676 lines
27 KiB
Python
676 lines
27 KiB
Python
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 scipy.integrate import quad, dblquad, tplquad, nquad
|
|
from scipy.special import erf, erfc
|
|
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, error_tolerance=1.5e-8):
|
|
value, err = value_and_err
|
|
assert_allclose(value, tabled_value, atol=err, rtol=0)
|
|
if error_tolerance is not None:
|
|
assert_array_less(err, error_tolerance)
|
|
|
|
|
|
def get_clib_test_routine(name, restype, *argtypes):
|
|
ptr = getattr(clib_test, name)
|
|
return ctypes.cast(ptr, ctypes.CFUNCTYPE(restype, *argtypes))
|
|
|
|
|
|
class TestCtypesQuad:
|
|
def setup_method(self):
|
|
if sys.platform == 'win32':
|
|
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
|
|
pytest.skip("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:
|
|
pytest.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:
|
|
pytest.raises(ValueError, quad, func, 0, pi)
|
|
|
|
|
|
class TestMultivariateCtypesQuad:
|
|
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:
|
|
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, error_tolerance=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.)
|
|
|
|
@pytest.mark.parametrize(
|
|
"x_lower, x_upper, y_lower, y_upper, expected",
|
|
[
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-inf, 0] for all n.
|
|
(-np.inf, 0, -np.inf, 0, np.pi / 4),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-inf, -1] for each n (one at a time).
|
|
(-np.inf, -1, -np.inf, 0, np.pi / 4 * erfc(1)),
|
|
(-np.inf, 0, -np.inf, -1, np.pi / 4 * erfc(1)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-inf, -1] for all n.
|
|
(-np.inf, -1, -np.inf, -1, np.pi / 4 * (erfc(1) ** 2)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-inf, 1] for each n (one at a time).
|
|
(-np.inf, 1, -np.inf, 0, np.pi / 4 * (erf(1) + 1)),
|
|
(-np.inf, 0, -np.inf, 1, np.pi / 4 * (erf(1) + 1)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-inf, 1] for all n.
|
|
(-np.inf, 1, -np.inf, 1, np.pi / 4 * ((erf(1) + 1) ** 2)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain Dx = [-inf, -1] and Dy = [-inf, 1].
|
|
(-np.inf, -1, -np.inf, 1, np.pi / 4 * ((erf(1) + 1) * erfc(1))),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain Dx = [-inf, 1] and Dy = [-inf, -1].
|
|
(-np.inf, 1, -np.inf, -1, np.pi / 4 * ((erf(1) + 1) * erfc(1))),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [0, inf] for all n.
|
|
(0, np.inf, 0, np.inf, np.pi / 4),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [1, inf] for each n (one at a time).
|
|
(1, np.inf, 0, np.inf, np.pi / 4 * erfc(1)),
|
|
(0, np.inf, 1, np.inf, np.pi / 4 * erfc(1)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [1, inf] for all n.
|
|
(1, np.inf, 1, np.inf, np.pi / 4 * (erfc(1) ** 2)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-1, inf] for each n (one at a time).
|
|
(-1, np.inf, 0, np.inf, np.pi / 4 * (erf(1) + 1)),
|
|
(0, np.inf, -1, np.inf, np.pi / 4 * (erf(1) + 1)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-1, inf] for all n.
|
|
(-1, np.inf, -1, np.inf, np.pi / 4 * ((erf(1) + 1) ** 2)),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain Dx = [-1, inf] and Dy = [1, inf].
|
|
(-1, np.inf, 1, np.inf, np.pi / 4 * ((erf(1) + 1) * erfc(1))),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain Dx = [1, inf] and Dy = [-1, inf].
