You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
2025 lines
73 KiB
Python
2025 lines
73 KiB
Python
"""
|
|
Test SciPy functions versus mpmath, if available.
|
|
|
|
"""
|
|
import numpy as np
|
|
from numpy.testing import assert_, assert_allclose
|
|
from numpy import pi
|
|
import pytest
|
|
import itertools
|
|
|
|
from distutils.version import LooseVersion
|
|
|
|
import scipy.special as sc
|
|
from scipy.special._testutils import (
|
|
MissingModule, check_version, FuncData,
|
|
assert_func_equal)
|
|
from scipy.special._mptestutils import (
|
|
Arg, FixedArg, ComplexArg, IntArg, assert_mpmath_equal,
|
|
nonfunctional_tooslow, trace_args, time_limited, exception_to_nan,
|
|
inf_to_nan)
|
|
from scipy.special._ufuncs import (
|
|
_sinpi, _cospi, _lgam1p, _lanczos_sum_expg_scaled, _log1pmx,
|
|
_igam_fac)
|
|
|
|
try:
|
|
import mpmath # type: ignore[import]
|
|
except ImportError:
|
|
mpmath = MissingModule('mpmath')
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# expi
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.10')
|
|
def test_expi_complex():
|
|
dataset = []
|
|
for r in np.logspace(-99, 2, 10):
|
|
for p in np.linspace(0, 2*np.pi, 30):
|
|
z = r*np.exp(1j*p)
|
|
dataset.append((z, complex(mpmath.ei(z))))
|
|
dataset = np.array(dataset, dtype=np.complex_)
|
|
|
|
FuncData(sc.expi, dataset, 0, 1).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# expn
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_expn_large_n():
|
|
# Test the transition to the asymptotic regime of n.
|
|
dataset = []
|
|
for n in [50, 51]:
|
|
for x in np.logspace(0, 4, 200):
|
|
with mpmath.workdps(100):
|
|
dataset.append((n, x, float(mpmath.expint(n, x))))
|
|
dataset = np.asarray(dataset)
|
|
|
|
FuncData(sc.expn, dataset, (0, 1), 2, rtol=1e-13).check()
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# hyp0f1
|
|
# ------------------------------------------------------------------------------
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_hyp0f1_gh5764():
|
|
# Do a small and somewhat systematic test that runs quickly
|
|
dataset = []
|
|
axis = [-99.5, -9.5, -0.5, 0.5, 9.5, 99.5]
|
|
for v in axis:
|
|
for x in axis:
|
|
for y in axis:
|
|
z = x + 1j*y
|
|
# mpmath computes the answer correctly at dps ~ 17 but
|
|
# fails for 20 < dps < 120 (uses a different method);
|
|
# set the dps high enough that this isn't an issue
|
|
with mpmath.workdps(120):
|
|
res = complex(mpmath.hyp0f1(v, z))
|
|
dataset.append((v, z, res))
|
|
dataset = np.array(dataset)
|
|
|
|
FuncData(lambda v, z: sc.hyp0f1(v.real, z), dataset, (0, 1), 2,
|
|
rtol=1e-13).check()
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_hyp0f1_gh_1609():
|
|
# this is a regression test for gh-1609
|
|
vv = np.linspace(150, 180, 21)
|
|
af = sc.hyp0f1(vv, 0.5)
|
|
mf = np.array([mpmath.hyp0f1(v, 0.5) for v in vv])
|
|
assert_allclose(af, mf.astype(float), rtol=1e-12)
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# hyperu
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '1.1.0')
|
|
def test_hyperu_around_0():
|
|
dataset = []
|
|
# DLMF 13.2.14-15 test points.
|
|
for n in np.arange(-5, 5):
|
|
for b in np.linspace(-5, 5, 20):
|
|
a = -n
|
|
dataset.append((a, b, 0, float(mpmath.hyperu(a, b, 0))))
|
|
a = -n + b - 1
|
|
dataset.append((a, b, 0, float(mpmath.hyperu(a, b, 0))))
|
|
# DLMF 13.2.16-22 test points.
|
|
for a in [-10.5, -1.5, -0.5, 0, 0.5, 1, 10]:
|
|
for b in [-1.0, -0.5, 0, 0.5, 1, 1.5, 2, 2.5]:
|
|
dataset.append((a, b, 0, float(mpmath.hyperu(a, b, 0))))
|
|
dataset = np.array(dataset)
|
|
|
|
FuncData(sc.hyperu, dataset, (0, 1, 2), 3, rtol=1e-15, atol=5e-13).check()
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# hyp2f1
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '1.0.0')
|
|
def test_hyp2f1_strange_points():
|
|
pts = [
|
|
(2, -1, -1, 0.7), # expected: 2.4
|
|
(2, -2, -2, 0.7), # expected: 3.87
|
|
]
|
|
pts += list(itertools.product([2, 1, -0.7, -1000], repeat=4))
|
|
pts = [
|
|
(a, b, c, x) for a, b, c, x in pts
|
|
if b == c and round(b) == b and b < 0 and b != -1000
|
|
]
|
|
kw = dict(eliminate=True)
|
|
dataset = [p + (float(mpmath.hyp2f1(*p, **kw)),) for p in pts]
|
|
dataset = np.array(dataset, dtype=np.float_)
|
|
|
|
FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check()
|
|
|
|
|
|
@check_version(mpmath, '0.13')
|
|
def test_hyp2f1_real_some_points():
|
|
pts = [
|
|
(1, 2, 3, 0),
|
|
(1./3, 2./3, 5./6, 27./32),
|
|
(1./4, 1./2, 3./4, 80./81),
|
|
(2,-2, -3, 3),
|
|
(2, -3, -2, 3),
|
|
(2, -1.5, -1.5, 3),
|
|
(1, 2, 3, 0),
|
|
(0.7235, -1, -5, 0.3),
|
|
(0.25, 1./3, 2, 0.999),
|
|
(0.25, 1./3, 2, -1),
|
|
(2, 3, 5, 0.99),
|
|
(3./2, -0.5, 3, 0.99),
|
|
(2, 2.5, -3.25, 0.999),
|
|
(-8, 18.016500331508873, 10.805295997850628, 0.90875647507000001),
|
|
(-10, 900, -10.5, 0.99),
|
|
(-10, 900, 10.5, 0.99),
|
|
(-1, 2, 1, 1.0),
|
|
(-1, 2, 1, -1.0),
|
|
(-3, 13, 5, 1.0),
|
|
(-3, 13, 5, -1.0),
|
|
(0.5, 1 - 270.5, 1.5, 0.999**2), # from issue 1561
|
|
]
|
|
dataset = [p + (float(mpmath.hyp2f1(*p)),) for p in pts]
|
|
dataset = np.array(dataset, dtype=np.float_)
|
|
|
|
with np.errstate(invalid='ignore'):
|
|
FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check()
|
|
|
|
|
|
@check_version(mpmath, '0.14')
|
|
def test_hyp2f1_some_points_2():
|
|
# Taken from mpmath unit tests -- this point failed for mpmath 0.13 but
|
|
# was fixed in their SVN since then
|
|
pts = [
|
|
(112, (51,10), (-9,10), -0.99999),
|
|
(10,-900,10.5,0.99),
|
|
(10,-900,-10.5,0.99),
|
|
]
|
|
|
|
def fev(x):
|
|
if isinstance(x, tuple):
|
|
return float(x[0]) / x[1]
|
|
else:
|
|
return x
|
|
|
|
dataset = [tuple(map(fev, p)) + (float(mpmath.hyp2f1(*p)),) for p in pts]
|
|
dataset = np.array(dataset, dtype=np.float_)
|
|
|
|
FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check()
|
|
|
|
|
|
@check_version(mpmath, '0.13')
|
|
def test_hyp2f1_real_some():
|
|
dataset = []
|
|
for a in [-10, -5, -1.8, 1.8, 5, 10]:
|
|
for b in [-2.5, -1, 1, 7.4]:
|
|
for c in [-9, -1.8, 5, 20.4]:
|
|
for z in [-10, -1.01, -0.99, 0, 0.6, 0.95, 1.5, 10]:
|
|
try:
|
|
v = float(mpmath.hyp2f1(a, b, c, z))
|
|
except Exception:
|
|
continue
|
|
dataset.append((a, b, c, z, v))
|
|
dataset = np.array(dataset, dtype=np.float_)
|
|
|
|
with np.errstate(invalid='ignore'):
|
|
FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-9,
|
|
ignore_inf_sign=True).check()
|
|
|
|
|
|
@check_version(mpmath, '0.12')
|
|
@pytest.mark.slow
|
|
def test_hyp2f1_real_random():
|
|
npoints = 500
|
|
dataset = np.zeros((npoints, 5), np.float_)
|
|
|
|
np.random.seed(1234)
|
|
dataset[:, 0] = np.random.pareto(1.5, npoints)
|
|
dataset[:, 1] = np.random.pareto(1.5, npoints)
|
|
dataset[:, 2] = np.random.pareto(1.5, npoints)
|
|
dataset[:, 3] = 2*np.random.rand(npoints) - 1
|
|
|
|
dataset[:, 0] *= (-1)**np.random.randint(2, npoints)
|
|
dataset[:, 1] *= (-1)**np.random.randint(2, npoints)
|
|
dataset[:, 2] *= (-1)**np.random.randint(2, npoints)
|
|
|
|
for ds in dataset:
|
|
if mpmath.__version__ < '0.14':
|
|
# mpmath < 0.14 fails for c too much smaller than a, b
|
|
if abs(ds[:2]).max() > abs(ds[2]):
|
|
ds[2] = abs(ds[:2]).max()
|
|
ds[4] = float(mpmath.hyp2f1(*tuple(ds[:4])))
|
|
|
|
FuncData(sc.hyp2f1, dataset, (0, 1, 2, 3), 4, rtol=1e-9).check()
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# erf (complex)
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.14')
|
|
def test_erf_complex():
|
|
# need to increase mpmath precision for this test
|
|
