('nd', 0.25, 1).ae(cosh(0.25)) + assert ellipfun('sc', 0.25, 1).ae(sinh(0.25)) + assert ellipfun('sd', 0.25, 1).ae(sinh(0.25)) + assert ellipfun('cd', 0.25, 1).ae(1) + assert ellipfun('cs', 0.25, 1).ae(csch(0.25)) + assert ellipfun('dc', 0.25, 1).ae(1) + assert ellipfun('ds', 0.25, 1).ae(csch(0.25)) + assert ellipfun('sn', 0.25, 0.5).ae(0.24615967096986145833) + assert ellipfun('cn', 0.25, 0.5).ae(0.96922928989378439337) + assert ellipfun('dn', 0.25, 0.5).ae(0.98473484156599474563) + assert ellipfun('ns', 0.25, 0.5).ae(4.0624038700573130369) + assert ellipfun('nc', 0.25, 0.5).ae(1.0317476065024692949) + assert ellipfun('nd', 0.25, 0.5).ae(1.0155017958029488665) + assert ellipfun('sc', 0.25, 0.5).ae(0.25397465134058993408) + assert ellipfun('sd', 0.25, 0.5).ae(0.24997558792415733063) + assert ellipfun('cd', 0.25, 0.5).ae(0.98425408443195497052) + assert ellipfun('cs', 0.25, 0.5).ae(3.9374008182374110826) + assert ellipfun('dc', 0.25, 0.5).ae(1.0159978158253033913) + assert ellipfun('ds', 0.25, 0.5).ae(4.0003906313579720593) + + + + +def test_calculate_nome(): + mp.dps = 100 + + q = calculate_nome(zero) + assert(q == zero) + + mp.dps = 25 + # used Mathematica's EllipticNomeQ[m] + math1 = [(mpf(1)/10, mpf('0.006584651553858370274473060')), + (mpf(2)/10, mpf('0.01394285727531826872146409')), + (mpf(3)/10, mpf('0.02227743615715350822901627')), + (mpf(4)/10, mpf('0.03188334731336317755064299')), + (mpf(5)/10, mpf('0.04321391826377224977441774')), + (mpf(6)/10, mpf('0.05702025781460967637754953')), + (mpf(7)/10, mpf('0.07468994353717944761143751')), + (mpf(8)/10, mpf('0.09927369733882489703607378')), + (mpf(9)/10, mpf('0.1401731269542615524091055')), + (mpf(9)/10, mpf('0.1401731269542615524091055'))] + + for i in math1: + m = i[0] + q = calculate_nome(sqrt(m)) + assert q.ae(i[1]) + + mp.dps = 15 + +def test_jtheta(): + mp.dps = 25 + + z = q = zero + for n in range(1,5): + value = jtheta(n, z, q) + assert(value == (n-1)//2) + + for q in [one, mpf(2)]: + for n in range(1,5): + pytest.raises(ValueError, lambda: jtheta(n, z, q)) + + z = one/10 + q = one/11 + + # Mathematical N[EllipticTheta[1, 1/10, 1/11], 25] + res = mpf('0.1069552990104042681962096') + result = jtheta(1, z, q) + assert(result.ae(res)) + + # Mathematica N[EllipticTheta[2, 1/10, 1/11], 25] + res = mpf('1.101385760258855791140606') + result = jtheta(2, z, q) + assert(result.ae(res)) + + # Mathematica N[EllipticTheta[3, 1/10, 1/11], 25] + res = mpf('1.178319743354331061795905') + result = jtheta(3, z, q) + assert(result.ae(res)) + + # Mathematica N[EllipticTheta[4, 1/10, 1/11], 25] + res = mpf('0.8219318954665153577314573') + result = jtheta(4, z, q) + assert(result.ae(res)) + + # test for sin zeros for jtheta(1, z, q) + # test for cos zeros for jtheta(2, z, q) + z1 = pi + z2 = pi/2 + for i in range(10): + qstring = str(random.random()) + q = mpf(qstring) + result = jtheta(1, z1, q) + assert(result.ae(0)) + result = jtheta(2, z2, q) + assert(result.ae(0)) + mp.dps = 15 + + +def test_jtheta_issue_79(): + # near the circle of covergence |q| = 1 the convergence slows + # down; for |q| > Q_LIM the theta functions raise ValueError + mp.dps = 30 + mp.dps += 30 + q = mpf(6)/10 - one/10**6 - mpf(8)/10 * j + mp.dps -= 30 + # Mathematica run first + # N[EllipticTheta[3, 1, 6/10 - 10^-6 - 8/10*I], 2000] + # then it works: + # N[EllipticTheta[3, 1, 6/10 - 10^-6 - 8/10*I], 30] + res = mpf('32.0031009628901652627099524264') + \ + mpf('16.6153027998236087899308935624') * j + result = jtheta(3, 1, q) + # check that for abs(q) > Q_LIM a ValueError exception is raised + mp.dps += 30 + q = mpf(6)/10 - one/10**7 - mpf(8)/10 * j + mp.dps -= 30 + pytest.raises(ValueError, lambda: jtheta(3, 1, q)) + + # bug reported in issue 79 + mp.dps = 100 + z = (1+j)/3 + q = mpf(368983957219251)/10**15 + mpf(636363636363636)/10**15 * j + # Mathematica N[EllipticTheta[1, z, q], 35] + res = mpf('2.4439389177990737589761828991467471') + \ + mpf('0.5446453005688226915290954851851490') *j + mp.dps = 30 + result = jtheta(1, z, q) + assert(result.ae(res)) + mp.dps = 80 + z = 3 + 4*j + q = 0.5 + 0.5*j + r1 = jtheta(1, z, q) + mp.dps = 15 + r2 = jtheta(1, z, q) + assert r1.ae(r2) + mp.dps = 80 + z = 3 + j + q1 = exp(j*3) + # longer test + # for n in range(1, 6) + for n in range(1, 2): + mp.dps = 80 + q = q1*(1 - mpf(1)/10**n) + r1 = jtheta(1, z, q) + mp.dps = 15 + r2 = jtheta(1, z, q) + assert r1.ae(r2) + mp.dps = 15 + # issue 79 about high derivatives + assert jtheta(3, 4.5, 0.25, 9).ae(1359.04892680683) + assert jtheta(3, 4.5, 0.25, 50).ae(-6.14832772630905e+33) + mp.dps = 50 + r = jtheta(3, 4.5, 0.25, 9) + assert r.ae('1359.048926806828939547859396600218966947753213803') + r = jtheta(3, 4.5, 0.25, 50) + assert r.ae('-6148327726309051673317975084654262.4119215720343656') + +def test_jtheta_identities(): + """ + Tests the some of the jacobi identidies found in Abramowitz, + Sec. 16.28, Pg. 576. The identities are tested to 1 part in 10^98. + """ + mp.dps = 110 + eps1 = ldexp(eps, 30) + + for i in range(10): + qstring = str(random.random()) + q = mpf(qstring) + + zstring = str(10*random.random()) + z = mpf(zstring) + # Abramowitz 16.28.1 + # v_1(z, q)**2 * v_4(0, q)**2 = v_3(z, q)**2 * v_2(0, q)**2 + # - v_2(z, q)**2 * v_3(0, q)**2 + term1 = (jtheta(1, z, q)**2) * (jtheta(4, zero, q)**2) + term2 = (jtheta(3, z, q)**2) * (jtheta(2, zero, q)**2) + term3 = (jtheta(2, z, q)**2) * (jtheta(3, zero, q)**2) + equality = term1 - term2 + term3 + assert(equality.ae(0, eps1)) + + zstring = str(100*random.random()) + z = mpf(zstring) + # Abramowitz 16.28.2 + # v_2(z, q)**2 * v_4(0, q)**2 = v_4(z, q)**2 * v_2(0, q)**2 + # - v_1(z, q)**2 * v_3(0, q)**2 + term1 = (jtheta(2, z, q)**2) * (jtheta(4, zero, q)**2) + term2 = (jtheta(4, z, q)**2) * (jtheta(2, zero, q)**2) + term3 = (jtheta(1, z, q)**2) * (jtheta(3, zero, q)**2) + equality = term1 - term2 + term3 + assert(equality.ae(0, eps1)) + + # Abramowitz 16.28.3 + # v_3(z, q)**2 * v_4(0, q)**2 = v_4(z, q)**2 * v_3(0, q)**2 + # - v_1(z, q)**2 * v_2(0, q)**2 + term1 = (jtheta(3, z, q)**2) * (jtheta(4, zero, q)**2) + term2 = (jtheta(4, z, q)**2) * (jtheta(3, zero, q)**2) + term3 = (jtheta(1, z, q)**2) * (jtheta(2, zero, q)**2) + equality = term1 - term2 + term3 + assert(equality.ae(0, eps1)) + + # Abramowitz 16.28.4 + # v_4(z, q)**2 * v_4(0, q)**2 = v_3(z, q)**2 * v_3(0, q)**2 + # - v_2(z, q)**2 * v_2(0, q)**2 + term1 = (jtheta(4, z, q)**2) * (jtheta(4, zero, q)**2) + term2 = (jtheta(3, z, q)**2) * (jtheta(3, zero, q)**2) + term3 = (jtheta(2, z, q)**2) * (jtheta(2, zero, q)**2) + equality = term1 - term2 + term3 + assert(equality.ae(0, eps1)) + + # Abramowitz 16.28.5 + # v_2(0, q)**4 + v_4(0, q)**4 == v_3(0, q)**4 + term1 = (jtheta(2, zero, q))**4 + term2 = (jtheta(4, zero, q))**4 + term3 = (jtheta(3, zero, q))**4 + equality = term1 + term2 - term3 + assert(equality.ae(0, eps1)) + mp.dps = 15 + +def test_jtheta_complex(): + mp.dps = 30 + z = mpf(1)/4 + j/8 + q = mpf(1)/3 + j/7 + # Mathematica N[EllipticTheta[1, 1/4 + I/8, 1/3 + I/7], 35] + res = mpf('0.31618034835986160705729105731678285') + \ + mpf('0.07542013825835103435142515194358975') * j + r = jtheta(1, z, q) + assert(mpc_ae(r, res)) + + # Mathematica N[EllipticTheta[2, 1/4 + I/8, 1/3 + I/7], 35] + res = mpf('1.6530986428239765928634711417951828') + \ + mpf('0.2015344864707197230526742145361455') * j + r = jtheta(2, z, q) + assert(mpc_ae(r, res)) + + # Mathematica N[EllipticTheta[3, 1/4 + I/8, 1/3 + I/7], 35] + res = mpf('1.6520564411784228184326012700348340') + \ + mpf('0.1998129119671271328684690067401823') * j + r = jtheta(3, z, q) + assert(mpc_ae(r, res)) + + # Mathematica N[EllipticTheta[4, 1/4 + I/8, 1/3 + I/7], 35] + res = mpf('0.37619082382228348252047624089973824') - \ + mpf('0.15623022130983652972686227200681074') * j + r = jtheta(4, z, q) + assert(mpc_ae(r, res)) + + # check some theta function identities + mp.dos = 100 + z = mpf(1)/4 + j/8 + q = mpf(1)/3 + j/7 + mp.dps += 10 + a = [0,0, jtheta(2, 0, q), jtheta(3, 0, q), jtheta(4, 0, q)] + t = [0, jtheta(1, z, q), jtheta(2, z, q), jtheta(3, z, q), jtheta(4, z, q)] + r = [(t[2]*a[4])**2 - (t[4]*a[2])**2 + (t[1] *a[3])**2, + (t[3]*a[4])**2 - (t[4]*a[3])**2 + (t[1] *a[2])**2, + (t[1]*a[4])**2 - (t[3]*a[2])**2 + (t[2] *a[3])**2, + (t[4]*a[4])**2 - (t[3]*a[3])**2 + (t[2] *a[2])**2, + a[2]**4 + a[4]**4 - a[3]**4] + mp.dps -= 10 + for x in r: + assert(mpc_ae(x, mpc(0))) + mp.dps = 15 + +def test_djtheta(): + mp.dps = 30 + + z = one/7 + j/3 + q = one/8 + j/5 + # Mathematica N[EllipticThetaPrime[1, 1/7 + I/3, 1/8 + I/5], 35] + res = mpf('1.5555195883277196036090928995803201') - \ + mpf('0.02439761276895463494054149673076275') * j + result = jtheta(1, z, q, 1) + assert(mpc_ae(result, res)) + + # Mathematica N[EllipticThetaPrime[2, 1/7 + I/3, 1/8 + I/5], 35] + res = mpf('0.19825296689470982332701283509685662') - \ + mpf('0.46038135182282106983251742935250009') * j + result = jtheta(2, z, q, 1) + assert(mpc_ae(result, res)) + + # Mathematica N[EllipticThetaPrime[3, 1/7 + I/3, 1/8 + I/5], 35] + res = mpf('0.36492498415476212680896699407390026') - \ + mpf('0.57743812698666990209897034525640369') * j + result = jtheta(3, z, q, 1) + assert(mpc_ae(result, res)) + + # Mathematica N[EllipticThetaPrime[4, 1/7 + I/3, 1/8 + I/5], 35] + res = mpf('-0.38936892528126996010818803742007352') + \ + mpf('0.66549886179739128256269617407313625') * j + result = jtheta(4, z, q, 1) + assert(mpc_ae(result, res)) + + for i in range(10): + q = (one*random.random() + j*random.random())/2 + # identity in Wittaker, Watson &21.41 + a = jtheta(1, 0, q, 1) + b = jtheta(2, 0, q)*jtheta(3, 0, q)*jtheta(4, 0, q) + assert(a.ae(b)) + + # test higher derivatives + mp.dps = 20 + for q,z in [(one/3, one/5), (one/3 + j/8, one/5), + (one/3, one/5 + j/8), (one/3 + j/7, one/5 + j/8)]: + for n in [1, 2, 3, 4]: + r = jtheta(n, z, q, 2) + r1 = diff(lambda zz: jtheta(n, zz, q), z, n=2) + assert r.ae(r1) + r = jtheta(n, z, q, 3) + r1 = diff(lambda zz: jtheta(n, zz, q), z, n=3) + assert r.ae(r1) + + # identity in Wittaker, Watson &21.41 + q = one/3 + z = zero + a = [0]*5 + a[1] = jtheta(1, z, q, 3)/jtheta(1, z, q, 1) + for n in [2,3,4]: + a[n] = jtheta(n, z, q, 2)/jtheta(n, z, q) + equality = a[2] + a[3] + a[4] - a[1] + assert(equality.ae(0)) + mp.dps = 15 + +def test_jsn(): + """ + Test some special cases of the sn(z, q) function. + """ + mp.dps = 100 + + # trival case + result = jsn(zero, zero) + assert(result == zero) + + # Abramowitz Table 16.5 + # + # sn(0, m) = 0 + + for i in range(10): + qstring = str(random.random()) + q = mpf(qstring) + + equality = jsn(zero, q) + assert(equality.ae(0)) + + # Abramowitz Table 16.6.1 + # + # sn(z, 0) = sin(z), m == 0 + # + # sn(z, 1) = tanh(z), m == 1 + # + # It would be nice to test these, but I find that they run + # in to numerical trouble. I'm currently treating as a boundary + # case for sn function. + + mp.dps = 25 + arg = one/10 + #N[JacobiSN[1/10, 2^-100], 25] + res = mpf('0.09983341664682815230681420') + m = ldexp(one, -100) + result = jsn(arg, m) + assert(result.ae(res)) + + # N[JacobiSN[1/10, 1/10], 25] + res = mpf('0.09981686718599080096451168') + result = jsn(arg, arg) + assert(result.ae(res)) + mp.dps = 15 + +def test_jcn(): + """ + Test some special cases of the cn(z, q) function. + """ + mp.dps = 100 + + # Abramowitz Table 16.5 + # cn(0, q) = 1 + qstring = str(random.random()) + q = mpf(qstring) + cn = jcn(zero, q) + assert(cn.ae(one)) + + # Abramowitz Table 16.6.2 + # + # cn(u, 0) = cos(u), m == 0 + # + # cn(u, 1) = sech(z), m == 1 + # + # It would be nice to test these, but I find that they run + # in to numerical trouble. I'm currently treating as a boundary + # case for cn function. + + mp.dps = 25 + arg = one/10 + m = ldexp(one, -100) + #N[JacobiCN[1/10, 2^-100], 25] + res = mpf('0.9950041652780257660955620') + result = jcn(arg, m) + assert(result.ae(res)) + + # N[JacobiCN[1/10, 1/10], 25] + res = mpf('0.9950058256237368748520459') + result = jcn(arg, arg) + assert(result.ae(res)) + mp.dps = 15 + +def test_jdn(): + """ + Test some special cases of the dn(z, q) function. + """ + mp.dps = 100 + + # Abramowitz Table 16.5 + # dn(0, q) = 1 + mstring = str(random.random()) + m = mpf(mstring) + + dn = jdn(zero, m) + assert(dn.ae(one)) + + mp.dps = 25 + # N[JacobiDN[1/10, 1/10], 25] + res = mpf('0.9995017055025556219713297') + arg = one/10 + result = jdn(arg, arg) + assert(result.ae(res)) + mp.dps = 15 + + +def test_sn_cn_dn_identities(): + """ + Tests the some of the jacobi elliptic function identities found + on Mathworld. Haven't found in Abramowitz. + """ + mp.dps = 100 + N = 5 + for i in range(N): + qstring = str(random.random()) + q = mpf(qstring) + zstring = str(100*random.random()) + z = mpf(zstring) + + # MathWorld + # sn(z, q)**2 + cn(z, q)**2 == 1 + term1 = jsn(z, q)**2 + term2 = jcn(z, q)**2 + equality = one - term1 - term2 + assert(equality.ae(0)) + + # MathWorld + # k**2 * sn(z, m)**2 + dn(z, m)**2 == 1 + for i in range(N): + mstring = str(random.random()) + m = mpf(qstring) + k = m.sqrt() + zstring = str(10*random.random()) + z = mpf(zstring) + term1 = k**2 * jsn(z, m)**2 + term2 = jdn(z, m)**2 + equality = one - term1 - term2 + assert(equality.ae(0)) + + + for i in range(N): + mstring = str(random.random()) + m = mpf(mstring) + k = m.sqrt() + zstring = str(random.random()) + z = mpf(zstring) + + # MathWorld + # k**2 * cn(z, m)**2 + (1 - k**2) = dn(z, m)**2 + term1 = k**2 * jcn(z, m)**2 + term2 = 1 - k**2 + term3 = jdn(z, m)**2 + equality = term3 - term1 - term2 + assert(equality.ae(0)) + + K = ellipk(k**2) + # Abramowitz Table 16.5 + # sn(K, m) = 1; K is K(k), first complete elliptic integral + r = jsn(K, m) + assert(r.ae(one)) + + # Abramowitz Table 16.5 + # cn(K, q) = 0; K is K(k), first complete elliptic integral + equality = jcn(K, m) + assert(equality.ae(0)) + + # Abramowitz Table 16.6.3 + # dn(z, 0) = 1, m == 0 + z = m + value = jdn(z, zero) + assert(value.ae(one)) + + mp.dps = 15 + +def test_sn_cn_dn_complex(): + mp.dps = 30 + # N[JacobiSN[1/4 + I/8, 1/3 + I/7], 35] in Mathematica + res = mpf('0.2495674401066275492326652143537') + \ + mpf('0.12017344422863833381301051702823') * j + u = mpf(1)/4 + j/8 + m = mpf(1)/3 + j/7 + r = jsn(u, m) + assert(mpc_ae(r, res)) + + #N[JacobiCN[1/4 + I/8, 1/3 + I/7], 35] + res = mpf('0.9762691700944007312693721148331') - \ + mpf('0.0307203994181623243583169154824')*j + r = jcn(u, m) + #assert r.real.ae(res.real) + #assert r.imag.ae(res.imag) + assert(mpc_ae(r, res)) + + #N[JacobiDN[1/4 + I/8, 1/3 + I/7], 35] + res = mpf('0.99639490163039577560547478589753039') - \ + mpf('0.01346296520008176393432491077244994')*j + r = jdn(u, m) + assert(mpc_ae(r, res)) + mp.dps = 15 + +def test_elliptic_integrals(): + # Test cases from Carlson's paper + mp.dps = 15 + assert elliprd(0,2,1).ae(1.7972103521033883112) + assert elliprd(2,3,4).ae(0.16510527294261053349) + assert elliprd(j,-j,2).ae(0.65933854154219768919) + assert elliprd(0,j,-j).ae(1.2708196271909686299 + 2.7811120159520578777j) + assert elliprd(0,j-1,j).ae(-1.8577235439239060056 - 0.96193450888838559989j) + assert elliprd(-2-j,-j,-1+j).ae(1.8249027393703805305 - 1.2218475784827035855j) + # extra test cases + assert elliprg(0,0,0) == 0 + assert elliprg(0,0,16).ae(2) + assert elliprg(0,16,0).ae(2) + assert elliprg(16,0,0).ae(2) + assert elliprg(1,4,0).ae(1.2110560275684595248036) + assert elliprg(1,0,4).ae(1.2110560275684595248036) + assert elliprg(0,4,1).ae(1.2110560275684595248036) + # should be symmetric -- fixes a bug present in the paper + x,y,z = 1,1j,-1+1j + assert elliprg(x,y,z).ae(0.64139146875812627545 + 0.58085463774808290907j) + assert elliprg(x,z,y).ae(0.64139146875812627545 + 0.58085463774808290907j) + assert elliprg(y,x,z).ae(0.64139146875812627545 + 0.58085463774808290907j) + assert elliprg(y,z,x).ae(0.64139146875812627545 + 0.58085463774808290907j) + assert elliprg(z,x,y).ae(0.64139146875812627545 + 0.58085463774808290907j) + assert elliprg(z,y,x).ae(0.64139146875812627545 + 0.58085463774808290907j) + + for n in [5, 15, 30, 60, 100]: + mp.dps = n + assert elliprf(1,2,0).ae('1.3110287771460599052324197949455597068413774757158115814084108519003952935352071251151477664807145467230678763') + assert elliprf(0.5,1,0).ae('1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897277771871') + assert elliprf(j,-j,0).ae('1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897277771871') + assert elliprf(j-1,j,0).ae(mpc('0.79612586584233913293056938229563057846592264089185680214929401744498956943287031832657642790719940442165621412', + '-1.2138566698364959864300942567386038975419875860741507618279563735753073152507112254567291141460317931258599889')) + assert elliprf(2,3,4).ae('0.58408284167715170669284916892566789240351359699303216166309375305508295130412919665541330837704050454472379308') + assert elliprf(j,-j,2).ae('1.0441445654064360931078658361850779139591660747973017593275012615517220315993723776182276555339288363064476126') + assert elliprf(j-1,j,1-j).ae(mpc('0.93912050218619371196624617169781141161485651998254431830645241993282941057500174238125105410055253623847335313', + '-0.53296252018635269264859303449447908970360344322834582313172115220559316331271520508208025270300138589669326136')) + assert elliprc(0,0.25).ae(+pi) + assert elliprc(2.25,2).ae(+ln2) + assert elliprc(0,j).ae(mpc('1.1107207345395915617539702475151734246536554223439225557713489017391086982748684776438317336911913093408525532', + '-1.1107207345395915617539702475151734246536554223439225557713489017391086982748684776438317336911913093408525532')) + assert elliprc(-j,j).ae(mpc('1.2260849569072198222319655083097718755633725139745941606203839524036426936825652935738621522906572884239069297', + '-0.34471136988767679699935618332997956653521218571295874986708834375026550946053920574015526038040124556716711353')) + assert elliprc(0.25,-2).ae(ln2/3) + assert elliprc(j,-1).ae(mpc('0.77778596920447389875196055840799837589537035343923012237628610795937014001905822029050288316217145443865649819', + '0.1983248499342877364755170948292130095921681309577950696116251029742793455964385947473103628983664877025779304')) + assert elliprj(0,1,2,3).ae('0.77688623778582332014190282640545501102298064276022952731669118325952563819813258230708177398475643634103990878') + assert elliprj(2,3,4,5).ae('0.14297579667156753833233879421985774801466647854232626336218889885463800128817976132826443904216546421431528308') + assert elliprj(2,3,4,-1+j).ae(mpc('0.13613945827770535203521374457913768360237593025944342652613569368333226052158214183059386307242563164036672709', + '-0.38207561624427164249600936454845112611060375760094156571007648297226090050927156176977091273224510621553615189')) + assert elliprj(j,-j,0,2).ae('1.6490011662710884518243257224860232300246792717163891216346170272567376981346412066066050103935109581019055806') + assert elliprj(-1+j,-1-j,1,2).ae('0.94148358841220238083044612133767270187474673547917988681610772381758628963408843935027667916713866133196845063') + assert elliprj(j,-j,0,1-j).ae(mpc('1.8260115229009316249372594065790946657011067182850435297162034335356430755397401849070610280860044610878657501', + '1.2290661908643471500163617732957042849283739403009556715926326841959667290840290081010472716420690899886276961')) + assert elliprj(-1+j,-1-j,1,-3+j).ae(mpc('-0.61127970812028172123588152373622636829986597243716610650831553882054127570542477508023027578037045504958619422', + '-1.0684038390006807880182112972232562745485871763154040245065581157751693730095703406209466903752930797510491155')) + assert elliprj(-1+j,-2-j,-j,-1+j).ae(mpc('1.8249027393703805304622013339009022294368078659619988943515764258335975852685224202567854526307030593012768954', + '-1.2218475784827035854568450371590419833166777535029296025352291308244564398645467465067845461070602841312456831')) + + assert elliprg(0,16,16).ae(+pi) + assert elliprg(2,3,4).ae('1.7255030280692277601061148835701141842692457170470456590515892070736643637303053506944907685301315299153040991') + assert elliprg(0,j,-j).ae('0.42360654239698954330324956174109581824072295516347109253028968632986700241706737986160014699730561497106114281') + assert elliprg(j-1,j,0).ae(mpc('0.44660591677018372656731970402124510811555212083508861036067729944477855594654762496407405328607219895053798354', + '0.70768352357515390073102719507612395221369717586839400605901402910893345301718731499237159587077682267374159282')) + assert elliprg(-j,j-1,j).ae(mpc('0.36023392184473309033675652092928695596803358846377334894215349632203382573844427952830064383286995172598964266', + '0.40348623401722113740956336997761033878615232917480045914551915169013722542827052849476969199578321834819903921')) + assert elliprg(0, mpf('0.0796'), 4).ae('1.0284758090288040009838871385180217366569777284430590125081211090574701293154645750017813190805144572673802094') + mp.dps = 15 + + # more test cases for the branch of ellippi / elliprj + assert elliprj(-1-0.5j, -10-6j, -10-3j, -5+10j).ae(0.128470516743927699 + 0.102175950778504625j, abs_eps=1e-8) + assert elliprj(1.987, 4.463 - 1.614j, 0, -3.965).ae(-0.341575118513811305 - 0.394703757004268486j, abs_eps=1e-8) + assert elliprj(0.3068, -4.037+0.632j, 1.654, -0.9609).ae(-1.14735199581485639 - 0.134450158867472264j, abs_eps=1e-8) + assert elliprj(0.3068, -4.037-0.632j, 1.654, -0.9609).ae(1.758765901861727 - 0.161002343366626892j, abs_eps=1e-5) + assert elliprj(0.3068, -4.037+0.0632j, 1.654, -0.9609).ae(-1.17157627949475577 - 0.069182614173988811j, abs_eps=1e-8) + assert elliprj(0.3068, -4.037+0.00632j, 1.654, -0.9609).ae(-1.17337595670549633 - 0.0623069224526925j, abs_eps=1e-8) + + # these require accurate integration + assert elliprj(0.3068, -4.037-0.0632j, 1.654, -0.9609).ae(1.77940452391261626 + 0.0388711305592447234j) + assert elliprj(0.3068, -4.037-0.00632j, 1.654, -0.9609).ae(1.77806722756403055 + 0.0592749824572262329j) + # issue #571 + assert ellippi(2.1 + 0.94j, 2.3 + 0.98j, 2.5 + 0.01j).ae(-0.40652414240811963438 + 2.1547659461404749309j) + + assert ellippi(2.0-1.0j, 2.0+1.0j).ae(1.8578723151271115 - 1.18642180609983531j) + assert ellippi(2.0-0.5j, 0.5+1.0j).ae(0.936761970766645807 - 1.61876787838890786j) + assert ellippi(2.0, 1.0+1.0j).ae(0.999881420735506708 - 2.4139272867045391j) + assert ellippi(2.0+1.0j, 2.0-1.0j).ae(1.8578723151271115 + 1.18642180609983531j) + assert ellippi(2.0+1.0j, 2.0).ae(2.78474654927885845 + 2.02204728966993314j) + +def test_issue_238(): + assert isnan(qfrom(m=nan)) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_fp.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_fp.py new file mode 100644 index 0000000000000000000000000000000000000000..99f3759c3071c4d55e0481472f3d16c1f5df1fef --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_fp.py @@ -0,0 +1,1671 @@ +""" +Easy-to-use test-generating code: + +cases = ''' +exp 2.25 +log 2.25 +''' + +from mpmath import * +mp.dps = 20 +for test in cases.splitlines(): + if not test: + continue + words = test.split() + fname = words[0] + args = words[1:] + argstr = ", ".join(args) + testline = "%s(%s)" % (fname, argstr) + ans = str(eval(testline)) + print " assert ae(fp.%s, %s)" % (testline, ans) + +""" + +from mpmath import fp + +def ae(x, y, tol=1e-12): + if x == y: + return True + return abs(x-y) <= tol*abs(y) + +def test_conj(): + assert fp.conj(4) == 4 + assert fp.conj(3+4j) == 3-4j + assert fp.fdot([1,2],[3,2+1j], conjugate=True) == 7-2j + +def test_fp_number_parts(): + assert ae(fp.arg(3), 0.0) + assert ae(fp.arg(-3), 3.1415926535897932385) + assert ae(fp.arg(3j), 1.5707963267948966192) + assert ae(fp.arg(-3j), -1.5707963267948966192) + assert ae(fp.arg(2+3j), 0.98279372324732906799) + assert ae(fp.arg(-1-1j), -2.3561944901923449288) + assert ae(fp.re(2.5), 2.5) + assert ae(fp.re(2.5+3j), 2.5) + assert ae(fp.im(2.5), 0.0) + assert ae(fp.im(2.5+3j), 3.0) + assert ae(fp.floor(2.5), 2.0) + assert ae(fp.floor(2), 2.0) + assert ae(fp.floor(2.0+0j), (2.0 + 0.0j)) + assert ae(fp.floor(-1.5-0.5j), (-2.0 - 1.0j)) + assert ae(fp.ceil(2.5), 3.0) + assert ae(fp.ceil(2), 2.0) + assert ae(fp.ceil(2.0+0j), (2.0 + 0.0j)) + assert ae(fp.ceil(-1.5-0.5j), (-1.0 + 0.0j)) + +def test_fp_cospi_sinpi(): + assert ae(fp.sinpi(0), 0.0) + assert ae(fp.sinpi(0.25), 0.7071067811865475244) + assert ae(fp.sinpi(0.5), 1.0) + assert ae(fp.sinpi(0.75), 0.7071067811865475244) + assert ae(fp.sinpi(1), 0.0) + assert ae(fp.sinpi(1.25), -0.7071067811865475244) + assert ae(fp.sinpi(1.5), -1.0) + assert ae(fp.sinpi(1.75), -0.7071067811865475244) + assert ae(fp.sinpi(2), 0.0) + assert ae(fp.sinpi(2.25), 0.7071067811865475244) + assert ae(fp.sinpi(0+3j), (0.0 + 6195.8238636085899556j)) + assert ae(fp.sinpi(0.25+3j), (4381.1091260582448033 + 4381.1090689950686908j)) + assert ae(fp.sinpi(0.5+3j), (6195.8239443081075259 + 0.0j)) + assert ae(fp.sinpi(0.75+3j), (4381.1091260582448033 - 4381.1090689950686908j)) + assert ae(fp.sinpi(1+3j), (0.0 - 6195.8238636085899556j)) + assert ae(fp.sinpi(1.25+3j), (-4381.1091260582448033 - 4381.1090689950686908j)) + assert ae(fp.sinpi(1.5+3j), (-6195.8239443081075259 + 0.0j)) + assert ae(fp.sinpi(1.75+3j), (-4381.1091260582448033 + 4381.1090689950686908j)) + assert ae(fp.sinpi(2+3j), (0.0 + 6195.8238636085899556j)) + assert ae(fp.sinpi(2.25+3j), (4381.1091260582448033 + 4381.1090689950686908j)) + assert ae(fp.sinpi(-0.75), -0.7071067811865475244) + assert ae(fp.sinpi(-1e-10), -3.1415926535897933529e-10) + assert ae(fp.sinpi(1e-10), 3.1415926535897933529e-10) + assert ae(fp.sinpi(1e-10+1e-10j), (3.141592653589793353e-10 + 3.1415926535897933528e-10j)) + assert ae(fp.sinpi(1e-10-1e-10j), (3.141592653589793353e-10 - 3.1415926535897933528e-10j)) + assert ae(fp.sinpi(-1e-10+1e-10j), (-3.141592653589793353e-10 + 3.1415926535897933528e-10j)) + assert ae(fp.sinpi(-1e-10-1e-10j), (-3.141592653589793353e-10 - 3.1415926535897933528e-10j)) + assert ae(fp.cospi(0), 1.0) + assert ae(fp.cospi(0.25), 0.7071067811865475244) + assert ae(fp.cospi(0.5), 0.0) + assert ae(fp.cospi(0.75), -0.7071067811865475244) + assert ae(fp.cospi(1), -1.0) + assert ae(fp.cospi(1.25), -0.7071067811865475244) + assert ae(fp.cospi(1.5), 0.0) + assert ae(fp.cospi(1.75), 0.7071067811865475244) + assert ae(fp.cospi(2), 1.0) + assert ae(fp.cospi(2.25), 0.7071067811865475244) + assert ae(fp.cospi(0+3j), (6195.8239443081075259 + 0.0j)) + assert ae(fp.cospi(0.25+3j), (4381.1091260582448033 - 4381.1090689950686908j)) + assert ae(fp.cospi(0.5+3j), (0.0 - 6195.8238636085899556j)) + assert ae(fp.cospi(0.75+3j), (-4381.1091260582448033 - 4381.1090689950686908j)) + assert ae(fp.cospi(1+3j), (-6195.8239443081075259 + 0.0j)) + assert ae(fp.cospi(1.25+3j), (-4381.1091260582448033 + 4381.1090689950686908j)) + assert ae(fp.cospi(1.5+3j), (0.0 + 6195.8238636085899556j)) + assert ae(fp.cospi(1.75+3j), (4381.1091260582448033 + 4381.1090689950686908j)) + assert ae(fp.cospi(2+3j), (6195.8239443081075259 + 0.0j)) + assert ae(fp.cospi(2.25+3j), (4381.1091260582448033 - 4381.1090689950686908j)) + assert ae(fp.cospi(-0.75), -0.7071067811865475244) + assert ae(fp.sinpi(-0.7), -0.80901699437494750611) + assert ae(fp.cospi(-0.7), -0.5877852522924730163) + assert ae(fp.cospi(-3+2j), (-267.74676148374822225 + 0.0j)) + assert ae(fp.sinpi(-3+2j), (0.0 - 267.74489404101651426j)) + assert ae(fp.sinpi(-0.7+2j), (-216.6116802292079471 - 157.37650009392034693j)) + assert ae(fp.cospi(-0.7+2j), (-157.37759774921754565 + 216.61016943630197336j)) + +def test_fp_expj(): + assert ae(fp.expj(0), (1.0 + 0.0j)) + assert ae(fp.expj(1), (0.5403023058681397174 + 0.84147098480789650665j)) + assert ae(fp.expj(2), (-0.416146836547142387 + 0.9092974268256816954j)) + assert ae(fp.expj(0.75), (0.73168886887382088631 + 0.68163876002333416673j)) + assert ae(fp.expj(2+3j), (-0.020718731002242879378 + 0.045271253156092975488j)) + assert ae(fp.expjpi(0), (1.0 + 0.0j)) + assert ae(fp.expjpi(1), (-1.0 + 0.0j)) + assert ae(fp.expjpi(2), (1.0 + 0.0j)) + assert ae(fp.expjpi(0.75), (-0.7071067811865475244 + 0.7071067811865475244j)) + assert ae(fp.expjpi(2+3j), (0.000080699517570304599239 + 0.0j)) + +def test_fp_bernoulli(): + assert ae(fp.bernoulli(0), 1.0) + assert ae(fp.bernoulli(1), -0.5) + assert ae(fp.bernoulli(2), 0.16666666666666666667) + assert ae(fp.bernoulli(10), 0.075757575757575757576) + assert ae(fp.bernoulli(11), 0.0) + +def test_fp_gamma(): + assert ae(fp.gamma(1), 1.0) + assert ae(fp.gamma(1.5), 0.88622692545275801365) + assert ae(fp.gamma(10), 362880.0) + assert ae(fp.gamma(-0.5), -3.5449077018110320546) + assert ae(fp.gamma(-7.1), 0.0016478244570263333622) + assert ae(fp.gamma(12.3), 83385367.899970000963) + assert ae(fp.gamma(2+0j), (1.0 + 0.0j)) + assert ae(fp.gamma(-2.5+0j), (-0.94530872048294188123 + 0.0j)) + assert ae(fp.gamma(3+4j), (0.0052255384713692141947 - 0.17254707929430018772j)) + assert ae(fp.gamma(-3-4j), (0.00001460997305874775607 - 0.000020760733311509070396j)) + assert ae(fp.fac(0), 1.0) + assert ae(fp.fac(1), 1.0) + assert ae(fp.fac(20), 2432902008176640000.0) + assert ae(fp.fac(-3.5), -0.94530872048294188123) + assert ae(fp.fac(2+3j), (-0.44011340763700171113 - 0.06363724312631702183j)) + assert ae(fp.loggamma(1.0), 0.0) + assert ae(fp.loggamma(2.0), 0.0) + assert ae(fp.loggamma(3.0), 0.69314718055994530942) + assert ae(fp.loggamma(7.25), 7.0521854507385394449) + assert ae(fp.loggamma(1000.0), 5905.2204232091812118) + assert ae(fp.loggamma(1e50), 1.1412925464970229298e+52) + assert ae(fp.loggamma(1e25+1e25j), (5.6125802751733671621e+26 + 5.7696599078528568383e+26j)) + assert ae(fp.loggamma(3+4j), (-1.7566267846037841105 + 4.7426644380346579282j)) + assert ae(fp.loggamma(-0.5), (1.2655121234846453965 - 3.1415926535897932385j)) + assert ae(fp.loggamma(-1.25), (1.3664317612369762346 - 6.2831853071795864769j)) + assert ae(fp.loggamma(-2.75), (0.0044878975359557733115 - 9.4247779607693797154j)) + assert ae(fp.loggamma(-3.5), (-1.3090066849930420464 - 12.566370614359172954j)) + assert ae(fp.loggamma(-4.5), (-2.8130840817693161197 - 15.707963267948966192j)) + assert ae(fp.loggamma(-2+3j), (-6.776523813485657093 - 4.568791367260286402j)) + assert ae(fp.loggamma(-1000.3), (-5912.8440347785205041 - 3144.7342462433830317j)) + assert ae(fp.loggamma(-100-100j), (-632.35117666833135562 - 158.37641469650352462j)) + assert ae(fp.loggamma(1e-10), 23.025850929882735237) + assert ae(fp.loggamma(-1e-10), (23.02585092999817837 - 3.1415926535897932385j)) + assert ae(fp.loggamma(1e-10j), (23.025850929940456804 - 1.5707963268526181857j)) + assert ae(fp.loggamma(1e-10j-1e-10), (22.679277339718205716 - 2.3561944902500664954j)) + +def test_fp_psi(): + assert ae(fp.psi(0, 3.7), 1.1671535393615114409) + assert ae(fp.psi(0, 0.5), -1.9635100260214234794) + assert ae(fp.psi(0, 1), -0.57721566490153286061) + assert ae(fp.psi(0, -2.5), 1.1031566406452431872) + assert ae(fp.psi(0, 12.9), 2.5179671503279156347) + assert ae(fp.psi(0, 100), 4.6001618527380874002) + assert ae(fp.psi(0, 2500.3), 7.8239660143238547877) + assert ae(fp.psi(0, 1e40), 92.103403719761827391) + assert ae(fp.psi(0, 1e200), 460.51701859880913677) + assert ae(fp.psi(0, 3.7+0j), (1.1671535393615114409 + 0.0j)) + assert ae(fp.psi(1, 3), 0.39493406684822643647) + assert ae(fp.psi(3, 2+3j), (-0.05383196209159972116 + 0.0076890935247364805218j)) + assert ae(fp.psi(4, -0.5+1j), (1.2719531355492328195 - 18.211833410936276774j)) + assert ae(fp.harmonic(0), 0.0) + assert ae(fp.harmonic(1), 1.0) + assert ae(fp.harmonic(2), 1.5) + assert ae(fp.harmonic(100), 5.1873775176396202608) + assert ae(fp.harmonic(-2.5), 1.2803723055467760478) + assert ae(fp.harmonic(2+3j), (1.9390425294578375875 + 0.87336044981834544043j)) + assert ae(fp.harmonic(-5-4j), (2.3725754822349437733 - 2.4160904444801621j)) + +def test_fp_zeta(): + assert ae(fp.zeta(1e100), 1.0) + assert ae(fp.zeta(3), 1.2020569031595942854) + assert ae(fp.zeta(2+0j), (1.6449340668482264365 + 0.0j)) + assert ae(fp.zeta(0.93), -13.713619351638164784) + assert ae(fp.zeta(1.74), 1.9796863545771774095) + assert ae(fp.zeta(0.0), -0.5) + assert ae(fp.zeta(-1.0), -0.083333333333333333333) + assert ae(fp.zeta(-2.0), 0.0) + assert ae(fp.zeta(-3.0), 0.0083333333333333333333) + assert ae(fp.zeta(-500.0), 0.0) + assert ae(fp.zeta(-7.4), 0.0036537321227995882447) + assert ae(fp.zeta(2.1), 1.5602165335033620158) + assert ae(fp.zeta(26.9), 1.0000000079854809935) + assert ae(fp.zeta(26), 1.0000000149015548284) + assert ae(fp.zeta(27), 1.0000000074507117898) + assert ae(fp.zeta(28), 1.0000000037253340248) + assert ae(fp.zeta(27.1), 1.000000006951755045) + assert ae(fp.zeta(32.7), 1.0000000001433243232) + assert ae(fp.zeta(100), 1.0) + assert ae(fp.altzeta(3.5), 0.92755357777394803511) + assert ae(fp.altzeta(1), 0.69314718055994530942) + assert ae(fp.altzeta(2), 0.82246703342411321824) + assert ae(fp.altzeta(0), 0.5) + assert ae(fp.zeta(-2+3j, 1), (0.13297115587929864827 + 0.12305330040458776494j)) + assert ae(fp.zeta(-2+3j, 5), (18.384866151867576927 - 11.377015110597711009j)) + assert ae(fp.zeta(1.0000000001), 9999999173.1735741337) + assert ae(fp.zeta(0.9999999999), -9999999172.0191428039) + assert ae(fp.zeta(1+0.000000001j), (0.57721566490153286061 - 999999999.99999993765j)) + assert ae(fp.primezeta(2.5+4j), (-0.16922458243438033385 - 0.010847965298387727811j)) + assert ae(fp.primezeta(4), 0.076993139764246844943) + assert ae(fp.riemannr(3.7), 2.3034079839110855717) + assert ae(fp.riemannr(8), 3.9011860449341499474) + assert ae(fp.riemannr(3+4j), (2.2369653314259991796 + 1.6339943856990281694j)) + +def test_fp_hyp2f1(): + assert ae(fp.hyp2f1(1, (3,2), 3.25, 5.0), (-0.46600275923108143059 - 0.74393667908854842325j)) + assert ae(fp.hyp2f1(1+1j, (3,2), 3.25, 5.0), (-5.9208875603806515987 - 2.3813557707889590686j)) + assert ae(fp.hyp2f1(1+1j, (3,2), 3.25, 2+3j), (0.17174552030925080445 + 0.19589781970539389999j)) + +def test_fp_erf(): + assert fp.erf(2) == fp.erf(2.0) == fp.erf(2.0+0.0j) + assert fp.erf(fp.inf) == 1.0 + assert fp.erf(fp.ninf) == -1.0 + assert ae(fp.erf(0), 0.0) + assert ae(fp.erf(-0), -0.0) + assert ae(fp.erf(0.3), 0.32862675945912741619) + assert ae(fp.erf(-0.3), -0.32862675945912741619) + assert ae(fp.erf(0.9), 0.79690821242283213966) + assert ae(fp.erf(-0.9), -0.79690821242283213966) + assert ae(fp.erf(1.0), 0.84270079294971486934) + assert ae(fp.erf(-1.0), -0.84270079294971486934) + assert ae(fp.erf(1.1), 0.88020506957408172966) + assert ae(fp.erf(-1.1), -0.88020506957408172966) + assert ae(fp.erf(8.5), 1.0) + assert ae(fp.erf(-8.5), -1.0) + assert ae(fp.erf(9.1), 1.0) + assert ae(fp.erf(-9.1), -1.0) + assert ae(fp.erf(20.0), 1.0) + assert ae(fp.erf(-20.0), -1.0) + assert ae(fp.erf(10000.0), 1.0) + assert ae(fp.erf(-10000.0), -1.0) + assert ae(fp.erf(1e+50), 1.0) + assert ae(fp.erf(-1e+50), -1.0) + assert ae(fp.erf(1j), 1.650425758797542876j) + assert ae(fp.erf(-1j), -1.650425758797542876j) + assert ae(fp.erf((2+3j)), (-20.829461427614568389 + 8.6873182714701631444j)) + assert ae(fp.erf(-(2+3j)), -(-20.829461427614568389 + 8.6873182714701631444j)) + assert ae(fp.erf((8+9j)), (-1072004.2525062051158 + 364149.91954310255423j)) + assert ae(fp.erf(-(8+9j)), -(-1072004.2525062051158 + 364149.91954310255423j)) + assert fp.erfc(fp.inf) == 0.0 + assert fp.erfc(fp.ninf) == 2.0 + assert fp.erfc(0) == 1 + assert fp.erfc(-0.0) == 1 + assert fp.erfc(0+0j) == 1 + assert ae(fp.erfc(0.3), 0.67137324054087258381) + assert ae(fp.erfc(-0.3), 1.3286267594591274162) + assert ae(fp.erfc(0.9), 0.20309178757716786034) + assert ae(fp.erfc(-0.9), 1.7969082124228321397) + assert ae(fp.erfc(1.0), 0.15729920705028513066) + assert ae(fp.erfc(-1.0), 1.8427007929497148693) + assert ae(fp.erfc(1.1), 0.11979493042591827034) + assert ae(fp.erfc(-1.1), 1.8802050695740817297) + assert ae(fp.erfc(8.5), 2.7623240713337714461e-33) + assert ae(fp.erfc(-8.5), 2.0) + assert ae(fp.erfc(9.1), 6.6969004279886077452e-38) + assert ae(fp.erfc(-9.1), 2.0) + assert ae(fp.erfc(20.0), 5.3958656116079009289e-176) + assert ae(fp.erfc(-20.0), 2.0) + assert ae(fp.erfc(10000.0), 0.0) + assert ae(fp.erfc(-10000.0), 2.0) + assert ae(fp.erfc(1e+50), 0.0) + assert ae(fp.erfc(-1e+50), 2.0) + assert ae(fp.erfc(1j), (1.0 - 1.650425758797542876j)) + assert ae(fp.erfc(-1j), (1.0 + 1.650425758797542876j)) + assert ae(fp.erfc((2+3j)), (21.829461427614568389 - 8.6873182714701631444j), 1e-13) + assert ae(fp.erfc(-(2+3j)), (-19.829461427614568389 + 8.6873182714701631444j), 1e-13) + assert ae(fp.erfc((8+9j)), (1072005.2525062051158 - 364149.91954310255423j)) + assert ae(fp.erfc(-(8+9j)), (-1072003.2525062051158 + 364149.91954310255423j)) + assert ae(fp.erfc(20+0j), (5.3958656116079009289e-176 + 0.0j)) + +def test_fp_lambertw(): + assert ae(fp.lambertw(0.0), 0.0) + assert ae(fp.lambertw(1.0), 0.567143290409783873) + assert ae(fp.lambertw(7.5), 1.5662309537823875394) + assert ae(fp.lambertw(-0.25), -0.35740295618138890307) + assert ae(fp.lambertw(-10.0), (1.3699809685212708156 + 2.140194527074713196j)) + assert ae(fp.lambertw(0+0j), (0.0 + 0.0j)) + assert ae(fp.lambertw(4+0j), (1.2021678731970429392 + 0.0j)) + assert ae(fp.lambertw(1000.5), 5.2500227450408980127) + assert ae(fp.lambertw(1e100), 224.84310644511850156) + assert ae(fp.lambertw(-1000.0), (5.1501630246362515223 + 2.6641981432905204596j)) + assert ae(fp.lambertw(1e-10), 9.9999999990000003645e-11) + assert ae(fp.lambertw(1e-10j), (1.0000000000000000728e-20 + 1.0000000000000000364e-10j)) + assert ae(fp.lambertw(3+4j), (1.2815618061237758782 + 0.53309522202097107131j)) + assert ae(fp.lambertw(-3-4j), (1.0750730665692549276 - 1.3251023817343588823j)) + assert ae(fp.lambertw(10000+1000j), (7.2361526563371602186 + 0.087567810943839352034j)) + assert ae(fp.lambertw(0.0, -1), -fp.inf) + assert ae(fp.lambertw(1.0, -1), (-1.5339133197935745079 - 4.3751851530618983855j)) + assert ae(fp.lambertw(7.5, -1), (0.44125668415098614999 - 4.8039842008452390179j)) + assert ae(fp.lambertw(-0.25, -1), -2.1532923641103496492) + assert ae(fp.lambertw(-10.0, -1), (1.3699809685212708156 - 2.140194527074713196j)) + assert ae(fp.lambertw(0+0j, -1), -fp.inf) + assert ae(fp.lambertw(4+0j, -1), (-0.15730793189620765317 - 4.6787800704666656212j)) + assert ae(fp.lambertw(1000.5, -1), (4.9153765415404024736 - 5.4465682700815159569j)) + assert ae(fp.lambertw(1e100, -1), (224.84272130101601052 - 6.2553713838167244141j)) + assert ae(fp.lambertw(-1000.0, -1), (5.1501630246362515223 - 2.6641981432905204596j)) + assert ae(fp.lambertw(1e-10, -1), (-26.303186778379041521 - 3.2650939117038283975j)) + assert ae(fp.lambertw(1e-10j, -1), (-26.297238779529035028 - 1.6328071613455765135j)) + assert ae(fp.lambertw(3+4j, -1), (0.25856740686699741676 - 3.8521166861614355895j)) + assert ae(fp.lambertw(-3-4j, -1), (-0.32028750204310768396 - 6.8801677192091972343j)) + assert ae(fp.lambertw(10000+1000j, -1), (7.0255308742285435567 - 5.5177506835734067601j)) + assert ae(fp.lambertw(0.0, 2), -fp.inf) + assert ae(fp.lambertw(1.0, 2), (-2.4015851048680028842 + 10.776299516115070898j)) + assert ae(fp.lambertw(7.5, 2), (-0.38003357962843791529 + 10.960916473368746184j)) + assert ae(fp.lambertw(-0.25, 2), (-4.0558735269061511898 + 13.852334658567271386j)) + assert ae(fp.lambertw(-10.0, 2), (-0.34479123764318858696 + 14.112740596763592363j)) + assert ae(fp.lambertw(0+0j, 2), -fp.inf) + assert ae(fp.lambertw(4+0j, 2), (-1.0070343323804262788 + 10.903476551861683082j)) + assert ae(fp.lambertw(1000.5, 2), (4.4076185165459395295 + 11.365524591091402177j)) + assert ae(fp.lambertw(1e100, 2), (224.84156762724875878 + 12.510785262632255672j)) + assert ae(fp.lambertw(-1000.0, 2), (4.1984245610246530756 + 14.420478573754313845j)) + assert ae(fp.lambertw(1e-10, 2), (-26.362258095445866488 + 9.7800247407031482519j)) + assert ae(fp.lambertw(1e-10j, 2), (-26.384250801683084252 + 11.403535950607739763j)) + assert ae(fp.lambertw(3+4j, 2), (-0.86554679943333993562 + 11.849956798331992027j)) + assert ae(fp.lambertw(-3-4j, 2), (-0.55792273874679112639 + 8.7173627024159324811j)) + assert ae(fp.lambertw(10000+1000j, 2), (6.6223802254585662734 + 11.61348646825020766j)) + +def test_fp_stress_ei_e1(): + # Can be tightened on recent Pythons with more accurate math/cmath + ATOL = 1e-13 + PTOL = 1e-12 + v = fp.e1(1.1641532182693481445e-10) + assert ae(v, 22.296641293693077672, tol=ATOL) + assert type(v) is float + v = fp.e1(0.25) + assert ae(v, 1.0442826344437381945, tol=ATOL) + assert type(v) is float + v = fp.e1(1.0) + assert ae(v, 0.21938393439552027368, tol=ATOL) + assert type(v) is float + v = fp.e1(2.0) + assert ae(v, 0.048900510708061119567, tol=ATOL) + assert type(v) is float + v = fp.e1(5.0) + assert ae(v, 0.0011482955912753257973, tol=ATOL) + assert type(v) is float + v = fp.e1(20.0) + assert ae(v, 9.8355252906498816904e-11, tol=ATOL) + assert type(v) is float + v = fp.e1(30.0) + assert ae(v, 3.0215520106888125448e-15, tol=ATOL) + assert type(v) is float + v = fp.e1(40.0) + assert ae(v, 1.0367732614516569722e-19, tol=ATOL) + assert type(v) is float + v = fp.e1(50.0) + assert ae(v, 3.7832640295504590187e-24, tol=ATOL) + assert type(v) is float + v = fp.e1(80.0) + assert ae(v, 2.2285432586884729112e-37, tol=ATOL) + assert type(v) is float + v = fp.e1((1.1641532182693481445e-10 + 0.0j)) + assert ae(v, (22.296641293693077672 + 0.0j), tol=ATOL) + assert