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Inzynierka/Lib/site-packages/scipy/special/tests/test_basic.py

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2023-06-02 12:51:02 +02:00
# this program corresponds to special.py
### Means test is not done yet
# E Means test is giving error (E)
# F Means test is failing (F)
# EF Means test is giving error and Failing
#! Means test is segfaulting
# 8 Means test runs forever
### test_besselpoly
### test_mathieu_a
### test_mathieu_even_coef
### test_mathieu_odd_coef
### test_modfresnelp
### test_modfresnelm
# test_pbdv_seq
### test_pbvv_seq
### test_sph_harm
import itertools
import platform
import sys
import numpy as np
from numpy import (array, isnan, r_, arange, finfo, pi, sin, cos, tan, exp,
log, zeros, sqrt, asarray, inf, nan_to_num, real, arctan, float_)
import pytest
from pytest import raises as assert_raises
from numpy.testing import (assert_equal, assert_almost_equal,
assert_array_equal, assert_array_almost_equal, assert_approx_equal,
assert_, assert_allclose, assert_array_almost_equal_nulp,
suppress_warnings)
from scipy import special
import scipy.special._ufuncs as cephes
from scipy.special import ellipe, ellipk, ellipkm1
from scipy.special import elliprc, elliprd, elliprf, elliprg, elliprj
from scipy.special import mathieu_odd_coef, mathieu_even_coef
from scipy.special._testutils import with_special_errors, \
assert_func_equal, FuncData
import math
class TestCephes:
def test_airy(self):
cephes.airy(0)
def test_airye(self):
cephes.airye(0)
def test_binom(self):
n = np.array([0.264, 4, 5.2, 17])
k = np.array([2, 0.4, 7, 3.3])
nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
).reshape(2, -1).T
rknown = np.array([[-0.097152, 0.9263051596159367, 0.01858423645695389,
-0.007581020651518199],[6, 2.0214389119675666, 0, 2.9827344527963846],
[10.92, 2.22993515861399, -0.00585728, 10.468891352063146],
[136, 3.5252179590758828, 19448, 1024.5526916174495]])
assert_func_equal(cephes.binom, rknown.ravel(), nk, rtol=1e-13)
# Test branches in implementation
np.random.seed(1234)
n = np.r_[np.arange(-7, 30), 1000*np.random.rand(30) - 500]
k = np.arange(0, 102)
nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
).reshape(2, -1).T
assert_func_equal(cephes.binom,
cephes.binom(nk[:,0], nk[:,1] * (1 + 1e-15)),
nk,
atol=1e-10, rtol=1e-10)
def test_binom_2(self):
# Test branches in implementation
np.random.seed(1234)
n = np.r_[np.logspace(1, 300, 20)]
k = np.arange(0, 102)
nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
).reshape(2, -1).T
assert_func_equal(cephes.binom,
cephes.binom(nk[:,0], nk[:,1] * (1 + 1e-15)),
nk,
atol=1e-10, rtol=1e-10)
def test_binom_exact(self):
@np.vectorize
def binom_int(n, k):
n = int(n)
k = int(k)
num = int(1)
den = int(1)
for i in range(1, k+1):
num *= i + n - k
den *= i
return float(num/den)
np.random.seed(1234)
n = np.arange(1, 15)
k = np.arange(0, 15)
nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
).reshape(2, -1).T
nk = nk[nk[:,0] >= nk[:,1]]
assert_func_equal(cephes.binom,
binom_int(nk[:,0], nk[:,1]),
nk,
atol=0, rtol=0)
def test_binom_nooverflow_8346(self):
# Test (binom(n, k) doesn't overflow prematurely */
dataset = [
(1000, 500, 2.70288240945436551e+299),
(1002, 501, 1.08007396880791225e+300),
(1004, 502, 4.31599279169058121e+300),
(1006, 503, 1.72468101616263781e+301),
(1008, 504, 6.89188009236419153e+301),
(1010, 505, 2.75402257948335448e+302),
(1012, 506, 1.10052048531923757e+303),
(1014, 507, 4.39774063758732849e+303),
(1016, 508, 1.75736486108312519e+304),
(1018, 509, 7.02255427788423734e+304),
(1020, 510, 2.80626776829962255e+305),
(1022, 511, 1.12140876377061240e+306),
(1024, 512, 4.48125455209897109e+306),
(1026, 513, 1.79075474304149900e+307),
(1028, 514, 7.15605105487789676e+307)
]
dataset = np.asarray(dataset)
FuncData(cephes.binom, dataset, (0, 1), 2, rtol=1e-12).check()
def test_bdtr(self):
assert_equal(cephes.bdtr(1,1,0.5),1.0)
def test_bdtri(self):
assert_equal(cephes.bdtri(1,3,0.5),0.5)
def test_bdtrc(self):
assert_equal(cephes.bdtrc(1,3,0.5),0.5)
def test_bdtrin(self):
assert_equal(cephes.bdtrin(1,0,1),5.0)
def test_bdtrik(self):
cephes.bdtrik(1,3,0.5)
def test_bei(self):
assert_equal(cephes.bei(0),0.0)
def test_beip(self):
assert_equal(cephes.beip(0),0.0)
def test_ber(self):
assert_equal(cephes.ber(0),1.0)
def test_berp(self):
assert_equal(cephes.berp(0),0.0)
def test_besselpoly(self):
assert_equal(cephes.besselpoly(0,0,0),1.0)
def test_beta(self):
assert_equal(cephes.beta(1,1),1.0)
assert_allclose(cephes.beta(-100.3, 1e-200), cephes.gamma(1e-200))
assert_allclose(cephes.beta(0.0342, 171), 24.070498359873497,
rtol=1e-13, atol=0)
def test_betainc(self):
assert_equal(cephes.betainc(1,1,1),1.0)
assert_allclose(cephes.betainc(0.0342, 171, 1e-10), 0.55269916901806648)
def test_betaln(self):
assert_equal(cephes.betaln(1,1),0.0)
assert_allclose(cephes.betaln(-100.3, 1e-200), cephes.gammaln(1e-200))
assert_allclose(cephes.betaln(0.0342, 170), 3.1811881124242447,
rtol=1e-14, atol=0)
def test_betaincinv(self):
assert_equal(cephes.betaincinv(1,1,1),1.0)
assert_allclose(cephes.betaincinv(0.0342, 171, 0.25),
8.4231316935498957e-21, rtol=3e-12, atol=0)
def test_beta_inf(self):
assert_(np.isinf(special.beta(-1, 2)))
def test_btdtr(self):
assert_equal(cephes.btdtr(1,1,1),1.0)
def test_btdtri(self):
assert_equal(cephes.btdtri(1,1,1),1.0)
def test_btdtria(self):
assert_equal(cephes.btdtria(1,1,1),5.0)
def test_btdtrib(self):
assert_equal(cephes.btdtrib(1,1,1),5.0)
def test_cbrt(self):
assert_approx_equal(cephes.cbrt(1),1.0)
def test_chdtr(self):
assert_equal(cephes.chdtr(1,0),0.0)
def test_chdtrc(self):
assert_equal(cephes.chdtrc(1,0),1.0)
def test_chdtri(self):
assert_equal(cephes.chdtri(1,1),0.0)
def test_chdtriv(self):
assert_equal(cephes.chdtriv(0,0),5.0)
def test_chndtr(self):
assert_equal(cephes.chndtr(0,1,0),0.0)
# Each row holds (x, nu, lam, expected_value)
# These values were computed using Wolfram Alpha with
# CDF[NoncentralChiSquareDistribution[nu, lam], x]
values = np.array([
[25.00, 20.0, 400, 4.1210655112396197139e-57],
[25.00, 8.00, 250, 2.3988026526832425878e-29],
[0.001, 8.00, 40., 5.3761806201366039084e-24],
[0.010, 8.00, 40., 5.45396231055999457039e-20],
[20.00, 2.00, 107, 1.39390743555819597802e-9],
[22.50, 2.00, 107, 7.11803307138105870671e-9],
[25.00, 2.00, 107, 3.11041244829864897313e-8],
[3.000, 2.00, 1.0, 0.62064365321954362734],
[350.0, 300., 10., 0.93880128006276407710],
[100.0, 13.5, 10., 0.99999999650104210949],
[700.0, 20.0, 400, 0.99999999925680650105],
[150.0, 13.5, 10., 0.99999999999999983046],
[160.0, 13.5, 10., 0.99999999999999999518], # 1.0
])
cdf = cephes.chndtr(values[:, 0], values[:, 1], values[:, 2])
assert_allclose(cdf, values[:, 3], rtol=1e-12)
assert_almost_equal(cephes.chndtr(np.inf, np.inf, 0), 2.0)
assert_almost_equal(cephes.chndtr(2, 1, np.inf), 0.0)
assert_(np.isnan(cephes.chndtr(np.nan, 1, 2)))
assert_(np.isnan(cephes.chndtr(5, np.nan, 2)))
assert_(np.isnan(cephes.chndtr(5, 1, np.nan)))
def test_chndtridf(self):
assert_equal(cephes.chndtridf(0,0,1),5.0)
def test_chndtrinc(self):
assert_equal(cephes.chndtrinc(0,1,0),5.0)
def test_chndtrix(self):
assert_equal(cephes.chndtrix(0,1,0),0.0)
def test_cosdg(self):
assert_equal(cephes.cosdg(0),1.0)
def test_cosm1(self):
assert_equal(cephes.cosm1(0),0.0)
def test_cotdg(self):
assert_almost_equal(cephes.cotdg(45),1.0)
def test_dawsn(self):
assert_equal(cephes.dawsn(0),0.0)
assert_allclose(cephes.dawsn(1.23), 0.50053727749081767)
def test_diric(self):
# Test behavior near multiples of 2pi. Regression test for issue
# described in gh-4001.
n_odd = [1, 5, 25]
x = np.array(2*np.pi + 5e-5).astype(np.float32)
assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=7)
x = np.array(2*np.pi + 1e-9).astype(np.float64)
assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=15)
x = np.array(2*np.pi + 1e-15).astype(np.float64)
assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=15)
if hasattr(np, 'float128'):
# No float128 available in 32-bit numpy
x = np.array(2*np.pi + 1e-12).astype(np.float128)
assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=19)
n_even = [2, 4, 24]
x = np.array(2*np.pi + 1e-9).astype(np.float64)
assert_almost_equal(special.diric(x, n_even), -1.0, decimal=15)
# Test at some values not near a multiple of pi
x = np.arange(0.2*np.pi, 1.0*np.pi, 0.2*np.pi)
octave_result = [0.872677996249965, 0.539344662916632,
0.127322003750035, -0.206011329583298]
assert_almost_equal(special.diric(x, 3), octave_result, decimal=15)
def test_diric_broadcasting(self):
x = np.arange(5)
n = np.array([1, 3, 7])
assert_(special.diric(x[:, np.newaxis], n).shape == (x.size, n.size))
def test_ellipe(self):
assert_equal(cephes.ellipe(1),1.0)
def test_ellipeinc(self):
assert_equal(cephes.ellipeinc(0,1),0.0)
def test_ellipj(self):
cephes.ellipj(0,1)
def test_ellipk(self):
assert_allclose(ellipk(0), pi/2)
def test_ellipkinc(self):
assert_equal(cephes.ellipkinc(0,0),0.0)
def test_erf(self):
assert_equal(cephes.erf(0), 0.0)
def test_erf_symmetry(self):
x = 5.905732037710919
assert_equal(cephes.erf(x) + cephes.erf(-x), 0.0)
def test_erfc(self):
assert_equal(cephes.erfc(0), 1.0)
def test_exp10(self):
assert_approx_equal(cephes.exp10(2),100.0)
def test_exp2(self):
assert_equal(cephes.exp2(2),4.0)
def test_expm1(self):
assert_equal(cephes.expm1(0),0.0)
assert_equal(cephes.expm1(np.inf), np.inf)
assert_equal(cephes.expm1(-np.inf), -1)
assert_equal(cephes.expm1(np.nan), np.nan)
def test_expm1_complex(self):
expm1 = cephes.expm1
assert_equal(expm1(0 + 0j), 0 + 0j)
assert_equal(expm1(complex(np.inf, 0)), complex(np.inf, 0))
assert_equal(expm1(complex(np.inf, 1)), complex(np.inf, np.inf))
assert_equal(expm1(complex(np.inf, 2)), complex(-np.inf, np.inf))
assert_equal(expm1(complex(np.inf, 4)), complex(-np.inf, -np.inf))
assert_equal(expm1(complex(np.inf, 5)), complex(np.inf, -np.inf))
assert_equal(expm1(complex(1, np.inf)), complex(np.nan, np.nan))
assert_equal(expm1(complex(0, np.inf)), complex(np.nan, np.nan))
assert_equal(expm1(complex(np.inf, np.inf)), complex(np.inf, np.nan))
assert_equal(expm1(complex(-np.inf, np.inf)), complex(-1, 0))
assert_equal(expm1(complex(-np.inf, np.nan)), complex(-1, 0))
assert_equal(expm1(complex(np.inf, np.nan)), complex(np.inf, np.nan))
assert_equal(expm1(complex(0, np.nan)), complex(np.nan, np.nan))
assert_equal(expm1(complex(1, np.nan)), complex(np.nan, np.nan))
assert_equal(expm1(complex(np.nan, 1)), complex(np.nan, np.nan))
assert_equal(expm1(complex(np.nan, np.nan)), complex(np.nan, np.nan))
@pytest.mark.xfail(reason='The real part of expm1(z) bad at these points')
def test_expm1_complex_hard(self):
# The real part of this function is difficult to evaluate when
# z.real = -log(cos(z.imag)).
y = np.array([0.1, 0.2, 0.3, 5, 11, 20])
x = -np.log(np.cos(y))
z = x + 1j*y
# evaluate using mpmath.expm1 with dps=1000
expected = np.array([-5.5507901846769623e-17+0.10033467208545054j,
2.4289354732893695e-18+0.20271003550867248j,
4.5235500262585768e-17+0.30933624960962319j,
7.8234305217489006e-17-3.3805150062465863j,
-1.3685191953697676e-16-225.95084645419513j,
8.7175620481291045e-17+2.2371609442247422j])
found = cephes.expm1(z)
# this passes.
assert_array_almost_equal_nulp(found.imag, expected.imag, 3)
# this fails.
assert_array_almost_equal_nulp(found.real, expected.real, 20)
def test_fdtr(self):
assert_equal(cephes.fdtr(1, 1, 0), 0.0)
# Computed using Wolfram Alpha: CDF[FRatioDistribution[1e-6, 5], 10]
assert_allclose(cephes.fdtr(1e-6, 5, 10), 0.9999940790193488,
rtol=1e-12)
def test_fdtrc(self):
assert_equal(cephes.fdtrc(1, 1, 0), 1.0)
# Computed using Wolfram Alpha:
# 1 - CDF[FRatioDistribution[2, 1/10], 1e10]
assert_allclose(cephes.fdtrc(2, 0.1, 1e10), 0.27223784621293512,
rtol=1e-12)
def test_fdtri(self):
assert_allclose(cephes.fdtri(1, 1, [0.499, 0.501]),
array([0.9937365, 1.00630298]), rtol=1e-6)
# From Wolfram Alpha:
# CDF[FRatioDistribution[1/10, 1], 3] = 0.8756751669632105666874...
p = 0.8756751669632105666874
assert_allclose(cephes.fdtri(0.1, 1, p), 3, rtol=1e-12)
@pytest.mark.xfail(reason='Returns nan on i686.')
