test_quadrature.py

import pytest
import numpy as np
from numpy import cos, sin, pi
from numpy.testing import (assert_equal, assert_almost_equal, assert_allclose,
                           assert_, suppress_warnings)

from scipy.integrate import (quadrature, romberg, romb, newton_cotes,
                             cumulative_trapezoid, cumtrapz, trapz, trapezoid,
                             quad, simpson, simps, fixed_quad, AccuracyWarning)
from scipy.integrate._quadrature import _qmc_quad as qmc_quad
from scipy import stats, special as sc


class TestFixedQuad:
    def test_scalar(self):
        n = 4
        expected = 1/(2*n)
        got, _ = fixed_quad(lambda x: x**(2*n - 1), 0, 1, n=n)
        # quadrature exact for this input
        assert_allclose(got, expected, rtol=1e-12)

    def test_vector(self):
        n = 4
        p = np.arange(1, 2*n)
        expected = 1/(p + 1)
        got, _ = fixed_quad(lambda x: x**p[:, None], 0, 1, n=n)
        assert_allclose(got, expected, rtol=1e-12)


class TestQuadrature:
    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 suppress_warnings() as sup:
            sup.filter(AccuracyWarning, "divmax .4. exceeded")
            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_simpson(self):
        y = np.arange(17)
        assert_equal(simpson(y), 128)
        assert_equal(simpson(y, dx=0.5), 64)
        assert_equal(simpson(y, x=np.linspace(0, 4, 17)), 32)

        y = np.arange(4)
        x = 2**y
        assert_equal(simpson(y, x=x, even='avg'), 13.875)
        assert_equal(simpson(y, x=x, even='first'), 13.75)
        assert_equal(simpson(y, x=x, even='last'), 14)

        # Tests for checking base case
        x = np.array([3])
        y = np.power(x, 2)
        assert_equal(simpson(y, x=x, axis=0), 0.0)
        assert_equal(simpson(y, x=x, axis=-1), 0.0)

        x = np.array([3, 3, 3, 3])
        y = np.power(x, 2)
        assert_equal(simpson(y, x=x, axis=0), 0.0)
        assert_equal(simpson(y, x=x, axis=-1), 0.0)

        x = np.array([[1, 2, 4, 8], [1, 2, 4, 8], [1, 2, 4, 8]])
        y = np.power(x, 2)
        zero_axis = [0.0, 0.0, 0.0, 0.0]
        default_axis = [175.75, 175.75, 175.75]
        assert_equal(simpson(y, x=x, axis=0), zero_axis)
        assert_equal(simpson(y, x=x, axis=-1), default_axis)

        x = np.array([[1, 2, 4, 8], [1, 2, 4, 8], [1, 8, 16, 32]])
        y = np.power(x, 2)
        zero_axis = [0.0, 136.0, 1088.0, 8704.0]
        default_axis = [175.75, 175.75, 11292.25]
        assert_equal(simpson(y, x=x, axis=0), zero_axis)
        assert_equal(simpson(y, x=x, axis=-1), default_axis)

    @pytest.mark.parametrize('droplast', [False, True])
    def test_simpson_2d_integer_no_x(self, droplast):
        # The inputs are 2d integer arrays.  The results should be
        # identical to the results when the inputs are floating point.
        y = np.array([[2, 2, 4, 4, 8, 8, -4, 5],
                      [4, 4, 2, -4, 10, 22, -2, 10]])
        if droplast:
            y = y[:, :-1]
        result = simpson(y, axis=-1)
        expected = simpson(np.array(y, dtype=np.float64), axis=-1)
        assert_equal(result, expected)

    def test_simps(self):
        # Basic coverage test for the alias
        y = np.arange(4)
        x = 2**y
        assert_equal(simpson(y, x=x, dx=0.5, even='first'),
                     simps(y, x=x, dx=0.5, even='first'))


class TestCumulative_trapezoid:
    def test_1d(self):
        x = np.linspace(-2, 2, num=5)
        y = x
        y_int = cumulative_trapezoid(y, x, initial=0)
        y_expected = [0., -1.5, -2., -1.5, 0.]
        assert_allclose(y_int, y_expected)

        y_int = cumulative_trapezoid(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 = cumulative_trapezoid(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 = cumulative_trapezoid(y, x, initial=3.45, axis=axis)
            assert_equal(y_int.shape, (3, 2, 4))
            y_int = cumulative_trapezoid(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 = cumulative_trapezoid(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 = cumulative_trapezoid(y)
        y_expected = [-1.5, -2., -1.5, 0.]
        assert_allclose(y_int, y_expected)

        y_int = cumulative_trapezoid(y, initial=1.23)
        y_expected = [1.23, -1.5, -2., -1.5, 0.]
        assert_allclose(y_int, y_expected)

        y_int = cumulative_trapezoid(y, dx=3)
        y_expected = [-4.5, -6., -4.5, 0.]
        assert_allclose(y_int, y_expected)

        y_int = cumulative_trapezoid(y, dx=3, initial=1.23)
        y_expected = [1.23, -4.5, -6., -4.5, 0.]
        assert_allclose(y_int, y_expected)