|
|
(1, np.inf, -1, np.inf, np.pi / 4 * ((erf(1) + 1) * erfc(1))),
|
|
# Multiple integration of a function in n = 2 variables: f(x, y, z)
|
|
# over domain D = [-inf, inf] for all n.
|
|
(-np.inf, np.inf, -np.inf, np.inf, np.pi)
|
|
]
|
|
)
|
|
def test_double_integral_improper(
|
|
self, x_lower, x_upper, y_lower, y_upper, expected
|
|
):
|
|
# The Gaussian Integral.
|
|
def f(x, y):
|
|
return np.exp(-x ** 2 - y ** 2)
|
|
|
|
assert_quad(
|
|
dblquad(f, x_lower, x_upper, y_lower, y_upper),
|
|
expected,
|
|
error_tolerance=3e-8
|
|
)
|
|
|
|
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))
|
|
|
|
@pytest.mark.parametrize(
|
|
"x_lower, x_upper, y_lower, y_upper, z_lower, z_upper, expected",
|
|
[
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, 0] for all n.
|
|
(-np.inf, 0, -np.inf, 0, -np.inf, 0, (np.pi ** (3 / 2)) / 8),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, -1] for each n (one at a time).
|
|
(-np.inf, -1, -np.inf, 0, -np.inf, 0,
|
|
(np.pi ** (3 / 2)) / 8 * erfc(1)),
|
|
(-np.inf, 0, -np.inf, -1, -np.inf, 0,
|
|
(np.pi ** (3 / 2)) / 8 * erfc(1)),
|
|
(-np.inf, 0, -np.inf, 0, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * erfc(1)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, -1] for each n (two at a time).
|
|
(-np.inf, -1, -np.inf, -1, -np.inf, 0,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)),
|
|
(-np.inf, -1, -np.inf, 0, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)),
|
|
(-np.inf, 0, -np.inf, -1, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, -1] for all n.
|
|
(-np.inf, -1, -np.inf, -1, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 3)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = [-inf, -1] and Dy = Dz = [-inf, 1].
|
|
(-np.inf, -1, -np.inf, 1, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dy = [-inf, -1] and Dz = [-inf, 1].
|
|
(-np.inf, -1, -np.inf, -1, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dz = [-inf, -1] and Dy = [-inf, 1].
|
|
(-np.inf, -1, -np.inf, 1, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = [-inf, 1] and Dy = Dz = [-inf, -1].
|
|
(-np.inf, 1, -np.inf, -1, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dy = [-inf, 1] and Dz = [-inf, -1].
|
|
(-np.inf, 1, -np.inf, 1, -np.inf, -1,
|
|
(np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dz = [-inf, 1] and Dy = [-inf, -1].
|
|
(-np.inf, 1, -np.inf, -1, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, 1] for each n (one at a time).
|
|
(-np.inf, 1, -np.inf, 0, -np.inf, 0,
|
|
(np.pi ** (3 / 2)) / 8 * (erf(1) + 1)),
|
|
(-np.inf, 0, -np.inf, 1, -np.inf, 0,
|
|
(np.pi ** (3 / 2)) / 8 * (erf(1) + 1)),
|
|
(-np.inf, 0, -np.inf, 0, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * (erf(1) + 1)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, 1] for each n (two at a time).
|
|
(-np.inf, 1, -np.inf, 1, -np.inf, 0,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)),
|
|
(-np.inf, 1, -np.inf, 0, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)),
|
|
(-np.inf, 0, -np.inf, 1, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, 1] for all n.
|
|
(-np.inf, 1, -np.inf, 1, -np.inf, 1,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 3)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [0, inf] for all n.
|
|
(0, np.inf, 0, np.inf, 0, np.inf, (np.pi ** (3 / 2)) / 8),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [1, inf] for each n (one at a time).
|
|
(1, np.inf, 0, np.inf, 0, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * erfc(1)),
|
|
(0, np.inf, 1, np.inf, 0, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * erfc(1)),
|
|
(0, np.inf, 0, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * erfc(1)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [1, inf] for each n (two at a time).
|
|
(1, np.inf, 1, np.inf, 0, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)),
|
|
(1, np.inf, 0, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)),
|
|
(0, np.inf, 1, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 2)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [1, inf] for all n.
|
|
(1, np.inf, 1, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erfc(1) ** 3)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-1, inf] for each n (one at a time).