old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec
|
|
try:
|
|
mpmath.mp.dps = 70
|
|
x1, y1 = np.meshgrid(np.linspace(-10, 1, 31), np.linspace(-10, 1, 11))
|
|
x2, y2 = np.meshgrid(np.logspace(-80, .8, 31), np.logspace(-80, .8, 11))
|
|
points = np.r_[x1.ravel(),x2.ravel()] + 1j*np.r_[y1.ravel(), y2.ravel()]
|
|
|
|
assert_func_equal(sc.erf, lambda x: complex(mpmath.erf(x)), points,
|
|
vectorized=False, rtol=1e-13)
|
|
assert_func_equal(sc.erfc, lambda x: complex(mpmath.erfc(x)), points,
|
|
vectorized=False, rtol=1e-13)
|
|
finally:
|
|
mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# lpmv
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.15')
|
|
def test_lpmv():
|
|
pts = []
|
|
for x in [-0.99, -0.557, 1e-6, 0.132, 1]:
|
|
pts.extend([
|
|
(1, 1, x),
|
|
(1, -1, x),
|
|
(-1, 1, x),
|
|
(-1, -2, x),
|
|
(1, 1.7, x),
|
|
(1, -1.7, x),
|
|
(-1, 1.7, x),
|
|
(-1, -2.7, x),
|
|
(1, 10, x),
|
|
(1, 11, x),
|
|
(3, 8, x),
|
|
(5, 11, x),
|
|
(-3, 8, x),
|
|
(-5, 11, x),
|
|
(3, -8, x),
|
|
(5, -11, x),
|
|
(-3, -8, x),
|
|
(-5, -11, x),
|
|
(3, 8.3, x),
|
|
(5, 11.3, x),
|
|
(-3, 8.3, x),
|
|
(-5, 11.3, x),
|
|
(3, -8.3, x),
|
|
(5, -11.3, x),
|
|
(-3, -8.3, x),
|
|
(-5, -11.3, x),
|
|
])
|
|
|
|
def mplegenp(nu, mu, x):
|
|
if mu == int(mu) and x == 1:
|
|
# mpmath 0.17 gets this wrong
|
|
if mu == 0:
|
|
return 1
|
|
else:
|
|
return 0
|
|
return mpmath.legenp(nu, mu, x)
|
|
|
|
dataset = [p + (mplegenp(p[1], p[0], p[2]),) for p in pts]
|
|
dataset = np.array(dataset, dtype=np.float_)
|
|
|
|
def evf(mu, nu, x):
|
|
return sc.lpmv(mu.astype(int), nu, x)
|
|
|
|
with np.errstate(invalid='ignore'):
|
|
FuncData(evf, dataset, (0,1,2), 3, rtol=1e-10, atol=1e-14).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# beta
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.15')
|
|
def test_beta():
|
|
np.random.seed(1234)
|
|
|
|
b = np.r_[np.logspace(-200, 200, 4),
|
|
np.logspace(-10, 10, 4),
|
|
np.logspace(-1, 1, 4),
|
|
np.arange(-10, 11, 1),
|
|
np.arange(-10, 11, 1) + 0.5,
|
|
-1, -2.3, -3, -100.3, -10003.4]
|
|
a = b
|
|
|
|
ab = np.array(np.broadcast_arrays(a[:,None], b[None,:])).reshape(2, -1).T
|
|
|
|
old_dps, old_prec = mpmath.mp.dps, mpmath.mp.prec
|
|
try:
|
|
mpmath.mp.dps = 400
|
|
|
|
assert_func_equal(sc.beta,
|
|
lambda a, b: float(mpmath.beta(a, b)),
|
|
ab,
|
|
vectorized=False,
|
|
rtol=1e-10,
|
|
ignore_inf_sign=True)
|
|
|
|
assert_func_equal(
|
|
sc.betaln,
|
|
lambda a, b: float(mpmath.log(abs(mpmath.beta(a, b)))),
|
|
ab,
|
|
vectorized=False,
|
|
rtol=1e-10)
|
|
finally:
|
|
mpmath.mp.dps, mpmath.mp.prec = old_dps, old_prec
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# loggamma
|
|
# ------------------------------------------------------------------------------
|
|
|
|
LOGGAMMA_TAYLOR_RADIUS = 0.2
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_loggamma_taylor_transition():
|
|
# Make sure there isn't a big jump in accuracy when we move from
|
|
# using the Taylor series to using the recurrence relation.
|
|
|
|
r = LOGGAMMA_TAYLOR_RADIUS + np.array([-0.1, -0.01, 0, 0.01, 0.1])
|
|
theta = np.linspace(0, 2*np.pi, 20)
|
|
r, theta = np.meshgrid(r, theta)
|
|
dz = r*np.exp(1j*theta)
|
|
z = np.r_[1 + dz, 2 + dz].flatten()
|
|
|
|
dataset = [(z0, complex(mpmath.loggamma(z0))) for z0 in z]
|
|
dataset = np.array(dataset)
|
|
|
|
FuncData(sc.loggamma, dataset, 0, 1, rtol=5e-14).check()
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_loggamma_taylor():
|
|
# Test around the zeros at z = 1, 2.
|
|
|
|
r = np.logspace(-16, np.log10(LOGGAMMA_TAYLOR_RADIUS), 10)
|
|
theta = np.linspace(0, 2*np.pi, 20)
|
|
r, theta = np.meshgrid(r, theta)
|
|
dz = r*np.exp(1j*theta)
|
|
z = np.r_[1 + dz, 2 + dz].flatten()
|
|
|
|
dataset = [(z0, complex(mpmath.loggamma(z0))) for z0 in z]
|
|
dataset = np.array(dataset)
|
|
|
|
FuncData(sc.loggamma, dataset, 0, 1, rtol=5e-14).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# rgamma
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
@pytest.mark.slow
|
|
def test_rgamma_zeros():
|
|
# Test around the zeros at z = 0, -1, -2, ..., -169. (After -169 we
|
|
# get values that are out of floating point range even when we're
|
|
# within 0.1 of the zero.)
|
|
|
|
# Can't use too many points here or the test takes forever.
|
|
dx = np.r_[-np.logspace(-1, -13, 3), 0, np.logspace(-13, -1, 3)]
|
|
dy = dx.copy()
|
|
dx, dy = np.meshgrid(dx, dy)
|
|
dz = dx + 1j*dy
|
|
zeros = np.arange(0, -170, -1).reshape(1, 1, -1)
|
|
z = (zeros + np.dstack((dz,)*zeros.size)).flatten()
|
|
with mpmath.workdps(100):
|
|
dataset = [(z0, complex(mpmath.rgamma(z0))) for z0 in z]
|
|
|
|
dataset = np.array(dataset)
|
|
FuncData(sc.rgamma, dataset, 0, 1, rtol=1e-12).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# digamma
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
@pytest.mark.slow
|
|
def test_digamma_roots():
|
|
# Test the special-cased roots for digamma.
|
|
root = mpmath.findroot(mpmath.digamma, 1.5)
|
|
roots = [float(root)]
|
|
root = mpmath.findroot(mpmath.digamma, -0.5)
|
|
roots.append(float(root))
|
|
roots = np.array(roots)
|
|
|
|
# If we test beyond a radius of 0.24 mpmath will take forever.
|
|
dx = np.r_[-0.24, -np.logspace(-1, -15, 10), 0, np.logspace(-15, -1, 10), 0.24]
|
|
dy = dx.copy()
|
|
dx, dy = np.meshgrid(dx, dy)
|
|
dz = dx + 1j*dy
|
|
z = (roots + np.dstack((dz,)*roots.size)).flatten()
|
|
with mpmath.workdps(30):
|
|
dataset = [(z0, complex(mpmath.digamma(z0))) for z0 in z]
|
|
|
|
dataset = np.array(dataset)
|
|
FuncData(sc.digamma, dataset, 0, 1, rtol=1e-14).check()
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_digamma_negreal():
|
|
# Test digamma around the negative real axis. Don't do this in
|
|
# TestSystematic because the points need some jiggering so that
|
|
# mpmath doesn't take forever.
|
|
|
|
digamma = exception_to_nan(mpmath.digamma)
|
|
|
|
x = -np.logspace(300, -30, 100)
|
|
y = np.r_[-np.logspace(0, -3, 5), 0, np.logspace(-3, 0, 5)]
|
|
x, y = np.meshgrid(x, y)
|
|
z = (x + 1j*y).flatten()
|
|
|
|
with mpmath.workdps(40):
|
|
dataset = [(z0, complex(digamma(z0))) for z0 in z]
|
|
dataset = np.asarray(dataset)
|
|
|
|
FuncData(sc.digamma, dataset, 0, 1, rtol=1e-13).check()
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_digamma_boundary():
|
|
# Check that there isn't a jump in accuracy when we switch from
|
|
# using the asymptotic series to the reflection formula.
|
|
|
|
x = -np.logspace(300, -30, 100)
|
|
y = np.array([-6.1, -5.9, 5.9, 6.1])
|
|
x, y = np.meshgrid(x, y)
|
|
z = (x + 1j*y).flatten()
|
|
|
|
with mpmath.workdps(30):
|
|
dataset = [(z0, complex(mpmath.digamma(z0))) for z0 in z]
|
|
dataset = np.asarray(dataset)
|
|
|
|
FuncData(sc.digamma, dataset, 0, 1, rtol=1e-13).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# gammainc
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
@pytest.mark.slow
|
|
def test_gammainc_boundary():
|
|
# Test the transition to the asymptotic series.
|
|
small = 20
|
|
a = np.linspace(0.5*small, 2*small, 50)
|
|
x = a.copy()
|
|
a, x = np.meshgrid(a, x)
|
|
a, x = a.flatten(), x.flatten()
|
|
with mpmath.workdps(100):
|
|
dataset = [(a0, x0, float(mpmath.gammainc(a0, b=x0, regularized=True)))
|
|
for a0, x0 in zip(a, x)]
|
|
dataset = np.array(dataset)
|
|
|
|
FuncData(sc.gammainc, dataset, (0, 1), 2, rtol=1e-12).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# spence
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
@pytest.mark.slow
|
|
def test_spence_circle():
