554 lines
15 KiB
Python
554 lines
15 KiB
Python
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from __future__ import division, print_function, absolute_import
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import sys
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try:
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from StringIO import StringIO
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except ImportError:
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from io import StringIO
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import numpy as np
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from numpy.testing import (assert_, assert_array_equal, assert_allclose,
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assert_equal)
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from pytest import raises as assert_raises
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from scipy.sparse import coo_matrix
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from scipy.special import erf
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from scipy.integrate._bvp import (modify_mesh, estimate_fun_jac,
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estimate_bc_jac, compute_jac_indices,
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construct_global_jac, solve_bvp)
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def exp_fun(x, y):
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return np.vstack((y[1], y[0]))
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def exp_fun_jac(x, y):
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df_dy = np.empty((2, 2, x.shape[0]))
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df_dy[0, 0] = 0
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df_dy[0, 1] = 1
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df_dy[1, 0] = 1
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df_dy[1, 1] = 0
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return df_dy
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def exp_bc(ya, yb):
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return np.hstack((ya[0] - 1, yb[0]))
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def exp_bc_complex(ya, yb):
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return np.hstack((ya[0] - 1 - 1j, yb[0]))
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def exp_bc_jac(ya, yb):
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dbc_dya = np.array([
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[1, 0],
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[0, 0]
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])
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dbc_dyb = np.array([
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[0, 0],
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[1, 0]
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])
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return dbc_dya, dbc_dyb
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def exp_sol(x):
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return (np.exp(-x) - np.exp(x - 2)) / (1 - np.exp(-2))
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def sl_fun(x, y, p):
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return np.vstack((y[1], -p[0]**2 * y[0]))
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def sl_fun_jac(x, y, p):
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n, m = y.shape
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df_dy = np.empty((n, 2, m))
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df_dy[0, 0] = 0
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df_dy[0, 1] = 1
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df_dy[1, 0] = -p[0]**2
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df_dy[1, 1] = 0
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df_dp = np.empty((n, 1, m))
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df_dp[0, 0] = 0
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df_dp[1, 0] = -2 * p[0] * y[0]
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return df_dy, df_dp
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def sl_bc(ya, yb, p):
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return np.hstack((ya[0], yb[0], ya[1] - p[0]))
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def sl_bc_jac(ya, yb, p):
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dbc_dya = np.zeros((3, 2))
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dbc_dya[0, 0] = 1
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dbc_dya[2, 1] = 1
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dbc_dyb = np.zeros((3, 2))
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dbc_dyb[1, 0] = 1
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dbc_dp = np.zeros((3, 1))
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dbc_dp[2, 0] = -1
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return dbc_dya, dbc_dyb, dbc_dp
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def sl_sol(x, p):
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return np.sin(p[0] * x)
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def emden_fun(x, y):
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return np.vstack((y[1], -y[0]**5))
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def emden_fun_jac(x, y):
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df_dy = np.empty((2, 2, x.shape[0]))
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df_dy[0, 0] = 0