def test_fdtri_mysterious_failure(self):
assert_allclose(cephes.fdtri(1, 1, 0.5), 1)
def test_fdtridfd(self):
assert_equal(cephes.fdtridfd(1,0,0),5.0)
def test_fresnel(self):
assert_equal(cephes.fresnel(0),(0.0,0.0))
def test_gamma(self):
assert_equal(cephes.gamma(5),24.0)
def test_gammainccinv(self):
assert_equal(cephes.gammainccinv(5,1),0.0)
def test_gammaln(self):
cephes.gammaln(10)
def test_gammasgn(self):
vals = np.array([-4, -3.5, -2.3, 1, 4.2], np.float64)
assert_array_equal(cephes.gammasgn(vals), np.sign(cephes.rgamma(vals)))
def test_gdtr(self):
assert_equal(cephes.gdtr(1,1,0),0.0)
def test_gdtr_inf(self):
assert_equal(cephes.gdtr(1,1,np.inf),1.0)
def test_gdtrc(self):
assert_equal(cephes.gdtrc(1,1,0),1.0)
def test_gdtria(self):
assert_equal(cephes.gdtria(0,1,1),0.0)
def test_gdtrib(self):
cephes.gdtrib(1,0,1)
# assert_equal(cephes.gdtrib(1,0,1),5.0)
def test_gdtrix(self):
cephes.gdtrix(1,1,.1)
def test_hankel1(self):
cephes.hankel1(1,1)
def test_hankel1e(self):
cephes.hankel1e(1,1)
def test_hankel2(self):
cephes.hankel2(1,1)
def test_hankel2e(self):
cephes.hankel2e(1,1)
def test_hyp1f1(self):
assert_approx_equal(cephes.hyp1f1(1,1,1), exp(1.0))
assert_approx_equal(cephes.hyp1f1(3,4,-6), 0.026056422099537251095)
cephes.hyp1f1(1,1,1)
def test_hyp2f1(self):
assert_equal(cephes.hyp2f1(1,1,1,0),1.0)
def test_i0(self):
assert_equal(cephes.i0(0),1.0)
def test_i0e(self):
assert_equal(cephes.i0e(0),1.0)
def test_i1(self):
assert_equal(cephes.i1(0),0.0)
def test_i1e(self):
assert_equal(cephes.i1e(0),0.0)
def test_it2i0k0(self):
cephes.it2i0k0(1)
def test_it2j0y0(self):
cephes.it2j0y0(1)
def test_it2struve0(self):
cephes.it2struve0(1)
def test_itairy(self):
cephes.itairy(1)
def test_iti0k0(self):
assert_equal(cephes.iti0k0(0),(0.0,0.0))
def test_itj0y0(self):
assert_equal(cephes.itj0y0(0),(0.0,0.0))
def test_itmodstruve0(self):
assert_equal(cephes.itmodstruve0(0),0.0)
def test_itstruve0(self):
assert_equal(cephes.itstruve0(0),0.0)
def test_iv(self):
assert_equal(cephes.iv(1,0),0.0)
def _check_ive(self):
assert_equal(cephes.ive(1,0),0.0)
def test_j0(self):
assert_equal(cephes.j0(0),1.0)
def test_j1(self):
assert_equal(cephes.j1(0),0.0)
def test_jn(self):
assert_equal(cephes.jn(0,0),1.0)
def test_jv(self):
assert_equal(cephes.jv(0,0),1.0)
def _check_jve(self):
assert_equal(cephes.jve(0,0),1.0)
def test_k0(self):
cephes.k0(2)
def test_k0e(self):
cephes.k0e(2)
def test_k1(self):
cephes.k1(2)
def test_k1e(self):
cephes.k1e(2)
def test_kei(self):
cephes.kei(2)
def test_keip(self):
assert_equal(cephes.keip(0),0.0)
def test_ker(self):
cephes.ker(2)
def test_kerp(self):
cephes.kerp(2)
def _check_kelvin(self):
cephes.kelvin(2)
def test_kn(self):
cephes.kn(1,1)
def test_kolmogi(self):
assert_equal(cephes.kolmogi(1),0.0)
assert_(np.isnan(cephes.kolmogi(np.nan)))
def test_kolmogorov(self):
assert_equal(cephes.kolmogorov(0), 1.0)
def test_kolmogp(self):
assert_equal(cephes._kolmogp(0), -0.0)
def test_kolmogc(self):
assert_equal(cephes._kolmogc(0), 0.0)
def test_kolmogci(self):
assert_equal(cephes._kolmogci(0), 0.0)
assert_(np.isnan(cephes._kolmogci(np.nan)))
def _check_kv(self):
cephes.kv(1,1)
def _check_kve(self):
cephes.kve(1,1)
def test_log1p(self):
log1p = cephes.log1p
assert_equal(log1p(0), 0.0)
assert_equal(log1p(-1), -np.inf)
assert_equal(log1p(-2), np.nan)
assert_equal(log1p(np.inf), np.inf)
def test_log1p_complex(self):
log1p = cephes.log1p
c = complex
assert_equal(log1p(0 + 0j), 0 + 0j)
assert_equal(log1p(c(-1, 0)), c(-np.inf, 0))
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "invalid value encountered in multiply")
assert_allclose(log1p(c(1, np.inf)), c(np.inf, np.pi/2))
assert_equal(log1p(c(1, np.nan)), c(np.nan, np.nan))
assert_allclose(log1p(c(-np.inf, 1)), c(np.inf, np.pi))
assert_equal(log1p(c(np.inf, 1)), c(np.inf, 0))
assert_allclose(log1p(c(-np.inf, np.inf)), c(np.inf, 3*np.pi/4))
assert_allclose(log1p(c(np.inf, np.inf)), c(np.inf, np.pi/4))
assert_equal(log1p(c(np.inf, np.nan)), c(np.inf, np.nan))
assert_equal(log1p(c(-np.inf, np.nan)), c(np.inf, np.nan))
assert_equal(log1p(c(np.nan, np.inf)), c(np.inf, np.nan))
assert_equal(log1p(c(np.nan, 1)), c(np.nan, np.nan))
assert_equal(log1p(c(np.nan, np.nan)), c(np.nan, np.nan))
def test_lpmv(self):
assert_equal(cephes.lpmv(0,0,1),1.0)
def test_mathieu_a(self):
assert_equal(cephes.mathieu_a(1,0),1.0)
def test_mathieu_b(self):
assert_equal(cephes.mathieu_b(1,0),1.0)
def test_mathieu_cem(self):
assert_equal(cephes.mathieu_cem(1,0,0),(1.0,0.0))
# Test AMS 20.2.27
@np.vectorize
def ce_smallq(m, q, z):
z *= np.pi/180
if m == 0:
return 2**(-0.5) * (1 - .5*q*cos(2*z)) # + O(q^2)
elif m == 1:
return cos(z) - q/8 * cos(3*z) # + O(q^2)
elif m == 2:
return cos(2*z) - q*(cos(4*z)/12 - 1/4) # + O(q^2)
else:
return cos(m*z) - q*(cos((m+2)*z)/(4*(m+1)) - cos((m-2)*z)/(4*(m-1))) # + O(q^2)
m = np.arange(0, 100)
q = np.r_[0, np.logspace(-30, -9, 10)]
assert_allclose(cephes.mathieu_cem(m[:,None], q[None,:], 0.123)[0],
ce_smallq(m[:,None], q[None,:], 0.123),
rtol=1e-14, atol=0)
def test_mathieu_sem(self):
assert_equal(cephes.mathieu_sem(1,0,0),(0.0,1.0))
# Test AMS 20.2.27
@np.vectorize
def se_smallq(m, q, z):
z *= np.pi/180
if m == 1:
return sin(z) - q/8 * sin(3*z) # + O(q^2)
elif m == 2:
return sin(2*z) - q*sin(4*z)/12 # + O(q^2)
else:
return sin(m*z) - q*(sin((m+2)*z)/(4*(m+1)) - sin((m-2)*z)/(4*(m-1))) # + O(q^2)
m = np.arange(1, 100)
q = np.r_[0, np.logspace(-30, -9, 10)]
assert_allclose(cephes.mathieu_sem(m[:,None], q[None,:], 0.123)[0],
se_smallq(m[:,None], q[None,:], 0.123),
rtol=1e-14, atol=0)
def test_mathieu_modcem1(self):
assert_equal(cephes.mathieu_modcem1(1,0,0),(0.0,0.0))
def test_mathieu_modcem2(self):
cephes.mathieu_modcem2(1,1,1)
# Test reflection relation AMS 20.6.19
m = np.arange(0, 4)[:,None,None]
q = np.r_[np.logspace(-2, 2, 10)][None,:,None]
z = np.linspace(0, 1, 7)[None,None,:]
y1 = cephes.mathieu_modcem2(m, q, -z)[0]
fr = -cephes.mathieu_modcem2(m, q, 0)[0] / cephes.mathieu_modcem1(m, q, 0)[0]
y2 = -cephes.mathieu_modcem2(m, q, z)[0] - 2*fr*cephes.mathieu_modcem1(m, q, z)[0]
assert_allclose(y1, y2, rtol=1e-10)
def test_mathieu_modsem1(self):
assert_equal(cephes.mathieu_modsem1(1,0,0),(0.0,0.0))
def test_mathieu_modsem2(self):
cephes.mathieu_modsem2(1,1,1)
# Test reflection relation AMS 20.6.20
m = np.arange(1, 4)[:,None,None]
q = np.r_[np.logspace(-2, 2, 10)][None,:,None]
z = np.linspace(0, 1, 7)[None,None,:]
y1 = cephes.mathieu_modsem2(m, q, -z)[0]
fr = cephes.mathieu_modsem2(m, q, 0)[1] / cephes.mathieu_modsem1(m, q, 0)[1]
y2 = cephes.mathieu_modsem2(m, q, z)[0] - 2*fr*cephes.mathieu_modsem1(m, q, z)[0]
assert_allclose(y1, y2, rtol=1e-10)
def test_mathieu_overflow(self):
# Check that these return NaNs instead of causing a SEGV
assert_equal(cephes.mathieu_cem(10000, 0, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_sem(10000, 0, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_cem(10000, 1.5, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_sem(10000, 1.5, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_modcem1(10000, 1.5, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_modsem1(10000, 1.5, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_modcem2(10000, 1.5, 1.3), (np.nan, np.nan))
assert_equal(cephes.mathieu_modsem2(10000, 1.5, 1.3), (np.nan, np.nan))
def test_mathieu_ticket_1847(self):
# Regression test --- this call had some out-of-bounds access
# and could return nan occasionally
for k in range(60):
v = cephes.mathieu_modsem2(2, 100, -1)
# Values from ACM TOMS 804 (derivate by numerical differentiation)
assert_allclose(v[0], 0.1431742913063671074347, rtol=1e-10)
assert_allclose(v[1], 0.9017807375832909144719, rtol=1e-4)
def test_modfresnelm(self):
cephes.modfresnelm(0)
def test_modfresnelp(self):
cephes.modfresnelp(0)
def _check_modstruve(self):
assert_equal(cephes.modstruve(1,0),0.0)
def test_nbdtr(self):
assert_equal(cephes.nbdtr(1,1,1),1.0)
def test_nbdtrc(self):
assert_equal(cephes.nbdtrc(1,1,1),0.0)
def test_nbdtri(self):
assert_equal(cephes.nbdtri(1,1,1),1.0)
def __check_nbdtrik(self):
cephes.nbdtrik(1,.4,.5)
def test_nbdtrin(self):
assert_equal(cephes.nbdtrin(1,0,0),5.0)
def test_ncfdtr(self):
assert_equal(cephes.ncfdtr(1,1,1,0),0.0)
def test_ncfdtri(self):
assert_equal(cephes.ncfdtri(1, 1, 1, 0), 0.0)
f = [0.5, 1, 1.5]
p = cephes.ncfdtr(2, 3, 1.5, f)
assert_allclose(cephes.ncfdtri(2, 3, 1.5, p), f)
def test_ncfdtridfd(self):
dfd = [1, 2, 3]
p = cephes.ncfdtr(2, dfd, 0.25, 15)
assert_allclose(cephes.ncfdtridfd(2, p, 0.25, 15), dfd)
def test_ncfdtridfn(self):
dfn = [0.1, 1, 2, 3, 1e4]
p = cephes.ncfdtr(dfn, 2, 0.25, 15)
assert_allclose(cephes.ncfdtridfn(p, 2, 0.25, 15), dfn, rtol=1e-5)
def test_ncfdtrinc(self):
nc = [0.5, 1.5, 2.0]
p = cephes.ncfdtr(2, 3, nc, 15)
assert_allclose(cephes.ncfdtrinc(2, 3, p, 15), nc)
def test_nctdtr(self):
assert_equal(cephes.nctdtr(1,0,0),0.5)
assert_equal(cephes.nctdtr(9, 65536, 45), 0.0)
assert_approx_equal(cephes.nctdtr(np.inf, 1., 1.), 0.5, 5)
assert_(np.isnan(cephes.nctdtr(2., np.inf, 10.)))
assert_approx_equal(cephes.nctdtr(2., 1., np.inf), 1.)
assert_(np.isnan(cephes.nctdtr(np.nan, 1., 1.)))
assert_(np.isnan(cephes.nctdtr(2., np.nan, 1.)))
assert_(np.isnan(cephes.nctdtr(2., 1., np.nan)))
def __check_nctdtridf(self):
cephes.nctdtridf(1,0.5,0)
def test_nctdtrinc(self):
cephes.nctdtrinc(1,0,0)
def test_nctdtrit(self):
cephes.nctdtrit(.1,0.2,.5)
def test_nrdtrimn(self):
assert_approx_equal(cephes.nrdtrimn(0.5,1,1),1.0)
def test_nrdtrisd(self):
assert_allclose(cephes.nrdtrisd(0.5,0.5,0.5), 0.0,
atol=0, rtol=0)
def test_obl_ang1(self):
cephes.obl_ang1(1,1,1,0)
def test_obl_ang1_cv(self):
result = cephes.obl_ang1_cv(1,1,1,1,0)
assert_almost_equal(result[0],1.0)
assert_almost_equal(result[1],0.0)
def _check_obl_cv(self):
assert_equal(cephes.obl_cv(1,1,0),2.0)
def test_obl_rad1(self):
cephes.obl_rad1(1,1,1,0)
def test_obl_rad1_cv(self):
cephes.obl_rad1_cv(1,1,1,1,0)
def test_obl_rad2(self):
cephes.obl_rad2(1,1,1,0)
def test_obl_rad2_cv(self):
cephes.obl_rad2_cv(1,1,1,1,0)
def test_pbdv(self):
assert_equal(cephes.pbdv(1,0),(0.0,1.0))
def test_pbvv(self):
cephes.pbvv(1,0)
def test_pbwa(self):
cephes.pbwa(1,0)
def test_pdtr(self):
val = cephes.pdtr(0, 1)
assert_almost_equal(val, np.exp(-1))
# Edge case: m = 0.
val = cephes.pdtr([0, 1, 2], 0)
assert_array_equal(val, [1, 1, 1])
def test_pdtrc(self):
val = cephes.pdtrc(0, 1)
assert_almost_equal(val, 1 - np.exp(-1))
# Edge case: m = 0.
val = cephes.pdtrc([0, 1, 2], 0.0)
assert_array_equal(val, [0, 0, 0])
def test_pdtri(self):
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "floating point number truncated to an integer")
cephes.pdtri(0.5,0.5)
def test_pdtrik(self):
k = cephes.pdtrik(0.5, 1)
assert_almost_equal(cephes.gammaincc(k + 1, 1), 0.5)
# Edge case: m = 0 or very small.
k = cephes.pdtrik([[0], [0.25], [0.95]], [0, 1e-20, 1e-6])
assert_array_equal(k, np.zeros((3, 3)))
def test_pro_ang1(self):
cephes.pro_ang1(1,1,1,0)
def test_pro_ang1_cv(self):
assert_array_almost_equal(cephes.pro_ang1_cv(1,1,1,1,0),
array((1.0,0.0)))
def _check_pro_cv(self):
assert_equal(cephes.pro_cv(1,1,0),2.0)
def test_pro_rad1(self):
cephes.pro_rad1(1,1,1,0.1)
def test_pro_rad1_cv(self):
cephes.pro_rad1_cv(1,1,1,1,0)
def test_pro_rad2(self):
cephes.pro_rad2(1,1,1,0)
def test_pro_rad2_cv(self):
cephes.pro_rad2_cv(1,1,1,1,0)
def test_psi(self):
cephes.psi(1)
def test_radian(self):
assert_equal(cephes.radian(0,0,0),0)
def test_rgamma(self):
assert_equal(cephes.rgamma(1),1.0)
def test_round(self):
assert_equal(cephes.round(3.4),3.0)
assert_equal(cephes.round(-3.4),-3.0)
assert_equal(cephes.round(3.6),4.0)
assert_equal(cephes.round(-3.6),-4.0)
assert_equal(cephes.round(3.5),4.0)
assert_equal(cephes.round(-3.5),-4.0)
def test_shichi(self):
cephes.shichi(1)
def test_sici(self):
cephes.sici(1)
s, c = cephes.sici(np.inf)
assert_almost_equal(s, np.pi * 0.5)
assert_almost_equal(c, 0)
s, c = cephes.sici(-np.inf)
assert_almost_equal(s, -np.pi * 0.5)
assert_(np.isnan(c), "cosine integral(-inf) is not nan")
def test_sindg(self):
assert_equal(cephes.sindg(90),1.0)
def test_smirnov(self):
assert_equal(cephes.smirnov(1,.1),0.9)
assert_(np.isnan(cephes.smirnov(1,np.nan)))
def test_smirnovp(self):
assert_equal(cephes._smirnovp(1, .1), -1)
assert_equal(cephes._smirnovp(2, 0.75), -2*(0.25)**(2-1))
assert_equal(cephes._smirnovp(3, 0.75), -3*(0.25)**(3-1))
assert_(np.isnan(cephes._smirnovp(1, np.nan)))
def test_smirnovc(self):
assert_equal(cephes._smirnovc(1,.1),0.1)
assert_(np.isnan(cephes._smirnovc(1,np.nan)))
x10 = np.linspace(0, 1, 11, endpoint=True)
assert_almost_equal(cephes._smirnovc(3, x10), 1-cephes.smirnov(3, x10))
x4 = np.linspace(0, 1, 5, endpoint=True)
assert_almost_equal(cephes._smirnovc(4, x4), 1-cephes.smirnov(4, x4))
def test_smirnovi(self):
assert_almost_equal(cephes.smirnov(1,cephes.smirnovi(1,0.4)),0.4)
assert_almost_equal(cephes.smirnov(1,cephes.smirnovi(1,0.6)),0.6)
assert_(np.isnan(cephes.smirnovi(1,np.nan)))
def test_smirnovci(self):
assert_almost_equal(cephes._smirnovc(1,cephes._smirnovci(1,0.4)),0.4)
assert_almost_equal(cephes._smirnovc(1,cephes._smirnovci(1,0.6)),0.6)
assert_(np.isnan(cephes._smirnovci(1,np.nan)))
def test_spence(self):
assert_equal(cephes.spence(1),0.0)
def test_stdtr(self):
assert_equal(cephes.stdtr(1,0),0.5)
assert_almost_equal(cephes.stdtr(1,1), 0.75)
assert_almost_equal(cephes.stdtr(1,2), 0.852416382349)
def test_stdtridf(self):
cephes.stdtridf(0.7,1)
def test_stdtrit(self):
cephes.stdtrit(1,0.7)
def test_struve(self):
assert_equal(cephes.struve(0,0),0.0)
def test_tandg(self):
assert_equal(cephes.tandg(45),1.0)
def test_tklmbda(self):
assert_almost_equal(cephes.tklmbda(1,1),1.0)
def test_y0(self):
cephes.y0(1)
def test_y1(self):
cephes.y1(1)
def test_yn(self):
cephes.yn(1,1)
def test_yv(self):
cephes.yv(1,1)
def _check_yve(self):
cephes.yve(1,1)
def test_wofz(self):
z = [complex(624.2,-0.26123), complex(-0.4,3.), complex(0.6,2.),
complex(-1.,1.), complex(-1.,-9.), complex(-1.,9.),
complex(-0.0000000234545,1.1234), complex(-3.,5.1),
complex(-53,30.1), complex(0.0,0.12345),
complex(11,1), complex(-22,-2), complex(9,-28),
complex(21,-33), complex(1e5,1e5), complex(1e14,1e14)
]
w = [
complex(-3.78270245518980507452677445620103199303131110e-7,
0.000903861276433172057331093754199933411710053155),
complex(0.1764906227004816847297495349730234591778719532788,
-0.02146550539468457616788719893991501311573031095617),
complex(0.2410250715772692146133539023007113781272362309451,