|
|
(-1, np.inf, 0, np.inf, 0, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erf(1) + 1)),
|
|
(0, np.inf, -1, np.inf, 0, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erf(1) + 1)),
|
|
(0, np.inf, 0, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (erf(1) + 1)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-1, inf] for each n (two at a time).
|
|
(-1, np.inf, -1, np.inf, 0, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)),
|
|
(-1, np.inf, 0, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)),
|
|
(0, np.inf, -1, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 2)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-1, inf] for all n.
|
|
(-1, np.inf, -1, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) ** 3)),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = [1, inf] and Dy = Dz = [-1, inf].
|
|
(1, np.inf, -1, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dy = [1, inf] and Dz = [-1, inf].
|
|
(1, np.inf, 1, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dz = [1, inf] and Dy = [-1, inf].
|
|
(1, np.inf, -1, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = [-1, inf] and Dy = Dz = [1, inf].
|
|
(-1, np.inf, 1, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * ((erf(1) + 1) * (erfc(1) ** 2))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dy = [-1, inf] and Dz = [1, inf].
|
|
(-1, np.inf, -1, np.inf, 1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain Dx = Dz = [-1, inf] and Dy = [1, inf].
|
|
(-1, np.inf, 1, np.inf, -1, np.inf,
|
|
(np.pi ** (3 / 2)) / 8 * (((erf(1) + 1) ** 2) * erfc(1))),
|
|
# Multiple integration of a function in n = 3 variables: f(x, y, z)
|
|
# over domain D = [-inf, inf] for all n.
|
|
(-np.inf, np.inf, -np.inf, np.inf, -np.inf, np.inf,
|
|
np.pi ** (3 / 2)),
|
|
],
|
|
)
|
|
def test_triple_integral_improper(
|
|
self,
|
|
x_lower,
|
|
x_upper,
|
|
y_lower,
|
|
y_upper,
|
|
z_lower,
|
|
z_upper,
|
|
expected
|
|
):
|
|
# The Gaussian Integral.
|
|
def f(x, y, z):
|
|
return np.exp(-x ** 2 - y ** 2 - z ** 2)
|
|
|
|
assert_quad(
|
|
tplquad(f, x_lower, x_upper, y_lower, y_upper, z_lower, z_upper),
|
|
expected,
|
|
error_tolerance=6e-8
|
|
)
|
|
|
|
def test_complex(self):
|
|
def tfunc(x):
|
|
return np.exp(1j*x)
|
|
|
|
assert np.allclose(
|
|
quad(tfunc, 0, np.pi/2, complex_func=True)[0],
|
|
1+1j)
|
|
|
|
# We consider a divergent case in order to force quadpack
|
|
# to return an error message. The output is compared
|
|
# against what is returned by explicit integration
|
|
# of the parts.
|
|
kwargs = {'a': 0, 'b': np.inf, 'full_output': True,
|
|
'weight': 'cos', 'wvar': 1}
|
|
res_c = quad(tfunc, complex_func=True, **kwargs)
|
|
res_r = quad(lambda x: np.real(np.exp(1j*x)),
|
|
complex_func=False,
|
|
**kwargs)
|
|
res_i = quad(lambda x: np.imag(np.exp(1j*x)),
|
|
complex_func=False,
|
|
**kwargs)
|
|
|
|
np.testing.assert_equal(res_c[0], res_r[0] + 1j*res_i[0])
|
|
np.testing.assert_equal(res_c[1], res_r[1] + 1j*res_i[1])
|
|
|
|
assert len(res_c[2]['real']) == len(res_r[2:]) == 3
|
|
assert res_c[2]['real'][2] == res_r[4]
|
|
assert res_c[2]['real'][1] == res_r[3]
|
|
assert res_c[2]['real'][0]['lst'] == res_r[2]['lst']
|
|
|
|
assert len(res_c[2]['imag']) == len(res_i[2:]) == 1
|
|
assert res_c[2]['imag'][0]['lst'] == res_i[2]['lst']
|
|
|
|
|
|
class TestNQuad:
|
|
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:
|
|
nquad(lambda x, y: x * y, [[0, 1], [0, 1]], opts={'epsrel': 0.0001})
|
|
except TypeError:
|
|
assert False
|
|
|