    def test_cumtrapz(self):
        # Basic coverage test for the alias
        x = np.arange(3 * 2 * 4).reshape(3, 2, 4)
        y = x
        assert_allclose(cumulative_trapezoid(y, x, dx=0.5, axis=0, initial=0),
                        cumtrapz(y, x, dx=0.5, axis=0, initial=0),
                        rtol=1e-14)


class TestTrapezoid():
    """This function is tested in NumPy more extensive, just do some
    basic due diligence here."""
    def test_trapezoid(self):
        y = np.arange(17)
        assert_equal(trapezoid(y), 128)
        assert_equal(trapezoid(y, dx=0.5), 64)
        assert_equal(trapezoid(y, x=np.linspace(0, 4, 17)), 32)

        y = np.arange(4)
        x = 2**y
        assert_equal(trapezoid(y, x=x, dx=0.1), 13.5)

    def test_trapz(self):
        # Basic coverage test for the alias
        y = np.arange(4)
        x = 2**y
        assert_equal(trapezoid(y, x=x, dx=0.5, axis=0),
                     trapz(y, x=x, dx=0.5, axis=0))


class TestQMCQuad():
    def test_input_validation(self):
        message = "`func` must be callable."
        with pytest.raises(TypeError, match=message):
            qmc_quad("a duck", [0, 0], [1, 1])

        message = "`func` must evaluate the integrand at points..."
        with pytest.raises(ValueError, match=message):
            qmc_quad(lambda: 1, [0, 0], [1, 1])

        def func(x):
            assert x.ndim == 1
            return np.sum(x)
        message = "Exception encountered when attempting vectorized call..."
        with pytest.warns(UserWarning, match=message):
            qmc_quad(func, [0, 0], [1, 1])

        message = "`n_points` must be an integer."
        with pytest.raises(TypeError, match=message):
            qmc_quad(lambda x: 1, [0, 0], [1, 1], n_points=1024.5)

        message = "`n_estimates` must be an integer."
        with pytest.raises(TypeError, match=message):
            qmc_quad(lambda x: 1, [0, 0], [1, 1], n_estimates=8.5)

        message = "`qrng` must be an instance of scipy.stats.qmc.QMCEngine."
        with pytest.raises(TypeError, match=message):
            qmc_quad(lambda x: 1, [0, 0], [1, 1], qrng="a duck")

        message = "`qrng` must be initialized with dimensionality equal to "
        with pytest.raises(ValueError, match=message):
            qmc_quad(lambda x: 1, [0, 0], [1, 1], qrng=stats.qmc.Sobol(1))

        message = r"`log` must be boolean \(`True` or `False`\)."
        with pytest.raises(TypeError, match=message):
            qmc_quad(lambda x: 1, [0, 0], [1, 1], log=10)

    def basic_test(self, n_points=2**8, n_estimates=8, signs=np.ones(2)):

        ndim = 2
        mean = np.zeros(ndim)
        cov = np.eye(ndim)

        def func(x):
            return stats.multivariate_normal.pdf(x, mean, cov)

        rng = np.random.default_rng(2879434385674690281)
        qrng = stats.qmc.Sobol(ndim, seed=rng)
        a = np.zeros(ndim)
        b = np.ones(ndim) * signs
        res = qmc_quad(func, a, b, n_points=n_points,
                       n_estimates=n_estimates, args=(mean, cov), qrng=qrng)
        ref = stats.multivariate_normal.cdf(b, mean, cov, lower_limit=a)
        atol = sc.stdtrit(n_estimates-1, 0.995) * res.standard_error  # 99% CI
        assert_allclose(res.integral, ref, atol=atol)
        assert np.prod(signs)*res.integral > 0

        rng = np.random.default_rng(2879434385674690281)
        qrng = stats.qmc.Sobol(ndim, seed=rng)
        logres = qmc_quad(lambda *args: np.log(func(*args)), a, b,
                          n_points=n_points, n_estimates=n_estimates,
                          args=(mean, cov), log=True, qrng=qrng)
        assert_allclose(np.exp(logres.integral), res.integral)
        assert np.imag(logres.integral) == (np.pi if np.prod(signs) < 0 else 0)

    @pytest.mark.parametrize("n_points", [2**8, 2**12])
    @pytest.mark.parametrize("n_estimates", [8, 16])
    def test_basic(self, n_points, n_estimates):
        self.basic_test(n_points, n_estimates)

    @pytest.mark.parametrize("signs", [[1, 1], [-1, -1], [-1, 1], [1, -1]])
    def test_sign(self, signs):
        self.basic_test(signs=signs)

    @pytest.mark.parametrize("log", [False, True])
    def test_zero(self, log):
        message = "A lower limit was equal to an upper limit, so"
        with pytest.warns(UserWarning, match=message):
            res = qmc_quad(lambda x: 1, [0, 0], [0, 1], log=log)
        assert res.integral == (-np.inf if log else 0)
        assert res.standard_error == 0

    def test_flexible_input(self):
        # check that qrng is not required
        # also checks that for 1d problems, a and b can be scalars
        def func(x):
            return stats.norm.pdf(x, scale=2)

        res = qmc_quad(func, 0, 1)
        ref = stats.norm.cdf(1, scale=2) - stats.norm.cdf(0, scale=2)
        assert_allclose(res.integral, ref, 1e-2)