0.06087579663428089745895459735240964093522265589350),
complex(0.30474420525691259245713884106959496013413834051768,
-0.20821893820283162728743734725471561394145872072738),
complex(7.317131068972378096865595229600561710140617977e34,
8.321873499714402777186848353320412813066170427e34),
complex(0.0615698507236323685519612934241429530190806818395,
-0.00676005783716575013073036218018565206070072304635),
complex(0.3960793007699874918961319170187598400134746631,
-5.593152259116644920546186222529802777409274656e-9),
complex(0.08217199226739447943295069917990417630675021771804,
-0.04701291087643609891018366143118110965272615832184),
complex(0.00457246000350281640952328010227885008541748668738,
-0.00804900791411691821818731763401840373998654987934),
complex(0.8746342859608052666092782112565360755791467973338452,
0.),
complex(0.00468190164965444174367477874864366058339647648741,
0.0510735563901306197993676329845149741675029197050),
complex(-0.0023193175200187620902125853834909543869428763219,
-0.025460054739731556004902057663500272721780776336),
complex(9.11463368405637174660562096516414499772662584e304,
3.97101807145263333769664875189354358563218932e305),
complex(-4.4927207857715598976165541011143706155432296e281,
-2.8019591213423077494444700357168707775769028e281),
complex(2.820947917809305132678577516325951485807107151e-6,
2.820947917668257736791638444590253942253354058e-6),
complex(2.82094791773878143474039725787438662716372268e-15,
2.82094791773878143474039725773333923127678361e-15)
]
assert_func_equal(cephes.wofz, w, z, rtol=1e-13)
class TestAiry:
def test_airy(self):
# This tests the airy function to ensure 8 place accuracy in computation
x = special.airy(.99)
assert_array_almost_equal(x,array([0.13689066,-0.16050153,1.19815925,0.92046818]),8)
x = special.airy(.41)
assert_array_almost_equal(x,array([0.25238916,-.23480512,0.80686202,0.51053919]),8)
x = special.airy(-.36)
assert_array_almost_equal(x,array([0.44508477,-0.23186773,0.44939534,0.48105354]),8)
def test_airye(self):
a = special.airye(0.01)
b = special.airy(0.01)
b1 = [None]*4
for n in range(2):
b1[n] = b[n]*exp(2.0/3.0*0.01*sqrt(0.01))
for n in range(2,4):
b1[n] = b[n]*exp(-abs(real(2.0/3.0*0.01*sqrt(0.01))))
assert_array_almost_equal(a,b1,6)
def test_bi_zeros(self):
bi = special.bi_zeros(2)
bia = (array([-1.17371322, -3.2710930]),
array([-2.29443968, -4.07315509]),
array([-0.45494438, 0.39652284]),
array([0.60195789, -0.76031014]))
assert_array_almost_equal(bi,bia,4)
bi = special.bi_zeros(5)
assert_array_almost_equal(bi[0],array([-1.173713222709127,
-3.271093302836352,
-4.830737841662016,
-6.169852128310251,
-7.376762079367764]),11)
assert_array_almost_equal(bi[1],array([-2.294439682614122,
-4.073155089071828,
-5.512395729663599,
-6.781294445990305,
-7.940178689168587]),10)
assert_array_almost_equal(bi[2],array([-0.454944383639657,
0.396522836094465,
-0.367969161486959,
0.349499116831805,
-0.336026240133662]),11)
assert_array_almost_equal(bi[3],array([0.601957887976239,
-0.760310141492801,
0.836991012619261,
-0.88947990142654,
0.929983638568022]),10)
def test_ai_zeros(self):
ai = special.ai_zeros(1)
assert_array_almost_equal(ai,(array([-2.33810741]),
array([-1.01879297]),
array([0.5357]),
array([0.7012])),4)
def test_ai_zeros_big(self):
z, zp, ai_zpx, aip_zx = special.ai_zeros(50000)
ai_z, aip_z, _, _ = special.airy(z)
ai_zp, aip_zp, _, _ = special.airy(zp)
ai_envelope = 1/abs(z)**(1./4)
aip_envelope = abs(zp)**(1./4)
# Check values
assert_allclose(ai_zpx, ai_zp, rtol=1e-10)
assert_allclose(aip_zx, aip_z, rtol=1e-10)
# Check they are zeros
assert_allclose(ai_z/ai_envelope, 0, atol=1e-10, rtol=0)
assert_allclose(aip_zp/aip_envelope, 0, atol=1e-10, rtol=0)
# Check first zeros, DLMF 9.9.1
assert_allclose(z[:6],
[-2.3381074105, -4.0879494441, -5.5205598281,
-6.7867080901, -7.9441335871, -9.0226508533], rtol=1e-10)
assert_allclose(zp[:6],
[-1.0187929716, -3.2481975822, -4.8200992112,
-6.1633073556, -7.3721772550, -8.4884867340], rtol=1e-10)
def test_bi_zeros_big(self):
z, zp, bi_zpx, bip_zx = special.bi_zeros(50000)
_, _, bi_z, bip_z = special.airy(z)
_, _, bi_zp, bip_zp = special.airy(zp)
bi_envelope = 1/abs(z)**(1./4)
bip_envelope = abs(zp)**(1./4)
# Check values
assert_allclose(bi_zpx, bi_zp, rtol=1e-10)
assert_allclose(bip_zx, bip_z, rtol=1e-10)
# Check they are zeros
assert_allclose(bi_z/bi_envelope, 0, atol=1e-10, rtol=0)
assert_allclose(bip_zp/bip_envelope, 0, atol=1e-10, rtol=0)
# Check first zeros, DLMF 9.9.2
assert_allclose(z[:6],
[-1.1737132227, -3.2710933028, -4.8307378417,
-6.1698521283, -7.3767620794, -8.4919488465], rtol=1e-10)
assert_allclose(zp[:6],
[-2.2944396826, -4.0731550891, -5.5123957297,
-6.7812944460, -7.9401786892, -9.0195833588], rtol=1e-10)
class TestAssocLaguerre:
def test_assoc_laguerre(self):
a1 = special.genlaguerre(11,1)
a2 = special.assoc_laguerre(.2,11,1)
assert_array_almost_equal(a2,a1(.2),8)
a2 = special.assoc_laguerre(1,11,1)
assert_array_almost_equal(a2,a1(1),8)
class TestBesselpoly:
def test_besselpoly(self):
pass
class TestKelvin:
def test_bei(self):
mbei = special.bei(2)
assert_almost_equal(mbei, 0.9722916273066613,5) # this may not be exact
def test_beip(self):
mbeip = special.beip(2)
assert_almost_equal(mbeip,0.91701361338403631,5) # this may not be exact
def test_ber(self):
mber = special.ber(2)
assert_almost_equal(mber,0.75173418271380821,5) # this may not be exact
def test_berp(self):
mberp = special.berp(2)
assert_almost_equal(mberp,-0.49306712470943909,5) # this may not be exact
def test_bei_zeros(self):
# Abramowitz & Stegun, Table 9.12
bi = special.bei_zeros(5)
assert_array_almost_equal(bi,array([5.02622,
9.45541,
13.89349,
18.33398,
22.77544]),4)
def test_beip_zeros(self):
bip = special.beip_zeros(5)
assert_array_almost_equal(bip,array([3.772673304934953,
8.280987849760042,
12.742147523633703,
17.193431752512542,
21.641143941167325]),8)
def test_ber_zeros(self):
ber = special.ber_zeros(5)
assert_array_almost_equal(ber,array([2.84892,
7.23883,
11.67396,
16.11356,
20.55463]),4)
def test_berp_zeros(self):
brp = special.berp_zeros(5)
assert_array_almost_equal(brp,array([6.03871,
10.51364,
14.96844,
19.41758,
23.86430]),4)
def test_kelvin(self):
mkelv = special.kelvin(2)
assert_array_almost_equal(mkelv,(special.ber(2) + special.bei(2)*1j,
special.ker(2) + special.kei(2)*1j,
special.berp(2) + special.beip(2)*1j,
special.kerp(2) + special.keip(2)*1j),8)
def test_kei(self):
mkei = special.kei(2)
assert_almost_equal(mkei,-0.20240006776470432,5)
def test_keip(self):
mkeip = special.keip(2)
assert_almost_equal(mkeip,0.21980790991960536,5)
def test_ker(self):
mker = special.ker(2)
assert_almost_equal(mker,-0.041664513991509472,5)
def test_kerp(self):
mkerp = special.kerp(2)
assert_almost_equal(mkerp,-0.10660096588105264,5)
def test_kei_zeros(self):
kei = special.kei_zeros(5)
assert_array_almost_equal(kei,array([3.91467,
8.34422,
12.78256,
17.22314,
21.66464]),4)
def test_keip_zeros(self):
keip = special.keip_zeros(5)
assert_array_almost_equal(keip,array([4.93181,
9.40405,
13.85827,
18.30717,
22.75379]),4)
# numbers come from 9.9 of A&S pg. 381
def test_kelvin_zeros(self):
tmp = special.kelvin_zeros(5)
berz,beiz,kerz,keiz,berpz,beipz,kerpz,keipz = tmp
assert_array_almost_equal(berz,array([2.84892,
7.23883,
11.67396,
16.11356,
20.55463]),4)
assert_array_almost_equal(beiz,array([5.02622,
9.45541,
13.89349,
18.33398,
22.77544]),4)
assert_array_almost_equal(kerz,array([1.71854,
6.12728,
10.56294,
15.00269,
19.44382]),4)
assert_array_almost_equal(keiz,array([3.91467,
8.34422,
12.78256,
17.22314,
21.66464]),4)
assert_array_almost_equal(berpz,array([6.03871,
10.51364,
14.96844,
19.41758,
23.86430]),4)
assert_array_almost_equal(beipz,array([3.77267,
# table from 1927 had 3.77320
# but this is more accurate
8.28099,
12.74215,
17.19343,
21.64114]),4)
assert_array_almost_equal(kerpz,array([2.66584,
7.17212,
11.63218,
16.08312,
20.53068]),4)
assert_array_almost_equal(keipz,array([4.93181,
9.40405,
13.85827,
18.30717,
22.75379]),4)
def test_ker_zeros(self):
ker = special.ker_zeros(5)
assert_array_almost_equal(ker,array([1.71854,
6.12728,
10.56294,
15.00269,
19.44381]),4)
def test_kerp_zeros(self):
kerp = special.kerp_zeros(5)
assert_array_almost_equal(kerp,array([2.66584,
7.17212,
11.63218,
16.08312,
20.53068]),4)
class TestBernoulli:
def test_bernoulli(self):
brn = special.bernoulli(5)
assert_array_almost_equal(brn,array([1.0000,
-0.5000,
0.1667,
0.0000,
-0.0333,
0.0000]),4)
class TestBeta:
def test_beta(self):
bet = special.beta(2,4)
betg = (special.gamma(2)*special.gamma(4))/special.gamma(6)
assert_almost_equal(bet,betg,8)
def test_betaln(self):
betln = special.betaln(2,4)
bet = log(abs(special.beta(2,4)))
assert_almost_equal(betln,bet,8)
def test_betainc(self):
btinc = special.betainc(1,1,.2)
assert_almost_equal(btinc,0.2,8)
def test_betaincinv(self):
y = special.betaincinv(2,4,.5)
comp = special.betainc(2,4,y)
assert_almost_equal(comp,.5,5)
class TestCombinatorics:
def test_comb(self):
assert_array_almost_equal(special.comb([10, 10], [3, 4]), [120., 210.])
assert_almost_equal(special.comb(10, 3), 120.)
assert_equal(special.comb(10, 3, exact=True), 120)
assert_equal(special.comb(10, 3, exact=True, repetition=True), 220)
assert_allclose([special.comb(20, k, exact=True) for k in range(21)],
special.comb(20, list(range(21))), atol=1e-15)
ii = np.iinfo(int).max + 1
assert_equal(special.comb(ii, ii-1, exact=True), ii)
expected = 100891344545564193334812497256
assert special.comb(100, 50, exact=True) == expected
@pytest.mark.parametrize("repetition", [True, False])
@pytest.mark.parametrize("legacy", [True, False])
@pytest.mark.parametrize("k", [3.5, 3])
@pytest.mark.parametrize("N", [4.5, 4])
def test_comb_legacy(self, N, k, legacy, repetition):
# test is only relevant for exact=True
if legacy and (N != int(N) or k != int(k)):
with pytest.warns(
DeprecationWarning,
match=r"Non-integer arguments are currently being cast to",
):
result = special.comb(N, k, exact=True, legacy=legacy,
repetition=repetition)
else:
result = special.comb(N, k, exact=True, legacy=legacy,
repetition=repetition)
if legacy:
# for exact=True and legacy=True, cast input arguments, else don't
if repetition:
# the casting in legacy mode happens AFTER transforming N & k,
# so rounding can change (e.g. both floats, but sum to int);
# hence we need to emulate the repetition-transformation here
N, k = int(N + k - 1), int(k)
repetition = False
else:
N, k = int(N), int(k)
# expected result is the same as with exact=False
expected = special.comb(N, k, legacy=legacy, repetition=repetition)
assert_equal(result, expected)
def test_comb_with_np_int64(self):
n = 70
k = 30
np_n = np.int64(n)
np_k = np.int64(k)
res_np = special.comb(np_n, np_k, exact=True)
res_py = special.comb(n, k, exact=True)
assert res_np == res_py
def test_comb_zeros(self):
assert_equal(special.comb(2, 3, exact=True), 0)
assert_equal(special.comb(-1, 3, exact=True), 0)
assert_equal(special.comb(2, -1, exact=True), 0)
assert_equal(special.comb(2, -1, exact=False), 0)
assert_array_almost_equal(special.comb([2, -1, 2, 10], [3, 3, -1, 3]),
[0., 0., 0., 120.])
def test_perm(self):
assert_array_almost_equal(special.perm([10, 10], [3, 4]), [720., 5040.])
assert_almost_equal(special.perm(10, 3), 720.)
assert_equal(special.perm(10, 3, exact=True), 720)
def test_perm_zeros(self):
assert_equal(special.perm(2, 3, exact=True), 0)
assert_equal(special.perm(-1, 3, exact=True), 0)
assert_equal(special.perm(2, -1, exact=True), 0)
assert_equal(special.perm(2, -1, exact=False), 0)
assert_array_almost_equal(special.perm([2, -1, 2, 10], [3, 3, -1, 3]),
[0., 0., 0., 720.])
class TestTrigonometric:
def test_cbrt(self):
cb = special.cbrt(27)
cbrl = 27**(1.0/3.0)
assert_approx_equal(cb,cbrl)
def test_cbrtmore(self):
cb1 = special.cbrt(27.9)
cbrl1 = 27.9**(1.0/3.0)
assert_almost_equal(cb1,cbrl1,8)
def test_cosdg(self):
cdg = special.cosdg(90)
cdgrl = cos(pi/2.0)
assert_almost_equal(cdg,cdgrl,8)
def test_cosdgmore(self):
cdgm = special.cosdg(30)
cdgmrl = cos(pi/6.0)
assert_almost_equal(cdgm,cdgmrl,8)
def test_cosm1(self):
cs = (special.cosm1(0),special.cosm1(.3),special.cosm1(pi/10))
csrl = (cos(0)-1,cos(.3)-1,cos(pi/10)-1)
assert_array_almost_equal(cs,csrl,8)
def test_cotdg(self):
ct = special.cotdg(30)
ctrl = tan(pi/6.0)**(-1)
assert_almost_equal(ct,ctrl,8)
def test_cotdgmore(self):
ct1 = special.cotdg(45)
ctrl1 = tan(pi/4.0)**(-1)
assert_almost_equal(ct1,ctrl1,8)
def test_specialpoints(self):
assert_almost_equal(special.cotdg(45), 1.0, 14)
assert_almost_equal(special.cotdg(-45), -1.0, 14)
assert_almost_equal(special.cotdg(90), 0.0, 14)
assert_almost_equal(special.cotdg(-90), 0.0, 14)
assert_almost_equal(special.cotdg(135), -1.0, 14)
assert_almost_equal(special.cotdg(-135), 1.0, 14)
assert_almost_equal(special.cotdg(225), 1.0, 14)
assert_almost_equal(special.cotdg(-225), -1.0, 14)
assert_almost_equal(special.cotdg(270), 0.0, 14)
assert_almost_equal(special.cotdg(-270), 0.0, 14)
assert_almost_equal(special.cotdg(315), -1.0, 14)
assert_almost_equal(special.cotdg(-315), 1.0, 14)
assert_almost_equal(special.cotdg(765), 1.0, 14)
def test_sinc(self):
# the sinc implementation and more extensive sinc tests are in numpy
assert_array_equal(special.sinc([0]), 1)
assert_equal(special.sinc(0.0), 1.0)
def test_sindg(self):
sn = special.sindg(90)
assert_equal(sn,1.0)
def test_sindgmore(self):
snm = special.sindg(30)
snmrl = sin(pi/6.0)
assert_almost_equal(snm,snmrl,8)
snm1 = special.sindg(45)
snmrl1 = sin(pi/4.0)
assert_almost_equal(snm1,snmrl1,8)
class TestTandg:
def test_tandg(self):
tn = special.tandg(30)
tnrl = tan(pi/6.0)
assert_almost_equal(tn,tnrl,8)
def test_tandgmore(self):
tnm = special.tandg(45)
tnmrl = tan(pi/4.0)
assert_almost_equal(tnm,tnmrl,8)
tnm1 = special.tandg(60)
tnmrl1 = tan(pi/3.0)
assert_almost_equal(tnm1,tnmrl1,8)
def test_specialpoints(self):
assert_almost_equal(special.tandg(0), 0.0, 14)
assert_almost_equal(special.tandg(45), 1.0, 14)
assert_almost_equal(special.tandg(-45), -1.0, 14)
assert_almost_equal(special.tandg(135), -1.0, 14)
assert_almost_equal(special.tandg(-135), 1.0, 14)
assert_almost_equal(special.tandg(180), 0.0, 14)
assert_almost_equal(special.tandg(-180), 0.0, 14)
assert_almost_equal(special.tandg(225), 1.0, 14)
assert_almost_equal(special.tandg(-225), -1.0, 14)
assert_almost_equal(special.tandg(315), -1.0, 14)
assert_almost_equal(special.tandg(-315), 1.0, 14)
class TestEllip:
def test_ellipj_nan(self):
"""Regression test for #912."""
