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from __future__ import division, print_function, absolute_import
import warnings
import numpy as np
from numpy import cos, sin, pi
from numpy.testing import TestCase, run_module_suite, assert_equal, \
assert_almost_equal, assert_allclose, assert_
from scipy.integrate import (quadrature, romberg, romb, newton_cotes,
cumtrapz, quad, simps)
from scipy.integrate.quadrature import AccuracyWarning
class TestQuadrature(TestCase):
def quad(self, x, a, b, args):
raise NotImplementedError
def test_quadrature(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return cos(n*x-z*sin(x))/pi
val, err = quadrature(myfunc, 0, pi, (2, 1.8))
table_val = 0.30614353532540296487
assert_almost_equal(val, table_val, decimal=7)
def test_quadrature_rtol(self):
def myfunc(x, n, z): # Bessel function integrand
return 1e90 * cos(n*x-z*sin(x))/pi
val, err = quadrature(myfunc, 0, pi, (2, 1.8), rtol=1e-10)
table_val = 1e90 * 0.30614353532540296487
assert_allclose(val, table_val, rtol=1e-10)
def test_quadrature_miniter(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return cos(n*x-z*sin(x))/pi
table_val = 0.30614353532540296487
for miniter in [5, 52]:
val, err = quadrature(myfunc, 0, pi, (2, 1.8), miniter=miniter)
assert_almost_equal(val, table_val, decimal=7)
assert_(err < 1.0)
def test_quadrature_single_args(self):
def myfunc(x, n):
return 1e90 * cos(n*x-1.8*sin(x))/pi
val, err = quadrature(myfunc, 0, pi, args=2, rtol=1e-10)
table_val = 1e90 * 0.30614353532540296487
assert_allclose(val, table_val, rtol=1e-10)
def test_romberg(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return cos(n*x-z*sin(x))/pi
val = romberg(myfunc, 0, pi, args=(2, 1.8))
table_val = 0.30614353532540296487
assert_almost_equal(val, table_val, decimal=7)
def test_romberg_rtol(self):
# Typical function with two extra arguments:
def myfunc(x, n, z): # Bessel function integrand
return 1e19*cos(n*x-z*sin(x))/pi
val = romberg(myfunc, 0, pi, args=(2, 1.8), rtol=1e-10)
table_val = 1e19*0.30614353532540296487
assert_allclose(val, table_val, rtol=1e-10)
def test_romb(self):
assert_equal(romb(np.arange(17)), 128)
def test_romb_gh_3731(self):
# Check that romb makes maximal use of data points
x = np.arange(2**4+1)
y = np.cos(0.2*x)
val = romb(y)
val2, err = quad(lambda x: np.cos(0.2*x), x.min(), x.max())
assert_allclose(val, val2, rtol=1e-8, atol=0)
# should be equal to romb with 2**k+1 samples
with warnings.catch_warnings():
warnings.filterwarnings('ignore', category=AccuracyWarning)
val3 = romberg(lambda x: np.cos(0.2*x), x.min(), x.max(),
divmax=4)
assert_allclose(val, val3, rtol=1e-12, atol=0)
def test_non_dtype(self):
# Check that we work fine with functions returning float
import math
valmath = romberg(math.sin, 0, 1)
expected_val = 0.45969769413185085
assert_almost_equal(valmath, expected_val, decimal=7)
def test_newton_cotes(self):
"""Test the first few degrees, for evenly spaced points."""
n = 1
wts, errcoff = newton_cotes(n, 1)
assert_equal(wts, n*np.array([0.5, 0.5]))
assert_almost_equal(errcoff, -n**3/12.0)
n = 2
wts, errcoff = newton_cotes(n, 1)
assert_almost_equal(wts, n*np.array([1.0, 4.0, 1.0])/6.0)
assert_almost_equal(errcoff, -n**5/2880.0)
n = 3
wts, errcoff = newton_cotes(n, 1)
assert_almost_equal(wts, n*np.array([1.0, 3.0, 3.0, 1.0])/8.0)
assert_almost_equal(errcoff, -n**5/6480.0)
n = 4
wts, errcoff = newton_cotes(n, 1)
assert_almost_equal(wts, n*np.array([7.0, 32.0, 12.0, 32.0, 7.0])/90.0)
assert_almost_equal(errcoff, -n**7/1935360.0)
def test_newton_cotes2(self):
"""Test newton_cotes with points that are not evenly spaced."""