ae(v.real, 22.296641293693077672, tol=PTOL) + assert v.imag == 0 + v = fp.e1((0.25 + 0.0j)) + assert ae(v, (1.0442826344437381945 + 0.0j), tol=ATOL) + assert ae(v.real, 1.0442826344437381945, tol=PTOL) + assert v.imag == 0 + v = fp.e1((1.0 + 0.0j)) + assert ae(v, (0.21938393439552027368 + 0.0j), tol=ATOL) + assert ae(v.real, 0.21938393439552027368, tol=PTOL) + assert v.imag == 0 + v = fp.e1((2.0 + 0.0j)) + assert ae(v, (0.048900510708061119567 + 0.0j), tol=ATOL) + assert ae(v.real, 0.048900510708061119567, tol=PTOL) + assert v.imag == 0 + v = fp.e1((5.0 + 0.0j)) + assert ae(v, (0.0011482955912753257973 + 0.0j), tol=ATOL) + assert ae(v.real, 0.0011482955912753257973, tol=PTOL) + assert v.imag == 0 + v = fp.e1((20.0 + 0.0j)) + assert ae(v, (9.8355252906498816904e-11 + 0.0j), tol=ATOL) + assert ae(v.real, 9.8355252906498816904e-11, tol=PTOL) + assert v.imag == 0 + v = fp.e1((30.0 + 0.0j)) + assert ae(v, (3.0215520106888125448e-15 + 0.0j), tol=ATOL) + assert ae(v.real, 3.0215520106888125448e-15, tol=PTOL) + assert v.imag == 0 + v = fp.e1((40.0 + 0.0j)) + assert ae(v, (1.0367732614516569722e-19 + 0.0j), tol=ATOL) + assert ae(v.real, 1.0367732614516569722e-19, tol=PTOL) + assert v.imag == 0 + v = fp.e1((50.0 + 0.0j)) + assert ae(v, (3.7832640295504590187e-24 + 0.0j), tol=ATOL) + assert ae(v.real, 3.7832640295504590187e-24, tol=PTOL) + assert v.imag == 0 + v = fp.e1((80.0 + 0.0j)) + assert ae(v, (2.2285432586884729112e-37 + 0.0j), tol=ATOL) + assert ae(v.real, 2.2285432586884729112e-37, tol=PTOL) + assert v.imag == 0 + v = fp.e1((4.6566128730773925781e-10 + 1.1641532182693481445e-10j)) + assert ae(v, (20.880034622014215597 - 0.24497866301044883237j), tol=ATOL) + assert ae(v.real, 20.880034622014215597, tol=PTOL) + assert ae(v.imag, -0.24497866301044883237, tol=PTOL) + v = fp.e1((1.0 + 0.25j)) + assert ae(v, (0.19731063945004229095 - 0.087366045774299963672j), tol=ATOL) + assert ae(v.real, 0.19731063945004229095, tol=PTOL) + assert ae(v.imag, -0.087366045774299963672, tol=PTOL) + v = fp.e1((4.0 + 1.0j)) + assert ae(v, (0.0013106173980145506944 - 0.0034542480199350626699j), tol=ATOL) + assert ae(v.real, 0.0013106173980145506944, tol=PTOL) + assert ae(v.imag, -0.0034542480199350626699, tol=PTOL) + v = fp.e1((8.0 + 2.0j)) + assert ae(v, (-0.000022278049065270225945 - 0.000029191940456521555288j), tol=ATOL) + assert ae(v.real, -0.000022278049065270225945, tol=PTOL) + assert ae(v.imag, -0.000029191940456521555288, tol=PTOL) + v = fp.e1((20.0 + 5.0j)) + assert ae(v, (4.7711374515765346894e-11 + 8.2902652405126947359e-11j), tol=ATOL) + assert ae(v.real, 4.7711374515765346894e-11, tol=PTOL) + assert ae(v.imag, 8.2902652405126947359e-11, tol=PTOL) + v = fp.e1((80.0 + 20.0j)) + assert ae(v, (3.8353473865788235787e-38 - 2.129247592349605139e-37j), tol=ATOL) + assert ae(v.real, 3.8353473865788235787e-38, tol=PTOL) + assert ae(v.imag, -2.129247592349605139e-37, tol=PTOL) + v = fp.e1((120.0 + 30.0j)) + assert ae(v, (2.3836002337480334716e-55 + 5.6704043587126198306e-55j), tol=ATOL) + assert ae(v.real, 2.3836002337480334716e-55, tol=PTOL) + assert ae(v.imag, 5.6704043587126198306e-55, tol=PTOL) + v = fp.e1((160.0 + 40.0j)) + assert ae(v, (-1.6238022898654510661e-72 - 1.104172355572287367e-72j), tol=ATOL) + assert ae(v.real, -1.6238022898654510661e-72, tol=PTOL) + assert ae(v.imag, -1.104172355572287367e-72, tol=PTOL) + v = fp.e1((200.0 + 50.0j)) + assert ae(v, (6.6800061461666228487e-90 + 1.4473816083541016115e-91j), tol=ATOL) + assert ae(v.real, 6.6800061461666228487e-90, tol=PTOL) + assert ae(v.imag, 1.4473816083541016115e-91, tol=PTOL) + v = fp.e1((320.0 + 80.0j)) + assert ae(v, (4.2737871527778786157e-143 + 3.1789935525785660314e-142j), tol=ATOL) + assert ae(v.real, 4.2737871527778786157e-143, tol=PTOL) + assert ae(v.imag, 3.1789935525785660314e-142, tol=PTOL) + v = fp.e1((1.1641532182693481445e-10 + 1.1641532182693481445e-10j)) + assert ae(v, (21.950067703413105017 - 0.7853981632810329878j), tol=ATOL) + assert ae(v.real, 21.950067703413105017, tol=PTOL) + assert ae(v.imag, -0.7853981632810329878, tol=PTOL) + v = fp.e1((0.25 + 0.25j)) + assert ae(v, (0.71092525792923287894 - 0.56491812441304194711j), tol=ATOL) + assert ae(v.real, 0.71092525792923287894, tol=PTOL) + assert ae(v.imag, -0.56491812441304194711, tol=PTOL) + v = fp.e1((1.0 + 1.0j)) + assert ae(v, (0.00028162445198141832551 - 0.17932453503935894015j), tol=ATOL) + assert ae(v.real, 0.00028162445198141832551, tol=PTOL) + assert ae(v.imag, -0.17932453503935894015, tol=PTOL) + v = fp.e1((2.0 + 2.0j)) + assert ae(v, (-0.033767089606562004246 - 0.018599414169750541925j), tol=ATOL) + assert ae(v.real, -0.033767089606562004246, tol=PTOL) + assert ae(v.imag, -0.018599414169750541925, tol=PTOL) + v = fp.e1((5.0 + 5.0j)) + assert ae(v, (0.0007266506660356393891 + 0.00047102780163522245054j), tol=ATOL) + assert ae(v.real, 0.0007266506660356393891, tol=PTOL) + assert ae(v.imag, 0.00047102780163522245054, tol=PTOL) + v = fp.e1((20.0 + 20.0j)) + assert ae(v, (-2.3824537449367396579e-11 - 6.6969873156525615158e-11j), tol=ATOL) + assert ae(v.real, -2.3824537449367396579e-11, tol=PTOL) + assert ae(v.imag, -6.6969873156525615158e-11, tol=PTOL) + v = fp.e1((30.0 + 30.0j)) + assert ae(v, (1.7316045841744061617e-15 + 1.3065678019487308689e-15j), tol=ATOL) + assert ae(v.real, 1.7316045841744061617e-15, tol=PTOL) + assert ae(v.imag, 1.3065678019487308689e-15, tol=PTOL) + v = fp.e1((40.0 + 40.0j)) + assert ae(v, (-7.4001043002899232182e-20 - 4.991847855336816304e-21j), tol=ATOL) + assert ae(v.real, -7.4001043002899232182e-20, tol=PTOL) + assert ae(v.imag, -4.991847855336816304e-21, tol=PTOL) + v = fp.e1((50.0 + 50.0j)) + assert ae(v, (2.3566128324644641219e-24 - 1.3188326726201614778e-24j), tol=ATOL) + assert ae(v.real, 2.3566128324644641219e-24, tol=PTOL) + assert ae(v.imag, -1.3188326726201614778e-24, tol=PTOL) + v = fp.e1((80.0 + 80.0j)) + assert ae(v, (9.8279750572186526673e-38 + 1.243952841288868831e-37j), tol=ATOL) + assert ae(v.real, 9.8279750572186526673e-38, tol=PTOL) + assert ae(v.imag, 1.243952841288868831e-37, tol=PTOL) + v = fp.e1((1.1641532182693481445e-10 + 4.6566128730773925781e-10j)) + assert ae(v, (20.880034621664969632 - 1.3258176632023711778j), tol=ATOL) + assert ae(v.real, 20.880034621664969632, tol=PTOL) + assert ae(v.imag, -1.3258176632023711778, tol=PTOL) + v = fp.e1((0.25 + 1.0j)) + assert ae(v, (-0.16868306393667788761 - 0.4858011885947426971j), tol=ATOL) + assert ae(v.real, -0.16868306393667788761, tol=PTOL) + assert ae(v.imag, -0.4858011885947426971, tol=PTOL) + v = fp.e1((1.0 + 4.0j)) + assert ae(v, (0.03373591813926547318 + 0.073523452241083821877j), tol=ATOL) + assert ae(v.real, 0.03373591813926547318, tol=PTOL) + assert ae(v.imag, 0.073523452241083821877, tol=PTOL) + v = fp.e1((2.0 + 8.0j)) + assert ae(v, (-0.015392833434733785143 - 0.0031747121557605415914j), tol=ATOL) + assert ae(v.real, -0.015392833434733785143, tol=PTOL) + assert ae(v.imag, -0.0031747121557605415914, tol=PTOL) + v = fp.e1((5.0 + 20.0j)) + assert ae(v, (-0.00024419662286542966525 - 0.00021008322966152755674j), tol=ATOL) + assert ae(v.real, -0.00024419662286542966525, tol=PTOL) + assert ae(v.imag, -0.00021008322966152755674, tol=PTOL) + v = fp.e1((20.0 + 80.0j)) + assert ae(v, (2.3255552781051330088e-11 + 8.9463918891349438007e-12j), tol=ATOL) + assert ae(v.real, 2.3255552781051330088e-11, tol=PTOL) + assert ae(v.imag, 8.9463918891349438007e-12, tol=PTOL) + v = fp.e1((30.0 + 120.0j)) + assert ae(v, (-2.7068919097124652332e-16 - 7.0477762411705130239e-16j), tol=ATOL) + assert ae(v.real, -2.7068919097124652332e-16, tol=PTOL) + assert ae(v.imag, -7.0477762411705130239e-16, tol=PTOL) + v = fp.e1((40.0 + 160.0j)) + assert ae(v, (-1.1695597827678024687e-20 + 2.2907401455645736661e-20j), tol=ATOL) + assert ae(v.real, -1.1695597827678024687e-20, tol=PTOL) + assert ae(v.imag, 2.2907401455645736661e-20, tol=PTOL) + v = fp.e1((50.0 + 200.0j)) + assert ae(v, (9.0323746914410162531e-25 - 2.3950601790033530935e-25j), tol=ATOL) + assert ae(v.real, 9.0323746914410162531e-25, tol=PTOL) + assert ae(v.imag, -2.3950601790033530935e-25, tol=PTOL) + v = fp.e1((80.0 + 320.0j)) + assert ae(v, (3.4819106748728063576e-38 - 4.215653005615772724e-38j), tol=ATOL) + assert ae(v.real, 3.4819106748728063576e-38, tol=PTOL) + assert ae(v.imag, -4.215653005615772724e-38, tol=PTOL) + v = fp.e1((0.0 + 1.1641532182693481445e-10j)) + assert ae(v, (22.29664129357666235 - 1.5707963266784812974j), tol=ATOL) + assert ae(v.real, 22.29664129357666235, tol=PTOL) + assert ae(v.imag, -1.5707963266784812974, tol=PTOL) + v = fp.e1((0.0 + 0.25j)) + assert ae(v, (0.82466306258094565309 - 1.3216627564751394551j), tol=ATOL) + assert ae(v.real, 0.82466306258094565309, tol=PTOL) + assert ae(v.imag, -1.3216627564751394551, tol=PTOL) + v = fp.e1((0.0 + 1.0j)) + assert ae(v, (-0.33740392290096813466 - 0.62471325642771360429j), tol=ATOL) + assert ae(v.real, -0.33740392290096813466, tol=PTOL) + assert ae(v.imag, -0.62471325642771360429, tol=PTOL) + v = fp.e1((0.0 + 2.0j)) + assert ae(v, (-0.4229808287748649957 + 0.034616650007798229345j), tol=ATOL) + assert ae(v.real, -0.4229808287748649957, tol=PTOL) + assert ae(v.imag, 0.034616650007798229345, tol=PTOL) + v = fp.e1((0.0 + 5.0j)) + assert ae(v, (0.19002974965664387862 - 0.020865081850222481957j), tol=ATOL) + assert ae(v.real, 0.19002974965664387862, tol=PTOL) + assert ae(v.imag, -0.020865081850222481957, tol=PTOL) + v = fp.e1((0.0 + 20.0j)) + assert ae(v, (-0.04441982084535331654 - 0.022554625751456779068j), tol=ATOL) + assert ae(v.real, -0.04441982084535331654, tol=PTOL) + assert ae(v.imag, -0.022554625751456779068, tol=PTOL) + v = fp.e1((0.0 + 30.0j)) + assert ae(v, (0.033032417282071143779 - 0.0040397867645455082476j), tol=ATOL) + assert ae(v.real, 0.033032417282071143779, tol=PTOL) + assert ae(v.imag, -0.0040397867645455082476, tol=PTOL) + v = fp.e1((0.0 + 40.0j)) + assert ae(v, (-0.019020007896208766962 + 0.016188792559887887544j), tol=ATOL) + assert ae(v.real, -0.019020007896208766962, tol=PTOL) + assert ae(v.imag, 0.016188792559887887544, tol=PTOL) + v = fp.e1((0.0 + 50.0j)) + assert ae(v, (0.0056283863241163054402 - 0.019179254308960724503j), tol=ATOL) + assert ae(v.real, 0.0056283863241163054402, tol=PTOL) + assert ae(v.imag, -0.019179254308960724503, tol=PTOL) + v = fp.e1((0.0 + 80.0j)) + assert ae(v, (0.012402501155070958192 + 0.0015345601175906961199j), tol=ATOL) + assert ae(v.real, 0.012402501155070958192, tol=PTOL) + assert ae(v.imag, 0.0015345601175906961199, tol=PTOL) + v = fp.e1((-1.1641532182693481445e-10 + 4.6566128730773925781e-10j)) + assert ae(v, (20.880034621432138988 - 1.8157749894560994861j), tol=ATOL) + assert ae(v.real, 20.880034621432138988, tol=PTOL) + assert ae(v.imag, -1.8157749894560994861, tol=PTOL) + v = fp.e1((-0.25 + 1.0j)) + assert ae(v, (-0.59066621214766308594 - 0.74474454765205036972j), tol=ATOL) + assert ae(v.real, -0.59066621214766308594, tol=PTOL) + assert ae(v.imag, -0.74474454765205036972, tol=PTOL) + v = fp.e1((-1.0 + 4.0j)) + assert ae(v, (0.49739047283060471093 + 0.41543605404038863174j), tol=ATOL) + assert ae(v.real, 0.49739047283060471093, tol=PTOL) + assert ae(v.imag, 0.41543605404038863174, tol=PTOL) + v = fp.e1((-2.0 + 8.0j)) + assert ae(v, (-0.8705211147733730969 + 0.24099328498605539667j), tol=ATOL) + assert ae(v.real, -0.8705211147733730969, tol=PTOL) + assert ae(v.imag, 0.24099328498605539667, tol=PTOL) + v = fp.e1((-5.0 + 20.0j)) + assert ae(v, (-7.0789514293925893007 - 1.6102177171960790536j), tol=ATOL) + assert ae(v.real, -7.0789514293925893007, tol=PTOL) + assert ae(v.imag, -1.6102177171960790536, tol=PTOL) + v = fp.e1((-20.0 + 80.0j)) + assert ae(v, (5855431.4907298084434 - 720920.93315409165707j), tol=ATOL) + assert ae(v.real, 5855431.4907298084434, tol=PTOL) + assert ae(v.imag, -720920.93315409165707, tol=PTOL) + v = fp.e1((-30.0 + 120.0j)) + assert ae(v, (-65402491644.703470747 - 56697658399.657460294j), tol=ATOL) + assert ae(v.real, -65402491644.703470747, tol=PTOL) + assert ae(v.imag, -56697658399.657460294, tol=PTOL) + v = fp.e1((-40.0 + 160.0j)) + assert ae(v, (25504929379604.776769 + 1429035198630573.2463j), tol=ATOL) + assert ae(v.real, 25504929379604.776769, tol=PTOL) + assert ae(v.imag, 1429035198630573.2463, tol=PTOL) + v = fp.e1((-50.0 + 200.0j)) + assert ae(v, (18437746526988116954.0 - 17146362239046152345.0j), tol=ATOL) + assert ae(v.real, 18437746526988116954.0, tol=PTOL) + assert ae(v.imag, -17146362239046152345.0, tol=PTOL) + v = fp.e1((-80.0 + 320.0j)) + assert ae(v, (3.3464697299634526706e+31 - 1.6473152633843023919e+32j), tol=ATOL) + assert ae(v.real, 3.3464697299634526706e+31, tol=PTOL) + assert ae(v.imag, -1.6473152633843023919e+32, tol=PTOL) + v = fp.e1((-4.6566128730773925781e-10 + 1.1641532182693481445e-10j)) + assert ae(v, (20.880034621082893023 - 2.8966139903465137624j), tol=ATOL) + assert ae(v.real, 20.880034621082893023, tol=PTOL) + assert ae(v.imag, -2.8966139903465137624, tol=PTOL) + v = fp.e1((-1.0 + 0.25j)) + assert ae(v, (-1.8942716983721074932 - 2.4689102827070540799j), tol=ATOL) + assert ae(v.real, -1.8942716983721074932, tol=PTOL) + assert ae(v.imag, -2.4689102827070540799, tol=PTOL) + v = fp.e1((-4.0 + 1.0j)) + assert ae(v, (-14.806699492675420438 + 9.1384225230837893776j), tol=ATOL) + assert ae(v.real, -14.806699492675420438, tol=PTOL) + assert ae(v.imag, 9.1384225230837893776, tol=PTOL) + v = fp.e1((-8.0 + 2.0j)) + assert ae(v, (54.633252667426386294 + 413.20318163814670688j), tol=ATOL) + assert ae(v.real, 54.633252667426386294, tol=PTOL) + assert ae(v.imag, 413.20318163814670688, tol=PTOL) + v = fp.e1((-20.0 + 5.0j)) + assert ae(v, (-711836.97165402624643 - 24745250.939695900956j), tol=ATOL) + assert ae(v.real, -711836.97165402624643, tol=PTOL) + assert ae(v.imag, -24745250.939695900956, tol=PTOL) + v = fp.e1((-80.0 + 20.0j)) + assert ae(v, (-4.2139911108612653091e+32 + 5.3367124741918251637e+32j), tol=ATOL) + assert ae(v.real, -4.2139911108612653091e+32, tol=PTOL) + assert ae(v.imag, 5.3367124741918251637e+32, tol=PTOL) + v = fp.e1((-120.0 + 30.0j)) + assert ae(v, (9.7760616203707508892e+48 - 1.058257682317195792e+50j), tol=ATOL) + assert ae(v.real, 9.7760616203707508892e+48, tol=PTOL) + assert ae(v.imag, -1.058257682317195792e+50, tol=PTOL) + v = fp.e1((-160.0 + 40.0j)) + assert ae(v, (8.7065541466623638861e+66 + 1.6577106725141739889e+67j), tol=ATOL) + assert ae(v.real, 8.7065541466623638861e+66, tol=PTOL) + assert ae(v.imag, 1.6577106725141739889e+67, tol=PTOL) + v = fp.e1((-200.0 + 50.0j)) + assert ae(v, (-3.070744996327018106e+84 - 1.7243244846769415903e+84j), tol=ATOL) + assert ae(v.real, -3.070744996327018106e+84, tol=PTOL) + assert ae(v.imag, -1.7243244846769415903e+84, tol=PTOL) + v = fp.e1((-320.0 + 80.0j)) + assert ae(v, (9.9960598637998647276e+135 - 2.6855081527595608863e+136j), tol=ATOL) + assert ae(v.real, 9.9960598637998647276e+135, tol=PTOL) + assert ae(v.imag, -2.6855081527595608863e+136, tol=PTOL) + v = fp.e1(-1.1641532182693481445e-10) + assert ae(v, (22.296641293460247028 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 22.296641293460247028, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-0.25) + assert ae(v, (0.54254326466191372953 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 0.54254326466191372953, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-1.0) + assert ae(v, (-1.8951178163559367555 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -1.8951178163559367555, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-2.0) + assert ae(v, (-4.9542343560018901634 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -4.9542343560018901634, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-5.0) + assert ae(v, (-40.185275355803177455 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -40.185275355803177455, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-20.0) + assert ae(v, (-25615652.66405658882 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -25615652.66405658882, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-30.0) + assert ae(v, (-368973209407.27419706 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -368973209407.27419706, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-40.0) + assert ae(v, (-6039718263611241.5784 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -6039718263611241.5784, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-50.0) + assert ae(v, (-1.0585636897131690963e+20 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -1.0585636897131690963e+20, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1(-80.0) + assert ae(v, (-7.0146000049047999696e+32 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -7.0146000049047999696e+32, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-1.1641532182693481445e-10 + 0.0j)) + assert ae(v, (22.296641293460247028 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 22.296641293460247028, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-0.25 + 0.0j)) + assert ae(v, (0.54254326466191372953 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 0.54254326466191372953, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-1.0 + 0.0j)) + assert ae(v, (-1.8951178163559367555 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -1.8951178163559367555, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-2.0 + 0.0j)) + assert ae(v, (-4.9542343560018901634 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -4.9542343560018901634, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-5.0 + 0.0j)) + assert ae(v, (-40.185275355803177455 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -40.185275355803177455, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-20.0 + 0.0j)) + assert ae(v, (-25615652.66405658882 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -25615652.66405658882, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-30.0 + 0.0j)) + assert ae(v, (-368973209407.27419706 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -368973209407.27419706, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-40.0 + 0.0j)) + assert ae(v, (-6039718263611241.5784 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -6039718263611241.5784, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-50.0 + 0.0j)) + assert ae(v, (-1.0585636897131690963e+20 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -1.0585636897131690963e+20, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-80.0 + 0.0j)) + assert ae(v, (-7.0146000049047999696e+32 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -7.0146000049047999696e+32, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.e1((-4.6566128730773925781e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (20.880034621082893023 + 2.8966139903465137624j), tol=ATOL) + assert ae(v.real, 20.880034621082893023, tol=PTOL) + assert ae(v.imag, 2.8966139903465137624, tol=PTOL) + v = fp.e1((-1.0 - 0.25j)) + assert ae(v, (-1.8942716983721074932 + 2.4689102827070540799j), tol=ATOL) + assert ae(v.real, -1.8942716983721074932, tol=PTOL) + assert ae(v.imag, 2.4689102827070540799, tol=PTOL) + v = fp.e1((-4.0 - 1.0j)) + assert ae(v, (-14.806699492675420438 - 9.1384225230837893776j), tol=ATOL) + assert ae(v.real, -14.806699492675420438, tol=PTOL) + assert ae(v.imag, -9.1384225230837893776, tol=PTOL) + v = fp.e1((-8.0 - 2.0j)) + assert ae(v, (54.633252667426386294 - 413.20318163814670688j), tol=ATOL) + assert ae(v.real, 54.633252667426386294, tol=PTOL) + assert ae(v.imag, -413.20318163814670688, tol=PTOL) + v = fp.e1((-20.0 - 5.0j)) + assert ae(v, (-711836.97165402624643 + 24745250.939695900956j), tol=ATOL) + assert ae(v.real, -711836.97165402624643, tol=PTOL) + assert ae(v.imag, 24745250.939695900956, tol=PTOL) + v = fp.e1((-80.0 - 20.0j)) + assert ae(v, (-4.2139911108612653091e+32 - 5.3367124741918251637e+32j), tol=ATOL) + assert ae(v.real, -4.2139911108612653091e+32, tol=PTOL) + assert ae(v.imag, -5.3367124741918251637e+32, tol=PTOL) + v = fp.e1((-120.0 - 30.0j)) + assert ae(v, (9.7760616203707508892e+48 + 1.058257682317195792e+50j), tol=ATOL) + assert ae(v.real, 9.7760616203707508892e+48, tol=PTOL) + assert ae(v.imag, 1.058257682317195792e+50, tol=PTOL) + v = fp.e1((-160.0 - 40.0j)) + assert ae(v, (8.7065541466623638861e+66 - 1.6577106725141739889e+67j), tol=ATOL) + assert ae(v.real, 8.7065541466623638861e+66, tol=PTOL) + assert ae(v.imag, -1.6577106725141739889e+67, tol=PTOL) + v = fp.e1((-200.0 - 50.0j)) + assert ae(v, (-3.070744996327018106e+84 + 1.7243244846769415903e+84j), tol=ATOL) + assert ae(v.real, -3.070744996327018106e+84, tol=PTOL) + assert ae(v.imag, 1.7243244846769415903e+84, tol=PTOL) + v = fp.e1((-320.0 - 80.0j)) + assert ae(v, (9.9960598637998647276e+135 + 2.6855081527595608863e+136j), tol=ATOL) + assert ae(v.real, 9.9960598637998647276e+135, tol=PTOL) + assert ae(v.imag, 2.6855081527595608863e+136, tol=PTOL) + v = fp.e1((-1.1641532182693481445e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (21.950067703180274374 + 2.356194490075929607j), tol=ATOL) + assert ae(v.real, 21.950067703180274374, tol=PTOL) + assert ae(v.imag, 2.356194490075929607, tol=PTOL) + v = fp.e1((-0.25 - 0.25j)) + assert ae(v, (0.21441047326710323254 + 2.0732153554307936389j), tol=ATOL) + assert ae(v.real, 0.21441047326710323254, tol=PTOL) + assert ae(v.imag, 2.0732153554307936389, tol=PTOL) + v = fp.e1((-1.0 - 1.0j)) + assert ae(v, (-1.7646259855638540684 + 0.7538228020792708192j), tol=ATOL) + assert ae(v.real, -1.7646259855638540684, tol=PTOL) + assert ae(v.imag, 0.7538228020792708192, tol=PTOL) + v = fp.e1((-2.0 - 2.0j)) + assert ae(v, (-1.8920781621855474089 - 2.1753697842428647236j), tol=ATOL) + assert ae(v.real, -1.8920781621855474089, tol=PTOL) + assert ae(v.imag, -2.1753697842428647236, tol=PTOL) + v = fp.e1((-5.0 - 5.0j)) + assert ae(v, (13.470936071475245856 + 18.464085049321024206j), tol=ATOL) + assert ae(v.real, 13.470936071475245856, tol=PTOL) + assert ae(v.imag, 18.464085049321024206, tol=PTOL) + v = fp.e1((-20.0 - 20.0j)) + assert ae(v, (-16589317.398788971896 - 5831702.3296441771206j), tol=ATOL) + assert ae(v.real, -16589317.398788971896, tol=PTOL) + assert ae(v.imag, -5831702.3296441771206, tol=PTOL) + v = fp.e1((-30.0 - 30.0j)) + assert ae(v, (154596484273.69322527 + 204179357837.41389696j), tol=ATOL) + assert ae(v.real, 154596484273.69322527, tol=PTOL) + assert ae(v.imag, 204179357837.41389696, tol=PTOL) + v = fp.e1((-40.0 - 40.0j)) + assert ae(v, (-287512180321448.45408 - 4203502407932314.974j), tol=ATOL) + assert ae(v.real, -287512180321448.45408, tol=PTOL) + assert ae(v.imag, -4203502407932314.974, tol=PTOL) + v = fp.e1((-50.0 - 50.0j)) + assert ae(v, (-36128528616649268826.0 + 64648801861338741963.0j), tol=ATOL) + assert ae(v.real, -36128528616649268826.0, tol=PTOL) + assert ae(v.imag, 64648801861338741963.0, tol=PTOL) + v = fp.e1((-80.0 - 80.0j)) + assert ae(v, (3.8674816337930010217e+32 + 3.0540709639658071041e+32j), tol=ATOL) + assert ae(v.real, 3.8674816337930010217e+32, tol=PTOL) + assert ae(v.imag, 3.0540709639658071041e+32, tol=PTOL) + v = fp.e1((-1.1641532182693481445e-10 - 4.6566128730773925781e-10j)) + assert ae(v, (20.880034621432138988 + 1.8157749894560994861j), tol=ATOL) + assert ae(v.real, 20.880034621432138988, tol=PTOL) + assert ae(v.imag, 1.8157749894560994861, tol=PTOL) + v = fp.e1((-0.25 - 1.0j)) + assert ae(v, (-0.59066621214766308594 + 0.74474454765205036972j), tol=ATOL) + assert ae(v.real, -0.59066621214766308594, tol=PTOL) + assert ae(v.imag, 0.74474454765205036972, tol=PTOL) + v = fp.e1((-1.0 - 4.0j)) + assert ae(v, (0.49739047283060471093 - 0.41543605404038863174j), tol=ATOL) + assert ae(v.real, 0.49739047283060471093, tol=PTOL) + assert ae(v.imag, -0.41543605404038863174, tol=PTOL) + v = fp.e1((-2.0 - 8.0j)) + assert ae(v, (-0.8705211147733730969 - 0.24099328498605539667j), tol=ATOL) + assert ae(v.real, -0.8705211147733730969, tol=PTOL) + assert ae(v.imag, -0.24099328498605539667, tol=PTOL) + v = fp.e1((-5.0 - 20.0j)) + assert ae(v, (-7.0789514293925893007 + 1.6102177171960790536j), tol=ATOL) + assert ae(v.real, -7.0789514293925893007, tol=PTOL) + assert ae(v.imag, 1.6102177171960790536, tol=PTOL) + v = fp.e1((-20.0 - 80.0j)) + assert ae(v, (5855431.4907298084434 + 720920.93315409165707j), tol=ATOL) + assert ae(v.real, 5855431.4907298084434, tol=PTOL) + assert ae(v.imag, 720920.93315409165707, tol=PTOL) + v = fp.e1((-30.0 - 120.0j)) + assert ae(v, (-65402491644.703470747 + 56697658399.657460294j), tol=ATOL) + assert ae(v.real, -65402491644.703470747, tol=PTOL) + assert ae(v.imag, 56697658399.657460294, tol=PTOL) + v = fp.e1((-40.0 - 160.0j)) + assert ae(v, (25504929379604.776769 - 1429035198630573.2463j), tol=ATOL) + assert ae(v.real, 25504929379604.776769, tol=PTOL) + assert ae(v.imag, -1429035198630573.2463, tol=PTOL) + v = fp.e1((-50.0 - 200.0j)) + assert ae(v, (18437746526988116954.0 + 17146362239046152345.0j), tol=ATOL) + assert ae(v.real, 18437746526988116954.0, tol=PTOL) + assert ae(v.imag, 17146362239046152345.0, tol=PTOL) + v = fp.e1((-80.0 - 320.0j)) + assert ae(v, (3.3464697299634526706e+31 + 1.6473152633843023919e+32j), tol=ATOL) + assert ae(v.real, 3.3464697299634526706e+31, tol=PTOL) + assert ae(v.imag, 1.6473152633843023919e+32, tol=PTOL) + v = fp.e1((0.0 - 1.1641532182693481445e-10j)) + assert ae(v, (22.29664129357666235 + 1.5707963266784812974j), tol=ATOL) + assert ae(v.real, 22.29664129357666235, tol=PTOL) + assert ae(v.imag, 1.5707963266784812974, tol=PTOL) + v = fp.e1((0.0 - 0.25j)) + assert ae(v, (0.82466306258094565309 + 1.3216627564751394551j), tol=ATOL) + assert ae(v.real, 0.82466306258094565309, tol=PTOL) + assert ae(v.imag, 1.3216627564751394551, tol=PTOL) + v = fp.e1((0.0 - 1.0j)) + assert ae(v, (-0.33740392290096813466 + 0.62471325642771360429j), tol=ATOL) + assert ae(v.real, -0.33740392290096813466, tol=PTOL) + assert ae(v.imag, 0.62471325642771360429, tol=PTOL) + v = fp.e1((0.0 - 2.0j)) + assert ae(v, (-0.4229808287748649957 - 0.034616650007798229345j), tol=ATOL) + assert ae(v.real, -0.4229808287748649957, tol=PTOL) + assert ae(v.imag, -0.034616650007798229345, tol=PTOL) + v = fp.e1((0.0 - 5.0j)) + assert ae(v, (0.19002974965664387862 + 0.020865081850222481957j), tol=ATOL) + assert ae(v.real, 0.19002974965664387862, tol=PTOL) + assert ae(v.imag, 0.020865081850222481957, tol=PTOL) + v = fp.e1((0.0 - 20.0j)) + assert ae(v, (-0.04441982084535331654 + 0.022554625751456779068j), tol=ATOL) + assert ae(v.real, -0.04441982084535331654, tol=PTOL) + assert ae(v.imag, 0.022554625751456779068, tol=PTOL) + v = fp.e1((0.0 - 30.0j)) + assert ae(v, (0.033032417282071143779 + 0.0040397867645455082476j), tol=ATOL) + assert ae(v.real, 0.033032417282071143779, tol=PTOL) + assert ae(v.imag, 0.0040397867645455082476, tol=PTOL) + v = fp.e1((0.0 - 40.0j)) + assert ae(v, (-0.019020007896208766962 - 0.016188792559887887544j), tol=ATOL) + assert ae(v.real, -0.019020007896208766962, tol=PTOL) + assert ae(v.imag, -0.016188792559887887544, tol=PTOL) + v = fp.e1((0.0 - 50.0j)) + assert ae(v, (0.0056283863241163054402 + 0.019179254308960724503j), tol=ATOL) + assert ae(v.real, 0.0056283863241163054402, tol=PTOL) + assert ae(v.imag, 0.019179254308960724503, tol=PTOL) + v = fp.e1((0.0 - 80.0j)) + assert ae(v, (0.012402501155070958192 - 0.0015345601175906961199j), tol=ATOL) + assert ae(v.real, 0.012402501155070958192, tol=PTOL) + assert ae(v.imag, -0.0015345601175906961199, tol=PTOL) + v = fp.e1((1.1641532182693481445e-10 - 4.6566128730773925781e-10j)) + assert ae(v, (20.880034621664969632 + 1.3258176632023711778j), tol=ATOL) + assert ae(v.real, 20.880034621664969632, tol=PTOL) + assert ae(v.imag, 1.3258176632023711778, tol=PTOL) + v = fp.e1((0.25 - 1.0j)) + assert ae(v, (-0.16868306393667788761 + 0.4858011885947426971j), tol=ATOL) + assert ae(v.real, -0.16868306393667788761, tol=PTOL) + assert ae(v.imag, 0.4858011885947426971, tol=PTOL) + v = fp.e1((1.0 - 4.0j)) + assert ae(v, (0.03373591813926547318 - 0.073523452241083821877j), tol=ATOL) + assert ae(v.real, 0.03373591813926547318, tol=PTOL) + assert ae(v.imag, -0.073523452241083821877, tol=PTOL) + v = fp.e1((2.0 - 8.0j)) + assert ae(v, (-0.015392833434733785143 + 0.0031747121557605415914j), tol=ATOL) + assert ae(v.real, -0.015392833434733785143, tol=PTOL) + assert ae(v.imag, 0.0031747121557605415914, tol=PTOL) + v = fp.e1((5.0 - 20.0j)) + assert ae(v, (-0.00024419662286542966525 + 0.00021008322966152755674j), tol=ATOL) + assert ae(v.real, -0.00024419662286542966525, tol=PTOL) + assert ae(v.imag, 0.00021008322966152755674, tol=PTOL) + v = fp.e1((20.0 - 80.0j)) + assert ae(v, (2.3255552781051330088e-11 - 8.9463918891349438007e-12j), tol=ATOL) + assert ae(v.real, 2.3255552781051330088e-11, tol=PTOL) + assert ae(v.imag, -8.9463918891349438007e-12, tol=PTOL) + v = fp.e1((30.0 - 120.0j)) + assert ae(v, (-2.7068919097124652332e-16 + 7.0477762411705130239e-16j), tol=ATOL) + assert ae(v.real, -2.7068919097124652332e-16, tol=PTOL) + assert ae(v.imag, 7.0477762411705130239e-16, tol=PTOL) + v = fp.e1((40.0 - 160.0j)) + assert ae(v, (-1.1695597827678024687e-20 - 2.2907401455645736661e-20j), tol=ATOL) + assert ae(v.real, -1.1695597827678024687e-20, tol=PTOL) + assert ae(v.imag, -2.2907401455645736661e-20, tol=PTOL) + v = fp.e1((50.0 - 200.0j)) + assert ae(v, (9.0323746914410162531e-25 + 2.3950601790033530935e-25j), tol=ATOL) + assert ae(v.real, 9.0323746914410162531e-25, tol=PTOL) + assert ae(v.imag, 2.3950601790033530935e-25, tol=PTOL) + v = fp.e1((80.0 - 320.0j)) + assert ae(v, (3.4819106748728063576e-38 + 4.215653005615772724e-38j), tol=ATOL) + assert ae(v.real, 3.4819106748728063576e-38, tol=PTOL) + assert ae(v.imag, 4.215653005615772724e-38, tol=PTOL) + v = fp.e1((1.1641532182693481445e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (21.950067703413105017 + 0.7853981632810329878j), tol=ATOL) + assert ae(v.real, 21.950067703413105017, tol=PTOL) + assert ae(v.imag, 0.7853981632810329878, tol=PTOL) + v = fp.e1((0.25 - 0.25j)) + assert ae(v, (0.71092525792923287894 + 0.56491812441304194711j), tol=ATOL) + assert ae(v.real, 0.71092525792923287894, tol=PTOL) + assert ae(v.imag, 0.56491812441304194711, tol=PTOL) + v = fp.e1((1.0 - 1.0j)) + assert ae(v, (0.00028162445198141832551 + 0.17932453503935894015j), tol=ATOL) + assert ae(v.real, 0.00028162445198141832551, tol=PTOL) + assert ae(v.imag, 0.17932453503935894015, tol=PTOL) + v = fp.e1((2.0 - 2.0j)) + assert ae(v, (-0.033767089606562004246 + 0.018599414169750541925j), tol=ATOL) + assert ae(v.real, -0.033767089606562004246, tol=PTOL) + assert ae(v.imag, 0.018599414169750541925, tol=PTOL) + v = fp.e1((5.0 - 5.0j)) + assert ae(v, (0.0007266506660356393891 - 0.00047102780163522245054j), tol=ATOL) + assert ae(v.real, 0.0007266506660356393891, tol=PTOL) + assert ae(v.imag, -0.00047102780163522245054, tol=PTOL) + v = fp.e1((20.0 - 20.0j)) + assert ae(v, (-2.3824537449367396579e-11 + 6.6969873156525615158e-11j), tol=ATOL) + assert ae(v.real, -2.3824537449367396579e-11, tol=PTOL) + assert ae(v.imag, 6.6969873156525615158e-11, tol=PTOL) + v = fp.e1((30.0 - 30.0j)) + assert ae(v, (1.7316045841744061617e-15 - 1.3065678019487308689e-15j), tol=ATOL) + assert ae(v.real, 1.7316045841744061617e-15, tol=PTOL) + assert ae(v.imag, -1.3065678019487308689e-15, tol=PTOL) + v = fp.e1((40.0 - 40.0j)) + assert ae(v, (-7.4001043002899232182e-20 + 4.991847855336816304e-21j), tol=ATOL) + assert ae(v.real, -7.4001043002899232182e-20, tol=PTOL) + assert ae(v.imag, 4.991847855336816304e-21, tol=PTOL) + v = fp.e1((50.0 - 50.0j)) + assert ae(v, (2.3566128324644641219e-24 + 1.3188326726201614778e-24j), tol=ATOL) + assert ae(v.real, 2.3566128324644641219e-24, tol=PTOL) + assert ae(v.imag, 1.3188326726201614778e-24, tol=PTOL) + v = fp.e1((80.0 - 80.0j)) + assert ae(v, (9.8279750572186526673e-38 - 1.243952841288868831e-37j), tol=ATOL) + assert ae(v.real, 9.8279750572186526673e-38, tol=PTOL) + assert ae(v.imag, -1.243952841288868831e-37, tol=PTOL) + v = fp.e1((4.6566128730773925781e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (20.880034622014215597 + 0.24497866301044883237j), tol=ATOL) + assert ae(v.real, 20.880034622014215597, tol=PTOL) + assert ae(v.imag, 0.24497866301044883237, tol=PTOL) + v = fp.e1((1.0 - 0.25j)) + assert ae(v, (0.19731063945004229095 + 0.087366045774299963672j), tol=ATOL) + assert ae(v.real, 0.19731063945004229095, tol=PTOL) + assert ae(v.imag, 0.087366045774299963672, tol=PTOL) + v = fp.e1((4.0 - 1.0j)) + assert ae(v, (0.0013106173980145506944 + 0.0034542480199350626699j), tol=ATOL) + assert ae(v.real, 0.0013106173980145506944, tol=PTOL) + assert ae(v.imag, 0.0034542480199350626699, tol=PTOL) + v = fp.e1((8.0 - 2.0j)) + assert ae(v, (-0.000022278049065270225945 + 0.000029191940456521555288j), tol=ATOL) + assert ae(v.real, -0.000022278049065270225945, tol=PTOL) + assert ae(v.imag, 0.000029191940456521555288, tol=PTOL) + v = fp.e1((20.0 - 5.0j)) + assert ae(v, (4.7711374515765346894e-11 - 8.2902652405126947359e-11j), tol=ATOL) + assert ae(v.real, 4.7711374515765346894e-11, tol=PTOL) + assert ae(v.imag, -8.2902652405126947359e-11, tol=PTOL) + v = fp.e1((80.0 - 20.0j)) + assert ae(v, (3.8353473865788235787e-38 + 2.129247592349605139e-37j), tol=ATOL) + assert ae(v.real, 3.8353473865788235787e-38, tol=PTOL) + assert ae(v.imag, 2.129247592349605139e-37, tol=PTOL) + v = fp.e1((120.0 - 30.0j)) + assert ae(v, (2.3836002337480334716e-55 - 5.6704043587126198306e-55j), tol=ATOL) + assert ae(v.real, 2.3836002337480334716e-55, tol=PTOL) + assert ae(v.imag, -5.6704043587126198306e-55, tol=PTOL) + v = fp.e1((160.0 - 40.0j)) + assert ae(v, (-1.6238022898654510661e-72 + 1.104172355572287367e-72j), tol=ATOL) + assert ae(v.real, -1.6238022898654510661e-72, tol=PTOL) + assert ae(v.imag, 1.104172355572287367e-72, tol=PTOL) + v = fp.e1((200.0 - 50.0j)) + assert ae(v, (6.6800061461666228487e-90 - 1.4473816083541016115e-91j), tol=ATOL) + assert ae(v.real, 6.6800061461666228487e-90, tol=PTOL) + assert ae(v.imag, -1.4473816083541016115e-91, tol=PTOL) + v = fp.e1((320.0 - 80.0j)) + assert ae(v, (4.2737871527778786157e-143 - 3.1789935525785660314e-142j), tol=ATOL) + assert ae(v.real, 4.2737871527778786157e-143, tol=PTOL) + assert ae(v.imag, -3.1789935525785660314e-142, tol=PTOL) + v = fp.ei(1.1641532182693481445e-10) + assert ae(v, -22.296641293460247028, tol=ATOL) + assert type(v) is float + v = fp.ei(0.25) + assert ae(v, -0.54254326466191372953, tol=ATOL) + assert type(v) is float + v = fp.ei(1.0) + assert ae(v, 1.8951178163559367555, tol=ATOL) + assert type(v) is float + v = fp.ei(2.0) + assert ae(v, 4.9542343560018901634, tol=ATOL) + assert type(v) is float + v = fp.ei(5.0) + assert ae(v, 40.185275355803177455, tol=ATOL) + assert type(v) is float + v = fp.ei(20.0) + assert ae(v, 25615652.66405658882, tol=ATOL) + assert type(v) is float + v = fp.ei(30.0) + assert ae(v, 368973209407.27419706, tol=ATOL) + assert type(v) is float + v = fp.ei(40.0) + assert ae(v, 6039718263611241.5784, tol=ATOL) + assert type(v) is float + v = fp.ei(50.0) + assert ae(v, 1.0585636897131690963e+20, tol=ATOL) + assert type(v) is float + v = fp.ei(80.0) + assert ae(v, 7.0146000049047999696e+32, tol=ATOL) + assert type(v) is float + v = fp.ei((1.1641532182693481445e-10 + 0.0j)) + assert ae(v, (-22.296641293460247028 + 0.0j), tol=ATOL) + assert ae(v.real, -22.296641293460247028, tol=PTOL) + assert v.imag == 0 + v = fp.ei((0.25 + 0.0j)) + assert ae(v, (-0.54254326466191372953 + 0.0j), tol=ATOL) + assert ae(v.real, -0.54254326466191372953, tol=PTOL) + assert v.imag == 0 + v = fp.ei((1.0 + 0.0j)) + assert ae(v, (1.8951178163559367555 + 0.0j), tol=ATOL) + assert ae(v.real, 1.8951178163559367555, tol=PTOL) + assert v.imag == 0 + v = fp.ei((2.0 + 0.0j)) + assert ae(v, (4.9542343560018901634 + 0.0j), tol=ATOL) + assert ae(v.real, 4.9542343560018901634, tol=PTOL) + assert v.imag == 0 + v = fp.ei((5.0 + 0.0j)) + assert ae(v, (40.185275355803177455 + 0.0j), tol=ATOL) + assert ae(v.real, 40.185275355803177455, tol=PTOL) + assert v.imag == 0 + v = fp.ei((20.0 + 0.0j)) + assert ae(v, (25615652.66405658882 + 0.0j), tol=ATOL) + assert ae(v.real, 25615652.66405658882, tol=PTOL) + assert v.imag == 0 + v = fp.ei((30.0 + 0.0j)) + assert ae(v, (368973209407.27419706 + 0.0j), tol=ATOL) + assert ae(v.real, 368973209407.27419706, tol=PTOL) + assert v.imag == 0 + v = fp.ei((40.0 + 0.0j)) + assert ae(v, (6039718263611241.5784 + 0.0j), tol=ATOL) + assert ae(v.real, 6039718263611241.5784, tol=PTOL) + assert v.imag == 0 + v = fp.ei((50.0 + 0.0j)) + assert ae(v, (1.0585636897131690963e+20 + 0.0j), tol=ATOL) + assert ae(v.real, 1.0585636897131690963e+20, tol=PTOL) + assert v.imag == 0 + v = fp.ei((80.0 + 0.0j)) + assert ae(v, (7.0146000049047999696e+32 + 0.0j), tol=ATOL) + assert ae(v.real, 7.0146000049047999696e+32, tol=PTOL) + assert v.imag == 0 + v = fp.ei((4.6566128730773925781e-10 + 1.1641532182693481445e-10j)) + assert ae(v, (-20.880034621082893023 + 0.24497866324327947603j), tol=ATOL) + assert ae(v.real, -20.880034621082893023, tol=PTOL) + assert ae(v.imag, 0.24497866324327947603, tol=PTOL) + v = fp.ei((1.0 + 0.25j)) + assert ae(v, (1.8942716983721074932 + 0.67268237088273915854j), tol=ATOL) + assert ae(v.real, 1.8942716983721074932, tol=PTOL) + assert ae(v.imag, 0.67268237088273915854, tol=PTOL) + v = fp.ei((4.0 + 1.0j)) + assert ae(v, (14.806699492675420438 + 12.280015176673582616j), tol=ATOL) + assert ae(v.real, 14.806699492675420438, tol=PTOL) + assert ae(v.imag, 12.280015176673582616, tol=PTOL) + v = fp.ei((8.0 + 2.0j)) + assert ae(v, (-54.633252667426386294 + 416.34477429173650012j), tol=ATOL) + assert ae(v.real, -54.633252667426386294, tol=PTOL) + assert ae(v.imag, 416.34477429173650012, tol=PTOL) + v = fp.ei((20.0 + 5.0j)) + assert ae(v, (711836.97165402624643 - 24745247.798103247366j), tol=ATOL) + assert ae(v.real, 711836.97165402624643, tol=PTOL) + assert ae(v.imag, -24745247.798103247366, tol=PTOL) + v = fp.ei((80.0 + 20.0j)) + assert ae(v, (4.2139911108612653091e+32 + 5.3367124741918251637e+32j), tol=ATOL) + assert ae(v.real, 4.2139911108612653091e+32, tol=PTOL) + assert ae(v.imag, 5.3367124741918251637e+32, tol=PTOL) + v = fp.ei((120.0 + 30.0j)) + assert ae(v, (-9.7760616203707508892e+48 - 1.058257682317195792e+50j), tol=ATOL) + assert ae(v.real, -9.7760616203707508892e+48, tol=PTOL) + assert ae(v.imag, -1.058257682317195792e+50, tol=PTOL) + v = fp.ei((160.0 + 40.0j)) + assert ae(v, (-8.7065541466623638861e+66 + 1.6577106725141739889e+67j), tol=ATOL) + assert ae(v.real, -8.7065541466623638861e+66, tol=PTOL) + assert ae(v.imag, 1.6577106725141739889e+67, tol=PTOL) + v = fp.ei((200.0 + 50.0j)) + assert ae(v, (3.070744996327018106e+84 - 1.7243244846769415903e+84j), tol=ATOL) + assert ae(v.real, 3.070744996327018106e+84, tol=PTOL) + assert ae(v.imag, -1.7243244846769415903e+84, tol=PTOL) + v = fp.ei((320.0 + 80.0j)) + assert ae(v, (-9.9960598637998647276e+135 - 2.6855081527595608863e+136j), tol=ATOL) + assert ae(v.real, -9.9960598637998647276e+135, tol=PTOL) + assert ae(v.imag, -2.6855081527595608863e+136, tol=PTOL) + v = fp.ei((1.1641532182693481445e-10 + 1.1641532182693481445e-10j)) + assert ae(v, (-21.950067703180274374 + 0.78539816351386363145j), tol=ATOL) + assert ae(v.real, -21.950067703180274374, tol=PTOL) + assert ae(v.imag, 0.78539816351386363145, tol=PTOL) + v = fp.ei((0.25 + 0.25j)) + assert ae(v, (-0.21441047326710323254 + 1.0683772981589995996j), tol=ATOL) + assert ae(v.real, -0.21441047326710323254, tol=PTOL) + assert ae(v.imag, 1.0683772981589995996, tol=PTOL) + v = fp.ei((1.0 + 1.0j)) + assert ae(v, (1.7646259855638540684 + 2.3877698515105224193j), tol=ATOL) + assert ae(v.real, 1.7646259855638540684, tol=PTOL) + assert ae(v.imag, 2.3877698515105224193, tol=PTOL) + v = fp.ei((2.0 + 2.0j)) + assert ae(v, (1.8920781621855474089 + 5.3169624378326579621j), tol=ATOL) + assert ae(v.real, 1.8920781621855474089, tol=PTOL) + assert ae(v.imag, 5.3169624378326579621, tol=PTOL) + v = fp.ei((5.0 + 5.0j)) + assert ae(v, (-13.470936071475245856 - 15.322492395731230968j), tol=ATOL) + assert ae(v.real, -13.470936071475245856, tol=PTOL) + assert ae(v.imag, -15.322492395731230968, tol=PTOL) + v = fp.ei((20.0 + 20.0j)) + assert ae(v, (16589317.398788971896 + 5831705.4712368307104j), tol=ATOL) + assert ae(v.real, 16589317.398788971896, tol=PTOL) + assert ae(v.imag, 5831705.4712368307104, tol=PTOL) + v = fp.ei((30.0 + 30.0j)) + assert ae(v, (-154596484273.69322527 - 204179357834.2723043j), tol=ATOL) + assert ae(v.real, -154596484273.69322527, tol=PTOL) + assert ae(v.imag, -204179357834.2723043, tol=PTOL) + v = fp.ei((40.0 + 40.0j)) + assert ae(v, (287512180321448.45408 + 4203502407932318.1156j), tol=ATOL) + assert ae(v.real, 287512180321448.45408, tol=PTOL) + assert ae(v.imag, 4203502407932318.1156, tol=PTOL) + v = fp.ei((50.0 + 50.0j)) + assert ae(v, (36128528616649268826.0 - 64648801861338741960.0j), tol=ATOL) + assert ae(v.real, 36128528616649268826.0, tol=PTOL) + assert ae(v.imag, -64648801861338741960.0, tol=PTOL) + v = fp.ei((80.0 + 80.0j)) + assert ae(v, (-3.8674816337930010217e+32 - 3.0540709639658071041e+32j), tol=ATOL) + assert ae(v.real, -3.8674816337930010217e+32, tol=PTOL) + assert ae(v.imag, -3.0540709639658071041e+32, tol=PTOL) + v = fp.ei((1.1641532182693481445e-10 + 