from __future__ import division, print_function, absolute_import
import sys
import math
import numpy as np
from numpy import sqrt, cos, sin, arctan, exp, log, pi, Inf
from numpy.testing import (assert_,
assert_allclose, assert_array_less, assert_almost_equal)
import pytest
from pytest import raises as assert_raises
from scipy.integrate import quad, dblquad, tplquad, nquad
from scipy._lib.six import xrange
from scipy._lib._ccallback import LowLevelCallable
import ctypes
import ctypes.util
from scipy._lib._ccallback_c import sine_ctypes
import scipy.integrate._test_multivariate as clib_test
def assert_quad(value_and_err, tabled_value, errTol=1.5e-8):
value, err = value_and_err
assert_allclose(value, tabled_value, atol=err, rtol=0)
if errTol is not None:
assert_array_less(err, errTol)
def get_clib_test_routine(name, restype, *argtypes):
ptr = getattr(clib_test, name)
return ctypes.cast(ptr, ctypes.CFUNCTYPE(restype, *argtypes))
class TestCtypesQuad(object):
def setup_method(self):
if sys.platform == 'win32':
if sys.version_info < (3, 5):
files = [ctypes.util.find_msvcrt()]
else:
files = ['api-ms-win-crt-math-l1-1-0.dll']
elif sys.platform == 'darwin':
files = ['libm.dylib']
else:
files = ['libm.so', 'libm.so.6']
for file in files:
try:
self.lib = ctypes.CDLL(file)
break
except OSError:
pass
else:
# This test doesn't work on some Linux platforms (Fedora for
# example) that put an ld script in libm.so - see gh-5370
self.skipTest("Ctypes can't import libm.so")
restype = ctypes.c_double
argtypes = (ctypes.c_double,)
for name in ['sin', 'cos', 'tan']:
func = getattr(self.lib, name)
func.restype = restype
func.argtypes = argtypes
def test_typical(self):
assert_quad(quad(self.lib.sin, 0, 5), quad(math.sin, 0, 5)[0])
assert_quad(quad(self.lib.cos, 0, 5), quad(math.cos, 0, 5)[0])
assert_quad(quad(self.lib.tan, 0, 1), quad(math.tan, 0, 1)[0])
def test_ctypes_sine(self):
quad(LowLevelCallable(sine_ctypes), 0, 1)
def test_ctypes_variants(self):
sin_0 = get_clib_test_routine('_sin_0', ctypes.c_double,
ctypes.c_double, ctypes.c_void_p)
sin_1 = get_clib_test_routine('_sin_1', ctypes.c_double,
ctypes.c_int, ctypes.POINTER(ctypes.c_double),
ctypes.c_void_p)
sin_2 = get_clib_test_routine('_sin_2', ctypes.c_double,
ctypes.c_double)
sin_3 = get_clib_test_routine('_sin_3', ctypes.c_double,
ctypes.c_int, ctypes.POINTER(ctypes.c_double))
sin_4 = get_clib_test_routine('_sin_3', ctypes.c_double,
ctypes.c_int, ctypes.c_double)
all_sigs = [sin_0, sin_1, sin_2, sin_3, sin_4]
legacy_sigs = [sin_2, sin_4]
legacy_only_sigs = [sin_4]
# LowLevelCallables work for new signatures
for j, func in enumerate(all_sigs):
callback = LowLevelCallable(func)
if func in legacy_only_sigs:
assert_raises(ValueError, quad, callback, 0, pi)
else:
assert_allclose(quad(callback, 0, pi)[0], 2.0)
# Plain ctypes items work only for legacy signatures
for j, func in enumerate(legacy_sigs):
if func in legacy_sigs:
assert_allclose(quad(func, 0, pi)[0], 2.0)
else:
assert_raises(ValueError, quad, func, 0, pi)
class TestMultivariateCtypesQuad(object):
def setup_method(self):
restype = ctypes.c_double
argtypes = (ctypes.c_int, ctypes.c_double)
for name in ['_multivariate_typical', '_multivariate_indefinite',
'_multivariate_sin']:
func = get_clib_test_routine(name, restype, *argtypes)
setattr(self, name, func)
def test_typical(self):
# 1) Typical function with two extra arguments:
assert_quad(quad(self._multivariate_typical, 0, pi, (2, 1.8)),
0.30614353532540296487)
def test_indefinite(self):
# 2) Infinite integration limits --- Euler's constant
assert_quad(quad(self._multivariate_indefinite, 0, Inf),
0.577215664901532860606512)
def test_threadsafety(self):
# Ensure multivariate ctypes are threadsafe
def threadsafety(y):
return y + quad(self._multivariate_sin, 0, 1)[0]
assert_quad(quad(threadsafety, 0, 1), 0.9596976941318602)
class TestQuad(object):
def test_typical(self):
# 1) Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return cos(n*x-z*sin(x))/pi
assert_quad(quad(myfunc, 0, pi, (2, 1.8)), 0.30614353532540296487)
def test_indefinite(self):
# 2) Infinite integration limits --- Euler's constant
def myfunc(x): # Euler's constant integrand
return -exp(-x)*log(x)
assert_quad(quad(myfunc, 0, Inf), 0.577215664901532860606512)
def test_singular(self):
