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Python

"""
Test Scipy functions versus mpmath, if available.
"""
from __future__ import division, print_function, absolute_import
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._lib.six import with_metaclass
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
except ImportError:
mpmath = MissingModule('mpmath')
_is_32bit_platform = np.intp(0).itemsize < 8
# ------------------------------------------------------------------------------
# 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)
# ------------------------------------------------------------------------------
# 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_)
olderr = np.seterr(invalid='ignore')
try:
FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-10).check()
finally:
np.seterr(**olderr)
@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_)
olderr = np.seterr(invalid='ignore')
try:
FuncData(sc.hyp2f1, dataset, (0,1,2,3), 4, rtol=1e-9,
ignore_inf_sign=True).check()
finally:
np.seterr(**olderr)
@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)
olderr = np.seterr(invalid='ignore')
try:
FuncData(evf, dataset, (0,1,2), 3, rtol=1e-10, atol=1e-14).check()
finally:
np.seterr(**olderr)
# ------------------------------------------------------------------------------
# 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 = []
for z0 in z:
dataset.append((z0, complex(mpmath.loggamma(z0))))
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 = []
for z0 in z:
dataset.append((z0, complex(mpmath.loggamma(z0))))
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()
dataset = []
with mpmath.workdps(100):
for z0 in z:
dataset.append((z0, complex(mpmath.rgamma(z0))))
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()
dataset = []
with mpmath.workdps(30):
for z0 in z:
dataset.append((z0, complex(mpmath.digamma(z0))))
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()
dataset = []
with mpmath.workdps(40):
for z0 in z:
res = digamma(z0)
dataset.append((z0, complex(res)))
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()
dataset = []
with mpmath.workdps(30):
for z0 in z:
res = mpmath.digamma(z0)
dataset.append((z0, complex(res)))
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()
dataset = []
with mpmath.workdps(100):
for a0, x0 in zip(a, x):
dataset.append((a0, x0, float(mpmath.gammainc(a0, b=x0, regularized=True))))
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 = []
for z0 in z:
dataset.append((z0, spence(z0)))
dataset = np.array(dataset)
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 = []
for z0 in z:
dataset.append((z0, complex(mpmath.sinpi(z0))))
dataset = np.array(dataset)
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 = []
for z0 in z:
dataset.append((z0, complex(mpmath.cospi(z0))))
dataset = np.array(dataset)
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 = []
for z0 in z:
dataset.append((z0, complex(_mpmath_wrightomega(z0, 25))))
dataset = np.asarray(dataset)
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 = []
for z0 in z:
dataset.append((z0, complex(_mpmath_wrightomega(z0, 25))))
dataset = np.asarray(dataset)
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 = []
for z0 in z:
dataset.append((z0, complex(_mpmath_wrightomega(z0, 25))))
dataset = np.asarray(dataset)
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 = []
for z0 in z:
dataset.append((z0, complex(mpmath.lambertw(z0))))
dataset = np.asarray(dataset)
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
olderr = np.seterr(invalid='ignore')
try:
r = np.inf * np.sign(r)
finally:
np.seterr(**olderr)
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.
@pytest.mark.xfail(run=False)
def test_hyp1f1(self):
assert_mpmath_equal(inf_to_nan(sc.hyp1f1),
exception_to_nan(lambda a, b, x: mpmath.hyp1f1(a, b, x, **HYPERKW)),
[Arg(-1e5, 1e5), Arg(-1e5, 1e5), Arg()],
n=2000)
@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(condition=_is_32bit_platform,
reason="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(condition=_is_32bit_platform, reason="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):
def rgamma(x):
if x < -8000:
return np.inf
else:
v = mpmath.rgamma(x)
return v
# n=500 (non-xslow default) fails for one bad point
assert_mpmath_equal(sc.rgamma,
rgamma,
[Arg()],
n=5000,
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(self):
assert_mpmath_equal(sc.wrightomega,
lambda z: _mpmath_wrightomega(z, 25),
[ComplexArg()], rtol=1e-14, nan_ok=False)
def test_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_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)