4.6566128730773925781e-10j)) + assert ae(v, (-20.880034621432138988 + 1.3258176641336937524j), tol=ATOL) + assert ae(v.real, -20.880034621432138988, tol=PTOL) + assert ae(v.imag, 1.3258176641336937524, tol=PTOL) + v = fp.ei((0.25 + 1.0j)) + assert ae(v, (0.59066621214766308594 + 2.3968481059377428687j), tol=ATOL) + assert ae(v.real, 0.59066621214766308594, tol=PTOL) + assert ae(v.imag, 2.3968481059377428687, tol=PTOL) + v = fp.ei((1.0 + 4.0j)) + assert ae(v, (-0.49739047283060471093 + 3.5570287076301818702j), tol=ATOL) + assert ae(v.real, -0.49739047283060471093, tol=PTOL) + assert ae(v.imag, 3.5570287076301818702, tol=PTOL) + v = fp.ei((2.0 + 8.0j)) + assert ae(v, (0.8705211147733730969 + 3.3825859385758486351j), tol=ATOL) + assert ae(v.real, 0.8705211147733730969, tol=PTOL) + assert ae(v.imag, 3.3825859385758486351, tol=PTOL) + v = fp.ei((5.0 + 20.0j)) + assert ae(v, (7.0789514293925893007 + 1.5313749363937141849j), tol=ATOL) + assert ae(v.real, 7.0789514293925893007, tol=PTOL) + assert ae(v.imag, 1.5313749363937141849, tol=PTOL) + v = fp.ei((20.0 + 80.0j)) + assert ae(v, (-5855431.4907298084434 - 720917.79156143806727j), tol=ATOL) + assert ae(v.real, -5855431.4907298084434, tol=PTOL) + assert ae(v.imag, -720917.79156143806727, tol=PTOL) + v = fp.ei((30.0 + 120.0j)) + assert ae(v, (65402491644.703470747 - 56697658396.51586764j), tol=ATOL) + assert ae(v.real, 65402491644.703470747, tol=PTOL) + assert ae(v.imag, -56697658396.51586764, tol=PTOL) + v = fp.ei((40.0 + 160.0j)) + assert ae(v, (-25504929379604.776769 + 1429035198630576.3879j), tol=ATOL) + assert ae(v.real, -25504929379604.776769, tol=PTOL) + assert ae(v.imag, 1429035198630576.3879, tol=PTOL) + v = fp.ei((50.0 + 200.0j)) + assert ae(v, (-18437746526988116954.0 - 17146362239046152342.0j), tol=ATOL) + assert ae(v.real, -18437746526988116954.0, tol=PTOL) + assert ae(v.imag, -17146362239046152342.0, tol=PTOL) + v = fp.ei((80.0 + 320.0j)) + assert ae(v, (-3.3464697299634526706e+31 - 1.6473152633843023919e+32j), tol=ATOL) + assert ae(v.real, -3.3464697299634526706e+31, tol=PTOL) + assert ae(v.imag, -1.6473152633843023919e+32, tol=PTOL) + v = fp.ei((0.0 + 1.1641532182693481445e-10j)) + assert ae(v, (-22.29664129357666235 + 1.5707963269113119411j), tol=ATOL) + assert ae(v.real, -22.29664129357666235, tol=PTOL) + assert ae(v.imag, 1.5707963269113119411, tol=PTOL) + v = fp.ei((0.0 + 0.25j)) + assert ae(v, (-0.82466306258094565309 + 1.8199298971146537833j), tol=ATOL) + assert ae(v.real, -0.82466306258094565309, tol=PTOL) + assert ae(v.imag, 1.8199298971146537833, tol=PTOL) + v = fp.ei((0.0 + 1.0j)) + assert ae(v, (0.33740392290096813466 + 2.5168793971620796342j), tol=ATOL) + assert ae(v.real, 0.33740392290096813466, tol=PTOL) + assert ae(v.imag, 2.5168793971620796342, tol=PTOL) + v = fp.ei((0.0 + 2.0j)) + assert ae(v, (0.4229808287748649957 + 3.1762093035975914678j), tol=ATOL) + assert ae(v.real, 0.4229808287748649957, tol=PTOL) + assert ae(v.imag, 3.1762093035975914678, tol=PTOL) + v = fp.ei((0.0 + 5.0j)) + assert ae(v, (-0.19002974965664387862 + 3.1207275717395707565j), tol=ATOL) + assert ae(v.real, -0.19002974965664387862, tol=PTOL) + assert ae(v.imag, 3.1207275717395707565, tol=PTOL) + v = fp.ei((0.0 + 20.0j)) + assert ae(v, (0.04441982084535331654 + 3.1190380278383364594j), tol=ATOL) + assert ae(v.real, 0.04441982084535331654, tol=PTOL) + assert ae(v.imag, 3.1190380278383364594, tol=PTOL) + v = fp.ei((0.0 + 30.0j)) + assert ae(v, (-0.033032417282071143779 + 3.1375528668252477302j), tol=ATOL) + assert ae(v.real, -0.033032417282071143779, tol=PTOL) + assert ae(v.imag, 3.1375528668252477302, tol=PTOL) + v = fp.ei((0.0 + 40.0j)) + assert ae(v, (0.019020007896208766962 + 3.157781446149681126j), tol=ATOL) + assert ae(v.real, 0.019020007896208766962, tol=PTOL) + assert ae(v.imag, 3.157781446149681126, tol=PTOL) + v = fp.ei((0.0 + 50.0j)) + assert ae(v, (-0.0056283863241163054402 + 3.122413399280832514j), tol=ATOL) + assert ae(v.real, -0.0056283863241163054402, tol=PTOL) + assert ae(v.imag, 3.122413399280832514, tol=PTOL) + v = fp.ei((0.0 + 80.0j)) + assert ae(v, (-0.012402501155070958192 + 3.1431272137073839346j), tol=ATOL) + assert ae(v.real, -0.012402501155070958192, tol=PTOL) + assert ae(v.imag, 3.1431272137073839346, tol=PTOL) + v = fp.ei((-1.1641532182693481445e-10 + 4.6566128730773925781e-10j)) + assert ae(v, (-20.880034621664969632 + 1.8157749903874220607j), tol=ATOL) + assert ae(v.real, -20.880034621664969632, tol=PTOL) + assert ae(v.imag, 1.8157749903874220607, tol=PTOL) + v = fp.ei((-0.25 + 1.0j)) + assert ae(v, (0.16868306393667788761 + 2.6557914649950505414j), tol=ATOL) + assert ae(v.real, 0.16868306393667788761, tol=PTOL) + assert ae(v.imag, 2.6557914649950505414, tol=PTOL) + v = fp.ei((-1.0 + 4.0j)) + assert ae(v, (-0.03373591813926547318 + 3.2151161058308770603j), tol=ATOL) + assert ae(v.real, -0.03373591813926547318, tol=PTOL) + assert ae(v.imag, 3.2151161058308770603, tol=PTOL) + v = fp.ei((-2.0 + 8.0j)) + assert ae(v, (0.015392833434733785143 + 3.1384179414340326969j), tol=ATOL) + assert ae(v.real, 0.015392833434733785143, tol=PTOL) + assert ae(v.imag, 3.1384179414340326969, tol=PTOL) + v = fp.ei((-5.0 + 20.0j)) + assert ae(v, (0.00024419662286542966525 + 3.1413825703601317109j), tol=ATOL) + assert ae(v.real, 0.00024419662286542966525, tol=PTOL) + assert ae(v.imag, 3.1413825703601317109, tol=PTOL) + v = fp.ei((-20.0 + 80.0j)) + assert ae(v, (-2.3255552781051330088e-11 + 3.1415926535987396304j), tol=ATOL) + assert ae(v.real, -2.3255552781051330088e-11, tol=PTOL) + assert ae(v.imag, 3.1415926535987396304, tol=PTOL) + v = fp.ei((-30.0 + 120.0j)) + assert ae(v, (2.7068919097124652332e-16 + 3.1415926535897925337j), tol=ATOL) + assert ae(v.real, 2.7068919097124652332e-16, tol=PTOL) + assert ae(v.imag, 3.1415926535897925337, tol=PTOL) + v = fp.ei((-40.0 + 160.0j)) + assert ae(v, (1.1695597827678024687e-20 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 1.1695597827678024687e-20, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-50.0 + 200.0j)) + assert ae(v, (-9.0323746914410162531e-25 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -9.0323746914410162531e-25, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-80.0 + 320.0j)) + assert ae(v, (-3.4819106748728063576e-38 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -3.4819106748728063576e-38, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-4.6566128730773925781e-10 + 1.1641532182693481445e-10j)) + assert ae(v, (-20.880034622014215597 + 2.8966139905793444061j), tol=ATOL) + assert ae(v.real, -20.880034622014215597, tol=PTOL) + assert ae(v.imag, 2.8966139905793444061, tol=PTOL) + v = fp.ei((-1.0 + 0.25j)) + assert ae(v, (-0.19731063945004229095 + 3.0542266078154932748j), tol=ATOL) + assert ae(v.real, -0.19731063945004229095, tol=PTOL) + assert ae(v.imag, 3.0542266078154932748, tol=PTOL) + v = fp.ei((-4.0 + 1.0j)) + assert ae(v, (-0.0013106173980145506944 + 3.1381384055698581758j), tol=ATOL) + assert ae(v.real, -0.0013106173980145506944, tol=PTOL) + assert ae(v.imag, 3.1381384055698581758, tol=PTOL) + v = fp.ei((-8.0 + 2.0j)) + assert ae(v, (0.000022278049065270225945 + 3.1415634616493367169j), tol=ATOL) + assert ae(v.real, 0.000022278049065270225945, tol=PTOL) + assert ae(v.imag, 3.1415634616493367169, tol=PTOL) + v = fp.ei((-20.0 + 5.0j)) + assert ae(v, (-4.7711374515765346894e-11 + 3.1415926536726958909j), tol=ATOL) + assert ae(v.real, -4.7711374515765346894e-11, tol=PTOL) + assert ae(v.imag, 3.1415926536726958909, tol=PTOL) + v = fp.ei((-80.0 + 20.0j)) + assert ae(v, (-3.8353473865788235787e-38 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -3.8353473865788235787e-38, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-120.0 + 30.0j)) + assert ae(v, (-2.3836002337480334716e-55 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -2.3836002337480334716e-55, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-160.0 + 40.0j)) + assert ae(v, (1.6238022898654510661e-72 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 1.6238022898654510661e-72, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-200.0 + 50.0j)) + assert ae(v, (-6.6800061461666228487e-90 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -6.6800061461666228487e-90, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei((-320.0 + 80.0j)) + assert ae(v, (-4.2737871527778786157e-143 + 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -4.2737871527778786157e-143, tol=PTOL) + assert ae(v.imag, 3.1415926535897932385, tol=PTOL) + v = fp.ei(-1.1641532182693481445e-10) + assert ae(v, -22.296641293693077672, tol=ATOL) + assert type(v) is float + v = fp.ei(-0.25) + assert ae(v, -1.0442826344437381945, tol=ATOL) + assert type(v) is float + v = fp.ei(-1.0) + assert ae(v, -0.21938393439552027368, tol=ATOL) + assert type(v) is float + v = fp.ei(-2.0) + assert ae(v, -0.048900510708061119567, tol=ATOL) + assert type(v) is float + v = fp.ei(-5.0) + assert ae(v, -0.0011482955912753257973, tol=ATOL) + assert type(v) is float + v = fp.ei(-20.0) + assert ae(v, -9.8355252906498816904e-11, tol=ATOL) + assert type(v) is float + v = fp.ei(-30.0) + assert ae(v, -3.0215520106888125448e-15, tol=ATOL) + assert type(v) is float + v = fp.ei(-40.0) + assert ae(v, -1.0367732614516569722e-19, tol=ATOL) + assert type(v) is float + v = fp.ei(-50.0) + assert ae(v, -3.7832640295504590187e-24, tol=ATOL) + assert type(v) is float + v = fp.ei(-80.0) + assert ae(v, -2.2285432586884729112e-37, tol=ATOL) + assert type(v) is float + v = fp.ei((-1.1641532182693481445e-10 + 0.0j)) + assert ae(v, (-22.296641293693077672 + 0.0j), tol=ATOL) + assert ae(v.real, -22.296641293693077672, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-0.25 + 0.0j)) + assert ae(v, (-1.0442826344437381945 + 0.0j), tol=ATOL) + assert ae(v.real, -1.0442826344437381945, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-1.0 + 0.0j)) + assert ae(v, (-0.21938393439552027368 + 0.0j), tol=ATOL) + assert ae(v.real, -0.21938393439552027368, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-2.0 + 0.0j)) + assert ae(v, (-0.048900510708061119567 + 0.0j), tol=ATOL) + assert ae(v.real, -0.048900510708061119567, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-5.0 + 0.0j)) + assert ae(v, (-0.0011482955912753257973 + 0.0j), tol=ATOL) + assert ae(v.real, -0.0011482955912753257973, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-20.0 + 0.0j)) + assert ae(v, (-9.8355252906498816904e-11 + 0.0j), tol=ATOL) + assert ae(v.real, -9.8355252906498816904e-11, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-30.0 + 0.0j)) + assert ae(v, (-3.0215520106888125448e-15 + 0.0j), tol=ATOL) + assert ae(v.real, -3.0215520106888125448e-15, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-40.0 + 0.0j)) + assert ae(v, (-1.0367732614516569722e-19 + 0.0j), tol=ATOL) + assert ae(v.real, -1.0367732614516569722e-19, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-50.0 + 0.0j)) + assert ae(v, (-3.7832640295504590187e-24 + 0.0j), tol=ATOL) + assert ae(v.real, -3.7832640295504590187e-24, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-80.0 + 0.0j)) + assert ae(v, (-2.2285432586884729112e-37 + 0.0j), tol=ATOL) + assert ae(v.real, -2.2285432586884729112e-37, tol=PTOL) + assert v.imag == 0 + v = fp.ei((-4.6566128730773925781e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (-20.880034622014215597 - 2.8966139905793444061j), tol=ATOL) + assert ae(v.real, -20.880034622014215597, tol=PTOL) + assert ae(v.imag, -2.8966139905793444061, tol=PTOL) + v = fp.ei((-1.0 - 0.25j)) + assert ae(v, (-0.19731063945004229095 - 3.0542266078154932748j), tol=ATOL) + assert ae(v.real, -0.19731063945004229095, tol=PTOL) + assert ae(v.imag, -3.0542266078154932748, tol=PTOL) + v = fp.ei((-4.0 - 1.0j)) + assert ae(v, (-0.0013106173980145506944 - 3.1381384055698581758j), tol=ATOL) + assert ae(v.real, -0.0013106173980145506944, tol=PTOL) + assert ae(v.imag, -3.1381384055698581758, tol=PTOL) + v = fp.ei((-8.0 - 2.0j)) + assert ae(v, (0.000022278049065270225945 - 3.1415634616493367169j), tol=ATOL) + assert ae(v.real, 0.000022278049065270225945, tol=PTOL) + assert ae(v.imag, -3.1415634616493367169, tol=PTOL) + v = fp.ei((-20.0 - 5.0j)) + assert ae(v, (-4.7711374515765346894e-11 - 3.1415926536726958909j), tol=ATOL) + assert ae(v.real, -4.7711374515765346894e-11, tol=PTOL) + assert ae(v.imag, -3.1415926536726958909, tol=PTOL) + v = fp.ei((-80.0 - 20.0j)) + assert ae(v, (-3.8353473865788235787e-38 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -3.8353473865788235787e-38, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-120.0 - 30.0j)) + assert ae(v, (-2.3836002337480334716e-55 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -2.3836002337480334716e-55, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-160.0 - 40.0j)) + assert ae(v, (1.6238022898654510661e-72 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 1.6238022898654510661e-72, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-200.0 - 50.0j)) + assert ae(v, (-6.6800061461666228487e-90 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -6.6800061461666228487e-90, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-320.0 - 80.0j)) + assert ae(v, (-4.2737871527778786157e-143 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -4.2737871527778786157e-143, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-1.1641532182693481445e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (-21.950067703413105017 - 2.3561944903087602507j), tol=ATOL) + assert ae(v.real, -21.950067703413105017, tol=PTOL) + assert ae(v.imag, -2.3561944903087602507, tol=PTOL) + v = fp.ei((-0.25 - 0.25j)) + assert ae(v, (-0.71092525792923287894 - 2.5766745291767512913j), tol=ATOL) + assert ae(v.real, -0.71092525792923287894, tol=PTOL) + assert ae(v.imag, -2.5766745291767512913, tol=PTOL) + v = fp.ei((-1.0 - 1.0j)) + assert ae(v, (-0.00028162445198141832551 - 2.9622681185504342983j), tol=ATOL) + assert ae(v.real, -0.00028162445198141832551, tol=PTOL) + assert ae(v.imag, -2.9622681185504342983, tol=PTOL) + v = fp.ei((-2.0 - 2.0j)) + assert ae(v, (0.033767089606562004246 - 3.1229932394200426965j), tol=ATOL) + assert ae(v.real, 0.033767089606562004246, tol=PTOL) + assert ae(v.imag, -3.1229932394200426965, tol=PTOL) + v = fp.ei((-5.0 - 5.0j)) + assert ae(v, (-0.0007266506660356393891 - 3.1420636813914284609j), tol=ATOL) + assert ae(v.real, -0.0007266506660356393891, tol=PTOL) + assert ae(v.imag, -3.1420636813914284609, tol=PTOL) + v = fp.ei((-20.0 - 20.0j)) + assert ae(v, (2.3824537449367396579e-11 - 3.1415926535228233653j), tol=ATOL) + assert ae(v.real, 2.3824537449367396579e-11, tol=PTOL) + assert ae(v.imag, -3.1415926535228233653, tol=PTOL) + v = fp.ei((-30.0 - 30.0j)) + assert ae(v, (-1.7316045841744061617e-15 - 3.141592653589794545j), tol=ATOL) + assert ae(v.real, -1.7316045841744061617e-15, tol=PTOL) + assert ae(v.imag, -3.141592653589794545, tol=PTOL) + v = fp.ei((-40.0 - 40.0j)) + assert ae(v, (7.4001043002899232182e-20 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 7.4001043002899232182e-20, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-50.0 - 50.0j)) + assert ae(v, (-2.3566128324644641219e-24 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -2.3566128324644641219e-24, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-80.0 - 80.0j)) + assert ae(v, (-9.8279750572186526673e-38 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -9.8279750572186526673e-38, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-1.1641532182693481445e-10 - 4.6566128730773925781e-10j)) + assert ae(v, (-20.880034621664969632 - 1.8157749903874220607j), tol=ATOL) + assert ae(v.real, -20.880034621664969632, tol=PTOL) + assert ae(v.imag, -1.8157749903874220607, tol=PTOL) + v = fp.ei((-0.25 - 1.0j)) + assert ae(v, (0.16868306393667788761 - 2.6557914649950505414j), tol=ATOL) + assert ae(v.real, 0.16868306393667788761, tol=PTOL) + assert ae(v.imag, -2.6557914649950505414, tol=PTOL) + v = fp.ei((-1.0 - 4.0j)) + assert ae(v, (-0.03373591813926547318 - 3.2151161058308770603j), tol=ATOL) + assert ae(v.real, -0.03373591813926547318, tol=PTOL) + assert ae(v.imag, -3.2151161058308770603, tol=PTOL) + v = fp.ei((-2.0 - 8.0j)) + assert ae(v, (0.015392833434733785143 - 3.1384179414340326969j), tol=ATOL) + assert ae(v.real, 0.015392833434733785143, tol=PTOL) + assert ae(v.imag, -3.1384179414340326969, tol=PTOL) + v = fp.ei((-5.0 - 20.0j)) + assert ae(v, (0.00024419662286542966525 - 3.1413825703601317109j), tol=ATOL) + assert ae(v.real, 0.00024419662286542966525, tol=PTOL) + assert ae(v.imag, -3.1413825703601317109, tol=PTOL) + v = fp.ei((-20.0 - 80.0j)) + assert ae(v, (-2.3255552781051330088e-11 - 3.1415926535987396304j), tol=ATOL) + assert ae(v.real, -2.3255552781051330088e-11, tol=PTOL) + assert ae(v.imag, -3.1415926535987396304, tol=PTOL) + v = fp.ei((-30.0 - 120.0j)) + assert ae(v, (2.7068919097124652332e-16 - 3.1415926535897925337j), tol=ATOL) + assert ae(v.real, 2.7068919097124652332e-16, tol=PTOL) + assert ae(v.imag, -3.1415926535897925337, tol=PTOL) + v = fp.ei((-40.0 - 160.0j)) + assert ae(v, (1.1695597827678024687e-20 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, 1.1695597827678024687e-20, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-50.0 - 200.0j)) + assert ae(v, (-9.0323746914410162531e-25 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -9.0323746914410162531e-25, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((-80.0 - 320.0j)) + assert ae(v, (-3.4819106748728063576e-38 - 3.1415926535897932385j), tol=ATOL) + assert ae(v.real, -3.4819106748728063576e-38, tol=PTOL) + assert ae(v.imag, -3.1415926535897932385, tol=PTOL) + v = fp.ei((0.0 - 1.1641532182693481445e-10j)) + assert ae(v, (-22.29664129357666235 - 1.5707963269113119411j), tol=ATOL) + assert ae(v.real, -22.29664129357666235, tol=PTOL) + assert ae(v.imag, -1.5707963269113119411, tol=PTOL) + v = fp.ei((0.0 - 0.25j)) + assert ae(v, (-0.82466306258094565309 - 1.8199298971146537833j), tol=ATOL) + assert ae(v.real, -0.82466306258094565309, tol=PTOL) + assert ae(v.imag, -1.8199298971146537833, tol=PTOL) + v = fp.ei((0.0 - 1.0j)) + assert ae(v, (0.33740392290096813466 - 2.5168793971620796342j), tol=ATOL) + assert ae(v.real, 0.33740392290096813466, tol=PTOL) + assert ae(v.imag, -2.5168793971620796342, tol=PTOL) + v = fp.ei((0.0 - 2.0j)) + assert ae(v, (0.4229808287748649957 - 3.1762093035975914678j), tol=ATOL) + assert ae(v.real, 0.4229808287748649957, tol=PTOL) + assert ae(v.imag, -3.1762093035975914678, tol=PTOL) + v = fp.ei((0.0 - 5.0j)) + assert ae(v, (-0.19002974965664387862 - 3.1207275717395707565j), tol=ATOL) + assert ae(v.real, -0.19002974965664387862, tol=PTOL) + assert ae(v.imag, -3.1207275717395707565, tol=PTOL) + v = fp.ei((0.0 - 20.0j)) + assert ae(v, (0.04441982084535331654 - 3.1190380278383364594j), tol=ATOL) + assert ae(v.real, 0.04441982084535331654, tol=PTOL) + assert ae(v.imag, -3.1190380278383364594, tol=PTOL) + v = fp.ei((0.0 - 30.0j)) + assert ae(v, (-0.033032417282071143779 - 3.1375528668252477302j), tol=ATOL) + assert ae(v.real, -0.033032417282071143779, tol=PTOL) + assert ae(v.imag, -3.1375528668252477302, tol=PTOL) + v = fp.ei((0.0 - 40.0j)) + assert ae(v, (0.019020007896208766962 - 3.157781446149681126j), tol=ATOL) + assert ae(v.real, 0.019020007896208766962, tol=PTOL) + assert ae(v.imag, -3.157781446149681126, tol=PTOL) + v = fp.ei((0.0 - 50.0j)) + assert ae(v, (-0.0056283863241163054402 - 3.122413399280832514j), tol=ATOL) + assert ae(v.real, -0.0056283863241163054402, tol=PTOL) + assert ae(v.imag, -3.122413399280832514, tol=PTOL) + v = fp.ei((0.0 - 80.0j)) + assert ae(v, (-0.012402501155070958192 - 3.1431272137073839346j), tol=ATOL) + assert ae(v.real, -0.012402501155070958192, tol=PTOL) + assert ae(v.imag, -3.1431272137073839346, tol=PTOL) + v = fp.ei((1.1641532182693481445e-10 - 4.6566128730773925781e-10j)) + assert ae(v, (-20.880034621432138988 - 1.3258176641336937524j), tol=ATOL) + assert ae(v.real, -20.880034621432138988, tol=PTOL) + assert ae(v.imag, -1.3258176641336937524, tol=PTOL) + v = fp.ei((0.25 - 1.0j)) + assert ae(v, (0.59066621214766308594 - 2.3968481059377428687j), tol=ATOL) + assert ae(v.real, 0.59066621214766308594, tol=PTOL) + assert ae(v.imag, -2.3968481059377428687, tol=PTOL) + v = fp.ei((1.0 - 4.0j)) + assert ae(v, (-0.49739047283060471093 - 3.5570287076301818702j), tol=ATOL) + assert ae(v.real, -0.49739047283060471093, tol=PTOL) + assert ae(v.imag, -3.5570287076301818702, tol=PTOL) + v = fp.ei((2.0 - 8.0j)) + assert ae(v, (0.8705211147733730969 - 3.3825859385758486351j), tol=ATOL) + assert ae(v.real, 0.8705211147733730969, tol=PTOL) + assert ae(v.imag, -3.3825859385758486351, tol=PTOL) + v = fp.ei((5.0 - 20.0j)) + assert ae(v, (7.0789514293925893007 - 1.5313749363937141849j), tol=ATOL) + assert ae(v.real, 7.0789514293925893007, tol=PTOL) + assert ae(v.imag, -1.5313749363937141849, tol=PTOL) + v = fp.ei((20.0 - 80.0j)) + assert ae(v, (-5855431.4907298084434 + 720917.79156143806727j), tol=ATOL) + assert ae(v.real, -5855431.4907298084434, tol=PTOL) + assert ae(v.imag, 720917.79156143806727, tol=PTOL) + v = fp.ei((30.0 - 120.0j)) + assert ae(v, (65402491644.703470747 + 56697658396.51586764j), tol=ATOL) + assert ae(v.real, 65402491644.703470747, tol=PTOL) + assert ae(v.imag, 56697658396.51586764, tol=PTOL) + v = fp.ei((40.0 - 160.0j)) + assert ae(v, (-25504929379604.776769 - 1429035198630576.3879j), tol=ATOL) + assert ae(v.real, -25504929379604.776769, tol=PTOL) + assert ae(v.imag, -1429035198630576.3879, tol=PTOL) + v = fp.ei((50.0 - 200.0j)) + assert ae(v, (-18437746526988116954.0 + 17146362239046152342.0j), tol=ATOL) + assert ae(v.real, -18437746526988116954.0, tol=PTOL) + assert ae(v.imag, 17146362239046152342.0, tol=PTOL) + v = fp.ei((80.0 - 320.0j)) + assert ae(v, (-3.3464697299634526706e+31 + 1.6473152633843023919e+32j), tol=ATOL) + assert ae(v.real, -3.3464697299634526706e+31, tol=PTOL) + assert ae(v.imag, 1.6473152633843023919e+32, tol=PTOL) + v = fp.ei((1.1641532182693481445e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (-21.950067703180274374 - 0.78539816351386363145j), tol=ATOL) + assert ae(v.real, -21.950067703180274374, tol=PTOL) + assert ae(v.imag, -0.78539816351386363145, tol=PTOL) + v = fp.ei((0.25 - 0.25j)) + assert ae(v, (-0.21441047326710323254 - 1.0683772981589995996j), tol=ATOL) + assert ae(v.real, -0.21441047326710323254, tol=PTOL) + assert ae(v.imag, -1.0683772981589995996, tol=PTOL) + v = fp.ei((1.0 - 1.0j)) + assert ae(v, (1.7646259855638540684 - 2.3877698515105224193j), tol=ATOL) + assert ae(v.real, 1.7646259855638540684, tol=PTOL) + assert ae(v.imag, -2.3877698515105224193, tol=PTOL) + v = fp.ei((2.0 - 2.0j)) + assert ae(v, (1.8920781621855474089 - 5.3169624378326579621j), tol=ATOL) + assert ae(v.real, 1.8920781621855474089, tol=PTOL) + assert ae(v.imag, -5.3169624378326579621, tol=PTOL) + v = fp.ei((5.0 - 5.0j)) + assert ae(v, (-13.470936071475245856 + 15.322492395731230968j), tol=ATOL) + assert ae(v.real, -13.470936071475245856, tol=PTOL) + assert ae(v.imag, 15.322492395731230968, tol=PTOL) + v = fp.ei((20.0 - 20.0j)) + assert ae(v, (16589317.398788971896 - 5831705.4712368307104j), tol=ATOL) + assert ae(v.real, 16589317.398788971896, tol=PTOL) + assert ae(v.imag, -5831705.4712368307104, tol=PTOL) + v = fp.ei((30.0 - 30.0j)) + assert ae(v, (-154596484273.69322527 + 204179357834.2723043j), tol=ATOL) + assert ae(v.real, -154596484273.69322527, tol=PTOL) + assert ae(v.imag, 204179357834.2723043, tol=PTOL) + v = fp.ei((40.0 - 40.0j)) + assert ae(v, (287512180321448.45408 - 4203502407932318.1156j), tol=ATOL) + assert ae(v.real, 287512180321448.45408, tol=PTOL) + assert ae(v.imag, -4203502407932318.1156, tol=PTOL) + v = fp.ei((50.0 - 50.0j)) + assert ae(v, (36128528616649268826.0 + 64648801861338741960.0j), tol=ATOL) + assert ae(v.real, 36128528616649268826.0, tol=PTOL) + assert ae(v.imag, 64648801861338741960.0, tol=PTOL) + v = fp.ei((80.0 - 80.0j)) + assert ae(v, (-3.8674816337930010217e+32 + 3.0540709639658071041e+32j), tol=ATOL) + assert ae(v.real, -3.8674816337930010217e+32, tol=PTOL) + assert ae(v.imag, 3.0540709639658071041e+32, tol=PTOL) + v = fp.ei((4.6566128730773925781e-10 - 1.1641532182693481445e-10j)) + assert ae(v, (-20.880034621082893023 - 0.24497866324327947603j), tol=ATOL) + assert ae(v.real, -20.880034621082893023, tol=PTOL) + assert ae(v.imag, -0.24497866324327947603, tol=PTOL) + v = fp.ei((1.0 - 0.25j)) + assert ae(v, (1.8942716983721074932 - 0.67268237088273915854j), tol=ATOL) + assert ae(v.real, 1.8942716983721074932, tol=PTOL) + assert ae(v.imag, -0.67268237088273915854, tol=PTOL) + v = fp.ei((4.0 - 1.0j)) + assert ae(v, (14.806699492675420438 - 12.280015176673582616j), tol=ATOL) + assert ae(v.real, 14.806699492675420438, tol=PTOL) + assert ae(v.imag, -12.280015176673582616, tol=PTOL) + v = fp.ei((8.0 - 2.0j)) + assert ae(v, (-54.633252667426386294 - 416.34477429173650012j), tol=ATOL) + assert ae(v.real, -54.633252667426386294, tol=PTOL) + assert ae(v.imag, -416.34477429173650012, tol=PTOL) + v = fp.ei((20.0 - 5.0j)) + assert ae(v, (711836.97165402624643 + 24745247.798103247366j), tol=ATOL) + assert ae(v.real, 711836.97165402624643, tol=PTOL) + assert ae(v.imag, 24745247.798103247366, tol=PTOL) + v = fp.ei((80.0 - 20.0j)) + assert ae(v, (4.2139911108612653091e+32 - 5.3367124741918251637e+32j), tol=ATOL) + assert ae(v.real, 4.2139911108612653091e+32, tol=PTOL) + assert ae(v.imag, -5.3367124741918251637e+32, tol=PTOL) + v = fp.ei((120.0 - 30.0j)) + assert ae(v, (-9.7760616203707508892e+48 + 1.058257682317195792e+50j), tol=ATOL) + assert ae(v.real, -9.7760616203707508892e+48, tol=PTOL) + assert ae(v.imag, 1.058257682317195792e+50, tol=PTOL) + v = fp.ei((160.0 - 40.0j)) + assert ae(v, (-8.7065541466623638861e+66 - 1.6577106725141739889e+67j), tol=ATOL) + assert ae(v.real, -8.7065541466623638861e+66, tol=PTOL) + assert ae(v.imag, -1.6577106725141739889e+67, tol=PTOL) + v = fp.ei((200.0 - 50.0j)) + assert ae(v, (3.070744996327018106e+84 + 1.7243244846769415903e+84j), tol=ATOL) + assert ae(v.real, 3.070744996327018106e+84, tol=PTOL) + assert ae(v.imag, 1.7243244846769415903e+84, tol=PTOL) + v = fp.ei((320.0 - 80.0j)) + assert ae(v, (-9.9960598637998647276e+135 + 2.6855081527595608863e+136j), tol=ATOL) + assert ae(v.real, -9.9960598637998647276e+135, tol=PTOL) + assert ae(v.imag, 2.6855081527595608863e+136, tol=PTOL) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_functions.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_functions.py new file mode 100644 index 0000000000000000000000000000000000000000..3bfe852f008173eb636c147abc83d71dbdd4d23a --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_functions.py @@ -0,0 +1,920 @@ +from mpmath.libmp import * +from mpmath import * +import random +import time +import math +import cmath + +def mpc_ae(a, b, eps=eps): + res = True + res = res and a.real.ae(b.real, eps) + res = res and a.imag.ae(b.imag, eps) + return res + +#---------------------------------------------------------------------------- +# Constants and functions +# + +tpi = "3.1415926535897932384626433832795028841971693993751058209749445923078\ +1640628620899862803482534211706798" +te = "2.71828182845904523536028747135266249775724709369995957496696762772407\ +663035354759457138217852516642743" +tdegree = "0.017453292519943295769236907684886127134428718885417254560971914\ +4017100911460344944368224156963450948221" +teuler = "0.5772156649015328606065120900824024310421593359399235988057672348\ +84867726777664670936947063291746749516" +tln2 = "0.693147180559945309417232121458176568075500134360255254120680009493\ +393621969694715605863326996418687542" +tln10 = "2.30258509299404568401799145468436420760110148862877297603332790096\ +757260967735248023599720508959829834" +tcatalan = "0.91596559417721901505460351493238411077414937428167213426649811\ +9621763019776254769479356512926115106249" +tkhinchin = "2.6854520010653064453097148354817956938203822939944629530511523\ +4555721885953715200280114117493184769800" +tglaisher = "1.2824271291006226368753425688697917277676889273250011920637400\ +2174040630885882646112973649195820237439420646" +tapery = "1.2020569031595942853997381615114499907649862923404988817922715553\ +4183820578631309018645587360933525815" +tphi = "1.618033988749894848204586834365638117720309179805762862135448622705\ +26046281890244970720720418939113748475" +tmertens = "0.26149721284764278375542683860869585905156664826119920619206421\ +3924924510897368209714142631434246651052" +ttwinprime = "0.660161815846869573927812110014555778432623360284733413319448\ +423335405642304495277143760031413839867912" + +def test_constants(): + for prec in [3, 7, 10, 15, 20, 37, 80, 100, 29]: + mp.dps = prec + assert pi == mpf(tpi) + assert e == mpf(te) + assert degree == mpf(tdegree) + assert euler == mpf(teuler) + assert ln2 == mpf(tln2) + assert ln10 == mpf(tln10) + assert catalan == mpf(tcatalan) + assert khinchin == mpf(tkhinchin) + assert glaisher == mpf(tglaisher) + assert phi == mpf(tphi) + if prec < 50: + assert mertens == mpf(tmertens) + assert twinprime == mpf(ttwinprime) + mp.dps = 15 + assert pi >= -1 + assert pi > 2 + assert pi > 3 + assert pi < 4 + +def test_exact_sqrts(): + for i in range(20000): + assert sqrt(mpf(i*i)) == i + random.seed(1) + for prec in [100, 300, 1000, 10000]: + mp.dps = prec + for i in range(20): + A = random.randint(10**(prec//2-2), 10**(prec//2-1)) + assert sqrt(mpf(A*A)) == A + mp.dps = 15 + for i in range(100): + for a in [1, 8, 25, 112307]: + assert sqrt(mpf((a*a, 2*i))) == mpf((a, i)) + assert sqrt(mpf((a*a, -2*i))) == mpf((a, -i)) + +def test_sqrt_rounding(): + for i in [2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15]: + i = from_int(i) + for dps in [7, 15, 83, 106, 2000]: + mp.dps = dps + a = mpf_pow_int(mpf_sqrt(i, mp.prec, round_down), 2, mp.prec, round_down) + b = mpf_pow_int(mpf_sqrt(i, mp.prec, round_up), 2, mp.prec, round_up) + assert mpf_lt(a, i) + assert mpf_gt(b, i) + random.seed(1234) + prec = 100 + for rnd in [round_down, round_nearest, round_ceiling]: + for i in range(100): + a = mpf_rand(prec) + b = mpf_mul(a, a) + assert mpf_sqrt(b, prec, rnd) == a + # Test some extreme cases + mp.dps = 100 + a = mpf(9) + 1e-90 + b = mpf(9) - 1e-90 + mp.dps = 15 + assert sqrt(a, rounding='d') == 3 + assert sqrt(a, rounding='n') == 3 + assert sqrt(a, rounding='u') > 3 + assert sqrt(b, rounding='d') < 3 + assert sqrt(b, rounding='n') == 3 + assert sqrt(b, rounding='u') == 3 + # A worst case, from the MPFR test suite + assert sqrt(mpf('7.0503726185518891')) == mpf('2.655253776675949') + +def test_float_sqrt(): + mp.dps = 15 + # These should round identically + for x in [0, 1e-7, 0.1, 0.5, 1, 2, 3, 4, 5, 0.333, 76.19]: + assert sqrt(mpf(x)) == float(x)**0.5 + assert sqrt(-1) == 1j + assert sqrt(-2).ae(cmath.sqrt(-2)) + assert sqrt(-3).ae(cmath.sqrt(-3)) + assert sqrt(-100).ae(cmath.sqrt(-100)) + assert sqrt(1j).ae(cmath.sqrt(1j)) + assert sqrt(-1j).ae(cmath.sqrt(-1j)) + assert sqrt(math.pi + math.e*1j).ae(cmath.sqrt(math.pi + math.e*1j)) + assert sqrt(math.pi - math.e*1j).ae(cmath.sqrt(math.pi - math.e*1j)) + +def test_hypot(): + assert hypot(0, 0) == 0 + assert hypot(0, 0.33) == mpf(0.33) + assert hypot(0.33, 0) == mpf(0.33) + assert hypot(-0.33, 0) == mpf(0.33) + assert hypot(3, 4) == mpf(5) + +def test_exact_cbrt(): + for i in range(0, 20000, 200): + assert cbrt(mpf(i*i*i)) == i + random.seed(1) + for prec in [100, 300, 1000, 10000]: + mp.dps = prec + A = random.randint(10**(prec//2-2), 10**(prec//2-1)) + assert cbrt(mpf(A*A*A)) == A + mp.dps = 15 + +def test_exp(): + assert exp(0) == 1 + assert exp(10000).ae(mpf('8.8068182256629215873e4342')) + assert exp(-10000).ae(mpf('1.1354838653147360985e-4343')) + a = exp(mpf((1, 8198646019315405, -53, 53))) + assert(a.bc == bitcount(a.man)) + mp.prec = 67 + a = exp(mpf((1, 1781864658064754565, -60, 61))) + assert(a.bc == bitcount(a.man)) + mp.prec = 53 + assert exp(ln2 * 10).ae(1024) + assert exp(2+2j).ae(cmath.exp(2+2j)) + +def test_issue_73(): + mp.dps = 512 + a = exp(-1) + b = exp(1) + mp.dps = 15 + assert (+a).ae(0.36787944117144233) + assert (+b).ae(2.7182818284590451) + +def test_log(): + mp.dps = 15 + assert log(1) == 0 + for x in [0.5, 1.5, 2.0, 3.0, 100, 10**50, 1e-50]: + assert log(x).ae(math.log(x)) + assert log(x, x) == 1 + assert log(1024, 2) == 10 + assert log(10**1234, 10) == 1234 + assert log(2+2j).ae(cmath.log(2+2j)) + # Accuracy near 1 + assert (log(0.6+0.8j).real*10**17).ae(2.2204460492503131) + assert (log(0.6-0.8j).real*10**17).ae(2.2204460492503131) + assert (log(0.8-0.6j).real*10**17).ae(2.2204460492503131) + assert (log(1+1e-8j).real*10**16).ae(0.5) + assert (log(1-1e-8j).real*10**16).ae(0.5) + assert (log(-1+1e-8j).real*10**16).ae(0.5) + assert (log(-1-1e-8j).real*10**16).ae(0.5) + assert (log(1j+1e-8).real*10**16).ae(0.5) + assert (log(1j-1e-8).real*10**16).ae(0.5) + assert (log(-1j+1e-8).real*10**16).ae(0.5) + assert (log(-1j-1e-8).real*10**16).ae(0.5) + assert (log(1+1e-40j).real*10**80).ae(0.5) + assert (log(1j+1e-40).real*10**80).ae(0.5) + # Huge + assert log(ldexp(1.234,10**20)).ae(log(2)*1e20) + assert log(ldexp(1.234,10**200)).ae(log(2)*1e200) + # Some special values + assert log(mpc(0,0)) == mpc(-inf,0) + assert isnan(log(mpc(nan,0)).real) + assert isnan(log(mpc(nan,0)).imag) + assert isnan(log(mpc(0,nan)).real) + assert isnan(log(mpc(0,nan)).imag) + assert isnan(log(mpc(nan,1)).real) + assert isnan(log(mpc(nan,1)).imag) + assert isnan(log(mpc(1,nan)).real) + assert isnan(log(mpc(1,nan)).imag) + +def test_trig_hyperb_basic(): + for x in (list(range(100)) + list(range(-100,0))): + t = x / 4.1 + assert cos(mpf(t)).ae(math.cos(t)) + assert sin(mpf(t)).ae(math.sin(t)) + assert tan(mpf(t)).ae(math.tan(t)) + assert cosh(mpf(t)).ae(math.cosh(t)) + assert sinh(mpf(t)).ae(math.sinh(t)) + assert tanh(mpf(t)).ae(math.tanh(t)) + assert sin(1+1j).ae(cmath.sin(1+1j)) + assert sin(-4-3.6j).ae(cmath.sin(-4-3.6j)) + assert cos(1+1j).ae(cmath.cos(1+1j)) + assert cos(-4-3.6j).ae(cmath.cos(-4-3.6j)) + +def test_degrees(): + assert cos(0*degree) == 1 + assert cos(90*degree).ae(0) + assert cos(180*degree).ae(-1) + assert cos(270*degree).ae(0) + assert cos(360*degree).ae(1) + assert sin(0*degree) == 0 + assert sin(90*degree).ae(1) + assert sin(180*degree).ae(0) + assert sin(270*degree).ae(-1) + assert sin(360*degree).ae(0) + +def random_complexes(N): + random.seed(1) + a = [] + for i in range(N): + x1 = random.uniform(-10, 10) + y1 = random.uniform(-10, 10) + x2 = random.uniform(-10, 10) + y2 = random.uniform(-10, 10) + z1 = complex(x1, y1) + z2 = complex(x2, y2) + a.append((z1, z2)) + return a + +def test_complex_powers(): + for dps in [15, 30, 100]: + # Check accuracy for complex square root + mp.dps = dps + a = mpc(1j)**0.5 + assert a.real == a.imag == mpf(2)**0.5 / 2 + mp.dps = 15 + random.seed(1) + for (z1, z2) in random_complexes(100): + assert (mpc(z1)**mpc(z2)).ae(z1**z2, 1e-12) + assert (e**(-pi*1j)).ae(-1) + mp.dps = 50 + assert (e**(-pi*1j)).ae(-1) + mp.dps = 15 + +def test_complex_sqrt_accuracy(): + def test_mpc_sqrt(lst): + for a, b in lst: + z = mpc(a + j*b) + assert mpc_ae(sqrt(z*z), z) + z = mpc(-a + j*b) + assert mpc_ae(sqrt(z*z), -z) + z = mpc(a - j*b) + assert mpc_ae(sqrt(z*z), z) + z = mpc(-a - j*b) + assert mpc_ae(sqrt(z*z), -z) + random.seed(2) + N = 10 + mp.dps = 30 + dps = mp.dps + test_mpc_sqrt([(random.uniform(0, 10),random.uniform(0, 10)) for i in range(N)]) + test_mpc_sqrt([(i + 0.1, (i + 0.2)*10**i) for i in range(N)]) + mp.dps = 15 + +def test_atan(): + mp.dps = 15 + assert atan(-2.3).ae(math.atan(-2.3)) + assert atan(1e-50) == 1e-50 + assert atan(1e50).ae(pi/2) + assert atan(-1e-50) == -1e-50 + assert atan(-1e50).ae(-pi/2) + assert atan(10**1000).ae(pi/2) + for dps in [25, 70, 100, 300, 1000]: + mp.dps = dps + assert (4*atan(1)).ae(pi) + mp.dps = 15 + pi2 = pi/2 + assert atan(mpc(inf,-1)).ae(pi2) + assert atan(mpc(inf,0)).ae(pi2) + assert atan(mpc(inf,1)).ae(pi2) + assert atan(mpc(1,inf)).ae(pi2) + assert atan(mpc(0,inf)).ae(pi2) + assert atan(mpc(-1,inf)).ae(-pi2) + assert atan(mpc(-inf,1)).ae(-pi2) + assert atan(mpc(-inf,0)).ae(-pi2) + assert atan(mpc(-inf,-1)).ae(-pi2) + assert atan(mpc(-1,-inf)).ae(-pi2) + assert atan(mpc(0,-inf)).ae(-pi2) + assert atan(mpc(1,-inf)).ae(pi2) + +def test_atan2(): + mp.dps = 15 + assert atan2(1,1).ae(pi/4) + assert atan2(1,-1).ae(3*pi/4) + assert atan2(-1,-1).ae(-3*pi/4) + assert atan2(-1,1).ae(-pi/4) + assert atan2(-1,0).ae(-pi/2) + assert atan2(1,0).ae(pi/2) + assert atan2(0,0) == 0 + assert atan2(inf,0).ae(pi/2) + assert atan2(-inf,0).ae(-pi/2) + assert isnan(atan2(inf,inf)) + assert isnan(atan2(-inf,inf)) + assert isnan(atan2(inf,-inf)) + assert isnan(atan2(3,nan)) + assert isnan(atan2(nan,3)) + assert isnan(atan2(0,nan)) + assert isnan(atan2(nan,0)) + assert atan2(0,inf) == 0 + assert atan2(0,-inf).ae(pi) + assert atan2(10,inf) == 0 + assert atan2(-10,inf) == 0 + assert atan2(-10,-inf).ae(-pi) + assert atan2(10,-inf).ae(pi) + assert atan2(inf,10).ae(pi/2) + assert atan2(inf,-10).ae(pi/2) + assert atan2(-inf,10).ae(-pi/2) + assert atan2(-inf,-10).ae(-pi/2) + +def test_areal_inverses(): + assert asin(mpf(0)) == 0 + assert asinh(mpf(0)) == 0 + assert acosh(mpf(1)) == 0 + assert isinstance(asin(mpf(0.5)), mpf) + assert isinstance(asin(mpf(2.0)), mpc) + assert isinstance(acos(mpf(0.5)), mpf) + assert isinstance(acos(mpf(2.0)), mpc) + assert isinstance(atanh(mpf(0.1)), mpf) + assert isinstance(atanh(mpf(1.1)), mpc) + + random.seed(1) + for i in range(50): + x = random.uniform(0, 1) + assert asin(mpf(x)).ae(math.asin(x)) + assert acos(mpf(x)).ae(math.acos(x)) + + x = random.uniform(-10, 10) + assert asinh(mpf(x)).ae(cmath.asinh(x).real) + assert isinstance(asinh(mpf(x)), mpf) + x = random.uniform(1, 10) + assert acosh(mpf(x)).ae(cmath.acosh(x).real) + assert isinstance(acosh(mpf(x)), mpf) + x = random.uniform(-10, 0.999) + assert isinstance(acosh(mpf(x)), mpc) + + x = random.uniform(-1, 1) + assert atanh(mpf(x)).ae(cmath.atanh(x).real) + assert isinstance(atanh(mpf(x)), mpf) + + dps = mp.dps + mp.dps = 300 + assert isinstance(asin(0.5), mpf) + mp.dps = 1000 + assert asin(1).ae(pi/2) + assert asin(-1).ae(-pi/2) + mp.dps = dps + +def test_invhyperb_inaccuracy(): + mp.dps = 15 + assert (asinh(1e-5)*10**5).ae(0.99999999998333333) + assert (asinh(1e-10)*10**10).ae(1) + assert (asinh(1e-50)*10**50).ae(1) + assert (asinh(-1e-5)*10**5).ae(-0.99999999998333333) + assert (asinh(-1e-10)*10**10).ae(-1) + assert (asinh(-1e-50)*10**50).ae(-1) + assert asinh(10**20).ae(46.744849040440862) + assert asinh(-10**20).ae(-46.744849040440862) + assert (tanh(1e-10)*10**10).ae(1) + assert (tanh(-1e-10)*10**10).ae(-1) + assert (atanh(1e-10)*10**10).ae(1) + assert (atanh(-1e-10)*10**10).ae(-1) + +def test_complex_functions(): + for x in (list(range(10)) + list(range(-10,0))): + for y in (list(range(10)) + list(range(-10,0))): + z = complex(x, y)/4.3 + 0.01j + assert exp(mpc(z)).ae(cmath.exp(z)) + assert log(mpc(z)).ae(cmath.log(z)) + assert cos(mpc(z)).ae(cmath.cos(z)) + assert sin(mpc(z)).ae(cmath.sin(z)) + assert tan(mpc(z)).ae(cmath.tan(z)) + assert sinh(mpc(z)).ae(cmath.sinh(z)) + assert cosh(mpc(z)).ae(cmath.cosh(z)) + assert tanh(mpc(z)).ae(cmath.tanh(z)) + +def test_complex_inverse_functions(): + mp.dps = 15 + iv.dps = 15 + for (z1, z2) in random_complexes(30): + # apparently cmath uses a different branch, so we + # can't use it for comparison + assert sinh(asinh(z1)).ae(z1) + # + assert acosh(z1).ae(cmath.acosh(z1)) + assert atanh(z1).ae(cmath.atanh(z1)) + assert atan(z1).ae(cmath.atan(z1)) + # the reason we set a big eps here is that the cmath + # functions are inaccurate + assert asin(z1).ae(cmath.asin(z1), rel_eps=1e-12) + assert acos(z1).ae(cmath.acos(z1), rel_eps=1e-12) + one = mpf(1) + for i in range(-9, 10, 3): + for k in range(-9, 10, 3): + a = 0.9*j*10**k + 0.8*one*10**i + b = cos(acos(a)) + assert b.ae(a) + b = sin(asin(a)) + assert b.ae(a) + one = mpf(1) + err = 2*10**-15 + for i in range(-9, 9, 3): + for k in range(-9, 9, 3): + a = -0.9*10**k + j*0.8*one*10**i + b = cosh(acosh(a)) + assert b.ae(a, err) + b = sinh(asinh(a)) + assert b.ae(a, err) + +def test_reciprocal_functions(): + assert sec(3).ae(-1.01010866590799375) + assert csc(3).ae(7.08616739573718592) + assert cot(3).ae(-7.01525255143453347) + assert sech(3).ae(0.0993279274194332078) + assert csch(3).ae(0.0998215696688227329) + assert coth(3).ae(1.00496982331368917) + assert asec(3).ae(1.23095941734077468) + assert acsc(3).ae(0.339836909454121937) + assert acot(3).ae(0.321750554396642193) + assert asech(0.5).ae(1.31695789692481671) + assert acsch(3).ae(0.327450150237258443) + assert acoth(3).ae(0.346573590279972655) + assert acot(0).ae(1.5707963267948966192) + assert acoth(0).ae(1.5707963267948966192j) + +def test_ldexp(): + mp.dps = 15 + assert ldexp(mpf(2.5), 0) == 2.5 + assert ldexp(mpf(2.5), -1) == 1.25 + assert ldexp(mpf(2.5), 2) == 10 + assert ldexp(mpf('inf'), 3) == mpf('inf') + +def test_frexp(): + mp.dps = 15 + assert frexp(0) == (0.0, 0) + assert frexp(9) == (0.5625, 4) + assert frexp(1) == (0.5, 1) + assert frexp(0.2) == (0.8, -2) + assert frexp(1000) == (0.9765625, 10) + +def