|
|
# The trickiest region for spence is around the circle |z - 1| = 1,
|
|
# so test that region carefully.
|
|
|
|
def spence(z):
|
|
return complex(mpmath.polylog(2, 1 - z))
|
|
|
|
r = np.linspace(0.5, 1.5)
|
|
theta = np.linspace(0, 2*pi)
|
|
z = (1 + np.outer(r, np.exp(1j*theta))).flatten()
|
|
dataset = np.asarray([(z0, spence(z0)) for z0 in z])
|
|
|
|
FuncData(sc.spence, dataset, 0, 1, rtol=1e-14).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# sinpi and cospi
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_sinpi_zeros():
|
|
eps = np.finfo(float).eps
|
|
dx = np.r_[-np.logspace(0, -13, 3), 0, np.logspace(-13, 0, 3)]
|
|
dy = dx.copy()
|
|
dx, dy = np.meshgrid(dx, dy)
|
|
dz = dx + 1j*dy
|
|
zeros = np.arange(-100, 100, 1).reshape(1, 1, -1)
|
|
z = (zeros + np.dstack((dz,)*zeros.size)).flatten()
|
|
dataset = np.asarray([(z0, complex(mpmath.sinpi(z0)))
|
|
for z0 in z])
|
|
FuncData(_sinpi, dataset, 0, 1, rtol=2*eps).check()
|
|
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_cospi_zeros():
|
|
eps = np.finfo(float).eps
|
|
dx = np.r_[-np.logspace(0, -13, 3), 0, np.logspace(-13, 0, 3)]
|
|
dy = dx.copy()
|
|
dx, dy = np.meshgrid(dx, dy)
|
|
dz = dx + 1j*dy
|
|
zeros = (np.arange(-100, 100, 1) + 0.5).reshape(1, 1, -1)
|
|
z = (zeros + np.dstack((dz,)*zeros.size)).flatten()
|
|
dataset = np.asarray([(z0, complex(mpmath.cospi(z0)))
|
|
for z0 in z])
|
|
|
|
FuncData(_cospi, dataset, 0, 1, rtol=2*eps).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# ellipj
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@check_version(mpmath, '0.19')
|
|
def test_dn_quarter_period():
|
|
def dn(u, m):
|
|
return sc.ellipj(u, m)[2]
|
|
|
|
def mpmath_dn(u, m):
|
|
return float(mpmath.ellipfun("dn", u=u, m=m))
|
|
|
|
m = np.linspace(0, 1, 20)
|
|
du = np.r_[-np.logspace(-1, -15, 10), 0, np.logspace(-15, -1, 10)]
|
|
dataset = []
|
|
for m0 in m:
|
|
u0 = float(mpmath.ellipk(m0))
|
|
for du0 in du:
|
|
p = u0 + du0
|
|
dataset.append((p, m0, mpmath_dn(p, m0)))
|
|
dataset = np.asarray(dataset)
|
|
|
|
FuncData(dn, dataset, (0, 1), 2, rtol=1e-10).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# Wright Omega
|
|
# ------------------------------------------------------------------------------
|
|
|
|
def _mpmath_wrightomega(z, dps):
|
|
with mpmath.workdps(dps):
|
|
z = mpmath.mpc(z)
|
|
unwind = mpmath.ceil((z.imag - mpmath.pi)/(2*mpmath.pi))
|
|
res = mpmath.lambertw(mpmath.exp(z), unwind)
|
|
return res
|
|
|
|
|
|
@pytest.mark.slow
|
|
@check_version(mpmath, '0.19')
|
|
def test_wrightomega_branch():
|
|
x = -np.logspace(10, 0, 25)
|
|
picut_above = [np.nextafter(np.pi, np.inf)]
|
|
picut_below = [np.nextafter(np.pi, -np.inf)]
|
|
npicut_above = [np.nextafter(-np.pi, np.inf)]
|
|
npicut_below = [np.nextafter(-np.pi, -np.inf)]
|
|
for i in range(50):
|
|
picut_above.append(np.nextafter(picut_above[-1], np.inf))
|
|
picut_below.append(np.nextafter(picut_below[-1], -np.inf))
|
|
npicut_above.append(np.nextafter(npicut_above[-1], np.inf))
|
|
npicut_below.append(np.nextafter(npicut_below[-1], -np.inf))
|
|
y = np.hstack((picut_above, picut_below, npicut_above, npicut_below))
|
|
x, y = np.meshgrid(x, y)
|
|
z = (x + 1j*y).flatten()
|
|
|
|
dataset = np.asarray([(z0, complex(_mpmath_wrightomega(z0, 25)))
|
|
for z0 in z])
|
|
|
|
FuncData(sc.wrightomega, dataset, 0, 1, rtol=1e-8).check()
|
|
|
|
|
|
@pytest.mark.slow
|
|
@check_version(mpmath, '0.19')
|
|
def test_wrightomega_region1():
|
|
# This region gets less coverage in the TestSystematic test
|
|
x = np.linspace(-2, 1)
|
|
y = np.linspace(1, 2*np.pi)
|
|
x, y = np.meshgrid(x, y)
|
|
z = (x + 1j*y).flatten()
|
|
|
|
dataset = np.asarray([(z0, complex(_mpmath_wrightomega(z0, 25)))
|
|
for z0 in z])
|
|
|
|
FuncData(sc.wrightomega, dataset, 0, 1, rtol=1e-15).check()
|
|
|
|
|
|
@pytest.mark.slow
|
|
@check_version(mpmath, '0.19')
|
|
def test_wrightomega_region2():
|
|
# This region gets less coverage in the TestSystematic test
|
|
x = np.linspace(-2, 1)
|
|
y = np.linspace(-2*np.pi, -1)
|
|
x, y = np.meshgrid(x, y)
|
|
z = (x + 1j*y).flatten()
|
|
|
|
dataset = np.asarray([(z0, complex(_mpmath_wrightomega(z0, 25)))
|
|
for z0 in z])
|
|
|
|
FuncData(sc.wrightomega, dataset, 0, 1, rtol=1e-15).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# lambertw
|
|
# ------------------------------------------------------------------------------
|
|
|
|
@pytest.mark.slow
|
|
@check_version(mpmath, '0.19')
|
|
def test_lambertw_smallz():
|
|
x, y = np.linspace(-1, 1, 25), np.linspace(-1, 1, 25)
|
|
x, y = np.meshgrid(x, y)
|
|
z = (x + 1j*y).flatten()
|
|
|
|
dataset = np.asarray([(z0, complex(mpmath.lambertw(z0)))
|
|
for z0 in z])
|
|
|
|
FuncData(sc.lambertw, dataset, 0, 1, rtol=1e-13).check()
|
|
|
|
|
|
# ------------------------------------------------------------------------------
|
|
# Systematic tests
|
|
# ------------------------------------------------------------------------------
|
|
|
|
HYPERKW = dict(maxprec=200, maxterms=200)
|
|
|
|
|
|
@pytest.mark.slow
|
|
@check_version(mpmath, '0.17')
|
|
class TestSystematic(object):
|
|
|
|
def test_airyai(self):
|
|
# oscillating function, limit range
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[0],
|
|
mpmath.airyai,
|
|
[Arg(-1e8, 1e8)],
|
|
rtol=1e-5)
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[0],
|
|
mpmath.airyai,
|
|
[Arg(-1e3, 1e3)])
|
|
|
|
def test_airyai_complex(self):
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[0],
|
|
mpmath.airyai,
|
|
[ComplexArg()])
|
|
|
|
def test_airyai_prime(self):
|
|
# oscillating function, limit range
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[1], lambda z:
|
|
mpmath.airyai(z, derivative=1),
|
|
[Arg(-1e8, 1e8)],
|
|
rtol=1e-5)
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[1], lambda z:
|
|
mpmath.airyai(z, derivative=1),
|
|
[Arg(-1e3, 1e3)])
|
|
|
|
def test_airyai_prime_complex(self):
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[1], lambda z:
|
|
mpmath.airyai(z, derivative=1),
|
|
[ComplexArg()])
|
|
|
|
def test_airybi(self):
|
|
# oscillating function, limit range
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[2], lambda z:
|
|
mpmath.airybi(z),
|
|
[Arg(-1e8, 1e8)],
|
|
rtol=1e-5)
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[2], lambda z:
|
|
mpmath.airybi(z),
|
|
[Arg(-1e3, 1e3)])
|
|
|
|
def test_airybi_complex(self):
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[2], lambda z:
|
|
mpmath.airybi(z),
|
|
[ComplexArg()])
|
|
|
|
def test_airybi_prime(self):
|
|
# oscillating function, limit range
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[3], lambda z:
|
|
mpmath.airybi(z, derivative=1),
|
|
[Arg(-1e8, 1e8)],
|
|
rtol=1e-5)
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[3], lambda z:
|
|
mpmath.airybi(z, derivative=1),
|
|
[Arg(-1e3, 1e3)])
|
|
|
|
def test_airybi_prime_complex(self):
|
|
assert_mpmath_equal(lambda z: sc.airy(z)[3], lambda z:
|
|
mpmath.airybi(z, derivative=1),
|
|
[ComplexArg()])
|
|
|
|
def test_bei(self):
|
|
assert_mpmath_equal(sc.bei,
|
|
exception_to_nan(lambda z: mpmath.bei(0, z, **HYPERKW)),
|
|
[Arg(-1e3, 1e3)])
|
|
|
|
def test_ber(self):
|
|
assert_mpmath_equal(sc.ber,
|
|
exception_to_nan(lambda z: mpmath.ber(0, z, **HYPERKW)),
|
|
[Arg(-1e3, 1e3)])
|
|
|
|
def test_bernoulli(self):
|
|
assert_mpmath_equal(lambda n: sc.bernoulli(int(n))[int(n)],
|
|
lambda n: float(mpmath.bernoulli(int(n))),
|
|
[IntArg(0, 13000)],
|
|
rtol=1e-9, n=13000)
|
|
|
|
def test_besseli(self):
|
|
assert_mpmath_equal(sc.iv,
|
|
exception_to_nan(lambda v, z: mpmath.besseli(v, z, **HYPERKW)),
|
|
[Arg(-1e100, 1e100), Arg()],
|
|
atol=1e-270)
|
|
|
|
def test_besseli_complex(self):
|
|
assert_mpmath_equal(lambda v, z: sc.iv(v.real, z),
|
|
exception_to_nan(lambda v, z: mpmath.besseli(v, z, **HYPERKW)),
|
|
[Arg(-1e100, 1e100), ComplexArg()])
|
|
|
|
def test_besselj(self):
|
|
assert_mpmath_equal(sc.jv,
|
|
exception_to_nan(lambda v, z: mpmath.besselj(v, z, **HYPERKW)),
|
|
[Arg(-1e100, 1e100), Arg(-1e3, 1e3)],
|
|
ignore_inf_sign=True)
|
|
|
|
# loss of precision at large arguments due to oscillation
|
|
assert_mpmath_equal(sc.jv,
|
|
exception_to_nan(lambda v, z: mpmath.besselj(v, z, **HYPERKW)),
|
|
[Arg(-1e100, 1e100), Arg(-1e8, 1e8)],
|
|
ignore_inf_sign=True,
|
|
rtol=1e-5)
|
|
|
|
def test_besselj_complex(self):
|
|
assert_mpmath_equal(lambda v, z: sc.jv(v.real, z),
|
|
exception_to_nan(lambda v, z: mpmath.besselj(v, z, **HYPERKW)),
|
|
[Arg(), ComplexArg()])
|
|
|
|
def test_besselk(self):
|
|
assert_mpmath_equal(sc.kv,
|
|
mpmath.besselk,
|
|
[Arg(-200, 200), Arg(0, np.inf)],
|
|
nan_ok=False, rtol=1e-12)
|
|
|
|
def test_besselk_int(self):
|
|
assert_mpmath_equal(sc.kn,
|
|
mpmath.besselk,
|
|
[IntArg(-200, 200), Arg(0, np.inf)],
|
|
nan_ok=False, rtol=1e-12)
|
|
|
|
def test_besselk_complex(self):
|
|
assert_mpmath_equal(lambda v, z: sc.kv(v.real, z),
|
|
exception_to_nan(lambda v, z: mpmath.besselk(v, z, **HYPERKW)),
|
|
[Arg(-1e100, 1e100), ComplexArg()])
|
|
|
|
def test_bessely(self):
|
|
def mpbessely(v, x):
|
|
r = float(mpmath.bessely(v, x, **HYPERKW))
|
|
if abs(r) > 1e305:
|
|
# overflowing to inf a bit earlier is OK
|
|
r = np.inf * np.sign(r)
|
|
if abs(r) == 0 and x == 0:
|
|
# invalid result from mpmath, point x=0 is a divergence
|
|
return np.nan
|
|
return r
|
|
assert_mpmath_equal(sc.yv,
|
|
exception_to_nan(mpbessely),
|
|
[Arg(-1e100, 1e100), Arg(-1e8, 1e8)],
|
|
n=5000)
|
|
|
|
def test_bessely_complex(self):
|
|
def mpbessely(v, x):
|
|
r = complex(mpmath.bessely(v, x, **HYPERKW))
|
|
if abs(r) > 1e305:
|
|
# overflowing to inf a bit earlier is OK
|
|
with np.errstate(invalid='ignore'):
|
|
r = np.inf * np.sign(r)
|
|
return r
|
|
assert_mpmath_equal(lambda v, z: sc.yv(v.real, z),
|
|
exception_to_nan(mpbessely),
|
|
[Arg(), ComplexArg()],
|
|
n=15000)
|
|
|
|
def test_bessely_int(self):
|
|
def mpbessely(v, x):
|
|
r = float(mpmath.bessely(v, x))
|
|
if abs(r) == 0 and x == 0:
|
|
# invalid result from mpmath, point x=0 is a divergence
|
|
return np.nan
|
|
return r
|
|
assert_mpmath_equal(lambda v, z: sc.yn(int(v), z),
|
|
exception_to_nan(mpbessely),
|
|
[IntArg(-1000, 1000), Arg(-1e8, 1e8)])
|
|
|
|
def test_beta(self):
|
|
bad_points = []
|
|
|
|
def beta(a, b, nonzero=False):
|
|
if a < -1e12 or b < -1e12:
|
|
# Function is defined here only at integers, but due
|
|
# to loss of precision this is numerically
|
|
# ill-defined. Don't compare values here.
|
|
return np.nan
|
|
if (a < 0 or b < 0) and (abs(float(a + b)) % 1) == 0:
|
|
# close to a zero of the function: mpmath and scipy
|
|
# will not round here the same, so the test needs to be
|
|
# run with an absolute tolerance
|
|
if nonzero:
|
|
bad_points.append((float(a), float(b)))
|
|
return np.nan
|
|
return mpmath.beta(a, b)
|
|
|
|
assert_mpmath_equal(sc.beta,
|
|
lambda a, b: beta(a, b, nonzero=True),
|
|
[Arg(), Arg()],
|
|
dps=400,
|
|
ignore_inf_sign=True)
|
|
|
|
assert_mpmath_equal(sc.beta,
|
|
beta,
|
|
np.array(bad_points),
|
|
dps=400,
|
|
ignore_inf_sign=True,
|
|
atol=1e-11)
|
|
|
|
def test_betainc(self):
|
|
assert_mpmath_equal(sc.betainc,
|
|
time_limited()(exception_to_nan(lambda a, b, x: mpmath.betainc(a, b, 0, x, regularized=True))),
|
|
[Arg(), Arg(), Arg()])
|
|
|
|
def test_binom(self):
|
|
bad_points = []
|
|
|
|
def binomial(n, k, nonzero=False):
|
|
if abs(k) > 1e8*(abs(n) + 1):
|
|
# The binomial is rapidly oscillating in this region,
|
|
# and the function is numerically ill-defined. Don't
|
|
# compare values here.
|
|
return np.nan
|
|
if n < k and abs(float(n-k) - np.round(float(n-k))) < 1e-15:
|
|
# close to a zero of the function: mpmath and scipy
|
|
# will not round here the same, so the test needs to be
|
|
# run with an absolute tolerance
|
|
if nonzero:
|
|
bad_points.append((float(n), float(k)))
|
|
return np.nan
|
|
return mpmath.binomial(n, k)
|
|
|
|
assert_mpmath_equal(sc.binom,
|
|
lambda n, k: binomial(n, k, nonzero=True),
|
|
[Arg(), Arg()],
|
|
dps=400)
|
|
|
|
assert_mpmath_equal(sc.binom,
|
|
binomial,
|
|
np.array(bad_points),
|
|
dps=400,
|
|
atol=1e-14)
|
|
|
|
def test_chebyt_int(self):
|
|
assert_mpmath_equal(lambda n, x: sc.eval_chebyt(int(n), x),
|
|
exception_to_nan(lambda n, x: mpmath.chebyt(n, x, **HYPERKW)),
|
|
[IntArg(), Arg()], dps=50)
|
|
|
|
@pytest.mark.xfail(run=False, reason="some cases in hyp2f1 not fully accurate")
|
|
def test_chebyt(self):
|
|
assert_mpmath_equal(sc.eval_chebyt,
|
|
lambda n, x: time_limited()(exception_to_nan(mpmath.chebyt))(n, x, **HYPERKW),
|
|
[Arg(-101, 101), Arg()], n=10000)
|
|
|
|
def test_chebyu_int(self):
|
|
assert_mpmath_equal(lambda n, x: sc.eval_chebyu(int(n), x),
|
|
exception_to_nan(lambda n, x: mpmath.chebyu(n, x, **HYPERKW)),
|
|
[IntArg(), Arg()], dps=50)
|
|
|
|
@pytest.mark.xfail(run=False, reason="some cases in hyp2f1 not fully accurate")
|
|
def test_chebyu(self):
|
|
assert_mpmath_equal(sc.eval_chebyu,
|
|
lambda n, x: time_limited()(exception_to_nan(mpmath.chebyu))(n, x, **HYPERKW),
|
|
[Arg(-101, 101), Arg()])
|
|
|
|
def test_chi(self):
|
|
def chi(x):
|
|
return sc.shichi(x)[1]
|
|
assert_mpmath_equal(chi, mpmath.chi, [Arg()])
|
|
# check asymptotic series cross-over
|
|
assert_mpmath_equal(chi, mpmath.chi, [FixedArg([88 - 1e-9, 88, 88 + 1e-9])])
|
|
|
|
def test_chi_complex(self):
|
|
def chi(z):
|
|
return sc.shichi(z)[1]
|
|
# chi oscillates as Im[z] -> +- inf, so limit range
|
|
assert_mpmath_equal(chi,
|
|
mpmath.chi,
|
|
[ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))],
|
|
rtol=1e-12)
|
|
|
|
def test_ci(self):
|
|
def ci(x):
|
|
return sc.sici(x)[1]
|
|
# oscillating function: limit range
|
|
assert_mpmath_equal(ci,
|
|
mpmath.ci,
|
|
[Arg(-1e8, 1e8)])
|
|
|
|
def test_ci_complex(self):
|
|
def ci(z):
|
|
return sc.sici(z)[1]
|
|
# ci oscillates as Re[z] -> +- inf, so limit range
|
|
assert_mpmath_equal(ci,
|
|
mpmath.ci,
|
|
[ComplexArg(complex(-1e8, -np.inf), complex(1e8, np.inf))],
|
|
rtol=1e-8)
|
|
|
|
def test_cospi(self):
|
|
eps = np.finfo(float).eps
|
|
assert_mpmath_equal(_cospi,
|
|
mpmath.cospi,
|
|
[Arg()], nan_ok=False, rtol=eps)
|
|
|
|
def test_cospi_complex(self):
|
|
assert_mpmath_equal(_cospi,
|
|
mpmath.cospi,
|
|
[ComplexArg()], nan_ok=False, rtol=1e-13)
|
|
|
|
def test_digamma(self):
|
|
assert_mpmath_equal(sc.digamma,
|
|
exception_to_nan(mpmath.digamma),
|
|
[Arg()], rtol=1e-12, dps=50)
|
|
|
|
def test_digamma_complex(self):