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df_dy[0, 1] = 1
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df_dy[1, 0] = -5 * y[0]**4
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df_dy[1, 1] = 0
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return df_dy
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def emden_bc(ya, yb):
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return np.array([ya[1], yb[0] - (3/4)**0.5])
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def emden_bc_jac(ya, yb):
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dbc_dya = np.array([
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[0, 1],
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[0, 0]
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])
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dbc_dyb = np.array([
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[0, 0],
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[1, 0]
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])
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return dbc_dya, dbc_dyb
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def emden_sol(x):
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return (1 + x**2/3)**-0.5
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def undefined_fun(x, y):
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return np.zeros_like(y)
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def undefined_bc(ya, yb):
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return np.array([ya[0], yb[0] - 1])
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def big_fun(x, y):
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f = np.zeros_like(y)
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f[::2] = y[1::2]
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return f
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def big_bc(ya, yb):
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return np.hstack((ya[::2], yb[::2] - 1))
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def big_sol(x, n):
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y = np.ones((2 * n, x.size))
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y[::2] = x
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return x
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def shock_fun(x, y):
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eps = 1e-3
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return np.vstack((
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y[1],
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-(x * y[1] + eps * np.pi**2 * np.cos(np.pi * x) +
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np.pi * x * np.sin(np.pi * x)) / eps
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))
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def shock_bc(ya, yb):
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return np.array([ya[0] + 2, yb[0]])
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def shock_sol(x):
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eps = 1e-3
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k = np.sqrt(2 * eps)
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return np.cos(np.pi * x) + erf(x / k) / erf(1 / k)
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def test_modify_mesh():
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x = np.array([0, 1, 3, 9], dtype=float)
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x_new = modify_mesh(x, np.array([0]), np.array([2]))
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assert_array_equal(x_new, np.array([0, 0.5, 1, 3, 5, 7, 9]))
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x = np.array([-6, -3, 0, 3, 6], dtype=float)
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x_new = modify_mesh(x, np.array([1], dtype=int), np.array([0, 2, 3]))
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assert_array_equal(x_new, [-6, -5, -4, -3, -1.5, 0, 1, 2, 3, 4, 5, 6])
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def test_compute_fun_jac():
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x = np.linspace(0, 1, 5)
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y = np.empty((2, x.shape[0]))
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y[0] = 0.01
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y[1] = 0.02
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p = np.array([])
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df_dy, df_dp = estimate_fun_jac(lambda x, y, p: exp_fun(x, y), x, y, p)
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df_dy_an = exp_fun_jac(x, y)
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assert_allclose(df_dy, df_dy_an)
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assert_(df_dp is None)
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x = np.linspace(0, np.pi, 5)
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y = np.empty((2, x.shape[0]))
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y[0] = np.sin(x)
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y[1] = np.cos(x)
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p = np.array([1.0])
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df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p)
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df_dy_an, df_dp_an = sl_fun_jac(x, y, p)
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assert_allclose(df_dy, df_dy_an)
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assert_allclose(df_dp, df_dp_an)