test_aliases(): + assert ln(7) == log(7) + assert log10(3.75) == log(3.75,10) + assert degrees(5.6) == 5.6 / degree + assert radians(5.6) == 5.6 * degree + assert power(-1,0.5) == j + assert fmod(25,7) == 4.0 and isinstance(fmod(25,7), mpf) + +def test_arg_sign(): + assert arg(3) == 0 + assert arg(-3).ae(pi) + assert arg(j).ae(pi/2) + assert arg(-j).ae(-pi/2) + assert arg(0) == 0 + assert isnan(atan2(3,nan)) + assert isnan(atan2(nan,3)) + assert isnan(atan2(0,nan)) + assert isnan(atan2(nan,0)) + assert isnan(atan2(nan,nan)) + assert arg(inf) == 0 + assert arg(-inf).ae(pi) + assert isnan(arg(nan)) + #assert arg(inf*j).ae(pi/2) + assert sign(0) == 0 + assert sign(3) == 1 + assert sign(-3) == -1 + assert sign(inf) == 1 + assert sign(-inf) == -1 + assert isnan(sign(nan)) + assert sign(j) == j + assert sign(-3*j) == -j + assert sign(1+j).ae((1+j)/sqrt(2)) + +def test_misc_bugs(): + # test that this doesn't raise an exception + mp.dps = 1000 + log(1302) + mp.dps = 15 + +def test_arange(): + assert arange(10) == [mpf('0.0'), mpf('1.0'), mpf('2.0'), mpf('3.0'), + mpf('4.0'), mpf('5.0'), mpf('6.0'), mpf('7.0'), + mpf('8.0'), mpf('9.0')] + assert arange(-5, 5) == [mpf('-5.0'), mpf('-4.0'), mpf('-3.0'), + mpf('-2.0'), mpf('-1.0'), mpf('0.0'), + mpf('1.0'), mpf('2.0'), mpf('3.0'), mpf('4.0')] + assert arange(0, 1, 0.1) == [mpf('0.0'), mpf('0.10000000000000001'), + mpf('0.20000000000000001'), + mpf('0.30000000000000004'), + mpf('0.40000000000000002'), + mpf('0.5'), mpf('0.60000000000000009'), + mpf('0.70000000000000007'), + mpf('0.80000000000000004'), + mpf('0.90000000000000002')] + assert arange(17, -9, -3) == [mpf('17.0'), mpf('14.0'), mpf('11.0'), + mpf('8.0'), mpf('5.0'), mpf('2.0'), + mpf('-1.0'), mpf('-4.0'), mpf('-7.0')] + assert arange(0.2, 0.1, -0.1) == [mpf('0.20000000000000001')] + assert arange(0) == [] + assert arange(1000, -1) == [] + assert arange(-1.23, 3.21, -0.0000001) == [] + +def test_linspace(): + assert linspace(2, 9, 7) == [mpf('2.0'), mpf('3.166666666666667'), + mpf('4.3333333333333339'), mpf('5.5'), mpf('6.666666666666667'), + mpf('7.8333333333333339'), mpf('9.0')] + assert linspace(2, 9, 7, endpoint=0) == [mpf('2.0'), mpf('3.0'), mpf('4.0'), + mpf('5.0'), mpf('6.0'), mpf('7.0'), mpf('8.0')] + assert linspace(2, 7, 1) == [mpf(2)] + +def test_float_cbrt(): + mp.dps = 30 + for a in arange(0,10,0.1): + assert cbrt(a*a*a).ae(a, eps) + assert cbrt(-1).ae(0.5 + j*sqrt(3)/2) + one_third = mpf(1)/3 + for a in arange(0,10,2.7) + [0.1 + 10**5]: + a = mpc(a + 1.1j) + r1 = cbrt(a) + mp.dps += 10 + r2 = pow(a, one_third) + mp.dps -= 10 + assert r1.ae(r2, eps) + mp.dps = 100 + for n in range(100, 301, 100): + w = 10**n + j*10**-3 + z = w*w*w + r = cbrt(z) + assert mpc_ae(r, w, eps) + mp.dps = 15 + +def test_root(): + mp.dps = 30 + random.seed(1) + a = random.randint(0, 10000) + p = a*a*a + r = nthroot(mpf(p), 3) + assert r == a + for n in range(4, 10): + p = p*a + assert nthroot(mpf(p), n) == a + mp.dps = 40 + for n in range(10, 5000, 100): + for a in [random.random()*10000, random.random()*10**100]: + r = nthroot(a, n) + r1 = pow(a, mpf(1)/n) + assert r.ae(r1) + r = nthroot(a, -n) + r1 = pow(a, -mpf(1)/n) + assert r.ae(r1) + # XXX: this is broken right now + # tests for nthroot rounding + for rnd in ['nearest', 'up', 'down']: + mp.rounding = rnd + for n in [-5, -3, 3, 5]: + prec = 50 + for i in range(10): + mp.prec = prec + a = rand() + mp.prec = 2*prec + b = a**n + mp.prec = prec + r = nthroot(b, n) + assert r == a + mp.dps = 30 + for n in range(3, 21): + a = (random.random() + j*random.random()) + assert nthroot(a, n).ae(pow(a, mpf(1)/n)) + assert mpc_ae(nthroot(a, n), pow(a, mpf(1)/n)) + a = (random.random()*10**100 + j*random.random()) + r = nthroot(a, n) + mp.dps += 4 + r1 = pow(a, mpf(1)/n) + mp.dps -= 4 + assert r.ae(r1) + assert mpc_ae(r, r1, eps) + r = nthroot(a, -n) + mp.dps += 4 + r1 = pow(a, -mpf(1)/n) + mp.dps -= 4 + assert r.ae(r1) + assert mpc_ae(r, r1, eps) + mp.dps = 15 + assert nthroot(4, 1) == 4 + assert nthroot(4, 0) == 1 + assert nthroot(4, -1) == 0.25 + assert nthroot(inf, 1) == inf + assert nthroot(inf, 2) == inf + assert nthroot(inf, 3) == inf + assert nthroot(inf, -1) == 0 + assert nthroot(inf, -2) == 0 + assert nthroot(inf, -3) == 0 + assert nthroot(j, 1) == j + assert nthroot(j, 0) == 1 + assert nthroot(j, -1) == -j + assert isnan(nthroot(nan, 1)) + assert isnan(nthroot(nan, 0)) + assert isnan(nthroot(nan, -1)) + assert isnan(nthroot(inf, 0)) + assert root(2,3) == nthroot(2,3) + assert root(16,4,0) == 2 + assert root(16,4,1) == 2j + assert root(16,4,2) == -2 + assert root(16,4,3) == -2j + assert root(16,4,4) == 2 + assert root(-125,3,1) == -5 + +def test_issue_136(): + for dps in [20, 80]: + mp.dps = dps + r = nthroot(mpf('-1e-20'), 4) + assert r.ae(mpf(10)**(-5) * (1 + j) * mpf(2)**(-0.5)) + mp.dps = 80 + assert nthroot('-1e-3', 4).ae(mpf(10)**(-3./4) * (1 + j)/sqrt(2)) + assert nthroot('-1e-6', 4).ae((1 + j)/(10 * sqrt(20))) + # Check that this doesn't take eternity to compute + mp.dps = 20 + assert nthroot('-1e100000000', 4).ae((1+j)*mpf('1e25000000')/sqrt(2)) + mp.dps = 15 + +def test_mpcfun_real_imag(): + mp.dps = 15 + x = mpf(0.3) + y = mpf(0.4) + assert exp(mpc(x,0)) == exp(x) + assert exp(mpc(0,y)) == mpc(cos(y),sin(y)) + assert cos(mpc(x,0)) == cos(x) + assert sin(mpc(x,0)) == sin(x) + assert cos(mpc(0,y)) == cosh(y) + assert sin(mpc(0,y)) == mpc(0,sinh(y)) + assert cospi(mpc(x,0)) == cospi(x) + assert sinpi(mpc(x,0)) == sinpi(x) + assert cospi(mpc(0,y)).ae(cosh(pi*y)) + assert sinpi(mpc(0,y)).ae(mpc(0,sinh(pi*y))) + c, s = cospi_sinpi(mpc(x,0)) + assert c == cospi(x) + assert s == sinpi(x) + c, s = cospi_sinpi(mpc(0,y)) + assert c.ae(cosh(pi*y)) + assert s.ae(mpc(0,sinh(pi*y))) + c, s = cos_sin(mpc(x,0)) + assert c == cos(x) + assert s == sin(x) + c, s = cos_sin(mpc(0,y)) + assert c == cosh(y) + assert s == mpc(0,sinh(y)) + +def test_perturbation_rounding(): + mp.dps = 100 + a = pi/10**50 + b = -pi/10**50 + c = 1 + a + d = 1 + b + mp.dps = 15 + assert exp(a) == 1 + assert exp(a, rounding='c') > 1 + assert exp(b, rounding='c') == 1 + assert exp(a, rounding='f') == 1 + assert exp(b, rounding='f') < 1 + assert cos(a) == 1 + assert cos(a, rounding='c') == 1 + assert cos(b, rounding='c') == 1 + assert cos(a, rounding='f') < 1 + assert cos(b, rounding='f') < 1 + for f in [sin, atan, asinh, tanh]: + assert f(a) == +a + assert f(a, rounding='c') > a + assert f(a, rounding='f') < a + assert f(b) == +b + assert f(b, rounding='c') > b + assert f(b, rounding='f') < b + for f in [asin, tan, sinh, atanh]: + assert f(a) == +a + assert f(b) == +b + assert f(a, rounding='c') > a + assert f(b, rounding='c') > b + assert f(a, rounding='f') < a + assert f(b, rounding='f') < b + assert ln(c) == +a + assert ln(d) == +b + assert ln(c, rounding='c') > a + assert ln(c, rounding='f') < a + assert ln(d, rounding='c') > b + assert ln(d, rounding='f') < b + assert cosh(a) == 1 + assert cosh(b) == 1 + assert cosh(a, rounding='c') > 1 + assert cosh(b, rounding='c') > 1 + assert cosh(a, rounding='f') == 1 + assert cosh(b, rounding='f') == 1 + +def test_integer_parts(): + assert floor(3.2) == 3 + assert ceil(3.2) == 4 + assert floor(3.2+5j) == 3+5j + assert ceil(3.2+5j) == 4+5j + +def test_complex_parts(): + assert fabs('3') == 3 + assert fabs(3+4j) == 5 + assert re(3) == 3 + assert re(1+4j) == 1 + assert im(3) == 0 + assert im(1+4j) == 4 + assert conj(3) == 3 + assert conj(3+4j) == 3-4j + assert mpf(3).conjugate() == 3 + +def test_cospi_sinpi(): + assert sinpi(0) == 0 + assert sinpi(0.5) == 1 + assert sinpi(1) == 0 + assert sinpi(1.5) == -1 + assert sinpi(2) == 0 + assert sinpi(2.5) == 1 + assert sinpi(-0.5) == -1 + assert cospi(0) == 1 + assert cospi(0.5) == 0 + assert cospi(1) == -1 + assert cospi(1.5) == 0 + assert cospi(2) == 1 + assert cospi(2.5) == 0 + assert cospi(-0.5) == 0 + assert cospi(100000000000.25).ae(sqrt(2)/2) + a = cospi(2+3j) + assert a.real.ae(cos((2+3j)*pi).real) + assert a.imag == 0 + b = sinpi(2+3j) + assert b.imag.ae(sin((2+3j)*pi).imag) + assert b.real == 0 + mp.dps = 35 + x1 = mpf(10000) - mpf('1e-15') + x2 = mpf(10000) + mpf('1e-15') + x3 = mpf(10000.5) - mpf('1e-15') + x4 = mpf(10000.5) + mpf('1e-15') + x5 = mpf(10001) - mpf('1e-15') + x6 = mpf(10001) + mpf('1e-15') + x7 = mpf(10001.5) - mpf('1e-15') + x8 = mpf(10001.5) + mpf('1e-15') + mp.dps = 15 + M = 10**15 + assert (sinpi(x1)*M).ae(-pi) + assert (sinpi(x2)*M).ae(pi) + assert (cospi(x3)*M).ae(pi) + assert (cospi(x4)*M).ae(-pi) + assert (sinpi(x5)*M).ae(pi) + assert (sinpi(x6)*M).ae(-pi) + assert (cospi(x7)*M).ae(-pi) + assert (cospi(x8)*M).ae(pi) + assert 0.999 < cospi(x1, rounding='d') < 1 + assert 0.999 < cospi(x2, rounding='d') < 1 + assert 0.999 < sinpi(x3, rounding='d') < 1 + assert 0.999 < sinpi(x4, rounding='d') < 1 + assert -1 < cospi(x5, rounding='d') < -0.999 + assert -1 < cospi(x6, rounding='d') < -0.999 + assert -1 < sinpi(x7, rounding='d') < -0.999 + assert -1 < sinpi(x8, rounding='d') < -0.999 + assert (sinpi(1e-15)*M).ae(pi) + assert (sinpi(-1e-15)*M).ae(-pi) + assert cospi(1e-15) == 1 + assert cospi(1e-15, rounding='d') < 1 + +def test_expj(): + assert expj(0) == 1 + assert expj(1).ae(exp(j)) + assert expj(j).ae(exp(-1)) + assert expj(1+j).ae(exp(j*(1+j))) + assert expjpi(0) == 1 + assert expjpi(1).ae(exp(j*pi)) + assert expjpi(j).ae(exp(-pi)) + assert expjpi(1+j).ae(exp(j*pi*(1+j))) + assert expjpi(-10**15 * j).ae('2.22579818340535731e+1364376353841841') + +def test_sinc(): + assert sinc(0) == sincpi(0) == 1 + assert sinc(inf) == sincpi(inf) == 0 + assert sinc(-inf) == sincpi(-inf) == 0 + assert sinc(2).ae(0.45464871341284084770) + assert sinc(2+3j).ae(0.4463290318402435457-2.7539470277436474940j) + assert sincpi(2) == 0 + assert sincpi(1.5).ae(-0.212206590789193781) + +def test_fibonacci(): + mp.dps = 15 + assert [fibonacci(n) for n in range(-5, 10)] == \ + [5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34] + assert fib(2.5).ae(1.4893065462657091) + assert fib(3+4j).ae(-5248.51130728372 - 14195.962288353j) + assert fib(1000).ae(4.3466557686937455e+208) + assert str(fib(10**100)) == '6.24499112864607e+2089876402499787337692720892375554168224592399182109535392875613974104853496745963277658556235103534' + mp.dps = 2100 + a = fib(10000) + assert a % 10**10 == 9947366875 + mp.dps = 15 + assert fibonacci(inf) == inf + assert fib(3+0j) == 2 + +def test_call_with_dps(): + mp.dps = 15 + assert abs(exp(1, dps=30)-e(dps=35)) < 1e-29 + +def test_tanh(): + mp.dps = 15 + assert tanh(0) == 0 + assert tanh(inf) == 1 + assert tanh(-inf) == -1 + assert isnan(tanh(nan)) + assert tanh(mpc('inf', '0')) == 1 + +def test_atanh(): + mp.dps = 15 + assert atanh(0) == 0 + assert atanh(0.5).ae(0.54930614433405484570) + assert atanh(-0.5).ae(-0.54930614433405484570) + assert atanh(1) == inf + assert atanh(-1) == -inf + assert isnan(atanh(nan)) + assert isinstance(atanh(1), mpf) + assert isinstance(atanh(-1), mpf) + # Limits at infinity + jpi2 = j*pi/2 + assert atanh(inf).ae(-jpi2) + assert atanh(-inf).ae(jpi2) + assert atanh(mpc(inf,-1)).ae(-jpi2) + assert atanh(mpc(inf,0)).ae(-jpi2) + assert atanh(mpc(inf,1)).ae(jpi2) + assert atanh(mpc(1,inf)).ae(jpi2) + assert atanh(mpc(0,inf)).ae(jpi2) + assert atanh(mpc(-1,inf)).ae(jpi2) + assert atanh(mpc(-inf,1)).ae(jpi2) + assert atanh(mpc(-inf,0)).ae(jpi2) + assert atanh(mpc(-inf,-1)).ae(-jpi2) + assert atanh(mpc(-1,-inf)).ae(-jpi2) + assert atanh(mpc(0,-inf)).ae(-jpi2) + assert atanh(mpc(1,-inf)).ae(-jpi2) + +def test_expm1(): + mp.dps = 15 + assert expm1(0) == 0 + assert expm1(3).ae(exp(3)-1) + assert expm1(inf) == inf + assert expm1(1e-50).ae(1e-50) + assert (expm1(1e-10)*1e10).ae(1.00000000005) + +def test_log1p(): + mp.dps = 15 + assert log1p(0) == 0 + assert log1p(3).ae(log(1+3)) + assert log1p(inf) == inf + assert log1p(1e-50).ae(1e-50) + assert (log1p(1e-10)*1e10).ae(0.99999999995) + +def test_powm1(): + mp.dps = 15 + assert powm1(2,3) == 7 + assert powm1(-1,2) == 0 + assert powm1(-1,0) == 0 + assert powm1(-2,0) == 0 + assert powm1(3+4j,0) == 0 + assert powm1(0,1) == -1 + assert powm1(0,0) == 0 + assert powm1(1,0) == 0 + assert powm1(1,2) == 0 + assert powm1(1,3+4j) == 0 + assert powm1(1,5) == 0 + assert powm1(j,4) == 0 + assert powm1(-j,4) == 0 + assert (powm1(2,1e-100)*1e100).ae(ln2) + assert powm1(2,'1e-100000000000') != 0 + assert (powm1(fadd(1,1e-100,exact=True), 5)*1e100).ae(5) + +def test_unitroots(): + assert unitroots(1) == [1] + assert unitroots(2) == [1, -1] + a, b, c = unitroots(3) + assert a == 1 + assert b.ae(-0.5 + 0.86602540378443864676j) + assert c.ae(-0.5 - 0.86602540378443864676j) + assert unitroots(1, primitive=True) == [1] + assert unitroots(2, primitive=True) == [-1] + assert unitroots(3, primitive=True) == unitroots(3)[1:] + assert unitroots(4, primitive=True) == [j, -j] + assert len(unitroots(17, primitive=True)) == 16 + assert len(unitroots(16, primitive=True)) == 8 + +def test_cyclotomic(): + mp.dps = 15 + assert [cyclotomic(n,1) for n in range(31)] == [1,0,2,3,2,5,1,7,2,3,1,11,1,13,1,1,2,17,1,19,1,1,1,23,1,5,1,3,1,29,1] + assert [cyclotomic(n,-1) for n in range(31)] == [1,-2,0,1,2,1,3,1,2,1,5,1,1,1,7,1,2,1,3,1,1,1,11,1,1,1,13,1,1,1,1] + assert [cyclotomic(n,j) for n in range(21)] == [1,-1+j,1+j,j,0,1,-j,j,2,-j,1,j,3,1,-j,1,2,1,j,j,5] + assert [cyclotomic(n,-j) for n in range(21)] == [1,-1-j,1-j,-j,0,1,j,-j,2,j,1,-j,3,1,j,1,2,1,-j,-j,5] + assert cyclotomic(1624,j) == 1 + assert cyclotomic(33600,j) == 1 + u = sqrt(j, prec=500) + assert cyclotomic(8, u).ae(0) + assert cyclotomic(30, u).ae(5.8284271247461900976) + assert cyclotomic(2040, u).ae(1) + assert cyclotomic(0,2.5) == 1 + assert cyclotomic(1,2.5) == 2.5-1 + assert cyclotomic(2,2.5) == 2.5+1 + assert cyclotomic(3,2.5) == 2.5**2 + 2.5 + 1 + assert cyclotomic(7,2.5) == 406.234375 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_functions2.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_functions2.py new file mode 100644 index 0000000000000000000000000000000000000000..2b2d57fcec9be0db4d921b013f24fd6a5e0e9930 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_functions2.py @@ -0,0 +1,2384 @@ +import math +import pytest +from mpmath import * + +def test_bessel(): + mp.dps = 15 + assert j0(1).ae(0.765197686557966551) + assert j0(pi).ae(-0.304242177644093864) + assert j0(1000).ae(0.0247866861524201746) + assert j0(-25).ae(0.0962667832759581162) + assert j1(1).ae(0.440050585744933516) + assert j1(pi).ae(0.284615343179752757) + assert j1(1000).ae(0.00472831190708952392) + assert j1(-25).ae(0.125350249580289905) + assert besselj(5,1).ae(0.000249757730211234431) + assert besselj(5+0j,1).ae(0.000249757730211234431) + assert besselj(5,pi).ae(0.0521411843671184747) + assert besselj(5,1000).ae(0.00502540694523318607) + assert besselj(5,-25).ae(0.0660079953984229934) + assert besselj(-3,2).ae(-0.128943249474402051) + assert besselj(-4,2).ae(0.0339957198075684341) + assert besselj(3,3+2j).ae(0.424718794929639595942 + 0.625665327745785804812j) + assert besselj(0.25,4).ae(-0.374760630804249715) + assert besselj(1+2j,3+4j).ae(0.319247428741872131 - 0.669557748880365678j) + assert (besselj(3, 10**10) * 10**5).ae(0.76765081748139204023) + assert bessely(-0.5, 0) == 0 + assert bessely(0.5, 0) == -inf + assert bessely(1.5, 0) == -inf + assert bessely(0,0) == -inf + assert bessely(-0.4, 0) == -inf + assert bessely(-0.6, 0) == inf + assert bessely(-1, 0) == inf + assert bessely(-1.4, 0) == inf + assert bessely(-1.6, 0) == -inf + assert bessely(-1, 0) == inf + assert bessely(-2, 0) == -inf + assert bessely(-3, 0) == inf + assert bessely(0.5, 0) == -inf + assert bessely(1, 0) == -inf + assert bessely(1.5, 0) == -inf + assert bessely(2, 0) == -inf + assert bessely(2.5, 0) == -inf + assert bessely(3, 0) == -inf + assert bessely(0,0.5).ae(-0.44451873350670655715) + assert bessely(1,0.5).ae(-1.4714723926702430692) + assert bessely(-1,0.5).ae(1.4714723926702430692) + assert bessely(3.5,0.5).ae(-138.86400867242488443) + assert bessely(0,3+4j).ae(4.6047596915010138655-8.8110771408232264208j) + assert bessely(0,j).ae(-0.26803248203398854876+1.26606587775200833560j) + assert (bessely(3, 10**10) * 10**5).ae(0.21755917537013204058) + assert besseli(0,0) == 1 + assert besseli(1,0) == 0 + assert besseli(2,0) == 0 + assert besseli(-1,0) == 0 + assert besseli(-2,0) == 0 + assert besseli(0,0.5).ae(1.0634833707413235193) + assert besseli(1,0.5).ae(0.25789430539089631636) + assert besseli(-1,0.5).ae(0.25789430539089631636) + assert besseli(3.5,0.5).ae(0.00068103597085793815863) + assert besseli(0,3+4j).ae(-3.3924877882755196097-1.3239458916287264815j) + assert besseli(0,j).ae(besselj(0,1)) + assert (besseli(3, 10**10) * mpf(10)**(-4342944813)).ae(4.2996028505491271875) + assert besselk(0,0) == inf + assert besselk(1,0) == inf + assert besselk(2,0) == inf + assert besselk(-1,0) == inf + assert besselk(-2,0) == inf + assert besselk(0,0.5).ae(0.92441907122766586178) + assert besselk(1,0.5).ae(1.6564411200033008937) + assert besselk(-1,0.5).ae(1.6564411200033008937) + assert besselk(3.5,0.5).ae(207.48418747548460607) + assert besselk(0,3+4j).ae(-0.007239051213570155013+0.026510418350267677215j) + assert besselk(0,j).ae(-0.13863371520405399968-1.20196971531720649914j) + assert (besselk(3, 10**10) * mpf(10)**4342944824).ae(1.1628981033356187851) + # test for issue 331, bug reported by Michael Hartmann + for n in range(10,100,10): + mp.dps = n + assert besseli(91.5,24.7708).ae("4.00830632138673963619656140653537080438462342928377020695738635559218797348548092636896796324190271316137982810144874264e-41") + +def test_bessel_zeros(): + mp.dps = 15 + assert besseljzero(0,1).ae(2.40482555769577276869) + assert besseljzero(2,1).ae(5.1356223018406825563) + assert besseljzero(1,50).ae(157.86265540193029781) + assert besseljzero(10,1).ae(14.475500686554541220) + assert besseljzero(0.5,3).ae(9.4247779607693797153) + assert besseljzero(2,1,1).ae(3.0542369282271403228) + assert besselyzero(0,1).ae(0.89357696627916752158) + assert besselyzero(2,1).ae(3.3842417671495934727) + assert besselyzero(1,50).ae(156.29183520147840108) + assert besselyzero(10,1).ae(12.128927704415439387) + assert besselyzero(0.5,3).ae(7.8539816339744830962) + assert besselyzero(2,1,1).ae(5.0025829314460639452) + +def test_hankel(): + mp.dps = 15 + assert hankel1(0,0.5).ae(0.93846980724081290423-0.44451873350670655715j) + assert hankel1(1,0.5).ae(0.2422684576748738864-1.4714723926702430692j) + assert hankel1(-1,0.5).ae(-0.2422684576748738864+1.4714723926702430692j) + assert hankel1(1.5,0.5).ae(0.0917016996256513026-2.5214655504213378514j) + assert hankel1(1.5,3+4j).ae(0.0066806866476728165382-0.0036684231610839127106j) + assert hankel2(0,0.5).ae(0.93846980724081290423+0.44451873350670655715j) + assert hankel2(1,0.5).ae(0.2422684576748738864+1.4714723926702430692j) + assert hankel2(-1,0.5).ae(-0.2422684576748738864-1.4714723926702430692j) + assert hankel2(1.5,0.5).ae(0.0917016996256513026+2.5214655504213378514j) + assert hankel2(1.5,3+4j).ae(14.783528526098567526-7.397390270853446512j) + +def test_struve(): + mp.dps = 15 + assert struveh(2,3).ae(0.74238666967748318564) + assert struveh(-2.5,3).ae(0.41271003220971599344) + assert struvel(2,3).ae(1.7476573277362782744) + assert struvel(-2.5,3).ae(1.5153394466819651377) + +def test_whittaker(): + mp.dps = 15 + assert whitm(2,3,4).ae(49.753745589025246591) + assert whitw(2,3,4).ae(14.111656223052932215) + +def test_kelvin(): + mp.dps = 15 + assert ber(2,3).ae(0.80836846563726819091) + assert ber(3,4).ae(-0.28262680167242600233) + assert ber(-3,2).ae(-0.085611448496796363669) + assert bei(2,3).ae(-0.89102236377977331571) + assert bei(-3,2).ae(-0.14420994155731828415) + assert ker(2,3).ae(0.12839126695733458928) + assert ker(-3,2).ae(-0.29802153400559142783) + assert ker(0.5,3).ae(-0.085662378535217097524) + assert kei(2,3).ae(0.036804426134164634000) + assert kei(-3,2).ae(0.88682069845786731114) + assert kei(0.5,3).ae(0.013633041571314302948) + +def test_hyper_misc(): + mp.dps = 15 + assert hyp0f1(1,0) == 1 + assert hyp1f1(1,2,0) == 1 + assert hyp1f2(1,2,3,0) == 1 + assert hyp2f1(1,2,3,0) == 1 + assert hyp2f2(1,2,3,4,0) == 1 + assert hyp2f3(1,2,3,4,5,0) == 1 + # Degenerate case: 0F0 + assert hyper([],[],0) == 1 + assert hyper([],[],-2).ae(exp(-2)) + # Degenerate case: 1F0 + assert hyper([2],[],1.5) == 4 + # + assert hyp2f1((1,3),(2,3),(5,6),mpf(27)/32).ae(1.6) + assert hyp2f1((1,4),(1,2),(3,4),mpf(80)/81).ae(1.8) + assert hyp2f1((2,3),(1,1),(3,2),(2+j)/3).ae(1.327531603558679093+0.439585080092769253j) + mp.dps = 25 + v = mpc('1.2282306665029814734863026', '-0.1225033830118305184672133') + assert hyper([(3,4),2+j,1],[1,5,j/3],mpf(1)/5+j/8).ae(v) + mp.dps = 15 + +def test_elliptic_integrals(): + mp.dps = 15 + assert ellipk(0).ae(pi/2) + assert ellipk(0.5).ae(gamma(0.25)**2/(4*sqrt(pi))) + assert ellipk(1) == inf + assert ellipk(1+0j) == inf + assert ellipk(-1).ae('1.3110287771460599052') + assert ellipk(-2).ae('1.1714200841467698589') + assert isinstance(ellipk(-2), mpf) + assert isinstance(ellipe(-2), mpf) + assert ellipk(-50).ae('0.47103424540873331679') + mp.dps = 30 + n1 = +fraction(99999,100000) + n2 = +fraction(100001,100000) + mp.dps = 15 + assert ellipk(n1).ae('7.1427724505817781901') + assert ellipk(n2).ae(mpc('7.1427417367963090109', '-1.5707923998261688019')) + assert ellipe(n1).ae('1.0000332138990829170') + v = ellipe(n2) + assert v.real.ae('0.999966786328145474069137') + assert (v.imag*10**6).ae('7.853952181727432') + assert ellipk(2).ae(mpc('1.3110287771460599052', '-1.3110287771460599052')) + assert ellipk(50).ae(mpc('0.22326753950210985451', '-0.47434723226254522087')) + assert ellipk(3+4j).ae(mpc('0.91119556380496500866', '0.63133428324134524388')) + assert ellipk(3-4j).ae(mpc('0.91119556380496500866', '-0.63133428324134524388')) + assert ellipk(-3+4j).ae(mpc('0.95357894880405122483', '0.23093044503746114444')) + assert ellipk(-3-4j).ae(mpc('0.95357894880405122483', '-0.23093044503746114444')) + assert isnan(ellipk(nan)) + assert isnan(ellipe(nan)) + assert ellipk(inf) == 0 + assert isinstance(ellipk(inf), mpc) + assert ellipk(-inf) == 0 + assert ellipk(1+0j) == inf + assert ellipe(0).ae(pi/2) + assert ellipe(0.5).ae(pi**(mpf(3)/2)/gamma(0.25)**2 +gamma(0.25)**2/(8*sqrt(pi))) + assert ellipe(1) == 1 + assert ellipe(1+0j) == 1 + assert ellipe(inf) == mpc(0,inf) + assert ellipe(-inf) == inf + assert ellipe(3+4j).ae(1.4995535209333469543-1.5778790079127582745j) + assert ellipe(3-4j).ae(1.4995535209333469543+1.5778790079127582745j) + assert ellipe(-3+4j).ae(2.5804237855343377803-0.8306096791000413778j) + assert ellipe(-3-4j).ae(2.5804237855343377803+0.8306096791000413778j) + assert ellipe(2).ae(0.59907011736779610372+0.59907011736779610372j) + assert ellipe('1e-1000000000').ae(pi/2) + assert ellipk('1e-1000000000').ae(pi/2) + assert ellipe(-pi).ae(2.4535865983838923) + mp.dps = 50 + assert ellipk(1/pi).ae('1.724756270009501831744438120951614673874904182624739673') + assert ellipe(1/pi).ae('1.437129808135123030101542922290970050337425479058225712') + assert ellipk(-10*pi).ae('0.5519067523886233967683646782286965823151896970015484512') + assert ellipe(-10*pi).ae('5.926192483740483797854383268707108012328213431657645509') + v = ellipk(pi) + assert v.real.ae('0.973089521698042334840454592642137667227167622330325225') + assert v.imag.ae('-1.156151296372835303836814390793087600271609993858798016') + v = ellipe(pi) + assert v.real.ae('0.4632848917264710404078033487934663562998345622611263332') + assert v.imag.ae('1.0637961621753130852473300451583414489944099504180510966') + mp.dps = 15 + +def test_exp_integrals(): + mp.dps = 15 + x = +e + z = e + sqrt(3)*j + assert ei(x).ae(8.21168165538361560) + assert li(x).ae(1.89511781635593676) + assert si(x).ae(1.82104026914756705) + assert ci(x).ae(0.213958001340379779) + assert shi(x).ae(4.11520706247846193) + assert chi(x).ae(4.09647459290515367) + assert fresnels(x).ae(0.437189718149787643) + assert fresnelc(x).ae(0.401777759590243012) + assert airyai(x).ae(0.0108502401568586681) + assert airybi(x).ae(8.98245748585468627) + assert ei(z).ae(3.72597969491314951 + 7.34213212314224421j) + assert li(z).ae(2.28662658112562502 + 1.50427225297269364j) + assert si(z).ae(2.48122029237669054 + 0.12684703275254834j) + assert ci(z).ae(0.169255590269456633 - 0.892020751420780353j) + assert shi(z).ae(1.85810366559344468 + 3.66435842914920263j) + assert chi(z).ae(1.86787602931970484 + 3.67777369399304159j) + assert fresnels(z/3).ae(0.034534397197008182 + 0.754859844188218737j) + assert fresnelc(z/3).ae(1.261581645990027372 + 0.417949198775061893j) + assert airyai(z).ae(-0.0162552579839056062 - 0.0018045715700210556j) + assert airybi(z).ae(-4.98856113282883371 + 2.08558537872180623j) + assert li(0) == 0.0 + assert li(1) == -inf + assert li(inf) == inf + assert isinstance(li(0.7), mpf) + assert si(inf).ae(pi/2) + assert si(-inf).ae(-pi/2) + assert ci(inf) == 0 + assert ci(0) == -inf + assert isinstance(ei(-0.7), mpf) + assert airyai(inf) == 0 + assert airybi(inf) == inf + assert airyai(-inf) == 0 + assert airybi(-inf) == 0 + assert fresnels(inf) == 0.5 + assert fresnelc(inf) == 0.5 + assert fresnels(-inf) == -0.5 + assert fresnelc(-inf) == -0.5 + assert shi(0) == 0 + assert shi(inf) == inf + assert shi(-inf) == -inf + assert chi(0) == -inf + assert chi(inf) == inf + +def test_ei(): + mp.dps = 15 + assert ei(0) == -inf + assert ei(inf) == inf + assert ei(-inf) == -0.0 + assert ei(20+70j).ae(6.1041351911152984397e6 - 2.7324109310519928872e6j) + # tests for the asymptotic expansion + # values checked with Mathematica ExpIntegralEi + mp.dps = 50 + r = ei(20000) + s = '3.8781962825045010930273870085501819470698476975019e+8681' + assert str(r) == s + r = ei(-200) + s = '-6.8852261063076355977108174824557929738368086933303e-90' + assert str(r) == s + r =ei(20000 + 10*j) + sre = '-3.255138234032069402493850638874410725961401274106e+8681' + sim = '-2.1081929993474403520785942429469187647767369645423e+8681' + assert str(r.real) == sre and str(r.imag) == sim + mp.dps = 15 + # More asymptotic expansions + assert chi(-10**6+100j).ae('1.3077239389562548386e+434288 + 7.6808956999707408158e+434287j') + assert shi(-10**6+100j).ae('-1.3077239389562548386e+434288 - 7.6808956999707408158e+434287j') + mp.dps = 15 + assert ei(10j).ae(-0.0454564330044553726+3.2291439210137706686j) + assert ei(100j).ae(-0.0051488251426104921+3.1330217936839529126j) + u = ei(fmul(10**20, j, exact=True)) + assert u.real.ae(-6.4525128526578084421345e-21, abs_eps=0, rel_eps=8*eps) + assert u.imag.ae(pi) + assert ei(-10j).ae(-0.0454564330044553726-3.2291439210137706686j) + assert ei(-100j).ae(-0.0051488251426104921-3.1330217936839529126j) + u = ei(fmul(-10**20, j, exact=True)) + assert u.real.ae(-6.4525128526578084421345e-21, abs_eps=0, rel_eps=8*eps) + assert u.imag.ae(-pi) + assert ei(10+10j).ae(-1576.1504265768517448+436.9192317011328140j) + u = ei(-10+10j) + assert u.real.ae(7.6698978415553488362543e-7, abs_eps=0, rel_eps=8*eps) + assert u.imag.ae(3.141595611735621062025) + +def test_e1(): + mp.dps = 15 + assert e1(0) == inf + assert e1(inf) == 0 + assert e1(-inf) == mpc(-inf, -pi) + assert e1(10j).ae(0.045456433004455372635 + 0.087551267423977430100j) + assert e1(100j).ae(0.0051488251426104921444 - 0.0085708599058403258790j) + assert e1(fmul(10**20, j, exact=True)).ae(6.4525128526578084421e-21 - 7.6397040444172830039e-21j, abs_eps=0, rel_eps=8*eps) + assert e1(-10j).ae(0.045456433004455372635 - 0.087551267423977430100j) + assert e1(-100j).ae(0.0051488251426104921444 + 0.0085708599058403258790j) + assert e1(fmul(-10**20, j, exact=True)).ae(6.4525128526578084421e-21 + 7.6397040444172830039e-21j, abs_eps=0, rel_eps=8*eps) + +def test_expint(): + mp.dps = 15 + assert expint(0,0) == inf + assert expint(0,1).ae(1/e) + assert expint(0,1.5).ae(2/exp(1.5)/3) + assert expint(1,1).ae(-ei(-1)) + assert expint(2,0).ae(1) + assert expint(3,0).ae(1/2.) + assert expint(4,0).ae(1/3.) + assert expint(-2, 0.5).ae(26/sqrt(e)) + assert expint(-1,-1) == 0 + assert expint(-2,-1).ae(-e) + assert expint(5.5, 0).ae(2/9.) + assert expint(2.00000001,0).ae(100000000./100000001) + assert expint(2+3j,4-j).ae(0.0023461179581675065414+0.0020395540604713669262j) + assert expint('1.01', '1e-1000').ae(99.9999999899412802) + assert expint('1.000000000001', 3.5).ae(0.00697013985754701819446) + assert expint(2,3).ae(3*ei(-3)+exp(-3)) + assert (expint(10,20)*10**10).ae(0.694439055541231353) + assert expint(3,inf) == 0 + assert expint(3.2,inf) == 0 + assert expint(3.2+2j,inf) == 0 + assert expint(1,3j).ae(-0.11962978600800032763 + 0.27785620120457163717j) + assert expint(1,3).ae(0.013048381094197037413) + assert expint(1,-3).ae(-ei(3)-pi*j) + #assert expint(3) == expint(1,3) + assert expint(1,-20).ae(-25615652.66405658882 - 3.1415926535897932385j) + assert expint(1000000,0).ae(1./999999) + assert expint(0,2+3j).ae(-0.025019798357114678171 + 0.027980439405104419040j) + assert expint(-1,2+3j).ae(-0.022411973626262070419 + 0.038058922011377716932j) + assert expint(-1.5,0) == inf + +def test_trig_integrals(): + mp.dps = 30 + assert si(mpf(1)/1000000).ae('0.000000999999999999944444444444446111') + assert ci(mpf(1)/1000000).ae('-13.2382948930629912435014366276') + assert si(10**10).ae('1.5707963267075846569685111517747537') + assert ci(10**10).ae('-4.87506025174822653785729773959e-11') + assert si(10**100).ae(pi/2) + assert (ci(10**100)*10**100).ae('-0.372376123661276688262086695553') + assert si(-3) == -si(3) + assert ci(-3).ae(ci(3) + pi*j) + # Test complex structure + mp.dps = 15 + assert mp.ci(50).ae(-0.0056283863241163054402) + assert mp.ci(50+2j).ae(-0.018378282946133067149+0.070352808023688336193j) + assert mp.ci(20j).ae(1.28078263320282943611e7+1.5707963267949j) + assert mp.ci(-2+20j).ae(-4.050116856873293505e6+1.207476188206989909e7j) + assert mp.ci(-50+2j).ae(-0.0183782829461330671+3.0712398455661049023j) + assert mp.ci(-50).ae(-0.0056283863241163054+3.1415926535897932385j) + assert mp.ci(-50-2j).ae(-0.0183782829461330671-3.0712398455661049023j) + assert mp.ci(-2-20j).ae(-4.050116856873293505e6-1.207476188206989909e7j) + assert mp.ci(-20j).ae(1.28078263320282943611e7-1.5707963267949j) + assert mp.ci(50-2j).ae(-0.018378282946133067149-0.070352808023688336193j) + assert mp.si(50).ae(1.5516170724859358947) + assert mp.si(50+2j).ae(1.497884414277228461-0.017515007378437448j) + assert mp.si(20j).ae(1.2807826332028294459e7j) + assert mp.si(-2+20j).ae(-1.20747603112735722103e7-4.050116856873293554e6j) + assert mp.si(-50+2j).ae(-1.497884414277228461-0.017515007378437448j) + assert mp.si(-50).ae(-1.5516170724859358947) + assert mp.si(-50-2j).ae(-1.497884414277228461+0.017515007378437448j) + assert mp.si(-2-20j).ae(-1.20747603112735722103e7+4.050116856873293554e6j) + assert mp.si(-20j).ae(-1.2807826332028294459e7j) + assert mp.si(50-2j).ae(1.497884414277228461+0.017515007378437448j) + assert mp.chi(50j).ae(-0.0056283863241163054+1.5707963267948966192j) + assert mp.chi(-2+50j).ae(-0.0183782829461330671+1.6411491348185849554j) + assert mp.chi(-20).ae(1.28078263320282943611e7+3.1415926535898j) + assert mp.chi(-20-2j).ae(-4.050116856873293505e6+1.20747571696809187053e7j) + assert mp.chi(-2-50j).ae(-0.0183782829461330671-1.6411491348185849554j) + assert mp.chi(-50j).ae(-0.0056283863241163054-1.5707963267948966192j) + assert mp.chi(2-50j).ae(-0.0183782829461330671-1.500443518771208283j) + assert mp.chi(20-2j).ae(-4.050116856873293505e6-1.20747603112735722951e7j) + assert mp.chi(20).ae(1.2807826332028294361e7) + assert mp.chi(2+50j).ae(-0.0183782829461330671+1.500443518771208283j) + assert mp.shi(50j).ae(1.5516170724859358947j) + assert mp.shi(-2+50j).ae(0.017515007378437448+1.497884414277228461j) + assert mp.shi(-20).ae(-1.2807826332028294459e7) + assert mp.shi(-20-2j).ae(4.050116856873293554e6-1.20747603112735722103e7j) + assert mp.shi(-2-50j).ae(0.017515007378437448-1.497884414277228461j) + assert mp.shi(-50j).ae(-1.5516170724859358947j) + assert mp.shi(2-50j).ae(-0.017515007378437448-1.497884414277228461j) + assert mp.shi(20-2j).ae(-4.050116856873293554e6-1.20747603112735722103e7j) + assert mp.shi(20).ae(1.2807826332028294459e7) + assert mp.shi(2+50j).ae(-0.017515007378437448+1.497884414277228461j) + def ae(x,y,tol=1e-12): + return abs(x-y) <= abs(y)*tol + assert fp.ci(fp.inf) == 0 + assert ae(fp.ci(fp.ninf), fp.pi*1j) + assert ae(fp.si(fp.inf), fp.pi/2) + assert ae(fp.si(fp.ninf), -fp.pi/2) + assert fp.si(0) == 0 + assert ae(fp.ci(50), -0.0056283863241163054402) + assert ae(fp.ci(50+2j), -0.018378282946133067149+0.070352808023688336193j) + assert ae(fp.ci(20j), 1.28078263320282943611e7+1.5707963267949j) + assert ae(fp.ci(-2+20j), -4.050116856873293505e6+1.207476188206989909e7j) + assert ae(fp.ci(-50+2j), -0.0183782829461330671+3.0712398455661049023j) + assert ae(fp.ci(-50), -0.0056283863241163054+3.1415926535897932385j) + assert ae(fp.ci(-50-2j), -0.0183782829461330671-3.0712398455661049023j) + assert ae(fp.ci(-2-20j), -4.050116856873293505e6-1.207476188206989909e7j) + assert ae(fp.ci(-20j), 1.28078263320282943611e7-1.5707963267949j) + assert ae(fp.ci(50-2j), -0.018378282946133067149-0.070352808023688336193j) + assert ae(fp.si(50), 1.5516170724859358947) + assert ae(fp.si(50+2j), 1.497884414277228461-0.017515007378437448j) + assert ae(fp.si(20j), 1.2807826332028294459e7j) + assert ae(fp.si(-2+20j), -1.20747603112735722103e7-4.050116856873293554e6j) + assert ae(fp.si(-50+2j), -1.497884414277228461-0.017515007378437448j) + assert ae(fp.si(-50), -1.5516170724859358947) + assert ae(fp.si(-50-2j), -1.497884414277228461+0.017515007378437448j) + assert ae(fp.si(-2-20j), -1.20747603112735722103e7+4.050116856873293554e6j) + assert ae(fp.si(-20j), -1.2807826332028294459e7j) + assert ae(fp.si(50-2j), 1.497884414277228461+0.017515007378437448j) + assert ae(fp.chi(50j), -0.0056283863241163054+1.5707963267948966192j) + assert ae(fp.chi(-2+50j), -0.0183782829461330671+1.6411491348185849554j) + assert ae(fp.chi(-20), 1.28078263320282943611e7+3.1415926535898j) + assert ae(fp.chi(-20-2j), -4.050116856873293505e6+1.20747571696809187053e7j) + assert ae(fp.chi(-2-50j), -0.0183782829461330671-1.6411491348185849554j) + assert ae(fp.chi(-50j), -0.0056283863241163054-1.5707963267948966192j) + assert ae(fp.chi(2-50j), -0.0183782829461330671-1.500443518771208283j) + assert ae(fp.chi(20-2j), -4.050116856873293505e6-1.20747603112735722951e7j) + assert ae(fp.chi(20), 1.2807826332028294361e7) + assert ae(fp.chi(2+50j), -0.0183782829461330671+1.500443518771208283j) + assert ae(fp.shi(50j), 1.5516170724859358947j) + assert ae(fp.shi(-2+50j), 0.017515007378437448+1.497884414277228461j) + assert ae(fp.shi(-20), -1.2807826332028294459e7) + assert ae(fp.shi(-20-2j), 4.050116856873293554e6-1.20747603112735722103e7j) + assert ae(fp.shi(-2-50j), 0.017515007378437448-1.497884414277228461j) + assert ae(fp.shi(-50j), -1.5516170724859358947j) + assert ae(fp.shi(2-50j), -0.017515007378437448-1.497884414277228461j) + assert ae(fp.shi(20-2j), -4.050116856873293554e6-1.20747603112735722103e7j) + assert ae(fp.shi(20), 1.2807826332028294459e7) + assert ae(fp.shi(2+50j), -0.017515007378437448+1.497884414277228461j) + +def test_airy(): + mp.dps = 15 + assert (airyai(10)*10**10).ae(1.1047532552898687) + assert (airybi(10)/10**9).ae(0.45564115354822515) + assert (airyai(1000)*10**9158).ae(9.306933063179556004) + assert (airybi(1000)/10**9154).ae(5.4077118391949465477) + assert airyai(-1000).ae(0.055971895773019918842) + assert airybi(-1000).ae(-0.083264574117080633012) + assert (airyai(100+100j)*10**188).ae(2.9099582462207032076 + 2.353013591706178756j) + assert (airybi(100+100j)/10**185).ae(1.7086751714463652039 - 3.1416590020830804578j) + +def test_hyper_0f1(): + mp.dps = 15 + v = 8.63911136507950465 + assert hyper([],[(1,3)],1.5).ae(v) + assert hyper([],[1/3.],1.5).ae(v) + assert hyp0f1(1/3.,1.5).ae(v) + assert hyp0f1((1,3),1.5).ae(v) + # Asymptotic expansion + assert hyp0f1(3,1e9).ae('4.9679055380347771271e+27455') + assert hyp0f1(3,1e9j).ae('-2.1222788784457702157e+19410 + 5.0840597555401854116e+19410j') + +def test_hyper_1f1(): + mp.dps = 15 + v = 1.2917526488617656673 + assert hyper([(1,2)],[(3,2)],0.7).ae(v) + assert hyper([(1,2)],[(3,2)],0.7+0j).ae(v) + assert hyper([0.5],[(3,2)],0.7).ae(v) + assert hyper([0.5],[1.5],0.7).ae(v) + assert hyper([0.5],[(3,2)],0.7+0j).ae(v) + assert hyper([0.5],[1.5],0.7+0j).ae(v) + assert hyper([(1,2)],[1.5+0j],0.7).ae(v) + assert hyper([0.5+0j],[1.5],0.7).ae(v) + assert hyper([0.5+0j],[1.5+0j],0.7+0j).ae(v) + assert hyp1f1(0.5,1.5,0.7).ae(v) + assert hyp1f1((1,2),1.5,0.7).ae(v) + # Asymptotic expansion + assert hyp1f1(2,3,1e10).ae('2.1555012157015796988e+4342944809') + assert (hyp1f1(2,3,1e10j)*10**10).ae(-0.97501205020039745852 - 1.7462392454512132074j) + # Shouldn't use asymptotic expansion + assert hyp1f1(-2, 1, 10000).ae(49980001) + # Bug + assert hyp1f1(1j,fraction(1,3),0.415-69.739j).ae(25.857588206024346592 + 15.738060264515292063j) + +def test_hyper_2f1(): + mp.dps = 15 + v = 1.0652207633823291032 + assert hyper([(1,2), (3,4)], [2], 0.3).ae(v) + assert hyper([(1,2), 0.75], [2], 0.3).ae(v) + assert hyper([0.5, 0.75], [2.0], 0.3).ae(v) + assert hyper([0.5, 0.75], [2.0], 0.3+0j).ae(v) + assert hyper([0.5+0j, (3,4)], [2.0], 0.3+0j).ae(v) + assert hyper([0.5+0j, (3,4)], [2.0], 0.3).ae(v) + assert hyper([0.5, (3,4)], [2.0+0j], 0.3).ae(v) + assert hyper([0.5+0j, 0.75+0j], [2.0+0j], 0.3+0j).ae(v) + v = 1.09234681096223231717 + 0.18104859169479360380j + assert hyper([(1,2),0.75+j], [2], 0.5).ae(v) + assert hyper([0.5,0.75+j], [2.0], 0.5).ae(v) + assert hyper([0.5,0.75+j], [2.0], 0.5+0j).ae(v) + assert hyper([0.5,0.75+j], [2.0+0j], 0.5+0j).ae(v) + v = 0.9625 - 0.125j + assert hyper([(3,2),-1],[4], 0.1+j/3).ae(v) + assert hyper([1.5,-1.0],[4], 0.1+j/3).ae(v) + assert hyper([1.5,-1.0],[4+0j], 0.1+j/3).ae(v) + assert hyper([1.5+0j,-1.0+0j],[4+0j], 0.1+j/3).ae(v) + v = 1.02111069501693445001 - 0.50402252613466859521j + assert hyper([(2,10),(3,10)],[(4,10)],1.5).ae(v) + assert hyper([0.2,(3,10)],[0.4+0j],1.5).ae(v) + assert hyper([0.2,(3,10)],[0.4+0j],1.5+0j).ae(v) + v = 0.76922501362865848528 + 0.32640579593235886194j + assert hyper([(2,10),(3,10)],[(4,10)],4+2j).ae(v) + assert hyper([0.2,(3,10)],[0.4+0j],4+2j).ae(v) + assert hyper([0.2,(3,10)],[(4,10)],4+2j).ae(v) + +def test_hyper_2f1_hard(): + mp.dps = 15 + # Singular cases + assert hyp2f1(2,-1,-1,3).ae(7) + assert hyp2f1(2,-1,-1,3,eliminate_all=True).ae(0.25) + assert hyp2f1(2,-2,-2,3).ae(34) + assert hyp2f1(2,-2,-2,3,eliminate_all=True).ae(0.25) + assert hyp2f1(2,-2,-3,3) == 14 + assert hyp2f1(2,-3,-2,3) == inf + assert hyp2f1(2,-1.5,-1.5,3) == 0.25 + assert hyp2f1(1,2,3,0) == 1 + assert hyp2f1(0,1,0,0) == 1 + assert hyp2f1(0,0,0,0) == 1 + assert isnan(hyp2f1(1,1,0,0)) + assert hyp2f1(2,-1,-5, 0.25+0.25j).ae(1.1+0.1j) + assert hyp2f1(2,-5,-5, 0.25+0.25j, eliminate=False).ae(163./128 + 125./128*j) + assert hyp2f1(0.7235, -1, -5, 0.3).ae(1.04341) + assert hyp2f1(0.7235, -5, -5, 0.3, eliminate=False).ae(1.2939225017815903812) + assert hyp2f1(-1,-2,4,1) == 1.5 + assert hyp2f1(1,2,-3,1) == inf + assert hyp2f1(-2,-2,1,1) == 6 + assert hyp2f1(1,-2,-4,1).ae(5./3) + assert hyp2f1(0,-6,-4,1) == 1 + assert hyp2f1(0,-3,-4,1) == 1 + assert hyp2f1(0,0,0,1) == 1 + assert hyp2f1(1,0,0,1,eliminate=False) == 1 + assert hyp2f1(1,1,0,1) == inf + assert hyp2f1(1,-6,-4,1) == inf + assert hyp2f1(-7.2,-0.5,-4.5,1) == 0 + assert hyp2f1(-7.2,-1,-2,1).ae(-2.6) + assert hyp2f1(1,-0.5,-4.5, 1) == inf + assert hyp2f1(1,0.5,-4.5, 1) == -inf + # Check evaluation on / close to unit circle + z = exp(j*pi/3) + w = (nthroot(2,3)+1)*exp(j*pi/12)/nthroot(3,4)**3 + assert hyp2f1('1/2','1/6','1/3', z).ae(w) + assert hyp2f1('1/2','1/6','1/3', z.conjugate()).ae(w.conjugate()) + assert hyp2f1(0.25, (1,3), 2, '0.999').ae(1.06826449496030635) + assert hyp2f1(0.25, (1,3), 2, '1.001').ae(1.06867299254830309446-0.00001446586793975874j) + assert hyp2f1(0.25, (1,3), 2, -1).ae(0.96656584492524351673) + assert hyp2f1(0.25, (1,3), 2, j).ae(0.99041766248982072266+0.03777135604180735522j) + assert hyp2f1(2,3,5,'0.99').ae(27.699347904322690602) + assert hyp2f1((3,2),-0.5,3,'0.99').ae(0.68403036843911661388) + assert hyp2f1(2,3,5,1j).ae(0.37290667145974386127+0.59210004902748285917j) + assert fsum([hyp2f1((7,10),(2,3),(-1,2), 0.95*exp(j*k)) for k in range(1,15)]).ae(52.851400204289452922+6.244285013912953225j) + assert fsum([hyp2f1((7,10),(2,3),(-1,2), 1.05*exp(j*k)) for k in range(1,15)]).ae(54.506013786220655330-3.000118813413217097j) + assert fsum([hyp2f1((7,10),(2,3),(-1,2), exp(j*k)) for k in range(1,15)]).ae(55.792077935955314887+1.731986485778500241j) + assert hyp2f1(2,2.5,-3.25,0.999).ae(218373932801217082543180041.33) + # Branches + assert hyp2f1(1,1,2,1.01).ae(4.5595744415723676911-3.1104877758314784539j) + assert hyp2f1(1,1,2,1.01+0.1j).ae(2.4149427480552782484+1.4148224796836938829j) + assert hyp2f1(1,1,2,3+4j).ae(0.14576709331407297807+0.48379185417980360773j) + assert hyp2f1(1,1,2,4).ae(-0.27465307216702742285 - 0.78539816339744830962j) + assert hyp2f1(1,1,2,-4).ae(0.40235947810852509365) + # Other: + # Cancellation with a large parameter involved (bug reported on sage-devel) + assert