x = np.array([0.0, 1.5, 2.0])
y = x**2
wts, errcoff = newton_cotes(x)
exact_integral = 8.0/3
numeric_integral = np.dot(wts, y)
assert_almost_equal(numeric_integral, exact_integral)
x = np.array([0.0, 1.4, 2.1, 3.0])
y = x**2
wts, errcoff = newton_cotes(x)
exact_integral = 9.0
numeric_integral = np.dot(wts, y)
assert_almost_equal(numeric_integral, exact_integral)
def test_simps(self):
y = np.arange(17)
assert_equal(simps(y), 128)
assert_equal(simps(y, dx=0.5), 64)
assert_equal(simps(y, x=np.linspace(0, 4, 17)), 32)
y = np.arange(4)
x = 2**y
assert_equal(simps(y, x=x, even='avg'), 13.875)
assert_equal(simps(y, x=x, even='first'), 13.75)
assert_equal(simps(y, x=x, even='last'), 14)
class TestCumtrapz(TestCase):
def test_1d(self):
x = np.linspace(-2, 2, num=5)
y = x
y_int = cumtrapz(y, x, initial=0)
y_expected = [0., -1.5, -2., -1.5, 0.]
assert_allclose(y_int, y_expected)
y_int = cumtrapz(y, x, initial=None)
assert_allclose(y_int, y_expected[1:])
def test_y_nd_x_nd(self):
x = np.arange(3 * 2 * 4).reshape(3, 2, 4)
y = x
y_int = cumtrapz(y, x, initial=0)
y_expected = np.array([[[0., 0.5, 2., 4.5],
[0., 4.5, 10., 16.5]],
[[0., 8.5, 18., 28.5],
[0., 12.5, 26., 40.5]],
[[0., 16.5, 34., 52.5],
[0., 20.5, 42., 64.5]]])
assert_allclose(y_int, y_expected)
# Try with all axes
shapes = [(2, 2, 4), (3, 1, 4), (3, 2, 3)]
for axis, shape in zip([0, 1, 2], shapes):
y_int = cumtrapz(y, x, initial=3.45, axis=axis)
assert_equal(y_int.shape, (3, 2, 4))
y_int = cumtrapz(y, x, initial=None, axis=axis)
assert_equal(y_int.shape, shape)
def test_y_nd_x_1d(self):
y = np.arange(3 * 2 * 4).reshape(3, 2, 4)
x = np.arange(4)**2
# Try with all axes
ys_expected = (
np.array([[[4., 5., 6., 7.],
[8., 9., 10., 11.]],
[[40., 44., 48., 52.],
[56., 60., 64., 68.]]]),
np.array([[[2., 3., 4., 5.]],
[[10., 11., 12., 13.]],
[[18., 19., 20., 21.]]]),
np.array([[[0.5, 5., 17.5],
[4.5, 21., 53.5]],
[[8.5, 37., 89.5],
[12.5, 53., 125.5]],
[[16.5, 69., 161.5],
[20.5, 85., 197.5]]]))
for axis, y_expected in zip([0, 1, 2], ys_expected):
y_int = cumtrapz(y, x=x[:y.shape[axis]], axis=axis, initial=None)
assert_allclose(y_int, y_expected)
def test_x_none(self):
y = np.linspace(-2, 2, num=5)
y_int = cumtrapz(y)
y_expected = [-1.5, -2., -1.5, 0.]
assert_allclose(y_int, y_expected)
y_int = cumtrapz(y, initial=1.23)
y_expected = [1.23, -1.5, -2., -1.5, 0.]
assert_allclose(y_int, y_expected)
y_int = cumtrapz(y, dx=3)
y_expected = [-4.5, -6., -4.5, 0.]
assert_allclose(y_int, y_expected)
y_int = cumtrapz(y, dx=3, initial=1.23)
y_expected = [1.23, -4.5, -6., -4.5, 0.]
assert_allclose(y_int, y_expected)
if __name__ == "__main__":
run_module_suite()
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