from __future__ import division, print_function, absolute_import from itertools import product from numpy.testing import (assert_, assert_allclose, assert_equal, assert_no_warnings) from pytest import raises as assert_raises from scipy._lib._numpy_compat import suppress_warnings import numpy as np from scipy.optimize._numdiff import group_columns from scipy.integrate import solve_ivp, RK23, RK45, Radau, BDF, LSODA from scipy.integrate import OdeSolution from scipy.integrate._ivp.common import num_jac from scipy.integrate._ivp.base import ConstantDenseOutput from scipy.sparse import coo_matrix, csc_matrix def fun_linear(t, y): return np.array([-y[0] - 5 * y[1], y[0] + y[1]]) def jac_linear(): return np.array([[-1, -5], [1, 1]]) def sol_linear(t): return np.vstack((-5 * np.sin(2 * t), 2 * np.cos(2 * t) + np.sin(2 * t))) def fun_rational(t, y): return np.array([y[1] / t, y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1))]) def fun_rational_vectorized(t, y): return np.vstack((y[1] / t, y[1] * (y[0] + 2 * y[1] - 1) / (t * (y[0] - 1)))) def jac_rational(t, y): return np.array([ [0, 1 / t], [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2), (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))] ]) def jac_rational_sparse(t, y): return csc_matrix([ [0, 1 / t], [-2 * y[1] ** 2 / (t * (y[0] - 1) ** 2), (y[0] + 4 * y[1] - 1) / (t * (y[0] - 1))] ]) def sol_rational(t): return np.asarray((t / (t + 10), 10 * t / (t + 10) ** 2)) def fun_medazko(t, y): n = y.shape[0] // 2 k = 100 c = 4 phi = 2 if t <= 5 else 0 y = np.hstack((phi, 0, y, y[-2])) d = 1 / n j = np.arange(n) + 1 alpha = 2 * (j * d - 1) ** 3 / c ** 2 beta = (j * d - 1) ** 4 / c ** 2 j_2_p1 = 2 * j + 2 j_2_m3 = 2 * j - 2 j_2_m1 = 2 * j j_2 = 2 * j + 1 f = np.empty(2 * n) f[::2] = (alpha * (y[j_2_p1] - y[j_2_m3]) / (2 * d) + beta * (y[j_2_m3] - 2 * y[j_2_m1] + y[j_2_p1]) / d ** 2 - k * y[j_2_m1] * y[j_2]) f[1::2] = -k * y[j_2] * y[j_2_m1] return f def medazko_sparsity(n): cols = [] rows = [] i = np.arange(n) * 2 cols.append(i[1:]) rows.append(i[1:] - 2) cols.append(i) rows.append(i) cols.append(i) rows.append(i + 1) cols.append(i[:-1]) rows.append(i[:-1] + 2) i = np.arange(n) * 2 + 1 cols.append(i) rows.append(i) cols.append(i) rows.append(i - 1) cols = np.hstack(cols) rows = np.hstack(rows) return coo_matrix((np.ones_like(cols), (cols, rows))) def fun_complex(t, y): return -y def jac_complex(t, y): return -np.eye(y.shape[0]) def jac_complex_sparse(t, y): return csc_matrix(jac_complex(t, y)) def sol_complex(t): y = (0.5 + 1j) * np.exp(-t) return y.reshape((1, -1)) def compute_error(y, y_true, rtol, atol): e = (y - y_true) / (atol + rtol * np.abs(y_true)) return np.sqrt(np.sum(np.real(e * e.conj()), axis=0) / e.shape[0]) def test_integration(): rtol = 1e-3 atol = 1e-6 y0 = [1/3, 2/9] for vectorized, method, t_span, jac in product( [False, True], ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA'], [[5, 9], [5, 1]], [None, jac_rational, jac_rational_sparse]): if vectorized: fun = fun_rational_vectorized else: fun = fun_rational with suppress_warnings() as sup: sup.filter(UserWarning, "The following arguments have no effect for a chosen solver: `jac`") res = solve_ivp(fun, t_span, y0, rtol=rtol, atol=atol, method=method, dense_output=True, jac=jac, vectorized=vectorized) assert_equal(res.t[0], t_span[0]) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) assert_(res.nfev < 40) if method in ['RK23', 'RK45', 'LSODA']: assert_equal(res.njev, 0) assert_equal(res.nlu, 0) else: assert_(0 < res.njev < 3) assert_(0 < res.nlu < 10) y_true = sol_rational(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 5)) tc = np.linspace(*t_span) yc_true = sol_rational(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, rtol, atol) assert_(np.all(e < 5)) tc = (t_span[0] + t_span[-1]) / 2 yc_true = sol_rational(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, rtol, atol) assert_(np.all(e < 5)) # LSODA for some reasons doesn't pass the polynomial through the # previous points exactly after the order change. It might be some # bug in LSOSA implementation or maybe we missing something. if method != 'LSODA': assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15) def