hyp2f1(112, (51,10), (-9,10), -0.99999).ae(-1.6241361047970862961e-24, abs_eps=0, rel_eps=eps*16) + +def test_hyper_3f2_etc(): + assert hyper([1,2,3],[1.5,8],-1).ae(0.67108992351533333030) + assert hyper([1,2,3,4],[5,6,7], -1).ae(0.90232988035425506008) + assert hyper([1,2,3],[1.25,5], 1).ae(28.924181329701905701) + assert hyper([1,2,3,4],[5,6,7],5).ae(1.5192307344006649499-1.1529845225075537461j) + assert hyper([1,2,3,4,5],[6,7,8,9],-1).ae(0.96288759462882357253) + assert hyper([1,2,3,4,5],[6,7,8,9],1).ae(1.0428697385885855841) + assert hyper([1,2,3,4,5],[6,7,8,9],5).ae(1.33980653631074769423-0.07143405251029226699j) + assert hyper([1,2.79,3.08,4.37],[5.2,6.1,7.3],5).ae(1.0996321464692607231-1.7748052293979985001j) + assert hyper([1,1,1],[1,2],1) == inf + assert hyper([1,1,1],[2,(101,100)],1).ae(100.01621213528313220) + # slow -- covered by doctests + #assert hyper([1,1,1],[2,3],0.9999).ae(1.2897972005319693905) + +def test_hyper_u(): + mp.dps = 15 + assert hyperu(2,-3,0).ae(0.05) + assert hyperu(2,-3.5,0).ae(4./99) + assert hyperu(2,0,0) == 0.5 + assert hyperu(-5,1,0) == -120 + assert hyperu(-5,2,0) == inf + assert hyperu(-5,-2,0) == 0 + assert hyperu(7,7,3).ae(0.00014681269365593503986) #exp(3)*gammainc(-6,3) + assert hyperu(2,-3,4).ae(0.011836478100271995559) + assert hyperu(3,4,5).ae(1./125) + assert hyperu(2,3,0.0625) == 256 + assert hyperu(-1,2,0.25+0.5j) == -1.75+0.5j + assert hyperu(0.5,1.5,7.25).ae(2/sqrt(29)) + assert hyperu(2,6,pi).ae(0.55804439825913399130) + assert (hyperu((3,2),8,100+201j)*10**4).ae(-0.3797318333856738798 - 2.9974928453561707782j) + assert (hyperu((5,2),(-1,2),-5000)*10**10).ae(-5.6681877926881664678j) + # XXX: fails because of undetected cancellation in low level series code + # Alternatively: could use asymptotic series here, if convergence test + # tweaked back to recognize this one + #assert (hyperu((5,2),(-1,2),-500)*10**7).ae(-1.82526906001593252847j) + +def test_hyper_2f0(): + mp.dps = 15 + assert hyper([1,2],[],3) == hyp2f0(1,2,3) + assert hyp2f0(2,3,7).ae(0.0116108068639728714668 - 0.0073727413865865802130j) + assert hyp2f0(2,3,0) == 1 + assert hyp2f0(0,0,0) == 1 + assert hyp2f0(-1,-1,1).ae(2) + assert hyp2f0(-4,1,1.5).ae(62.5) + assert hyp2f0(-4,1,50).ae(147029801) + assert hyp2f0(-4,1,0.0001).ae(0.99960011997600240000) + assert hyp2f0(0.5,0.25,0.001).ae(1.0001251174078538115) + assert hyp2f0(0.5,0.25,3+4j).ae(0.85548875824755163518 + 0.21636041283392292973j) + # Important: cancellation check + assert hyp2f0((1,6),(5,6),-0.02371708245126284498).ae(0.996785723120804309) + # Should be exact; polynomial case + assert hyp2f0(-2,1,0.5+0.5j,zeroprec=200) == 0 + assert hyp2f0(1,-2,0.5+0.5j,zeroprec=200) == 0 + # There used to be a bug in thresholds that made one of the following hang + for d in [15, 50, 80]: + mp.dps = d + assert hyp2f0(1.5, 0.5, 0.009).ae('1.006867007239309717945323585695344927904000945829843527398772456281301440034218290443367270629519483 + 1.238277162240704919639384945859073461954721356062919829456053965502443570466701567100438048602352623e-46j') + +def test_hyper_1f2(): + mp.dps = 15 + assert hyper([1],[2,3],4) == hyp1f2(1,2,3,4) + a1,b1,b2 = (1,10),(2,3),1./16 + assert hyp1f2(a1,b1,b2,10).ae(298.7482725554557568) + assert hyp1f2(a1,b1,b2,100).ae(224128961.48602947604) + assert hyp1f2(a1,b1,b2,1000).ae(1.1669528298622675109e+27) + assert hyp1f2(a1,b1,b2,10000).ae(2.4780514622487212192e+86) + assert hyp1f2(a1,b1,b2,100000).ae(1.3885391458871523997e+274) + assert hyp1f2(a1,b1,b2,1000000).ae('9.8851796978960318255e+867') + assert hyp1f2(a1,b1,b2,10**7).ae('1.1505659189516303646e+2746') + assert hyp1f2(a1,b1,b2,10**8).ae('1.4672005404314334081e+8685') + assert hyp1f2(a1,b1,b2,10**20).ae('3.6888217332150976493e+8685889636') + assert hyp1f2(a1,b1,b2,10*j).ae(-16.163252524618572878 - 44.321567896480184312j) + assert hyp1f2(a1,b1,b2,100*j).ae(61938.155294517848171 + 637349.45215942348739j) + assert hyp1f2(a1,b1,b2,1000*j).ae(8455057657257695958.7 + 6261969266997571510.6j) + assert hyp1f2(a1,b1,b2,10000*j).ae(-8.9771211184008593089e+60 + 4.6550528111731631456e+59j) + assert hyp1f2(a1,b1,b2,100000*j).ae(2.6398091437239324225e+193 + 4.1658080666870618332e+193j) + assert hyp1f2(a1,b1,b2,1000000*j).ae('3.5999042951925965458e+613 + 1.5026014707128947992e+613j') + assert hyp1f2(a1,b1,b2,10**7*j).ae('-8.3208715051623234801e+1939 - 3.6752883490851869429e+1941j') + assert hyp1f2(a1,b1,b2,10**8*j).ae('2.0724195707891484454e+6140 - 1.3276619482724266387e+6141j') + assert hyp1f2(a1,b1,b2,10**20*j).ae('-1.1734497974795488504e+6141851462 + 1.1498106965385471542e+6141851462j') + +def test_hyper_2f3(): + mp.dps = 15 + assert hyper([1,2],[3,4,5],6) == hyp2f3(1,2,3,4,5,6) + a1,a2,b1,b2,b3 = (1,10),(2,3),(3,10), 2, 1./16 + # Check asymptotic expansion + assert hyp2f3(a1,a2,b1,b2,b3,10).ae(128.98207160698659976) + assert hyp2f3(a1,a2,b1,b2,b3,1000).ae(6.6309632883131273141e25) + assert hyp2f3(a1,a2,b1,b2,b3,10000).ae(4.6863639362713340539e84) + assert hyp2f3(a1,a2,b1,b2,b3,100000).ae(8.6632451236103084119e271) + assert hyp2f3(a1,a2,b1,b2,b3,10**6).ae('2.0291718386574980641e865') + assert hyp2f3(a1,a2,b1,b2,b3,10**7).ae('7.7639836665710030977e2742') + assert hyp2f3(a1,a2,b1,b2,b3,10**8).ae('3.2537462584071268759e8681') + assert hyp2f3(a1,a2,b1,b2,b3,10**20).ae('1.2966030542911614163e+8685889627') + assert hyp2f3(a1,a2,b1,b2,b3,10*j).ae(-18.551602185587547854 - 13.348031097874113552j) + assert hyp2f3(a1,a2,b1,b2,b3,100*j).ae(78634.359124504488695 + 74459.535945281973996j) + assert hyp2f3(a1,a2,b1,b2,b3,1000*j).ae(597682550276527901.59 - 65136194809352613.078j) + assert hyp2f3(a1,a2,b1,b2,b3,10000*j).ae(-1.1779696326238582496e+59 + 1.2297607505213133872e+59j) + assert hyp2f3(a1,a2,b1,b2,b3,100000*j).ae(2.9844228969804380301e+191 + 7.5587163231490273296e+190j) + assert hyp2f3(a1,a2,b1,b2,b3,1000000*j).ae('7.4859161049322370311e+610 - 2.8467477015940090189e+610j') + assert hyp2f3(a1,a2,b1,b2,b3,10**7*j).ae('-1.7477645579418800826e+1938 - 1.7606522995808116405e+1938j') + assert hyp2f3(a1,a2,b1,b2,b3,10**8*j).ae('-1.6932731942958401784e+6137 - 2.4521909113114629368e+6137j') + assert hyp2f3(a1,a2,b1,b2,b3,10**20*j).ae('-2.0988815677627225449e+6141851451 + 5.7708223542739208681e+6141851452j') + +def test_hyper_2f2(): + mp.dps = 15 + assert hyper([1,2],[3,4],5) == hyp2f2(1,2,3,4,5) + a1,a2,b1,b2 = (3,10),4,(1,2),1./16 + assert hyp2f2(a1,a2,b1,b2,10).ae(448225936.3377556696) + assert hyp2f2(a1,a2,b1,b2,10000).ae('1.2012553712966636711e+4358') + assert hyp2f2(a1,a2,b1,b2,-20000).ae(-0.04182343755661214626) + assert hyp2f2(a1,a2,b1,b2,10**20).ae('1.1148680024303263661e+43429448190325182840') + +def test_orthpoly(): + mp.dps = 15 + assert jacobi(-4,2,3,0.7).ae(22800./4913) + assert jacobi(3,2,4,5.5) == 4133.125 + assert jacobi(1.5,5/6.,4,0).ae(-1.0851951434075508417) + assert jacobi(-2, 1, 2, 4).ae(-0.16) + assert jacobi(2, -1, 2.5, 4).ae(34.59375) + #assert jacobi(2, -1, 2, 4) == 28.5 + assert legendre(5, 7) == 129367 + assert legendre(0.5,0).ae(0.53935260118837935667) + assert legendre(-1,-1) == 1 + assert legendre(0,-1) == 1 + assert legendre(0, 1) == 1 + assert legendre(1, -1) == -1 + assert legendre(7, 1) == 1 + assert legendre(7, -1) == -1 + assert legendre(8,1.5).ae(15457523./32768) + assert legendre(j,-j).ae(2.4448182735671431011 + 0.6928881737669934843j) + assert chebyu(5,1) == 6 + assert chebyt(3,2) == 26 + assert legendre(3.5,-1) == inf + assert legendre(4.5,-1) == -inf + assert legendre(3.5+1j,-1) == mpc(inf,inf) + assert legendre(4.5+1j,-1) == mpc(-inf,-inf) + assert laguerre(4, -2, 3).ae(-1.125) + assert laguerre(3, 1+j, 0.5).ae(0.2291666666666666667 + 2.5416666666666666667j) + +def test_hermite(): + mp.dps = 15 + assert hermite(-2, 0).ae(0.5) + assert hermite(-1, 0).ae(0.88622692545275801365) + assert hermite(0, 0).ae(1) + assert hermite(1, 0) == 0 + assert hermite(2, 0).ae(-2) + assert hermite(0, 2).ae(1) + assert hermite(1, 2).ae(4) + assert hermite(1, -2).ae(-4) + assert hermite(2, -2).ae(14) + assert hermite(0.5, 0).ae(0.69136733903629335053) + assert hermite(9, 0) == 0 + assert hermite(4,4).ae(3340) + assert hermite(3,4).ae(464) + assert hermite(-4,4).ae(0.00018623860287512396181) + assert hermite(-3,4).ae(0.0016540169879668766270) + assert hermite(9, 2.5j).ae(13638725j) + assert hermite(9, -2.5j).ae(-13638725j) + assert hermite(9, 100).ae(511078883759363024000) + assert hermite(9, -100).ae(-511078883759363024000) + assert hermite(9, 100j).ae(512922083920643024000j) + assert hermite(9, -100j).ae(-512922083920643024000j) + assert hermite(-9.5, 2.5j).ae(-2.9004951258126778174e-6 + 1.7601372934039951100e-6j) + assert hermite(-9.5, -2.5j).ae(-2.9004951258126778174e-6 - 1.7601372934039951100e-6j) + assert hermite(-9.5, 100).ae(1.3776300722767084162e-22, abs_eps=0, rel_eps=eps) + assert hermite(-9.5, -100).ae('1.3106082028470671626e4355') + assert hermite(-9.5, 100j).ae(-9.7900218581864768430e-23 - 9.7900218581864768430e-23j, abs_eps=0, rel_eps=eps) + assert hermite(-9.5, -100j).ae(-9.7900218581864768430e-23 + 9.7900218581864768430e-23j, abs_eps=0, rel_eps=eps) + assert hermite(2+3j, -1-j).ae(851.3677063883687676 - 1496.4373467871007997j) + +def test_gegenbauer(): + mp.dps = 15 + assert gegenbauer(1,2,3).ae(12) + assert gegenbauer(2,3,4).ae(381) + assert gegenbauer(0,0,0) == 0 + assert gegenbauer(2,-1,3) == 0 + assert gegenbauer(-7, 0.5, 3).ae(8989) + assert gegenbauer(1, -0.5, 3).ae(-3) + assert gegenbauer(1, -1.5, 3).ae(-9) + assert gegenbauer(1, -0.5, 3).ae(-3) + assert gegenbauer(-0.5, -0.5, 3).ae(-2.6383553159023906245) + assert gegenbauer(2+3j, 1-j, 3+4j).ae(14.880536623203696780 + 20.022029711598032898j) + #assert gegenbauer(-2, -0.5, 3).ae(-12) + +def test_legenp(): + mp.dps = 15 + assert legenp(2,0,4) == legendre(2,4) + assert legenp(-2, -1, 0.5).ae(0.43301270189221932338) + assert legenp(-2, -1, 0.5, type=3).ae(0.43301270189221932338j) + assert legenp(-2, 1, 0.5).ae(-0.86602540378443864676) + assert legenp(2+j, 3+4j, -j).ae(134742.98773236786148 + 429782.72924463851745j) + assert legenp(2+j, 3+4j, -j, type=3).ae(802.59463394152268507 - 251.62481308942906447j) + assert legenp(2,4,3).ae(0) + assert legenp(2,4,3,type=3).ae(0) + assert legenp(2,1,0.5).ae(-1.2990381056766579701) + assert legenp(2,1,0.5,type=3).ae(1.2990381056766579701j) + assert legenp(3,2,3).ae(-360) + assert legenp(3,3,3).ae(240j*2**0.5) + assert legenp(3,4,3).ae(0) + assert legenp(0,0.5,2).ae(0.52503756790433198939 - 0.52503756790433198939j) + assert legenp(-1,-0.5,2).ae(0.60626116232846498110 + 0.60626116232846498110j) + assert legenp(-2,0.5,2).ae(1.5751127037129959682 - 1.5751127037129959682j) + assert legenp(-2,0.5,-0.5).ae(-0.85738275810499171286) + +def test_legenq(): + mp.dps = 15 + f = legenq + # Evaluation at poles + assert isnan(f(3,2,1)) + assert isnan(f(3,2,-1)) + assert isnan(f(3,2,1,type=3)) + assert isnan(f(3,2,-1,type=3)) + # Evaluation at 0 + assert f(0,1,0,type=2).ae(-1) + assert f(-2,2,0,type=2,zeroprec=200).ae(0) + assert f(1.5,3,0,type=2).ae(-2.2239343475841951023) + assert f(0,1,0,type=3).ae(j) + assert f(-2,2,0,type=3,zeroprec=200).ae(0) + assert f(1.5,3,0,type=3).ae(2.2239343475841951022*(1-1j)) + # Standard case, degree 0 + assert f(0,0,-1.5).ae(-0.8047189562170501873 + 1.5707963267948966192j) + assert f(0,0,-0.5).ae(-0.54930614433405484570) + assert f(0,0,0,zeroprec=200).ae(0) + assert f(0,0,0.5).ae(0.54930614433405484570) + assert f(0,0,1.5).ae(0.8047189562170501873 - 1.5707963267948966192j) + assert f(0,0,-1.5,type=3).ae(-0.80471895621705018730) + assert f(0,0,-0.5,type=3).ae(-0.5493061443340548457 - 1.5707963267948966192j) + assert f(0,0,0,type=3).ae(-1.5707963267948966192j) + assert f(0,0,0.5,type=3).ae(0.5493061443340548457 - 1.5707963267948966192j) + assert f(0,0,1.5,type=3).ae(0.80471895621705018730) + # Standard case, degree 1 + assert f(1,0,-1.5).ae(0.2070784343255752810 - 2.3561944901923449288j) + assert f(1,0,-0.5).ae(-0.72534692783297257715) + assert f(1,0,0).ae(-1) + assert f(1,0,0.5).ae(-0.72534692783297257715) + assert f(1,0,1.5).ae(0.2070784343255752810 - 2.3561944901923449288j) + # Standard case, degree 2 + assert f(2,0,-1.5).ae(-0.0635669991240192885 + 4.5160394395353277803j) + assert f(2,0,-0.5).ae(0.81866326804175685571) + assert f(2,0,0,zeroprec=200).ae(0) + assert f(2,0,0.5).ae(-0.81866326804175685571) + assert f(2,0,1.5).ae(0.0635669991240192885 - 4.5160394395353277803j) + # Misc orders and degrees + assert f(2,3,1.5,type=2).ae(-5.7243340223994616228j) + assert f(2,3,1.5,type=3).ae(-5.7243340223994616228) + assert f(2,3,0.5,type=2).ae(-12.316805742712016310) + assert f(2,3,0.5,type=3).ae(-12.316805742712016310j) + assert f(2,3,-1.5,type=2).ae(-5.7243340223994616228j) + assert f(2,3,-1.5,type=3).ae(5.7243340223994616228) + assert f(2,3,-0.5,type=2).ae(-12.316805742712016310) + assert f(2,3,-0.5,type=3).ae(-12.316805742712016310j) + assert f(2+3j, 3+4j, 0.5, type=3).ae(0.0016119404873235186807 - 0.0005885900510718119836j) + assert f(2+3j, 3+4j, -1.5, type=3).ae(0.008451400254138808670 + 0.020645193304593235298j) + assert f(-2.5,1,-1.5).ae(3.9553395527435335749j) + assert f(-2.5,1,-0.5).ae(1.9290561746445456908) + assert f(-2.5,1,0).ae(1.2708196271909686299) + assert f(-2.5,1,0.5).ae(-0.31584812990742202869) + assert f(-2.5,1,1.5).ae(-3.9553395527435335742 + 0.2993235655044701706j) + assert f(-2.5,1,-1.5,type=3).ae(0.29932356550447017254j) + assert f(-2.5,1,-0.5,type=3).ae(-0.3158481299074220287 - 1.9290561746445456908j) + assert f(-2.5,1,0,type=3).ae(1.2708196271909686292 - 1.2708196271909686299j) + assert f(-2.5,1,0.5,type=3).ae(1.9290561746445456907 + 0.3158481299074220287j) + assert f(-2.5,1,1.5,type=3).ae(-0.29932356550447017254) + +def test_agm(): + mp.dps = 15 + assert agm(0,0) == 0 + assert agm(0,1) == 0 + assert agm(1,1) == 1 + assert agm(7,7) == 7 + assert agm(j,j) == j + assert (1/agm(1,sqrt(2))).ae(0.834626841674073186) + assert agm(1,2).ae(1.4567910310469068692) + assert agm(1,3).ae(1.8636167832448965424) + assert agm(1,j).ae(0.599070117367796104+0.599070117367796104j) + assert agm(2) == agm(1,2) + assert agm(-3,4).ae(0.63468509766550907+1.3443087080896272j) + +def test_gammainc(): + mp.dps = 15 + assert gammainc(2,5).ae(6*exp(-5)) + assert gammainc(2,0,5).ae(1-6*exp(-5)) + assert gammainc(2,3,5).ae(-6*exp(-5)+4*exp(-3)) + assert gammainc(-2.5,-0.5).ae(-0.9453087204829418812-5.3164237738936178621j) + assert gammainc(0,2,4).ae(0.045121158298212213088) + assert gammainc(0,3).ae(0.013048381094197037413) + assert gammainc(0,2+j,1-j).ae(0.00910653685850304839-0.22378752918074432574j) + assert gammainc(0,1-j).ae(0.00028162445198141833+0.17932453503935894015j) + assert gammainc(3,4,5,True).ae(0.11345128607046320253) + assert gammainc(3.5,0,inf).ae(gamma(3.5)) + assert gammainc(-150.5,500).ae('6.9825435345798951153e-627') + assert gammainc(-150.5,800).ae('4.6885137549474089431e-788') + assert gammainc(-3.5, -20.5).ae(0.27008820585226911 - 1310.31447140574997636j) + assert gammainc(-3.5, -200.5).ae(0.27008820585226911 - 5.3264597096208368435e76j) # XXX real part + assert gammainc(0,0,2) == inf + assert gammainc(1,b=1).ae(0.6321205588285576784) + assert gammainc(3,2,2) == 0 + assert gammainc(2,3+j,3-j).ae(-0.28135485191849314194j) + assert gammainc(4+0j,1).ae(5.8860710587430771455) + # GH issue #301 + assert gammainc(-1,-1).ae(-0.8231640121031084799 + 3.1415926535897932385j) + assert gammainc(-2,-1).ae(1.7707229202810768576 - 1.5707963267948966192j) + assert gammainc(-3,-1).ae(-1.4963349162467073643 + 0.5235987755982988731j) + assert gammainc(-4,-1).ae(1.05365418617643814992 - 0.13089969389957471827j) + # Regularized upper gamma + assert isnan(gammainc(0, 0, regularized=True)) + assert gammainc(-1, 0, regularized=True) == inf + assert gammainc(1, 0, regularized=True) == 1 + assert gammainc(0, 5, regularized=True) == 0 + assert gammainc(0, 2+3j, regularized=True) == 0 + assert gammainc(0, 5000, regularized=True) == 0 + assert gammainc(0, 10**30, regularized=True) == 0 + assert gammainc(-1, 5, regularized=True) == 0 + assert gammainc(-1, 5000, regularized=True) == 0 + assert gammainc(-1, 10**30, regularized=True) == 0 + assert gammainc(-1, -5, regularized=True) == 0 + assert gammainc(-1, -5000, regularized=True) == 0 + assert gammainc(-1, -10**30, regularized=True) == 0 + assert gammainc(-1, 3+4j, regularized=True) == 0 + assert gammainc(1, 5, regularized=True).ae(exp(-5)) + assert gammainc(1, 5000, regularized=True).ae(exp(-5000)) + assert gammainc(1, 10**30, regularized=True).ae(exp(-10**30)) + assert gammainc(1, 3+4j, regularized=True).ae(exp(-3-4j)) + assert gammainc(-1000000,2).ae('1.3669297209397347754e-301037', abs_eps=0, rel_eps=8*eps) + assert gammainc(-1000000,2,regularized=True) == 0 + assert gammainc(-1000000,3+4j).ae('-1.322575609404222361e-698979 - 4.9274570591854533273e-698978j', abs_eps=0, rel_eps=8*eps) + assert gammainc(-1000000,3+4j,regularized=True) == 0 + assert gammainc(2+3j, 4+5j, regularized=True).ae(0.085422013530993285774-0.052595379150390078503j) + assert gammainc(1000j, 1000j, regularized=True).ae(0.49702647628921131761 + 0.00297355675013575341j) + # Generalized + assert gammainc(3,4,2) == -gammainc(3,2,4) + assert gammainc(4, 2, 3).ae(1.2593494302978947396) + assert gammainc(4, 2, 3, regularized=True).ae(0.20989157171631578993) + assert gammainc(0, 2, 3).ae(0.035852129613864082155) + assert gammainc(0, 2, 3, regularized=True) == 0 + assert gammainc(-1, 2, 3).ae(0.015219822548487616132) + assert gammainc(-1, 2, 3, regularized=True) == 0 + assert gammainc(0, 2, 3).ae(0.035852129613864082155) + assert gammainc(0, 2, 3, regularized=True) == 0 + # Should use upper gammas + assert gammainc(5, 10000, 12000).ae('1.1359381951461801687e-4327', abs_eps=0, rel_eps=8*eps) + # Should use lower gammas + assert gammainc(10000, 2, 3).ae('8.1244514125995785934e4765') + # GH issue 306 + assert gammainc(3,-1-1j) == 0 + assert gammainc(3,-1+1j) == 0 + assert gammainc(2,-1) == 0 + assert gammainc(2,-1+0j) == 0 + assert gammainc(2+0j,-1) == 0 + +def test_gammainc_expint_n(): + # These tests are intended to check all cases of the low-level code + # for upper gamma and expint with small integer index. + # Need to cover positive/negative arguments; small/large/huge arguments + # for both positive and negative indices, as well as indices 0 and 1 + # which may be special-cased + mp.dps = 15 + assert expint(-3,3.5).ae(0.021456366563296693987) + assert expint(-2,3.5).ae(0.014966633183073309405) + assert expint(-1,3.5).ae(0.011092916359219041088) + assert expint(0,3.5).ae(0.0086278238349481430685) + assert expint(1,3.5).ae(0.0069701398575483929193) + assert expint(2,3.5).ae(0.0058018939208991255223) + assert expint(3,3.5).ae(0.0049453773495857807058) + assert expint(-3,-3.5).ae(-4.6618170604073311319) + assert expint(-2,-3.5).ae(-5.5996974157555515963) + assert expint(-1,-3.5).ae(-6.7582555017739415818) + assert expint(0,-3.5).ae(-9.4615577024835182145) + assert expint(1,-3.5).ae(-13.925353995152335292 - 3.1415926535897932385j) + assert expint(2,-3.5).ae(-15.62328702434085977 - 10.995574287564276335j) + assert expint(3,-3.5).ae(-10.783026313250347722 - 19.242255003237483586j) + assert expint(-3,350).ae(2.8614825451252838069e-155, abs_eps=0, rel_eps=8*eps) + assert expint(-2,350).ae(2.8532837224504675901e-155, abs_eps=0, rel_eps=8*eps) + assert expint(-1,350).ae(2.8451316155828634555e-155, abs_eps=0, rel_eps=8*eps) + assert expint(0,350).ae(2.8370258275042797989e-155, abs_eps=0, rel_eps=8*eps) + assert expint(1,350).ae(2.8289659656701459404e-155, abs_eps=0, rel_eps=8*eps) + assert expint(2,350).ae(2.8209516419468505006e-155, abs_eps=0, rel_eps=8*eps) + assert expint(3,350).ae(2.8129824725501272171e-155, abs_eps=0, rel_eps=8*eps) + assert expint(-3,-350).ae(-2.8528796154044839443e+149) + assert expint(-2,-350).ae(-2.8610072121701264351e+149) + assert expint(-1,-350).ae(-2.8691813842677537647e+149) + assert expint(0,-350).ae(-2.8774025343659421709e+149) + u = expint(1,-350) + assert u.ae(-2.8856710698020863568e+149) + assert u.imag.ae(-3.1415926535897932385) + u = expint(2,-350) + assert u.ae(-2.8939874026504650534e+149) + assert u.imag.ae(-1099.5574287564276335) + u = expint(3,-350) + assert u.ae(-2.9023519497915044349e+149) + assert u.imag.ae(-192422.55003237483586) + assert expint(-3,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(-2,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(-1,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(0,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(1,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(2,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(3,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert expint(-3,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') + assert expint(-2,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') + assert expint(-1,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') + assert expint(0,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') + u = expint(1,-350000000000000000000000) + assert u.ae('-3.7805306852415755699e+152003068666138139677871') + assert u.imag.ae(-3.1415926535897932385) + u = expint(2,-350000000000000000000000) + assert u.imag.ae(-1.0995574287564276335e+24) + assert u.ae('-3.7805306852415755699e+152003068666138139677871') + u = expint(3,-350000000000000000000000) + assert u.imag.ae(-1.9242255003237483586e+47) + assert u.ae('-3.7805306852415755699e+152003068666138139677871') + # Small case; no branch cut + assert gammainc(-3,3.5).ae(0.00010020262545203707109) + assert gammainc(-2,3.5).ae(0.00040370427343557393517) + assert gammainc(-1,3.5).ae(0.0016576839773997501492) + assert gammainc(0,3.5).ae(0.0069701398575483929193) + assert gammainc(1,3.5).ae(0.03019738342231850074) + assert gammainc(2,3.5).ae(0.13588822540043325333) + assert gammainc(3,3.5).ae(0.64169439772426814072) + # Small case; with branch cut + assert gammainc(-3,-3.5).ae(0.03595832954467563286 + 0.52359877559829887308j) + assert gammainc(-2,-3.5).ae(-0.88024704597962022221 - 1.5707963267948966192j) + assert gammainc(-1,-3.5).ae(4.4637962926688170771 + 3.1415926535897932385j) + assert gammainc(0,-3.5).ae(-13.925353995152335292 - 3.1415926535897932385j) + assert gammainc(1,-3.5).ae(33.115451958692313751) + assert gammainc(2,-3.5).ae(-82.788629896730784377) + assert gammainc(3,-3.5).ae(240.08702670051927469) + # Asymptotic case; no branch cut + assert gammainc(-3,350).ae(6.5424095113340358813e-163, abs_eps=0, rel_eps=8*eps) + assert gammainc(-2,350).ae(2.296312222489899769e-160, abs_eps=0, rel_eps=8*eps) + assert gammainc(-1,350).ae(8.059861834133858573e-158, abs_eps=0, rel_eps=8*eps) + assert gammainc(0,350).ae(2.8289659656701459404e-155, abs_eps=0, rel_eps=8*eps) + assert gammainc(1,350).ae(9.9295903962649792963e-153, abs_eps=0, rel_eps=8*eps) + assert gammainc(2,350).ae(3.485286229089007733e-150, abs_eps=0, rel_eps=8*eps) + assert gammainc(3,350).ae(1.2233453960006379793e-147, abs_eps=0, rel_eps=8*eps) + # Asymptotic case; branch cut + u = gammainc(-3,-350) + assert u.ae(6.7889565783842895085e+141) + assert u.imag.ae(0.52359877559829887308) + u = gammainc(-2,-350) + assert u.ae(-2.3692668977889832121e+144) + assert u.imag.ae(-1.5707963267948966192) + u = gammainc(-1,-350) + assert u.ae(8.2685354361441858669e+146) + assert u.imag.ae(3.1415926535897932385) + u = gammainc(0,-350) + assert u.ae(-2.8856710698020863568e+149) + assert u.imag.ae(-3.1415926535897932385) + u = gammainc(1,-350) + assert u.ae(1.0070908870280797598e+152) + assert u.imag == 0 + u = gammainc(2,-350) + assert u.ae(-3.5147471957279983618e+154) + assert u.imag == 0 + u = gammainc(3,-350) + assert u.ae(1.2266568422179417091e+157) + assert u.imag == 0 + # Extreme asymptotic case + assert gammainc(-3,350000000000000000000000).ae('5.0362468738874738859e-152003068666138139677990', abs_eps=0, rel_eps=8*eps) + assert gammainc(-2,350000000000000000000000).ae('1.7626864058606158601e-152003068666138139677966', abs_eps=0, rel_eps=8*eps) + assert gammainc(-1,350000000000000000000000).ae('6.1694024205121555102e-152003068666138139677943', abs_eps=0, rel_eps=8*eps) + assert gammainc(0,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) + assert gammainc(1,350000000000000000000000).ae('7.5575179651273905e-152003068666138139677896', abs_eps=0, rel_eps=8*eps) + assert gammainc(2,350000000000000000000000).ae('2.645131287794586675e-152003068666138139677872', abs_eps=0, rel_eps=8*eps) + assert gammainc(3,350000000000000000000000).ae('9.2579595072810533625e-152003068666138139677849', abs_eps=0, rel_eps=8*eps) + u = gammainc(-3,-350000000000000000000000) + assert u.ae('8.8175642804468234866e+152003068666138139677800') + assert u.imag.ae(0.52359877559829887308) + u = gammainc(-2,-350000000000000000000000) + assert u.ae('-3.0861474981563882203e+152003068666138139677824') + assert u.imag.ae(-1.5707963267948966192) + u = gammainc(-1,-350000000000000000000000) + assert u.ae('1.0801516243547358771e+152003068666138139677848') + assert u.imag.ae(3.1415926535897932385) + u = gammainc(0,-350000000000000000000000) + assert u.ae('-3.7805306852415755699e+152003068666138139677871') + assert u.imag.ae(-3.1415926535897932385) + assert gammainc(1,-350000000000000000000000).ae('1.3231857398345514495e+152003068666138139677895') + assert gammainc(2,-350000000000000000000000).ae('-4.6311500894209300731e+152003068666138139677918') + assert gammainc(3,-350000000000000000000000).ae('1.6209025312973255256e+152003068666138139677942') + +def test_incomplete_beta(): + mp.dps = 15 + assert betainc(-2,-3,0.5,0.75).ae(63.4305673311255413583969) + assert betainc(4.5,0.5+2j,2.5,6).ae(0.2628801146130621387903065 + 0.5162565234467020592855378j) + assert betainc(4,5,0,6).ae(90747.77142857142857142857) + +def test_erf(): + mp.dps = 15 + assert erf(0) == 0 + assert erf(1).ae(0.84270079294971486934) + assert erf(3+4j).ae(-120.186991395079444098 - 27.750337293623902498j) + assert erf(-4-3j).ae(-0.99991066178539168236 + 0.00004972026054496604j) + assert erf(pi).ae(0.99999112385363235839) + assert erf(1j).ae(1.6504257587975428760j) + assert erf(-1j).ae(-1.6504257587975428760j) + assert isinstance(erf(1), mpf) + assert isinstance(erf(-1), mpf) + assert isinstance(erf(0), mpf) + assert isinstance(erf(0j), mpc) + assert erf(inf) == 1 + assert erf(-inf) == -1 + assert erfi(0) == 0 + assert erfi(1/pi).ae(0.371682698493894314) + assert erfi(inf) == inf + assert erfi(-inf) == -inf + assert erf(1+0j) == erf(1) + assert erfc(1+0j) == erfc(1) + assert erf(0.2+0.5j).ae(1 - erfc(0.2+0.5j)) + assert erfc(0) == 1 + assert erfc(1).ae(1-erf(1)) + assert erfc(-1).ae(1-erf(-1)) + assert erfc(1/pi).ae(1-erf(1/pi)) + assert erfc(-10) == 2 + assert erfc(-1000000) == 2 + assert erfc(-inf) == 2 + assert erfc(inf) == 0 + assert isnan(erfc(nan)) + assert (erfc(10**4)*mpf(10)**43429453).ae('3.63998738656420') + assert erf(8+9j).ae(-1072004.2525062051158 + 364149.91954310255423j) + assert erfc(8+9j).ae(1072005.2525062051158 - 364149.91954310255423j) + assert erfc(-8-9j).ae(-1072003.2525062051158 + 364149.91954310255423j) + mp.dps = 50 + # This one does not use the asymptotic series + assert (erfc(10)*10**45).ae('2.0884875837625447570007862949577886115608181193212') + # This one does + assert (erfc(50)*10**1088).ae('2.0709207788416560484484478751657887929322509209954') + mp.dps = 15 + assert str(erfc(10**50)) == '3.66744826532555e-4342944819032518276511289189166050822943970058036665661144537831658646492088707747292249493384317534' + assert erfinv(0) == 0 + assert erfinv(0.5).ae(0.47693627620446987338) + assert erfinv(-0.5).ae(-0.47693627620446987338) + assert erfinv(1) == inf + assert erfinv(-1) == -inf + assert erf(erfinv(0.95)).ae(0.95) + assert erf(erfinv(0.999999999995)).ae(0.999999999995) + assert erf(erfinv(-0.999999999995)).ae(-0.999999999995) + mp.dps = 50 + assert erf(erfinv('0.99999999999999999999999999999995')).ae('0.99999999999999999999999999999995') + assert erf(erfinv('0.999999999999999999999999999999995')).ae('0.999999999999999999999999999999995') + assert erf(erfinv('-0.999999999999999999999999999999995')).ae('-0.999999999999999999999999999999995') + mp.dps = 15 + # Complex asymptotic expansions + v = erfc(50j) + assert v.real == 1 + assert v.imag.ae('-6.1481820666053078736e+1083') + assert erfc(-100+5j).ae(2) + assert (erfc(100+5j)*10**4335).ae(2.3973567853824133572 - 3.9339259530609420597j) + assert erfc(100+100j).ae(0.00065234366376857698698 - 0.0039357263629214118437j) + +def test_pdf(): + mp.dps = 15 + assert npdf(-inf) == 0 + assert npdf(inf) == 0 + assert npdf(5,0,2).ae(npdf(5+4,4,2)) + assert quadts(lambda x: npdf(x,-0.5,0.8), [-inf, inf]) == 1 + assert ncdf(0) == 0.5 + assert ncdf(3,3) == 0.5 + assert ncdf(-inf) == 0 + assert ncdf(inf) == 1 + assert ncdf(10) == 1 + # Verify that this is computed accurately + assert (ncdf(-10)*10**24).ae(7.619853024160526) + +def test_lambertw(): + mp.dps = 15 + assert lambertw(0) == 0 + assert lambertw(0+0j) == 0 + assert lambertw(inf) == inf + assert isnan(lambertw(nan)) + assert lambertw(inf,1).real == inf + assert lambertw(inf,1).imag.ae(2*pi) + assert lambertw(-inf,1).real == inf + assert lambertw(-inf,1).imag.ae(3*pi) + assert lambertw(0,-1) == -inf + assert lambertw(0,1) == -inf + assert lambertw(0,3) == -inf + assert lambertw(e).ae(1) + assert lambertw(1).ae(0.567143290409783873) + assert lambertw(-pi/2).ae(j*pi/2) + assert lambertw(-log(2)/2).ae(-log(2)) + assert lambertw(0.25).ae(0.203888354702240164) + assert lambertw(-0.25).ae(-0.357402956181388903) + assert lambertw(-1./10000,0).ae(-0.000100010001500266719) + assert lambertw(-0.25,-1).ae(-2.15329236411034965) + assert lambertw(0.25,-1).ae(-3.00899800997004620-4.07652978899159763j) + assert lambertw(-0.25,-1).ae(-2.15329236411034965) + assert lambertw(0.25,1).ae(-3.00899800997004620+4.07652978899159763j) + assert lambertw(-0.25,1).ae(-3.48973228422959210+7.41405453009603664j) + assert lambertw(-4).ae(0.67881197132094523+1.91195078174339937j) + assert lambertw(-4,1).ae(-0.66743107129800988+7.76827456802783084j) + assert lambertw(-4,-1).ae(0.67881197132094523-1.91195078174339937j) + assert lambertw(1000).ae(5.24960285240159623) + assert lambertw(1000,1).ae(4.91492239981054535+5.44652615979447070j) + assert lambertw(1000,-1).ae(4.91492239981054535-5.44652615979447070j) + assert lambertw(1000,5).ae(3.5010625305312892+29.9614548941181328j) + assert lambertw(3+4j).ae(1.281561806123775878+0.533095222020971071j) + assert lambertw(-0.4+0.4j).ae(-0.10396515323290657+0.61899273315171632j) + assert lambertw(3+4j,1).ae(-0.11691092896595324+5.61888039871282334j) + assert lambertw(3+4j,-1).ae(0.25856740686699742-3.85211668616143559j) + assert lambertw(-0.5,-1).ae(-0.794023632344689368-0.770111750510379110j) + assert lambertw(-1./10000,1).ae(-11.82350837248724344+6.80546081842002101j) + assert lambertw(-1./10000,-1).ae(-11.6671145325663544) + assert lambertw(-1./10000,-2).ae(-11.82350837248724344-6.80546081842002101j) + assert lambertw(-1./100000,4).ae(-14.9186890769540539+26.1856750178782046j) + assert lambertw(-1./100000,5).ae(-15.0931437726379218666+32.5525721210262290086j) + assert lambertw((2+j)/10).ae(0.173704503762911669+0.071781336752835511j) + assert lambertw((2+j)/10,1).ae(-3.21746028349820063+4.56175438896292539j) + assert lambertw((2+j)/10,-1).ae(-3.03781405002993088-3.53946629633505737j) + assert lambertw((2+j)/10,4).ae(-4.6878509692773249+23.8313630697683291j) + assert lambertw(-(2+j)/10).ae(-0.226933772515757933-0.164986470020154580j) + assert lambertw(-(2+j)/10,1).ae(-2.43569517046110001+0.76974067544756289j) + assert lambertw(-(2+j)/10,-1).ae(-3.54858738151989450-6.91627921869943589j) + assert lambertw(-(2+j)/10,4).ae(-4.5500846928118151+20.6672982215434637j) + mp.dps = 50 + assert lambertw(pi).ae('1.073658194796149172092178407024821347547745350410314531') + mp.dps = 15 + # Former bug in generated branch + assert lambertw(-0.5+0.002j).ae(-0.78917138132659918344 + 0.76743539379990327749j) + assert lambertw(-0.5-0.002j).ae(-0.78917138132659918344 - 0.76743539379990327749j) + assert lambertw(-0.448+0.4j).ae(-0.11855133765652382241 + 0.66570534313583423116j) + assert lambertw(-0.448-0.4j).ae(-0.11855133765652382241 - 0.66570534313583423116j) + assert lambertw(-0.65475+0.0001j).ae(-0.61053421111385310898+1.0396534993944097723803j) + # Huge branch index + w = lambertw(1,10**20) + assert w.real.ae(-47.889578926290259164) + assert w.imag.ae(6.2831853071795864769e+20) + +def test_lambertw_hard(): + def check(x,y): + y = convert(y) + type_ok = True + if isinstance(y, mpf): + type_ok = isinstance(x, mpf) + real_ok = abs(x.real-y.real) <= abs(y.real)*8*eps + imag_ok = abs(x.imag-y.imag) <= abs(y.imag)*8*eps + #print x, y, abs(x.real-y.real), abs(x.imag-y.imag) + return real_ok and imag_ok + # Evaluation near 0 + mp.dps = 15 + assert check(lambertw(1e-10), 9.999999999000000000e-11) + assert check(lambertw(-1e-10), -1.000000000100000000e-10) + assert check(lambertw(1e-10j), 9.999999999999999999733e-21 + 9.99999999999999999985e-11j) + assert check(lambertw(-1e-10j), 9.999999999999999999733e-21 - 9.99999999999999999985e-11j) + assert check(lambertw(1e-10,1), -26.303186778379041559 + 3.265093911703828397j) + assert check(lambertw(-1e-10,1), -26.326236166739163892 + 6.526183280686333315j) + assert check(lambertw(1e-10j,1), -26.312931726911421551 + 4.896366881798013421j) + assert check(lambertw(-1e-10j,1), -26.297238779529035066 + 1.632807161345576513j) + assert check(lambertw(1e-10,-1), -26.303186778379041559 - 3.265093911703828397j) + assert check(lambertw(-1e-10,-1), -26.295238819246925694) + assert check(lambertw(1e-10j,-1), -26.297238779529035028 - 1.6328071613455765135j) + assert check(lambertw(-1e-10j,-1), -26.312931726911421551 - 4.896366881798013421j) + # Test evaluation very close to the branch point -1/e + # on the -1, 0, and 1 branches + add = lambda x, y: fadd(x,y,exact=True) + sub = lambda x, y: fsub(x,y,exact=True) + addj = lambda x, y: fadd(x,fmul(y,1j,exact=True),exact=True) + subj = lambda x, y: fadd(x,fmul(y,-1j,exact=True),exact=True) + mp.dps = 1500 + a = -1/e + 10*eps + d3 = mpf('1e-3') + d10 = mpf('1e-10') + d20 = mpf('1e-20') + d40 = mpf('1e-40') + d80 = mpf('1e-80') + d300 = mpf('1e-300') + d1000 = mpf('1e-1000') + mp.dps = 15 + # ---- Branch 0 ---- + # -1/e + eps + assert check(lambertw(add(a,d3)), -0.92802015005456704876) + assert check(lambertw(add(a,d10)), -0.99997668374140088071) + assert check(lambertw(add(a,d20)), -0.99999999976683560186) + assert lambertw(add(a,d40)) == -1 + assert lambertw(add(a,d80)) == -1 + assert lambertw(add(a,d300)) == -1 + assert lambertw(add(a,d1000)) == -1 + # -1/e - eps + assert check(lambertw(sub(a,d3)), -0.99819016149860989001+0.07367191188934638577j) + assert check(lambertw(sub(a,d10)), -0.9999999998187812114595992+0.0000233164398140346109194j) + assert check(lambertw(sub(a,d20)), -0.99999999999999999998187+2.331643981597124203344e-10j) + assert check(lambertw(sub(a,d40)), -1.0+2.33164398159712420336e-20j) + assert check(lambertw(sub(a,d80)), -1.0+2.33164398159712420336e-40j) + assert check(lambertw(sub(a,d300)), -1.0+2.33164398159712420336e-150j) + assert check(lambertw(sub(a,d1000)), mpc(-1,'2.33164398159712420336e-500')) + # -1/e + eps*j + assert check(lambertw(addj(a,d3)), -0.94790387486938526634+0.05036819639190132490j) + assert check(lambertw(addj(a,d10)), -0.9999835127872943680999899+0.0000164870314895821225256j) + assert check(lambertw(addj(a,d20)), -0.999999999835127872929987+1.64872127051890935830e-10j) + assert check(lambertw(addj(a,d40)), -0.9999999999999999999835+1.6487212707001281468305e-20j) + assert check(lambertw(addj(a,d80)), -1.0 + 1.64872127070012814684865e-40j) + assert check(lambertw(addj(a,d300)), -1.0 + 1.64872127070012814684865e-150j) + assert check(lambertw(addj(a,d1000)), mpc(-1.0,'1.64872127070012814684865e-500')) + # -1/e - eps*j + assert check(lambertw(subj(a,d3)), -0.94790387486938526634-0.05036819639190132490j) + assert check(lambertw(subj(a,d10)), -0.9999835127872943680999899-0.0000164870314895821225256j) + assert check(lambertw(subj(a,d20)), -0.999999999835127872929987-1.64872127051890935830e-10j) + assert check(lambertw(subj(a,d40)), -0.9999999999999999999835-1.6487212707001281468305e-20j) + assert check(lambertw(subj(a,d80)), -1.0 - 1.64872127070012814684865e-40j) + assert check(lambertw(subj(a,d300)), -1.0 - 1.64872127070012814684865e-150j) + assert check(lambertw(subj(a,d1000)), mpc(-1.0,'-1.64872127070012814684865e-500')) + # ---- Branch 1 ---- + assert check(lambertw(addj(a,d3),1), -3.088501303219933378005990 + 7.458676867597474813950098j) + assert check(lambertw(addj(a,d80),1), -3.088843015613043855957087 + 7.461489285654254556906117j) + assert check(lambertw(addj(a,d300),1), -3.088843015613043855957087 + 7.461489285654254556906117j) + assert check(lambertw(addj(a,d1000),1), -3.088843015613043855957087 + 7.461489285654254556906117j) + assert check(lambertw(subj(a,d3),1), -1.0520914180450129534365906 + 0.0539925638125450525673175j) + assert check(lambertw(subj(a,d10),1), -1.0000164872127056318529390 + 0.000016487393927159250398333077j) + assert check(lambertw(subj(a,d20),1), -1.0000000001648721270700128 + 1.64872127088134693542628e-10j) + assert check(lambertw(subj(a,d40),1), -1.000000000000000000016487 + 1.64872127070012814686677e-20j) + assert check(lambertw(subj(a,d80),1), -1.0 + 1.64872127070012814684865e-40j) + assert check(lambertw(subj(a,d300),1), -1.0 + 1.64872127070012814684865e-150j) + assert check(lambertw(subj(a,d1000),1), mpc(-1.0, '1.64872127070012814684865e-500')) + # ---- Branch -1 ---- + # -1/e + eps + assert check(lambertw(add(a,d3),-1), -1.075608941186624989414945) + assert check(lambertw(add(a,d10),-1), -1.000023316621036696460620) + assert check(lambertw(add(a,d20),-1), -1.000000000233164398177834) + assert lambertw(add(a,d40),-1) == -1 + assert lambertw(add(a,d80),-1) == -1 + assert lambertw(add(a,d300),-1) == -1 + assert lambertw(add(a,d1000),-1) == -1 + # -1/e - eps + assert check(lambertw(sub(a,d3),-1), -0.99819016149860989001-0.07367191188934638577j) + assert check(lambertw(sub(a,d10),-1), -0.9999999998187812114595992-0.0000233164398140346109194j) + assert check(lambertw(sub(a,d20),-1), -0.99999999999999999998187-2.331643981597124203344e-10j) + assert check(lambertw(sub(a,d40),-1), -1.0-2.33164398159712420336e-20j) + assert check(lambertw(sub(a,d80),-1), -1.0-2.33164398159712420336e-40j) + assert check(lambertw(sub(a,d300),-1), -1.0-2.33164398159712420336e-150j) + assert check(lambertw(sub(a,d1000),-1), mpc(-1,'-2.33164398159712420336e-500')) + # -1/e + eps*j + assert check(lambertw(addj(a,d3),-1), -1.0520914180450129534365906 - 0.0539925638125450525673175j) + assert check(lambertw(addj(a,d10),-1), -1.0000164872127056318529390 - 0.0000164873939271592503983j) + assert check(lambertw(addj(a,d20),-1), -1.0000000001648721270700 - 1.64872127088134693542628e-10j) + assert check(lambertw(addj(a,d40),-1), -1.00000000000000000001648 - 1.6487212707001281468667726e-20j) + assert check(lambertw(addj(a,d80),-1), -1.0 - 1.64872127070012814684865e-40j) + assert check(lambertw(addj(a,d300),-1), -1.0 - 1.64872127070012814684865e-150j) + assert check(lambertw(addj(a,d1000),-1), mpc(-1.0,'-1.64872127070012814684865e-500')) + # -1/e - eps*j + assert check(lambertw(subj(a,d3),-1), -3.088501303219933378005990-7.458676867597474813950098j) + assert check(lambertw(subj(a,d10),-1), -3.088843015579260686911033-7.461489285372968780020716j) + assert check(lambertw(subj(a,d20),-1), -3.088843015613043855953708-7.461489285654254556877988j) + assert check(lambertw(subj(a,d40),-1), -3.088843015613043855957087-7.461489285654254556906117j) + assert check(lambertw(subj(a,d80),-1), -3.088843015613043855957087 - 7.461489285654254556906117j) + assert check(lambertw(subj(a,d300),-1), -3.088843015613043855957087 - 7.461489285654254556906117j) + assert check(lambertw(subj(a,d1000),-1), -3.088843015613043855957087 - 7.461489285654254556906117j) + # One more case, testing higher precision + mp.dps = 500 + x = -1/e + mpf('1e-13') + ans = "-0.99999926266961377166355784455394913638782494543377383"\ + "744978844374498153493943725364881490261187530235150668593869563"\ + "168276697689459394902153960200361935311512317183678882" + mp.dps = 15 + assert lambertw(x).ae(ans) + mp.dps = 50 + assert lambertw(x).ae(ans) + mp.dps = 150 + assert lambertw(x).ae(ans) + +def test_meijerg(): + mp.dps = 15 + assert meijerg([[2,3],[1]],[[0.5,2],[3,4]], 2.5).ae(4.2181028074787439386) + assert meijerg([[],[1+j]],[[1],[1]], 3+4j).ae(271.46290321152464592 - 703.03330399954820169j) + assert meijerg([[0.25],[1]],[[0.5],[2]],0) == 0 + assert meijerg([[0],[]],[[0,0,'1/3','2/3'], []], '2/27').ae(2.2019391389653314120) + # Verify 1/z series being used + assert meijerg([[-3],[-0.5]], [[-1],[-2.5]], -0.5).ae(-1.338096165935754898687431) + assert meijerg([[1-(-1)],[1-(-2.5)]], [[1-(-3)],[1-(-0.5)]], -2.0).ae(-1.338096165935754898687431) + assert meijerg([[-3],[-0.5]], [[-1],[-2.5]], -1).ae(-(pi+4)/(4*pi)) + a = 2.5 + b = 1.25 + for z in [mpf(0.25), mpf(2)]: + x1 = hyp1f1(a,b,z) + x2 = gamma(b)/gamma(a)*meijerg([[1-a],[]],[[0],[1-b]],-z) + x3 = gamma(b)/gamma(a)*meijerg([[1-0],[1-(1-b)]],[[1-(1-a)],[]],-1/z) + assert x1.ae(x2) + assert x1.ae(x3) + +def