|
|
# Test on a cut plane because mpmath will hang. See
|
|
# test_digamma_negreal for tests on the negative real axis.
|
|
def param_filter(z):
|
|
return np.where((z.real < 0) & (np.abs(z.imag) < 1.12), False, True)
|
|
|
|
assert_mpmath_equal(sc.digamma,
|
|
exception_to_nan(mpmath.digamma),
|
|
[ComplexArg()], rtol=1e-13, dps=40,
|
|
param_filter=param_filter)
|
|
|
|
def test_e1(self):
|
|
assert_mpmath_equal(sc.exp1,
|
|
mpmath.e1,
|
|
[Arg()], rtol=1e-14)
|
|
|
|
def test_e1_complex(self):
|
|
# E_1 oscillates as Im[z] -> +- inf, so limit range
|
|
assert_mpmath_equal(sc.exp1,
|
|
mpmath.e1,
|
|
[ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))],
|
|
rtol=1e-11)
|
|
|
|
# Check cross-over region
|
|
assert_mpmath_equal(sc.exp1,
|
|
mpmath.e1,
|
|
(np.linspace(-50, 50, 171)[:, None] +
|
|
np.r_[0, np.logspace(-3, 2, 61),
|
|
-np.logspace(-3, 2, 11)]*1j).ravel(),
|
|
rtol=1e-11)
|
|
assert_mpmath_equal(sc.exp1,
|
|
mpmath.e1,
|
|
(np.linspace(-50, -35, 10000) + 0j),
|
|
rtol=1e-11)
|
|
|
|
def test_exprel(self):
|
|
assert_mpmath_equal(sc.exprel,
|
|
lambda x: mpmath.expm1(x)/x if x != 0 else mpmath.mpf('1.0'),
|
|
[Arg(a=-np.log(np.finfo(np.double).max), b=np.log(np.finfo(np.double).max))])
|
|
assert_mpmath_equal(sc.exprel,
|
|
lambda x: mpmath.expm1(x)/x if x != 0 else mpmath.mpf('1.0'),
|
|
np.array([1e-12, 1e-24, 0, 1e12, 1e24, np.inf]), rtol=1e-11)
|
|
assert_(np.isinf(sc.exprel(np.inf)))
|
|
assert_(sc.exprel(-np.inf) == 0)
|
|
|
|
def test_expm1_complex(self):
|
|
# Oscillates as a function of Im[z], so limit range to avoid loss of precision
|
|
assert_mpmath_equal(sc.expm1,
|
|
mpmath.expm1,
|
|
[ComplexArg(complex(-np.inf, -1e7), complex(np.inf, 1e7))])
|
|
|
|
def test_log1p_complex(self):
|
|
assert_mpmath_equal(sc.log1p,
|
|
lambda x: mpmath.log(x+1),
|
|
[ComplexArg()], dps=60)
|
|
|
|
def test_log1pmx(self):
|
|
assert_mpmath_equal(_log1pmx,
|
|
lambda x: mpmath.log(x + 1) - x,
|
|
[Arg()], dps=60, rtol=1e-14)
|
|
|
|
def test_ei(self):
|
|
assert_mpmath_equal(sc.expi,
|
|
mpmath.ei,
|
|
[Arg()],
|
|
rtol=1e-11)
|
|
|
|
def test_ei_complex(self):
|
|
# Ei oscillates as Im[z] -> +- inf, so limit range
|
|
assert_mpmath_equal(sc.expi,
|
|
mpmath.ei,
|
|
[ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))],
|
|
rtol=1e-9)
|
|
|
|
def test_ellipe(self):
|
|
assert_mpmath_equal(sc.ellipe,
|
|
mpmath.ellipe,
|
|
[Arg(b=1.0)])
|
|
|
|
def test_ellipeinc(self):
|
|
assert_mpmath_equal(sc.ellipeinc,
|
|
mpmath.ellipe,
|
|
[Arg(-1e3, 1e3), Arg(b=1.0)])
|
|
|
|
def test_ellipeinc_largephi(self):
|
|
assert_mpmath_equal(sc.ellipeinc,
|
|
mpmath.ellipe,
|
|
[Arg(), Arg()])
|
|
|
|
def test_ellipf(self):
|
|
assert_mpmath_equal(sc.ellipkinc,
|
|
mpmath.ellipf,
|
|
[Arg(-1e3, 1e3), Arg()])
|
|
|
|
def test_ellipf_largephi(self):
|
|
assert_mpmath_equal(sc.ellipkinc,
|
|
mpmath.ellipf,
|
|
[Arg(), Arg()])
|
|
|
|
def test_ellipk(self):
|
|
assert_mpmath_equal(sc.ellipk,
|
|
mpmath.ellipk,
|
|
[Arg(b=1.0)])
|
|
assert_mpmath_equal(sc.ellipkm1,
|
|
lambda m: mpmath.ellipk(1 - m),
|
|
[Arg(a=0.0)],
|
|
dps=400)
|
|
|
|
def test_ellipkinc(self):
|
|
def ellipkinc(phi, m):
|
|
return mpmath.ellippi(0, phi, m)
|
|
assert_mpmath_equal(sc.ellipkinc,
|
|
ellipkinc,
|
|
[Arg(-1e3, 1e3), Arg(b=1.0)],
|
|
ignore_inf_sign=True)
|
|
|
|
def test_ellipkinc_largephi(self):
|
|
def ellipkinc(phi, m):
|
|
return mpmath.ellippi(0, phi, m)
|
|
assert_mpmath_equal(sc.ellipkinc,
|
|
ellipkinc,
|
|
[Arg(), Arg(b=1.0)],
|
|
ignore_inf_sign=True)
|
|
|
|
def test_ellipfun_sn(self):
|
|
def sn(u, m):
|
|
# mpmath doesn't get the zero at u = 0--fix that
|
|
if u == 0:
|
|
return 0
|
|
else:
|
|
return mpmath.ellipfun("sn", u=u, m=m)
|
|
|
|
# Oscillating function --- limit range of first argument; the
|
|
# loss of precision there is an expected numerical feature
|
|
# rather than an actual bug
|
|
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[0],
|
|
sn,
|
|
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
|
|
rtol=1e-8)
|
|
|
|
def test_ellipfun_cn(self):
|
|
# see comment in ellipfun_sn
|
|
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[1],
|
|
lambda u, m: mpmath.ellipfun("cn", u=u, m=m),
|
|
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
|
|
rtol=1e-8)
|
|
|
|
def test_ellipfun_dn(self):
|
|
# see comment in ellipfun_sn
|
|
assert_mpmath_equal(lambda u, m: sc.ellipj(u, m)[2],
|
|
lambda u, m: mpmath.ellipfun("dn", u=u, m=m),
|
|
[Arg(-1e6, 1e6), Arg(a=0, b=1)],
|
|
rtol=1e-8)
|
|
|
|
def test_erf(self):
|
|
assert_mpmath_equal(sc.erf,
|
|
lambda z: mpmath.erf(z),
|
|
[Arg()])
|
|
|
|
def test_erf_complex(self):
|
|
assert_mpmath_equal(sc.erf,
|
|
lambda z: mpmath.erf(z),
|
|
[ComplexArg()], n=200)
|
|
|
|
def test_erfc(self):
|
|
assert_mpmath_equal(sc.erfc,
|
|
exception_to_nan(lambda z: mpmath.erfc(z)),
|
|
[Arg()], rtol=1e-13)
|
|
|
|
def test_erfc_complex(self):
|
|
assert_mpmath_equal(sc.erfc,
|
|
exception_to_nan(lambda z: mpmath.erfc(z)),
|
|
[ComplexArg()], n=200)
|
|
|
|
def test_erfi(self):
|
|
assert_mpmath_equal(sc.erfi,
|
|
mpmath.erfi,
|
|
[Arg()], n=200)
|
|
|
|
def test_erfi_complex(self):
|
|
assert_mpmath_equal(sc.erfi,
|
|
mpmath.erfi,
|
|
[ComplexArg()], n=200)
|
|
|
|
def test_ndtr(self):
|
|
assert_mpmath_equal(sc.ndtr,
|
|
exception_to_nan(lambda z: mpmath.ncdf(z)),
|
|
[Arg()], n=200)
|
|
|
|
def test_ndtr_complex(self):
|
|
assert_mpmath_equal(sc.ndtr,
|
|
lambda z: mpmath.erfc(-z/np.sqrt(2.))/2.,
|
|
[ComplexArg(a=complex(-10000, -10000), b=complex(10000, 10000))], n=400)
|
|
|
|
def test_log_ndtr(self):
|
|
assert_mpmath_equal(sc.log_ndtr,
|
|
exception_to_nan(lambda z: mpmath.log(mpmath.ncdf(z))),
|
|
[Arg()], n=600, dps=300)
|
|
|
|
def test_log_ndtr_complex(self):
|
|
assert_mpmath_equal(sc.log_ndtr,
|
|
exception_to_nan(lambda z: mpmath.log(mpmath.erfc(-z/np.sqrt(2.))/2.)),
|
|
[ComplexArg(a=complex(-10000, -100),
|
|
b=complex(10000, 100))], n=200, dps=300)
|
|
|
|
def test_eulernum(self):
|
|
assert_mpmath_equal(lambda n: sc.euler(n)[-1],
|
|
mpmath.eulernum,
|
|
[IntArg(1, 10000)], n=10000)
|
|
|
|
def test_expint(self):
|
|
assert_mpmath_equal(sc.expn,
|
|
mpmath.expint,
|
|
[IntArg(0, 200), Arg(0, np.inf)],
|
|
rtol=1e-13, dps=160)
|
|
|
|
def test_fresnels(self):
|
|
def fresnels(x):
|
|
return sc.fresnel(x)[0]
|
|
assert_mpmath_equal(fresnels,
|
|
mpmath.fresnels,
|
|
[Arg()])
|
|
|
|
def test_fresnelc(self):
|
|
def fresnelc(x):
|
|
return sc.fresnel(x)[1]
|
|
assert_mpmath_equal(fresnelc,
|
|
mpmath.fresnelc,
|
|
[Arg()])
|
|
|
|
def test_gamma(self):
|
|
assert_mpmath_equal(sc.gamma,
|
|
exception_to_nan(mpmath.gamma),
|
|
[Arg()])
|
|
|
|
def test_gamma_complex(self):
|
|
assert_mpmath_equal(sc.gamma,
|
|
exception_to_nan(mpmath.gamma),
|
|
[ComplexArg()], rtol=5e-13)
|
|
|
|
def test_gammainc(self):
|
|
# Larger arguments are tested in test_data.py:test_local
|
|
assert_mpmath_equal(sc.gammainc,
|
|
lambda z, b: mpmath.gammainc(z, b=b, regularized=True),
|
|
[Arg(0, 1e4, inclusive_a=False), Arg(0, 1e4)],
|
|
nan_ok=False, rtol=1e-11)
|
|
|
|
def test_gammaincc(self):
|
|
# Larger arguments are tested in test_data.py:test_local
|
|
assert_mpmath_equal(sc.gammaincc,
|
|
lambda z, a: mpmath.gammainc(z, a=a, regularized=True),
|
|
[Arg(0, 1e4, inclusive_a=False), Arg(0, 1e4)],
|
|
nan_ok=False, rtol=1e-11)
|
|
|
|
def test_gammaln(self):