test_integration_complex(): rtol = 1e-3 atol = 1e-6 y0 = [0.5 + 1j] t_span = [0, 1] tc = np.linspace(t_span[0], t_span[1]) for method, jac in product(['RK23', 'RK45', 'BDF'], [None, jac_complex, jac_complex_sparse]): with suppress_warnings() as sup: sup.filter(UserWarning, "The following arguments have no effect for a chosen solver: `jac`") res = solve_ivp(fun_complex, t_span, y0, method=method, dense_output=True, rtol=rtol, atol=atol, jac=jac) assert_equal(res.t[0], t_span[0]) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) assert_(res.nfev < 25) if method == 'BDF': assert_equal(res.njev, 1) assert_(res.nlu < 6) else: assert_equal(res.njev, 0) assert_equal(res.nlu, 0) y_true = sol_complex(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 5)) yc_true = sol_complex(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, rtol, atol) assert_(np.all(e < 5)) def test_integration_sparse_difference(): n = 200 t_span = [0, 20] y0 = np.zeros(2 * n) y0[1::2] = 1 sparsity = medazko_sparsity(n) for method in ['BDF', 'Radau']: res = solve_ivp(fun_medazko, t_span, y0, method=method, jac_sparsity=sparsity) assert_equal(res.t[0], t_span[0]) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) assert_allclose(res.y[78, -1], 0.233994e-3, rtol=1e-2) assert_allclose(res.y[79, -1], 0, atol=1e-3) assert_allclose(res.y[148, -1], 0.359561e-3, rtol=1e-2) assert_allclose(res.y[149, -1], 0, atol=1e-3) assert_allclose(res.y[198, -1], 0.117374129e-3, rtol=1e-2) assert_allclose(res.y[199, -1], 0.6190807e-5, atol=1e-3) assert_allclose(res.y[238, -1], 0, atol=1e-3) assert_allclose(res.y[239, -1], 0.9999997, rtol=1e-2) def test_integration_const_jac(): rtol = 1e-3 atol = 1e-6 y0 = [0, 2] t_span = [0, 2] J = jac_linear() J_sparse = csc_matrix(J) for method, jac in product(['Radau', 'BDF'], [J, J_sparse]): res = solve_ivp(fun_linear, t_span, y0, rtol=rtol, atol=atol, method=method, dense_output=True, jac=jac) assert_equal(res.t[0], t_span[0]) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) assert_(res.nfev < 100) assert_equal(res.njev, 0) assert_(0 < res.nlu < 15) y_true = sol_linear(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 10)) tc = np.linspace(*t_span) yc_true = sol_linear(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, rtol, atol) assert_(np.all(e < 15)) assert_allclose(res.sol(res.t), res.y, rtol=1e-14, atol=1e-14) def test_events(): def event_rational_1(t, y): return y[0] - y[1] ** 0.7 def event_rational_2(t, y): return y[1] ** 0.6 - y[0] def event_rational_3(t, y): return t - 7.4 event_rational_3.terminal = True for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']: res = solve_ivp(fun_rational, [5, 8], [1/3, 2/9], method=method, events=(event_rational_1, event_rational_2)) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 1) assert_equal(res.t_events[1].size, 1) assert_(5.3 < res.t_events[0][0] < 5.7) assert_(7.3 < res.t_events[1][0] < 7.7) event_rational_1.direction = 1 event_rational_2.direction = 1 res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, events=(event_rational_1, event_rational_2)) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 1) assert_equal(res.t_events[1].size, 0) assert_(5.3 < res.t_events[0][0] < 5.7) event_rational_1.direction = -1 event_rational_2.direction = -1 res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, events=(event_rational_1, event_rational_2)) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 0) assert_equal(res.t_events[1].size, 1) assert_(7.3 < res.t_events[1][0] < 7.7) event_rational_1.direction = 0 event_rational_2.direction = 0 res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, events=(event_rational_1, event_rational_2, event_rational_3), dense_output=True) assert_equal(res.status, 1) assert_equal(res.t_events[0].size, 1) assert_equal(res.t_events[1].size, 0) assert_equal(res.t_events[2].size, 1) assert_(5.3 < res.t_events[0][0] < 5.7) assert_(7.3 < res.t_events[2][0] < 7.5) res = solve_ivp(fun_rational, [5, 8], [1 / 3, 2 / 9], method=method, events=event_rational_1, dense_output=True) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 1) assert_(5.3 < res.t_events[0][0] < 5.7) # Also test that termination by event doesn't break