special.ellipj(0.5, np.nan)
def test_ellipj(self):
el = special.ellipj(0.2,0)
rel = [sin(0.2),cos(0.2),1.0,0.20]
assert_array_almost_equal(el,rel,13)
def test_ellipk(self):
elk = special.ellipk(.2)
assert_almost_equal(elk,1.659623598610528,11)
assert_equal(special.ellipkm1(0.0), np.inf)
assert_equal(special.ellipkm1(1.0), pi/2)
assert_equal(special.ellipkm1(np.inf), 0.0)
assert_equal(special.ellipkm1(np.nan), np.nan)
assert_equal(special.ellipkm1(-1), np.nan)
assert_allclose(special.ellipk(-10), 0.7908718902387385)
def test_ellipkinc(self):
elkinc = special.ellipkinc(pi/2,.2)
elk = special.ellipk(0.2)
assert_almost_equal(elkinc,elk,15)
alpha = 20*pi/180
phi = 45*pi/180
m = sin(alpha)**2
elkinc = special.ellipkinc(phi,m)
assert_almost_equal(elkinc,0.79398143,8)
# From pg. 614 of A & S
assert_equal(special.ellipkinc(pi/2, 0.0), pi/2)
assert_equal(special.ellipkinc(pi/2, 1.0), np.inf)
assert_equal(special.ellipkinc(pi/2, -np.inf), 0.0)
assert_equal(special.ellipkinc(pi/2, np.nan), np.nan)
assert_equal(special.ellipkinc(pi/2, 2), np.nan)
assert_equal(special.ellipkinc(0, 0.5), 0.0)
assert_equal(special.ellipkinc(np.inf, 0.5), np.inf)
assert_equal(special.ellipkinc(-np.inf, 0.5), -np.inf)
assert_equal(special.ellipkinc(np.inf, np.inf), np.nan)
assert_equal(special.ellipkinc(np.inf, -np.inf), np.nan)
assert_equal(special.ellipkinc(-np.inf, -np.inf), np.nan)
assert_equal(special.ellipkinc(-np.inf, np.inf), np.nan)
assert_equal(special.ellipkinc(np.nan, 0.5), np.nan)
assert_equal(special.ellipkinc(np.nan, np.nan), np.nan)
assert_allclose(special.ellipkinc(0.38974112035318718, 1), 0.4, rtol=1e-14)
assert_allclose(special.ellipkinc(1.5707, -10), 0.79084284661724946)
def test_ellipkinc_2(self):
# Regression test for gh-3550
# ellipkinc(phi, mbad) was NaN and mvals[2:6] were twice the correct value
mbad = 0.68359375000000011
phi = 0.9272952180016123
m = np.nextafter(mbad, 0)
mvals = []
for j in range(10):
mvals.append(m)
m = np.nextafter(m, 1)
f = special.ellipkinc(phi, mvals)
assert_array_almost_equal_nulp(f, np.full_like(f, 1.0259330100195334), 1)
# this bug also appears at phi + n * pi for at least small n
f1 = special.ellipkinc(phi + pi, mvals)
assert_array_almost_equal_nulp(f1, np.full_like(f1, 5.1296650500976675), 2)
def test_ellipkinc_singular(self):
# ellipkinc(phi, 1) has closed form and is finite only for phi in (-pi/2, pi/2)
xlog = np.logspace(-300, -17, 25)
xlin = np.linspace(1e-17, 0.1, 25)
xlin2 = np.linspace(0.1, pi/2, 25, endpoint=False)
assert_allclose(special.ellipkinc(xlog, 1), np.arcsinh(np.tan(xlog)), rtol=1e14)
assert_allclose(special.ellipkinc(xlin, 1), np.arcsinh(np.tan(xlin)), rtol=1e14)
assert_allclose(special.ellipkinc(xlin2, 1), np.arcsinh(np.tan(xlin2)), rtol=1e14)
assert_equal(special.ellipkinc(np.pi/2, 1), np.inf)
assert_allclose(special.ellipkinc(-xlog, 1), np.arcsinh(np.tan(-xlog)), rtol=1e14)
assert_allclose(special.ellipkinc(-xlin, 1), np.arcsinh(np.tan(-xlin)), rtol=1e14)
assert_allclose(special.ellipkinc(-xlin2, 1), np.arcsinh(np.tan(-xlin2)), rtol=1e14)
assert_equal(special.ellipkinc(-np.pi/2, 1), np.inf)
def test_ellipe(self):
ele = special.ellipe(.2)
assert_almost_equal(ele,1.4890350580958529,8)
assert_equal(special.ellipe(0.0), pi/2)
assert_equal(special.ellipe(1.0), 1.0)
assert_equal(special.ellipe(-np.inf), np.inf)
assert_equal(special.ellipe(np.nan), np.nan)
assert_equal(special.ellipe(2), np.nan)
assert_allclose(special.ellipe(-10), 3.6391380384177689)
def test_ellipeinc(self):
eleinc = special.ellipeinc(pi/2,.2)
ele = special.ellipe(0.2)
assert_almost_equal(eleinc,ele,14)
# pg 617 of A & S
alpha, phi = 52*pi/180,35*pi/180
m = sin(alpha)**2
eleinc = special.ellipeinc(phi,m)
assert_almost_equal(eleinc, 0.58823065, 8)
assert_equal(special.ellipeinc(pi/2, 0.0), pi/2)
assert_equal(special.ellipeinc(pi/2, 1.0), 1.0)
assert_equal(special.ellipeinc(pi/2, -np.inf), np.inf)
assert_equal(special.ellipeinc(pi/2, np.nan), np.nan)
assert_equal(special.ellipeinc(pi/2, 2), np.nan)
assert_equal(special.ellipeinc(0, 0.5), 0.0)
assert_equal(special.ellipeinc(np.inf, 0.5), np.inf)
assert_equal(special.ellipeinc(-np.inf, 0.5), -np.inf)
assert_equal(special.ellipeinc(np.inf, -np.inf), np.inf)
assert_equal(special.ellipeinc(-np.inf, -np.inf), -np.inf)
assert_equal(special.ellipeinc(np.inf, np.inf), np.nan)
assert_equal(special.ellipeinc(-np.inf, np.inf), np.nan)
assert_equal(special.ellipeinc(np.nan, 0.5), np.nan)
assert_equal(special.ellipeinc(np.nan, np.nan), np.nan)
assert_allclose(special.ellipeinc(1.5707, -10), 3.6388185585822876)
def test_ellipeinc_2(self):
# Regression test for gh-3550
# ellipeinc(phi, mbad) was NaN and mvals[2:6] were twice the correct value
mbad = 0.68359375000000011
phi = 0.9272952180016123
m = np.nextafter(mbad, 0)
mvals = []
for j in range(10):
mvals.append(m)
m = np.nextafter(m, 1)
f = special.ellipeinc(phi, mvals)
assert_array_almost_equal_nulp(f, np.full_like(f, 0.84442884574781019), 2)
# this bug also appears at phi + n * pi for at least small n
f1 = special.ellipeinc(phi + pi, mvals)
assert_array_almost_equal_nulp(f1, np.full_like(f1, 3.3471442287390509), 4)
class TestEllipCarlson(object):
"""Test for Carlson elliptic integrals ellipr[cdfgj].
The special values used in these tests can be found in Sec. 3 of Carlson
(1994), https://arxiv.org/abs/math/9409227
"""
def test_elliprc(self):
assert_allclose(elliprc(1, 1), 1)
assert elliprc(1, inf) == 0.0
assert isnan(elliprc(1, 0))
assert elliprc(1, complex(1, inf)) == 0.0
args = array([[0.0, 0.25],
[2.25, 2.0],
[0.0, 1.0j],
[-1.0j, 1.0j],
[0.25, -2.0],
[1.0j, -1.0]])
expected_results = array([np.pi,
np.log(2.0),
1.1107207345396 * (1.0-1.0j),
1.2260849569072-0.34471136988768j,
np.log(2.0) / 3.0,
0.77778596920447+0.19832484993429j])
for i, arr in enumerate(args):
assert_allclose(elliprc(*arr), expected_results[i])
def test_elliprd(self):
assert_allclose(elliprd(1, 1, 1), 1)
assert_allclose(elliprd(0, 2, 1) / 3.0, 0.59907011736779610371)
assert elliprd(1, 1, inf) == 0.0
assert np.isinf(elliprd(1, 1, 0))
assert np.isinf(elliprd(1, 1, complex(0, 0)))
assert np.isinf(elliprd(0, 1, complex(0, 0)))
assert isnan(elliprd(1, 1, -np.finfo(np.double).tiny / 2.0))
assert isnan(elliprd(1, 1, complex(-1, 0)))
args = array([[0.0, 2.0, 1.0],
[2.0, 3.0, 4.0],
[1.0j, -1.0j, 2.0],
[0.0, 1.0j, -1.0j],
[0.0, -1.0+1.0j, 1.0j],
[-2.0-1.0j, -1.0j, -1.0+1.0j]])
expected_results = array([1.7972103521034,
0.16510527294261,
0.65933854154220,
1.2708196271910+2.7811120159521j,
-1.8577235439239-0.96193450888839j,
1.8249027393704-1.2218475784827j])
for i, arr in enumerate(args):
assert_allclose(elliprd(*arr), expected_results[i])
def test_elliprf(self):
assert_allclose(elliprf(1, 1, 1), 1)
assert_allclose(elliprf(0, 1, 2), 1.31102877714605990523)
assert elliprf(1, inf, 1) == 0.0
assert np.isinf(elliprf(0, 1, 0))
assert isnan(elliprf(1, 1, -1))
assert elliprf(complex(inf), 0, 1) == 0.0
assert isnan(elliprf(1, 1, complex(-inf, 1)))
args = array([[1.0, 2.0, 0.0],
[1.0j, -1.0j, 0.0],
[0.5, 1.0, 0.0],
[-1.0+1.0j, 1.0j, 0.0],
[2.0, 3.0, 4.0],
[1.0j, -1.0j, 2.0],
[-1.0+1.0j, 1.0j, 1.0-1.0j]])
expected_results = array([1.3110287771461,
1.8540746773014,
1.8540746773014,
0.79612586584234-1.2138566698365j,
0.58408284167715,
1.0441445654064,
0.93912050218619-0.53296252018635j])
for i, arr in enumerate(args):
assert_allclose(elliprf(*arr), expected_results[i])
def test_elliprg(self):
assert_allclose(elliprg(1, 1, 1), 1)
assert_allclose(elliprg(0, 0, 1), 0.5)
assert_allclose(elliprg(0, 0, 0), 0)
assert np.isinf(elliprg(1, inf, 1))
assert np.isinf(elliprg(complex(inf), 1, 1))
args = array([[0.0, 16.0, 16.0],
[2.0, 3.0, 4.0],
[0.0, 1.0j, -1.0j],
[-1.0+1.0j, 1.0j, 0.0],
[-1.0j, -1.0+1.0j, 1.0j],
[0.0, 0.0796, 4.0]])
expected_results = array([np.pi,
1.7255030280692,
0.42360654239699,
0.44660591677018+0.70768352357515j,
0.36023392184473+0.40348623401722j,
1.0284758090288])
for i, arr in enumerate(args):
assert_allclose(elliprg(*arr), expected_results[i])
def test_elliprj(self):
assert_allclose(elliprj(1, 1, 1, 1), 1)
assert elliprj(1, 1, inf, 1) == 0.0
assert isnan(elliprj(1, 0, 0, 0))
assert isnan(elliprj(-1, 1, 1, 1))
assert elliprj(1, 1, 1, inf) == 0.0
args = array([[0.0, 1.0, 2.0, 3.0],
[2.0, 3.0, 4.0, 5.0],
[2.0, 3.0, 4.0, -1.0+1.0j],
[1.0j, -1.0j, 0.0, 2.0],
[-1.0+1.0j, -1.0-1.0j, 1.0, 2.0],
[1.0j, -1.0j, 0.0, 1.0-1.0j],
[-1.0+1.0j, -1.0-1.0j, 1.0, -3.0+1.0j],
[2.0, 3.0, 4.0, -0.5], # Cauchy principal value
[2.0, 3.0, 4.0, -5.0]]) # Cauchy principal value
expected_results = array([0.77688623778582,
0.14297579667157,
0.13613945827771-0.38207561624427j,
1.6490011662711,
0.94148358841220,
1.8260115229009+1.2290661908643j,
-0.61127970812028-1.0684038390007j,
0.24723819703052, # Cauchy principal value
-0.12711230042964]) # Caucny principal value
for i, arr in enumerate(args):
assert_allclose(elliprj(*arr), expected_results[i])
@pytest.mark.xfail(reason="Insufficient accuracy on 32-bit")
def test_elliprj_hard(self):
assert_allclose(elliprj(6.483625725195452e-08,
1.1649136528196886e-27,
3.6767340167168e+13,
0.493704617023468),
8.63426920644241857617477551054e-6,
rtol=5e-15, atol=1e-20)
assert_allclose(elliprj(14.375105857849121,
9.993988969725365e-11,
1.72844262269944e-26,
5.898871222598245e-06),
829774.1424801627252574054378691828,
rtol=5e-15, atol=1e-20)
class TestEllipLegendreCarlsonIdentities(object):
"""Test identities expressing the Legendre elliptic integrals in terms
of Carlson's symmetric integrals. These identities can be found
in the DLMF https://dlmf.nist.gov/19.25#i .
"""
def setup_class(self):
self.m_n1_1 = np.arange(-1., 1., 0.01)
# For double, this is -(2**1024)
self.max_neg = finfo(float_).min
# Lots of very negative numbers
self.very_neg_m = -1. * 2.**arange(-1 +
np.log2(-self.max_neg), 0.,
-1.)
self.ms_up_to_1 = np.concatenate(([self.max_neg],
self.very_neg_m,
self.m_n1_1))
def test_k(self):
"""Test identity:
K(m) = R_F(0, 1-m, 1)
"""
m = self.ms_up_to_1
assert_allclose(ellipk(m), elliprf(0., 1.-m, 1.))
def test_km1(self):
"""Test identity:
K(m) = R_F(0, 1-m, 1)
But with the ellipkm1 function
"""
# For double, this is 2**-1022
tiny = finfo(float_).tiny
# All these small powers of 2, up to 2**-1
m1 = tiny * 2.**arange(0., -np.log2(tiny))
assert_allclose(ellipkm1(m1), elliprf(0., m1, 1.))
def test_e(self):
"""Test identity:
E(m) = 2*R_G(0, 1-k^2, 1)
"""
m = self.ms_up_to_1
assert_allclose(ellipe(m), 2.*elliprg(0., 1.-m, 1.))
class TestErf:
def test_erf(self):
er = special.erf(.25)
assert_almost_equal(er,0.2763263902,8)
def test_erf_zeros(self):
erz = special.erf_zeros(5)
erzr = array([1.45061616+1.88094300j,
2.24465928+2.61657514j,
2.83974105+3.17562810j,
3.33546074+3.64617438j,
3.76900557+4.06069723j])
assert_array_almost_equal(erz,erzr,4)
def _check_variant_func(self, func, other_func, rtol, atol=0):
np.random.seed(1234)
n = 10000
x = np.random.pareto(0.02, n) * (2*np.random.randint(0, 2, n) - 1)
y = np.random.pareto(0.02, n) * (2*np.random.randint(0, 2, n) - 1)
z = x + 1j*y
with np.errstate(all='ignore'):
w = other_func(z)
w_real = other_func(x).real
mask = np.isfinite(w)
w = w[mask]
z = z[mask]
mask = np.isfinite(w_real)
w_real = w_real[mask]
x = x[mask]
# test both real and complex variants
assert_func_equal(func, w, z, rtol=rtol, atol=atol)
assert_func_equal(func, w_real, x, rtol=rtol, atol=atol)
def test_erfc_consistent(self):
self._check_variant_func(
cephes.erfc,
lambda z: 1 - cephes.erf(z),
rtol=1e-12,
atol=1e-14 # <- the test function loses precision
)
def test_erfcx_consistent(self):
self._check_variant_func(
cephes.erfcx,
lambda z: np.exp(z*z) * cephes.erfc(z),
rtol=1e-12
)
def test_erfi_consistent(self):
self._check_variant_func(
cephes.erfi,
lambda z: -1j * cephes.erf(1j*z),
rtol=1e-12
)
def test_dawsn_consistent(self):
self._check_variant_func(
cephes.dawsn,
lambda z: sqrt(pi)/2 * np.exp(-z*z) * cephes.erfi(z),
rtol=1e-12
)
def test_erf_nan_inf(self):
vals = [np.nan, -np.inf, np.inf]
expected = [np.nan, -1, 1]
assert_allclose(special.erf(vals), expected, rtol=1e-15)
def test_erfc_nan_inf(self):
vals = [np.nan, -np.inf, np.inf]
expected = [np.nan, 2, 0]
assert_allclose(special.erfc(vals), expected, rtol=1e-15)
def test_erfcx_nan_inf(self):
vals = [np.nan, -np.inf, np.inf]
expected = [np.nan, np.inf, 0]
assert_allclose(special.erfcx(vals), expected, rtol=1e-15)
def test_erfi_nan_inf(self):
vals = [np.nan, -np.inf, np.inf]
expected = [np.nan, -np.inf, np.inf]
assert_allclose(special.erfi(vals), expected, rtol=1e-15)
def test_dawsn_nan_inf(self):
vals = [np.nan, -np.inf, np.inf]
expected = [np.nan, -0.0, 0.0]
assert_allclose(special.dawsn(vals), expected, rtol=1e-15)
def test_wofz_nan_inf(self):
vals = [np.nan, -np.inf, np.inf]
expected = [np.nan + np.nan * 1.j, 0.-0.j, 0.+0.j]
assert_allclose(special.wofz(vals), expected, rtol=1e-15)
class TestEuler:
def test_euler(self):
eu0 = special.euler(0)
eu1 = special.euler(1)
eu2 = special.euler(2) # just checking segfaults
assert_allclose(eu0, [1], rtol=1e-15)
assert_allclose(eu1, [1, 0], rtol=1e-15)
assert_allclose(eu2, [1, 0, -1], rtol=1e-15)
eu24 = special.euler(24)
mathworld = [1,1,5,61,1385,50521,2702765,199360981,
19391512145,2404879675441,
370371188237525,69348874393137901,
15514534163557086905]
correct = zeros((25,),'d')
for k in range(0,13):
if (k % 2):
correct[2*k] = -float(mathworld[k])
else:
correct[2*k] = float(mathworld[k])
with np.errstate(all='ignore'):
err = nan_to_num((eu24-correct)/correct)
errmax = max(err)
assert_almost_equal(errmax, 0.0, 14)
class TestExp:
def test_exp2(self):
ex = special.exp2(2)
exrl = 2**2
assert_equal(ex,exrl)
def test_exp2more(self):
exm = special.exp2(2.5)
exmrl = 2**(2.5)
assert_almost_equal(exm,exmrl,8)
def test_exp10(self):
ex = special.exp10(2)
exrl = 10**2
assert_approx_equal(ex,exrl)
def test_exp10more(self):
exm = special.exp10(2.5)
exmrl = 10**(2.5)
assert_almost_equal(exm,exmrl,8)
def test_expm1(self):
ex = (special.expm1(2),special.expm1(3),special.expm1(4))
exrl = (exp(2)-1,exp(3)-1,exp(4)-1)
assert_array_almost_equal(ex,exrl,8)
def test_expm1more(self):
ex1 = (special.expm1(2),special.expm1(2.1),special.expm1(2.2))
exrl1 = (exp(2)-1,exp(2.1)-1,exp(2.2)-1)
assert_array_almost_equal(ex1,exrl1,8)
class TestFactorialFunctions:
def test_factorial(self):
# Some known values, float math
assert_array_almost_equal(special.factorial(0), 1)
assert_array_almost_equal(special.factorial(1), 1)
assert_array_almost_equal(special.factorial(2), 2)
assert_array_almost_equal([6., 24., 120.],
special.factorial([3, 4, 5], exact=False))
assert_array_almost_equal(special.factorial([[5, 3], [4, 3]]),
[[120, 6], [24, 6]])
# Some known values, integer math
assert_equal(special.factorial(0, exact=True), 1)
assert_equal(special.factorial(1, exact=True), 1)
assert_equal(special.factorial(2, exact=True), 2)
assert_equal(special.factorial(5, exact=True), 120)
assert_equal(special.factorial(15, exact=True), 1307674368000)
# ndarray shape is maintained
assert_equal(special.factorial([7, 4, 15, 10], exact=True),
[5040, 24, 1307674368000, 3628800])
assert_equal(special.factorial([[5, 3], [4, 3]], True),
[[120, 6], [24, 6]])
# object arrays
assert_equal(special.factorial(np.arange(-3, 22), True),
special.factorial(np.arange(-3, 22), False))
# int64 array
assert_equal(special.factorial(np.arange(-3, 15), True),
special.factorial(np.arange(-3, 15), False))
# int32 array
assert_equal(special.factorial(np.arange(-3, 5), True),
special.factorial(np.arange(-3, 5), False))
# Consistent output for n < 0
for exact in (True, False):
assert_array_equal(0, special.factorial(-3, exact))
assert_array_equal([1, 2, 0, 0],
special.factorial([1, 2, -5, -4], exact))
for n in range(0, 22):
# Compare all with math.factorial
correct = math.factorial(n)
assert_array_equal(correct, special.factorial(n, True))
assert_array_equal(correct, special.factorial([n], True)[0])
assert_allclose(float(correct), special.factorial(n, False))
assert_allclose(float(correct), special.factorial([n], False)[0])
# Compare exact=True vs False, scalar vs array
assert_array_equal(special.factorial(n, True),
special.factorial(n, False))
assert_array_equal(special.factorial([n], True),
special.factorial([n], False))
@pytest.mark.parametrize('x, exact', [
(1, True),
(1, False),
(np.array(1), True),
(np.array(1), False),
])
def test_factorial_0d_return_type(self, x, exact):
assert np.isscalar(special.factorial(x, exact=exact))
def test_factorial2(self):
assert_array_almost_equal([105., 384., 945.],
special.factorial2([7, 8, 9], exact=False))
assert_equal(special.factorial2(7, exact=True), 105)
def test_factorialk(self):
assert_equal(special.factorialk(5, 1, exact=True), 120)
assert_equal(special.factorialk(5, 3, exact=True), 10)
@pytest.mark.parametrize('x, exact', [
(np.nan, True),
(np.nan, False),
(np.array([np.nan]), True),
(np.array([np.nan]), False),
])
def test_nan_inputs(self, x, exact):
result = special.factorial(x, exact=exact)
assert_(np.isnan(result))
