4321 lines
168 KiB
Python
4321 lines
168 KiB
Python
# this program corresponds to special.py
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### Means test is not done yet
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# E Means test is giving error (E)
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# F Means test is failing (F)
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# EF Means test is giving error and Failing
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#! Means test is segfaulting
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# 8 Means test runs forever
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### test_besselpoly
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### test_mathieu_a
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### test_mathieu_even_coef
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### test_mathieu_odd_coef
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### test_modfresnelp
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### test_modfresnelm
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# test_pbdv_seq
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### test_pbvv_seq
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### test_sph_harm
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import functools
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import itertools
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import operator
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import platform
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import sys
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import numpy as np
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from numpy import (array, isnan, r_, arange, finfo, pi, sin, cos, tan, exp,
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log, zeros, sqrt, asarray, inf, nan_to_num, real, arctan, double,
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array_equal)
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import pytest
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from pytest import raises as assert_raises
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from numpy.testing import (assert_equal, assert_almost_equal,
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assert_array_equal, assert_array_almost_equal, assert_approx_equal,
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assert_, assert_allclose, assert_array_almost_equal_nulp,
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suppress_warnings)
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from scipy import special
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import scipy.special._ufuncs as cephes
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from scipy.special import ellipe, ellipk, ellipkm1
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from scipy.special import elliprc, elliprd, elliprf, elliprg, elliprj
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from scipy.special import mathieu_odd_coef, mathieu_even_coef, stirling2
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from scipy._lib.deprecation import _NoValue
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from scipy._lib._util import np_long, np_ulong
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from scipy.special._basic import _FACTORIALK_LIMITS_64BITS, \
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_FACTORIALK_LIMITS_32BITS
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from scipy.special._testutils import with_special_errors, \
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assert_func_equal, FuncData
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import math
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class TestCephes:
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def test_airy(self):
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cephes.airy(0)
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def test_airye(self):
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cephes.airye(0)
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def test_binom(self):
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n = np.array([0.264, 4, 5.2, 17])
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k = np.array([2, 0.4, 7, 3.3])
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nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
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).reshape(2, -1).T
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rknown = np.array([[-0.097152, 0.9263051596159367, 0.01858423645695389,
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-0.007581020651518199],[6, 2.0214389119675666, 0, 2.9827344527963846],
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[10.92, 2.22993515861399, -0.00585728, 10.468891352063146],
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[136, 3.5252179590758828, 19448, 1024.5526916174495]])
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assert_func_equal(cephes.binom, rknown.ravel(), nk, rtol=1e-13)
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# Test branches in implementation
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np.random.seed(1234)
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n = np.r_[np.arange(-7, 30), 1000*np.random.rand(30) - 500]
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k = np.arange(0, 102)
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nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
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).reshape(2, -1).T
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assert_func_equal(cephes.binom,
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cephes.binom(nk[:,0], nk[:,1] * (1 + 1e-15)),
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nk,
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atol=1e-10, rtol=1e-10)
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def test_binom_2(self):
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# Test branches in implementation
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np.random.seed(1234)
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n = np.r_[np.logspace(1, 300, 20)]
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k = np.arange(0, 102)
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nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
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).reshape(2, -1).T
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assert_func_equal(cephes.binom,
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cephes.binom(nk[:,0], nk[:,1] * (1 + 1e-15)),
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nk,
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atol=1e-10, rtol=1e-10)
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def test_binom_exact(self):
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@np.vectorize
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def binom_int(n, k):
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n = int(n)
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k = int(k)
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num = 1
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den = 1
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for i in range(1, k+1):
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num *= i + n - k
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den *= i
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return float(num/den)
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np.random.seed(1234)
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n = np.arange(1, 15)
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k = np.arange(0, 15)
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nk = np.array(np.broadcast_arrays(n[:,None], k[None,:])
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).reshape(2, -1).T
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nk = nk[nk[:,0] >= nk[:,1]]
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assert_func_equal(cephes.binom,
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binom_int(nk[:,0], nk[:,1]),
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nk,
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atol=0, rtol=0)
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def test_binom_nooverflow_8346(self):
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# Test (binom(n, k) doesn't overflow prematurely */
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dataset = [
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(1000, 500, 2.70288240945436551e+299),
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(1002, 501, 1.08007396880791225e+300),
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(1004, 502, 4.31599279169058121e+300),
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(1006, 503, 1.72468101616263781e+301),
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(1008, 504, 6.89188009236419153e+301),
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(1010, 505, 2.75402257948335448e+302),
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(1012, 506, 1.10052048531923757e+303),
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(1014, 507, 4.39774063758732849e+303),
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(1016, 508, 1.75736486108312519e+304),
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(1018, 509, 7.02255427788423734e+304),
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(1020, 510, 2.80626776829962255e+305),
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(1022, 511, 1.12140876377061240e+306),
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(1024, 512, 4.48125455209897109e+306),
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(1026, 513, 1.79075474304149900e+307),
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(1028, 514, 7.15605105487789676e+307)
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]
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dataset = np.asarray(dataset)
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FuncData(cephes.binom, dataset, (0, 1), 2, rtol=1e-12).check()
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def test_bdtr(self):
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assert_equal(cephes.bdtr(1,1,0.5),1.0)
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def test_bdtri(self):
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assert_equal(cephes.bdtri(1,3,0.5),0.5)
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def test_bdtrc(self):
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assert_equal(cephes.bdtrc(1,3,0.5),0.5)
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def test_bdtrin(self):
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assert_equal(cephes.bdtrin(1,0,1),5.0)
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def test_bdtrik(self):
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cephes.bdtrik(1,3,0.5)
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def test_bei(self):
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assert_equal(cephes.bei(0),0.0)
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def test_beip(self):
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assert_equal(cephes.beip(0),0.0)
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def test_ber(self):
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assert_equal(cephes.ber(0),1.0)
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def test_berp(self):
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assert_equal(cephes.berp(0),0.0)
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def test_besselpoly(self):
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assert_equal(cephes.besselpoly(0,0,0),1.0)
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def test_btdtr(self):
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with pytest.deprecated_call(match='deprecated in SciPy 1.12.0'):
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y = special.btdtr(1, 1, 1)
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assert_equal(y, 1.0)
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def test_btdtri(self):
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with pytest.deprecated_call(match='deprecated in SciPy 1.12.0'):