interpolants. tc = np.linspace(res.t[0], res.t[-1]) yc_true = sol_rational(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, 1e-3, 1e-6) assert_(np.all(e < 5)) # Test in backward direction. event_rational_1.direction = 0 event_rational_2.direction = 0 for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']: res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, events=(event_rational_1, event_rational_2)) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 1) assert_equal(res.t_events[1].size, 1) assert_(5.3 < res.t_events[0][0] < 5.7) assert_(7.3 < res.t_events[1][0] < 7.7) event_rational_1.direction = -1 event_rational_2.direction = -1 res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, events=(event_rational_1, event_rational_2)) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 1) assert_equal(res.t_events[1].size, 0) assert_(5.3 < res.t_events[0][0] < 5.7) event_rational_1.direction = 1 event_rational_2.direction = 1 res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, events=(event_rational_1, event_rational_2)) assert_equal(res.status, 0) assert_equal(res.t_events[0].size, 0) assert_equal(res.t_events[1].size, 1) assert_(7.3 < res.t_events[1][0] < 7.7) event_rational_1.direction = 0 event_rational_2.direction = 0 res = solve_ivp(fun_rational, [8, 5], [4/9, 20/81], method=method, events=(event_rational_1, event_rational_2, event_rational_3), dense_output=True) assert_equal(res.status, 1) assert_equal(res.t_events[0].size, 0) assert_equal(res.t_events[1].size, 1) assert_equal(res.t_events[2].size, 1) assert_(7.3 < res.t_events[1][0] < 7.7) assert_(7.3 < res.t_events[2][0] < 7.5) # Also test that termination by event doesn't break interpolants. tc = np.linspace(res.t[-1], res.t[0]) yc_true = sol_rational(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, 1e-3, 1e-6) assert_(np.all(e < 5)) def test_max_step(): rtol = 1e-3 atol = 1e-6 y0 = [1/3, 2/9] for method in [RK23, RK45, Radau, BDF, LSODA]: for t_span in ([5, 9], [5, 1]): res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, max_step=0.5, atol=atol, method=method, dense_output=True) assert_equal(res.t[0], t_span[0]) assert_equal(res.t[-1], t_span[-1]) assert_(np.all(np.abs(np.diff(res.t)) <= 0.5)) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) y_true = sol_rational(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 5)) tc = np.linspace(*t_span) yc_true = sol_rational(tc) yc = res.sol(tc) e = compute_error(yc, yc_true, rtol, atol) assert_(np.all(e < 5)) # See comment in test_integration. if method is not LSODA: assert_allclose(res.sol(res.t), res.y, rtol=1e-15, atol=1e-15) assert_raises(ValueError, method, fun_rational, t_span[0], y0, t_span[1], max_step=-1) if method is not LSODA: solver = method(fun_rational, t_span[0], y0, t_span[1], rtol=rtol, atol=atol, max_step=1e-20) message = solver.step() assert_equal(solver.status, 'failed') assert_("step size is less" in message) assert_raises(RuntimeError, solver.step) def test_t_eval(): rtol = 1e-3 atol = 1e-6 y0 = [1/3, 2/9] for t_span in ([5, 9], [5, 1]): t_eval = np.linspace(t_span[0], t_span[1], 10) res = solve_ivp(fun_rational, t_span, y0, rtol=rtol, atol=atol, t_eval=t_eval) assert_equal(res.t, t_eval) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) y_true = sol_rational(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 5)) t_eval = [5, 5.01, 7, 8, 8.01, 9] res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol, t_eval=t_eval) assert_equal(res.t, t_eval) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) y_true = sol_rational(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 5)) t_eval = [5, 4.99, 3, 1.5, 1.1, 1.01, 1] res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol, t_eval=t_eval) assert_equal(res.t, t_eval) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) t_eval = [5.01, 7, 8, 8.01] res = solve_ivp(fun_rational, [5, 9], y0, rtol=rtol, atol=atol, t_eval=t_eval) assert_equal(res.t, t_eval) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) y_true = sol_rational(res.t) e = compute_error(res.y, y_true, rtol, atol) assert_(np.all(e < 5)) t_eval = [4.99, 3, 1.5, 1.1, 1.01] res = solve_ivp(fun_rational, [5, 