test_appellf1(): + mp.dps = 15 + assert appellf1(2,-2,1,1,2,3).ae(-1.75) + assert appellf1(2,1,-2,1,2,3).ae(-8) + assert appellf1(2,1,-2,1,0.5,0.25).ae(1.5) + assert appellf1(-2,1,3,2,3,3).ae(19) + assert appellf1(1,2,3,4,0.5,0.125).ae( 1.53843285792549786518) + +def test_coulomb(): + # Note: most tests are doctests + # Test for a bug: + mp.dps = 15 + assert coulombg(mpc(-5,0),2,3).ae(20.087729487721430394) + +def test_hyper_param_accuracy(): + mp.dps = 15 + As = [n+1e-10 for n in range(-5,-1)] + Bs = [n+1e-10 for n in range(-12,-5)] + assert hyper(As,Bs,10).ae(-381757055858.652671927) + assert legenp(0.5, 100, 0.25).ae(-2.4124576567211311755e+144) + assert (hyp1f1(1000,1,-100)*10**24).ae(5.2589445437370169113) + assert (hyp2f1(10, -900, 10.5, 0.99)*10**24).ae(1.9185370579660768203) + assert (hyp2f1(1000,1.5,-3.5,-1.5)*10**385).ae(-2.7367529051334000764) + assert hyp2f1(-5, 10, 3, 0.5, zeroprec=500) == 0 + assert (hyp1f1(-10000, 1000, 100)*10**424).ae(-3.1046080515824859974) + assert (hyp2f1(1000,1.5,-3.5,-0.75,maxterms=100000)*10**231).ae(-4.0534790813913998643) + assert legenp(2, 3, 0.25) == 0 + pytest.raises(ValueError, lambda: hypercomb(lambda a: [([],[],[],[],[a],[-a],0.5)], [3])) + assert hypercomb(lambda a: [([],[],[],[],[a],[-a],0.5)], [3], infprec=200) == inf + assert meijerg([[],[]],[[0,0,0,0],[]],0.1).ae(1.5680822343832351418) + assert (besselk(400,400)*10**94).ae(1.4387057277018550583) + mp.dps = 5 + (hyp1f1(-5000.5, 1500, 100)*10**185).ae(8.5185229673381935522) + (hyp1f1(-5000, 1500, 100)*10**185).ae(9.1501213424563944311) + mp.dps = 15 + (hyp1f1(-5000.5, 1500, 100)*10**185).ae(8.5185229673381935522) + (hyp1f1(-5000, 1500, 100)*10**185).ae(9.1501213424563944311) + assert hyp0f1(fadd(-20,'1e-100',exact=True), 0.25).ae(1.85014429040102783e+49) + assert hyp0f1((-20*10**100+1, 10**100), 0.25).ae(1.85014429040102783e+49) + +def test_hypercomb_zero_pow(): + # check that 0^0 = 1 + assert hypercomb(lambda a: (([0],[a],[],[],[],[],0),), [0]) == 1 + assert meijerg([[-1.5],[]],[[0],[-0.75]],0).ae(1.4464090846320771425) + +def test_spherharm(): + mp.dps = 15 + t = 0.5; r = 0.25 + assert spherharm(0,0,t,r).ae(0.28209479177387814347) + assert spherharm(1,-1,t,r).ae(0.16048941205971996369 - 0.04097967481096344271j) + assert spherharm(1,0,t,r).ae(0.42878904414183579379) + assert spherharm(1,1,t,r).ae(-0.16048941205971996369 - 0.04097967481096344271j) + assert spherharm(2,-2,t,r).ae(0.077915886919031181734 - 0.042565643022253962264j) + assert spherharm(2,-1,t,r).ae(0.31493387233497459884 - 0.08041582001959297689j) + assert spherharm(2,0,t,r).ae(0.41330596756220761898) + assert spherharm(2,1,t,r).ae(-0.31493387233497459884 - 0.08041582001959297689j) + assert spherharm(2,2,t,r).ae(0.077915886919031181734 + 0.042565643022253962264j) + assert spherharm(3,-3,t,r).ae(0.033640236589690881646 - 0.031339125318637082197j) + assert spherharm(3,-2,t,r).ae(0.18091018743101461963 - 0.09883168583167010241j) + assert spherharm(3,-1,t,r).ae(0.42796713930907320351 - 0.10927795157064962317j) + assert spherharm(3,0,t,r).ae(0.27861659336351639787) + assert spherharm(3,1,t,r).ae(-0.42796713930907320351 - 0.10927795157064962317j) + assert spherharm(3,2,t,r).ae(0.18091018743101461963 + 0.09883168583167010241j) + assert spherharm(3,3,t,r).ae(-0.033640236589690881646 - 0.031339125318637082197j) + assert spherharm(0,-1,t,r) == 0 + assert spherharm(0,-2,t,r) == 0 + assert spherharm(0,1,t,r) == 0 + assert spherharm(0,2,t,r) == 0 + assert spherharm(1,2,t,r) == 0 + assert spherharm(1,3,t,r) == 0 + assert spherharm(1,-2,t,r) == 0 + assert spherharm(1,-3,t,r) == 0 + assert spherharm(2,3,t,r) == 0 + assert spherharm(2,4,t,r) == 0 + assert spherharm(2,-3,t,r) == 0 + assert spherharm(2,-4,t,r) == 0 + assert spherharm(3,4.5,0.5,0.25).ae(-22.831053442240790148 + 10.910526059510013757j) + assert spherharm(2+3j, 1-j, 1+j, 3+4j).ae(-2.6582752037810116935 - 1.0909214905642160211j) + assert spherharm(-6,2.5,t,r).ae(0.39383644983851448178 + 0.28414687085358299021j) + assert spherharm(-3.5, 3, 0.5, 0.25).ae(0.014516852987544698924 - 0.015582769591477628495j) + assert spherharm(-3, 3, 0.5, 0.25) == 0 + assert spherharm(-6, 3, 0.5, 0.25).ae(-0.16544349818782275459 - 0.15412657723253924562j) + assert spherharm(-6, 1.5, 0.5, 0.25).ae(0.032208193499767402477 + 0.012678000924063664921j) + assert spherharm(3,0,0,1).ae(0.74635266518023078283) + assert spherharm(3,-2,0,1) == 0 + assert spherharm(3,-2,1,1).ae(-0.16270707338254028971 - 0.35552144137546777097j) + +def test_qfunctions(): + mp.dps = 15 + assert qp(2,3,100).ae('2.7291482267247332183e2391') + +def test_issue_239(): + mp.prec = 150 + x = ldexp(2476979795053773,-52) + assert betainc(206, 385, 0, 0.55, 1).ae('0.99999999999999999999996570910644857895771110649954') + mp.dps = 15 + pytest.raises(ValueError, lambda: hyp2f1(-5,5,0.5,0.5)) + +# Extra stress testing for Bessel functions +# Reference zeros generated with the aid of scipy.special +# jn_zero, jnp_zero, yn_zero, ynp_zero + +V = 15 +M = 15 + +jn_small_zeros = \ +[[2.4048255576957728, + 5.5200781102863106, + 8.6537279129110122, + 11.791534439014282, + 14.930917708487786, + 18.071063967910923, + 21.211636629879259, + 24.352471530749303, + 27.493479132040255, + 30.634606468431975, + 33.775820213573569, + 36.917098353664044, + 40.058425764628239, + 43.19979171317673, + 46.341188371661814], + [3.8317059702075123, + 7.0155866698156188, + 10.173468135062722, + 13.323691936314223, + 16.470630050877633, + 19.615858510468242, + 22.760084380592772, + 25.903672087618383, + 29.046828534916855, + 32.189679910974404, + 35.332307550083865, + 38.474766234771615, + 41.617094212814451, + 44.759318997652822, + 47.901460887185447], + [5.1356223018406826, + 8.4172441403998649, + 11.619841172149059, + 14.795951782351261, + 17.959819494987826, + 21.116997053021846, + 24.270112313573103, + 27.420573549984557, + 30.569204495516397, + 33.7165195092227, + 36.86285651128381, + 40.008446733478192, + 43.153453778371463, + 46.297996677236919, + 49.442164110416873], + [6.3801618959239835, + 9.7610231299816697, + 13.015200721698434, + 16.223466160318768, + 19.409415226435012, + 22.582729593104442, + 25.748166699294978, + 28.908350780921758, + 32.064852407097709, + 35.218670738610115, + 38.370472434756944, + 41.520719670406776, + 44.669743116617253, + 47.817785691533302, + 50.965029906205183], + [7.5883424345038044, + 11.064709488501185, + 14.37253667161759, + 17.615966049804833, + 20.826932956962388, + 24.01901952477111, + 27.199087765981251, + 30.371007667117247, + 33.537137711819223, + 36.699001128744649, + 39.857627302180889, + 43.01373772335443, + 46.167853512924375, + 49.320360686390272, + 52.471551398458023], + [8.771483815959954, + 12.338604197466944, + 15.700174079711671, + 18.980133875179921, + 22.217799896561268, + 25.430341154222704, + 28.626618307291138, + 31.811716724047763, + 34.988781294559295, + 38.159868561967132, + 41.326383254047406, + 44.489319123219673, + 47.649399806697054, + 50.80716520300633, + 53.963026558378149], + [9.9361095242176849, + 13.589290170541217, + 17.003819667816014, + 20.320789213566506, + 23.58608443558139, + 26.820151983411405, + 30.033722386570469, + 33.233041762847123, + 36.422019668258457, + 39.603239416075404, + 42.778481613199507, + 45.949015998042603, + 49.11577372476426, + 52.279453903601052, + 55.440592068853149], + [11.086370019245084, + 14.821268727013171, + 18.287582832481726, + 21.641541019848401, + 24.934927887673022, + 28.191188459483199, + 31.42279419226558, + 34.637089352069324, + 37.838717382853611, + 41.030773691585537, + 44.21540850526126, + 47.394165755570512, + 50.568184679795566, + 53.738325371963291, + 56.905249991978781], + 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62.248409689597653, + 65.457606670836759], + [18.774423978290318, + 23.06220035979272, + 26.872520985976736, + 30.479680663499762, + 33.971869047372436, + 37.390118854896324, + 40.757072537673599, + 44.086572292170345, + 47.387688809191869, + 50.66667461073936, + 53.928009929563275, + 57.175005343085052, + 60.410169281219877, + 63.635442539153021, + 66.85235358587768]] + +@pytest.mark.slow +def test_bessel_zeros_extra(): + mp.dps = 15 + for v in range(V): + for m in range(1,M+1): + print(v, m, "of", V, M) + # Twice to test cache (if used) + assert besseljzero(v,m).ae(jn_small_zeros[v][m-1]) + assert besseljzero(v,m).ae(jn_small_zeros[v][m-1]) + assert besseljzero(v,m,1).ae(jnp_small_zeros[v][m-1]) + assert besseljzero(v,m,1).ae(jnp_small_zeros[v][m-1]) + assert besselyzero(v,m).ae(yn_small_zeros[v][m-1]) + assert besselyzero(v,m).ae(yn_small_zeros[v][m-1]) + assert besselyzero(v,m,1).ae(ynp_small_zeros[v][m-1]) + assert besselyzero(v,m,1).ae(ynp_small_zeros[v][m-1]) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_gammazeta.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_gammazeta.py new file mode 100644 index 0000000000000000000000000000000000000000..6a18a7964d746561dfd5f81177cd78ccc46d2a5d --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_gammazeta.py @@ -0,0 +1,698 @@ +from mpmath import * +from mpmath.libmp import round_up, from_float, mpf_zeta_int + +def test_zeta_int_bug(): + assert mpf_zeta_int(0, 10) == from_float(-0.5) + +def test_bernoulli(): + assert bernfrac(0) == (1,1) + assert bernfrac(1) == (-1,2) + assert bernfrac(2) == (1,6) + assert bernfrac(3) == (0,1) + assert bernfrac(4) == (-1,30) + assert bernfrac(5) == (0,1) + assert bernfrac(6) == (1,42) + assert bernfrac(8) == (-1,30) + assert bernfrac(10) == (5,66) + assert bernfrac(12) == (-691,2730) + assert bernfrac(18) == (43867,798) + p, q = bernfrac(228) + assert p % 10**10 == 164918161 + assert q == 625170 + p, q = bernfrac(1000) + assert p % 10**10 == 7950421099 + assert q == 342999030 + mp.dps = 15 + assert bernoulli(0) == 1 + assert bernoulli(1) == -0.5 + assert bernoulli(2).ae(1./6) + assert bernoulli(3) == 0 + assert bernoulli(4).ae(-1./30) + assert bernoulli(5) == 0 + assert bernoulli(6).ae(1./42) + assert str(bernoulli(10)) == '0.0757575757575758' + assert str(bernoulli(234)) == '7.62772793964344e+267' + assert str(bernoulli(10**5)) == '-5.82229431461335e+376755' + assert str(bernoulli(10**8+2)) == '1.19570355039953e+676752584' + + mp.dps = 50 + assert str(bernoulli(10)) == '0.075757575757575757575757575757575757575757575757576' + assert str(bernoulli(234)) == '7.6277279396434392486994969020496121553385863373331e+267' + assert str(bernoulli(10**5)) == '-5.8222943146133508236497045360612887555320691004308e+376755' + assert str(bernoulli(10**8+2)) == '1.1957035503995297272263047884604346914602088317782e+676752584' + + mp.dps = 1000 + assert bernoulli(10).ae(mpf(5)/66) + + mp.dps = 50000 + assert bernoulli(10).ae(mpf(5)/66) + + mp.dps = 15 + +def test_bernpoly_eulerpoly(): + mp.dps = 15 + assert bernpoly(0,-1).ae(1) + assert bernpoly(0,0).ae(1) + assert bernpoly(0,'1/2').ae(1) + assert bernpoly(0,'3/4').ae(1) + assert bernpoly(0,1).ae(1) + assert bernpoly(0,2).ae(1) + assert bernpoly(1,-1).ae('-3/2') + assert bernpoly(1,0).ae('-1/2') + assert bernpoly(1,'1/2').ae(0) + assert bernpoly(1,'3/4').ae('1/4') + assert bernpoly(1,1).ae('1/2') + assert bernpoly(1,2).ae('3/2') + assert bernpoly(2,-1).ae('13/6') + assert bernpoly(2,0).ae('1/6') + assert bernpoly(2,'1/2').ae('-1/12') + assert bernpoly(2,'3/4').ae('-1/48') + assert bernpoly(2,1).ae('1/6') + assert bernpoly(2,2).ae('13/6') + assert bernpoly(3,-1).ae(-3) + assert bernpoly(3,0).ae(0) + assert bernpoly(3,'1/2').ae(0) + assert bernpoly(3,'3/4').ae('-3/64') + assert bernpoly(3,1).ae(0) + assert bernpoly(3,2).ae(3) + assert bernpoly(4,-1).ae('119/30') + assert bernpoly(4,0).ae('-1/30') + assert bernpoly(4,'1/2').ae('7/240') + assert bernpoly(4,'3/4').ae('7/3840') + assert bernpoly(4,1).ae('-1/30') + assert bernpoly(4,2).ae('119/30') + assert bernpoly(5,-1).ae(-5) + assert bernpoly(5,0).ae(0) + assert bernpoly(5,'1/2').ae(0) + assert bernpoly(5,'3/4').ae('25/1024') + assert bernpoly(5,1).ae(0) + assert bernpoly(5,2).ae(5) + assert bernpoly(10,-1).ae('665/66') + assert bernpoly(10,0).ae('5/66') + assert bernpoly(10,'1/2').ae('-2555/33792') + assert bernpoly(10,'3/4').ae('-2555/34603008') + assert bernpoly(10,1).ae('5/66') + assert bernpoly(10,2).ae('665/66') + assert bernpoly(11,-1).ae(-11) + assert bernpoly(11,0).ae(0) + assert bernpoly(11,'1/2').ae(0) + assert bernpoly(11,'3/4').ae('-555731/4194304') + assert bernpoly(11,1).ae(0) + assert bernpoly(11,2).ae(11) + assert eulerpoly(0,-1).ae(1) + assert eulerpoly(0,0).ae(1) + assert eulerpoly(0,'1/2').ae(1) + assert eulerpoly(0,'3/4').ae(1) + assert eulerpoly(0,1).ae(1) + assert eulerpoly(0,2).ae(1) + assert eulerpoly(1,-1).ae('-3/2') + assert eulerpoly(1,0).ae('-1/2') + assert eulerpoly(1,'1/2').ae(0) + assert eulerpoly(1,'3/4').ae('1/4') + assert eulerpoly(1,1).ae('1/2') + assert eulerpoly(1,2).ae('3/2') + assert eulerpoly(2,-1).ae(2) + assert eulerpoly(2,0).ae(0) + assert eulerpoly(2,'1/2').ae('-1/4') + assert eulerpoly(2,'3/4').ae('-3/16') + assert eulerpoly(2,1).ae(0) + assert eulerpoly(2,2).ae(2) + assert eulerpoly(3,-1).ae('-9/4') + assert eulerpoly(3,0).ae('1/4') + assert eulerpoly(3,'1/2').ae(0) + assert eulerpoly(3,'3/4').ae('-11/64') + assert eulerpoly(3,1).ae('-1/4') + assert eulerpoly(3,2).ae('9/4') + assert eulerpoly(4,-1).ae(2) + assert eulerpoly(4,0).ae(0) + assert eulerpoly(4,'1/2').ae('5/16') + assert eulerpoly(4,'3/4').ae('57/256') + assert eulerpoly(4,1).ae(0) + assert eulerpoly(4,2).ae(2) + assert eulerpoly(5,-1).ae('-3/2') + assert eulerpoly(5,0).ae('-1/2') + assert eulerpoly(5,'1/2').ae(0) + assert eulerpoly(5,'3/4').ae('361/1024') + assert eulerpoly(5,1).ae('1/2') + assert eulerpoly(5,2).ae('3/2') + assert eulerpoly(10,-1).ae(2) + assert eulerpoly(10,0).ae(0) + assert eulerpoly(10,'1/2').ae('-50521/1024') + assert eulerpoly(10,'3/4').ae('-36581523/1048576') + assert eulerpoly(10,1).ae(0) + assert eulerpoly(10,2).ae(2) + assert eulerpoly(11,-1).ae('-699/4') + assert eulerpoly(11,0).ae('691/4') + assert eulerpoly(11,'1/2').ae(0) + assert eulerpoly(11,'3/4').ae('-512343611/4194304') + assert eulerpoly(11,1).ae('-691/4') + assert eulerpoly(11,2).ae('699/4') + # Potential accuracy issues + assert bernpoly(10000,10000).ae('5.8196915936323387117e+39999') + assert bernpoly(200,17.5).ae(3.8048418524583064909e244) + assert eulerpoly(200,17.5).ae(-3.7309911582655785929e275) + +def test_gamma(): + mp.dps = 15 + assert gamma(0.25).ae(3.6256099082219083119) + assert gamma(0.0001).ae(9999.4228832316241908) + assert gamma(300).ae('1.0201917073881354535e612') + assert gamma(-0.5).ae(-3.5449077018110320546) + assert gamma(-7.43).ae(0.00026524416464197007186) + #assert gamma(Rational(1,2)) == gamma(0.5) + #assert gamma(Rational(-7,3)).ae(gamma(mpf(-7)/3)) + assert gamma(1+1j).ae(0.49801566811835604271 - 0.15494982830181068512j) + assert gamma(-1+0.01j).ae(-0.422733904013474115 + 99.985883082635367436j) + assert gamma(20+30j).ae(-1453876687.5534810 + 1163777777.8031573j) + # Should always give exact factorials when they can + # be represented as mpfs under the current working precision + fact = 1 + for i in range(1, 18): + assert gamma(i) == fact + fact *= i + for dps in [170, 600]: + fact = 1 + mp.dps = dps + for i in range(1, 105): + assert gamma(i) == fact + fact *= i + mp.dps = 100 + assert gamma(0.5).ae(sqrt(pi)) + mp.dps = 15 + assert factorial(0) == fac(0) == 1 + assert factorial(3) == 6 + assert isnan(gamma(nan)) + assert gamma(1100).ae('4.8579168073569433667e2866') + assert rgamma(0) == 0 + assert rgamma(-1) == 0 + assert rgamma(2) == 1.0 + assert rgamma(3) == 0.5 + assert loggamma(2+8j).ae(-8.5205176753667636926 + 10.8569497125597429366j) + assert loggamma('1e10000').ae('2.302485092994045684017991e10004') + assert loggamma('1e10000j').ae(mpc('-1.570796326794896619231322e10000','2.302485092994045684017991e10004')) + +def test_fac2(): + mp.dps = 15 + assert [fac2(n) for n in range(10)] == [1,1,2,3,8,15,48,105,384,945] + assert fac2(-5).ae(1./3) + assert fac2(-11).ae(-1./945) + assert fac2(50).ae(5.20469842636666623e32) + assert fac2(0.5+0.75j).ae(0.81546769394688069176-0.34901016085573266889j) + assert fac2(inf) == inf + assert isnan(fac2(-inf)) + +def test_gamma_quotients(): + mp.dps = 15 + h = 1e-8 + ep = 1e-4 + G = gamma + assert gammaprod([-1],[-3,-4]) == 0 + assert gammaprod([-1,0],[-5]) == inf + assert abs(gammaprod([-1],[-2]) - G(-1+h)/G(-2+h)) < 1e-4 + assert abs(gammaprod([-4,-3],[-2,0]) - G(-4+h)*G(-3+h)/G(-2+h)/G(0+h)) < 1e-4 + assert rf(3,0) == 1 + assert rf(2.5,1) == 2.5 + assert rf(-5,2) == 20 + assert rf(j,j).ae(gamma(2*j)/gamma(j)) + assert rf('-255.5815971722918','-0.5119253100282322').ae('-0.1952720278805729485') # issue 421 + assert ff(-2,0) == 1 + assert ff(-2,1) == -2 + assert ff(4,3) == 24 + assert ff(3,4) == 0 + assert binomial(0,0) == 1 + assert binomial(1,0) == 1 + assert binomial(0,-1) == 0 + assert binomial(3,2) == 3 + assert binomial(5,2) == 10 + assert binomial(5,3) == 10 + assert binomial(5,5) == 1 + assert binomial(-1,0) == 1 + assert binomial(-2,-4) == 3 + assert binomial(4.5, 1.5) == 6.5625 + assert binomial(1100,1) == 1100 + assert binomial(1100,2) == 604450 + assert beta(1,1) == 1 + assert beta(0,0) == inf + assert beta(3,0) == inf + assert beta(-1,-1) == inf + assert beta(1.5,1).ae(2/3.) + assert beta(1.5,2.5).ae(pi/16) + assert (10**15*beta(10,100)).ae(2.3455339739604649879) + assert beta(inf,inf) == 0 + assert isnan(beta(-inf,inf)) + assert isnan(beta(-3,inf)) + assert isnan(beta(0,inf)) + assert beta(inf,0.5) == beta(0.5,inf) == 0 + assert beta(inf,-1.5) == inf + assert beta(inf,-0.5) == -inf + assert beta(1+2j,-1-j/2).ae(1.16396542451069943086+0.08511695947832914640j) + assert beta(-0.5,0.5) == 0 + assert beta(-3,3).ae(-1/3.) + assert beta('-255.5815971722918','-0.5119253100282322').ae('18.157330562703710339') # issue 421 + +def test_zeta(): + mp.dps = 15 + assert zeta(2).ae(pi**2 / 6) + assert zeta(2.0).ae(pi**2 / 6) + assert zeta(mpc(2)).ae(pi**2 / 6) + assert zeta(100).ae(1) + assert zeta(0).ae(-0.5) + assert zeta(0.5).ae(-1.46035450880958681) + assert zeta(-1).ae(-mpf(1)/12) + assert zeta(-2) == 0 + assert zeta(-3).ae(mpf(1)/120) + assert zeta(-4) == 0 + assert zeta(-100) == 0 + assert isnan(zeta(nan)) + assert zeta(1e-30).ae(-0.5) + assert zeta(-1e-30).ae(-0.5) + # Zeros in the critical strip + assert zeta(mpc(0.5, 14.1347251417346937904)).ae(0) + assert zeta(mpc(0.5, 21.0220396387715549926)).ae(0) + assert zeta(mpc(0.5, 25.0108575801456887632)).ae(0) + assert zeta(mpc(1e-30,1e-40)).ae(-0.5) + assert zeta(mpc(-1e-30,1e-40)).ae(-0.5) + mp.dps = 50 + im = '236.5242296658162058024755079556629786895294952121891237' + assert zeta(mpc(0.5, im)).ae(0, 1e-46) + mp.dps = 15 + # Complex reflection formula + assert (zeta(-60+3j) / 10**34).ae(8.6270183987866146+15.337398548226238j) + # issue #358 + assert zeta(0,0.5) == 0 + assert zeta(0,0) == 0.5 + assert zeta(0,0.5,1).ae(-0.34657359027997265) + # see issue #390 + assert zeta(-1.5,0.5j).ae(-0.13671400162512768475 + 0.11411333638426559139j) + +def test_altzeta(): + mp.dps = 15 + assert altzeta(-2) == 0 + assert altzeta(-4) == 0 + assert altzeta(-100) == 0 + assert altzeta(0) == 0.5 + assert altzeta(-1) == 0.25 + assert altzeta(-3) == -0.125 + assert altzeta(-5) == 0.25 + assert altzeta(-21) == 1180529130.25 + assert altzeta(1).ae(log(2)) + assert altzeta(2).ae(pi**2/12) + assert altzeta(10).ae(73*pi**10/6842880) + assert altzeta(50) < 1 + assert altzeta(60, rounding='d') < 1 + assert altzeta(60, rounding='u') == 1 + assert altzeta(10000, rounding='d') < 1 + assert altzeta(10000, rounding='u') == 1 + assert altzeta(3+0j) == altzeta(3) + s = 3+4j + assert altzeta(s).ae((1-2**(1-s))*zeta(s)) + s = -3+4j + assert altzeta(s).ae((1-2**(1-s))*zeta(s)) + assert altzeta(-100.5).ae(4.58595480083585913e+108) + assert altzeta(1.3).ae(0.73821404216623045) + assert altzeta(1e-30).ae(0.5) + assert altzeta(-1e-30).ae(0.5) + assert altzeta(mpc(1e-30,1e-40)).ae(0.5) + assert altzeta(mpc(-1e-30,1e-40)).ae(0.5) + +def test_zeta_huge(): + mp.dps = 15 + assert zeta(inf) == 1 + mp.dps = 50 + assert zeta(100).ae('1.0000000000000000000000000000007888609052210118073522') + assert zeta(40*pi).ae('1.0000000000000000000000000000000000000148407238666182') + mp.dps = 10000 + v = zeta(33000) + mp.dps = 15 + assert str(v-1) == '1.02363019598118e-9934' + assert zeta(pi*1000, rounding=round_up) > 1 + assert zeta(3000, rounding=round_up) > 1 + assert zeta(pi*1000) == 1 + assert zeta(3000) == 1 + +def test_zeta_negative(): + mp.dps = 150 + a = -pi*10**40 + mp.dps = 15 + assert str(zeta(a)) == '2.55880492708712e+1233536161668617575553892558646631323374078' + mp.dps = 50 + assert str(zeta(a)) == '2.5588049270871154960875033337384432038436330847333e+1233536161668617575553892558646631323374078' + mp.dps = 15 + +def test_polygamma(): + mp.dps = 15 + psi0 = lambda z: psi(0,z) + psi1 = lambda z: psi(1,z) + assert psi0(3) == psi(0,3) == digamma(3) + #assert psi2(3) == psi(2,3) == tetragamma(3) + #assert psi3(3) == psi(3,3) == pentagamma(3) + assert psi0(pi).ae(0.97721330794200673) + assert psi0(-pi).ae(7.8859523853854902) + assert psi0(-pi+1).ae(7.5676424992016996) + assert psi0(pi+j).ae(1.04224048313859376 + 0.35853686544063749j) + assert psi0(-pi-j).ae(1.3404026194821986 - 2.8824392476809402j) + assert findroot(psi0, 1).ae(1.4616321449683622) + assert psi0(1e-10).ae(-10000000000.57722) + assert psi0(1e-40).ae(-1.000000000000000e+40) + assert psi0(1e-10+1e-10j).ae(-5000000000.577215 + 5000000000.000000j) + assert psi0(1e-40+1e-40j).ae(-5.000000000000000e+39 + 5.000000000000000e+39j) + assert psi0(inf) == inf + assert psi1(inf) == 0 + assert psi(2,inf) == 0 + assert psi1(pi).ae(0.37424376965420049) + assert psi1(-pi).ae(53.030438740085385) + assert psi1(pi+j).ae(0.32935710377142464 - 0.12222163911221135j) + assert psi1(-pi-j).ae(-0.30065008356019703 + 0.01149892486928227j) + assert (10**6*psi(4,1+10*pi*j)).ae(-6.1491803479004446 - 0.3921316371664063j) + assert psi0(1+10*pi*j).ae(3.4473994217222650 + 1.5548808324857071j) + assert isnan(psi0(nan)) + assert isnan(psi0(-inf)) + assert psi0(-100.5).ae(4.615124601338064) + assert psi0(3+0j).ae(psi0(3)) + assert psi0(-100+3j).ae(4.6106071768714086321+3.1117510556817394626j) + assert isnan(psi(2,mpc(0,inf))) + assert isnan(psi(2,mpc(0,nan))) + assert isnan(psi(2,mpc(0,-inf))) + assert isnan(psi(2,mpc(1,inf))) + assert isnan(psi(2,mpc(1,nan))) + assert isnan(psi(2,mpc(1,-inf))) + assert isnan(psi(2,mpc(inf,inf))) + assert isnan(psi(2,mpc(nan,nan))) + assert isnan(psi(2,mpc(-inf,-inf))) + mp.dps = 30 + # issue #534 + assert digamma(-0.75+1j).ae(mpc('0.46317279488182026118963809283042317', '2.4821070143037957102007677817351115')) + mp.dps = 15 + +def test_polygamma_high_prec(): + mp.dps = 100 + assert str(psi(0,pi)) == "0.9772133079420067332920694864061823436408346099943256380095232865318105924777141317302075654362928734" + assert str(psi(10,pi)) == "-12.98876181434889529310283769414222588307175962213707170773803550518307617769657562747174101900659238" + +def test_polygamma_identities(): + mp.dps = 15 + psi0 = lambda z: psi(0,z) + psi1 = lambda z: psi(1,z) + psi2 = lambda z: psi(2,z) + assert psi0(0.5).ae(-euler-2*log(2)) + assert psi0(1).ae(-euler) + assert psi1(0.5).ae(0.5*pi**2) + assert psi1(1).ae(pi**2/6) + assert psi1(0.25).ae(pi**2 + 8*catalan) + assert psi2(1).ae(-2*apery) + mp.dps = 20 + u = -182*apery+4*sqrt(3)*pi**3 + mp.dps = 15 + assert psi(2,5/6.).ae(u) + assert psi(3,0.5).ae(pi**4) + +def test_foxtrot_identity(): + # A test of the complex digamma function. + # See http://mathworld.wolfram.com/FoxTrotSeries.html and + # http://mathworld.wolfram.com/DigammaFunction.html + psi0 = lambda z: psi(0,z) + mp.dps = 50 + a = (-1)**fraction(1,3) + b = (-1)**fraction(2,3) + x = -psi0(0.5*a) - psi0(-0.5*b) + psi0(0.5*(1+a)) + psi0(0.5*(1-b)) + y = 2*pi*sech(0.5*sqrt(3)*pi) + assert x.ae(y) + mp.dps = 15 + +def test_polygamma_high_order(): + mp.dps = 100 + assert str(psi(50, pi)) == "-1344100348958402765749252447726432491812.641985273160531055707095989227897753035823152397679626136483" + assert str(psi(50, pi + 14*e)) == "-0.00000000000000000189793739550804321623512073101895801993019919886375952881053090844591920308111549337295143780341396" + assert str(psi(50, pi + 14*e*j)) == ("(-0.0000000000000000522516941152169248975225472155683565752375889510631513244785" + "9377385233700094871256507814151956624433 - 0.00000000000000001813157041407010184" + "702414110218205348527862196327980417757665282244728963891298080199341480881811613j)") + mp.dps = 15 + assert str(psi(50, pi)) == "-1.34410034895841e+39" + assert str(psi(50, pi + 14*e)) == "-1.89793739550804e-18" + assert str(psi(50, pi + 14*e*j)) == "(-5.2251694115217e-17 - 1.81315704140701e-17j)" + +def test_harmonic(): + mp.dps = 15 + assert harmonic(0) == 0 + assert harmonic(1) == 1 + assert harmonic(2) == 1.5 + assert harmonic(3).ae(1. + 1./2 + 1./3) + assert harmonic(10**10).ae(23.603066594891989701) + assert harmonic(10**1000).ae(2303.162308658947) + assert harmonic(0.5).ae(2-2*log(2)) + assert harmonic(inf) == inf + assert harmonic(2+0j) == 1.5+0j + assert harmonic(1+2j).ae(1.4918071802755104+0.92080728264223022j) + +def test_gamma_huge_1(): + mp.dps = 500 + x = mpf(10**10) / 7 + mp.dps = 15 + assert str(gamma(x)) == "6.26075321389519e+12458010678" + mp.dps = 50 + assert str(gamma(x)) == "6.2607532138951929201303779291707455874010420783933e+12458010678" + mp.dps = 15 + +def test_gamma_huge_2(): + mp.dps = 500 + x = mpf(10**100) / 19 + mp.dps = 15 + assert str(gamma(x)) == (\ + "1.82341134776679e+5172997469323364168990133558175077136829182824042201886051511" + "9656908623426021308685461258226190190661") + mp.dps = 50 + assert str(gamma(x)) == (\ + "1.82341134776678875374414910350027596939980412984e+5172997469323364168990133558" + "1750771368291828240422018860515119656908623426021308685461258226190190661") + +def test_gamma_huge_3(): + mp.dps = 500 + x = 10**80 // 3 + 10**70*j / 7 + mp.dps = 15 + y = gamma(x) + assert str(y.real) == (\ + "-6.82925203918106e+2636286142112569524501781477865238132302397236429627932441916" + "056964386399485392600") + assert str(y.imag) == (\ + "8.54647143678418e+26362861421125695245017814778652381323023972364296279324419160" + "56964386399485392600") + mp.dps = 50 + y = gamma(x) + assert str(y.real) == (\ + "-6.8292520391810548460682736226799637356016538421817e+26362861421125695245017814" + "77865238132302397236429627932441916056964386399485392600") + assert str(y.imag) == (\ + "8.5464714367841748507479306948130687511711420234015e+263628614211256952450178147" + "7865238132302397236429627932441916056964386399485392600") + +def test_gamma_huge_4(): + x = 3200+11500j + mp.dps = 15 + assert str(gamma(x)) == \ + "(8.95783268539713e+5164 - 1.94678798329735e+5164j)" + mp.dps = 50 + assert str(gamma(x)) == (\ + "(8.9578326853971339570292952697675570822206567327092e+5164" + " - 1.9467879832973509568895402139429643650329524144794e+51" + "64j)") + mp.dps = 15 + +def test_gamma_huge_5(): + mp.dps = 500 + x = 10**60 * j / 3 + mp.dps = 15 + y = gamma(x) + assert str(y.real) == "-3.27753899634941e-227396058973640224580963937571892628368354580620654233316839" + assert str(y.imag) == "-7.1519888950416e-227396058973640224580963937571892628368354580620654233316841" + mp.dps = 50 + y = gamma(x) + assert str(y.real) == (\ + "-3.2775389963494132168950056995974690946983219123935e-22739605897364022458096393" + "7571892628368354580620654233316839") + assert str(y.imag) == (\ + "-7.1519888950415979749736749222530209713136588885897e-22739605897364022458096393" + "7571892628368354580620654233316841") + mp.dps = 15 + +def test_gamma_huge_7(): + mp.dps = 100 + a = 3 + j/mpf(10)**1000 + mp.dps = 15 + y = gamma(a) + assert str(y.real) == "2.0" + # wrong + #assert str(y.imag) == "2.16735365342606e-1000" + assert str(y.imag) == "1.84556867019693e-1000" + mp.dps = 50 + y = gamma(a) + assert str(y.real) == "2.0" + #assert str(y.imag) == "2.1673536534260596065418805612488708028522563689298e-1000" + assert str(y.imag) == "1.8455686701969342787869758198351951379156813281202e-1000" + +def test_stieltjes(): + mp.dps = 15 + assert stieltjes(0).ae(+euler) + mp.dps = 25 + assert stieltjes(1).ae('-0.07281584548367672486058637587') + assert stieltjes(2).ae('-0.009690363192872318484530386035') + assert stieltjes(3).ae('0.002053834420303345866160046543') + assert stieltjes(4).ae('0.002325370065467300057468170178') + mp.dps = 15 + assert stieltjes(1).ae(-0.07281584548367672486058637587) + assert stieltjes(2).ae(-0.009690363192872318484530386035) + assert stieltjes(3).ae(0.002053834420303345866160046543) + assert stieltjes(4).ae(0.0023253700654673000574681701775) + +def test_barnesg(): + mp.dps = 15 + assert barnesg(0) == barnesg(-1) == 0 + assert [superfac(i) for i in range(8)] == [1, 1, 2, 12, 288, 34560, 24883200, 125411328000] + assert str(superfac(1000)) == '3.24570818422368e+1177245' + assert isnan(barnesg(nan)) + assert isnan(superfac(nan)) + assert isnan(hyperfac(nan)) + assert barnesg(inf) == inf + assert superfac(inf) == inf + assert hyperfac(inf) == inf + assert isnan(superfac(-inf)) + assert barnesg(0.7).ae(0.8068722730141471) + assert barnesg(2+3j).ae(-0.17810213864082169+0.04504542715447838j) + assert [hyperfac(n) for n in range(7)] == [1, 1, 4, 108, 27648, 86400000, 4031078400000] + assert [hyperfac(n) for n in range(0,-7,-1)] == [1,1,-1,-4,108,27648,-86400000] + a = barnesg(-3+0j) + assert a == 0 and isinstance(a, mpc) + a = hyperfac(-3+0j) + assert a == -4 and isinstance(a, mpc) + +def test_polylog(): + mp.dps = 15 + zs = [mpmathify(z) for z in [0, 0.5, 0.99, 4, -0.5, -4, 1j, 3+4j]] + for z in zs: assert polylog(1, z).ae(-log(1-z)) + for z in zs: assert polylog(0, z).ae(z/(1-z)) + for z in zs: assert polylog(-1, z).ae(z/(1-z)**2) + for z in zs: assert polylog(-2, z).ae(z*(1+z)/(1-z)**3) + for z in zs: assert polylog(-3, z).ae(z*(1+4*z+z**2)/(1-z)**4) + assert polylog(3, 7).ae(5.3192579921456754382-5.9479244480803301023j) + assert polylog(3, -7).ae(-4.5693548977219423182) + assert polylog(2, 0.9).ae(1.2997147230049587252) + assert polylog(2, -0.9).ae(-0.75216317921726162037) + assert polylog(2, 0.9j).ae(-0.17177943786580149299+0.83598828572550503226j) + assert polylog(2, 1.1).ae(1.9619991013055685931-0.2994257606855892575j) + assert polylog(2, -1.1).ae(-0.89083809026228260587) + assert polylog(2, 1.1*sqrt(j)).ae(0.58841571107611387722+1.09962542118827026011j) + assert polylog(-2, 0.9).ae(1710) + assert polylog(-2, -0.9).ae(-90/6859.) + assert polylog(3, 0.9).ae(1.0496589501864398696) + assert polylog(-3, 0.9).ae(48690) + assert polylog(-3, -4).ae(-0.0064) + assert polylog(0.5+j/3, 0.5+j/2).ae(0.31739144796565650535 + 0.99255390416556261437j) + assert polylog(3+4j,1).ae(zeta(3+4j)) + assert polylog(3+4j,-1).ae(-altzeta(3+4j)) + # issue 390 + assert polylog(1.5, -48.910886523731889).ae(-6.272992229311817) + assert polylog(1.5, 200).ae(-8.349608319033686529 - 8.159694826434266042j) + assert polylog(-2+0j, -2).ae(mpf(1)/13.5) + assert polylog(-2+0j, 1.25).ae(-180) + +def test_bell_polyexp(): + mp.dps = 15 + # TODO: more tests for polyexp + assert (polyexp(0,1e-10)*10**10).ae(1.00000000005) + assert (polyexp(1,1e-10)*10**10).ae(1.0000000001) + assert polyexp(5,3j).ae(-607.7044517476176454+519.962786482001476087j) + assert polyexp(-1,3.5).ae(12.09537536175543444) + # bell(0,x) = 1 + assert bell(0,0) == 1 + assert bell(0,1) == 1 + assert bell(0,2) == 1 + assert bell(0,inf) == 1 + assert bell(0,-inf) == 1 + assert isnan(bell(0,nan)) + # bell(1,x) = x + assert bell(1,4) == 4 + assert bell(1,0) == 0 + assert bell(1,inf) == inf + assert bell(1,-inf) == -inf + assert isnan(bell(1,nan)) + # bell(2,x) = x*(1+x) + assert bell(2,-1) == 0 + assert bell(2,0) == 0 + # large orders / arguments + assert bell(10) == 115975 + assert bell(10,1) == 115975 + assert bell(10, -8) == 11054008 + assert bell(5,-50) == -253087550 + assert bell(50,-50).ae('3.4746902914629720259e74') + mp.dps = 80 + assert bell(50,-50) == 347469029146297202586097646631767227177164818163463279814268368579055777450 + assert bell(40,50) == 5575520134721105844739265207408344706846955281965031698187656176321717550 + assert bell(74) == 5006908024247925379707076470957722220463116781409659160159536981161298714301202 + mp.dps = 15 + assert bell(10,20j) == 7504528595600+15649605360020j + # continuity of the generalization + assert bell(0.5,0).ae(sinc(pi*0.5)) + +def test_primezeta(): + mp.dps = 15 + assert primezeta(0.9).ae(1.8388316154446882243 + 3.1415926535897932385j) + assert primezeta(4).ae(0.076993139764246844943) + assert primezeta(1) == inf + assert primezeta(inf) == 0 + assert isnan(primezeta(nan)) + +def test_rs_zeta(): + mp.dps = 15 + assert zeta(0.5+100000j).ae(1.0730320148577531321 + 5.7808485443635039843j) + assert zeta(0.75+100000j).ae(1.837852337251873704 + 1.9988492668661145358j) + assert zeta(0.5+1000000j, derivative=3).ae(1647.7744105852674733 - 1423.1270943036622097j) + assert zeta(1+1000000j, derivative=3).ae(3.4085866124523582894 - 18.179184721525947301j) + assert zeta(1+1000000j, derivative=1).ae(-0.10423479366985452134 - 0.74728992803359056244j) + assert zeta(0.5-1000000j, derivative=1).ae(11.636804066002521459 + 17.127254072212996004j) + # Additional sanity tests using fp arithmetic. + # Some more high-precision tests are found in the docstrings + def ae(x, y, tol=1e-6): + return abs(x-y) < tol*abs(y) + assert ae(fp.zeta(0.5-100000j), 1.0730320148577531321 - 5.7808485443635039843j) + assert ae(fp.zeta(0.75-100000j), 1.837852337251873704 - 1.9988492668661145358j) + assert ae(fp.zeta(0.5+1e6j), 0.076089069738227100006 + 2.8051021010192989554j) + assert ae(fp.zeta(0.5+1e6j, derivative=1), 11.636804066002521459 - 17.127254072212996004j) + assert ae(fp.zeta(1+1e6j), 0.94738726251047891048 + 0.59421999312091832833j) + assert ae(fp.zeta(1+1e6j, derivative=1), -0.10423479366985452134 - 0.74728992803359056244j) + assert ae(fp.zeta(0.5+100000j, derivative=1), 10.766962036817482375 - 30.92705282105996714j) + assert ae(fp.zeta(0.5+100000j, derivative=2), -119.40515625740538429 + 217.14780631141830251j) + assert ae(fp.zeta(0.5+100000j, derivative=3), 1129.7550282628460881 - 1685.4736895169690346j) + assert ae(fp.zeta(0.5+100000j, derivative=4), -10407.160819314958615 + 13777.786698628045085j) + assert ae(fp.zeta(0.75+100000j, derivative=1), -0.41742276699594321475 - 6.4453816275049955949j) + assert ae(fp.zeta(0.75+100000j, derivative=2), -9.214314279161977266 + 35.07290795337967899j) + assert ae(fp.zeta(0.75+100000j, derivative=3), 110.61331857820103469 - 236.87847130518129926j) + assert ae(fp.zeta(0.75+100000j, derivative=4), -1054.334275898559401 + 1769.9177890161596383j) + +def test_siegelz(): + mp.dps = 15 + assert siegelz(100000).ae(5.87959246868176504171) + assert siegelz(100000, derivative=2).ae(-54.1172711010126452832) + assert siegelz(100000, derivative=3).ae(-278.930831343966552538) + assert siegelz(100000+j,derivative=1).ae(678.214511857070283307-379.742160779916375413j) + + + +def test_zeta_near_1(): + # Test for a former bug in mpf_zeta and mpc_zeta + mp.dps = 15 + s1 = fadd(1, '1e-10', exact=True) + s2 = fadd(1, '-1e-10', exact=True) + s3 = fadd(1, '1e-10j', exact=True) + assert zeta(s1).ae(1.000000000057721566490881444e10) + assert zeta(s2).ae(-9.99999999942278433510574872e9) + z = zeta(s3) + assert z.real.ae(0.57721566490153286060) + assert z.imag.ae(-9.9999999999999999999927184e9) + mp.dps = 30 + s1 = fadd(1, '1e-50', exact=True) + s2 = fadd(1, '-1e-50', exact=True) + s3 = fadd(1, '1e-50j', exact=True) + assert zeta(s1).ae('1e50') + assert zeta(s2).ae('-1e50') + z = zeta(s3) + assert z.real.ae('0.57721566490153286060651209008240243104215933593992') + assert z.imag.ae('-1e50') diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_hp.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_hp.py new file mode 100644 index 0000000000000000000000000000000000000000..9eba0af798f64ac3f8d464e2d3bf231567a48c9b --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_hp.py @@ -0,0 +1,291 @@ +""" +Check that the output from irrational functions is accurate for +high-precision input, from 5 to 200 digits. The reference values were +verified with Mathematica. +""" + +import time +from mpmath import * + +precs = [5, 15, 28, 35, 57, 80, 100, 150, 200] + +# sqrt(3) + pi/2 +a = \ +"3.302847134363773912758768033145623809041389953497933538543279275605"\ +"841220051904536395163599428307109666700184672047856353516867399774243594"\ +"67433521615861420725323528325327484262075464241255915238845599752675" + +# e + 1/euler**2 +b = \ +"5.719681166601007617111261398629939965860873957353320734275716220045750"\ +"31474116300529519620938123730851145473473708966080207482581266469342214"\ +"824842256999042984813905047895479210702109260221361437411947323431" + +# sqrt(a) +sqrt_a = \ +"1.817373691447021556327498239690365674922395036495564333152483422755"\ +"144321726165582817927383239308173567921345318453306994746434073691275094"\ +"484777905906961689902608644112196725896908619756404253109722911487" + +# sqrt(a+b*i).real +sqrt_abi_real = \ +"2.225720098415113027729407777066107959851146508557282707197601407276"\ +"89160998185797504198062911768240808839104987021515555650875977724230130"\ +"3584116233925658621288393930286871862273400475179312570274423840384" + +# sqrt(a+b*i).imag +sqrt_abi_imag = \ +"1.2849057639084690902371581529110949983261182430040898147672052833653668"\ +"0629534491275114877090834296831373498336559849050755848611854282001250"\ +"1924311019152914021365263161630765255610885489295778894976075186" + +# log(a) +log_a = \ +"1.194784864491089550288313512105715261520511949410072046160598707069"\ +"4336653155025770546309137440687056366757650909754708302115204338077595203"\ +"83005773986664564927027147084436553262269459110211221152925732612" + +# log(a+b*i).real +log_abi_real = \ +"1.8877985921697018111624077550443297276844736840853590212962006811663"\ +"04949387789489704203167470111267581371396245317618589339274243008242708"\ +"014251531496104028712866224020066439049377679709216784954509456421" + +# log(a+b*i).imag +log_abi_imag = \ +"1.0471204952840802663567714297078763189256357109769672185219334169734948"\ +"4265809854092437285294686651806426649541504240470168212723133326542181"\ +"8300136462287639956713914482701017346851009323172531601894918640" + +# exp(a) +exp_a = \ +"27.18994224087168661137253262213293847994194869430518354305430976149"\ +"382792035050358791398632888885200049857986258414049540376323785711941636"\ +"100358982497583832083513086941635049329804685212200507288797531143" + +# exp(a+b*i).real +exp_abi_real = \ +"22.98606617170543596386921087657586890620262522816912505151109385026"\ +"40160179326569526152851983847133513990281518417211964710397233157168852"\ +"4963130831190142571659948419307628119985383887599493378056639916701" + +# exp(a+b*i).imag +exp_abi_imag = \ +"-14.523557450291489727214750571590272774669907424478129280902375851196283"\ +"3377162379031724734050088565710975758824441845278120105728824497308303"\ +"6065619788140201636218705414429933685889542661364184694108251449" + +# a**b +pow_a_b = \ +"928.7025342285568142947391505837660251004990092821305668257284426997"\ +"361966028275685583421197860603126498884545336686124793155581311527995550"\ +"580229264427202446131740932666832138634013168125809402143796691154" + +# (a**(a+b*i)).real +pow_a_abi_real = \ +"44.09156071394489511956058111704382592976814280267142206420038656267"\ +"67707916510652790502399193109819563864568986234654864462095231138500505"\ +"8197456514795059492120303477512711977915544927440682508821426093455" + +# (a**(a+b*i)).imag +pow_a_abi_imag = \ +"27.069371511573224750478105146737852141664955461266218367212527612279886"\ +"9322304536553254659049205414427707675802193810711302947536332040474573"\ +"8166261217563960235014674118610092944307893857862518964990092301" + +# ((a+b*i)**(a+b*i)).real +pow_abi_abi_real = \ +"-0.15171310677859590091001057734676423076527145052787388589334350524"\ +"8084195882019497779202452975350579073716811284169068082670778986235179"\ +"0813026562962084477640470612184016755250592698408112493759742219150452"\ + +# ((a+b*i)**(a+b*i)).imag +pow_abi_abi_imag = \ +"1.2697592504953448936553147870155987153192995316950583150964099070426"\ +"4736837932577176947632535475040521749162383347758827307504526525647759"\ +"97547638617201824468382194146854367480471892602963428122896045019902" + +# sin(a) +sin_a = \ +"-0.16055653857469062740274792907968048154164433772938156243509084009"\ +"38437090841460493108570147191289893388608611542655654723437248152535114"\ +"528368009465836614227575701220612124204622383149391870684288862269631" + +# sin(1000*a) +sin_1000a = \ +"-0.85897040577443833776358106803777589664322997794126153477060795801"\ +"09151695416961724733492511852267067419573754315098042850381158563024337"\ +"216458577140500488715469780315833217177634490142748614625281171216863" + +# sin(a+b*i) +sin_abi_real = \ +"-24.4696999681556977743346798696005278716053366404081910969773939630"\ +"7149215135459794473448465734589287491880563183624997435193637389884206"\ +"02151395451271809790360963144464736839412254746645151672423256977064" + +sin_abi_imag = \ +"-150.42505378241784671801405965872972765595073690984080160750785565810981"\ +"8314482499135443827055399655645954830931316357243750839088113122816583"\ +"7169201254329464271121058839499197583056427233866320456505060735" + +# cos +cos_a = \ +"-0.98702664499035378399332439243967038895709261414476495730788864004"\ +"05406821549361039745258003422386169330787395654908532996287293003581554"\ +"257037193284199198069707141161341820684198547572456183525659969145501" + +cos_1000a = \ +"-0.51202523570982001856195696460663971099692261342827540426136215533"\ +"52686662667660613179619804463250686852463876088694806607652218586060613"\ +"951310588158830695735537073667299449753951774916401887657320950496820" + +# tan +tan_a = \ +"0.162666873675188117341401059858835168007137819495998960250142156848"\ +"639654718809412181543343168174807985559916643549174530459883826451064966"\ +"7996119428949951351938178809444268785629011625179962457123195557310" + +tan_abi_real = \ +"6.822696615947538488826586186310162599974827139564433912601918442911"\ +"1026830824380070400102213741875804368044342309515353631134074491271890"\ +"467615882710035471686578162073677173148647065131872116479947620E-6" + +tan_abi_imag = \ +"0.9999795833048243692245661011298447587046967777739649018690797625964167"\ +"1446419978852235960862841608081413169601038230073129482874832053357571"\ +"62702259309150715669026865777947502665936317953101462202542168429" + + +def test_hp(): + for dps in precs: + mp.dps = dps + 8 + aa = mpf(a) + bb = mpf(b) + a1000 = 1000*mpf(a) + abi = mpc(aa, bb) + mp.dps = dps + assert (sqrt(3) + pi/2).ae(aa) + assert (e + 1/euler**2).ae(bb) + + assert sqrt(aa).ae(mpf(sqrt_a)) + assert sqrt(abi).ae(mpc(sqrt_abi_real, sqrt_abi_imag)) + + assert log(aa).ae(mpf(log_a)) + assert log(abi).ae(mpc(log_abi_real, log_abi_imag)) + + assert exp(aa).ae(mpf(exp_a)) + assert exp(abi).ae(mpc(exp_abi_real, exp_abi_imag)) + + assert (aa**bb).ae(mpf(pow_a_b)) + assert (aa**abi).ae(mpc(pow_a_abi_real, pow_a_abi_imag)) + assert (abi**abi).ae(mpc(pow_abi_abi_real, pow_abi_abi_imag)) + + assert sin(a).ae(mpf(sin_a)) + assert sin(a1000).ae(mpf(sin_1000a)) + assert sin(abi).ae(mpc(sin_abi_real, sin_abi_imag)) + + assert cos(a).ae(mpf(cos_a)) + assert cos(a1000).ae(mpf(cos_1000a)) + + assert tan(a).ae(mpf(tan_a)) + assert tan(abi).ae(mpc(tan_abi_real, tan_abi_imag)) + + # check that complex cancellation is avoided so that both + # real and imaginary parts have high relative accuracy. + # abs_eps should be 0, but has to be set to 1e-205 to pass the + # 200-digit case, probably due to slight inaccuracy in the + # precomputed input + assert (tan(abi).real).ae(mpf(tan_abi_real), abs_eps=1e-205) + assert (tan(abi).imag).ae(mpf(tan_abi_imag), abs_eps=1e-205) + mp.dps = 460 + assert str(log(3))[-20:] == '02166121184001409826' + mp.dps = 15 + +# Since str(a) can differ in the last digit from rounded a, and I want +# to compare the last digits of big numbers with the results in Mathematica, +# I made this hack to get the last 20 digits of rounded a + +def last_digits(a): + r = repr(a) + s = str(a) + #dps = mp.dps + #mp.dps += 3 + m = 10 + r = r.replace(s[:-m],'') + r = r.replace("mpf('",'').replace("')",'') + num0 = 0 + for c in r: + if c == '0': + num0 += 1 + else: + break + b = float(int(r))/10**(len(r) - m) + if b >= 10**m - 0.5: # pragma: no cover + raise NotImplementedError + n = int(round(b)) + sn = str(n) + s = s[:-m] + '0'*num0 + sn + return s[-20:] + +# values checked with Mathematica +def test_log_hp(): + mp.dps = 2000 + a = mpf(10)**15000/3 + r = log(a) + res = last_digits(r) + # Mathematica N[Log[10^15000/3], 2000] + # ...7443804441768333470331 + assert res == '43804441768333470331' + + # see issue 145 + r = log(mpf(3)/2) + # Mathematica N[Log[3/2], 2000] + # ...69653749808140753263288 + res = last_digits(r) + assert res == '53749808140753263288' + + mp.dps = 10000 + r = log(2) + res = last_digits(r) + # Mathematica N[Log[2], 10000] + # ...695615913401856601359655561 + assert res == '13401856601359655561' + r = log(mpf(10)**10/3) + res = last_digits(r) + # Mathematica N[Log[10^10/3], 10000] + # ...587087654020631943060007154 + assert res == '54020631943060007154', res + r = log(mpf(10)**100/3) + res = last_digits(r) + # Mathematica N[Log[10^100/3], 10000] + # ,,,59246336539088351652334666 + assert res == '36539088351652334666', res + mp.dps += 10 + a = 1 - mpf(1)/10**10 + mp.dps -= 10 + r = log(a) + res = last_digits(r) + # ...3310334360482956137216724048322957404 + # 372167240483229574038733026370 + # Mathematica N[Log[1 - 10^-10]*10^10, 10000] + # ...60482956137216724048322957404 + assert res == '37216724048322957404', res + mp.dps = 10000 + mp.dps += 100 + a = 1 + mpf(1)/10**100 + mp.dps -= 100 + + r = log(a) + res = last_digits(+r) + # Mathematica N[Log[1 + 10^-100]*10^10, 10030] + # ...3994733877377412241546890854692521568292338268273 10^-91 + assert res == '39947338773774122415', res + + mp.dps = 15 + +def test_exp_hp(): + mp.dps = 4000 + r = exp(mpf(1)/10) + # IntegerPart[N[Exp[1/10] * 10^4000, 4000]] + # ...92167105162069688129 + assert int(r * 10**mp.dps) % 10**20 == 92167105162069688129 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_identify.