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y = special.btdtri(1, 1, 1)
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assert_equal(y, 1.0)
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def test_btdtria(self):
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assert_equal(cephes.btdtria(1,1,1),5.0)
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def test_btdtrib(self):
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assert_equal(cephes.btdtrib(1,1,1),5.0)
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def test_cbrt(self):
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assert_approx_equal(cephes.cbrt(1),1.0)
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def test_chdtr(self):
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assert_equal(cephes.chdtr(1,0),0.0)
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def test_chdtrc(self):
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assert_equal(cephes.chdtrc(1,0),1.0)
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def test_chdtri(self):
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assert_equal(cephes.chdtri(1,1),0.0)
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def test_chdtriv(self):
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assert_equal(cephes.chdtriv(0,0),5.0)
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def test_chndtr(self):
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assert_equal(cephes.chndtr(0,1,0),0.0)
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# Each row holds (x, nu, lam, expected_value)
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# These values were computed using Wolfram Alpha with
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# CDF[NoncentralChiSquareDistribution[nu, lam], x]
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values = np.array([
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[25.00, 20.0, 400, 4.1210655112396197139e-57],
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[25.00, 8.00, 250, 2.3988026526832425878e-29],
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[0.001, 8.00, 40., 5.3761806201366039084e-24],
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[0.010, 8.00, 40., 5.45396231055999457039e-20],
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[20.00, 2.00, 107, 1.39390743555819597802e-9],
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[22.50, 2.00, 107, 7.11803307138105870671e-9],
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[25.00, 2.00, 107, 3.11041244829864897313e-8],
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[3.000, 2.00, 1.0, 0.62064365321954362734],
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[350.0, 300., 10., 0.93880128006276407710],
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[100.0, 13.5, 10., 0.99999999650104210949],
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[700.0, 20.0, 400, 0.99999999925680650105],
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[150.0, 13.5, 10., 0.99999999999999983046],
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[160.0, 13.5, 10., 0.99999999999999999518], # 1.0
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])
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cdf = cephes.chndtr(values[:, 0], values[:, 1], values[:, 2])
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assert_allclose(cdf, values[:, 3], rtol=1e-12)
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assert_almost_equal(cephes.chndtr(np.inf, np.inf, 0), 2.0)
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assert_almost_equal(cephes.chndtr(2, 1, np.inf), 0.0)
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assert_(np.isnan(cephes.chndtr(np.nan, 1, 2)))
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assert_(np.isnan(cephes.chndtr(5, np.nan, 2)))
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assert_(np.isnan(cephes.chndtr(5, 1, np.nan)))
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def test_chndtridf(self):
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assert_equal(cephes.chndtridf(0,0,1),5.0)
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def test_chndtrinc(self):
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assert_equal(cephes.chndtrinc(0,1,0),5.0)
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def test_chndtrix(self):
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assert_equal(cephes.chndtrix(0,1,0),0.0)
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def test_cosdg(self):
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assert_equal(cephes.cosdg(0),1.0)
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def test_cosm1(self):
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assert_equal(cephes.cosm1(0),0.0)
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def test_cotdg(self):
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assert_almost_equal(cephes.cotdg(45),1.0)
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def test_dawsn(self):
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assert_equal(cephes.dawsn(0),0.0)
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assert_allclose(cephes.dawsn(1.23), 0.50053727749081767)
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def test_diric(self):
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# Test behavior near multiples of 2pi. Regression test for issue
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# described in gh-4001.
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n_odd = [1, 5, 25]
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x = np.array(2*np.pi + 5e-5).astype(np.float32)
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assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=7)
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x = np.array(2*np.pi + 1e-9).astype(np.float64)
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assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=15)
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x = np.array(2*np.pi + 1e-15).astype(np.float64)
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assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=15)
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if hasattr(np, 'float128'):
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# No float128 available in 32-bit numpy
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x = np.array(2*np.pi + 1e-12).astype(np.float128)
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assert_almost_equal(special.diric(x, n_odd), 1.0, decimal=19)
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n_even = [2, 4, 24]
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x = np.array(2*np.pi + 1e-9).astype(np.float64)
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assert_almost_equal(special.diric(x, n_even), -1.0, decimal=15)
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# Test at some values not near a multiple of pi
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x = np.arange(0.2*np.pi, 1.0*np.pi, 0.2*np.pi)
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octave_result = [0.872677996249965, 0.539344662916632,
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0.127322003750035, -0.206011329583298]
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assert_almost_equal(special.diric(x, 3), octave_result, decimal=15)
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def test_diric_broadcasting(self):
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x = np.arange(5)
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n = np.array([1, 3, 7])
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assert_(special.diric(x[:, np.newaxis], n).shape == (x.size, n.size))
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def test_ellipe(self):
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assert_equal(cephes.ellipe(1),1.0)
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def test_ellipeinc(self):
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assert_equal(cephes.ellipeinc(0,1),0.0)
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def test_ellipj(self):
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cephes.ellipj(0,1)
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def test_ellipk(self):
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assert_allclose(ellipk(0), pi/2)
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def test_ellipkinc(self):
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assert_equal(cephes.ellipkinc(0,0),0.0)
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def test_erf(self):
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assert_equal(cephes.erf(0), 0.0)
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def test_erf_symmetry(self):
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x = 5.905732037710919
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assert_equal(cephes.erf(x) + cephes.erf(-x), 0.0)
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def test_erfc(self):
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assert_equal(cephes.erfc(0), 1.0)
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def test_exp10(self):
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assert_approx_equal(cephes.exp10(2),100.0)
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def test_exp2(self):
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assert_equal(cephes.exp2(2),4.0)
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def test_expm1(self):
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assert_equal(cephes.expm1(0),0.0)
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assert_equal(cephes.expm1(np.inf), np.inf)
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assert_equal(cephes.expm1(-np.inf), -1)
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assert_equal(cephes.expm1(np.nan), np.nan)
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def test_expm1_complex(self):
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expm1 = cephes.expm1
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assert_equal(expm1(0 + 0j), 0 + 0j)
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assert_equal(expm1(complex(np.inf, 0)), complex(np.inf, 0))
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assert_equal(expm1(complex(np.inf, 1)), complex(np.inf, np.inf))
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assert_equal(expm1(complex(np.inf, 2)), complex(-np.inf, np.inf))
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assert_equal(expm1(complex(np.inf, 4)), complex(-np.inf, -np.inf))
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assert_equal(expm1(complex(np.inf, 5)), complex(np.inf, -np.inf))
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assert_equal(expm1(complex(1, np.inf)), complex(np.nan, np.nan))
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assert_equal(expm1(complex(0, np.inf)), complex(np.nan, np.nan))
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assert_equal(expm1(complex(np.inf, np.inf)), complex(np.inf, np.nan))
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assert_equal(expm1(complex(-np.inf, np.inf)), complex(-1, 0))
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assert_equal(expm1(complex(-np.inf, np.nan)), complex(-1, 0))
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assert_equal(expm1(complex(np.inf, np.nan)), complex(np.inf, np.nan))
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assert_equal(expm1(complex(0, np.nan)), complex(np.nan, np.nan))
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assert_equal(expm1(complex(1, np.nan)), complex(np.nan, np.nan))
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assert_equal(expm1(complex(np.nan, 1)), complex(np.nan, np.nan))
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assert_equal(expm1(complex(np.nan, np.nan)), complex(np.nan, np.nan))
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@pytest.mark.xfail(reason='The real part of expm1(z) bad at these points')
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def test_expm1_complex_hard(self):
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# The real part of this function is difficult to evaluate when
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# z.real = -log(cos(z.imag)).