|
|
# The real part of loggamma is log(|gamma(z)|).
|
|
def f(z):
|
|
return mpmath.loggamma(z).real
|
|
|
|
assert_mpmath_equal(sc.gammaln, exception_to_nan(f), [Arg()])
|
|
|
|
@pytest.mark.xfail(run=False)
|
|
def test_gegenbauer(self):
|
|
assert_mpmath_equal(sc.eval_gegenbauer,
|
|
exception_to_nan(mpmath.gegenbauer),
|
|
[Arg(-1e3, 1e3), Arg(), Arg()])
|
|
|
|
def test_gegenbauer_int(self):
|
|
# Redefine functions to deal with numerical + mpmath issues
|
|
def gegenbauer(n, a, x):
|
|
# Avoid overflow at large `a` (mpmath would need an even larger
|
|
# dps to handle this correctly, so just skip this region)
|
|
if abs(a) > 1e100:
|
|
return np.nan
|
|
|
|
# Deal with n=0, n=1 correctly; mpmath 0.17 doesn't do these
|
|
# always correctly
|
|
if n == 0:
|
|
r = 1.0
|
|
elif n == 1:
|
|
r = 2*a*x
|
|
else:
|
|
r = mpmath.gegenbauer(n, a, x)
|
|
|
|
# Mpmath 0.17 gives wrong results (spurious zero) in some cases, so
|
|
# compute the value by perturbing the result
|
|
if float(r) == 0 and a < -1 and float(a) == int(float(a)):
|
|
r = mpmath.gegenbauer(n, a + mpmath.mpf('1e-50'), x)
|
|
if abs(r) < mpmath.mpf('1e-50'):
|
|
r = mpmath.mpf('0.0')
|
|
|
|
# Differing overflow thresholds in scipy vs. mpmath
|
|
if abs(r) > 1e270:
|
|
return np.inf
|
|
return r
|
|
|
|
def sc_gegenbauer(n, a, x):
|
|
r = sc.eval_gegenbauer(int(n), a, x)
|
|
# Differing overflow thresholds in scipy vs. mpmath
|
|
if abs(r) > 1e270:
|
|
return np.inf
|
|
return r
|
|
assert_mpmath_equal(sc_gegenbauer,
|
|
exception_to_nan(gegenbauer),
|
|
[IntArg(0, 100), Arg(-1e9, 1e9), Arg()],
|
|
n=40000, dps=100,
|
|
ignore_inf_sign=True, rtol=1e-6)
|
|
|
|
# Check the small-x expansion
|
|
assert_mpmath_equal(sc_gegenbauer,
|
|
exception_to_nan(gegenbauer),
|
|
[IntArg(0, 100), Arg(), FixedArg(np.logspace(-30, -4, 30))],
|
|
dps=100,
|
|
ignore_inf_sign=True)
|
|
|
|
@pytest.mark.xfail(run=False)
|
|
def test_gegenbauer_complex(self):
|
|
assert_mpmath_equal(lambda n, a, x: sc.eval_gegenbauer(int(n), a.real, x),
|
|
exception_to_nan(mpmath.gegenbauer),
|
|
[IntArg(0, 100), Arg(), ComplexArg()])
|
|
|
|
@nonfunctional_tooslow
|
|
def test_gegenbauer_complex_general(self):
|
|
assert_mpmath_equal(lambda n, a, x: sc.eval_gegenbauer(n.real, a.real, x),
|
|
exception_to_nan(mpmath.gegenbauer),
|
|
[Arg(-1e3, 1e3), Arg(), ComplexArg()])
|
|
|
|
def test_hankel1(self):
|
|
assert_mpmath_equal(sc.hankel1,
|
|
exception_to_nan(lambda v, x: mpmath.hankel1(v, x,
|
|
**HYPERKW)),
|
|
[Arg(-1e20, 1e20), Arg()])
|
|
|
|
def test_hankel2(self):
|
|
assert_mpmath_equal(sc.hankel2,
|
|
exception_to_nan(lambda v, x: mpmath.hankel2(v, x, **HYPERKW)),
|
|
[Arg(-1e20, 1e20), Arg()])
|
|
|
|
@pytest.mark.xfail(run=False, reason="issues at intermediately large orders")
|
|
def test_hermite(self):
|
|
assert_mpmath_equal(lambda n, x: sc.eval_hermite(int(n), x),
|
|
exception_to_nan(mpmath.hermite),
|
|
[IntArg(0, 10000), Arg()])
|
|
|
|
# hurwitz: same as zeta
|
|
|
|
def test_hyp0f1(self):
|
|
# mpmath reports no convergence unless maxterms is large enough
|
|
KW = dict(maxprec=400, maxterms=1500)
|
|
# n=500 (non-xslow default) fails for one bad point
|
|
assert_mpmath_equal(sc.hyp0f1,
|
|
lambda a, x: mpmath.hyp0f1(a, x, **KW),
|
|
[Arg(-1e7, 1e7), Arg(0, 1e5)],
|
|
n=5000)
|
|
# NB: The range of the second parameter ("z") is limited from below
|
|
# because of an overflow in the intermediate calculations. The way
|
|
# for fix it is to implement an asymptotic expansion for Bessel J
|
|
# (similar to what is implemented for Bessel I here).
|
|
|
|
def test_hyp0f1_complex(self):
|
|
assert_mpmath_equal(lambda a, z: sc.hyp0f1(a.real, z),
|
|
exception_to_nan(lambda a, x: mpmath.hyp0f1(a, x, **HYPERKW)),
|
|
[Arg(-10, 10), ComplexArg(complex(-120, -120), complex(120, 120))])
|
|
# NB: The range of the first parameter ("v") are limited by an overflow
|
|
# in the intermediate calculations. Can be fixed by implementing an
|
|
# asymptotic expansion for Bessel functions for large order.
|
|
|
|
def test_hyp1f1(self):
|
|
def mpmath_hyp1f1(a, b, x):
|
|
try:
|
|
return mpmath.hyp1f1(a, b, x)
|
|
except ZeroDivisionError:
|
|
return np.inf
|
|
|
|
assert_mpmath_equal(
|
|
sc.hyp1f1,
|
|
mpmath_hyp1f1,
|
|
[Arg(-50, 50), Arg(1, 50, inclusive_a=False), Arg(-50, 50)],
|
|
n=500,
|
|
nan_ok=False
|
|
)
|
|
|
|
@pytest.mark.xfail(run=False)
|
|
def test_hyp1f1_complex(self):
|
|
assert_mpmath_equal(inf_to_nan(lambda a, b, x: sc.hyp1f1(a.real, b.real, x)),
|
|
exception_to_nan(lambda a, b, x: mpmath.hyp1f1(a, b, x, **HYPERKW)),
|
|
[Arg(-1e3, 1e3), Arg(-1e3, 1e3), ComplexArg()],
|
|
n=2000)
|
|
|
|
@nonfunctional_tooslow
|
|
def test_hyp2f1_complex(self):
|
|
# SciPy's hyp2f1 seems to have performance and accuracy problems
|
|
assert_mpmath_equal(lambda a, b, c, x: sc.hyp2f1(a.real, b.real, c.real, x),
|
|
exception_to_nan(lambda a, b, c, x: mpmath.hyp2f1(a, b, c, x, **HYPERKW)),
|
|
[Arg(-1e2, 1e2), Arg(-1e2, 1e2), Arg(-1e2, 1e2), ComplexArg()],
|
|
n=10)
|
|
|
|
@pytest.mark.xfail(run=False)
|
|
def test_hyperu(self):
|
|
assert_mpmath_equal(sc.hyperu,
|
|
exception_to_nan(lambda a, b, x: mpmath.hyperu(a, b, x, **HYPERKW)),
|
|
[Arg(), Arg(), Arg()])
|
|
|
|
@pytest.mark.xfail_on_32bit("mpmath issue gh-342: unsupported operand mpz, long for pow")
|
|
def test_igam_fac(self):
|
|
def mp_igam_fac(a, x):
|
|
return mpmath.power(x, a)*mpmath.exp(-x)/mpmath.gamma(a)
|
|
|
|
assert_mpmath_equal(_igam_fac,
|
|
mp_igam_fac,
|
|
[Arg(0, 1e14, inclusive_a=False), Arg(0, 1e14)],
|
|
rtol=1e-10)
|
|
|
|
def test_j0(self):
|
|
# The Bessel function at large arguments is j0(x) ~ cos(x + phi)/sqrt(x)
|
|
# and at large arguments the phase of the cosine loses precision.
|
|
#
|
|
# This is numerically expected behavior, so we compare only up to
|
|
# 1e8 = 1e15 * 1e-7
|
|
assert_mpmath_equal(sc.j0,
|
|
mpmath.j0,
|
|
[Arg(-1e3, 1e3)])
|
|
assert_mpmath_equal(sc.j0,
|
|
mpmath.j0,
|
|
[Arg(-1e8, 1e8)],
|
|
rtol=1e-5)
|
|
|
|
def test_j1(self):
|
|
# See comment in test_j0
|
|
assert_mpmath_equal(sc.j1,
|
|
mpmath.j1,
|
|
[Arg(-1e3, 1e3)])
|
|
assert_mpmath_equal(sc.j1,
|
|
mpmath.j1,
|
|
[Arg(-1e8, 1e8)],
|
|
rtol=1e-5)
|
|
|
|
@pytest.mark.xfail(run=False)
|
|
def test_jacobi(self):
|
|
assert_mpmath_equal(sc.eval_jacobi,
|
|
exception_to_nan(lambda a, b, c, x: mpmath.jacobi(a, b, c, x, **HYPERKW)),
|
|
[Arg(), Arg(), Arg(), Arg()])
|
|
assert_mpmath_equal(lambda n, b, c, x: sc.eval_jacobi(int(n), b, c, x),
|
|
exception_to_nan(lambda a, b, c, x: mpmath.jacobi(a, b, c, x, **HYPERKW)),
|
|
[IntArg(), Arg(), Arg(), Arg()])
|
|
|
|
def test_jacobi_int(self):
|
|
# Redefine functions to deal with numerical + mpmath issues
|
|
def jacobi(n, a, b, x):
|
|
# Mpmath does not handle n=0 case always correctly
|
|
if n == 0:
|
|
return 1.0
|
|
return mpmath.jacobi(n, a, b, x)
|
|
assert_mpmath_equal(lambda n, a, b, x: sc.eval_jacobi(int(n), a, b, x),
|
|
lambda n, a, b, x: exception_to_nan(jacobi)(n, a, b, x, **HYPERKW),
|
|
[IntArg(), Arg(), Arg(), Arg()],
|
|
n=20000, dps=50)
|
|
|
|
def test_kei(self):
|
|
def kei(x):
|
|
if x == 0:
|
|
# work around mpmath issue at x=0
|
|
return -pi/4
|
|
return exception_to_nan(mpmath.kei)(0, x, **HYPERKW)
|
|
assert_mpmath_equal(sc.kei,
|
|
kei,
|
|
[Arg(-1e30, 1e30)], n=1000)
|
|
|
|
def test_ker(self):
|
|
assert_mpmath_equal(sc.ker,
|
|
exception_to_nan(lambda x: mpmath.ker(0, x, **HYPERKW)),
|
|
[Arg(-1e30, 1e30)], n=1000)
|
|
|
|
@nonfunctional_tooslow
|
|
def test_laguerre(self):
|
|
assert_mpmath_equal(trace_args(sc.eval_laguerre),
|
|
lambda n, x: exception_to_nan(mpmath.laguerre)(n, x, **HYPERKW),
|
|
[Arg(), Arg()])
|
|
|
|
def test_laguerre_int(self):
|
|
assert_mpmath_equal(lambda n, x: sc.eval_laguerre(int(n), x),
|
|
lambda n, x: exception_to_nan(mpmath.laguerre)(n, x, **HYPERKW),
|
|
[IntArg(), Arg()], n=20000)
|
|
|
|
@pytest.mark.xfail_on_32bit("see gh-3551 for bad points")
|
|
def test_lambertw_real(self):
|
|
assert_mpmath_equal(lambda x, k: sc.lambertw(x, int(k.real)),
|
|
lambda x, k: mpmath.lambertw(x, int(k.real)),
|
|
[ComplexArg(-np.inf, np.inf), IntArg(0, 10)],
|
|
rtol=1e-13, nan_ok=False)
|
|
|
|
def test_lanczos_sum_expg_scaled(self):
|
|
maxgamma = 171.624376956302725
|
|
e = np.exp(1)
|
|
g = 6.024680040776729583740234375
|
|
|
|
def gamma(x):
|
|
with np.errstate(over='ignore'):
|
|
fac = ((x + g - 0.5)/e)**(x - 0.5)
|
|
if fac != np.inf:
|
|
res = fac*_lanczos_sum_expg_scaled(x)
|
|
else:
|
|
fac = ((x + g - 0.5)/e)**(0.5*(x - 0.5))
|
|
res = fac*_lanczos_sum_expg_scaled(x)
|
|
res *= fac
|
|
return res
|
|
|
|
assert_mpmath_equal(gamma,
|
|
mpmath.gamma,
|
|
[Arg(0, maxgamma, inclusive_a=False)],
|
|
rtol=1e-13)
|
|
|
|
@nonfunctional_tooslow
|
|
def test_legendre(self):
|
|
assert_mpmath_equal(sc.eval_legendre,
|
|
mpmath.legendre,
|
|
[Arg(), Arg()])
|
|
|
|
def test_legendre_int(self):
|
|
assert_mpmath_equal(lambda n, x: sc.eval_legendre(int(n), x),
|
|
lambda n, x: exception_to_nan(mpmath.legendre)(n, x, **HYPERKW),
|
|
[IntArg(), Arg()],
|
|
n=20000)
|
|
|
|
# Check the small-x expansion
|
|
assert_mpmath_equal(lambda n, x: sc.eval_legendre(int(n), x),
|
|
lambda n, x: exception_to_nan(mpmath.legendre)(n, x, **HYPERKW),
|
|
[IntArg(), FixedArg(np.logspace(-30, -4, 20))])
|
|
|
|
def test_legenp(self):
|
|
def lpnm(n, m, z):
|
|
try:
|
|
v = sc.lpmn(m, n, z)[0][-1,-1]
|
|
except ValueError:
|
|
return np.nan
|
|
if abs(v) > 1e306:
|
|
# harmonize overflow to inf
|
|
v = np.inf * np.sign(v.real)
|
|
return v
|
|
|
|
def lpnm_2(n, m, z):
|
|
v = sc.lpmv(m, n, z)
|
|
if abs(v) > 1e306:
|
|
# harmonize overflow to inf
|
|
v = np.inf * np.sign(v.real)
|
|
return v
|
|
|
|
def legenp(n, m, z):
|
|
if (z == 1 or z == -1) and int(n) == n:
|
|
# Special case (mpmath may give inf, we take the limit by
|
|
# continuity)
|
|
if m == 0:
|
|
if n < 0:
|
|
n = -n - 1
|
|
return mpmath.power(mpmath.sign(z), n)
|
|
else:
|
|
return 0
|
|
|
|
if abs(z) < 1e-15:
|
|
# mpmath has bad performance here
|
|
return np.nan
|
|
|
|
typ = 2 if abs(z) < 1 else 3
|
|
v = exception_to_nan(mpmath.legenp)(n, m, z, type=typ)
|
|
|
|
if abs(v) > 1e306:
|
|
# harmonize overflow to inf
|
|
v = mpmath.inf * mpmath.sign(v.real)
|
|
|
|
return v
|
|
|
|
assert_mpmath_equal(lpnm,
|
|
legenp,
|
|
[IntArg(-100, 100), IntArg(-100, 100), Arg()])
|
|
|
|
assert_mpmath_equal(lpnm_2,
|
|
legenp,
|
|
[IntArg(-100, 100), Arg(-100, 100), Arg(-1, 1)],
|
|
atol=1e-10)
|
|
|
|
def test_legenp_complex_2(self):
|
|
def clpnm(n, m, z):
|
|
try:
|
|
return sc.clpmn(m.real, n.real, z, type=2)[0][-1,-1]
|
|
except ValueError:
|
|
return np.nan
|
|
|
|
def legenp(n, m, z):
|
|
if abs(z) < 1e-15:
|
|
# mpmath has bad performance here
|
|
return np.nan
|
|
return exception_to_nan(mpmath.legenp)(int(n.real), int(m.real), z, type=2)
|
|
|
|
# mpmath is quite slow here
|
|
x = np.array([-2, -0.99, -0.5, 0, 1e-5, 0.5, 0.99, 20, 2e3])
|
|
y = np.array([-1e3, -0.5, 0.5, 1.3])
|
|
z = (x[:,None] + 1j*y[None,:]).ravel()
|
|
|
|
assert_mpmath_equal(clpnm,
|
|
legenp,
|
|
[FixedArg([-2, -1, 0, 1, 2, 10]), FixedArg([-2, -1, 0, 1, 2, 10]), FixedArg(z)],
|
|
rtol=1e-6,
|
|
n=500)
|
|
|
|
def test_legenp_complex_3(self):
|
|
def clpnm(n, m, z):
|
|
try:
|
|
return sc.clpmn(m.real, n.real, z, type=3)[0][-1,-1]
|
|
except ValueError:
|
|
return np.nan
|
|
|
|
def legenp(n, m, z):
|
|
if abs(z) < 1e-15:
|
|
# mpmath has bad performance here
|
|
return np.nan
|
|
return exception_to_nan(mpmath.legenp)(int(n.real), int(m.real), z, type=3)
|
|
|
|
# mpmath is quite slow here
|
|
x = np.array([-2, -0.99, -0.5, 0, 1e-5, 0.5, 0.99, 20, 2e3])
|
|
y = np.array([-1e3, -0.5, 0.5, 1.3])
|
|
z = (x[:,None] + 1j*y[None,:]).ravel()
|
|
|
|
assert_mpmath_equal(clpnm,
|
|
legenp,
|
|
[FixedArg([-2, -1, 0, 1, 2, 10]), FixedArg([-2, -1, 0, 1, 2, 10]), FixedArg(z)],
|
|
rtol=1e-6,
|
|
n=500)
|
|
|
|
@pytest.mark.xfail(run=False, reason="apparently picks wrong function at |z| > 1")
|
|
def test_legenq(self):
|
|
def lqnm(n, m, z):
|
|
return sc.lqmn(m, n, z)[0][-1,-1]
|
|
|
|
def legenq(n, m, z):
|
|
if abs(z) < 1e-15:
|
|
# mpmath has bad performance here
|
|
return np.nan
|
|
return exception_to_nan(mpmath.legenq)(n, m, z, type=2)
|
|
|
|
assert_mpmath_equal(lqnm,
|
|
legenq,
|
|
[IntArg(0, 100), IntArg(0, 100), Arg()])
|
|
|
|
@nonfunctional_tooslow
|
|
def test_legenq_complex(self):
|
|
def lqnm(n, m, z):
|
|
return sc.lqmn(int(m.real), int(n.real), z)[0][-1,-1]
|
|
|
|
def legenq(n, m, z):
|
|
if abs(z) < 1e-15:
|
|
# mpmath has bad performance here
|
|
return np.nan
|
|
return exception_to_nan(mpmath.legenq)(int(n.real), int(m.real), z, type=2)
|
|
|
|
assert_mpmath_equal(lqnm,
|
|
legenq,
|
|
[IntArg(0, 100), IntArg(0, 100), ComplexArg()],
|
|
n=100)
|
|
|
|
def test_lgam1p(self):
|
|
def param_filter(x):
|
|
# Filter the poles
|
|
return np.where((np.floor(x) == x) & (x <= 0), False, True)
|
|
|
|
def mp_lgam1p(z):
|
|
# The real part of loggamma is log(|gamma(z)|)
|
|
return mpmath.loggamma(1 + z).real
|
|
|
|
assert_mpmath_equal(_lgam1p,
|
|
mp_lgam1p,
|
|
[Arg()], rtol=1e-13, dps=100,
|
|
param_filter=param_filter)
|
|
|
|
def test_loggamma(self):
|
|
def mpmath_loggamma(z):
|
|
try:
|
|
res = mpmath.loggamma(z)
|
|
except ValueError:
|
|
res = complex(np.nan, np.nan)
|
|
return res
|
|
|
|
assert_mpmath_equal(sc.loggamma,
|
|
mpmath_loggamma,
|
|
[ComplexArg()], nan_ok=False,
|
|
distinguish_nan_and_inf=False, rtol=5e-14)
|
|
|
|
@pytest.mark.xfail(run=False)
|
|
def test_pcfd(self):
|
|
def pcfd(v, x):
|
|
return sc.pbdv(v, x)[0]
|
|
assert_mpmath_equal(pcfd,
|
|
exception_to_nan(lambda v, x: mpmath.pcfd(v, x, **HYPERKW)),
|
|
[Arg(), Arg()])
|
|
|
|
@pytest.mark.xfail(run=False, reason="it's not the same as the mpmath function --- maybe different definition?")
|
|
def test_pcfv(self):
|
|
def pcfv(v, x):
|
|
return sc.pbvv(v, x)[0]
|
|
assert_mpmath_equal(pcfv,
|
|
lambda v, x: time_limited()(exception_to_nan(mpmath.pcfv))(v, x, **HYPERKW),
|
|
[Arg(), Arg()], n=1000)
|
|
|
|
def test_pcfw(self):
|
|
def pcfw(a, x):
|
|
return sc.pbwa(a, x)[0]
|
|
|
|
def dpcfw(a, x):
|
|
return sc.pbwa(a, x)[1]
|
|
|
|
def mpmath_dpcfw(a, x):
|
|
return mpmath.diff(mpmath.pcfw, (a, x), (0, 1))