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_identify.py new file mode 100644 index 0000000000000000000000000000000000000000..f75ab0bc4f04ecb614011e7f4599989465cab785 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_identify.py @@ -0,0 +1,19 @@ +from mpmath import * + +def test_pslq(): + mp.dps = 15 + assert pslq([3*pi+4*e/7, pi, e, log(2)]) == [7, -21, -4, 0] + assert pslq([4.9999999999999991, 1]) == [1, -5] + assert pslq([2,1]) == [1, -2] + +def test_identify(): + mp.dps = 20 + assert identify(zeta(4), ['log(2)', 'pi**4']) == '((1/90)*pi**4)' + mp.dps = 15 + assert identify(exp(5)) == 'exp(5)' + assert identify(exp(4)) == 'exp(4)' + assert identify(log(5)) == 'log(5)' + assert identify(exp(3*pi), ['pi']) == 'exp((3*pi))' + assert identify(3, full=True) == ['3', '3', '1/(1/3)', 'sqrt(9)', + '1/sqrt((1/9))', '(sqrt(12)/2)**2', '1/(sqrt(12)/6)**2'] + assert identify(pi+1, {'a':+pi}) == '(1 + 1*a)' diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_interval.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_interval.py new file mode 100644 index 0000000000000000000000000000000000000000..251fd8b7ddb00074e8ae27cce4a01d8f4f8fe151 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_interval.py @@ -0,0 +1,453 @@ +from mpmath import * + +def test_interval_identity(): + iv.dps = 15 + assert mpi(2) == mpi(2, 2) + assert mpi(2) != mpi(-2, 2) + assert not (mpi(2) != mpi(2, 2)) + assert mpi(-1, 1) == mpi(-1, 1) + assert str(mpi('0.1')) == "[0.099999999999999991673, 0.10000000000000000555]" + assert repr(mpi('0.1')) == "mpi('0.099999999999999992', '0.10000000000000001')" + u = mpi(-1, 3) + assert -1 in u + assert 2 in u + assert 3 in u + assert -1.1 not in u + assert 3.1 not in u + assert mpi(-1, 3) in u + assert mpi(0, 1) in u + assert mpi(-1.1, 2) not in u + assert mpi(2.5, 3.1) not in u + w = mpi(-inf, inf) + assert mpi(-5, 5) in w + assert mpi(2, inf) in w + assert mpi(0, 2) in mpi(0, 10) + assert not (3 in mpi(-inf, 0)) + +def test_interval_hash(): + assert hash(mpi(3)) == hash(3) + assert hash(mpi(3.25)) == hash(3.25) + assert hash(mpi(3,4)) == hash(mpi(3,4)) + assert hash(iv.mpc(3)) == hash(3) + assert hash(iv.mpc(3,4)) == hash(3+4j) + assert hash(iv.mpc((1,3),(2,4))) == hash(iv.mpc((1,3),(2,4))) + +def test_interval_arithmetic(): + iv.dps = 15 + assert mpi(2) + mpi(3,4) == mpi(5,6) + assert mpi(1, 2)**2 == mpi(1, 4) + assert mpi(1) + mpi(0, 1e-50) == mpi(1, mpf('1.0000000000000002')) + x = 1 / (1 / mpi(3)) + assert x.a < 3 < x.b + x = mpi(2) ** mpi(0.5) + iv.dps += 5 + sq = iv.sqrt(2) + iv.dps -= 5 + assert x.a < sq < x.b + assert mpi(1) / mpi(1, inf) + assert mpi(2, 3) / inf == mpi(0, 0) + assert mpi(0) / inf == 0 + assert mpi(0) / 0 == mpi(-inf, inf) + assert mpi(inf) / 0 == mpi(-inf, inf) + assert mpi(0) * inf == mpi(-inf, inf) + assert 1 / mpi(2, inf) == mpi(0, 0.5) + assert str((mpi(50, 50) * mpi(-10, -10)) / 3) == \ + '[-166.66666666666668561, -166.66666666666665719]' + assert mpi(0, 4) ** 3 == mpi(0, 64) + assert mpi(2,4).mid == 3 + iv.dps = 30 + a = mpi(iv.pi) + iv.dps = 15 + b = +a + assert b.a < a.a + assert b.b > a.b + a = mpi(iv.pi) + assert a == +a + assert abs(mpi(-1,2)) == mpi(0,2) + assert abs(mpi(0.5,2)) == mpi(0.5,2) + assert abs(mpi(-3,2)) == mpi(0,3) + assert abs(mpi(-3,-0.5)) == mpi(0.5,3) + assert mpi(0) * mpi(2,3) == mpi(0) + assert mpi(2,3) * mpi(0) == mpi(0) + assert mpi(1,3).delta == 2 + assert mpi(1,2) - mpi(3,4) == mpi(-3,-1) + assert mpi(-inf,0) - mpi(0,inf) == mpi(-inf,0) + assert mpi(-inf,0) - mpi(-inf,inf) == mpi(-inf,inf) + assert mpi(0,inf) - mpi(-inf,1) == mpi(-1,inf) + +def test_interval_mul(): + assert mpi(-1, 0) * inf == mpi(-inf, 0) + assert mpi(-1, 0) * -inf == mpi(0, inf) + assert mpi(0, 1) * inf == mpi(0, inf) + assert mpi(0, 1) * mpi(0, inf) == mpi(0, inf) + assert mpi(-1, 1) * inf == mpi(-inf, inf) + assert mpi(-1, 1) * mpi(0, inf) == mpi(-inf, inf) + assert mpi(-1, 1) * mpi(-inf, inf) == mpi(-inf, inf) + assert mpi(-inf, 0) * mpi(0, 1) == mpi(-inf, 0) + assert mpi(-inf, 0) * mpi(0, 0) * mpi(-inf, 0) + assert mpi(-inf, 0) * mpi(-inf, inf) == mpi(-inf, inf) + assert mpi(-5,0)*mpi(-32,28) == mpi(-140,160) + assert mpi(2,3) * mpi(-1,2) == mpi(-3,6) + # Should be undefined? + assert mpi(inf, inf) * 0 == mpi(-inf, inf) + assert mpi(-inf, -inf) * 0 == mpi(-inf, inf) + assert mpi(0) * mpi(-inf,2) == mpi(-inf,inf) + assert mpi(0) * mpi(-2,inf) == mpi(-inf,inf) + assert mpi(-2,inf) * mpi(0) == mpi(-inf,inf) + assert mpi(-inf,2) * mpi(0) == mpi(-inf,inf) + +def test_interval_pow(): + assert mpi(3)**2 == mpi(9, 9) + assert mpi(-3)**2 == mpi(9, 9) + assert mpi(-3, 1)**2 == mpi(0, 9) + assert mpi(-3, -1)**2 == mpi(1, 9) + assert mpi(-3, -1)**3 == mpi(-27, -1) + assert mpi(-3, 1)**3 == mpi(-27, 1) + assert mpi(-2, 3)**2 == mpi(0, 9) + assert mpi(-3, 2)**2 == mpi(0, 9) + assert mpi(4) ** -1 == mpi(0.25, 0.25) + assert mpi(-4) ** -1 == mpi(-0.25, -0.25) + assert mpi(4) ** -2 == mpi(0.0625, 0.0625) + assert mpi(-4) ** -2 == mpi(0.0625, 0.0625) + assert mpi(0, 1) ** inf == mpi(0, 1) + assert mpi(0, 1) ** -inf == mpi(1, inf) + assert mpi(0, inf) ** inf == mpi(0, inf) + assert mpi(0, inf) ** -inf == mpi(0, inf) + assert mpi(1, inf) ** inf == mpi(1, inf) + assert mpi(1, inf) ** -inf == mpi(0, 1) + assert mpi(2, 3) ** 1 == mpi(2, 3) + assert mpi(2, 3) ** 0 == 1 + assert mpi(1,3) ** mpi(2) == mpi(1,9) + +def test_interval_sqrt(): + assert mpi(4) ** 0.5 == mpi(2) + +def test_interval_div(): + assert mpi(0.5, 1) / mpi(-1, 0) == mpi(-inf, -0.5) + assert mpi(0, 1) / mpi(0, 1) == mpi(0, inf) + assert mpi(inf, inf) / mpi(inf, inf) == mpi(0, inf) + assert mpi(inf, inf) / mpi(2, inf) == mpi(0, inf) + assert mpi(inf, inf) / mpi(2, 2) == mpi(inf, inf) + assert mpi(0, inf) / mpi(2, inf) == mpi(0, inf) + assert mpi(0, inf) / mpi(2, 2) == mpi(0, inf) + assert mpi(2, inf) / mpi(2, 2) == mpi(1, inf) + assert mpi(2, inf) / mpi(2, inf) == mpi(0, inf) + assert mpi(-4, 8) / mpi(1, inf) == mpi(-4, 8) + assert mpi(-4, 8) / mpi(0.5, inf) == mpi(-8, 16) + assert mpi(-inf, 8) / mpi(0.5, inf) == mpi(-inf, 16) + assert mpi(-inf, inf) / mpi(0.5, inf) == mpi(-inf, inf) + assert mpi(8, inf) / mpi(0.5, inf) == mpi(0, inf) + assert mpi(-8, inf) / mpi(0.5, inf) == mpi(-16, inf) + assert mpi(-4, 8) / mpi(inf, inf) == mpi(0, 0) + assert mpi(0, 8) / mpi(inf, inf) == mpi(0, 0) + assert mpi(0, 0) / mpi(inf, inf) == mpi(0, 0) + assert mpi(-inf, 0) / mpi(inf, inf) == mpi(-inf, 0) + assert mpi(-inf, 8) / mpi(inf, inf) == mpi(-inf, 0) + assert mpi(-inf, inf) / mpi(inf, inf) == mpi(-inf, inf) + assert mpi(-8, inf) / mpi(inf, inf) == mpi(0, inf) + assert mpi(0, inf) / mpi(inf, inf) == mpi(0, inf) + assert mpi(8, inf) / mpi(inf, inf) == mpi(0, inf) + assert mpi(inf, inf) / mpi(inf, inf) == mpi(0, inf) + assert mpi(-1, 2) / mpi(0, 1) == mpi(-inf, +inf) + assert mpi(0, 1) / mpi(0, 1) == mpi(0.0, +inf) + assert mpi(-1, 0) / mpi(0, 1) == mpi(-inf, 0.0) + assert mpi(-0.5, -0.25) / mpi(0, 1) == mpi(-inf, -0.25) + assert mpi(0.5, 1) / mpi(0, 1) == mpi(0.5, +inf) + assert mpi(0.5, 4) / mpi(0, 1) == mpi(0.5, +inf) + assert mpi(-1, -0.5) / mpi(0, 1) == mpi(-inf, -0.5) + assert mpi(-4, -0.5) / mpi(0, 1) == mpi(-inf, -0.5) + assert mpi(-1, 2) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(0, 1) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(-1, 0) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(-0.5, -0.25) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(0.5, 1) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(0.5, 4) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(-1, -0.5) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(-4, -0.5) / mpi(-2, 0.5) == mpi(-inf, +inf) + assert mpi(-1, 2) / mpi(-1, 0) == mpi(-inf, +inf) + assert mpi(0, 1) / mpi(-1, 0) == mpi(-inf, 0.0) + assert mpi(-1, 0) / mpi(-1, 0) == mpi(0.0, +inf) + assert mpi(-0.5, -0.25) / mpi(-1, 0) == mpi(0.25, +inf) + assert mpi(0.5, 1) / mpi(-1, 0) == mpi(-inf, -0.5) + assert mpi(0.5, 4) / mpi(-1, 0) == mpi(-inf, -0.5) + assert mpi(-1, -0.5) / mpi(-1, 0) == mpi(0.5, +inf) + assert mpi(-4, -0.5) / mpi(-1, 0) == mpi(0.5, +inf) + assert mpi(-1, 2) / mpi(0.5, 1) == mpi(-2.0, 4.0) + assert mpi(0, 1) / mpi(0.5, 1) == mpi(0.0, 2.0) + assert mpi(-1, 0) / mpi(0.5, 1) == mpi(-2.0, 0.0) + assert mpi(-0.5, -0.25) / mpi(0.5, 1) == mpi(-1.0, -0.25) + assert mpi(0.5, 1) / mpi(0.5, 1) == mpi(0.5, 2.0) + assert mpi(0.5, 4) / mpi(0.5, 1) == mpi(0.5, 8.0) + assert mpi(-1, -0.5) / mpi(0.5, 1) == mpi(-2.0, -0.5) + assert mpi(-4, -0.5) / mpi(0.5, 1) == mpi(-8.0, -0.5) + assert mpi(-1, 2) / mpi(-2, -0.5) == mpi(-4.0, 2.0) + assert mpi(0, 1) / mpi(-2, -0.5) == mpi(-2.0, 0.0) + assert mpi(-1, 0) / mpi(-2, -0.5) == mpi(0.0, 2.0) + assert mpi(-0.5, -0.25) / mpi(-2, -0.5) == mpi(0.125, 1.0) + assert mpi(0.5, 1) / mpi(-2, -0.5) == mpi(-2.0, -0.25) + assert mpi(0.5, 4) / mpi(-2, -0.5) == mpi(-8.0, -0.25) + assert mpi(-1, -0.5) / mpi(-2, -0.5) == mpi(0.25, 2.0) + assert mpi(-4, -0.5) / mpi(-2, -0.5) == mpi(0.25, 8.0) + # Should be undefined? + assert mpi(0, 0) / mpi(0, 0) == mpi(-inf, inf) + assert mpi(0, 0) / mpi(0, 1) == mpi(-inf, inf) + +def test_interval_cos_sin(): + iv.dps = 15 + cos = iv.cos + sin = iv.sin + tan = iv.tan + pi = iv.pi + # Around 0 + assert cos(mpi(0)) == 1 + assert sin(mpi(0)) == 0 + assert cos(mpi(0,1)) == mpi(0.54030230586813965399, 1.0) + assert sin(mpi(0,1)) == mpi(0, 0.8414709848078966159) + assert cos(mpi(1,2)) == mpi(-0.4161468365471424069, 0.54030230586813976501) + assert sin(mpi(1,2)) == mpi(0.84147098480789650488, 1.0) + assert sin(mpi(1,2.5)) == mpi(0.59847214410395643824, 1.0) + assert cos(mpi(-1, 1)) == mpi(0.54030230586813965399, 1.0) + assert cos(mpi(-1, 0.5)) == mpi(0.54030230586813965399, 1.0) + assert cos(mpi(-1, 1.5)) == mpi(0.070737201667702906405, 1.0) + assert sin(mpi(-1,1)) == mpi(-0.8414709848078966159, 0.8414709848078966159) + assert sin(mpi(-1,0.5)) == mpi(-0.8414709848078966159, 0.47942553860420300538) + assert mpi(-0.8414709848078966159, 1.00000000000000002e-100) in sin(mpi(-1,1e-100)) + assert mpi(-2.00000000000000004e-100, 1.00000000000000002e-100) in sin(mpi(-2e-100,1e-100)) + # Same interval + assert cos(mpi(2, 2.5)) + assert cos(mpi(3.5, 4)) == mpi(-0.93645668729079634129, -0.65364362086361182946) + assert cos(mpi(5, 5.5)) == mpi(0.28366218546322624627, 0.70866977429126010168) + assert mpi(0.59847214410395654927, 0.90929742682568170942) in sin(mpi(2, 2.5)) + assert sin(mpi(3.5, 4)) == mpi(-0.75680249530792831347, -0.35078322768961983646) + assert sin(mpi(5, 5.5)) == mpi(-0.95892427466313856499, -0.70554032557039181306) + # Higher roots + iv.dps = 55 + w = 4*10**50 + mpi(0.5) + for p in [15, 40, 80]: + iv.dps = p + assert 0 in sin(4*mpi(pi)) + assert 0 in sin(4*10**50*mpi(pi)) + assert 0 in cos((4+0.5)*mpi(pi)) + assert 0 in cos(w*mpi(pi)) + assert 1 in cos(4*mpi(pi)) + assert 1 in cos(4*10**50*mpi(pi)) + iv.dps = 15 + assert cos(mpi(2,inf)) == mpi(-1,1) + assert sin(mpi(2,inf)) == mpi(-1,1) + assert cos(mpi(-inf,2)) == mpi(-1,1) + assert sin(mpi(-inf,2)) == mpi(-1,1) + u = tan(mpi(0.5,1)) + assert mpf(u.a).ae(mp.tan(0.5)) + assert mpf(u.b).ae(mp.tan(1)) + v = iv.cot(mpi(0.5,1)) + assert mpf(v.a).ae(mp.cot(1)) + assert mpf(v.b).ae(mp.cot(0.5)) + # Sanity check of evaluation at n*pi and (n+1/2)*pi + for n in range(-5,7,2): + x = iv.cos(n*iv.pi) + assert -1 in x + assert x >= -1 + assert x != -1 + x = iv.sin((n+0.5)*iv.pi) + assert -1 in x + assert x >= -1 + assert x != -1 + for n in range(-6,8,2): + x = iv.cos(n*iv.pi) + assert 1 in x + assert x <= 1 + if n: + assert x != 1 + x = iv.sin((n+0.5)*iv.pi) + assert 1 in x + assert x <= 1 + assert x != 1 + for n in range(-6,7): + x = iv.cos((n+0.5)*iv.pi) + assert x.a < 0 < x.b + x = iv.sin(n*iv.pi) + if n: + assert x.a < 0 < x.b + +def test_interval_complex(): + # TODO: many more tests + iv.dps = 15 + mp.dps = 15 + assert iv.mpc(2,3) == 2+3j + assert iv.mpc(2,3) != 2+4j + assert iv.mpc(2,3) != 1+3j + assert 1+3j in iv.mpc([1,2],[3,4]) + assert 2+5j not in iv.mpc([1,2],[3,4]) + assert iv.mpc(1,2) + 1j == 1+3j + assert iv.mpc([1,2],[2,3]) + 2+3j == iv.mpc([3,4],[5,6]) + assert iv.mpc([2,4],[4,8]) / 2 == iv.mpc([1,2],[2,4]) + assert iv.mpc([1,2],[2,4]) * 2j == iv.mpc([-8,-4],[2,4]) + assert iv.mpc([2,4],[4,8]) / 2j == iv.mpc([2,4],[-2,-1]) + assert iv.exp(2+3j).ae(mp.exp(2+3j)) + assert iv.log(2+3j).ae(mp.log(2+3j)) + assert (iv.mpc(2,3) ** iv.mpc(0.5,2)).ae(mp.mpc(2,3) ** mp.mpc(0.5,2)) + assert 1j in (iv.mpf(-1) ** 0.5) + assert 1j in (iv.mpc(-1) ** 0.5) + assert abs(iv.mpc(0)) == 0 + assert abs(iv.mpc(inf)) == inf + assert abs(iv.mpc(3,4)) == 5 + assert abs(iv.mpc(4)) == 4 + assert abs(iv.mpc(0,4)) == 4 + assert abs(iv.mpc(0,[2,3])) == iv.mpf([2,3]) + assert abs(iv.mpc(0,[-3,2])) == iv.mpf([0,3]) + assert abs(iv.mpc([3,5],[4,12])) == iv.mpf([5,13]) + assert abs(iv.mpc([3,5],[-4,12])) == iv.mpf([3,13]) + assert iv.mpc(2,3) ** 0 == 1 + assert iv.mpc(2,3) ** 1 == (2+3j) + assert iv.mpc(2,3) ** 2 == (2+3j)**2 + assert iv.mpc(2,3) ** 3 == (2+3j)**3 + assert iv.mpc(2,3) ** 4 == (2+3j)**4 + assert iv.mpc(2,3) ** 5 == (2+3j)**5 + assert iv.mpc(2,2) ** (-1) == (2+2j) ** (-1) + assert iv.mpc(2,2) ** (-2) == (2+2j) ** (-2) + assert iv.cos(2).ae(mp.cos(2)) + assert iv.sin(2).ae(mp.sin(2)) + assert iv.cos(2+3j).ae(mp.cos(2+3j)) + assert iv.sin(2+3j).ae(mp.sin(2+3j)) + +def test_interval_complex_arg(): + mp.dps = 15 + iv.dps = 15 + assert iv.arg(3) == 0 + assert iv.arg(0) == 0 + assert iv.arg([0,3]) == 0 + assert iv.arg(-3).ae(pi) + assert iv.arg(2+3j).ae(iv.arg(2+3j)) + z = iv.mpc([-2,-1],[3,4]) + t = iv.arg(z) + assert t.a.ae(mp.arg(-1+4j)) + assert t.b.ae(mp.arg(-2+3j)) + z = iv.mpc([-2,1],[3,4]) + t = iv.arg(z) + assert t.a.ae(mp.arg(1+3j)) + assert t.b.ae(mp.arg(-2+3j)) + z = iv.mpc([1,2],[3,4]) + t = iv.arg(z) + assert t.a.ae(mp.arg(2+3j)) + assert t.b.ae(mp.arg(1+4j)) + z = iv.mpc([1,2],[-2,3]) + t = iv.arg(z) + assert t.a.ae(mp.arg(1-2j)) + assert t.b.ae(mp.arg(1+3j)) + z = iv.mpc([1,2],[-4,-3]) + t = iv.arg(z) + assert t.a.ae(mp.arg(1-4j)) + assert t.b.ae(mp.arg(2-3j)) + z = iv.mpc([-1,2],[-4,-3]) + t = iv.arg(z) + assert t.a.ae(mp.arg(-1-3j)) + assert t.b.ae(mp.arg(2-3j)) + z = iv.mpc([-2,-1],[-4,-3]) + t = iv.arg(z) + assert t.a.ae(mp.arg(-2-3j)) + assert t.b.ae(mp.arg(-1-4j)) + z = iv.mpc([-2,-1],[-3,3]) + t = iv.arg(z) + assert t.a.ae(-mp.pi) + assert t.b.ae(mp.pi) + z = iv.mpc([-2,2],[-3,3]) + t = iv.arg(z) + assert t.a.ae(-mp.pi) + assert t.b.ae(mp.pi) + +def test_interval_ae(): + iv.dps = 15 + x = iv.mpf([1,2]) + assert x.ae(1) is None + assert x.ae(1.5) is None + assert x.ae(2) is None + assert x.ae(2.01) is False + assert x.ae(0.99) is False + x = iv.mpf(3.5) + assert x.ae(3.5) is True + assert x.ae(3.5+1e-15) is True + assert x.ae(3.5-1e-15) is True + assert x.ae(3.501) is False + assert x.ae(3.499) is False + assert x.ae(iv.mpf([3.5,3.501])) is None + assert x.ae(iv.mpf([3.5,4.5+1e-15])) is None + +def test_interval_nstr(): + iv.dps = n = 30 + x = mpi(1, 2) + # FIXME: error_dps should not be necessary + assert iv.nstr(x, n, mode='plusminus', error_dps=6) == '1.5 +- 0.5' + assert iv.nstr(x, n, mode='plusminus', use_spaces=False, error_dps=6) == '1.5+-0.5' + assert iv.nstr(x, n, mode='percent') == '1.5 (33.33%)' + assert iv.nstr(x, n, mode='brackets', use_spaces=False) == '[1.0,2.0]' + assert iv.nstr(x, n, mode='brackets' , brackets=('<', '>')) == '<1.0, 2.0>' + x = mpi('5.2582327113062393041', '5.2582327113062749951') + assert iv.nstr(x, n, mode='diff') == '5.2582327113062[393041, 749951]' + assert iv.nstr(iv.cos(mpi(1)), n, mode='diff', use_spaces=False) == '0.54030230586813971740093660744[2955,3053]' + assert iv.nstr(mpi('1e123', '1e129'), n, mode='diff') == '[1.0e+123, 1.0e+129]' + exp = iv.exp + assert iv.nstr(iv.exp(mpi('5000.1')), n, mode='diff') == '3.2797365856787867069110487[0926, 1191]e+2171' + iv.dps = 15 + +def test_mpi_from_str(): + iv.dps = 15 + assert iv.convert('1.5 +- 0.5') == mpi(mpf('1.0'), mpf('2.0')) + assert mpi(1, 2) in iv.convert('1.5 (33.33333333333333333333333333333%)') + assert iv.convert('[1, 2]') == mpi(1, 2) + assert iv.convert('1[2, 3]') == mpi(12, 13) + assert iv.convert('1.[23,46]e-8') == mpi('1.23e-8', '1.46e-8') + assert iv.convert('12[3.4,5.9]e4') == mpi('123.4e+4', '125.9e4') + +def test_interval_gamma(): + mp.dps = 15 + iv.dps = 15 + # TODO: need many more tests + assert iv.rgamma(0) == 0 + assert iv.fac(0) == 1 + assert iv.fac(1) == 1 + assert iv.fac(2) == 2 + assert iv.fac(3) == 6 + assert iv.gamma(0) == [-inf,inf] + assert iv.gamma(1) == 1 + assert iv.gamma(2) == 1 + assert iv.gamma(3) == 2 + assert -3.5449077018110320546 in iv.gamma(-0.5) + assert iv.loggamma(1) == 0 + assert iv.loggamma(2) == 0 + assert 0.69314718055994530942 in iv.loggamma(3) + # Test tight log-gamma endpoints based on monotonicity + xs = [iv.mpc([2,3],[1,4]), + iv.mpc([2,3],[-4,-1]), + iv.mpc([2,3],[-1,4]), + iv.mpc([2,3],[-4,1]), + iv.mpc([2,3],[-4,4]), + iv.mpc([-3,-2],[2,4]), + iv.mpc([-3,-2],[-4,-2])] + for x in xs: + ys = [mp.loggamma(mp.mpc(x.a,x.c)), + mp.loggamma(mp.mpc(x.b,x.c)), + mp.loggamma(mp.mpc(x.a,x.d)), + mp.loggamma(mp.mpc(x.b,x.d))] + if 0 in x.imag: + ys += [mp.loggamma(x.a), mp.loggamma(x.b)] + min_real = min([y.real for y in ys]) + max_real = max([y.real for y in ys]) + min_imag = min([y.imag for y in ys]) + max_imag = max([y.imag for y in ys]) + z = iv.loggamma(x) + assert z.a.ae(min_real) + assert z.b.ae(max_real) + assert z.c.ae(min_imag) + assert z.d.ae(max_imag) + +def test_interval_conversions(): + mp.dps = 15 + iv.dps = 15 + for a, b in ((-0.0, 0), (0.0, 0.5), (1.0, 1), \ + ('-inf', 20.5), ('-inf', float(sqrt(2)))): + r = mpi(a, b) + assert int(r.b) == int(b) + assert float(r.a) == float(a) + assert float(r.b) == float(b) + assert complex(r.a) == complex(a) + assert complex(r.b) == complex(b) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_levin.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_levin.py new file mode 100644 index 0000000000000000000000000000000000000000..b14855df4de1a45da27080dcd239267842a4ac7a --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_levin.py @@ -0,0 +1,153 @@ +#!/usr/bin/python +# -*- coding: utf-8 -*- + +from mpmath import mp +from mpmath import libmp + +xrange = libmp.backend.xrange + +# Attention: +# These tests run with 15-20 decimal digits precision. For higher precision the +# working precision must be raised. + +def test_levin_0(): + mp.dps = 17 + eps = mp.mpf(mp.eps) + with mp.extraprec(2 * mp.prec): + L = mp.levin(method = "levin", variant = "u") + S, s, n = [], 0, 1 + while 1: + s += mp.one / (n * n) + n += 1 + S.append(s) + v, e = L.update_psum(S) + if e < eps: + break + if n > 1000: raise RuntimeError("iteration limit exceeded") + eps = mp.exp(0.9 * mp.log(eps)) + err = abs(v - mp.pi ** 2 / 6) + assert err < eps + w = mp.nsum(lambda n: 1/(n * n), [1, mp.inf], method = "levin", levin_variant = "u") + err = abs(v - w) + assert err < eps + +def test_levin_1(): + mp.dps = 17 + eps = mp.mpf(mp.eps) + with mp.extraprec(2 * mp.prec): + L = mp.levin(method = "levin", variant = "v") + A, n = [], 1 + while 1: + s = mp.mpf(n) ** (2 + 3j) + n += 1 + A.append(s) + v, e = L.update(A) + if e < eps: + break + if n > 1000: raise RuntimeError("iteration limit exceeded") + eps = mp.exp(0.9 * mp.log(eps)) + err = abs(v - mp.zeta(-2-3j)) + assert err < eps + w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v") + err = abs(v - w) + assert err < eps + +def test_levin_2(): + # [2] A. Sidi - "Pratical Extrapolation Methods" p.373 + mp.dps = 17 + z=mp.mpf(10) + eps = mp.mpf(mp.eps) + with mp.extraprec(2 * mp.prec): + L = mp.levin(method = "sidi", variant = "t") + n = 0 + while 1: + s = (-1)**n * mp.fac(n) * z ** (-n) + v, e = L.step(s) + n += 1 + if e < eps: + break + if n > 1000: raise RuntimeError("iteration limit exceeded") + eps = mp.exp(0.9 * mp.log(eps)) + exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf]) + # there is also a symbolic expression for the integral: + # exact = z * mp.exp(z) * mp.expint(1,z) + err = abs(v - exact) + assert err < eps + w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t") + assert err < eps + +def test_levin_3(): + mp.dps = 17 + z=mp.mpf(2) + eps = mp.mpf(mp.eps) + with mp.extraprec(7*mp.prec): # we need copious amount of precision to sum this highly divergent series + L = mp.levin(method = "levin", variant = "t") + n, s = 0, 0 + while 1: + s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)) + n += 1 + v, e = L.step_psum(s) + if e < eps: + break + if n > 1000: raise RuntimeError("iteration limit exceeded") + eps = mp.exp(0.8 * mp.log(eps)) + exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi) + # there is also a symbolic expression for the integral: + # exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi)) + err = abs(v - exact) + assert err < eps + w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)), [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)]) + err = abs(v - w) + assert err < eps + +def test_levin_nsum(): + mp.dps = 17 + + with mp.extraprec(mp.prec): + z = mp.mpf(10) ** (-10) + a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "l") - 1 / z + assert abs(a - mp.euler) < 1e-10 + + eps = mp.exp(0.8 * mp.log(mp.eps)) + + a = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "sidi") + assert abs(a - mp.log(2)) < eps + + z = 2 + 1j + f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n)) + v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)]) + exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z) + assert abs(exact - v) < eps + +def test_cohen_alt_0(): + mp.dps = 17 + AC = mp.cohen_alt() + S, s, n = [], 0, 1 + while 1: + s += -((-1) ** n) * mp.one / (n * n) + n += 1 + S.append(s) + v, e = AC.update_psum(S) + if e < mp.eps: + break + if n > 1000: raise RuntimeError("iteration limit exceeded") + eps = mp.exp(0.9 * mp.log(mp.eps)) + err = abs(v - mp.pi ** 2 / 12) + assert err < eps + +def test_cohen_alt_1(): + mp.dps = 17 + A = [] + AC = mp.cohen_alt() + n = 1 + while 1: + A.append( mp.loggamma(1 + mp.one / (2 * n - 1))) + A.append(-mp.loggamma(1 + mp.one / (2 * n))) + n += 1 + v, e = AC.update(A) + if e < mp.eps: + break + if n > 1000: raise RuntimeError("iteration limit exceeded") + v = mp.exp(v) + err = abs(v - 1.06215090557106) + assert err < 1e-12 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_linalg.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_linalg.py new file mode 100644 index 0000000000000000000000000000000000000000..14256a79f8953d3e4ef8b296258560d48204f547 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_linalg.py @@ -0,0 +1,332 @@ +# TODO: don't use round + +from __future__ import division + +import pytest +from mpmath import * +xrange = libmp.backend.xrange + +# XXX: these shouldn't be visible(?) +LU_decomp = mp.LU_decomp +L_solve = mp.L_solve +U_solve = mp.U_solve +householder = mp.householder +improve_solution = mp.improve_solution + +A1 = matrix([[3, 1, 6], + [2, 1, 3], + [1, 1, 1]]) +b1 = [2, 7, 4] + +A2 = matrix([[ 2, -1, -1, 2], + [ 6, -2, 3, -1], + [-4, 2, 3, -2], + [ 2, 0, 4, -3]]) +b2 = [3, -3, -2, -1] + +A3 = matrix([[ 1, 0, -1, -1, 0], + [ 0, 1, 1, 0, -1], + [ 4, -5, 2, 0, 0], + [ 0, 0, -2, 9,-12], + [ 0, 5, 0, 0, 12]]) +b3 = [0, 0, 0, 0, 50] + +A4 = matrix([[10.235, -4.56, 0., -0.035, 5.67], + [-2.463, 1.27, 3.97, -8.63, 1.08], + [-6.58, 0.86, -0.257, 9.32, -43.6 ], + [ 9.83, 7.39, -17.25, 0.036, 24.86], + [-9.31, 34.9, 78.56, 1.07, 65.8 ]]) +b4 = [8.95, 20.54, 7.42, 5.60, 58.43] + +A5 = matrix([[ 1, 2, -4], + [-2, -3, 5], + [ 3, 5, -8]]) + +A6 = matrix([[ 1.377360, 2.481400, 5.359190], + [ 2.679280, -1.229560, 25.560210], + [-1.225280+1.e6, 9.910180, -35.049900-1.e6]]) +b6 = [23.500000, -15.760000, 2.340000] + +A7 = matrix([[1, -0.5], + [2, 1], + [-2, 6]]) +b7 = [3, 2, -4] + +A8 = matrix([[1, 2, 3], + [-1, 0, 1], + [-1, -2, -1], + [1, 0, -1]]) +b8 = [1, 2, 3, 4] + +A9 = matrix([[ 4, 2, -2], + [ 2, 5, -4], + [-2, -4, 5.5]]) +b9 = [10, 16, -15.5] + +A10 = matrix([[1.0 + 1.0j, 2.0, 2.0], + [4.0, 5.0, 6.0], + [7.0, 8.0, 9.0]]) +b10 = [1.0, 1.0 + 1.0j, 1.0] + + +def test_LU_decomp(): + A = A3.copy() + b = b3 + A, p = LU_decomp(A) + y = L_solve(A, b, p) + x = U_solve(A, y) + assert p == [2, 1, 2, 3] + assert [round(i, 14) for i in x] == [3.78953107960742, 2.9989094874591098, + -0.081788440567070006, 3.8713195201744801, 2.9171210468920399] + A = A4.copy() + b = b4 + A, p = LU_decomp(A) + y = L_solve(A, b, p) + x = U_solve(A, y) + assert p == [0, 3, 4, 3] + assert [round(i, 14) for i in x] == [2.6383625899619201, 2.6643834462368399, + 0.79208015947958998, -2.5088376454101899, -1.0567657691375001] + A = randmatrix(3) + bak = A.copy() + LU_decomp(A, overwrite=1) + assert A != bak + +def test_inverse(): + for A in [A1, A2, A5]: + inv = inverse(A) + assert mnorm(A*inv - eye(A.rows), 1) < 1.e-14 + +def test_householder(): + mp.dps = 15 + A, b = A8, b8 + H, p, x, r = householder(extend(A, b)) + assert H == matrix( + [[mpf('3.0'), mpf('-2.0'), mpf('-1.0'), 0], + [-1.0,mpf('3.333333333333333'),mpf('-2.9999999999999991'),mpf('2.0')], + [-1.0, mpf('-0.66666666666666674'),mpf('2.8142135623730948'), + mpf('-2.8284271247461898')], + [1.0, mpf('-1.3333333333333333'),mpf('-0.20000000000000018'), + mpf('4.2426406871192857')]]) + assert p == [-2, -2, mpf('-1.4142135623730949')] + assert round(norm(r, 2), 10) == 4.2426406870999998 + + y = [102.102, 58.344, 36.463, 24.310, 17.017, 12.376, 9.282, 7.140, 5.610, + 4.488, 3.6465, 3.003] + + def coeff(n): + # similiar to Hilbert matrix + A = [] + for i in range(1, 13): + A.append([1. / (i + j - 1) for j in range(1, n + 1)]) + return matrix(A) + + residuals = [] + refres = [] + for n in range(2, 7): + A = coeff(n) + H, p, x, r = householder(extend(A, y)) + x = matrix(x) + y = matrix(y) + residuals.append(norm(r, 2)) + refres.append(norm(residual(A, x, y), 2)) + assert [round(res, 10) for res in residuals] == [15.1733888877, + 0.82378073210000002, 0.302645887, 0.0260109244, + 0.00058653999999999998] + assert norm(matrix(residuals) - matrix(refres), inf) < 1.e-13 + + def hilbert_cmplx(n): + # Complexified Hilbert matrix + A = hilbert(2*n,n) + v = randmatrix(2*n, 2, min=-1, max=1) + v = v.apply(lambda x: exp(1J*pi()*x)) + A = diag(v[:,0])*A*diag(v[:n,1]) + return A + + residuals_cmplx = [] + refres_cmplx = [] + for n in range(2, 10): + A = hilbert_cmplx(n) + H, p, x, r = householder(A.copy()) + residuals_cmplx.append(norm(r, 2)) + refres_cmplx.append(norm(residual(A[:,:n-1], x, A[:,n-1]), 2)) + assert norm(matrix(residuals_cmplx) - matrix(refres_cmplx), inf) < 1.e-13 + +def test_factorization(): + A = randmatrix(5) + P, L, U = lu(A) + assert mnorm(P*A - L*U, 1) < 1.e-15 + +def test_solve(): + assert norm(residual(A6, lu_solve(A6, b6), b6), inf) < 1.e-10 + assert norm(residual(A7, lu_solve(A7, b7), b7), inf) < 1.5 + assert norm(residual(A8, lu_solve(A8, b8), b8), inf) <= 3 + 1.e-10 + assert norm(residual(A6, qr_solve(A6, b6)[0], b6), inf) < 1.e-10 + assert norm(residual(A7, qr_solve(A7, b7)[0], b7), inf) < 1.5 + assert norm(residual(A8, qr_solve(A8, b8)[0], b8), 2) <= 4.3 + assert norm(residual(A10, lu_solve(A10, b10), b10), 2) < 1.e-10 + assert norm(residual(A10, qr_solve(A10, b10)[0], b10), 2) < 1.e-10 + +def test_solve_overdet_complex(): + A = matrix([[1, 2j], [3, 4j], [5, 6]]) + b = matrix([1 + j, 2, -j]) + assert norm(residual(A, lu_solve(A, b), b)) < 1.0208 + +def test_singular(): + mp.dps = 15 + A = [[5.6, 1.2], [7./15, .1]] + B = repr(zeros(2)) + b = [1, 2] + for i in ['lu_solve(%s, %s)' % (A, b), 'lu_solve(%s, %s)' % (B, b), + 'qr_solve(%s, %s)' % (A, b), 'qr_solve(%s, %s)' % (B, b)]: + pytest.raises((ZeroDivisionError, ValueError), lambda: eval(i)) + +def test_cholesky(): + assert fp.cholesky(fp.matrix(A9)) == fp.matrix([[2, 0, 0], [1, 2, 0], [-1, -3/2, 3/2]]) + x = fp.cholesky_solve(A9, b9) + assert fp.norm(fp.residual(A9, x, b9), fp.inf) == 0 + +def test_det(): + assert det(A1) == 1 + assert round(det(A2), 14) == 8 + assert round(det(A3)) == 1834 + assert round(det(A4)) == 4443376 + assert det(A5) == 1 + assert round(det(A6)) == 78356463 + assert det(zeros(3)) == 0 + +def test_cond(): + mp.dps = 15 + A = matrix([[1.2969, 0.8648], [0.2161, 0.1441]]) + assert cond(A, lambda x: mnorm(x,1)) == mpf('327065209.73817754') + assert cond(A, lambda x: mnorm(x,inf)) == mpf('327065209.73817754') + assert cond(A, lambda x: mnorm(x,'F')) == mpf('249729266.80008656') + +@extradps(50) +def test_precision(): + A = randmatrix(10, 10) + assert mnorm(inverse(inverse(A)) - A, 1) < 1.e-45 + +def test_interval_matrix(): + mp.dps = 15 + iv.dps = 15 + a = iv.matrix([['0.1','0.3','1.0'],['7.1','5.5','4.8'],['3.2','4.4','5.6']]) + b = iv.matrix(['4','0.6','0.5']) + c = iv.lu_solve(a, b) + assert c[0].delta < 1e-13 + assert c[1].delta < 1e-13 + assert c[2].delta < 1e-13 + assert 5.25823271130625686059275 in c[0] + assert -13.155049396267837541163 in c[1] + assert 7.42069154774972557628979 in c[2] + +def test_LU_cache(): + A = randmatrix(3) + LU = LU_decomp(A) + assert A._LU == LU_decomp(A) + A[0,0] = -1000 + assert A._LU is None + +def test_improve_solution(): + A = randmatrix(5, min=1e-20, max=1e20) + b = randmatrix(5, 1, min=-1000, max=1000) + x1 = lu_solve(A, b) + randmatrix(5, 1, min=-1e-5, max=1.e-5) + x2 = improve_solution(A, x1, b) + assert norm(residual(A, x2, b), 2) < norm(residual(A, x1, b), 2) + +def test_exp_pade(): + for i in range(3): + dps = 15 + extra = 15 + mp.dps = dps + extra + dm = 0 + N = 3 + dg = range(1,N+1) + a = diag(dg) + expa = diag([exp(x) for x in dg]) + # choose a random matrix not close to be singular + # to avoid adding too much extra precision in computing + # m**-1 * M * m + while abs(dm) < 0.01: + m = randmatrix(N) + dm = det(m) + m = m/dm + a1 = m**-1 * a * m + e2 = m**-1 * expa * m + mp.dps = dps + e1 = expm(a1, method='pade') + mp.dps = dps + extra + d = e2 - e1 + #print d + mp.dps = dps + assert norm(d, inf).ae(0) + mp.dps = 15 + +def test_qr(): + mp.dps = 15 # used default value for dps + lowlimit = -9 # lower limit of matrix element value + uplimit = 9 # uppter limit of matrix element value + maxm = 4 # max matrix size + flg = False # toggle to create real vs complex matrix + zero = mpf('0.0') + + for k in xrange(0,10): + exdps = 0 + mode = 'full' + flg = bool(k % 2) + + # generate arbitrary matrix size (2 to maxm) + num1 = nint(maxm*rand()) + num2 = nint(maxm*rand()) + m = int(max(num1, num2)) + n = int(min(num1, num2)) + + # create matrix + A = mp.matrix(m,n) + + # populate matrix values with arbitrary integers + if flg: + flg = False + dtype = 'complex' + for j in xrange(0,n): + for i in xrange(0,m): + val = nint(lowlimit + (uplimit-lowlimit)*rand()) + val2 = nint(lowlimit + (uplimit-lowlimit)*rand()) + A[i,j] = mpc(val, val2) + else: + flg = True + dtype = 'real' + for j in xrange(0,n): + for i in xrange(0,m): + val = nint(lowlimit + (uplimit-lowlimit)*rand()) + A[i,j] = mpf(val) + + # perform A -> QR decomposition + Q, R = qr(A, mode, edps = exdps) + + #print('\n\n A = \n', nstr(A, 4)) + #print('\n Q = \n', nstr(Q, 4)) + #print('\n R = \n', nstr(R, 4)) + #print('\n Q*R = \n', nstr(Q*R, 4)) + + maxnorm = mpf('1.0E-11') + n1 = norm(A - Q * R) + #print '\n Norm of A - Q * R = ', n1 + assert n1 <= maxnorm + + if dtype == 'real': + n1 = norm(eye(m) - Q.T * Q) + #print ' Norm of I - Q.T * Q = ', n1 + assert n1 <= maxnorm + + n1 = norm(eye(m) - Q * Q.T) + #print ' Norm of I - Q * Q.T = ', n1 + assert n1 <= maxnorm + + if dtype == 'complex': + n1 = norm(eye(m) - Q.T * Q.conjugate()) + #print ' Norm of I - Q.T * Q.conjugate() = ', n1 + assert n1 <= maxnorm + + n1 = norm(eye(m) - Q.conjugate() * Q.T) + #print ' Norm of I - Q.conjugate() * Q.T = ', n1 + assert n1 <= maxnorm diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_matrices.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_matrices.py new file mode 100644 index 0000000000000000000000000000000000000000..1547b90664dba66a98a7f026a04a4ed1aa1ed3b4 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_matrices.py @@ -0,0 +1,253 @@ +import pytest +import sys +from mpmath import * + +def test_matrix_basic(): + A1 = matrix(3) + for i in range(3): + A1[i,i] = 1 + assert A1 == eye(3) + assert A1 == matrix(A1) + A2 = matrix(3, 2) + assert not A2._matrix__data + A3 = matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) + assert list(A3) == list(range(1, 10)) + A3[1,1] = 0 + assert not (1, 1) in A3._matrix__data + A4 = matrix([[1, 2, 3], [4, 5, 6]]) + A5 = matrix([[6, -1], [3, 2], [0, -3]]) + assert A4 * A5 == matrix([[12, -6], [39, -12]]) + assert A1 * A3 == A3 * A1 == A3 + pytest.raises(ValueError, lambda: A2*A2) + l = [[10, 20, 30], [40, 0, 60], [70, 80, 90]] + A6 = matrix(l) + assert A6.tolist() == l + assert A6 == eval(repr(A6)) + A6 = fp.matrix(A6) + assert A6 == eval(repr(A6)) + assert A6*1j == eval(repr(A6*1j)) + assert A3 * 10 == 10 * A3 == A6 + assert A2.rows == 3 + assert A2.cols == 2 + A3.rows = 2 + A3.cols = 2 + assert len(A3._matrix__data) == 3 + assert A4 + A4 == 2*A4 + pytest.raises(ValueError, lambda: A4 + A2) + assert sum(A1 - A1) == 0 + A7 = matrix([[1, 2], [3, 4], [5, 6], [7, 8]]) + x = matrix([10, -10]) + assert A7*x == matrix([-10, -10, -10, -10]) + A8 = ones(5) + assert sum((A8 + 1) - (2 - zeros(5))) == 0 + assert (1 + ones(4)) / 2 - 1 == zeros(4) + assert eye(3)**10 == eye(3) + pytest.raises(ValueError, lambda: A7**2) + A9 = randmatrix(3) + A10 = matrix(A9) + A9[0,0] = -100 + assert A9 != A10 + assert nstr(A9) + +def test_matmul(): + """ + Test the PEP465 "@" matrix multiplication syntax. + To avoid syntax errors when importing this file in Python 3.5 and below, we have to use exec() - sorry for that. + """ + # TODO remove exec() wrapper as soon as we drop support for Python <= 3.5 + if sys.hexversion < 0x30500f0: + # we are on Python < 3.5 + pytest.skip("'@' (__matmul__) is only supported in Python 3.5 or newer") + A4 = matrix([[1, 2, 3], [4, 5, 6]]) + A5 = matrix([[6, -1], [3, 2], [0, -3]]) + exec("assert A4 @ A5 == A4 * A5") + +def test_matrix_slices(): + A = matrix([ [1, 2, 3], + [4, 5 ,6], + [7, 8 ,9]]) + V = matrix([1,2,3,4,5]) + + # Get slice + assert A[:,:] == A + assert A[:,1] == matrix([[2],[5],[8]]) + assert A[2,:] == matrix([[7, 8 ,9]]) + assert A[1:3,1:3] == matrix([[5,6],[8,9]]) + assert V[2:4] == matrix([3,4]) + pytest.raises(IndexError, lambda: A[:,1:6]) + + # Assign slice with matrix + A1 = matrix(3) + A1[:,:] = A + assert A1[:,:] == matrix([[1, 2, 3], + [4, 5 ,6], + [7, 8 ,9]]) + A1[0,:] = matrix([[10, 11, 12]]) + assert A1 == matrix([ [10, 11, 12], + [4, 5 ,6], + [7, 8 ,9]]) + A1[:,2] = matrix([[13], [14], [15]]) + assert A1 == matrix([ [10, 11, 13], + [4, 5 ,14], + [7, 8 ,15]]) + A1[:2,:2] = matrix([[16, 17], [18 , 19]]) + assert A1 == matrix([ [16, 17, 13], + [18, 19 ,14], + [7, 8 ,15]]) + V[1:3] = 10 + assert V == matrix([1,10,10,4,5]) + with pytest.raises(ValueError): + A1[2,:] = A[:,1] + + with pytest.raises(IndexError): + A1[2,1:20] = A[:,:] + + # Assign slice with scalar + A1[:,2] = 10 + assert A1 == matrix([ [16, 17, 10], + [18, 19 ,10], + [7, 8 ,10]]) + A1[:,:] = 40 + for x in A1: + assert x == 40 + + +def test_matrix_power(): + A = matrix([[1, 2], [3, 4]]) + assert A**2 == A*A + assert A**3 == A*A*A + assert A**-1 == inverse(A) + assert A**-2 == inverse(A*A) + +def test_matrix_transform(): + A = matrix([[1, 2], [3, 4], [5, 6]]) + assert A.T == A.transpose() == matrix([[1, 3, 5], [2, 4, 6]]) + swap_row(A, 1, 2) + assert A == matrix([[1, 2], [5, 6], [3, 4]]) + l = [1, 2] + swap_row(l, 0, 1) + assert l == [2, 1] + assert extend(eye(3), [1,2,3]) == matrix([[1,0,0,1],[0,1,0,2],[0,0,1,3]]) + +def test_matrix_conjugate(): + A = matrix([[1 + j, 0], [2, j]]) + assert A.conjugate() == matrix([[mpc(1, -1), 0], [2, mpc(0, -1)]]) + assert A.transpose_conj() == A.H == matrix([[mpc(1, -1), 2], + [0, mpc(0, -1)]]) + +def test_matrix_creation(): + assert diag([1, 2, 3]) == matrix([[1, 0, 0], [0, 2, 0], [0, 0, 3]]) + A1 = ones(2, 3) + assert A1.rows == 2 and A1.cols == 3 + for a in A1: + assert a == 1 + A2 = zeros(3, 2) + assert A2.rows == 3 and A2.cols == 2 + for a in A2: + assert a == 0 + assert randmatrix(10) != randmatrix(10) + one = mpf(1) + assert hilbert(3) == matrix([[one, one/2, one/3], + [one/2, one/3, one/4], + [one/3, one/4, one/5]]) + +def test_norms(): + # matrix norms + A = matrix([[1, -2], [-3, -1], [2, 1]]) + assert mnorm(A,1) == 6 + assert mnorm(A,inf) == 4 + assert mnorm(A,'F') == sqrt(20) + # vector norms + assert norm(-3) == 3 + x = [1, -2, 7, -12] + assert norm(x, 1) == 22 + assert round(norm(x, 2), 10) == 14.0712472795 + assert round(norm(x, 10), 10) == 12.0054633727 + assert norm(x, inf) == 12 + +def test_vector(): + x = matrix([0, 1, 2, 3, 4]) + assert x == matrix([[0], [1], [2], [3], [4]]) + assert x[3] == 3 + assert len(x._matrix__data) == 4 + assert list(x) == list(range(5)) + x[0] = -10 + x[4] = 0 + assert x[0] == -10 + assert len(x) == len(x.T) == 5 + assert x.T*x == matrix([[114]]) + +def test_matrix_copy(): + A = ones(6) + B = A.copy() + C = +A + assert A == B + assert A == C + B[0,0] = 0 + assert A != B + C[0,0] = 42 + assert A != C + +def test_matrix_numpy(): + try: + import numpy + except ImportError: + return + l = [[1, 2], [3, 4], [5, 6]] + a = numpy.array(l) + assert matrix(l) == matrix(a) + +def test_interval_matrix_scalar_mult(): + """Multiplication of iv.matrix and any scalar type""" + a = mpi(-1, 1) + b = a + a * 2j + c = mpf(42) + d = c + c * 2j + e = 1.234 + f = fp.convert(e) + g = e + e * 3j + h = fp.convert(g) + M = iv.ones(1) + for x in [a, b, c, d, e, f, g, h]: + assert x * M == iv.matrix([x]) + assert M * x == iv.matrix([x]) + +@pytest.mark.xfail() +def test_interval_matrix_matrix_mult(): + """Multiplication of iv.matrix and other matrix types""" + A = ones(1) + B = fp.ones(1) + M = iv.ones(1) + for X in [A, B, M]: + assert X * M == iv.matrix(X) + assert X * M == X + assert M * X == iv.matrix(X) + assert M * X == X + +def test_matrix_conversion_to_iv(): + # Test that matrices with foreign datatypes are properly converted + for other_type_eye in [eye(3), fp.eye(3), iv.eye(3)]: + A = iv.matrix(other_type_eye) + B = iv.eye(3) + assert type(A[0,0]) == type(B[0,0]) + assert A.tolist() == B.tolist() + +def test_interval_matrix_mult_bug(): + # regression test for interval matrix multiplication: + # result must be nonzero-width and contain the exact result + x = convert('1.00000000000001') # note: this is implicitly rounded to some near mpf float value + A = matrix([[x]]) + B = iv.matrix(A) + C = iv.matrix([[x]]) + assert B == C + B = B * B + C = C * C + assert B == C + assert B[0, 0].delta > 1e-16 + assert B[0, 0].delta < 3e-16 + assert C[0, 0].delta > 1e-16 + assert C[0, 0].delta < 3e-16 + assert mp.mpf('1.00000000000001998401444325291756783368705994138804689654') in B[0, 0] + assert mp.mpf('1.00000000000001998401444325291756783368705994138804689654') in C[0, 0] + # the following caused an error before the bug was fixed + assert iv.matrix(mp.eye(2)) * (iv.ones(2) + mpi(1, 2)) == iv.matrix([[mpi(2, 3), mpi(2, 3)], [mpi(2, 3), mpi(2, 3)]]) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_mpmath.