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y = np.array([0.1, 0.2, 0.3, 5, 11, 20])
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x = -np.log(np.cos(y))
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z = x + 1j*y
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# evaluate using mpmath.expm1 with dps=1000
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expected = np.array([-5.5507901846769623e-17+0.10033467208545054j,
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2.4289354732893695e-18+0.20271003550867248j,
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4.5235500262585768e-17+0.30933624960962319j,
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7.8234305217489006e-17-3.3805150062465863j,
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-1.3685191953697676e-16-225.95084645419513j,
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8.7175620481291045e-17+2.2371609442247422j])
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found = cephes.expm1(z)
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# this passes.
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assert_array_almost_equal_nulp(found.imag, expected.imag, 3)
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# this fails.
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assert_array_almost_equal_nulp(found.real, expected.real, 20)
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def test_fdtr(self):
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assert_equal(cephes.fdtr(1, 1, 0), 0.0)
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# Computed using Wolfram Alpha: CDF[FRatioDistribution[1e-6, 5], 10]
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assert_allclose(cephes.fdtr(1e-6, 5, 10), 0.9999940790193488,
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rtol=1e-12)
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def test_fdtrc(self):
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assert_equal(cephes.fdtrc(1, 1, 0), 1.0)
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# Computed using Wolfram Alpha:
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# 1 - CDF[FRatioDistribution[2, 1/10], 1e10]
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assert_allclose(cephes.fdtrc(2, 0.1, 1e10), 0.27223784621293512,
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rtol=1e-12)
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def test_fdtri(self):
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assert_allclose(cephes.fdtri(1, 1, [0.499, 0.501]),
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array([0.9937365, 1.00630298]), rtol=1e-6)
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# From Wolfram Alpha:
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# CDF[FRatioDistribution[1/10, 1], 3] = 0.8756751669632105666874...
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p = 0.8756751669632105666874
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assert_allclose(cephes.fdtri(0.1, 1, p), 3, rtol=1e-12)
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@pytest.mark.xfail(reason='Returns nan on i686.')
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def test_fdtri_mysterious_failure(self):
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assert_allclose(cephes.fdtri(1, 1, 0.5), 1)
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def test_fdtridfd(self):
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assert_equal(cephes.fdtridfd(1,0,0),5.0)
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def test_fresnel(self):
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assert_equal(cephes.fresnel(0),(0.0,0.0))
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def test_gamma(self):
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assert_equal(cephes.gamma(5),24.0)
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def test_gammainccinv(self):
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assert_equal(cephes.gammainccinv(5,1),0.0)
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def test_gammaln(self):
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cephes.gammaln(10)
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def test_gammasgn(self):
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vals = np.array([-4, -3.5, -2.3, 1, 4.2], np.float64)
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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 test_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 test_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 test_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 test_kv(self):
|
|
cephes.kv(1,1)
|
|
|
|
def test_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:
|
|
# + O(q^2)
|
|
return 2**(-0.5) * (1 - .5*q*cos(2*z))
|
|
elif m == 1:
|
|
# + O(q^2)
|
|
return cos(z) - q/8 * cos(3*z)
|
|
elif m == 2:
|
|
# + O(q^2)
|
|
return cos(2*z) - q*(cos(4*z)/12 - 1/4)
|
|
else:
|
|
# + O(q^2)
|
|
return cos(m*z) - q*(cos((m+2)*z)/(4*(m+1)) - cos((m-2)*z)/(4*(m-1)))
|
|
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:
|
|
# + O(q^2)
|
|
return sin(z) - q/8 * sin(3*z)
|
|
elif m == 2:
|
|
# + O(q^2)
|
|
return sin(2*z) - q*sin(4*z)/12
|
|
else:
|
|
# + O(q^2)
|
|
return sin(m*z) - q*(sin((m+2)*z)/(4*(m+1)) - sin((m-2)*z)/(4*(m-1)))
|
|
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 test_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 test_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 test_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 test_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 test_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 test_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:
|
|
"""
|
|
Test beta and betaln.
|
|
"""
|
|
|
|
def test_beta(self):
|
|
assert_equal(special.beta(1, 1), 1.0)
|
|
assert_allclose(special.beta(-100.3, 1e-200), special.gamma(1e-200))
|
|
assert_allclose(special.beta(0.0342, 171), 24.070498359873497,
|
|
rtol=1e-13, atol=0)
|
|
|
|
bet = special.beta(2, 4)
|
|
betg = (special.gamma(2)*special.gamma(4))/special.gamma(6)
|
|
assert_allclose(bet, betg, rtol=1e-13)
|
|
|
|
def test_beta_inf(self):
|
|
assert_(np.isinf(special.beta(-1, 2)))
|
|
|
|
def test_betaln(self):
|
|
assert_equal(special.betaln(1, 1), 0.0)
|
|
assert_allclose(special.betaln(-100.3, 1e-200),
|
|
special.gammaln(1e-200))
|
|
assert_allclose(special.betaln(0.0342, 170), 3.1811881124242447,
|
|
rtol=1e-14, atol=0)
|
|
|
|
betln = special.betaln(2, 4)
|
|
bet = log(abs(special.beta(2, 4)))
|
|
assert_allclose(betln, bet, rtol=1e-13)
|
|
|
|
|
|
class TestBetaInc:
|
|
"""
|
|
Tests for betainc, betaincinv, betaincc, betainccinv.
|
|
"""
|
|
|
|
def test_a1_b1(self):