# GH-13122: special.factorial() argument should be an array of integers.
# On Python 3.10, math.factorial() reject float.
# On Python 3.9, a DeprecationWarning is emitted.
# A numpy array casts all integers to float if the array contains a
# single NaN.
@pytest.mark.skipif(sys.version_info >= (3, 10),
reason="Python 3.10+ math.factorial() requires int")
def test_mixed_nan_inputs(self):
x = np.array([np.nan, 1, 2, 3, np.nan])
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, "Using factorial\\(\\) with floats is deprecated")
result = special.factorial(x, exact=True)
assert_equal(np.array([np.nan, 1, 2, 6, np.nan]), result)
result = special.factorial(x, exact=False)
assert_equal(np.array([np.nan, 1, 2, 6, np.nan]), result)
class TestFresnel:
@pytest.mark.parametrize("z, s, c", [
# some positive value
(.5, 0.064732432859999287, 0.49234422587144644),
(.5 + .0j, 0.064732432859999287, 0.49234422587144644),
# negative half annulus
# https://github.com/scipy/scipy/issues/12309
# Reference values can be reproduced with
# https://www.wolframalpha.com/input/?i=FresnelS%5B-2.0+%2B+0.1i%5D
# https://www.wolframalpha.com/input/?i=FresnelC%5B-2.0+%2B+0.1i%5D
(
-2.0 + 0.1j,
-0.3109538687728942-0.0005870728836383176j,
-0.4879956866358554+0.10670801832903172j
),
(
-0.1 - 1.5j,
-0.03918309471866977+0.7197508454568574j,
0.09605692502968956-0.43625191013617465j
),
# a different algorithm kicks in for "large" values, i.e., |z| >= 4.5,
# make sure to test both float and complex values; a different
# algorithm is used
(6.0, 0.44696076, 0.49953147),
(6.0 + 0.0j, 0.44696076, 0.49953147),
(6.0j, -0.44696076j, 0.49953147j),
(-6.0 + 0.0j, -0.44696076, -0.49953147),
(-6.0j, 0.44696076j, -0.49953147j),
# inf
(np.inf, 0.5, 0.5),
(-np.inf, -0.5, -0.5),
])
def test_fresnel_values(self, z, s, c):
frs = array(special.fresnel(z))
assert_array_almost_equal(frs, array([s, c]), 8)
# values from pg 329 Table 7.11 of A & S
# slightly corrected in 4th decimal place
def test_fresnel_zeros(self):
szo, czo = special.fresnel_zeros(5)
assert_array_almost_equal(szo,
array([2.0093+0.2885j,
2.8335+0.2443j,
3.4675+0.2185j,
4.0026+0.2009j,
4.4742+0.1877j]),3)
assert_array_almost_equal(czo,
array([1.7437+0.3057j,
2.6515+0.2529j,
3.3204+0.2240j,
3.8757+0.2047j,
4.3611+0.1907j]),3)
vals1 = special.fresnel(szo)[0]
vals2 = special.fresnel(czo)[1]
assert_array_almost_equal(vals1,0,14)
assert_array_almost_equal(vals2,0,14)
def test_fresnelc_zeros(self):
szo, czo = special.fresnel_zeros(6)
frc = special.fresnelc_zeros(6)
assert_array_almost_equal(frc,czo,12)
def test_fresnels_zeros(self):
szo, czo = special.fresnel_zeros(5)
frs = special.fresnels_zeros(5)
assert_array_almost_equal(frs,szo,12)
class TestGamma:
def test_gamma(self):
gam = special.gamma(5)
assert_equal(gam,24.0)
def test_gammaln(self):
gamln = special.gammaln(3)
lngam = log(special.gamma(3))
assert_almost_equal(gamln,lngam,8)
def test_gammainccinv(self):
gccinv = special.gammainccinv(.5,.5)
gcinv = special.gammaincinv(.5,.5)
assert_almost_equal(gccinv,gcinv,8)
@with_special_errors
def test_gammaincinv(self):
y = special.gammaincinv(.4,.4)
x = special.gammainc(.4,y)
assert_almost_equal(x,0.4,1)
y = special.gammainc(10, 0.05)
x = special.gammaincinv(10, 2.5715803516000736e-20)
assert_almost_equal(0.05, x, decimal=10)
assert_almost_equal(y, 2.5715803516000736e-20, decimal=10)
x = special.gammaincinv(50, 8.20754777388471303050299243573393e-18)
assert_almost_equal(11.0, x, decimal=10)
@with_special_errors
def test_975(self):
# Regression test for ticket #975 -- switch point in algorithm
# check that things work OK at the point, immediately next floats
# around it, and a bit further away
pts = [0.25,
np.nextafter(0.25, 0), 0.25 - 1e-12,
np.nextafter(0.25, 1), 0.25 + 1e-12]
for xp in pts:
y = special.gammaincinv(.4, xp)
x = special.gammainc(0.4, y)
assert_allclose(x, xp, rtol=1e-12)
def test_rgamma(self):
rgam = special.rgamma(8)
rlgam = 1/special.gamma(8)
assert_almost_equal(rgam,rlgam,8)
def test_infinity(self):
assert_(np.isinf(special.gamma(-1)))
assert_equal(special.rgamma(-1), 0)
class TestHankel:
def test_negv1(self):
assert_almost_equal(special.hankel1(-3,2), -special.hankel1(3,2), 14)
def test_hankel1(self):
hank1 = special.hankel1(1,.1)
hankrl = (special.jv(1,.1) + special.yv(1,.1)*1j)
assert_almost_equal(hank1,hankrl,8)
def test_negv1e(self):
assert_almost_equal(special.hankel1e(-3,2), -special.hankel1e(3,2), 14)
def test_hankel1e(self):
hank1e = special.hankel1e(1,.1)
hankrle = special.hankel1(1,.1)*exp(-.1j)
assert_almost_equal(hank1e,hankrle,8)
def test_negv2(self):
assert_almost_equal(special.hankel2(-3,2), -special.hankel2(3,2), 14)
def test_hankel2(self):
hank2 = special.hankel2(1,.1)
hankrl2 = (special.jv(1,.1) - special.yv(1,.1)*1j)
assert_almost_equal(hank2,hankrl2,8)
def test_neg2e(self):
assert_almost_equal(special.hankel2e(-3,2), -special.hankel2e(3,2), 14)
def test_hankl2e(self):
hank2e = special.hankel2e(1,.1)
hankrl2e = special.hankel2e(1,.1)
assert_almost_equal(hank2e,hankrl2e,8)
class TestHyper:
def test_h1vp(self):
h1 = special.h1vp(1,.1)
h1real = (special.jvp(1,.1) + special.yvp(1,.1)*1j)
assert_almost_equal(h1,h1real,8)
def test_h2vp(self):
h2 = special.h2vp(1,.1)
h2real = (special.jvp(1,.1) - special.yvp(1,.1)*1j)
assert_almost_equal(h2,h2real,8)
def test_hyp0f1(self):
# scalar input
assert_allclose(special.hyp0f1(2.5, 0.5), 1.21482702689997, rtol=1e-12)
assert_allclose(special.hyp0f1(2.5, 0), 1.0, rtol=1e-15)
# float input, expected values match mpmath
x = special.hyp0f1(3.0, [-1.5, -1, 0, 1, 1.5])
expected = np.array([0.58493659229143, 0.70566805723127, 1.0,
1.37789689539747, 1.60373685288480])
assert_allclose(x, expected, rtol=1e-12)
# complex input
x = special.hyp0f1(3.0, np.array([-1.5, -1, 0, 1, 1.5]) + 0.j)
assert_allclose(x, expected.astype(complex), rtol=1e-12)
# test broadcasting
x1 = [0.5, 1.5, 2.5]
x2 = [0, 1, 0.5]
x = special.hyp0f1(x1, x2)
expected = [1.0, 1.8134302039235093, 1.21482702689997]
assert_allclose(x, expected, rtol=1e-12)
x = special.hyp0f1(np.row_stack([x1] * 2), x2)
assert_allclose(x, np.row_stack([expected] * 2), rtol=1e-12)
assert_raises(ValueError, special.hyp0f1,
np.row_stack([x1] * 3), [0, 1])
def test_hyp0f1_gh5764(self):
# Just checks the point that failed; there's a more systematic
# test in test_mpmath
res = special.hyp0f1(0.8, 0.5 + 0.5*1J)
# The expected value was generated using mpmath
assert_almost_equal(res, 1.6139719776441115 + 1J*0.80893054061790665)
def test_hyp1f1(self):
hyp1 = special.hyp1f1(.1,.1,.3)
assert_almost_equal(hyp1, 1.3498588075760032,7)
# test contributed by Moritz Deger (2008-05-29)
# https://github.com/scipy/scipy/issues/1186 (Trac #659)
# reference data obtained from mathematica [ a, b, x, m(a,b,x)]:
# produced with test_hyp1f1.nb
ref_data = array([[-8.38132975e+00, -1.28436461e+01, -2.91081397e+01, 1.04178330e+04],
[2.91076882e+00, -6.35234333e+00, -1.27083993e+01, 6.68132725e+00],
[-1.42938258e+01, 1.80869131e-01, 1.90038728e+01, 1.01385897e+05],
[5.84069088e+00, 1.33187908e+01, 2.91290106e+01, 1.59469411e+08],
[-2.70433202e+01, -1.16274873e+01, -2.89582384e+01, 1.39900152e+24],
[4.26344966e+00, -2.32701773e+01, 1.91635759e+01, 6.13816915e+21],
[1.20514340e+01, -3.40260240e+00, 7.26832235e+00, 1.17696112e+13],
[2.77372955e+01, -1.99424687e+00, 3.61332246e+00, 3.07419615e+13],
[1.50310939e+01, -2.91198675e+01, -1.53581080e+01, -3.79166033e+02],
[1.43995827e+01, 9.84311196e+00, 1.93204553e+01, 2.55836264e+10],
[-4.08759686e+00, 1.34437025e+01, -1.42072843e+01, 1.70778449e+01],
[8.05595738e+00, -1.31019838e+01, 1.52180721e+01, 3.06233294e+21],
[1.81815804e+01, -1.42908793e+01, 9.57868793e+00, -2.84771348e+20],
[-2.49671396e+01, 1.25082843e+01, -1.71562286e+01, 2.36290426e+07],
[2.67277673e+01, 1.70315414e+01, 6.12701450e+00, 7.77917232e+03],
[2.49565476e+01, 2.91694684e+01, 6.29622660e+00, 2.35300027e+02],
[6.11924542e+00, -1.59943768e+00, 9.57009289e+00, 1.32906326e+11],
[-1.47863653e+01, 2.41691301e+01, -1.89981821e+01, 2.73064953e+03],
[2.24070483e+01, -2.93647433e+00, 8.19281432e+00, -6.42000372e+17],
[8.04042600e-01, 1.82710085e+01, -1.97814534e+01, 5.48372441e-01],
[1.39590390e+01, 1.97318686e+01, 2.37606635e+00, 5.51923681e+00],
[-4.66640483e+00, -2.00237930e+01, 7.40365095e+00, 4.50310752e+00],
[2.76821999e+01, -6.36563968e+00, 1.11533984e+01, -9.28725179e+23],
[-2.56764457e+01, 1.24544906e+00, 1.06407572e+01, 1.25922076e+01],
[3.20447808e+00, 1.30874383e+01, 2.26098014e+01, 2.03202059e+04],
[-1.24809647e+01, 4.15137113e+00, -2.92265700e+01, 2.39621411e+08],
[2.14778108e+01, -2.35162960e+00, -1.13758664e+01, 4.46882152e-01],
[-9.85469168e+00, -3.28157680e+00, 1.67447548e+01, -1.07342390e+07],
[1.08122310e+01, -2.47353236e+01, -1.15622349e+01, -2.91733796e+03],
[-2.67933347e+01, -3.39100709e+00, 2.56006986e+01, -5.29275382e+09],
[-8.60066776e+00, -8.02200924e+00, 1.07231926e+01, 1.33548320e+06],
[-1.01724238e-01, -1.18479709e+01, -2.55407104e+01, 1.55436570e+00],
[-3.93356771e+00, 2.11106818e+01, -2.57598485e+01, 2.13467840e+01],
[3.74750503e+00, 1.55687633e+01, -2.92841720e+01, 1.43873509e-02],
[6.99726781e+00, 2.69855571e+01, -1.63707771e+01, 3.08098673e-02],
[-2.31996011e+01, 3.47631054e+00, 9.75119815e-01, 1.79971073e-02],
[2.38951044e+01, -2.91460190e+01, -2.50774708e+00, 9.56934814e+00],
[1.52730825e+01, 5.77062507e+00, 1.21922003e+01, 1.32345307e+09],
[1.74673917e+01, 1.89723426e+01, 4.94903250e+00, 9.90859484e+01],
[1.88971241e+01, 2.86255413e+01, 5.52360109e-01, 1.44165360e+00],
[1.02002319e+01, -1.66855152e+01, -2.55426235e+01, 6.56481554e+02],
[-1.79474153e+01, 1.22210200e+01, -1.84058212e+01, 8.24041812e+05],
[-1.36147103e+01, 1.32365492e+00, -7.22375200e+00, 9.92446491e+05],
[7.57407832e+00, 2.59738234e+01, -1.34139168e+01, 3.64037761e-02],
[2.21110169e+00, 1.28012666e+01, 1.62529102e+01, 1.33433085e+02],
[-2.64297569e+01, -1.63176658e+01, -1.11642006e+01, -2.44797251e+13],
[-2.46622944e+01, -3.02147372e+00, 8.29159315e+00, -3.21799070e+05],
[-1.37215095e+01, -1.96680183e+01, 2.91940118e+01, 3.21457520e+12],
[-5.45566105e+00, 2.81292086e+01, 1.72548215e-01, 9.66973000e-01],
[-1.55751298e+00, -8.65703373e+00, 2.68622026e+01, -3.17190834e+16],
[2.45393609e+01, -2.70571903e+01, 1.96815505e+01, 1.80708004e+37],
[5.77482829e+00, 1.53203143e+01, 2.50534322e+01, 1.14304242e+06],
[-1.02626819e+01, 2.36887658e+01, -2.32152102e+01, 7.28965646e+02],
[-1.30833446e+00, -1.28310210e+01, 1.87275544e+01, -9.33487904e+12],
[5.83024676e+00, -1.49279672e+01, 2.44957538e+01, -7.61083070e+27],
[-2.03130747e+01, 2.59641715e+01, -2.06174328e+01, 4.54744859e+04],
[1.97684551e+01, -2.21410519e+01, -2.26728740e+01, 3.53113026e+06],
[2.73673444e+01, 2.64491725e+01, 1.57599882e+01, 1.07385118e+07],
[5.73287971e+00, 1.21111904e+01, 1.33080171e+01, 2.63220467e+03],
[-2.82751072e+01, 2.08605881e+01, 9.09838900e+00, -6.60957033e-07],