# 3) Singular points in region of integration.
def myfunc(x):
if 0 < x < 2.5:
return sin(x)
elif 2.5 <= x <= 5.0:
return exp(-x)
else:
return 0.0
assert_quad(quad(myfunc, 0, 10, points=[2.5, 5.0]),
1 - cos(2.5) + exp(-2.5) - exp(-5.0))
def test_sine_weighted_finite(self):
# 4) Sine weighted integral (finite limits)
def myfunc(x, a):
return exp(a*(x-1))
ome = 2.0**3.4
assert_quad(quad(myfunc, 0, 1, args=20, weight='sin', wvar=ome),
(20*sin(ome)-ome*cos(ome)+ome*exp(-20))/(20**2 + ome**2))
def test_sine_weighted_infinite(self):
# 5) Sine weighted integral (infinite limits)
def myfunc(x, a):
return exp(-x*a)
a = 4.0
ome = 3.0
assert_quad(quad(myfunc, 0, Inf, args=a, weight='sin', wvar=ome),
ome/(a**2 + ome**2))
def test_cosine_weighted_infinite(self):
# 6) Cosine weighted integral (negative infinite limits)
def myfunc(x, a):
return exp(x*a)
a = 2.5
ome = 2.3
assert_quad(quad(myfunc, -Inf, 0, args=a, weight='cos', wvar=ome),
a/(a**2 + ome**2))
def test_algebraic_log_weight(self):
# 6) Algebraic-logarithmic weight.
def myfunc(x, a):
return 1/(1+x+2**(-a))
a = 1.5
assert_quad(quad(myfunc, -1, 1, args=a, weight='alg',
wvar=(-0.5, -0.5)),
pi/sqrt((1+2**(-a))**2 - 1))
def test_cauchypv_weight(self):
# 7) Cauchy prinicpal value weighting w(x) = 1/(x-c)
def myfunc(x, a):
return 2.0**(-a)/((x-1)**2+4.0**(-a))
a = 0.4
tabledValue = ((2.0**(-0.4)*log(1.5) -
2.0**(-1.4)*log((4.0**(-a)+16) / (4.0**(-a)+1)) -
arctan(2.0**(a+2)) -
arctan(2.0**a)) /
(4.0**(-a) + 1))
assert_quad(quad(myfunc, 0, 5, args=0.4, weight='cauchy', wvar=2.0),
tabledValue, errTol=1.9e-8)
def test_b_less_than_a(self):
def f(x, p, q):
return p * np.exp(-q*x)
val_1, err_1 = quad(f, 0, np.inf, args=(2, 3))
val_2, err_2 = quad(f, np.inf, 0, args=(2, 3))
assert_allclose(val_1, -val_2, atol=max(err_1, err_2))
def test_b_less_than_a_2(self):
def f(x, s):
return np.exp(-x**2 / 2 / s) / np.sqrt(2.*s)
val_1, err_1 = quad(f, -np.inf, np.inf, args=(2,))
val_2, err_2 = quad(f, np.inf, -np.inf, args=(2,))
assert_allclose(val_1, -val_2, atol=max(err_1, err_2))
def test_b_less_than_a_3(self):
def f(x):
return 1.0
val_1, err_1 = quad(f, 0, 1, weight='alg', wvar=(0, 0))
val_2, err_2 = quad(f, 1, 0, weight='alg', wvar=(0, 0))
assert_allclose(val_1, -val_2, atol=max(err_1, err_2))
def test_b_less_than_a_full_output(self):
def f(x):
return 1.0
res_1 = quad(f, 0, 1, weight='alg', wvar=(0, 0), full_output=True)
res_2 = quad(f, 1, 0, weight='alg', wvar=(0, 0), full_output=True)
err = max(res_1[1], res_2[1])
assert_allclose(res_1[0], -res_2[0], atol=err)
def test_double_integral(self):
# 8) Double Integral test
def simpfunc(y, x): # Note order of arguments.
return x+y
a, b = 1.0, 2.0
assert_quad(dblquad(simpfunc, a, b, lambda x: x, lambda x: 2*x),
5/6.0 * (b**3.0-a**3.0))
def test_double_integral2(self):
def func(x0, x1, t0, t1):
return x0 + x1 + t0 + t1
g = lambda x: x
h = lambda x: 2 * x
args = 1, 2
assert_quad(dblquad(func, 1, 2, g, h, args=args),35./6 + 9*.5)
def test_double_integral3(self):
def func(x0, x1):
return x0 + x1 + 1 + 2
assert_quad(dblquad(func, 1, 2, 1, 2),6.)