|
|
# betainc(1, 1, x) is x.
|
|
x = np.array([0, 0.25, 1])
|
|
assert_equal(special.betainc(1, 1, x), x)
|
|
assert_equal(special.betaincinv(1, 1, x), x)
|
|
assert_equal(special.betaincc(1, 1, x), 1 - x)
|
|
assert_equal(special.betainccinv(1, 1, x), 1 - x)
|
|
|
|
# Nontrivial expected values computed with mpmath:
|
|
# from mpmath import mp
|
|
# mp.dps = 100
|
|
# p = mp.betainc(a, b, 0, x, regularized=True)
|
|
#
|
|
# or, e.g.,
|
|
#
|
|
# p = 0.25
|
|
# a, b = 0.0342, 171
|
|
# x = mp.findroot(
|
|
# lambda t: mp.betainc(a, b, 0, t, regularized=True) - p,
|
|
# (8e-21, 9e-21),
|
|
# solver='anderson',
|
|
# )
|
|
#
|
|
@pytest.mark.parametrize(
|
|
'a, b, x, p',
|
|
[(2, 4, 0.3138101704556974, 0.5),
|
|
(0.0342, 171.0, 1e-10, 0.552699169018070910641),
|
|
# gh-3761:
|
|
(0.0342, 171, 8.42313169354797e-21, 0.25),
|
|
# gh-4244:
|
|
(0.0002742794749792665, 289206.03125, 1.639984034231756e-56,
|
|
0.9688708782196045),
|
|
# gh-12796:
|
|
(4, 99997, 0.0001947841578892121, 0.999995)])
|
|
def test_betainc_betaincinv(self, a, b, x, p):
|
|
p1 = special.betainc(a, b, x)
|
|
assert_allclose(p1, p, rtol=1e-15)
|
|
x1 = special.betaincinv(a, b, p)
|
|
assert_allclose(x1, x, rtol=5e-13)
|
|
|
|
# Expected values computed with mpmath:
|
|
# from mpmath import mp
|
|
# mp.dps = 100
|
|
# p = mp.betainc(a, b, x, 1, regularized=True)
|
|
@pytest.mark.parametrize('a, b, x, p',
|
|
[(2.5, 3.0, 0.25, 0.833251953125),
|
|
(7.5, 13.25, 0.375, 0.43298734645560368593),
|
|
(0.125, 7.5, 0.425, 0.0006688257851314237),
|
|
(0.125, 18.0, 1e-6, 0.72982359145096327654),
|
|
(0.125, 18.0, 0.996, 7.2745875538380150586e-46),
|
|
(0.125, 24.0, 0.75, 3.70853404816862016966e-17),
|
|
(16.0, 0.75, 0.99999999975,
|
|
5.4408759277418629909e-07),
|
|
# gh-4677 (numbers from stackoverflow question):
|
|
(0.4211959643503401, 16939.046996018118,
|
|
0.000815296167195521, 1e-7)])
|
|
def test_betaincc_betainccinv(self, a, b, x, p):
|
|
p1 = special.betaincc(a, b, x)
|
|
assert_allclose(p1, p, rtol=5e-15)
|
|
x1 = special.betainccinv(a, b, p)
|
|
assert_allclose(x1, x, rtol=8e-15)
|
|
|
|
@pytest.mark.parametrize(
|
|
'a, b, y, ref',
|
|
[(14.208308325339239, 14.208308325339239, 7.703145458496392e-307,
|
|
8.566004561846704e-23),
|
|
(14.0, 14.5, 1e-280, 2.9343915006642424e-21),
|
|
(3.5, 15.0, 4e-95, 1.3290751429289227e-28),
|
|
(10.0, 1.25, 2e-234, 3.982659092143654e-24),
|
|
(4.0, 99997.0, 5e-88, 3.309800566862242e-27)]
|
|
)
|
|
def test_betaincinv_tiny_y(self, a, b, y, ref):
|
|
# Test with extremely small y values. This test includes
|
|
# a regression test for an issue in the boost code;
|
|
# see https://github.com/boostorg/math/issues/961
|
|
#
|
|
# The reference values were computed with mpmath. For example,
|
|
#
|
|
# from mpmath import mp
|
|
# mp.dps = 1000
|
|
# a = 14.208308325339239
|
|
# p = 7.703145458496392e-307
|
|
# x = mp.findroot(lambda t: mp.betainc(a, a, 0, t,
|
|
# regularized=True) - p,
|
|
# x0=8.566e-23)
|
|
# print(float(x))
|
|
#
|
|
x = special.betaincinv(a, b, y)
|
|
assert_allclose(x, ref, rtol=1e-14)
|
|
|
|
@pytest.mark.parametrize('func', [special.betainc, special.betaincinv,
|
|
special.betaincc, special.betainccinv])
|
|
@pytest.mark.parametrize('args', [(-1.0, 2, 0.5), (0, 2, 0.5),
|
|
(1.5, -2.0, 0.5), (1.5, 0, 0.5),
|
|
(1.5, 2.0, -0.3), (1.5, 2.0, 1.1)])
|
|
def test_betainc_domain_errors(self, func, args):
|
|
with special.errstate(domain='raise'):
|
|
with pytest.raises(special.SpecialFunctionError, match='domain'):
|
|
special.betainc(*args)
|
|
|
|
|
|
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, _NoValue])
|
|
@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 is not _NoValue:
|
|
with pytest.warns(
|
|
DeprecationWarning,
|
|
match=r"Using 'legacy' keyword is deprecated"
|
|
):
|
|
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
|
|
with suppress_warnings() as sup:
|
|
if legacy is not _NoValue:
|
|
sup.filter(DeprecationWarning)
|
|
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.])
|
|
|
|
def test_positional_deprecation(self):
|
|
with pytest.deprecated_call(match="use keyword arguments"):
|
|
# from test_comb
|
|
special.comb([10, 10], [3, 4], False, False)
|
|
|
|
|
|
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:
|
|
"""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.float64).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:
|
|
"""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(double).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(double).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:
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
def test_factorialx_scalar_return_type(self, exact):
|
|
assert np.isscalar(special.factorial(1, exact=exact))
|
|
assert np.isscalar(special.factorial2(1, exact=exact))
|
|
assert np.isscalar(special.factorialk(1, 3, exact=exact))
|
|
|
|
@pytest.mark.parametrize("n", [-1, -2, -3])
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
def test_factorialx_negative(self, exact, n):
|
|
assert_equal(special.factorial(n, exact=exact), 0)
|
|
assert_equal(special.factorial2(n, exact=exact), 0)
|
|
assert_equal(special.factorialk(n, 3, exact=exact), 0)
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
def test_factorialx_negative_array(self, exact):
|
|
assert_func = assert_array_equal if exact else assert_allclose
|
|
# Consistent output for n < 0
|
|
assert_func(special.factorial([-5, -4, 0, 1], exact=exact),
|
|
[0, 0, 1, 1])
|
|
assert_func(special.factorial2([-5, -4, 0, 1], exact=exact),
|
|
[0, 0, 1, 1])
|
|
assert_func(special.factorialk([-5, -4, 0, 1], 3, exact=exact),
|
|
[0, 0, 1, 1])
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("content", [np.nan, None, np.datetime64('nat')],
|
|
ids=["NaN", "None", "NaT"])
|
|
def test_factorialx_nan(self, content, exact):
|
|
# scalar
|
|
assert special.factorial(content, exact=exact) is np.nan
|
|
assert special.factorial2(content, exact=exact) is np.nan
|
|
assert special.factorialk(content, 3, exact=exact) is np.nan
|
|
# array-like (initializes np.array with default dtype)
|
|
if content is not np.nan:
|
|
# None causes object dtype, which is not supported; as is datetime
|
|
with pytest.raises(ValueError, match="Unsupported datatype.*"):
|
|
special.factorial([content], exact=exact)
|
|
elif exact:
|
|
with pytest.raises(ValueError, match="factorial with `exact=Tr.*"):
|
|
special.factorial([content], exact=exact)
|
|
else:
|
|
assert np.isnan(special.factorial([content], exact=exact)[0])
|
|
# factorial{2,k} don't support array case due to dtype constraints
|
|
with pytest.raises(ValueError, match="factorial2 does not support.*"):
|
|
special.factorial2([content], exact=exact)
|
|
with pytest.raises(ValueError, match="factorialk does not support.*"):
|
|
special.factorialk([content], 3, exact=exact)
|
|
# array-case also tested in test_factorial{,2,k}_corner_cases
|
|
|
|
@pytest.mark.parametrize("levels", range(1, 5))
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