[1.87270691e+01, -1.74437016e+01, 1.52413599e+01, 6.59572851e+27],
[6.60681457e+00, -2.69449855e+00, 9.78972047e+00, -2.38587870e+12],
[1.20895561e+01, -2.51355765e+01, 2.30096101e+01, 7.58739886e+32],
[-2.44682278e+01, 2.10673441e+01, -1.36705538e+01, 4.54213550e+04],
[-4.50665152e+00, 3.72292059e+00, -4.83403707e+00, 2.68938214e+01],
[-7.46540049e+00, -1.08422222e+01, -1.72203805e+01, -2.09402162e+02],
[-2.00307551e+01, -7.50604431e+00, -2.78640020e+01, 4.15985444e+19],
[1.99890876e+01, 2.20677419e+01, -2.51301778e+01, 1.23840297e-09],
[2.03183823e+01, -7.66942559e+00, 2.10340070e+01, 1.46285095e+31],
[-2.90315825e+00, -2.55785967e+01, -9.58779316e+00, 2.65714264e-01],
[2.73960829e+01, -1.80097203e+01, -2.03070131e+00, 2.52908999e+02],
[-2.11708058e+01, -2.70304032e+01, 2.48257944e+01, 3.09027527e+08],
[2.21959758e+01, 4.00258675e+00, -1.62853977e+01, -9.16280090e-09],
[1.61661840e+01, -2.26845150e+01, 2.17226940e+01, -8.24774394e+33],
[-3.35030306e+00, 1.32670581e+00, 9.39711214e+00, -1.47303163e+01],
[7.23720726e+00, -2.29763909e+01, 2.34709682e+01, -9.20711735e+29],
[2.71013568e+01, 1.61951087e+01, -7.11388906e-01, 2.98750911e-01],
[8.40057933e+00, -7.49665220e+00, 2.95587388e+01, 6.59465635e+29],
[-1.51603423e+01, 1.94032322e+01, -7.60044357e+00, 1.05186941e+02],
[-8.83788031e+00, -2.72018313e+01, 1.88269907e+00, 1.81687019e+00],
[-1.87283712e+01, 5.87479570e+00, -1.91210203e+01, 2.52235612e+08],
[-5.61338513e-01, 2.69490237e+01, 1.16660111e-01, 9.97567783e-01],
[-5.44354025e+00, -1.26721408e+01, -4.66831036e+00, 1.06660735e-01],
[-2.18846497e+00, 2.33299566e+01, 9.62564397e+00, 3.03842061e-01],
[6.65661299e+00, -2.39048713e+01, 1.04191807e+01, 4.73700451e+13],
[-2.57298921e+01, -2.60811296e+01, 2.74398110e+01, -5.32566307e+11],
[-1.11431826e+01, -1.59420160e+01, -1.84880553e+01, -1.01514747e+02],
[6.50301931e+00, 2.59859051e+01, -2.33270137e+01, 1.22760500e-02],
[-1.94987891e+01, -2.62123262e+01, 3.90323225e+00, 1.71658894e+01],
[7.26164601e+00, -1.41469402e+01, 2.81499763e+01, -2.50068329e+31],
[-1.52424040e+01, 2.99719005e+01, -2.85753678e+01, 1.31906693e+04],
[5.24149291e+00, -1.72807223e+01, 2.22129493e+01, 2.50748475e+25],
[3.63207230e-01, -9.54120862e-02, -2.83874044e+01, 9.43854939e-01],
[-2.11326457e+00, -1.25707023e+01, 1.17172130e+00, 1.20812698e+00],
[2.48513582e+00, 1.03652647e+01, -1.84625148e+01, 6.47910997e-02],
[2.65395942e+01, 2.74794672e+01, 1.29413428e+01, 2.89306132e+05],
[-9.49445460e+00, 1.59930921e+01, -1.49596331e+01, 3.27574841e+02],
[-5.89173945e+00, 9.96742426e+00, 2.60318889e+01, -3.15842908e-01],
[-1.15387239e+01, -2.21433107e+01, -2.17686413e+01, 1.56724718e-01],
[-5.30592244e+00, -2.42752190e+01, 1.29734035e+00, 1.31985534e+00]])
for a,b,c,expected in ref_data:
result = special.hyp1f1(a,b,c)
assert_(abs(expected - result)/expected < 1e-4)
def test_hyp1f1_gh2957(self):
hyp1 = special.hyp1f1(0.5, 1.5, -709.7827128933)
hyp2 = special.hyp1f1(0.5, 1.5, -709.7827128934)
assert_almost_equal(hyp1, hyp2, 12)
def test_hyp1f1_gh2282(self):
hyp = special.hyp1f1(0.5, 1.5, -1000)
assert_almost_equal(hyp, 0.028024956081989643, 12)
def test_hyp2f1(self):
# a collection of special cases taken from AMS 55
values = [[0.5, 1, 1.5, 0.2**2, 0.5/0.2*log((1+0.2)/(1-0.2))],
[0.5, 1, 1.5, -0.2**2, 1./0.2*arctan(0.2)],
[1, 1, 2, 0.2, -1/0.2*log(1-0.2)],
[3, 3.5, 1.5, 0.2**2,
0.5/0.2/(-5)*((1+0.2)**(-5)-(1-0.2)**(-5))],
[-3, 3, 0.5, sin(0.2)**2, cos(2*3*0.2)],
[3, 4, 8, 1, special.gamma(8)*special.gamma(8-4-3)/special.gamma(8-3)/special.gamma(8-4)],
[3, 2, 3-2+1, -1, 1./2**3*sqrt(pi) *
special.gamma(1+3-2)/special.gamma(1+0.5*3-2)/special.gamma(0.5+0.5*3)],
[5, 2, 5-2+1, -1, 1./2**5*sqrt(pi) *
special.gamma(1+5-2)/special.gamma(1+0.5*5-2)/special.gamma(0.5+0.5*5)],
[4, 0.5+4, 1.5-2*4, -1./3, (8./9)**(-2*4)*special.gamma(4./3) *
special.gamma(1.5-2*4)/special.gamma(3./2)/special.gamma(4./3-2*4)],
# and some others
# ticket #424
[1.5, -0.5, 1.0, -10.0, 4.1300097765277476484],
# negative integer a or b, with c-a-b integer and x > 0.9
[-2,3,1,0.95,0.715],
[2,-3,1,0.95,-0.007],
[-6,3,1,0.95,0.0000810625],
[2,-5,1,0.95,-0.000029375],
# huge negative integers
(10, -900, 10.5, 0.99, 1.91853705796607664803709475658e-24),
(10, -900, -10.5, 0.99, 3.54279200040355710199058559155e-18),
]
for i, (a, b, c, x, v) in enumerate(values):
cv = special.hyp2f1(a, b, c, x)
assert_almost_equal(cv, v, 8, err_msg='test #%d' % i)
def test_hyperu(self):
val1 = special.hyperu(1,0.1,100)
assert_almost_equal(val1,0.0098153,7)
a,b = [0.3,0.6,1.2,-2.7],[1.5,3.2,-0.4,-3.2]
a,b = asarray(a), asarray(b)
z = 0.5
hypu = special.hyperu(a,b,z)
hprl = (pi/sin(pi*b))*(special.hyp1f1(a,b,z) /
(special.gamma(1+a-b)*special.gamma(b)) -
z**(1-b)*special.hyp1f1(1+a-b,2-b,z)
/ (special.gamma(a)*special.gamma(2-b)))
assert_array_almost_equal(hypu,hprl,12)
def test_hyperu_gh2287(self):
assert_almost_equal(special.hyperu(1, 1.5, 20.2),
0.048360918656699191, 12)
class TestBessel:
def test_itj0y0(self):
it0 = array(special.itj0y0(.2))
assert_array_almost_equal(it0,array([0.19933433254006822, -0.34570883800412566]),8)
def test_it2j0y0(self):
it2 = array(special.it2j0y0(.2))
assert_array_almost_equal(it2,array([0.0049937546274601858, -0.43423067011231614]),8)
def test_negv_iv(self):
assert_equal(special.iv(3,2), special.iv(-3,2))
def test_j0(self):
oz = special.j0(.1)
ozr = special.jn(0,.1)
assert_almost_equal(oz,ozr,8)
def test_j1(self):
o1 = special.j1(.1)
o1r = special.jn(1,.1)
assert_almost_equal(o1,o1r,8)
def test_jn(self):
jnnr = special.jn(1,.2)
assert_almost_equal(jnnr,0.099500832639235995,8)
def test_negv_jv(self):
assert_almost_equal(special.jv(-3,2), -special.jv(3,2), 14)
def test_jv(self):
values = [[0, 0.1, 0.99750156206604002],
[2./3, 1e-8, 0.3239028506761532e-5],
[2./3, 1e-10, 0.1503423854873779e-6],
[3.1, 1e-10, 0.1711956265409013e-32],
[2./3, 4.0, -0.2325440850267039],
]
for i, (v, x, y) in enumerate(values):
yc = special.jv(v, x)
assert_almost_equal(yc, y, 8, err_msg='test #%d' % i)
def test_negv_jve(self):
assert_almost_equal(special.jve(-3,2), -special.jve(3,2), 14)
def test_jve(self):
jvexp = special.jve(1,.2)
assert_almost_equal(jvexp,0.099500832639235995,8)
jvexp1 = special.jve(1,.2+1j)
z = .2+1j
jvexpr = special.jv(1,z)*exp(-abs(z.imag))
assert_almost_equal(jvexp1,jvexpr,8)
def test_jn_zeros(self):
jn0 = special.jn_zeros(0,5)
jn1 = special.jn_zeros(1,5)
assert_array_almost_equal(jn0,array([2.4048255577,
5.5200781103,
8.6537279129,
11.7915344391,
14.9309177086]),4)
assert_array_almost_equal(jn1,array([3.83171,
7.01559,
10.17347,
13.32369,
16.47063]),4)
jn102 = special.jn_zeros(102,5)
assert_allclose(jn102, array([110.89174935992040343,
117.83464175788308398,
123.70194191713507279,
129.02417238949092824,
134.00114761868422559]), rtol=1e-13)
jn301 = special.jn_zeros(301,5)
assert_allclose(jn301, array([313.59097866698830153,
323.21549776096288280,
331.22338738656748796,
338.39676338872084500,
345.03284233056064157]), rtol=1e-13)
def test_jn_zeros_slow(self):
jn0 = special.jn_zeros(0, 300)
assert_allclose(jn0[260-1], 816.02884495068867280, rtol=1e-13)
assert_allclose(jn0[280-1], 878.86068707124422606, rtol=1e-13)
assert_allclose(jn0[300-1], 941.69253065317954064, rtol=1e-13)
jn10 = special.jn_zeros(10, 300)
assert_allclose(jn10[260-1], 831.67668514305631151, rtol=1e-13)
assert_allclose(jn10[280-1], 894.51275095371316931, rtol=1e-13)
assert_allclose(jn10[300-1], 957.34826370866539775, rtol=1e-13)
jn3010 = special.jn_zeros(3010,5)
assert_allclose(jn3010, array([3036.86590780927,
3057.06598526482,
3073.66360690272,
3088.37736494778,
3101.86438139042]), rtol=1e-8)
def test_jnjnp_zeros(self):
jn = special.jn
def jnp(n, x):
return (jn(n-1,x) - jn(n+1,x))/2
for nt in range(1, 30):
z, n, m, t = special.jnjnp_zeros(nt)
for zz, nn, tt in zip(z, n, t):
if tt == 0:
assert_allclose(jn(nn, zz), 0, atol=1e-6)
elif tt == 1:
assert_allclose(jnp(nn, zz), 0, atol=1e-6)
else:
raise AssertionError("Invalid t return for nt=%d" % nt)
def test_jnp_zeros(self):
jnp = special.jnp_zeros(1,5)
assert_array_almost_equal(jnp, array([1.84118,
5.33144,
8.53632,
11.70600,
14.86359]),4)
jnp = special.jnp_zeros(443,5)
assert_allclose(special.jvp(443, jnp), 0, atol=1e-15)
def test_jnyn_zeros(self):
jnz = special.jnyn_zeros(1,5)
assert_array_almost_equal(jnz,(array([3.83171,
7.01559,
10.17347,
13.32369,
16.47063]),
array([1.84118,
5.33144,
8.53632,
11.70600,
14.86359]),
array([2.19714,
5.42968,
8.59601,
11.74915,
14.89744]),
array([3.68302,
6.94150,
10.12340,
13.28576,
16.44006])),5)
def test_jvp(self):
jvprim = special.jvp(2,2)
jv0 = (special.jv(1,2)-special.jv(3,2))/2
assert_almost_equal(jvprim,jv0,10)
def test_k0(self):
ozk = special.k0(.1)
ozkr = special.kv(0,.1)
assert_almost_equal(ozk,ozkr,8)
def test_k0e(self):
ozke = special.k0e(.1)
ozker = special.kve(0,.1)
assert_almost_equal(ozke,ozker,8)
def test_k1(self):
o1k = special.k1(.1)
o1kr = special.kv(1,.1)
assert_almost_equal(o1k,o1kr,8)
def test_k1e(self):
o1ke = special.k1e(.1)
o1ker = special.kve(1,.1)
assert_almost_equal(o1ke,o1ker,8)
def test_jacobi(self):
a = 5*np.random.random() - 1
b = 5*np.random.random() - 1
P0 = special.jacobi(0,a,b)
P1 = special.jacobi(1,a,b)
P2 = special.jacobi(2,a,b)
P3 = special.jacobi(3,a,b)
assert_array_almost_equal(P0.c,[1],13)
assert_array_almost_equal(P1.c,array([a+b+2,a-b])/2.0,13)
cp = [(a+b+3)*(a+b+4), 4*(a+b+3)*(a+2), 4*(a+1)*(a+2)]
p2c = [cp[0],cp[1]-2*cp[0],cp[2]-cp[1]+cp[0]]
assert_array_almost_equal(P2.c,array(p2c)/8.0,13)
cp = [(a+b+4)*(a+b+5)*(a+b+6),6*(a+b+4)*(a+b+5)*(a+3),
12*(a+b+4)*(a+2)*(a+3),8*(a+1)*(a+2)*(a+3)]
p3c = [cp[0],cp[1]-3*cp[0],cp[2]-2*cp[1]+3*cp[0],cp[3]-cp[2]+cp[1]-cp[0]]
assert_array_almost_equal(P3.c,array(p3c)/48.0,13)
def test_kn(self):
kn1 = special.kn(0,.2)
assert_almost_equal(kn1,1.7527038555281462,8)
def test_negv_kv(self):
assert_equal(special.kv(3.0, 2.2), special.kv(-3.0, 2.2))
def test_kv0(self):
kv0 = special.kv(0,.2)
assert_almost_equal(kv0, 1.7527038555281462, 10)
def test_kv1(self):
kv1 = special.kv(1,0.2)
assert_almost_equal(kv1, 4.775972543220472, 10)
def test_kv2(self):
kv2 = special.kv(2,0.2)
assert_almost_equal(kv2, 49.51242928773287, 10)
def test_kn_largeorder(self):
assert_allclose(special.kn(32, 1), 1.7516596664574289e+43)
def test_kv_largearg(self):
assert_equal(special.kv(0, 1e19), 0)
def test_negv_kve(self):
assert_equal(special.kve(3.0, 2.2), special.kve(-3.0, 2.2))
def test_kve(self):
kve1 = special.kve(0,.2)
kv1 = special.kv(0,.2)*exp(.2)
assert_almost_equal(kve1,kv1,8)
z = .2+1j
kve2 = special.kve(0,z)
kv2 = special.kv(0,z)*exp(z)
assert_almost_equal(kve2,kv2,8)
def test_kvp_v0n1(self):
z = 2.2
assert_almost_equal(-special.kv(1,z), special.kvp(0,z, n=1), 10)
def test_kvp_n1(self):
v = 3.
z = 2.2
xc = -special.kv(v+1,z) + v/z*special.kv(v,z)
x = special.kvp(v,z, n=1)
assert_almost_equal(xc, x, 10) # this function (kvp) is broken
def test_kvp_n2(self):
v = 3.