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x = np.linspace(0, 1, 10)
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y = np.empty((2, x.shape[0]))
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y[0] = (3/4)**0.5
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y[1] = 1e-4
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p = np.array([])
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df_dy, df_dp = estimate_fun_jac(lambda x, y, p: emden_fun(x, y), x, y, p)
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df_dy_an = emden_fun_jac(x, y)
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assert_allclose(df_dy, df_dy_an)
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assert_(df_dp is None)
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def test_compute_bc_jac():
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ya = np.array([-1.0, 2])
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yb = np.array([0.5, 3])
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p = np.array([])
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dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(
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lambda ya, yb, p: exp_bc(ya, yb), ya, yb, p)
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dbc_dya_an, dbc_dyb_an = exp_bc_jac(ya, yb)
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assert_allclose(dbc_dya, dbc_dya_an)
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assert_allclose(dbc_dyb, dbc_dyb_an)
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assert_(dbc_dp is None)
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ya = np.array([0.0, 1])
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yb = np.array([0.0, -1])
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p = np.array([0.5])
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dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(sl_bc, ya, yb, p)
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dbc_dya_an, dbc_dyb_an, dbc_dp_an = sl_bc_jac(ya, yb, p)
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assert_allclose(dbc_dya, dbc_dya_an)
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assert_allclose(dbc_dyb, dbc_dyb_an)
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assert_allclose(dbc_dp, dbc_dp_an)
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ya = np.array([0.5, 100])
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yb = np.array([-1000, 10.5])
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p = np.array([])
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dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(
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lambda ya, yb, p: emden_bc(ya, yb), ya, yb, p)
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dbc_dya_an, dbc_dyb_an = emden_bc_jac(ya, yb)
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assert_allclose(dbc_dya, dbc_dya_an)
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assert_allclose(dbc_dyb, dbc_dyb_an)
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assert_(dbc_dp is None)
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def test_compute_jac_indices():
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n = 2
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m = 4
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k = 2
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i, j = compute_jac_indices(n, m, k)
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s = coo_matrix((np.ones_like(i), (i, j))).toarray()
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s_true = np.array([
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[1, 1, 1, 1, 0, 0, 0, 0, 1, 1],
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[1, 1, 1, 1, 0, 0, 0, 0, 1, 1],
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[0, 0, 1, 1, 1, 1, 0, 0, 1, 1],
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[0, 0, 1, 1, 1, 1, 0, 0, 1, 1],
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[0, 0, 0, 0, 1, 1, 1, 1, 1, 1],
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[0, 0, 0, 0, 1, 1, 1, 1, 1, 1],
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[1, 1, 0, 0, 0, 0, 1, 1, 1, 1],
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[1, 1, 0, 0, 0, 0, 1, 1, 1, 1],
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[1, 1, 0, 0, 0, 0, 1, 1, 1, 1],
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[1, 1, 0, 0, 0, 0, 1, 1, 1, 1],
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])
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assert_array_equal(s, s_true)
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def test_compute_global_jac():
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n = 2
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m = 5
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k = 1
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i_jac, j_jac = compute_jac_indices(2, 5, 1)
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x = np.linspace(0, 1, 5)
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h = np.diff(x)
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y = np.vstack((np.sin(np.pi * x), np.pi * np.cos(np.pi * x)))
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p = np.array([3.0])
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f = sl_fun(x, y, p)
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x_middle = x[:-1] + 0.5 * h
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y_middle = 0.5 * (y[:, :-1] + y[:, 1:]) - h/8 * (f[:, 1:] - f[:, :-1])
