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