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_mpmath.py new file mode 100644 index 0000000000000000000000000000000000000000..9f1fe36ae9b1b0feca4677eeb90396bfa7ed8f7a --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_mpmath.py @@ -0,0 +1,7 @@ +from mpmath.libmp import * +from mpmath import * + +def test_newstyle_classes(): + for cls in [mp, fp, iv, mpf, mpc]: + for s in cls.__class__.__mro__: + assert isinstance(s, type) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_ode.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_ode.py new file mode 100644 index 0000000000000000000000000000000000000000..6b6dbffa79cfd4ca6dbf14f8591296ee48b16682 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_ode.py @@ -0,0 +1,73 @@ +#from mpmath.calculus import ODE_step_euler, ODE_step_rk4, odeint, arange +from mpmath import odefun, cos, sin, mpf, sinc, mp + +''' +solvers = [ODE_step_euler, ODE_step_rk4] + +def test_ode1(): + """ + Let's solve: + + x'' + w**2 * x = 0 + + i.e. x1 = x, x2 = x1': + + x1' = x2 + x2' = -x1 + """ + def derivs((x1, x2), t): + return x2, -x1 + + for solver in solvers: + t = arange(0, 3.1415926, 0.005) + sol = odeint(derivs, (0., 1.), t, solver) + x1 = [a[0] for a in sol] + x2 = [a[1] for a in sol] + # the result is x1 = sin(t), x2 = cos(t) + # let's just check the end points for t = pi + assert abs(x1[-1]) < 1e-2 + assert abs(x2[-1] - (-1)) < 1e-2 + +def test_ode2(): + """ + Let's solve: + + x' - x = 0 + + i.e. x = exp(x) + + """ + def derivs((x), t): + return x + + for solver in solvers: + t = arange(0, 1, 1e-3) + sol = odeint(derivs, (1.,), t, solver) + x = [a[0] for a in sol] + # the result is x = exp(t) + # let's just check the end point for t = 1, i.e. x = e + assert abs(x[-1] - 2.718281828) < 1e-2 +''' + +def test_odefun_rational(): + mp.dps = 15 + # A rational function + f = lambda t: 1/(1+mpf(t)**2) + g = odefun(lambda x, y: [-2*x*y[0]**2], 0, [f(0)]) + assert f(2).ae(g(2)[0]) + +def test_odefun_sinc_large(): + mp.dps = 15 + # Sinc function; test for large x + f = sinc + g = odefun(lambda x, y: [(cos(x)-y[0])/x], 1, [f(1)], tol=0.01, degree=5) + assert abs(f(100) - g(100)[0])/f(100) < 0.01 + +def test_odefun_harmonic(): + mp.dps = 15 + # Harmonic oscillator + f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0]) + for x in [0, 1, 2.5, 8, 3.7]: # we go back to 3.7 to check caching + c, s = f(x) + assert c.ae(cos(x)) + assert s.ae(sin(x)) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_pickle.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_pickle.py new file mode 100644 index 0000000000000000000000000000000000000000..c3d96e73a53603e0fa3f9525c5c0059725bdffb7 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_pickle.py @@ -0,0 +1,27 @@ +import os +import tempfile +import pickle + +from mpmath import * + +def pickler(obj): + fn = tempfile.mktemp() + + f = open(fn, 'wb') + pickle.dump(obj, f) + f.close() + + f = open(fn, 'rb') + obj2 = pickle.load(f) + f.close() + os.remove(fn) + + return obj2 + +def test_pickle(): + + obj = mpf('0.5') + assert obj == pickler(obj) + + obj = mpc('0.5','0.2') + assert obj == pickler(obj) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_power.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_power.py new file mode 100644 index 0000000000000000000000000000000000000000..7a2447a62c36f9e02df79b9a40a8603f8a69b1d8 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_power.py @@ -0,0 +1,156 @@ +from mpmath import * +from mpmath.libmp import * + +import random + +def test_fractional_pow(): + mp.dps = 15 + assert mpf(16) ** 2.5 == 1024 + assert mpf(64) ** 0.5 == 8 + assert mpf(64) ** -0.5 == 0.125 + assert mpf(16) ** -2.5 == 0.0009765625 + assert (mpf(10) ** 0.5).ae(3.1622776601683791) + assert (mpf(10) ** 2.5).ae(316.2277660168379) + assert (mpf(10) ** -0.5).ae(0.31622776601683794) + assert (mpf(10) ** -2.5).ae(0.0031622776601683794) + assert (mpf(10) ** 0.3).ae(1.9952623149688795) + assert (mpf(10) ** -0.3).ae(0.50118723362727224) + +def test_pow_integer_direction(): + """ + Test that inexact integer powers are rounded in the right + direction. + """ + random.seed(1234) + for prec in [10, 53, 200]: + for i in range(50): + a = random.randint(1<<(prec-1), 1< ab + + +def test_pow_epsilon_rounding(): + """ + Stress test directed rounding for powers with integer exponents. + Basically, we look at the following cases: + + >>> 1.0001 ** -5 # doctest: +SKIP + 0.99950014996500702 + >>> 0.9999 ** -5 # doctest: +SKIP + 1.000500150035007 + >>> (-1.0001) ** -5 # doctest: +SKIP + -0.99950014996500702 + >>> (-0.9999) ** -5 # doctest: +SKIP + -1.000500150035007 + + >>> 1.0001 ** -6 # doctest: +SKIP + 0.99940020994401269 + >>> 0.9999 ** -6 # doctest: +SKIP + 1.0006002100560125 + >>> (-1.0001) ** -6 # doctest: +SKIP + 0.99940020994401269 + >>> (-0.9999) ** -6 # doctest: +SKIP + 1.0006002100560125 + + etc. + + We run the tests with values a very small epsilon away from 1: + small enough that the result is indistinguishable from 1 when + rounded to nearest at the output precision. We check that the + result is not erroneously rounded to 1 in cases where the + rounding should be done strictly away from 1. + """ + + def powr(x, n, r): + return make_mpf(mpf_pow_int(x._mpf_, n, mp.prec, r)) + + for (inprec, outprec) in [(100, 20), (5000, 3000)]: + + mp.prec = inprec + + pos10001 = mpf(1) + mpf(2)**(-inprec+5) + pos09999 = mpf(1) - mpf(2)**(-inprec+5) + neg10001 = -pos10001 + neg09999 = -pos09999 + + mp.prec = outprec + r = round_up + assert powr(pos10001, 5, r) > 1 + assert powr(pos09999, 5, r) == 1 + assert powr(neg10001, 5, r) < -1 + assert powr(neg09999, 5, r) == -1 + assert powr(pos10001, 6, r) > 1 + assert powr(pos09999, 6, r) == 1 + assert powr(neg10001, 6, r) > 1 + assert powr(neg09999, 6, r) == 1 + + assert powr(pos10001, -5, r) == 1 + assert powr(pos09999, -5, r) > 1 + assert powr(neg10001, -5, r) == -1 + assert powr(neg09999, -5, r) < -1 + assert powr(pos10001, -6, r) == 1 + assert powr(pos09999, -6, r) > 1 + assert powr(neg10001, -6, r) == 1 + assert powr(neg09999, -6, r) > 1 + + r = round_down + assert powr(pos10001, 5, r) == 1 + assert powr(pos09999, 5, r) < 1 + assert powr(neg10001, 5, r) == -1 + assert powr(neg09999, 5, r) > -1 + assert powr(pos10001, 6, r) == 1 + assert powr(pos09999, 6, r) < 1 + assert powr(neg10001, 6, r) == 1 + assert powr(neg09999, 6, r) < 1 + + assert powr(pos10001, -5, r) < 1 + assert powr(pos09999, -5, r) == 1 + assert powr(neg10001, -5, r) > -1 + assert powr(neg09999, -5, r) == -1 + assert powr(pos10001, -6, r) < 1 + assert powr(pos09999, -6, r) == 1 + assert powr(neg10001, -6, r) < 1 + assert powr(neg09999, -6, r) == 1 + + r = round_ceiling + assert powr(pos10001, 5, r) > 1 + assert powr(pos09999, 5, r) == 1 + assert powr(neg10001, 5, r) == -1 + assert powr(neg09999, 5, r) > -1 + assert powr(pos10001, 6, r) > 1 + assert powr(pos09999, 6, r) == 1 + assert powr(neg10001, 6, r) > 1 + assert powr(neg09999, 6, r) == 1 + + assert powr(pos10001, -5, r) == 1 + assert powr(pos09999, -5, r) > 1 + assert powr(neg10001, -5, r) > -1 + assert powr(neg09999, -5, r) == -1 + assert powr(pos10001, -6, r) == 1 + assert powr(pos09999, -6, r) > 1 + assert powr(neg10001, -6, r) == 1 + assert powr(neg09999, -6, r) > 1 + + r = round_floor + assert powr(pos10001, 5, r) == 1 + assert powr(pos09999, 5, r) < 1 + assert powr(neg10001, 5, r) < -1 + assert powr(neg09999, 5, r) == -1 + assert powr(pos10001, 6, r) == 1 + assert powr(pos09999, 6, r) < 1 + assert powr(neg10001, 6, r) == 1 + assert powr(neg09999, 6, r) < 1 + + assert powr(pos10001, -5, r) < 1 + assert powr(pos09999, -5, r) == 1 + assert powr(neg10001, -5, r) == -1 + assert powr(neg09999, -5, r) < -1 + assert powr(pos10001, -6, r) < 1 + assert powr(pos09999, -6, r) == 1 + assert powr(neg10001, -6, r) < 1 + assert powr(neg09999, -6, r) == 1 + + mp.dps = 15 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_quad.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_quad.py new file mode 100644 index 0000000000000000000000000000000000000000..fc71c5f5ef9c0ecd876c988e7d033b321f065cdc --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_quad.py @@ -0,0 +1,95 @@ +import pytest +from mpmath import * + +def ae(a, b): + return abs(a-b) < 10**(-mp.dps+5) + +def test_basic_integrals(): + for prec in [15, 30, 100]: + mp.dps = prec + assert ae(quadts(lambda x: x**3 - 3*x**2, [-2, 4]), -12) + assert ae(quadgl(lambda x: x**3 - 3*x**2, [-2, 4]), -12) + assert ae(quadts(sin, [0, pi]), 2) + assert ae(quadts(sin, [0, 2*pi]), 0) + assert ae(quadts(exp, [-inf, -1]), 1/e) + assert ae(quadts(lambda x: exp(-x), [0, inf]), 1) + assert ae(quadts(lambda x: exp(-x*x), [-inf, inf]), sqrt(pi)) + assert ae(quadts(lambda x: 1/(1+x*x), [-1, 1]), pi/2) + assert ae(quadts(lambda x: 1/(1+x*x), [-inf, inf]), pi) + assert ae(quadts(lambda x: 2*sqrt(1-x*x), [-1, 1]), pi) + mp.dps = 15 + +def test_multiple_intervals(): + y,err = quad(lambda x: sign(x), [-0.5, 0.9, 1], maxdegree=2, error=True) + assert abs(y-0.5) < 2*err + +def test_quad_symmetry(): + assert quadts(sin, [-1, 1]) == 0 + assert quadgl(sin, [-1, 1]) == 0 + +def test_quad_infinite_mirror(): + # Check mirrored infinite interval + assert ae(quad(lambda x: exp(-x*x), [inf,-inf]), -sqrt(pi)) + assert ae(quad(lambda x: exp(x), [0,-inf]), -1) + +def test_quadgl_linear(): + assert quadgl(lambda x: x, [0, 1], maxdegree=1).ae(0.5) + +def test_complex_integration(): + assert quadts(lambda x: x, [0, 1+j]).ae(j) + +def test_quadosc(): + mp.dps = 15 + assert quadosc(lambda x: sin(x)/x, [0, inf], period=2*pi).ae(pi/2) + +# Double integrals +def test_double_trivial(): + assert ae(quadts(lambda x, y: x, [0, 1], [0, 1]), 0.5) + assert ae(quadts(lambda x, y: x, [-1, 1], [-1, 1]), 0.0) + +def test_double_1(): + assert ae(quadts(lambda x, y: cos(x+y/2), [-pi/2, pi/2], [0, pi]), 4) + +def test_double_2(): + assert ae(quadts(lambda x, y: (x-1)/((1-x*y)*log(x*y)), [0, 1], [0, 1]), euler) + +def test_double_3(): + assert ae(quadts(lambda x, y: 1/sqrt(1+x*x+y*y), [-1, 1], [-1, 1]), 4*log(2+sqrt(3))-2*pi/3) + +def test_double_4(): + assert ae(quadts(lambda x, y: 1/(1-x*x * y*y), [0, 1], [0, 1]), pi**2 / 8) + +def test_double_5(): + assert ae(quadts(lambda x, y: 1/(1-x*y), [0, 1], [0, 1]), pi**2 / 6) + +def test_double_6(): + assert ae(quadts(lambda x, y: exp(-(x+y)), [0, inf], [0, inf]), 1) + +def test_double_7(): + assert ae(quadts(lambda x, y: exp(-x*x-y*y), [-inf, inf], [-inf, inf]), pi) + + +# Test integrals from "Experimentation in Mathematics" by Borwein, +# Bailey & Girgensohn +def test_expmath_integrals(): + for prec in [15, 30, 50]: + mp.dps = prec + assert ae(quadts(lambda x: x/sinh(x), [0, inf]), pi**2 / 4) + assert ae(quadts(lambda x: log(x)**2 / (1+x**2), [0, inf]), pi**3 / 8) + assert ae(quadts(lambda x: (1+x**2)/(1+x**4), [0, inf]), pi/sqrt(2)) + assert ae(quadts(lambda x: log(x)/cosh(x)**2, [0, inf]), log(pi)-2*log(2)-euler) + assert ae(quadts(lambda x: log(1+x**3)/(1-x+x**2), [0, inf]), 2*pi*log(3)/sqrt(3)) + assert ae(quadts(lambda x: log(x)**2 / (x**2+x+1), [0, 1]), 8*pi**3 / (81*sqrt(3))) + assert ae(quadts(lambda x: log(cos(x))**2, [0, pi/2]), pi/2 * (log(2)**2+pi**2/12)) + assert ae(quadts(lambda x: x**2 / sin(x)**2, [0, pi/2]), pi*log(2)) + assert ae(quadts(lambda x: x**2/sqrt(exp(x)-1), [0, inf]), 4*pi*(log(2)**2 + pi**2/12)) + assert ae(quadts(lambda x: x*exp(-x)*sqrt(1-exp(-2*x)), [0, inf]), pi*(1+2*log(2))/8) + mp.dps = 15 + +# Do not reach full accuracy +@pytest.mark.xfail +def test_expmath_fail(): + assert ae(quadts(lambda x: sqrt(tan(x)), [0, pi/2]), pi*sqrt(2)/2) + assert ae(quadts(lambda x: atan(x)/(x*sqrt(1-x**2)), [0, 1]), pi*log(1+sqrt(2))/2) + assert ae(quadts(lambda x: log(1+x**2)/x**2, [0, 1]), pi/2-log(2)) + assert ae(quadts(lambda x: x**2/((1+x**4)*sqrt(1-x**4)), [0, 1]), pi/8) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_rootfinding.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_rootfinding.py new file mode 100644 index 0000000000000000000000000000000000000000..7c3c06463682eb1fd60efeb75b809bbb932a241c --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_rootfinding.py @@ -0,0 +1,91 @@ +import pytest +from mpmath import * +from mpmath.calculus.optimization import Secant, Muller, Bisection, Illinois, \ + Pegasus, Anderson, Ridder, ANewton, Newton, MNewton, MDNewton + +def test_findroot(): + # old tests, assuming secant + mp.dps = 15 + assert findroot(lambda x: 4*x-3, mpf(5)).ae(0.75) + assert findroot(sin, mpf(3)).ae(pi) + assert findroot(sin, (mpf(3), mpf(3.14))).ae(pi) + assert findroot(lambda x: x*x+1, mpc(2+2j)).ae(1j) + # test all solvers with 1 starting point + f = lambda x: cos(x) + for solver in [Newton, Secant, MNewton, Muller, ANewton]: + x = findroot(f, 2., solver=solver) + assert abs(f(x)) < eps + # test all solvers with interval of 2 points + for solver in [Secant, Muller, Bisection, Illinois, Pegasus, Anderson, + Ridder]: + x = findroot(f, (1., 2.), solver=solver) + assert abs(f(x)) < eps + # test types + f = lambda x: (x - 2)**2 + + assert isinstance(findroot(f, 1, tol=1e-10), mpf) + assert isinstance(iv.findroot(f, 1., tol=1e-10), iv.mpf) + assert isinstance(fp.findroot(f, 1, tol=1e-10), float) + assert isinstance(fp.findroot(f, 1+0j, tol=1e-10), complex) + + # issue 401 + with pytest.raises(ValueError): + with workprec(2): + findroot(lambda x: x**2 - 4456178*x + 60372201703370, + mpc(real='5.278e+13', imag='-5.278e+13')) + + # issue 192 + with pytest.raises(ValueError): + findroot(lambda x: -1, 0) + + # issue 387 + with pytest.raises(ValueError): + findroot(lambda p: (1 - p)**30 - 1, 0.9) + +def test_bisection(): + # issue 273 + assert findroot(lambda x: x**2-1,(0,2),solver='bisect') == 1 + +def test_mnewton(): + f = lambda x: polyval([1,3,3,1],x) + x = findroot(f, -0.9, solver='mnewton') + assert abs(f(x)) < eps + +def test_anewton(): + f = lambda x: (x - 2)**100 + x = findroot(f, 1., solver=ANewton) + assert abs(f(x)) < eps + +def test_muller(): + f = lambda x: (2 + x)**3 + 2 + x = findroot(f, 1., solver=Muller) + assert abs(f(x)) < eps + +def test_multiplicity(): + for i in range(1, 5): + assert multiplicity(lambda x: (x - 1)**i, 1) == i + assert multiplicity(lambda x: x**2, 1) == 0 + +def test_multidimensional(): + def f(*x): + return [3*x[0]**2-2*x[1]**2-1, x[0]**2-2*x[0]+x[1]**2+2*x[1]-8] + assert mnorm(jacobian(f, (1,-2)) - matrix([[6,8],[0,-2]]),1) < 1.e-7 + for x, error in MDNewton(mp, f, (1,-2), verbose=0, + norm=lambda x: norm(x, inf)): + pass + assert norm(f(*x), 2) < 1e-14 + # The Chinese mathematician Zhu Shijie was the very first to solve this + # nonlinear system 700 years ago + f1 = lambda x, y: -x + 2*y + f2 = lambda x, y: (x**2 + x*(y**2 - 2) - 4*y) / (x + 4) + f3 = lambda x, y: sqrt(x**2 + y**2) + def f(x, y): + f1x = f1(x, y) + return (f2(x, y) - f1x, f3(x, y) - f1x) + x = findroot(f, (10, 10)) + assert [int(round(i)) for i in x] == [3, 4] + +def test_trivial(): + assert findroot(lambda x: 0, 1) == 1 + assert findroot(lambda x: x, 0) == 0 + #assert findroot(lambda x, y: x + y, (1, -1)) == (1, -1) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_special.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_special.py new file mode 100644 index 0000000000000000000000000000000000000000..30825abd89ada00f937260cb51ef649546be7021 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_special.py @@ -0,0 +1,113 @@ +from mpmath import * + +def test_special(): + assert inf == inf + assert inf != -inf + assert -inf == -inf + assert inf != nan + assert nan != nan + assert isnan(nan) + assert --inf == inf + assert abs(inf) == inf + assert abs(-inf) == inf + assert abs(nan) != abs(nan) + + assert isnan(inf - inf) + assert isnan(inf + (-inf)) + assert isnan(-inf - (-inf)) + + assert isnan(inf + nan) + assert isnan(-inf + nan) + + assert mpf(2) + inf == inf + assert 2 + inf == inf + assert mpf(2) - inf == -inf + assert 2 - inf == -inf + + assert inf > 3 + assert 3 < inf + assert 3 > -inf + assert -inf < 3 + assert inf > mpf(3) + assert mpf(3) < inf + assert mpf(3) > -inf + assert -inf < mpf(3) + + assert not (nan < 3) + assert not (nan > 3) + + assert isnan(inf * 0) + assert isnan(-inf * 0) + assert inf * 3 == inf + assert inf * -3 == -inf + assert -inf * 3 == -inf + assert -inf * -3 == inf + assert inf * inf == inf + assert -inf * -inf == inf + + assert isnan(nan / 3) + assert inf / -3 == -inf + assert inf / 3 == inf + assert 3 / inf == 0 + assert -3 / inf == 0 + assert 0 / inf == 0 + assert isnan(inf / inf) + assert isnan(inf / -inf) + assert isnan(inf / nan) + + assert mpf('inf') == mpf('+inf') == inf + assert mpf('-inf') == -inf + assert isnan(mpf('nan')) + + assert isinf(inf) + assert isinf(-inf) + assert not isinf(mpf(0)) + assert not isinf(nan) + +def test_special_powers(): + assert inf**3 == inf + assert isnan(inf**0) + assert inf**-3 == 0 + assert (-inf)**2 == inf + assert (-inf)**3 == -inf + assert isnan((-inf)**0) + assert (-inf)**-2 == 0 + assert (-inf)**-3 == 0 + assert isnan(nan**5) + assert isnan(nan**0) + +def test_functions_special(): + assert exp(inf) == inf + assert exp(-inf) == 0 + assert isnan(exp(nan)) + assert log(inf) == inf + assert isnan(log(nan)) + assert isnan(sin(inf)) + assert isnan(sin(nan)) + assert atan(inf).ae(pi/2) + assert atan(-inf).ae(-pi/2) + assert isnan(sqrt(nan)) + assert sqrt(inf) == inf + +def test_convert_special(): + float_inf = 1e300 * 1e300 + float_ninf = -float_inf + float_nan = float_inf/float_ninf + assert mpf(3) * float_inf == inf + assert mpf(3) * float_ninf == -inf + assert isnan(mpf(3) * float_nan) + assert not (mpf(3) < float_nan) + assert not (mpf(3) > float_nan) + assert not (mpf(3) <= float_nan) + assert not (mpf(3) >= float_nan) + assert float(mpf('1e1000')) == float_inf + assert float(mpf('-1e1000')) == float_ninf + assert float(mpf('1e100000000000000000')) == float_inf + assert float(mpf('-1e100000000000000000')) == float_ninf + assert float(mpf('1e-100000000000000000')) == 0.0 + +def test_div_bug(): + assert isnan(nan/1) + assert isnan(nan/2) + assert inf/2 == inf + assert (-inf)/2 == -inf diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_str.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_str.py new file mode 100644 index 0000000000000000000000000000000000000000..569244f252c057ec1029b7efbd8b0ffbfbc47522 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_str.py @@ -0,0 +1,14 @@ +from mpmath import nstr, matrix, inf + +def test_nstr(): + m = matrix([[0.75, 0.190940654, -0.0299195971], + [0.190940654, 0.65625, 0.205663228], + [-0.0299195971, 0.205663228, 0.64453125e-20]]) + assert nstr(m, 4, min_fixed=-inf) == \ + '''[ 0.75 0.1909 -0.02992] +[ 0.1909 0.6563 0.2057] +[-0.02992 0.2057 0.000000000000000000006445]''' + assert nstr(m, 4) == \ + '''[ 0.75 0.1909 -0.02992] +[ 0.1909 0.6563 0.2057] +[-0.02992 0.2057 6.445e-21]''' diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_summation.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_summation.py new file mode 100644 index 0000000000000000000000000000000000000000..04ffd29f994e1e6310678eec292c0e03f2d6c725 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_summation.py @@ -0,0 +1,53 @@ +from mpmath import * + +def test_sumem(): + mp.dps = 15 + assert sumem(lambda k: 1/k**2.5, [50, 100]).ae(0.0012524505324784962) + assert sumem(lambda k: k**4 + 3*k + 1, [10, 100]).ae(2050333103) + +def test_nsum(): + mp.dps = 15 + assert nsum(lambda x: x**2, [1, 3]) == 14 + assert nsum(lambda k: 1/factorial(k), [0, inf]).ae(e) + assert nsum(lambda k: (-1)**(k+1) / k, [1, inf]).ae(log(2)) + assert nsum(lambda k: (-1)**(k+1) / k**2, [1, inf]).ae(pi**2 / 12) + assert nsum(lambda k: (-1)**k / log(k), [2, inf]).ae(0.9242998972229388) + assert nsum(lambda k: 1/k**2, [1, inf]).ae(pi**2 / 6) + assert nsum(lambda k: 2**k/fac(k), [0, inf]).ae(exp(2)) + assert nsum(lambda k: 1/k**2, [4, inf], method='e').ae(0.2838229557371153) + assert abs(fp.nsum(lambda k: 1/k**4, [1, fp.inf]) - 1.082323233711138) < 1e-5 + assert abs(fp.nsum(lambda k: 1/k**4, [1, fp.inf], method='e') - 1.082323233711138) < 1e-4 + +def test_nprod(): + mp.dps = 15 + assert nprod(lambda k: exp(1/k**2), [1,inf], method='r').ae(exp(pi**2/6)) + assert nprod(lambda x: x**2, [1, 3]) == 36 + +def test_fsum(): + mp.dps = 15 + assert fsum([]) == 0 + assert fsum([-4]) == -4 + assert fsum([2,3]) == 5 + assert fsum([1e-100,1]) == 1 + assert fsum([1,1e-100]) == 1 + assert fsum([1e100,1]) == 1e100 + assert fsum([1,1e100]) == 1e100 + assert fsum([1e-100,0]) == 1e-100 + assert fsum([1e-100,1e100,1e-100]) == 1e100 + assert fsum([2,1+1j,1]) == 4+1j + assert fsum([2,inf,3]) == inf + assert fsum([2,-1], absolute=1) == 3 + assert fsum([2,-1], squared=1) == 5 + assert fsum([1,1+j], squared=1) == 1+2j + assert fsum([1,3+4j], absolute=1) == 6 + assert fsum([1,2+3j], absolute=1, squared=1) == 14 + assert isnan(fsum([inf,-inf])) + assert fsum([inf,-inf], absolute=1) == inf + assert fsum([inf,-inf], squared=1) == inf + assert fsum([inf,-inf], absolute=1, squared=1) == inf + assert iv.fsum([1,mpi(2,3)]) == mpi(3,4) + +def test_fprod(): + mp.dps = 15 + assert fprod([]) == 1 + assert fprod([2,3]) == 6 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_trig.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_trig.py new file mode 100644 index 0000000000000000000000000000000000000000..c70a2a0ff4c44c784404ecdb15357d5b91a992d6 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_trig.py @@ -0,0 +1,136 @@ +from mpmath import * +from mpmath.libmp import * + +def test_trig_misc_hard(): + mp.prec = 53 + # Worst-case input for an IEEE double, from a paper by Kahan + x = ldexp(6381956970095103,797) + assert cos(x) == mpf('-4.6871659242546277e-19') + assert sin(x) == 1 + + mp.prec = 150 + a = mpf(10**50) + mp.prec = 53 + assert sin(a).ae(-0.7896724934293100827) + assert cos(a).ae(-0.6135286082336635622) + + # Check relative accuracy close to x = zero + assert sin(1e-100) == 1e-100 # when rounding to nearest + assert sin(1e-6).ae(9.999999999998333e-007, rel_eps=2e-15, abs_eps=0) + assert sin(1e-6j).ae(1.0000000000001666e-006j, rel_eps=2e-15, abs_eps=0) + assert sin(-1e-6j).ae(-1.0000000000001666e-006j, rel_eps=2e-15, abs_eps=0) + assert cos(1e-100) == 1 + assert cos(1e-6).ae(0.9999999999995) + assert cos(-1e-6j).ae(1.0000000000005) + assert tan(1e-100) == 1e-100 + assert tan(1e-6).ae(1.0000000000003335e-006, rel_eps=2e-15, abs_eps=0) + assert tan(1e-6j).ae(9.9999999999966644e-007j, rel_eps=2e-15, abs_eps=0) + assert tan(-1e-6j).ae(-9.9999999999966644e-007j, rel_eps=2e-15, abs_eps=0) + +def test_trig_near_zero(): + mp.dps = 15 + + for r in [round_nearest, round_down, round_up, round_floor, round_ceiling]: + assert sin(0, rounding=r) == 0 + assert cos(0, rounding=r) == 1 + + a = mpf('1e-100') + b = mpf('-1e-100') + + assert sin(a, rounding=round_nearest) == a + assert sin(a, rounding=round_down) < a + assert sin(a, rounding=round_floor) < a + assert sin(a, rounding=round_up) >= a + assert sin(a, rounding=round_ceiling) >= a + assert sin(b, rounding=round_nearest) == b + assert sin(b, rounding=round_down) > b + assert sin(b, rounding=round_floor) <= b + assert sin(b, rounding=round_up) <= b + assert sin(b, rounding=round_ceiling) > b + + assert cos(a, rounding=round_nearest) == 1 + assert cos(a, rounding=round_down) < 1 + assert cos(a, rounding=round_floor) < 1 + assert cos(a, rounding=round_up) == 1 + assert cos(a, rounding=round_ceiling) == 1 + assert cos(b, rounding=round_nearest) == 1 + assert cos(b, rounding=round_down) < 1 + assert cos(b, rounding=round_floor) < 1 + assert cos(b, rounding=round_up) == 1 + assert cos(b, rounding=round_ceiling) == 1 + + +def test_trig_near_n_pi(): + + mp.dps = 15 + a = [n*pi for n in [1, 2, 6, 11, 100, 1001, 10000, 100001]] + mp.dps = 135 + a.append(10**100 * pi) + mp.dps = 15 + + assert sin(a[0]) == mpf('1.2246467991473531772e-16') + assert sin(a[1]) == mpf('-2.4492935982947063545e-16') + assert sin(a[2]) == mpf('-7.3478807948841190634e-16') + assert sin(a[3]) == mpf('4.8998251578625894243e-15') + assert sin(a[4]) == mpf('1.9643867237284719452e-15') + assert sin(a[5]) == mpf('-8.8632615209684813458e-15') + assert sin(a[6]) == mpf('-4.8568235395684898392e-13') + assert sin(a[7]) == mpf('3.9087342299491231029e-11') + assert sin(a[8]) == mpf('-1.369235466754566993528e-36') + + r = round_nearest + assert cos(a[0], rounding=r) == -1 + assert cos(a[1], rounding=r) == 1 + assert cos(a[2], rounding=r) == 1 + assert cos(a[3], rounding=r) == -1 + assert cos(a[4], rounding=r) == 1 + assert cos(a[5], rounding=r) == -1 + assert cos(a[6], rounding=r) == 1 + assert cos(a[7], rounding=r) == -1 + assert cos(a[8], rounding=r) == 1 + + r = round_up + assert cos(a[0], rounding=r) == -1 + assert cos(a[1], rounding=r) == 1 + assert cos(a[2], rounding=r) == 1 + assert cos(a[3], rounding=r) == -1 + assert cos(a[4], rounding=r) == 1 + assert cos(a[5], rounding=r) == -1 + assert cos(a[6], rounding=r) == 1 + assert cos(a[7], rounding=r) == -1 + assert cos(a[8], rounding=r) == 1 + + r = round_down + assert cos(a[0], rounding=r) > -1 + assert cos(a[1], rounding=r) < 1 + assert cos(a[2], rounding=r) < 1 + assert cos(a[3], rounding=r) > -1 + assert cos(a[4], rounding=r) < 1 + assert cos(a[5], rounding=r) > -1 + assert cos(a[6], rounding=r) < 1 + assert cos(a[7], rounding=r) > -1 + assert cos(a[8], rounding=r) < 1 + + r = round_floor + assert cos(a[0], rounding=r) == -1 + assert cos(a[1], rounding=r) < 1 + assert cos(a[2], rounding=r) < 1 + assert cos(a[3], rounding=r) == -1 + assert cos(a[4], rounding=r) < 1 + assert cos(a[5], rounding=r) == -1 + assert cos(a[6], rounding=r) < 1 + assert cos(a[7], rounding=r) == -1 + assert cos(a[8], rounding=r) < 1 + + r = round_ceiling + assert cos(a[0], rounding=r) > -1 + assert cos(a[1], rounding=r) == 1 + assert cos(a[2], rounding=r) == 1 + assert cos(a[3], rounding=r) > -1 + assert cos(a[4], rounding=r) == 1 + assert cos(a[5], rounding=r) > -1 + assert cos(a[6], rounding=r) == 1 + assert cos(a[7], rounding=r) > -1 + assert cos(a[8], rounding=r) == 1 + + mp.dps = 15 diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_visualization.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_visualization.py new file mode 100644 index 0000000000000000000000000000000000000000..81ffd05194322f00e4c75dc02bc862b383468bff --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/test_visualization.py @@ -0,0 +1,32 @@ +""" +Limited tests of the visualization module. Right now it just makes +sure that passing custom Axes works. + +""" + +from mpmath import mp, fp + +def test_axes(): + try: + import matplotlib + version = matplotlib.__version__.split("-")[0] + version = version.split(".")[:2] + if [int(_) for _ in version] < [0,99]: + raise ImportError + import pylab + except ImportError: + print("\nSkipping test (pylab not available or too old version)\n") + return + fig = pylab.figure() + axes = fig.add_subplot(111) + for ctx in [mp, fp]: + ctx.plot(lambda x: x**2, [0, 3], axes=axes) + assert axes.get_xlabel() == 'x' + assert axes.get_ylabel() == 'f(x)' + + fig = pylab.figure() + axes = fig.add_subplot(111) + for ctx in [mp, fp]: + ctx.cplot(lambda z: z, [-2, 2], [-10, 10], axes=axes) + assert axes.get_xlabel() == 'Re(z)' + assert axes.get_ylabel() == 'Im(z)' diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/tests/torture.py b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/torture.py new file mode 100644 index 0000000000000000000000000000000000000000..845d5c6d7d017e51e1ed9a8fe3106cfa32fd967f --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/tests/torture.py @@ -0,0 +1,224 @@ +""" +Torture tests for asymptotics and high precision evaluation of +special functions. + +(Other torture tests may also be placed here.) + +Running this file (gmpy recommended!) takes several CPU minutes. +With Python 2.6+, multiprocessing is used automatically to run tests +in parallel if many cores are available. (A single test may take between +a second and several minutes; possibly more.) + +The idea: + +* We evaluate functions at positive, negative, imaginary, 45- and 135-degree + complex values with magnitudes between 10^-20 to 10^20, at precisions between + 5 and 150 digits (we can go even higher for fast functions). + +* Comparing the result from two different precision levels provides + a strong consistency check (particularly for functions that use + different algorithms at different precision levels). + +* That the computation finishes at all (without failure), within reasonable + time, provides a check that evaluation works at all: that the code runs, + that it doesn't get stuck in an infinite loop, and that it doesn't use + some extremely slowly algorithm where it could use a faster one. + +TODO: + +* Speed up those functions that take long to finish! +* Generalize to test more cases; more options. +* Implement a timeout mechanism. +* Some functions are notably absent, including the following: + * inverse trigonometric functions (some become inaccurate for complex arguments) + * ci, si (not implemented properly for large complex arguments) + * zeta functions (need to modify test not to try too large imaginary values) + * and others... + +""" + + +import sys, os +from timeit import default_timer as clock + +if "-nogmpy" in sys.argv: + sys.argv.remove('-nogmpy') + os.environ['MPMATH_NOGMPY'] = 'Y' + +filt = '' +if not sys.argv[-1].endswith(".py"): + filt = sys.argv[-1] + +from mpmath import * +from mpmath.libmp.backend import exec_ + +def test_asymp(f, maxdps=150, verbose=False, huge_range=False): + dps = [5,15,25,50,90,150,500,1500,5000,10000] + dps = [p for p in dps if p <= maxdps] + def check(x,y,p,inpt): + if abs(x-y)/abs(y) < workprec(20)(power)(10, -p+1): + return + print() + print("Error!") + print("Input:", inpt) + print("dps =", p) + print("Result 1:", x) + print("Result 2:", y) + print("Absolute error:", abs(x-y)) + print("Relative error:", abs(x-y)/abs(y)) + raise AssertionError + exponents = range(-20,20) + if huge_range: + exponents += [-1000, -100, -50, 50, 100, 1000] + for n in exponents: + if verbose: + sys.stdout.write(". ") + mp.dps = 25 + xpos = mpf(10)**n / 1.1287 + xneg = -xpos + ximag = xpos*j + xcomplex1 = xpos*(1+j) + xcomplex2 = xpos*(-1+j) + for i in range(len(dps)): + if verbose: + print("Testing dps = %s" % dps[i]) + mp.dps = dps[i] + new = f(xpos), f(xneg), f(ximag), f(xcomplex1), f(xcomplex2) + if i != 0: + p = dps[i-1] + check(prev[0], new[0], p, xpos) + check(prev[1], new[1], p, xneg) + check(prev[2], new[2], p, ximag) + check(prev[3], new[3], p, xcomplex1) + check(prev[4], new[4], p, xcomplex2) + prev = new + if verbose: + print() + +a1, a2, a3, a4, a5 = 1.5, -2.25, 3.125, 4, 2 + +def test_bernoulli_huge(): + p, q = bernfrac(9000) + assert p % 10**10 == 9636701091 + assert q == 4091851784687571609141381951327092757255270 + mp.dps = 15 + assert str(bernoulli(10**100)) == '-2.58183325604736e+987675256497386331227838638980680030172857347883537824464410652557820800494271520411283004120790908623' + mp.dps = 50 + assert str(bernoulli(10**100)) == '-2.5818332560473632073252488656039475548106223822913e+987675256497386331227838638980680030172857347883537824464410652557820800494271520411283004120790908623' + mp.dps = 15 + +cases = """\ +test_bernoulli_huge() +test_asymp(lambda z: +pi, maxdps=10000) +test_asymp(lambda z: +e, maxdps=10000) +test_asymp(lambda z: +ln2, maxdps=10000) +test_asymp(lambda z: +ln10, maxdps=10000) +test_asymp(lambda z: +phi, maxdps=10000) +test_asymp(lambda z: +catalan, maxdps=5000) +test_asymp(lambda z: +euler, maxdps=5000) +test_asymp(lambda z: +glaisher, maxdps=1000) +test_asymp(lambda z: +khinchin, maxdps=1000) +test_asymp(lambda z: +twinprime, maxdps=150) +test_asymp(lambda z: stieltjes(2), maxdps=150) +test_asymp(lambda z: +mertens, maxdps=150) +test_asymp(lambda z: +apery, maxdps=5000) +test_asymp(sqrt, maxdps=10000, huge_range=True) +test_asymp(cbrt, maxdps=5000, huge_range=True) +test_asymp(lambda z: root(z,4), maxdps=5000, huge_range=True) +test_asymp(lambda z: root(z,-5), maxdps=5000, huge_range=True) +test_asymp(exp, maxdps=5000, huge_range=True) +test_asymp(expm1, maxdps=1500) +test_asymp(ln, maxdps=5000, huge_range=True) +test_asymp(cosh, maxdps=5000) +test_asymp(sinh, maxdps=5000) +test_asymp(tanh, maxdps=1500) +test_asymp(sin, maxdps=5000, huge_range=True) +test_asymp(cos, maxdps=5000, huge_range=True) +test_asymp(tan, maxdps=1500) +test_asymp(agm, maxdps=1500, huge_range=True) +test_asymp(ellipk, maxdps=1500) +test_asymp(ellipe, maxdps=1500) +test_asymp(lambertw, huge_range=True) +test_asymp(lambda z: lambertw(z,-1)) +test_asymp(lambda z: lambertw(z,1)) +test_asymp(lambda z: lambertw(z,4)) +test_asymp(gamma) +test_asymp(loggamma) # huge_range=True ? +test_asymp(ei) +test_asymp(e1) +test_asymp(li, huge_range=True) +test_asymp(ci) +test_asymp(si) +test_asymp(chi) +test_asymp(shi) +test_asymp(erf) +test_asymp(erfc) +test_asymp(erfi) +test_asymp(lambda z: besselj(2, z)) +test_asymp(lambda z: bessely(2, z)) +test_asymp(lambda z: besseli(2, z)) +test_asymp(lambda z: besselk(2, z)) +test_asymp(lambda z: besselj(-2.25, z)) +test_asymp(lambda z: bessely(-2.25, z)) +test_asymp(lambda z: besseli(-2.25, z)) +test_asymp(lambda z: besselk(-2.25, z)) +test_asymp(airyai) +test_asymp(airybi) +test_asymp(lambda z: hyp0f1(a1, z)) +test_asymp(lambda z: hyp1f1(a1, a2, z)) +test_asymp(lambda z: hyp1f2(a1, a2, a3, z)) +test_asymp(lambda z: hyp2f0(a1, a2, z)) +test_asymp(lambda z: hyperu(a1, a2, z)) +test_asymp(lambda z: hyp2f1(a1, a2, a3, z)) +test_asymp(lambda z: hyp2f2(a1, a2, a3, a4, z)) +test_asymp(lambda z: hyp2f3(a1, a2, a3, a4, a5, z)) +test_asymp(lambda z: coulombf(a1, a2, z)) +test_asymp(lambda z: coulombg(a1, a2, z)) +test_asymp(lambda z: polylog(2,z)) +test_asymp(lambda z: polylog(3,z)) +test_asymp(lambda z: polylog(-2,z)) +test_asymp(lambda z: expint(4, z)) +test_asymp(lambda z: expint(-4, z)) +test_asymp(lambda z: expint(2.25, z)) +test_asymp(lambda z: gammainc(2.5, z, 5)) +test_asymp(lambda z: gammainc(2.5, 5, z)) +test_asymp(lambda z: hermite(3, z)) +test_asymp(lambda z: hermite(2.5, z)) +test_asymp(lambda z: legendre(3, z)) +test_asymp(lambda z: legendre(4, z)) +test_asymp(lambda z: legendre(2.5, z)) +test_asymp(lambda z: legenp(a1, a2, z)) +test_asymp(lambda z: legenq(a1, a2, z), maxdps=90) # abnormally slow +test_asymp(lambda z: jtheta(1, z, 0.5)) +test_asymp(lambda z: jtheta(2, z, 0.5)) +test_asymp(lambda z: jtheta(3, z, 0.5)) +test_asymp(lambda z: jtheta(4, z, 0.5)) +test_asymp(lambda z: jtheta(1, z, 0.5, 1)) +test_asymp(lambda z: jtheta(2, z, 0.5, 1)) +test_asymp(lambda z: jtheta(3, z, 0.5, 1)) +test_asymp(lambda z: jtheta(4, z, 0.5, 1)) +test_asymp(barnesg, maxdps=90) +""" + +def testit(line): + if filt in line: + print(line) + t1 = clock() + exec_(line, globals(), locals()) + t2 = clock() + elapsed = t2-t1 + print("Time:", elapsed, "for", line, "(OK)") + +if __name__ == '__main__': + try: + from multiprocessing import Pool + mapf = Pool(None).map + print("Running tests with multiprocessing") + except ImportError: + print("Not using multiprocessing") + mapf = map + t1 = clock() + tasks = cases.splitlines() + mapf(testit, tasks) + t2 = clock() + print("Cumulative wall time:", t2-t1) diff --git a/env-llmeval/lib/python3.10/site-packages/mpmath/usertools.py b/env-llmeval/lib/python3.10/site-packages/mpmath/usertools.py new file mode 100644 index 0000000000000000000000000000000000000000..8028a4c46f1c635a6857f1f2de48ac6675d3c6d3 --- /dev/null +++ b/env-llmeval/lib/python3.10/site-packages/mpmath/usertools.py @@ -0,0 +1,93 @@ + +def monitor(f, input='print', output='print'): + """ + Returns a wrapped copy of *f* that monitors evaluation by calling + *input* with every input (*args*, *kwargs*) passed to *f* and + *output* with every value returned from *f*. The default action + (specify using the special string value ``'print'``) is to print + inputs and outputs to stdout, along with the total evaluation + count:: + + >>> from mpmath import * + >>> mp.dps = 5; mp.pretty = False + >>> diff(monitor(exp), 1) # diff will eval f(x-h) and f(x+h) + in 0 (mpf('0.99999999906867742538452148'),) {} + out 0 mpf('2.7182818259274480055282064') + in 1 (mpf('1.0000000009313225746154785'),) {} + out 1 mpf('2.7182818309906424675501024') + mpf('2.7182808') + + To disable either the input or the output handler, you may + pass *None* as argument. + + Custom input and output handlers may be used e.g. to store + results for later analysis:: + + >>> mp.dps = 15 + >>> input = [] + >>> output = [] + >>> findroot(monitor(sin, input.append, output.append), 3.0) + mpf('3.1415926535897932') + >>> len(input) # Count number of evaluations + 9 + >>> print(input[3]); print(output[3]) + ((mpf('3.1415076583334066'),), {}) + 8.49952562843408e-5 + >>> print(input[4]); print(output[4]) + ((mpf('3.1415928201669122'),), {}) + -1.66577118985331e-7 + + """ + if not input: + input = lambda v: None + elif input == 'print': + incount = [0] + def input(value): + args, kwargs = value + print("in %s %r %r" % (incount[0], args, kwargs)) + incount[0] += 1 + if not output: + output = lambda v: None + elif output == 'print': + outcount = [0] + def output(value): + print("out %s %r" % (outcount[0], value)) + outcount[0] += 1 + def f_monitored(*args, **kwargs): + input((args, kwargs)) + v = f(*args, **kwargs) + output(v) + return v + return f_monitored + +def timing(f, *args, **kwargs): + """ + Returns time elapsed for evaluating ``f()``. Optionally arguments + may be passed to time the execution of ``f(*args, **kwargs)``. + + If the first call is very quick, ``f`` is called + repeatedly and the best time is returned. + """ + once = kwargs.get('once') + if 'once' in kwargs: + del kwargs['once'] + if args or kwargs: + if len(args) == 1 and not kwargs: + arg = args[0] + g = lambda: f(arg) + else: + g = lambda: f(*args, **kwargs) + else: + g = f + from timeit import default_timer as clock + t1=clock(); v=g(); t2=clock(); t=t2-t1 + if t > 0.05 or once: + return t + for i in range(3): + t1=clock(); + # Evaluate multiple times because the timer function + # has a significant overhead + g();g();g();g();g();g();g();g();g();g() + t2=clock() + t=min(t,(t2-t1)/10) + return t