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df_dy, df_dp = sl_fun_jac(x, y, p)
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df_dy_middle, df_dp_middle = sl_fun_jac(x_middle, y_middle, p)
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dbc_dya, dbc_dyb, dbc_dp = sl_bc_jac(y[:, 0], y[:, -1], p)
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J = construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle,
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df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp)
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J = J.toarray()
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def J_block(h, p):
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return np.array([
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[h**2*p**2/12 - 1, -0.5*h, -h**2*p**2/12 + 1, -0.5*h],
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[0.5*h*p**2, h**2*p**2/12 - 1, 0.5*h*p**2, 1 - h**2*p**2/12]
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])
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J_true = np.zeros((m * n + k, m * n + k))
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for i in range(m - 1):
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J_true[i * n: (i + 1) * n, i * n: (i + 2) * n] = J_block(h[i], p)
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J_true[:(m - 1) * n:2, -1] = p * h**2/6 * (y[0, :-1] - y[0, 1:])
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J_true[1:(m - 1) * n:2, -1] = p * (h * (y[0, :-1] + y[0, 1:]) +
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h**2/6 * (y[1, :-1] - y[1, 1:]))
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J_true[8, 0] = 1
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J_true[9, 8] = 1
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J_true[10, 1] = 1
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J_true[10, 10] = -1
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assert_allclose(J, J_true, rtol=1e-10)
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df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p)
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df_dy_middle, df_dp_middle = estimate_fun_jac(sl_fun, x_middle, y_middle, p)
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dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(sl_bc, y[:, 0], y[:, -1], p)
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J = construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle,
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df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp)
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J = J.toarray()
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assert_allclose(J, J_true, rtol=1e-8, atol=1e-9)
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def test_parameter_validation():
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x = [0, 1, 0.5]
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y = np.zeros((2, 3))
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assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y)
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x = np.linspace(0, 1, 5)
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y = np.zeros((2, 4))
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assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y)
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fun = lambda x, y, p: exp_fun(x, y)
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bc = lambda ya, yb, p: exp_bc(ya, yb)
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y = np.zeros((2, x.shape[0]))
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assert_raises(ValueError, solve_bvp, fun, bc, x, y, p=[1])
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def wrong_shape_fun(x, y):
|
||
|
|
return np.zeros(3)
|
||
|
|
|
||
|
|
assert_raises(ValueError, solve_bvp, wrong_shape_fun, bc, x, y)
|
||
|
|
|
||
|
|
S = np.array([[0, 0]])
|
||
|
|
assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y, S=S)
|
||
|
|
|
||
|
|
|
||
|
|
def test_no_params():
|
||
|
|
x = np.linspace(0, 1, 5)
|
||
|
|
x_test = np.linspace(0, 1, 100)
|
||
|
|
y = np.zeros((2, x.shape[0]))
|
||
|
|
for fun_jac in [None, exp_fun_jac]:
|
||
|
|
for bc_jac in [None, exp_bc_jac]:
|
||
|
|
sol = solve_bvp(exp_fun, exp_bc, x, y, fun_jac=fun_jac,
|
||
|
|
bc_jac=bc_jac)
|
||
|
|
|
||
|
|
assert_equal(sol.status, 0)
|
||
|
|
assert_(sol.success)
|
||
|
|
|
||
|
|
assert_equal(sol.x.size, 5)
|
||
|
|
|
||
|
|
sol_test = sol.sol(x_test)
|
||
|
|
|
||
|
|
assert_allclose(sol_test[0], exp_sol(x_test), atol=1e-5)
|
||
|
|
|
||
|
|
f_test = exp_fun(x_test, sol_test)
|
||
|
|
r = sol.sol(x_test, 1) - f_test
|
||
|
|
rel_res = r / (1 + np.abs(f_test))
|
||
|
|
norm_res = np.sum(rel_res**2, axis=0)**0.5
|
||
|
|
assert_(np.all(norm_res < 1e-3))
|
||
|
|
|
||
|
|
assert_(np.all(sol.rms_residuals < 1e-3))
|
||
|
|
assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10)
|
||
|
|
assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_with_params():
|
||
|
|
x = np.linspace(0, np.pi, 5)
|
||
|
|
x_test = np.linspace(0, np.pi, 100)
|
||
|
|
y = np.ones((2, x.shape[0]))
|
||
|
|
|
||
|
|
for fun_jac in [None, sl_fun_jac]:
|
||
|
|
for bc_jac in [None, sl_bc_jac]:
|
||
|
|
sol = solve_bvp(sl_fun, sl_bc, x, y, p=[0.5], fun_jac=fun_jac,
|
||
|
|
bc_jac=bc_jac)
|
||
|
|
|
||
|
|
assert_equal(sol.status, 0)
|
||
|
|
assert_(sol.success)
|
||
|
|
|
||
|
|
assert_(sol.x.size < 10)
|
||
|
|
|
||
|
|
assert_allclose(sol.p, [1], rtol=1e-4)
|
||
|
|
|
||
|
|