z = 2.2
xc = (z**2+v**2-v)/z**2 * special.kv(v,z) + special.kv(v+1,z)/z
x = special.kvp(v, z, n=2)
assert_almost_equal(xc, x, 10)
def test_y0(self):
oz = special.y0(.1)
ozr = special.yn(0,.1)
assert_almost_equal(oz,ozr,8)
def test_y1(self):
o1 = special.y1(.1)
o1r = special.yn(1,.1)
assert_almost_equal(o1,o1r,8)
def test_y0_zeros(self):
yo,ypo = special.y0_zeros(2)
zo,zpo = special.y0_zeros(2,complex=1)
all = r_[yo,zo]
allval = r_[ypo,zpo]
assert_array_almost_equal(abs(special.yv(0.0,all)),0.0,11)
assert_array_almost_equal(abs(special.yv(1,all)-allval),0.0,11)
def test_y1_zeros(self):
y1 = special.y1_zeros(1)
assert_array_almost_equal(y1,(array([2.19714]),array([0.52079])),5)
def test_y1p_zeros(self):
y1p = special.y1p_zeros(1,complex=1)
assert_array_almost_equal(y1p,(array([0.5768+0.904j]), array([-0.7635+0.5892j])),3)
def test_yn_zeros(self):
an = special.yn_zeros(4,2)
assert_array_almost_equal(an,array([5.64515, 9.36162]),5)
an = special.yn_zeros(443,5)
assert_allclose(an, [450.13573091578090314, 463.05692376675001542,
472.80651546418663566, 481.27353184725625838,
488.98055964441374646], rtol=1e-15)
def test_ynp_zeros(self):
ao = special.ynp_zeros(0,2)
assert_array_almost_equal(ao,array([2.19714133, 5.42968104]),6)
ao = special.ynp_zeros(43,5)
assert_allclose(special.yvp(43, ao), 0, atol=1e-15)
ao = special.ynp_zeros(443,5)
assert_allclose(special.yvp(443, ao), 0, atol=1e-9)
def test_ynp_zeros_large_order(self):
ao = special.ynp_zeros(443,5)
assert_allclose(special.yvp(443, ao), 0, atol=1e-14)
def test_yn(self):
yn2n = special.yn(1,.2)
assert_almost_equal(yn2n,-3.3238249881118471,8)
def test_negv_yv(self):
assert_almost_equal(special.yv(-3,2), -special.yv(3,2), 14)
def test_yv(self):
yv2 = special.yv(1,.2)
assert_almost_equal(yv2,-3.3238249881118471,8)
def test_negv_yve(self):
assert_almost_equal(special.yve(-3,2), -special.yve(3,2), 14)
def test_yve(self):
yve2 = special.yve(1,.2)
assert_almost_equal(yve2,-3.3238249881118471,8)
yve2r = special.yv(1,.2+1j)*exp(-1)
yve22 = special.yve(1,.2+1j)
assert_almost_equal(yve22,yve2r,8)
def test_yvp(self):
yvpr = (special.yv(1,.2) - special.yv(3,.2))/2.0
yvp1 = special.yvp(2,.2)
assert_array_almost_equal(yvp1,yvpr,10)
def _cephes_vs_amos_points(self):
"""Yield points at which to compare Cephes implementation to AMOS"""
# check several points, including large-amplitude ones
v = [-120, -100.3, -20., -10., -1., -.5, 0., 1., 12.49, 120., 301]
z = [-1300, -11, -10, -1, 1., 10., 200.5, 401., 600.5, 700.6, 1300,
10003]
yield from itertools.product(v, z)
# check half-integers; these are problematic points at least
# for cephes/iv
yield from itertools.product(0.5 + arange(-60, 60), [3.5])
def check_cephes_vs_amos(self, f1, f2, rtol=1e-11, atol=0, skip=None):
for v, z in self._cephes_vs_amos_points():
if skip is not None and skip(v, z):
continue
c1, c2, c3 = f1(v, z), f1(v,z+0j), f2(int(v), z)
if np.isinf(c1):
assert_(np.abs(c2) >= 1e300, (v, z))
elif np.isnan(c1):
assert_(c2.imag != 0, (v, z))
else:
assert_allclose(c1, c2, err_msg=(v, z), rtol=rtol, atol=atol)
if v == int(v):
assert_allclose(c3, c2, err_msg=(v, z),
rtol=rtol, atol=atol)
@pytest.mark.xfail(platform.machine() == 'ppc64le',
reason="fails on ppc64le")
def test_jv_cephes_vs_amos(self):
self.check_cephes_vs_amos(special.jv, special.jn, rtol=1e-10, atol=1e-305)
@pytest.mark.xfail(platform.machine() == 'ppc64le',
reason="fails on ppc64le")
def test_yv_cephes_vs_amos(self):
self.check_cephes_vs_amos(special.yv, special.yn, rtol=1e-11, atol=1e-305)
def test_yv_cephes_vs_amos_only_small_orders(self):
skipper = lambda v, z: (abs(v) > 50)
self.check_cephes_vs_amos(special.yv, special.yn, rtol=1e-11, atol=1e-305, skip=skipper)
def test_iv_cephes_vs_amos(self):
with np.errstate(all='ignore'):
self.check_cephes_vs_amos(special.iv, special.iv, rtol=5e-9, atol=1e-305)
@pytest.mark.slow
def test_iv_cephes_vs_amos_mass_test(self):
N = 1000000
np.random.seed(1)
v = np.random.pareto(0.5, N) * (-1)**np.random.randint(2, size=N)
x = np.random.pareto(0.2, N) * (-1)**np.random.randint(2, size=N)
imsk = (np.random.randint(8, size=N) == 0)
v[imsk] = v[imsk].astype(int)
with np.errstate(all='ignore'):
c1 = special.iv(v, x)
c2 = special.iv(v, x+0j)
# deal with differences in the inf and zero cutoffs
c1[abs(c1) > 1e300] = np.inf
c2[abs(c2) > 1e300] = np.inf
c1[abs(c1) < 1e-300] = 0
c2[abs(c2) < 1e-300] = 0
dc = abs(c1/c2 - 1)
dc[np.isnan(dc)] = 0
k = np.argmax(dc)
# Most error apparently comes from AMOS and not our implementation;
# there are some problems near integer orders there
assert_(dc[k] < 2e-7, (v[k], x[k], special.iv(v[k], x[k]), special.iv(v[k], x[k]+0j)))
def test_kv_cephes_vs_amos(self):
self.check_cephes_vs_amos(special.kv, special.kn, rtol=1e-9, atol=1e-305)
self.check_cephes_vs_amos(special.kv, special.kv, rtol=1e-9, atol=1e-305)
def test_ticket_623(self):
assert_allclose(special.jv(3, 4), 0.43017147387562193)
assert_allclose(special.jv(301, 1300), 0.0183487151115275)
assert_allclose(special.jv(301, 1296.0682), -0.0224174325312048)
def test_ticket_853(self):
"""Negative-order Bessels"""
# cephes
assert_allclose(special.jv(-1, 1), -0.4400505857449335)
assert_allclose(special.jv(-2, 1), 0.1149034849319005)
assert_allclose(special.yv(-1, 1), 0.7812128213002887)
assert_allclose(special.yv(-2, 1), -1.650682606816255)
assert_allclose(special.iv(-1, 1), 0.5651591039924851)
assert_allclose(special.iv(-2, 1), 0.1357476697670383)
assert_allclose(special.kv(-1, 1), 0.6019072301972347)
assert_allclose(special.kv(-2, 1), 1.624838898635178)
assert_allclose(special.jv(-0.5, 1), 0.43109886801837607952)
assert_allclose(special.yv(-0.5, 1), 0.6713967071418031)
assert_allclose(special.iv(-0.5, 1), 1.231200214592967)
assert_allclose(special.kv(-0.5, 1), 0.4610685044478945)
# amos
assert_allclose(special.jv(-1, 1+0j), -0.4400505857449335)
assert_allclose(special.jv(-2, 1+0j), 0.1149034849319005)
assert_allclose(special.yv(-1, 1+0j), 0.7812128213002887)
assert_allclose(special.yv(-2, 1+0j), -1.650682606816255)
assert_allclose(special.iv(-1, 1+0j), 0.5651591039924851)
assert_allclose(special.iv(-2, 1+0j), 0.1357476697670383)
assert_allclose(special.kv(-1, 1+0j), 0.6019072301972347)
assert_allclose(special.kv(-2, 1+0j), 1.624838898635178)
assert_allclose(special.jv(-0.5, 1+0j), 0.43109886801837607952)
assert_allclose(special.jv(-0.5, 1+1j), 0.2628946385649065-0.827050182040562j)
assert_allclose(special.yv(-0.5, 1+0j), 0.6713967071418031)
assert_allclose(special.yv(-0.5, 1+1j), 0.967901282890131+0.0602046062142816j)
assert_allclose(special.iv(-0.5, 1+0j), 1.231200214592967)
assert_allclose(special.iv(-0.5, 1+1j), 0.77070737376928+0.39891821043561j)
assert_allclose(special.kv(-0.5, 1+0j), 0.4610685044478945)
assert_allclose(special.kv(-0.5, 1+1j), 0.06868578341999-0.38157825981268j)
assert_allclose(special.jve(-0.5,1+0.3j), special.jv(-0.5, 1+0.3j)*exp(-0.3))
assert_allclose(special.yve(-0.5,1+0.3j), special.yv(-0.5, 1+0.3j)*exp(-0.3))
assert_allclose(special.ive(-0.5,0.3+1j), special.iv(-0.5, 0.3+1j)*exp(-0.3))
assert_allclose(special.kve(-0.5,0.3+1j), special.kv(-0.5, 0.3+1j)*exp(0.3+1j))
assert_allclose(special.hankel1(-0.5, 1+1j), special.jv(-0.5, 1+1j) + 1j*special.yv(-0.5,1+1j))
assert_allclose(special.hankel2(-0.5, 1+1j), special.jv(-0.5, 1+1j) - 1j*special.yv(-0.5,1+1j))
def test_ticket_854(self):
"""Real-valued Bessel domains"""
assert_(isnan(special.jv(0.5, -1)))
assert_(isnan(special.iv(0.5, -1)))
assert_(isnan(special.yv(0.5, -1)))
assert_(isnan(special.yv(1, -1)))
assert_(isnan(special.kv(0.5, -1)))
assert_(isnan(special.kv(1, -1)))
assert_(isnan(special.jve(0.5, -1)))
assert_(isnan(special.ive(0.5, -1)))
assert_(isnan(special.yve(0.5, -1)))
assert_(isnan(special.yve(1, -1)))
assert_(isnan(special.kve(0.5, -1)))
assert_(isnan(special.kve(1, -1)))
assert_(isnan(special.airye(-1)[0:2]).all(), special.airye(-1))
assert_(not isnan(special.airye(-1)[2:4]).any(), special.airye(-1))
def test_gh_7909(self):
assert_(special.kv(1.5, 0) == np.inf)
assert_(special.kve(1.5, 0) == np.inf)
def test_ticket_503(self):
"""Real-valued Bessel I overflow"""
assert_allclose(special.iv(1, 700), 1.528500390233901e302)
assert_allclose(special.iv(1000, 1120), 1.301564549405821e301)
def test_iv_hyperg_poles(self):
assert_allclose(special.iv(-0.5, 1), 1.231200214592967)
def iv_series(self, v, z, n=200):
k = arange(0, n).astype(float_)
r = (v+2*k)*log(.5*z) - special.gammaln(k+1) - special.gammaln(v+k+1)
r[isnan(r)] = inf
r = exp(r)
err = abs(r).max() * finfo(float_).eps * n + abs(r[-1])*10
return r.sum(), err
def test_i0_series(self):
for z in [1., 10., 200.5]:
value, err = self.iv_series(0, z)
assert_allclose(special.i0(z), value, atol=err, err_msg=z)
def test_i1_series(self):
for z in [1., 10., 200.5]:
value, err = self.iv_series(1, z)
assert_allclose(special.i1(z), value, atol=err, err_msg=z)
def test_iv_series(self):
for v in [-20., -10., -1., 0., 1., 12.49, 120.]:
for z in [1., 10., 200.5, -1+2j]:
value, err = self.iv_series(v, z)
assert_allclose(special.iv(v, z), value, atol=err, err_msg=(v, z))
def test_i0(self):
values = [[0.0, 1.0],
[1e-10, 1.0],
[0.1, 0.9071009258],
[0.5, 0.6450352706],
[1.0, 0.4657596077],
[2.5, 0.2700464416],
[5.0, 0.1835408126],
[20.0, 0.0897803119],
]
for i, (x, v) in enumerate(values):
cv = special.i0(x) * exp(-x)
assert_almost_equal(cv, v, 8, err_msg='test #%d' % i)
def test_i0e(self):
oize = special.i0e(.1)
oizer = special.ive(0,.1)
assert_almost_equal(oize,oizer,8)
def test_i1(self):
values = [[0.0, 0.0],
[1e-10, 0.4999999999500000e-10],
[0.1, 0.0452984468],
[0.5, 0.1564208032],
[1.0, 0.2079104154],
[5.0, 0.1639722669],
[20.0, 0.0875062222],
]
for i, (x, v) in enumerate(values):
cv = special.i1(x) * exp(-x)
assert_almost_equal(cv, v, 8, err_msg='test #%d' % i)
def test_i1e(self):
oi1e = special.i1e(.1)
oi1er = special.ive(1,.1)
assert_almost_equal(oi1e,oi1er,8)
def test_iti0k0(self):
iti0 = array(special.iti0k0(5))
assert_array_almost_equal(iti0,array([31.848667776169801, 1.5673873907283657]),5)
def test_it2i0k0(self):
it2k = special.it2i0k0(.1)
assert_array_almost_equal(it2k,array([0.0012503906973464409, 3.3309450354686687]),6)
def test_iv(self):
iv1 = special.iv(0,.1)*exp(-.1)
assert_almost_equal(iv1,0.90710092578230106,10)
def test_negv_ive(self):
assert_equal(special.ive(3,2), special.ive(-3,2))
def test_ive(self):
ive1 = special.ive(0,.1)
iv1 = special.iv(0,.1)*exp(-.1)
assert_almost_equal(ive1,iv1,10)
def test_ivp0(self):
assert_almost_equal(special.iv(1,2), special.ivp(0,2), 10)
def test_ivp(self):
y = (special.iv(0,2) + special.iv(2,2))/2
x = special.ivp(1,2)
assert_almost_equal(x,y,10)
class TestLaguerre:
def test_laguerre(self):
lag0 = special.laguerre(0)
lag1 = special.laguerre(1)
lag2 = special.laguerre(2)
lag3 = special.laguerre(3)
lag4 = special.laguerre(4)
lag5 = special.laguerre(5)
assert_array_almost_equal(lag0.c,[1],13)
assert_array_almost_equal(lag1.c,[-1,1],13)
assert_array_almost_equal(lag2.c,array([1,-4,2])/2.0,13)
assert_array_almost_equal(lag3.c,array([-1,9,-18,6])/6.0,13)
assert_array_almost_equal(lag4.c,array([1,-16,72,-96,24])/24.0,13)
assert_array_almost_equal(lag5.c,array([-1,25,-200,600,-600,120])/120.0,13)
def test_genlaguerre(self):
k = 5*np.random.random() - 0.9
lag0 = special.genlaguerre(0,k)
lag1 = special.genlaguerre(1,k)
lag2 = special.genlaguerre(2,k)
lag3 = special.genlaguerre(3,k)
assert_equal(lag0.c,[1])
assert_equal(lag1.c,[-1,k+1])
assert_almost_equal(lag2.c,array([1,-2*(k+2),(k+1.)*(k+2.)])/2.0)
assert_almost_equal(lag3.c,array([-1,3*(k+3),-3*(k+2)*(k+3),(k+1)*(k+2)*(k+3)])/6.0)
# Base polynomials come from Abrahmowitz and Stegan
class TestLegendre:
def test_legendre(self):
leg0 = special.legendre(0)
leg1 = special.legendre(1)
leg2 = special.legendre(2)
leg3 = special.legendre(3)
leg4 = special.legendre(4)
leg5 = special.legendre(5)
assert_equal(leg0.c, [1])
assert_equal(leg1.c, [1,0])
assert_almost_equal(leg2.c, array([3,0,-1])/2.0, decimal=13)
assert_almost_equal(leg3.c, array([5,0,-3,0])/2.0)
assert_almost_equal(leg4.c, array([35,0,-30,0,3])/8.0)
assert_almost_equal(leg5.c, array([63,0,-70,0,15,0])/8.0)
class TestLambda:
def test_lmbda(self):
lam = special.lmbda(1,.1)
lamr = (array([special.jn(0,.1), 2*special.jn(1,.1)/.1]),
array([special.jvp(0,.1), -2*special.jv(1,.1)/.01 + 2*special.jvp(1,.1)/.1]))
assert_array_almost_equal(lam,lamr,8)
class TestLog1p:
def test_log1p(self):
l1p = (special.log1p(10), special.log1p(11), special.log1p(12))
l1prl = (log(11), log(12), log(13))
assert_array_almost_equal(l1p,l1prl,8)
def test_log1pmore(self):
l1pm = (special.log1p(1), special.log1p(1.1), special.log1p(1.2))
l1pmrl = (log(2),log(2.1),log(2.2))
assert_array_almost_equal(l1pm,l1pmrl,8)
class TestLegendreFunctions:
def test_clpmn(self):
z = 0.5+0.3j
clp = special.clpmn(2, 2, z, 3)
assert_array_almost_equal(clp,
(array([[1.0000, z, 0.5*(3*z*z-1)],
[0.0000, sqrt(z*z-1), 3*z*sqrt(z*z-1)],
[0.0000, 0.0000, 3*(z*z-1)]]),
array([[0.0000, 1.0000, 3*z],
[0.0000, z/sqrt(z*z-1), 3*(2*z*z-1)/sqrt(z*z-1)],
[0.0000, 0.0000, 6*z]])),
7)
def test_clpmn_close_to_real_2(self):
eps = 1e-10
m = 1
n = 3
x = 0.5
clp_plus = special.clpmn(m, n, x+1j*eps, 2)[0][m, n]
clp_minus = special.clpmn(m, n, x-1j*eps, 2)[0][m, n]
assert_array_almost_equal(array([clp_plus, clp_minus]),
array([special.lpmv(m, n, x),
special.lpmv(m, n, x)]),
7)
def test_clpmn_close_to_real_3(self):
eps = 1e-10
m = 1
n = 3
x = 0.5
clp_plus = special.clpmn(m, n, x+1j*eps, 3)[0][m, n]
clp_minus = special.clpmn(m, n, x-1j*eps, 3)[0][m, n]
assert_array_almost_equal(array([clp_plus, clp_minus]),
array([special.lpmv(m, n, x)*np.exp(-0.5j*m*np.pi),
special.lpmv(m, n, x)*np.exp(0.5j*m*np.pi)]),
7)
def test_clpmn_across_unit_circle(self):
eps = 1e-7
m = 1
n = 1
x = 1j
for type in [2, 3]:
assert_almost_equal(special.clpmn(m, n, x+1j*eps, type)[0][m, n],
special.clpmn(m, n, x-1j*eps, type)[0][m, n], 6)
def test_inf(self):
for z in (1, -1):
for n in range(4):
for m in range(1, n):
lp = special.clpmn(m, n, z)
assert_(np.isinf(lp[1][1,1:]).all())
lp = special.lpmn(m, n, z)
assert_(np.isinf(lp[1][1,1:]).all())
def test_deriv_clpmn(self):
# data inside and outside of the unit circle
zvals = [0.5+0.5j, -0.5+0.5j, -0.5-0.5j, 0.5-0.5j,
1+1j, -1+1j, -1-1j, 1-1j]
m = 2
n = 3
for type in [2, 3]:
for z in zvals:
for h in [1e-3, 1e-3j]:
approx_derivative = (special.clpmn(m, n, z+0.5*h, type)[0]
- special.clpmn(m, n, z-0.5*h, type)[0])/h
assert_allclose(special.clpmn(m, n, z, type)[1],
approx_derivative,
rtol=1e-4)
def test_lpmn(self):
lp = special.lpmn(0,2,.5)
assert_array_almost_equal(lp,(array([[1.00000,
0.50000,
-0.12500]]),
array([[0.00000,
1.00000,
1.50000]])),4)
def test_lpn(self):
lpnf = special.lpn(2,.5)
assert_array_almost_equal(lpnf,(array([1.00000,
0.50000,
-0.12500]),
array([0.00000,
1.00000,
1.50000])),4)
def test_lpmv(self):
lp = special.lpmv(0,2,.5)
assert_almost_equal(lp,-0.125,7)
lp = special.lpmv(0,40,.001)
assert_almost_equal(lp,0.1252678976534484,7)
# XXX: this is outside the domain of the current implementation,
# so ensure it returns a NaN rather than a wrong answer.
with np.errstate(all='ignore'):
lp = special.lpmv(-1,-1,.001)
assert_(lp != 0 or np.isnan(lp))
def test_lqmn(self):
lqmnf = special.lqmn(0,2,.5)
lqf = special.lqn(2,.5)
assert_array_almost_equal(lqmnf[0][0],lqf[0],4)
assert_array_almost_equal(lqmnf[1][0],lqf[1],4)
def test_lqmn_gt1(self):
"""algorithm for real arguments changes at 1.0001
test against analytical result for m=2, n=1
"""
x0 = 1.0001
delta = 0.00002
for x in (x0-delta, x0+delta):
lq = special.lqmn(2, 1, x)[0][-1, -1]
expected = 2/(x*x-1)
assert_almost_equal(lq, expected)
def test_lqmn_shape(self):
a, b = special.lqmn(4, 4, 1.1)
assert_equal(a.shape, (5, 5))
assert_equal(b.shape, (5, 5))
a, b = special.lqmn(4, 0, 1.1)
assert_equal(a.shape, (5, 1))
assert_equal(b.shape, (5, 1))
def test_lqn(self):
lqf = special.lqn(2,.5)
assert_array_almost_equal(lqf,(array([0.5493, -0.7253, -0.8187]),
array([1.3333, 1.216, -0.8427])),4)
class TestMathieu:
def test_mathieu_a(self):
pass
def test_mathieu_even_coef(self):
special.mathieu_even_coef(2,5)
# Q not defined broken and cannot figure out proper reporting order
def test_mathieu_odd_coef(self):
# same problem as above
pass
class TestFresnelIntegral:
def test_modfresnelp(self):
pass
def test_modfresnelm(self):
pass
class TestOblCvSeq:
def test_obl_cv_seq(self):
obl = special.obl_cv_seq(0,3,1)
assert_array_almost_equal(obl,array([-0.348602,
1.393206,
5.486800,
11.492120]),5)
class TestParabolicCylinder:
def test_pbdn_seq(self):
pb = special.pbdn_seq(1,.1)
assert_array_almost_equal(pb,(array([0.9975,
0.0998]),
array([-0.0499,
0.9925])),4)
def test_pbdv(self):
special.pbdv(1,.2)
1/2*(.2)*special.pbdv(1,.2)[0] - special.pbdv(0,.2)[0]
def test_pbdv_seq(self):
pbn = special.pbdn_seq(1,.1)
pbv = special.pbdv_seq(1,.1)
assert_array_almost_equal(pbv,(real(pbn[0]),real(pbn[1])),4)
def test_pbdv_points(self):
# simple case
eta = np.linspace(-10, 10, 5)
z = 2**(eta/2)*np.sqrt(np.pi)/special.gamma(.5-.5*eta)
assert_allclose(special.pbdv(eta, 0.)[0], z, rtol=1e-14, atol=1e-14)
# some points
assert_allclose(special.pbdv(10.34, 20.44)[0], 1.3731383034455e-32, rtol=1e-12)
assert_allclose(special.pbdv(-9.53, 3.44)[0], 3.166735001119246e-8, rtol=1e-12)
def test_pbdv_gradient(self):
x = np.linspace(-4, 4, 8)[:,None]
eta = np.linspace(-10, 10, 5)[None,:]
p = special.pbdv(eta, x)
eps = 1e-7 + 1e-7*abs(x)
dp = (special.pbdv(eta, x + eps)[0] - special.pbdv(eta, x - eps)[0]) / eps / 2.