def test_factorialx_array_shape(self, levels, exact):
|
|
def _nest_me(x, k=1):
|
|
"""
|
|
Double x and nest it k times
|
|
|
|
For example:
|
|
>>> _nest_me([3, 4], 2)
|
|
[[[3, 4], [3, 4]], [[3, 4], [3, 4]]]
|
|
"""
|
|
if k == 0:
|
|
return x
|
|
else:
|
|
return _nest_me([x, x], k-1)
|
|
|
|
def _check(res, nucleus):
|
|
exp = np.array(_nest_me(nucleus, k=levels), dtype=object)
|
|
# test that ndarray shape is maintained
|
|
# need to cast to float due to numpy/numpy#21220
|
|
assert_allclose(res.astype(np.float64), exp.astype(np.float64))
|
|
|
|
n = np.array(_nest_me([5, 25], k=levels))
|
|
exp_nucleus = {1: [120, math.factorial(25)],
|
|
# correctness of factorial{2,k}() is tested elsewhere
|
|
2: [15, special.factorial2(25, exact=True)],
|
|
3: [10, special.factorialk(25, 3, exact=True)]}
|
|
|
|
_check(special.factorial(n, exact=exact), exp_nucleus[1])
|
|
_check(special.factorial2(n, exact=exact), exp_nucleus[2])
|
|
_check(special.factorialk(n, 3, exact=exact), exp_nucleus[3])
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("dim", range(0, 5))
|
|
def test_factorialx_array_dimension(self, dim, exact):
|
|
n = np.array(5, ndmin=dim)
|
|
exp = {1: 120, 2: 15, 3: 10}
|
|
assert_allclose(special.factorial(n, exact=exact),
|
|
np.array(exp[1], ndmin=dim))
|
|
assert_allclose(special.factorial2(n, exact=exact),
|
|
np.array(exp[2], ndmin=dim))
|
|
assert_allclose(special.factorialk(n, 3, exact=exact),
|
|
np.array(exp[3], ndmin=dim))
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("level", range(1, 5))
|
|
def test_factorialx_array_like(self, level, exact):
|
|
def _nest_me(x, k=1):
|
|
if k == 0:
|
|
return x
|
|
else:
|
|
return _nest_me([x], k-1)
|
|
|
|
n = _nest_me([5], k=level-1) # nested list
|
|
exp_nucleus = {1: 120, 2: 15, 3: 10}
|
|
assert_func = assert_array_equal if exact else assert_allclose
|
|
assert_func(special.factorial(n, exact=exact),
|
|
np.array(exp_nucleus[1], ndmin=level))
|
|
assert_func(special.factorial2(n, exact=exact),
|
|
np.array(exp_nucleus[2], ndmin=level))
|
|
assert_func(special.factorialk(n, 3, exact=exact),
|
|
np.array(exp_nucleus[3], ndmin=level))
|
|
|
|
# note that n=170 is the last integer such that factorial(n) fits float64
|
|
@pytest.mark.parametrize('n', range(30, 180, 10))
|
|
def test_factorial_accuracy(self, n):
|
|
# Compare exact=True vs False, i.e. that the accuracy of the
|
|
# approximation is better than the specified tolerance.
|
|
|
|
rtol = 6e-14 if sys.platform == 'win32' else 1e-15
|
|
# need to cast exact result to float due to numpy/numpy#21220
|
|
assert_allclose(float(special.factorial(n, exact=True)),
|
|
special.factorial(n, exact=False), rtol=rtol)
|
|
assert_allclose(special.factorial([n], exact=True).astype(float),
|
|
special.factorial([n], exact=False), rtol=rtol)
|
|
|
|
@pytest.mark.parametrize('n',
|
|
list(range(0, 22)) + list(range(30, 180, 10)))
|
|
def test_factorial_int_reference(self, n):
|
|
# 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])
|
|
|
|
rtol = 6e-14 if sys.platform == 'win32' else 1e-15
|
|
assert_allclose(float(correct), special.factorial(n, False),
|
|
rtol=rtol)
|
|
assert_allclose(float(correct), special.factorial([n], False)[0],
|
|
rtol=rtol)
|
|
|
|
def test_factorial_float_reference(self):
|
|
def _check(n, expected):
|
|
assert_allclose(special.factorial(n), expected)
|
|
assert_allclose(special.factorial([n])[0], expected)
|
|
# using floats with exact=True is deprecated for scalars...
|
|
with pytest.deprecated_call(match="Non-integer values.*"):
|
|
assert_allclose(special.factorial(n, exact=True), expected)
|
|
# ... and already an error for arrays
|
|
with pytest.raises(ValueError, match="factorial with `exact=Tr.*"):
|
|
special.factorial([n], exact=True)
|
|
|
|
# Reference values from mpmath for gamma(n+1)
|
|
_check(0.01, 0.994325851191506032181932988)
|
|
_check(1.11, 1.051609009483625091514147465)
|
|
_check(5.55, 314.9503192327208241614959052)
|
|
_check(11.1, 50983227.84411615655137170553)
|
|
_check(33.3, 2.493363339642036352229215273e+37)
|
|
_check(55.5, 9.479934358436729043289162027e+73)
|
|
_check(77.7, 3.060540559059579022358692625e+114)
|
|
_check(99.9, 5.885840419492871504575693337e+157)
|
|
# close to maximum for float64
|
|
_check(170.6243, 1.79698185749571048960082e+308)
|
|
|
|
@pytest.mark.parametrize("dtype", [np.int64, np.float64,
|
|
np.complex128, object])
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("dim", range(0, 5))
|
|
# test empty & non-empty arrays, with nans and mixed
|
|
@pytest.mark.parametrize("content",
|
|
[[], [1], [1.1], [np.nan], [np.nan, 1]],
|
|
ids=["[]", "[1]", "[1.1]", "[NaN]", "[NaN, 1]"])
|
|
def test_factorial_array_corner_cases(self, content, dim, exact, dtype):
|
|
if dtype == np.int64 and any(np.isnan(x) for x in content):
|
|
pytest.skip("impossible combination")
|
|
# np.array(x, ndim=0) will not be 0-dim. unless x is too
|
|
content = content if (dim > 0 or len(content) != 1) else content[0]
|
|
n = np.array(content, ndmin=dim, dtype=dtype)
|
|
result = None
|
|
if not content:
|
|
result = special.factorial(n, exact=exact)
|
|
elif not (np.issubdtype(n.dtype, np.integer)
|
|
or np.issubdtype(n.dtype, np.floating)):
|
|
with pytest.raises(ValueError, match="Unsupported datatype*"):
|
|
special.factorial(n, exact=exact)
|
|
elif exact and not np.issubdtype(n.dtype, np.integer):
|
|
with pytest.raises(ValueError, match="factorial with `exact=.*"):
|
|
special.factorial(n, exact=exact)
|
|
else:
|
|
# no error
|
|
result = special.factorial(n, exact=exact)
|
|
|
|
# assert_equal does not distinguish scalars and 0-dim arrays of the same value,
|
|
# see https://github.com/numpy/numpy/issues/24050
|
|
def assert_really_equal(x, y):
|
|
assert type(x) == type(y), f"types not equal: {type(x)}, {type(y)}"
|
|
assert_equal(x, y)
|
|
|
|
if result is not None:
|
|
# keep 0-dim.; otherwise n.ravel().ndim==1, even if n.ndim==0
|
|
n_flat = n.ravel() if n.ndim else n
|
|
ref = special.factorial(n_flat, exact=exact) if n.size else []
|
|
# expected result is empty if and only if n is empty,
|
|
# and has the same dtype & dimension as n
|
|
expected = np.array(ref, ndmin=dim, dtype=dtype)
|
|
assert_really_equal(result, expected)
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("n", [1, 1.1, 2 + 2j, np.nan, None],
|
|
ids=["1", "1.1", "2+2j", "NaN", "None"])
|
|
def test_factorial_scalar_corner_cases(self, n, exact):
|
|
if (n is None or n is np.nan or np.issubdtype(type(n), np.integer)
|
|
or np.issubdtype(type(n), np.floating)):
|
|
# no error
|
|
if (np.issubdtype(type(n), np.floating) and exact
|
|
and n is not np.nan):
|
|
with pytest.deprecated_call(match="Non-integer values.*"):
|
|
result = special.factorial(n, exact=exact)
|
|
else:
|
|
result = special.factorial(n, exact=exact)
|
|
exp = np.nan if n is np.nan or n is None else special.factorial(n)
|
|
assert_equal(result, exp)
|
|
else:
|
|
with pytest.raises(ValueError, match="Unsupported datatype*"):
|
|
special.factorial(n, exact=exact)