def test_triple_integral(self):
# 9) Triple Integral test
def simpfunc(z, y, x, t): # Note order of arguments.
return (x+y+z)*t
a, b = 1.0, 2.0
assert_quad(tplquad(simpfunc, a, b,
lambda x: x, lambda x: 2*x,
lambda x, y: x - y, lambda x, y: x + y,
(2.,)),
2*8/3.0 * (b**4.0 - a**4.0))
class TestNQuad(object):
def test_fixed_limits(self):
def func1(x0, x1, x2, x3):
val = (x0**2 + x1*x2 - x3**3 + np.sin(x0) +
(1 if (x0 - 0.2*x3 - 0.5 - 0.25*x1 > 0) else 0))
return val
def opts_basic(*args):
return {'points': [0.2*args[2] + 0.5 + 0.25*args[0]]}
res = nquad(func1, [[0, 1], [-1, 1], [.13, .8], [-.15, 1]],
opts=[opts_basic, {}, {}, {}], full_output=True)
assert_quad(res[:-1], 1.5267454070738635)
assert_(res[-1]['neval'] > 0 and res[-1]['neval'] < 4e5)
def test_variable_limits(self):
scale = .1
def func2(x0, x1, x2, x3, t0, t1):
val = (x0*x1*x3**2 + np.sin(x2) + 1 +
(1 if x0 + t1*x1 - t0 > 0 else 0))
return val
def lim0(x1, x2, x3, t0, t1):
return [scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) - 1,
scale * (x1**2 + x2 + np.cos(x3)*t0*t1 + 1) + 1]
def lim1(x2, x3, t0, t1):
return [scale * (t0*x2 + t1*x3) - 1,
scale * (t0*x2 + t1*x3) + 1]
def lim2(x3, t0, t1):
return [scale * (x3 + t0**2*t1**3) - 1,
scale * (x3 + t0**2*t1**3) + 1]
def lim3(t0, t1):
return [scale * (t0 + t1) - 1, scale * (t0 + t1) + 1]
def opts0(x1, x2, x3, t0, t1):
return {'points': [t0 - t1*x1]}
def opts1(x2, x3, t0, t1):
return {}
def opts2(x3, t0, t1):
return {}
def opts3(t0, t1):
return {}
res = nquad(func2, [lim0, lim1, lim2, lim3], args=(0, 0),
opts=[opts0, opts1, opts2, opts3])
assert_quad(res, 25.066666666666663)
def test_square_separate_ranges_and_opts(self):
def f(y, x):
return 1.0
assert_quad(nquad(f, [[-1, 1], [-1, 1]], opts=[{}, {}]), 4.0)
def test_square_aliased_ranges_and_opts(self):
def f(y, x):
return 1.0
r = [-1, 1]
opt = {}
assert_quad(nquad(f, [r, r], opts=[opt, opt]), 4.0)
def test_square_separate_fn_ranges_and_opts(self):
def f(y, x):
return 1.0
def fn_range0(*args):
return (-1, 1)
def fn_range1(*args):
return (-1, 1)
def fn_opt0(*args):
return {}
def fn_opt1(*args):
return {}
ranges = [fn_range0, fn_range1]
opts = [fn_opt0, fn_opt1]
assert_quad(nquad(f, ranges, opts=opts), 4.0)
def test_square_aliased_fn_ranges_and_opts(self):
def f(y, x):
return 1.0
def fn_range(*args):
return (-1, 1)
def fn_opt(*args):
return {}
ranges = [fn_range, fn_range]
opts = [fn_opt, fn_opt]
assert_quad(nquad(f, ranges, opts=opts), 4.0)
def test_matching_quad(self):
def func(x):
return x**2 + 1
res, reserr = quad(func, 0, 4)
res2, reserr2 = nquad(func, ranges=[[0, 4]])
assert_almost_equal(res, res2)
assert_almost_equal(reserr, reserr2)
def test_matching_dblquad(self):
def func2d(x0, x1):
return x0**2 + x1**3 - x0 * x1 + 1
res, reserr = dblquad(func2d, -2, 2, lambda x: -3, lambda x: 3)
res2, reserr2 = nquad(func2d, [[-3, 3], (-2, 2)])
assert_almost_equal(res, res2)
assert_almost_equal(reserr, reserr2)
def test_matching_tplquad(self):
def func3d(x0, x1, x2, c0, c1):
return x0**2 + c0 * x1**3 - x0 * x1 + 1 + c1 * np.sin(x2)
res = tplquad(func3d, -1, 2, lambda x: -2, lambda x: 2,
lambda x, y: -np.pi, lambda x, y: np.pi,
args=(2, 3))
res2 = nquad(func3d, [[-np.pi, np.pi], [-2, 2], (-1, 2)], args=(2, 3))
assert_almost_equal(res, res2)
def test_dict_as_opts(self):
try:
out = nquad(lambda x, y: x * y, [[0, 1], [0, 1]], opts={'epsrel': 0.0001})
except(TypeError):
assert False