sol_test = sol.sol(x_test)
|
||
|
|
|
||
|
|
assert_allclose(sol_test[0], sl_sol(x_test, [1]),
|
||
|
|
rtol=1e-4, atol=1e-4)
|
||
|
|
|
||
|
|
f_test = sl_fun(x_test, sol_test, [1])
|
||
|
|
r = sol.sol(x_test, 1) - f_test
|
||
|
|
rel_res = r / (1 + np.abs(f_test))
|
||
|
|
norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5
|
||
|
|
assert_(np.all(norm_res < 1e-3))
|
||
|
|
|
||
|
|
assert_(np.all(sol.rms_residuals < 1e-3))
|
||
|
|
assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10)
|
||
|
|
assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_singular_term():
|
||
|
|
x = np.linspace(0, 1, 10)
|
||
|
|
x_test = np.linspace(0.05, 1, 100)
|
||
|
|
y = np.empty((2, 10))
|
||
|
|
y[0] = (3/4)**0.5
|
||
|
|
y[1] = 1e-4
|
||
|
|
S = np.array([[0, 0], [0, -2]])
|
||
|
|
|
||
|
|
for fun_jac in [None, emden_fun_jac]:
|
||
|
|
for bc_jac in [None, emden_bc_jac]:
|
||
|
|
sol = solve_bvp(emden_fun, emden_bc, x, y, S=S, fun_jac=fun_jac,
|
||
|
|
bc_jac=bc_jac)
|
||
|
|
|
||
|
|
assert_equal(sol.status, 0)
|
||
|
|
assert_(sol.success)
|
||
|
|
|
||
|
|
assert_equal(sol.x.size, 10)
|
||
|
|
|
||
|
|
sol_test = sol.sol(x_test)
|
||
|
|
assert_allclose(sol_test[0], emden_sol(x_test), atol=1e-5)
|
||
|
|
|
||
|
|
f_test = emden_fun(x_test, sol_test) + S.dot(sol_test) / x_test
|
||
|
|
r = sol.sol(x_test, 1) - f_test
|
||
|
|
rel_res = r / (1 + np.abs(f_test))
|
||
|
|
norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5
|
||
|
|
|
||
|
|
assert_(np.all(norm_res < 1e-3))
|
||
|
|
assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10)
|
||
|
|
assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_complex():
|
||
|
|
# The test is essentially the same as test_no_params, but boundary
|
||
|
|
# conditions are turned into complex.
|
||
|
|
x = np.linspace(0, 1, 5)
|
||
|
|
x_test = np.linspace(0, 1, 100)
|
||
|
|
y = np.zeros((2, x.shape[0]), dtype=complex)
|
||
|
|
for fun_jac in [None, exp_fun_jac]:
|
||
|
|
for bc_jac in [None, exp_bc_jac]:
|
||
|
|
sol = solve_bvp(exp_fun, exp_bc_complex, x, y, fun_jac=fun_jac,
|
||
|
|
bc_jac=bc_jac)
|
||
|
|
|
||
|
|
assert_equal(sol.status, 0)
|
||
|
|
assert_(sol.success)
|
||
|
|
|
||
|
|
sol_test = sol.sol(x_test)
|
||
|
|
|
||
|
|
assert_allclose(sol_test[0].real, exp_sol(x_test), atol=1e-5)
|
||
|
|
assert_allclose(sol_test[0].imag, exp_sol(x_test), atol=1e-5)
|
||
|
|
|
||
|
|
f_test = exp_fun(x_test, sol_test)
|
||
|
|
r = sol.sol(x_test, 1) - f_test
|
||
|
|
rel_res = r / (1 + np.abs(f_test))
|
||
|
|
norm_res = np.sum(np.real(rel_res * np.conj(rel_res)),
|
||
|
|
axis=0) ** 0.5
|
||
|
|
assert_(np.all(norm_res < 1e-3))
|
||
|
|
|
||
|
|
assert_(np.all(sol.rms_residuals < 1e-3))
|
||
|
|
assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10)
|
||
|
|
assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_failures():
|
||
|
|
x = np.linspace(0, 1, 2)
|
||
|
|
y = np.zeros((2, x.size))
|
||
|
|
res = solve_bvp(exp_fun, exp_bc, x, y, tol=1e-5, max_nodes=5)
|
||
|
|
assert_equal(res.status, 1)
|
||
|
|
assert_(not res.success)
|
||
|
|
|
||
|
|
x = np.linspace(0, 1, 5)
|
||
|
|
y = np.zeros((2, x.size))
|
||
|
|
res = solve_bvp(undefined_fun, undefined_bc, x, y)
|
||
|
|
assert_equal(res.status, 2)
|
||
|
|
assert_(not res.success)
|
||
|
|
|
||
|
|
|
||
|
|
def test_big_problem():
|
||
|
|
n = 30
|
||
|
|
x = np.linspace(0, 1, 5)
|
||
|
|
y = np.zeros((2 * n, x.size))
|
||
|
|
sol = solve_bvp(big_fun, big_bc, x, y)
|
||
|
|
|
||
|
|
assert_equal(sol.status, 0)
|
||
|
|
assert_(sol.success)
|
||
|
|
|
||
|
|
sol_test = sol.sol(x)
|
||
|
|
|
||
|
|
assert_allclose(sol_test[0], big_sol(x, n))
|
||
|
|
|
||
|
|
f_test = big_fun(x, sol_test)
|
||
|
|
r = sol.sol(x, 1) - f_test
|
||
|
|
rel_res = r / (1 + np.abs(f_test))
|
||
|
|
norm_res = np.sum(np.real(rel_res * np.conj(rel_res)), axis=0) ** 0.5
|
||
|
|
assert_(np.all(norm_res < 1e-3))
|
||
|
|
|
||
|
|
assert_(np.all(sol.rms_residuals < 1e-3))
|
||
|
|
assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10)
|
||
|
|
assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_shock_layer():
|
||
|
|
x = np.linspace(-1, 1, 5)
|
||
|
|
x_test = np.linspace(-1, 1, 100)
|
||
|
|
y = np.zeros((2, x.size))
|
||
|
|
sol = solve_bvp(shock_fun, shock_bc, x, y)
|
||
|
|
|
||
|
|
assert_equal(sol.status, 0)
|
||
|
|
assert_(sol.success)
|
||
|
|
|
||
|
|
assert_(sol.x.size < 110)
|
||
|
|
|
||
|
|
sol_test = sol.sol(x_test)
|
||
|
|
assert_allclose(sol_test[0], shock_sol(x_test), rtol=1e-5, atol=1e-5)
|
||
|
|
|
||
|
|
f_test = shock_fun(x_test, sol_test)
|
||
|
|
r = sol.sol(x_test, 1) - f_test
|
||
|
|
rel_res = r / (1 + np.abs(f_test))
|
||
|
|
norm_res = np.sum(rel_res ** 2, axis=0) ** 0.5
|
||
|
|
|
||
|
|
assert_(np.all(norm_res < 1e-3))
|
||
|
|
assert_allclose(sol.sol(sol.x), sol.y, rtol=1e-10, atol=1e-10)
|
||
|
|
assert_allclose(sol.sol(sol.x, 1), sol.yp, rtol=1e-10, atol=1e-10)
|
||
|
|
|
||
|
|
|
||
|
|
def test_verbose():
|
||
|
|
# Smoke test that checks the printing does something and does not crash
|
||
|
|
x = np.linspace(0, 1, 5)
|
||
|
|
y = np.zeros((2, x.shape[0]))
|
||
|
|
for verbose in [0, 1, 2]:
|
||
|
|
old_stdout = sys.stdout
|
||
|
|
sys.stdout = StringIO()
|
||
|
|
try:
|
||
|
|
sol = solve_bvp(exp_fun, exp_bc, x, y, verbose=verbose)
|
||
|
|
text = sys.stdout.getvalue()
|
||
|
|
finally:
|
||
|
|
sys.stdout = old_stdout
|
||
|
|
|
||
|
|
assert_(sol.success)
|
||
|
|
if verbose == 0:
|
||
|
|
assert_(not text, text)
|
||
|
|
if verbose >= 1:
|
||
|
|
assert_("Solved in" in text, text)
|
||
|
|
if verbose >= 2:
|
||
|
|
assert_("Max residual" in text, text)
|
||
|
|
|