|
|
|
|
# use odd increment to make sure both odd & even numbers are tested!
|
|
@pytest.mark.parametrize('n', range(30, 180, 11))
|
|
def test_factorial2_accuracy(self, n):
|
|
# Compare exact=True vs False, i.e. that the accuracy of the
|
|
# approximation is better than the specified tolerance.
|
|
|
|
rtol = 2e-14 if sys.platform == 'win32' else 1e-15
|
|
# need to cast exact result to float due to numpy/numpy#21220
|
|
assert_allclose(float(special.factorial2(n, exact=True)),
|
|
special.factorial2(n, exact=False), rtol=rtol)
|
|
assert_allclose(special.factorial2([n], exact=True).astype(float),
|
|
special.factorial2([n], exact=False), rtol=rtol)
|
|
|
|
@pytest.mark.parametrize('n',
|
|
list(range(0, 22)) + list(range(30, 180, 11)))
|
|
def test_factorial2_int_reference(self, n):
|
|
# Compare all with correct value
|
|
|
|
# Cannot use np.product due to overflow
|
|
correct = functools.reduce(operator.mul, list(range(n, 0, -2)), 1)
|
|
|
|
assert_array_equal(correct, special.factorial2(n, True))
|
|
assert_array_equal(correct, special.factorial2([n], True)[0])
|
|
|
|
assert_allclose(float(correct), special.factorial2(n, False))
|
|
assert_allclose(float(correct), special.factorial2([n], False)[0])
|
|
|
|
@pytest.mark.parametrize("dtype", [np.int64, np.float64,
|
|
np.complex128, object])
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("dim", range(0, 5))
|
|
# test empty & non-empty arrays, with nans and mixed
|
|
@pytest.mark.parametrize("content", [[], [1], [np.nan], [np.nan, 1]],
|
|
ids=["[]", "[1]", "[NaN]", "[NaN, 1]"])
|
|
def test_factorial2_array_corner_cases(self, content, dim, exact, dtype):
|
|
if dtype == np.int64 and any(np.isnan(x) for x in content):
|
|
pytest.skip("impossible combination")
|
|
# np.array(x, ndim=0) will not be 0-dim. unless x is too
|
|
content = content if (dim > 0 or len(content) != 1) else content[0]
|
|
n = np.array(content, ndmin=dim, dtype=dtype)
|
|
if np.issubdtype(n.dtype, np.integer) or (not content):
|
|
# no error
|
|
result = special.factorial2(n, exact=exact)
|
|
# expected result is identical to n for exact=True resp. empty
|
|
# arrays (assert_allclose chokes on object), otherwise up to tol
|
|
func = assert_equal if exact or (not content) else assert_allclose
|
|
func(result, n)
|
|
else:
|
|
with pytest.raises(ValueError, match="factorial2 does not*"):
|
|
special.factorial2(n, 3)
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("n", [1, 1.1, 2 + 2j, np.nan, None],
|
|
ids=["1", "1.1", "2+2j", "NaN", "None"])
|
|
def test_factorial2_scalar_corner_cases(self, n, exact):
|
|
if n is None or n is np.nan or np.issubdtype(type(n), np.integer):
|
|
# no error
|
|
result = special.factorial2(n, exact=exact)
|
|
exp = np.nan if n is np.nan or n is None else special.factorial(n)
|
|
assert_equal(result, exp)
|
|
else:
|
|
with pytest.raises(ValueError, match="factorial2 does not*"):
|
|
special.factorial2(n, exact=exact)
|
|
|
|
@pytest.mark.parametrize("k", range(1, 5))
|
|
# note that n=170 is the last integer such that factorial(n) fits float64;
|
|
# use odd increment to make sure both odd & even numbers are tested
|
|
@pytest.mark.parametrize('n', range(170, 20, -29))
|
|
def test_factorialk_accuracy(self, n, k):