assert_allclose(p[1], dp, rtol=1e-6, atol=1e-6)
def test_pbvv_gradient(self):
x = np.linspace(-4, 4, 8)[:,None]
eta = np.linspace(-10, 10, 5)[None,:]
p = special.pbvv(eta, x)
eps = 1e-7 + 1e-7*abs(x)
dp = (special.pbvv(eta, x + eps)[0] - special.pbvv(eta, x - eps)[0]) / eps / 2.
assert_allclose(p[1], dp, rtol=1e-6, atol=1e-6)
class TestPolygamma:
# from Table 6.2 (pg. 271) of A&S
def test_polygamma(self):
poly2 = special.polygamma(2,1)
poly3 = special.polygamma(3,1)
assert_almost_equal(poly2,-2.4041138063,10)
assert_almost_equal(poly3,6.4939394023,10)
# Test polygamma(0, x) == psi(x)
x = [2, 3, 1.1e14]
assert_almost_equal(special.polygamma(0, x), special.psi(x))
# Test broadcasting
n = [0, 1, 2]
x = [0.5, 1.5, 2.5]
expected = [-1.9635100260214238, 0.93480220054467933,
-0.23620405164172739]
assert_almost_equal(special.polygamma(n, x), expected)
expected = np.row_stack([expected]*2)
assert_almost_equal(special.polygamma(n, np.row_stack([x]*2)),
expected)
assert_almost_equal(special.polygamma(np.row_stack([n]*2), x),
expected)
class TestProCvSeq:
def test_pro_cv_seq(self):
prol = special.pro_cv_seq(0,3,1)
assert_array_almost_equal(prol,array([0.319000,
2.593084,
6.533471,
12.514462]),5)
class TestPsi:
def test_psi(self):
ps = special.psi(1)
assert_almost_equal(ps,-0.57721566490153287,8)
class TestRadian:
def test_radian(self):
rad = special.radian(90,0,0)
assert_almost_equal(rad,pi/2.0,5)
def test_radianmore(self):
rad1 = special.radian(90,1,60)
assert_almost_equal(rad1,pi/2+0.0005816135199345904,5)
class TestRiccati:
def test_riccati_jn(self):
N, x = 2, 0.2
S = np.empty((N, N))
for n in range(N):
j = special.spherical_jn(n, x)
jp = special.spherical_jn(n, x, derivative=True)
S[0,n] = x*j
S[1,n] = x*jp + j
assert_array_almost_equal(S, special.riccati_jn(n, x), 8)
def test_riccati_yn(self):
N, x = 2, 0.2
C = np.empty((N, N))
for n in range(N):
y = special.spherical_yn(n, x)
yp = special.spherical_yn(n, x, derivative=True)
C[0,n] = x*y
C[1,n] = x*yp + y
assert_array_almost_equal(C, special.riccati_yn(n, x), 8)
class TestRound:
def test_round(self):
rnd = list(map(int,(special.round(10.1),special.round(10.4),special.round(10.5),special.round(10.6))))
# Note: According to the documentation, scipy.special.round is
# supposed to round to the nearest even number if the fractional
# part is exactly 0.5. On some platforms, this does not appear
# to work and thus this test may fail. However, this unit test is
# correctly written.
rndrl = (10,10,10,11)
assert_array_equal(rnd,rndrl)
def test_sph_harm():
# Tests derived from tables in
# https://en.wikipedia.org/wiki/Table_of_spherical_harmonics
sh = special.sph_harm
pi = np.pi
exp = np.exp
sqrt = np.sqrt
sin = np.sin
cos = np.cos
assert_array_almost_equal(sh(0,0,0,0),
0.5/sqrt(pi))
assert_array_almost_equal(sh(-2,2,0.,pi/4),
0.25*sqrt(15./(2.*pi)) *
(sin(pi/4))**2.)
assert_array_almost_equal(sh(-2,2,0.,pi/2),
0.25*sqrt(15./(2.*pi)))
assert_array_almost_equal(sh(2,2,pi,pi/2),
0.25*sqrt(15/(2.*pi)) *
exp(0+2.*pi*1j)*sin(pi/2.)**2.)
assert_array_almost_equal(sh(2,4,pi/4.,pi/3.),
(3./8.)*sqrt(5./(2.*pi)) *
exp(0+2.*pi/4.*1j) *
sin(pi/3.)**2. *
(7.*cos(pi/3.)**2.-1))
assert_array_almost_equal(sh(4,4,pi/8.,pi/6.),
(3./16.)*sqrt(35./(2.*pi)) *
exp(0+4.*pi/8.*1j)*sin(pi/6.)**4.)
def test_sph_harm_ufunc_loop_selection():
# see https://github.com/scipy/scipy/issues/4895
dt = np.dtype(np.complex128)
assert_equal(special.sph_harm(0, 0, 0, 0).dtype, dt)
assert_equal(special.sph_harm([0], 0, 0, 0).dtype, dt)
assert_equal(special.sph_harm(0, [0], 0, 0).dtype, dt)
assert_equal(special.sph_harm(0, 0, [0], 0).dtype, dt)
assert_equal(special.sph_harm(0, 0, 0, [0]).dtype, dt)
assert_equal(special.sph_harm([0], [0], [0], [0]).dtype, dt)
class TestStruve:
def _series(self, v, z, n=100):
"""Compute Struve function & error estimate from its power series."""
k = arange(0, n)
r = (-1)**k * (.5*z)**(2*k+v+1)/special.gamma(k+1.5)/special.gamma(k+v+1.5)
err = abs(r).max() * finfo(float_).eps * n
return r.sum(), err
def test_vs_series(self):
"""Check Struve function versus its power series"""
for v in [-20, -10, -7.99, -3.4, -1, 0, 1, 3.4, 12.49, 16]:
for z in [1, 10, 19, 21, 30]:
value, err = self._series(v, z)
assert_allclose(special.struve(v, z), value, rtol=0, atol=err), (v, z)
def test_some_values(self):
assert_allclose(special.struve(-7.99, 21), 0.0467547614113, rtol=1e-7)
assert_allclose(special.struve(-8.01, 21), 0.0398716951023, rtol=1e-8)
assert_allclose(special.struve(-3.0, 200), 0.0142134427432, rtol=1e-12)
assert_allclose(special.struve(-8.0, -41), 0.0192469727846, rtol=1e-11)
assert_equal(special.struve(-12, -41), -special.struve(-12, 41))
assert_equal(special.struve(+12, -41), -special.struve(+12, 41))
assert_equal(special.struve(-11, -41), +special.struve(-11, 41))
assert_equal(special.struve(+11, -41), +special.struve(+11, 41))
assert_(isnan(special.struve(-7.1, -1)))
assert_(isnan(special.struve(-10.1, -1)))
def test_regression_679(self):
"""Regression test for #679"""
assert_allclose(special.struve(-1.0, 20 - 1e-8), special.struve(-1.0, 20 + 1e-8))
assert_allclose(special.struve(-2.0, 20 - 1e-8), special.struve(-2.0, 20 + 1e-8))
assert_allclose(special.struve(-4.3, 20 - 1e-8), special.struve(-4.3, 20 + 1e-8))
def test_chi2_smalldf():
assert_almost_equal(special.chdtr(0.6,3), 0.957890536704110)
def test_ch2_inf():
assert_equal(special.chdtr(0.7,np.inf), 1.0)
def test_chi2c_smalldf():
assert_almost_equal(special.chdtrc(0.6,3), 1-0.957890536704110)
def test_chi2_inv_smalldf():
assert_almost_equal(special.chdtri(0.6,1-0.957890536704110), 3)
def test_agm_simple():
rtol = 1e-13
# Gauss's constant
assert_allclose(1/special.agm(1, np.sqrt(2)), 0.834626841674073186,
rtol=rtol)
# These values were computed using Wolfram Alpha, with the
# function ArithmeticGeometricMean[a, b].
agm13 = 1.863616783244897
agm15 = 2.604008190530940
agm35 = 3.936235503649555
assert_allclose(special.agm([[1], [3]], [1, 3, 5]),
[[1, agm13, agm15],
[agm13, 3, agm35]], rtol=rtol)
# Computed by the iteration formula using mpmath,
# with mpmath.mp.prec = 1000:
agm12 = 1.4567910310469068
assert_allclose(special.agm(1, 2), agm12, rtol=rtol)
assert_allclose(special.agm(2, 1), agm12, rtol=rtol)
assert_allclose(special.agm(-1, -2), -agm12, rtol=rtol)
assert_allclose(special.agm(24, 6), 13.458171481725614, rtol=rtol)
assert_allclose(special.agm(13, 123456789.5), 11111458.498599306,
rtol=rtol)
assert_allclose(special.agm(1e30, 1), 2.229223055945383e+28, rtol=rtol)
assert_allclose(special.agm(1e-22, 1), 0.030182566420169886, rtol=rtol)
assert_allclose(special.agm(1e150, 1e180), 2.229223055945383e+178,
rtol=rtol)
assert_allclose(special.agm(1e180, 1e-150), 2.0634722510162677e+177,
rtol=rtol)
assert_allclose(special.agm(1e-150, 1e-170), 3.3112619670463756e-152,
rtol=rtol)
fi = np.finfo(1.0)
assert_allclose(special.agm(fi.tiny, fi.max), 1.9892072050015473e+305,
rtol=rtol)
assert_allclose(special.agm(0.75*fi.max, fi.max), 1.564904312298045e+308,
rtol=rtol)
assert_allclose(special.agm(fi.tiny, 3*fi.tiny), 4.1466849866735005e-308,
rtol=rtol)
# zero, nan and inf cases.
assert_equal(special.agm(0, 0), 0)
assert_equal(special.agm(99, 0), 0)
assert_equal(special.agm(-1, 10), np.nan)
assert_equal(special.agm(0, np.inf), np.nan)
assert_equal(special.agm(np.inf, 0), np.nan)
assert_equal(special.agm(0, -np.inf), np.nan)
assert_equal(special.agm(-np.inf, 0), np.nan)
assert_equal(special.agm(np.inf, -np.inf), np.nan)
assert_equal(special.agm(-np.inf, np.inf), np.nan)
assert_equal(special.agm(1, np.nan), np.nan)
assert_equal(special.agm(np.nan, -1), np.nan)
assert_equal(special.agm(1, np.inf), np.inf)
assert_equal(special.agm(np.inf, 1), np.inf)
assert_equal(special.agm(-1, -np.inf), -np.inf)
assert_equal(special.agm(-np.inf, -1), -np.inf)
def test_legacy():
# Legacy behavior: truncating arguments to integers
with suppress_warnings() as sup:
sup.filter(RuntimeWarning, "floating point number truncated to an integer")
assert_equal(special.expn(1, 0.3), special.expn(1.8, 0.3))
assert_equal(special.nbdtrc(1, 2, 0.3), special.nbdtrc(1.8, 2.8, 0.3))
assert_equal(special.nbdtr(1, 2, 0.3), special.nbdtr(1.8, 2.8, 0.3))
assert_equal(special.nbdtri(1, 2, 0.3), special.nbdtri(1.8, 2.8, 0.3))
assert_equal(special.pdtri(1, 0.3), special.pdtri(1.8, 0.3))
assert_equal(special.kn(1, 0.3), special.kn(1.8, 0.3))
assert_equal(special.yn(1, 0.3), special.yn(1.8, 0.3))
assert_equal(special.smirnov(1, 0.3), special.smirnov(1.8, 0.3))
assert_equal(special.smirnovi(1, 0.3), special.smirnovi(1.8, 0.3))
@with_special_errors
def test_error_raising():
assert_raises(special.SpecialFunctionError, special.iv, 1, 1e99j)
def test_xlogy():
def xfunc(x, y):
with np.errstate(invalid='ignore'):
if x == 0 and not np.isnan(y):
return x
else:
return x*np.log(y)
z1 = np.asarray([(0,0), (0, np.nan), (0, np.inf), (1.0, 2.0)], dtype=float)
z2 = np.r_[z1, [(0, 1j), (1, 1j)]]
w1 = np.vectorize(xfunc)(z1[:,0], z1[:,1])
assert_func_equal(special.xlogy, w1, z1, rtol=1e-13, atol=1e-13)
w2 = np.vectorize(xfunc)(z2[:,0], z2[:,1])
assert_func_equal(special.xlogy, w2, z2, rtol=1e-13, atol=1e-13)
def test_xlog1py():
def xfunc(x, y):
with np.errstate(invalid='ignore'):
if x == 0 and not np.isnan(y):
return x
else:
return x * np.log1p(y)
z1 = np.asarray([(0,0), (0, np.nan), (0, np.inf), (1.0, 2.0),
(1, 1e-30)], dtype=float)
w1 = np.vectorize(xfunc)(z1[:,0], z1[:,1])
assert_func_equal(special.xlog1py, w1, z1, rtol=1e-13, atol=1e-13)
def test_entr():
def xfunc(x):
if x < 0:
return -np.inf
else:
return -special.xlogy(x, x)
values = (0, 0.5, 1.0, np.inf)
signs = [-1, 1]
arr = []
for sgn, v in itertools.product(signs, values):
arr.append(sgn * v)
z = np.array(arr, dtype=float)
w = np.vectorize(xfunc, otypes=[np.float64])(z)
assert_func_equal(special.entr, w, z, rtol=1e-13, atol=1e-13)
def test_kl_div():
def xfunc(x, y):
if x < 0 or y < 0 or (y == 0 and x != 0):
# extension of natural domain to preserve convexity
return np.inf
elif np.isposinf(x) or np.isposinf(y):
# limits within the natural domain
return np.inf
elif x == 0:
return y
else:
return special.xlogy(x, x/y) - x + y
values = (0, 0.5, 1.0)
signs = [-1, 1]
arr = []
for sgna, va, sgnb, vb in itertools.product(signs, values, signs, values):
arr.append((sgna*va, sgnb*vb))
z = np.array(arr, dtype=float)
w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1])
assert_func_equal(special.kl_div, w, z, rtol=1e-13, atol=1e-13)
def test_rel_entr():
def xfunc(x, y):
if x > 0 and y > 0:
return special.xlogy(x, x/y)
elif x == 0 and y >= 0:
return 0
else:
return np.inf
values = (0, 0.5, 1.0)
signs = [-1, 1]
arr = []
for sgna, va, sgnb, vb in itertools.product(signs, values, signs, values):
arr.append((sgna*va, sgnb*vb))
z = np.array(arr, dtype=float)
w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1])
assert_func_equal(special.rel_entr, w, z, rtol=1e-13, atol=1e-13)
def test_huber():
assert_equal(special.huber(-1, 1.5), np.inf)
assert_allclose(special.huber(2, 1.5), 0.5 * np.square(1.5))
assert_allclose(special.huber(2, 2.5), 2 * (2.5 - 0.5 * 2))
def xfunc(delta, r):
if delta < 0:
return np.inf
elif np.abs(r) < delta:
return 0.5 * np.square(r)
else:
return delta * (np.abs(r) - 0.5 * delta)
z = np.random.randn(10, 2)
w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1])
assert_func_equal(special.huber, w, z, rtol=1e-13, atol=1e-13)
def test_pseudo_huber():
def xfunc(delta, r):
if delta < 0:
return np.inf
elif (not delta) or (not r):
return 0
else:
return delta**2 * (np.sqrt(1 + (r/delta)**2) - 1)
z = np.array(np.random.randn(10, 2).tolist() + [[0, 0.5], [0.5, 0]])
w = np.vectorize(xfunc, otypes=[np.float64])(z[:,0], z[:,1])
assert_func_equal(special.pseudo_huber, w, z, rtol=1e-13, atol=1e-13)
def test_pseudo_huber_small_r():
delta = 1.0
r = 1e-18
y = special.pseudo_huber(delta, r)
# expected computed with mpmath:
# import mpmath
# mpmath.mp.dps = 200
# r = mpmath.mpf(1e-18)
# expected = float(mpmath.sqrt(1 + r**2) - 1)
expected = 5.0000000000000005e-37
assert_allclose(y, expected, rtol=1e-13)
def test_runtime_warning():
with pytest.warns(RuntimeWarning,
match=r'Too many predicted coefficients'):
mathieu_odd_coef(1000, 1000)
with pytest.warns(RuntimeWarning,
match=r'Too many predicted coefficients'):
mathieu_even_coef(1000, 1000)
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