import sys try: from StringIO import StringIO except ImportError: from io import StringIO import numpy as np from numpy.testing import (assert_, assert_array_equal, assert_allclose, assert_equal) from pytest import raises as assert_raises from scipy.sparse import coo_matrix from scipy.special import erf from scipy.integrate._bvp import (modify_mesh, estimate_fun_jac, estimate_bc_jac, compute_jac_indices, construct_global_jac, solve_bvp) def exp_fun(x, y): return np.vstack((y[1], y[0])) def exp_fun_jac(x, y): df_dy = np.empty((2, 2, x.shape[0])) df_dy[0, 0] = 0 df_dy[0, 1] = 1 df_dy[1, 0] = 1 df_dy[1, 1] = 0 return df_dy def exp_bc(ya, yb): return np.hstack((ya[0] - 1, yb[0])) def exp_bc_complex(ya, yb): return np.hstack((ya[0] - 1 - 1j, yb[0])) def exp_bc_jac(ya, yb): dbc_dya = np.array([ [1, 0], [0, 0] ]) dbc_dyb = np.array([ [0, 0], [1, 0] ]) return dbc_dya, dbc_dyb def exp_sol(x): return (np.exp(-x) - np.exp(x - 2)) / (1 - np.exp(-2)) def sl_fun(x, y, p): return np.vstack((y[1], -p[0]**2 * y[0])) def sl_fun_jac(x, y, p): n, m = y.shape df_dy = np.empty((n, 2, m)) df_dy[0, 0] = 0 df_dy[0, 1] = 1 df_dy[1, 0] = -p[0]**2 df_dy[1, 1] = 0 df_dp = np.empty((n, 1, m)) df_dp[0, 0] = 0 df_dp[1, 0] = -2 * p[0] * y[0] return df_dy, df_dp def sl_bc(ya, yb, p): return np.hstack((ya[0], yb[0], ya[1] - p[0])) def sl_bc_jac(ya, yb, p): dbc_dya = np.zeros((3, 2)) dbc_dya[0, 0] = 1 dbc_dya[2, 1] = 1 dbc_dyb = np.zeros((3, 2)) dbc_dyb[1, 0] = 1 dbc_dp = np.zeros((3, 1)) dbc_dp[2, 0] = -1 return dbc_dya, dbc_dyb, dbc_dp def sl_sol(x, p): return np.sin(p[0] * x) def emden_fun(x, y): return np.vstack((y[1], -y[0]**5)) def emden_fun_jac(x, y): df_dy = np.empty((2, 2, x.shape[0])) df_dy[0, 0] = 0 df_dy[0, 1] = 1 df_dy[1, 0] = -5 * y[0]**4 df_dy[1, 1] = 0 return df_dy def emden_bc(ya, yb): return np.array([ya[1], yb[0] - (3/4)**0.5]) def emden_bc_jac(ya, yb): dbc_dya = np.array([ [0, 1], [0, 0] ]) dbc_dyb = np.array([ [0, 0], [1, 0] ]) return dbc_dya, dbc_dyb def emden_sol(x): return (1 + x**2/3)**-0.5 def undefined_fun(x, y): return np.zeros_like(y) def undefined_bc(ya, yb): return np.array([ya[0], yb[0] - 1]) def big_fun(x, y): f = np.zeros_like(y) f[::2] = y[1::2] return f def big_bc(ya, yb): return np.hstack((ya[::2], yb[::2] - 1)) def big_sol(x, n): y = np.ones((2 * n, x.size)) y[::2] = x return x def big_fun_with_parameters(x, y, p): """ Big version of sl_fun, with two parameters. The two differential equations represented by sl_fun are broadcast to the number of rows of y, rotating between the parameters p[0] and p[1]. Here are the differential equations: dy[0]/dt = y[1] dy[1]/dt = -p[0]**2 * y[0] dy[2]/dt = y[3] dy[3]/dt = -p[1]**2 * y[2] dy[4]/dt = y[5] dy[5]/dt = -p[0]**2 * y[4] dy[6]/dt = y[7] dy[7]/dt = -p[1]**2 * y[6] . . . """ f = np.zeros_like(y) f[::2] = y[1::2] f[1::4] = -p[0]**2 * y[::4] f[3::4] = -p[1]**2 * y[2::4] return f def big_fun_with_parameters_jac(x, y, p): # big version of sl_fun_jac, with two parameters n, m = y.shape df_dy = np.zeros((n, n, m)) df_dy[range(0, n, 2), range(1, n, 2)] = 1 df_dy[range(1, n, 4), range(0, n, 4)] = -p[0]**2 df_dy[range(3, n, 4), range(2, n, 4)] = -p[1]**2 df_dp = np.zeros((n, 2, m)) df_dp[range(1, n, 4), 0] = -2 * p[0] * y[range(0, n, 4)] df_dp[range(3, n, 4), 1] = -2 * p[1] * y[range(2, n, 4)] return df_dy, df_dp def big_bc_with_parameters(ya, yb, p): # big version of sl_bc, with two parameters return np.hstack((ya[::2], yb[::2], ya[1] - p[0], ya[3] - p[1])) def big_bc_with_parameters_jac(ya, yb, p): # big version of sl_bc_jac, with two parameters n = ya.shape[0] dbc_dya = np.zeros((n + 2, n)) dbc_dyb = np.zeros((n + 2, n)) dbc_dya[range(n // 2), range(0, n, 2)] = 1 dbc_dyb[range(n // 2, n), range(0, n, 2)] = 1 dbc_dp = np.zeros((n + 2, 2)) dbc_dp[n, 0] = -1 dbc_dya[n, 1] = 1 dbc_dp[n + 1, 1] = -1 dbc_dya[n + 1, 3] = 1 return dbc_dya, dbc_dyb, dbc_dp def big_sol_with_parameters(x, p): # big version of sl_sol, with two parameters return np.vstack((np.sin(p[0] * x), np.sin(p[1] * x))) def shock_fun(x, y): eps = 1e-3 return np.vstack(( y[1], -(x * y[1] + eps * np.pi**2 * np.cos(np.pi * x) + np.pi * x * np.sin(np.pi * x)) / eps )) def shock_bc(ya, yb): return np.array([ya[0] + 2, yb[0]]) def shock_sol(x): eps = 1e-3 k = np.sqrt(2 * eps) return np.cos(np.pi * x) + erf(x / k) / erf(1 / k) def nonlin_bc_fun(x, y): # laplace eq. return np.stack([y[1], np.zeros_like(x)]) def nonlin_bc_bc(ya, yb): phiA, phipA = ya phiC, phipC = yb kappa, ioA, ioC, V, f = 1.64, 0.01, 1.0e-4, 0.5, 38.9 # Butler-Volmer Kinetics at Anode hA = 0.0-phiA-0.0 iA = ioA * (np.exp(f*hA) - np.exp(-f*hA)) res0 = iA + kappa * phipA # Butler-Volmer Kinetics at Cathode hC = V - phiC - 1.0 iC = ioC * (np.exp(f*hC) - np.exp(-f*hC)) res1 = iC - kappa*phipC return np.array([res0, res1]) def nonlin_bc_sol(x): return -0.13426436116763119 - 1.1308709 * x def test_modify_mesh(): x = np.array([0, 1, 3, 9], dtype=float) x_new = modify_mesh(x, np.array([0]), np.array([2])) assert_array_equal(x_new, np.array([0, 0.5, 1, 3, 5, 7, 9])) x = np.array([-6, -3, 0, 3, 6], dtype=float) x_new = modify_mesh(x, np.array([1], dtype=int), np.array([0, 2, 3])) assert_array_equal(x_new, [-6, -5, -4, -3, -1.5, 0, 1, 2, 3, 4, 5, 6]) def test_compute_fun_jac(): x = np.linspace(0, 1, 5) y = np.empty((2, x.shape[0])) y[0] = 0.01 y[1] = 0.02 p = np.array([]) df_dy, df_dp = estimate_fun_jac(lambda x, y, p: exp_fun(x, y), x, y, p) df_dy_an = exp_fun_jac(x, y) assert_allclose(df_dy, df_dy_an) assert_(df_dp is None) x = np.linspace(0, np.pi, 5) y = np.empty((2, x.shape[0])) y[0] = np.sin(x) y[1] = np.cos(x) p = np.array([1.0]) df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p) df_dy_an, df_dp_an = sl_fun_jac(x, y, p) assert_allclose(df_dy, df_dy_an) assert_allclose(df_dp, df_dp_an) x = np.linspace(0, 1, 10) y = np.empty((2, x.shape[0])) y[0] = (3/4)**0.5 y[1] = 1e-4 p = np.array([]) df_dy, df_dp = estimate_fun_jac(lambda x, y, p: emden_fun(x, y), x, y, p) df_dy_an = emden_fun_jac(x, y) assert_allclose(df_dy, df_dy_an) assert_(df_dp is None) def test_compute_bc_jac(): ya = np.array([-1.0, 2]) yb = np.array([0.5, 3]) p = np.array([]) dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac( lambda ya, yb, p: exp_bc(ya, yb), ya, yb, p) dbc_dya_an, dbc_dyb_an = exp_bc_jac(ya, yb) assert_allclose(dbc_dya, dbc_dya_an) assert_allclose(dbc_dyb, dbc_dyb_an) assert_(dbc_dp is None) ya = np.array([0.0, 1]) yb = np.array([0.0, -1]) p = np.array([0.5]) dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(sl_bc, ya, yb, p) dbc_dya_an, dbc_dyb_an, dbc_dp_an = sl_bc_jac(ya, yb, p) assert_allclose(dbc_dya, dbc_dya_an) assert_allclose(dbc_dyb, dbc_dyb_an) assert_allclose(dbc_dp, dbc_dp_an) ya = np.array([0.5, 100]) yb = np.array([-1000, 10.5]) p = np.array([]) dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac( lambda ya, yb, p: emden_bc(ya, yb), ya, yb, p) dbc_dya_an, dbc_dyb_an = emden_bc_jac(ya, yb) assert_allclose(dbc_dya, dbc_dya_an) assert_allclose(dbc_dyb, dbc_dyb_an) assert_(dbc_dp is None) def test_compute_jac_indices(): n = 2 m = 4 k = 2 i, j = compute_jac_indices(n, m, k) s = coo_matrix((np.ones_like(i), (i, j))).toarray() s_true = np.array([ [1, 1, 1, 1, 0, 0, 0, 0, 1, 1], [1, 1, 1, 1, 0, 0, 0, 0, 1, 1], [0, 0, 1, 1, 1, 1, 0, 0, 1, 1], [0, 0, 1, 1, 1, 1, 0, 0, 1, 1], [0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0, 1, 1, 1, 1], ]) assert_array_equal(s, s_true) def test_compute_global_jac(): n = 2 m = 5 k = 1 i_jac, j_jac = compute_jac_indices(2, 5, 1) x = np.linspace(0, 1, 5) h = np.diff(x) y = np.vstack((np.sin(np.pi * x), np.pi * np.cos(np.pi * x))) p = np.array([3.0]) f = sl_fun(x, y, p) x_middle = x[:-1] + 0.5 * h y_middle = 0.5 * (y[:, :-1] + y[:, 1:]) - h/8 * (f[:, 1:] - f[:, :-1]) df_dy, df_dp = sl_fun_jac(x, y, p) df_dy_middle, df_dp_middle = sl_fun_jac(x_middle, y_middle, p) dbc_dya, dbc_dyb, dbc_dp = sl_bc_jac(y[:, 0], y[:, -1], p) J = construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle, df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp) J = J.toarray() def J_block(h, p): return np.array([ [h**2*p**2/12 - 1, -0.5*h, -h**2*p**2/12 + 1, -0.5*h], [0.5*h*p**2, h**2*p**2/12 - 1, 0.5*h*p**2, 1 - h**2*p**2/12] ]) J_true = np.zeros((m * n + k, m * n + k)) for i in range(m - 1): J_true[i * n: (i + 1) * n, i * n: (i + 2) * n] = J_block(h[i], p[0]) J_true[:(m - 1) * n:2, -1] = p * h**2/6 * (y[0, :-1] - y[0, 1:]) J_true[1:(m - 1) * n:2, -1] = p * (h * (y[0, :-1] + y[0, 1:]) + h**2/6 * (y[1, :-1] - y[1, 1:])) J_true[8, 0] = 1 J_true[9, 8] = 1 J_true[10, 1] = 1 J_true[10, 10] = -1 assert_allclose(J, J_true, rtol=1e-10) df_dy, df_dp = estimate_fun_jac(sl_fun, x, y, p) df_dy_middle, df_dp_middle = estimate_fun_jac(sl_fun, x_middle, y_middle, p) dbc_dya, dbc_dyb, dbc_dp = estimate_bc_jac(sl_bc, y[:, 0], y[:, -1], p) J = construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle, df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp) J = J.toarray() assert_allclose(J, J_true, rtol=2e-8, atol=2e-8) def test_parameter_validation(): x = [0, 1, 0.5] y = np.zeros((2, 3)) assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y) x = np.linspace(0, 1, 5) y = np.zeros((2, 4)) assert_raises(ValueError, solve_bvp, exp_fun, exp_bc, x, y) fun = lambda x, y, p: exp_fun(x, y) bc = lambda ya, yb, p: exp_bc(ya, yb) y = np.zeros((2, x.shape[0])) assert_raises(ValueError, solve_bvp, fun, bc, x, y, p=[1]) 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_big_problem_with_parameters(): n = 30 x = np.linspace(0, np.pi, 5) x_test = np.linspace(0, np.pi, 100) y = np.ones((2 * n, x.size)) for fun_jac in [None, big_fun_with_parameters_jac]: for bc_jac in [None, big_bc_with_parameters_jac]: sol = solve_bvp(big_fun_with_parameters, big_bc_with_parameters, x, y, p=[0.5, 0.5], fun_jac=fun_jac, bc_jac=bc_jac) assert_equal(sol.status, 0) assert_(sol.success) assert_allclose(sol.p, [1, 1], rtol=1e-4) sol_test = sol.sol(x_test) for isol in range(0, n, 4): assert_allclose(sol_test[isol], big_sol_with_parameters(x_test, [1, 1])[0], rtol=1e-4, atol=1e-4) assert_allclose(sol_test[isol + 2], big_sol_with_parameters(x_test, [1, 1])[1], rtol=1e-4, atol=1e-4) f_test = big_fun_with_parameters(x_test, sol_test, [1, 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_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_nonlin_bc(): x = np.linspace(0, 0.1, 5) x_test = x y = np.zeros([2, x.size]) sol = solve_bvp(nonlin_bc_fun, nonlin_bc_bc, x, y) assert_equal(sol.status, 0) assert_(sol.success) assert_(sol.x.size < 8) sol_test = sol.sol(x_test) assert_allclose(sol_test[0], nonlin_bc_sol(x_test), rtol=1e-5, atol=1e-5) f_test = nonlin_bc_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)