|
|
# Compare exact=True vs False, i.e. that the accuracy of the
|
|
# approximation is better than the specified tolerance.
|
|
|
|
# need to cast exact result to float due to numpy/numpy#21220
|
|
assert_allclose(float(special.factorialk(n, k=k, exact=True)),
|
|
special.factorialk(n, k=k, exact=False))
|
|
assert_allclose(special.factorialk([n], k=k, exact=True).astype(float),
|
|
special.factorialk([n], k=k, exact=False))
|
|
|
|
@pytest.mark.parametrize('k', list(range(1, 5)) + [10, 20])
|
|
@pytest.mark.parametrize('n',
|
|
list(range(0, 22)) + list(range(22, 100, 11)))
|
|
def test_factorialk_int_reference(self, n, k):
|
|
# Compare all with correct value
|
|
|
|
# Would be nice to use np.product here, but that's
|
|
# broken on windows, see numpy/numpy#21219
|
|
correct = functools.reduce(operator.mul, list(range(n, 0, -k)), 1)
|
|
|
|
assert_array_equal(correct, special.factorialk(n, k, True))
|
|
assert_array_equal(correct, special.factorialk([n], k, True)[0])
|
|
|
|
assert_allclose(float(correct), special.factorialk(n, k, False))
|
|
assert_allclose(float(correct), special.factorialk([n], k, False)[0])
|
|
|
|
@pytest.mark.parametrize("dtype", [np.int64, np.float64,
|
|
np.complex128, object])
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("dim", range(0, 5))
|
|
# test empty & non-empty arrays, with nans and mixed
|
|
@pytest.mark.parametrize("content", [[], [1], [np.nan], [np.nan, 1]],
|
|
ids=["[]", "[1]", "[NaN]", "[NaN, 1]"])
|
|
def test_factorialk_array_corner_cases(self, content, dim, exact, dtype):
|
|
if dtype == np.int64 and any(np.isnan(x) for x in content):
|
|
pytest.skip("impossible combination")
|
|
# np.array(x, ndim=0) will not be 0-dim. unless x is too
|
|
content = content if (dim > 0 or len(content) != 1) else content[0]
|
|
n = np.array(content, ndmin=dim, dtype=dtype if exact else np.float64)
|
|
if np.issubdtype(n.dtype, np.integer) or (not content):
|
|
# no error; expected result is identical to n
|
|
assert_equal(special.factorialk(n, 3, exact=exact), n)
|
|
else:
|
|
with pytest.raises(ValueError, match="factorialk does not*"):
|
|
special.factorialk(n, 3, exact=exact)
|
|
|
|
@pytest.mark.parametrize("exact", [True, False, None])
|
|
@pytest.mark.parametrize("k", range(1, 5))
|
|
@pytest.mark.parametrize("n", [1, 1.1, 2 + 2j, np.nan, None],
|
|
ids=["1", "1.1", "2+2j", "NaN", "None"])
|
|
def test_factorialk_scalar_corner_cases(self, n, k, exact):
|
|
if n is None or n is np.nan or np.issubdtype(type(n), np.integer):
|
|
if exact is None:
|
|
with pytest.deprecated_call(match="factorialk will default.*"):
|
|
result = special.factorialk(n, k=k, exact=exact)
|
|
else:
|
|
# no error
|
|
result = special.factorialk(n, k=k, exact=exact)
|
|
|
|
nan_cond = n is np.nan or n is None
|
|
# factorialk(1, k) == 1 for all k
|
|
expected = np.nan if nan_cond else 1
|
|
assert_equal(result, expected)
|
|
else:
|
|
with pytest.raises(ValueError, match="factorialk does not*"):
|
|
with suppress_warnings() as sup:
|
|
sup.filter(DeprecationWarning, "factorialk will default")
|
|
special.factorialk(n, k=k, exact=exact)
|
|
|
|
@pytest.mark.parametrize("k", [0, 1.1, np.nan, "1"])
|
|
def test_factorialk_raises_k(self, k):
|
|
with pytest.raises(ValueError, match="k must be a positive integer*"):
|
|
special.factorialk(1, k)
|
|
|
|
@pytest.mark.parametrize("exact", [True, False])
|
|
@pytest.mark.parametrize("k", range(1, 12))
|
|
def test_factorialk_dtype(self, k, exact):
|
|
kw = {"k": k, "exact": exact}
|
|
if exact and k in _FACTORIALK_LIMITS_64BITS.keys():
|
|
n = np.array([_FACTORIALK_LIMITS_32BITS[k]])
|
|
assert_equal(special.factorialk(n, **kw).dtype, np_long)
|
|
assert_equal(special.factorialk(n + 1, **kw).dtype, np.int64)
|
|
# assert maximality of limits for given dtype
|
|
assert special.factorialk(n + 1, **kw) > np.iinfo(np.int32).max
|
|
|
|
n = np.array([_FACTORIALK_LIMITS_64BITS[k]])
|
|
assert_equal(special.factorialk(n, **kw).dtype, np.int64)
|
|
assert_equal(special.factorialk(n + 1, **kw).dtype, object)
|
|
assert special.factorialk(n + 1, **kw) > np.iinfo(np.int64).max
|
|
else:
|
|
n = np.array([_FACTORIALK_LIMITS_64BITS.get(k, 1)])
|
|
# for exact=True and k >= 10, we always return object;
|
|
# for exact=False it's always float
|
|
dtype = object if exact else np.float64
|
|
assert_equal(special.factorialk(n, **kw).dtype, dtype)
|
|
|
|
def test_factorial_mixed_nan_inputs(self):
|
|
x = np.array([np.nan, 1, 2, 3, np.nan])
|
|
expected = np.array([np.nan, 1, 2, 6, np.nan])
|
|
assert_equal(special.factorial(x, exact=False), expected)
|
|
with pytest.raises(ValueError, match="factorial with `exact=True.*"):
|
|
special.factorial(x, exact=True)
|
|
|
|
|
|
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.vstack([x1] * 2), x2)
|
|
assert_allclose(x, np.vstack([expected] * 2), rtol=1e-12)
|
|
assert_raises(ValueError, special.hyp0f1,
|
|
np.vstack([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):
|
|
def skipper(v, z):
|
|
return 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(np.int64)
|
|
|
|
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(double)
|
|
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(double).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)
|
|
|
|
@pytest.mark.parametrize('n', [1, 2, 3, 4, 5])
|
|
@pytest.mark.parametrize('zr', [0.5241717, 12.80232, -9.699001,
|
|
0.5122437, 0.1714377])
|
|
@pytest.mark.parametrize('zi', [9.766818, 0.2999083, 8.24726, -22.84843,
|
|
-0.8792666])
|
|
def test_lpn_against_clpmn(self, n, zr, zi):
|
|
reslpn = special.lpn(n, zr + zi*1j)
|
|
resclpmn = special.clpmn(0, n, zr+zi*1j)
|
|
assert_allclose(reslpn[0], resclpmn[0][0])
|
|
assert_allclose(reslpn[1], resclpmn[1][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)
|
|
|
|
def test_pbvv_seq(self):
|
|
res1, res2 = special.pbvv_seq(2, 3)
|
|
assert_allclose(res1, np.array([2.976319645712036,
|
|
1.358840996329579,
|
|
0.5501016716383508]))
|
|
assert_allclose(res2, np.array([3.105638472238475,
|
|
0.9380581512176672,
|
|
0.533688488872053]))
|
|
|
|
|
|
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.vstack([expected]*2)
|
|
assert_almost_equal(special.polygamma(n, np.vstack([x]*2)),
|
|
expected)
|
|
assert_almost_equal(special.polygamma(np.vstack([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(double).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)
|
|
|
|
|
|
class TestStirling2:
|
|
table = [
|
|
[1],
|
|
[0, 1],
|
|
[0, 1, 1],
|
|
[0, 1, 3, 1],
|
|
[0, 1, 7, 6, 1],
|
|
[0, 1, 15, 25, 10, 1],
|
|
[0, 1, 31, 90, 65, 15, 1],
|
|
[0, 1, 63, 301, 350, 140, 21, 1],
|
|
[0, 1, 127, 966, 1701, 1050, 266, 28, 1],
|
|
[0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1],
|
|
[0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1],
|
|
]
|
|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
|
|
(True, assert_equal, {}),
|
|
(False, assert_allclose, {'rtol': 1e-12})
|
|
])
|
|