|
|
|
|
# The Zhang and Jin implementation only uses Taylor series and
|
|
# is thus accurate in only a very small range.
|
|
assert_mpmath_equal(pcfw,
|
|
mpmath.pcfw,
|
|
[Arg(-5, 5), Arg(-5, 5)], rtol=2e-8, n=100)
|
|
|
|
assert_mpmath_equal(dpcfw,
|
|
mpmath_dpcfw,
|
|
[Arg(-5, 5), Arg(-5, 5)], rtol=2e-9, n=100)
|
|
|
|
@pytest.mark.xfail(run=False, reason="issues at large arguments (atol OK, rtol not) and <eps-close to z=0")
|
|
def test_polygamma(self):
|
|
assert_mpmath_equal(sc.polygamma,
|
|
time_limited()(exception_to_nan(mpmath.polygamma)),
|
|
[IntArg(0, 1000), Arg()])
|
|
|
|
def test_rgamma(self):
|
|
assert_mpmath_equal(
|
|
sc.rgamma,
|
|
mpmath.rgamma,
|
|
[Arg(-8000, np.inf)],
|
|
n=5000,
|
|
nan_ok=False,
|
|
ignore_inf_sign=True,
|
|
)
|
|
|
|
def test_rgamma_complex(self):
|
|
assert_mpmath_equal(sc.rgamma,
|
|
exception_to_nan(mpmath.rgamma),
|
|
[ComplexArg()], rtol=5e-13)
|
|
|
|
@pytest.mark.xfail(reason=("see gh-3551 for bad points on 32 bit "
|
|
"systems and gh-8095 for another bad "
|
|
"point"))
|
|
def test_rf(self):
|
|
if LooseVersion(mpmath.__version__) >= LooseVersion("1.0.0"):
|
|
# no workarounds needed
|
|
mppoch = mpmath.rf
|
|
else:
|
|
def mppoch(a, m):
|
|
# deal with cases where the result in double precision
|
|
# hits exactly a non-positive integer, but the
|
|
# corresponding extended-precision mpf floats don't
|
|
if float(a + m) == int(a + m) and float(a + m) <= 0:
|
|
a = mpmath.mpf(a)
|
|
m = int(a + m) - a
|
|
return mpmath.rf(a, m)
|
|
|
|
assert_mpmath_equal(sc.poch,
|
|
mppoch,
|
|
[Arg(), Arg()],
|
|
dps=400)
|
|
|
|
def test_sinpi(self):
|
|
eps = np.finfo(float).eps
|
|
assert_mpmath_equal(_sinpi, mpmath.sinpi,
|
|
[Arg()], nan_ok=False, rtol=eps)
|
|
|
|
def test_sinpi_complex(self):
|
|
assert_mpmath_equal(_sinpi, mpmath.sinpi,
|
|
[ComplexArg()], nan_ok=False, rtol=2e-14)
|
|
|
|
def test_shi(self):
|
|
def shi(x):
|
|
return sc.shichi(x)[0]
|
|
assert_mpmath_equal(shi, mpmath.shi, [Arg()])
|
|
# check asymptotic series cross-over
|
|
assert_mpmath_equal(shi, mpmath.shi, [FixedArg([88 - 1e-9, 88, 88 + 1e-9])])
|
|
|
|
def test_shi_complex(self):
|
|
def shi(z):
|
|
return sc.shichi(z)[0]
|
|
# shi oscillates as Im[z] -> +- inf, so limit range
|
|
assert_mpmath_equal(shi,
|
|
mpmath.shi,
|
|
[ComplexArg(complex(-np.inf, -1e8), complex(np.inf, 1e8))],
|
|
rtol=1e-12)
|
|
|
|
def test_si(self):
|
|
def si(x):
|
|
return sc.sici(x)[0]
|
|
assert_mpmath_equal(si, mpmath.si, [Arg()])
|
|
|
|
def test_si_complex(self):
|
|
def si(z):
|
|
return sc.sici(z)[0]
|
|
# si oscillates as Re[z] -> +- inf, so limit range
|
|
assert_mpmath_equal(si,
|
|
mpmath.si,
|
|
[ComplexArg(complex(-1e8, -np.inf), complex(1e8, np.inf))],
|
|
rtol=1e-12)
|
|
|
|
def test_spence(self):
|
|
# mpmath uses a different convention for the dilogarithm
|
|
def dilog(x):
|
|
return mpmath.polylog(2, 1 - x)
|
|
# Spence has a branch cut on the negative real axis
|
|
assert_mpmath_equal(sc.spence,
|
|
exception_to_nan(dilog),
|
|
[Arg(0, np.inf)], rtol=1e-14)
|
|
|
|
def test_spence_complex(self):
|
|
def dilog(z):
|
|
return mpmath.polylog(2, 1 - z)
|
|
assert_mpmath_equal(sc.spence,
|
|
exception_to_nan(dilog),
|
|
[ComplexArg()], rtol=1e-14)
|
|
|
|
def test_spherharm(self):
|
|
def spherharm(l, m, theta, phi):
|
|
if m > l:
|
|
return np.nan
|
|
return sc.sph_harm(m, l, phi, theta)
|
|
assert_mpmath_equal(spherharm,
|
|
mpmath.spherharm,
|
|
[IntArg(0, 100), IntArg(0, 100),
|
|
Arg(a=0, b=pi), Arg(a=0, b=2*pi)],
|
|
atol=1e-8, n=6000,
|
|
dps=150)
|
|
|
|
def test_struveh(self):
|
|
assert_mpmath_equal(sc.struve,
|
|
exception_to_nan(mpmath.struveh),
|
|
[Arg(-1e4, 1e4), Arg(0, 1e4)],
|
|
rtol=5e-10)
|
|
|
|
def test_struvel(self):
|
|
def mp_struvel(v, z):
|
|
if v < 0 and z < -v and abs(v) > 1000:
|
|
# larger DPS needed for correct results
|
|
old_dps = mpmath.mp.dps
|
|
try:
|
|
mpmath.mp.dps = 300
|
|
return mpmath.struvel(v, z)
|
|
finally:
|
|
mpmath.mp.dps = old_dps
|
|
return mpmath.struvel(v, z)
|
|
|
|
assert_mpmath_equal(sc.modstruve,
|
|
exception_to_nan(mp_struvel),
|
|
[Arg(-1e4, 1e4), Arg(0, 1e4)],
|
|
rtol=5e-10,
|
|
ignore_inf_sign=True)
|
|
|
|
def test_wrightomega_real(self):
|
|
def mpmath_wrightomega_real(x):
|
|
return mpmath.lambertw(mpmath.exp(x), mpmath.mpf('-0.5'))
|
|
|
|
# For x < -1000 the Wright Omega function is just 0 to double
|
|
# precision, and for x > 1e21 it is just x to double
|
|
# precision.
|
|
assert_mpmath_equal(
|
|
sc.wrightomega,
|
|
mpmath_wrightomega_real,
|
|
[Arg(-1000, 1e21)],
|
|
rtol=5e-15,
|
|
atol=0,
|
|
nan_ok=False,
|
|
)
|
|
|
|
def test_wrightomega(self):
|
|
assert_mpmath_equal(sc.wrightomega,
|
|
lambda z: _mpmath_wrightomega(z, 25),
|
|
[ComplexArg()], rtol=1e-14, nan_ok=False)
|
|
|
|
def test_hurwitz_zeta(self):
|
|
assert_mpmath_equal(sc.zeta,
|
|
exception_to_nan(mpmath.zeta),
|
|
[Arg(a=1, b=1e10, inclusive_a=False),
|
|
Arg(a=0, inclusive_a=False)])
|
|
|
|
def test_riemann_zeta(self):
|
|
assert_mpmath_equal(
|
|
sc.zeta,
|
|
mpmath.zeta,
|
|
[Arg(-100, 100)],
|
|
nan_ok=False,
|
|
rtol=1e-13,
|
|
)
|
|
|
|
def test_zetac(self):
|
|
assert_mpmath_equal(sc.zetac,
|
|
lambda x: mpmath.zeta(x) - 1,
|
|
[Arg(-100, 100)],
|
|
nan_ok=False, dps=45, rtol=1e-13)
|
|
|
|
def test_boxcox(self):
|
|
|
|
def mp_boxcox(x, lmbda):
|
|
x = mpmath.mp.mpf(x)
|
|
lmbda = mpmath.mp.mpf(lmbda)
|
|
if lmbda == 0:
|
|
return mpmath.mp.log(x)
|
|
else:
|
|
return mpmath.mp.powm1(x, lmbda) / lmbda
|
|
|
|
assert_mpmath_equal(sc.boxcox,
|
|
exception_to_nan(mp_boxcox),
|
|
[Arg(a=0, inclusive_a=False), Arg()],
|
|
n=200,
|
|
dps=60,
|
|
rtol=1e-13)
|
|
|
|
def test_boxcox1p(self):
|
|
|
|
def mp_boxcox1p(x, lmbda):
|
|
x = mpmath.mp.mpf(x)
|
|
lmbda = mpmath.mp.mpf(lmbda)
|
|
one = mpmath.mp.mpf(1)
|
|
if lmbda == 0:
|
|
return mpmath.mp.log(one + x)
|
|
else:
|
|
return mpmath.mp.powm1(one + x, lmbda) / lmbda
|
|
|
|
assert_mpmath_equal(sc.boxcox1p,
|
|
exception_to_nan(mp_boxcox1p),
|
|
[Arg(a=-1, inclusive_a=False), Arg()],
|
|
n=200,
|
|
dps=60,
|
|
rtol=1e-13)
|
|
|
|
def test_spherical_jn(self):
|
|
def mp_spherical_jn(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.besselj(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_jn(int(n), z),
|
|
exception_to_nan(mp_spherical_jn),
|
|
[IntArg(0, 200), Arg(-1e8, 1e8)],
|
|
dps=300)
|
|
|
|
def test_spherical_jn_complex(self):
|
|
def mp_spherical_jn(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.besselj(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_jn(int(n.real), z),
|
|
exception_to_nan(mp_spherical_jn),
|
|
[IntArg(0, 200), ComplexArg()])
|
|
|
|
def test_spherical_yn(self):
|
|
def mp_spherical_yn(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.bessely(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_yn(int(n), z),
|
|
exception_to_nan(mp_spherical_yn),
|
|
[IntArg(0, 200), Arg(-1e10, 1e10)],
|
|
dps=100)
|
|
|
|
def test_spherical_yn_complex(self):
|
|
def mp_spherical_yn(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.bessely(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_yn(int(n.real), z),
|
|
exception_to_nan(mp_spherical_yn),
|
|
[IntArg(0, 200), ComplexArg()])
|
|
|
|
def test_spherical_in(self):
|
|
def mp_spherical_in(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.besseli(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_in(int(n), z),
|
|
exception_to_nan(mp_spherical_in),
|
|
[IntArg(0, 200), Arg()],
|
|
dps=200, atol=10**(-278))
|
|
|
|
def test_spherical_in_complex(self):
|
|
def mp_spherical_in(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.besseli(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_in(int(n.real), z),
|
|
exception_to_nan(mp_spherical_in),
|
|
[IntArg(0, 200), ComplexArg()])
|
|
|
|
def test_spherical_kn(self):
|
|
def mp_spherical_kn(n, z):
|
|
out = (mpmath.besselk(n + mpmath.mpf(1)/2, z) *
|
|
mpmath.sqrt(mpmath.pi/(2*mpmath.mpmathify(z))))
|
|
if mpmath.mpmathify(z).imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_kn(int(n), z),
|
|
exception_to_nan(mp_spherical_kn),
|
|
[IntArg(0, 150), Arg()],
|
|
dps=100)
|
|
|
|
@pytest.mark.xfail(run=False, reason="Accuracy issues near z = -1 inherited from kv.")
|
|
def test_spherical_kn_complex(self):
|
|
def mp_spherical_kn(n, z):
|
|
arg = mpmath.mpmathify(z)
|
|
out = (mpmath.besselk(n + mpmath.mpf(1)/2, arg) /
|
|
mpmath.sqrt(2*arg/mpmath.pi))
|
|
if arg.imag == 0:
|
|
return out.real
|
|
else:
|
|
return out
|
|
|
|
assert_mpmath_equal(lambda n, z: sc.spherical_kn(int(n.real), z),
|
|
exception_to_nan(mp_spherical_kn),
|
|
[IntArg(0, 200), ComplexArg()],
|
|
dps=200)
|