def test_table_cases(self, is_exact, comp, kwargs):
|
|
for n in range(1, len(self.table)):
|
|
k_values = list(range(n+1))
|
|
row = self.table[n]
|
|
comp(row, stirling2([n], k_values, exact=is_exact), **kwargs)
|
|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
|
|
(True, assert_equal, {}),
|
|
(False, assert_allclose, {'rtol': 1e-12})
|
|
])
|
|
def test_valid_single_integer(self, is_exact, comp, kwargs):
|
|
comp(stirling2(0, 0, exact=is_exact), self.table[0][0], **kwargs)
|
|
comp(stirling2(4, 2, exact=is_exact), self.table[4][2], **kwargs)
|
|
# a single 2-tuple of integers as arguments must return an int and not
|
|
# an array whereas arrays of single values should return array
|
|
comp(stirling2(5, 3, exact=is_exact), 25, **kwargs)
|
|
comp(stirling2([5], [3], exact=is_exact), [25], **kwargs)
|
|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
|
|
(True, assert_equal, {}),
|
|
(False, assert_allclose, {'rtol': 1e-12})
|
|
])
|
|
def test_negative_integer(self, is_exact, comp, kwargs):
|
|
# negative integers for n or k arguments return 0
|
|
comp(stirling2(-1, -1, exact=is_exact), 0, **kwargs)
|
|
comp(stirling2(-1, 2, exact=is_exact), 0, **kwargs)
|
|
comp(stirling2(2, -1, exact=is_exact), 0, **kwargs)
|
|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
|
|
(True, assert_equal, {}),
|
|
(False, assert_allclose, {'rtol': 1e-12})
|
|
])
|
|
def test_array_inputs(self, is_exact, comp, kwargs):
|
|
ans = [self.table[10][3], self.table[10][4]]
|
|
comp(stirling2(asarray([10, 10]),
|
|
asarray([3, 4]),
|
|
exact=is_exact),
|
|
ans)
|
|
comp(stirling2([10, 10],
|
|
asarray([3, 4]),
|
|
exact=is_exact),
|
|
ans)
|
|
comp(stirling2(asarray([10, 10]),
|
|
[3, 4],
|
|
exact=is_exact),
|
|
ans)
|
|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
|
|
(True, assert_equal, {}),
|
|
(False, assert_allclose, {'rtol': 1e-13})
|
|
])
|
|
def test_mixed_values(self, is_exact, comp, kwargs):
|
|
# negative values-of either n or k-should return 0 for the entry
|
|
ans = [0, 1, 3, 25, 1050, 5880, 9330]
|
|
n = [-1, 0, 3, 5, 8, 10, 10]
|
|
k = [-2, 0, 2, 3, 5, 7, 3]
|
|
comp(stirling2(n, k, exact=is_exact), ans, **kwargs)
|
|
|
|
def test_correct_parity(self):
|
|
"""Test parity follows well known identity.
|
|
|
|
en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Parity
|
|
"""
|
|
n, K = 100, np.arange(101)
|
|
assert_equal(
|
|
stirling2(n, K, exact=True) % 2,
|
|
[math.comb(n - (k // 2) - 1, n - k) % 2 for k in K],
|
|
)
|
|
|
|
def test_big_numbers(self):
|
|
# via mpmath (bigger than 32bit)
|
|
ans = asarray([48063331393110, 48004081105038305])
|
|
n = [25, 30]
|
|
k = [17, 4]
|
|
assert array_equal(stirling2(n, k, exact=True), ans)
|
|
# bigger than 64 bit
|
|
ans = asarray([2801934359500572414253157841233849412,
|
|
14245032222277144547280648984426251])
|
|
n = [42, 43]
|
|
k = [17, 23]
|
|
assert array_equal(stirling2(n, k, exact=True), ans)
|
|
|
|
@pytest.mark.parametrize("N", [4.5, 3., 4+1j, "12", np.nan])
|
|
@pytest.mark.parametrize("K", [3.5, 3, "2", None])
|
|
@pytest.mark.parametrize("is_exact", [True, False])
|
|
def test_unsupported_input_types(self, N, K, is_exact):
|
|
# object, float, string, complex are not supported and raise TypeError
|
|
with pytest.raises(TypeError):
|
|
stirling2(N, K, exact=is_exact)
|
|
|
|
@pytest.mark.parametrize("is_exact", [True, False])
|
|
def test_numpy_array_int_object_dtype(self, is_exact):
|
|
# python integers with arbitrary precision are *not* allowed as
|
|
# object type in numpy arrays are inconsistent from api perspective
|
|
ans = asarray(self.table[4][1:])
|
|
n = asarray([4, 4, 4, 4], dtype=object)
|
|
k = asarray([1, 2, 3, 4], dtype=object)
|
|
with pytest.raises(TypeError):
|
|
array_equal(stirling2(n, k, exact=is_exact), ans)
|
|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
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(True, assert_equal, {}),
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(False, assert_allclose, {'rtol': 1e-13})
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|
])
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def test_numpy_array_unsigned_int_dtype(self, is_exact, comp, kwargs):
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# numpy unsigned integers are allowed as dtype in numpy arrays
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ans = asarray(self.table[4][1:])
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n = asarray([4, 4, 4, 4], dtype=np_ulong)
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k = asarray([1, 2, 3, 4], dtype=np_ulong)
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comp(stirling2(n, k, exact=False), ans, **kwargs)
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|
|
|
@pytest.mark.parametrize("is_exact, comp, kwargs", [
|
|
(True, assert_equal, {}),
|
|
(False, assert_allclose, {'rtol': 1e-13})
|
|
])
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|
def test_broadcasting_arrays_correctly(self, is_exact, comp, kwargs):
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|
# broadcasting is handled by stirling2
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|
# test leading 1s are replicated
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|
ans = asarray([[1, 15, 25, 10], [1, 7, 6, 1]]) # shape (2,4)
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n = asarray([[5, 5, 5, 5], [4, 4, 4, 4]]) # shape (2,4)
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k = asarray([1, 2, 3, 4]) # shape (4,)
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comp(stirling2(n, k, exact=is_exact), ans, **kwargs)
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|
# test that dims both mismatch broadcast correctly (5,1) & (6,)
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|
n = asarray([[4], [4], [4], [4], [4]])
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|
k = asarray([0, 1, 2, 3, 4, 5])
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|
ans = asarray([[0, 1, 7, 6, 1, 0] for _ in range(5)])
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|
comp(stirling2(n, k, exact=False), ans, **kwargs)
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|
|
|
def test_temme_rel_max_error(self):
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|
# python integers with arbitrary precision are *not* allowed as
|
|
# object type in numpy arrays are inconsistent from api perspective
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|
x = list(range(51, 101, 5))
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|
for n in x:
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|
k_entries = list(range(1, n+1))
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|
denom = stirling2([n], k_entries, exact=True)
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|
num = denom - stirling2([n], k_entries, exact=False)
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|
assert np.max(np.abs(num / denom)) < 2e-5
|