1], y0, rtol=rtol, atol=atol, t_eval=t_eval) assert_equal(res.t, t_eval) assert_(res.t_events is None) assert_(res.success) assert_equal(res.status, 0) t_eval = [4, 6] assert_raises(ValueError, solve_ivp, fun_rational, [5, 9], y0, rtol=rtol, atol=atol, t_eval=t_eval) def test_no_integration(): for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']: sol = solve_ivp(lambda t, y: -y, [4, 4], [2, 3], method=method, dense_output=True) assert_equal(sol.sol(4), [2, 3]) assert_equal(sol.sol([4, 5, 6]), [[2, 2, 2], [3, 3, 3]]) def test_no_integration_class(): for method in [RK23, RK45, Radau, BDF, LSODA]: solver = method(lambda t, y: -y, 0.0, [10.0, 0.0], 0.0) solver.step() assert_equal(solver.status, 'finished') sol = solver.dense_output() assert_equal(sol(0.0), [10.0, 0.0]) assert_equal(sol([0, 1, 2]), [[10, 10, 10], [0, 0, 0]]) solver = method(lambda t, y: -y, 0.0, [], np.inf) solver.step() assert_equal(solver.status, 'finished') sol = solver.dense_output() assert_equal(sol(100.0), []) assert_equal(sol([0, 1, 2]), np.empty((0, 3))) def test_empty(): def fun(t, y): return np.zeros((0,)) y0 = np.zeros((0,)) for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']: sol = assert_no_warnings(solve_ivp, fun, [0, 10], y0, method=method, dense_output=True) assert_equal(sol.sol(10), np.zeros((0,))) assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3))) for method in ['RK23', 'RK45', 'Radau', 'BDF', 'LSODA']: sol = assert_no_warnings(solve_ivp, fun, [0, np.inf], y0, method=method, dense_output=True) assert_equal(sol.sol(10), np.zeros((0,))) assert_equal(sol.sol([1, 2, 3]), np.zeros((0, 3))) def test_ConstantDenseOutput(): sol = ConstantDenseOutput(0, 1, np.array([1, 2])) assert_allclose(sol(1.5), [1, 2]) assert_allclose(sol([1, 1.5, 2]), [[1, 1, 1], [2, 2, 2]]) sol = ConstantDenseOutput(0, 1, np.array([])) assert_allclose(sol(1.5), np.empty(0)) assert_allclose(sol([1, 1.5, 2]), np.empty((0, 3))) def test_classes(): y0 = [1 / 3, 2 / 9] for cls in [RK23, RK45, Radau, BDF, LSODA]: solver = cls(fun_rational, 5, y0, np.inf) assert_equal(solver.n, 2) assert_equal(solver.status, 'running') assert_equal(solver.t_bound, np.inf) assert_equal(solver.direction, 1) assert_equal(solver.t, 5) assert_equal(solver.y, y0) assert_(solver.step_size is None) if cls is not LSODA: assert_(solver.nfev > 0) assert_(solver.njev >= 0) assert_equal(solver.nlu, 0) else: assert_equal(solver.nfev, 0) assert_equal(solver.njev, 0) assert_equal(solver.nlu, 0) assert_raises(RuntimeError, solver.dense_output) message = solver.step() assert_equal(solver.status, 'running') assert_equal(message, None) assert_equal(solver.n, 2) assert_equal(solver.t_bound, np.inf) assert_equal(solver.direction, 1) assert_(solver.t > 5) assert_(not np.all(np.equal(solver.y, y0))) assert_(solver.step_size > 0) assert_(solver.nfev > 0) assert_(solver.njev >= 0) assert_(solver.nlu >= 0) sol = solver.dense_output() assert_allclose(sol(5), y0, rtol=1e-15, atol=0) def test_OdeSolution(): ts = np.array([0, 2, 5], dtype=float) s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1])) s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1])) sol = OdeSolution(ts, [s1, s2]) assert_equal(sol(-1), [-1]) assert_equal(sol(1), [-1]) assert_equal(sol(2), [-1]) assert_equal(sol(3), [1]) assert_equal(sol(5), [1]) assert_equal(sol(6), [1]) assert_equal(sol([0, 6, -2, 1.5, 4.5, 2.5, 5, 5.5, 2]), np.array([[-1, 1, -1, -1, 1, 1, 1, 1, -1]])) ts = np.array([10, 4, -3]) s1 = ConstantDenseOutput(ts[0], ts[1], np.array([-1])) s2 = ConstantDenseOutput(ts[1], ts[2], np.array([1])) sol = OdeSolution(ts, [s1, s2]) assert_equal(sol(11), [-1]) assert_equal(sol(10), [-1]) assert_equal(sol(5), [-1]) assert_equal(sol(4), [-1]) assert_equal(sol(0), [1]) assert_equal(sol(-3), [1]) assert_equal(sol(-4), [1]) assert_equal(sol([12, -5, 10, -3, 6, 1, 4]), np.array([[-1, 1, -1, 1, -1, 1, -1]])) ts = np.array([1, 1]) s = ConstantDenseOutput(1, 1, np.array([10])) sol = OdeSolution(ts, [s]) assert_equal(sol(0), [10]) assert_equal(sol(1), [10]) assert_equal(sol(2), [10]) assert_equal(sol([2, 1, 0]), np.array([[10, 10, 10]])) def test_num_jac(): def fun(t, y): return np.vstack([ -0.04 * y[0] + 1e4 * y[1] * y[2], 0.04 * y[0] - 1e4 * y[1] * y[2] - 3e7 * y[1] ** 2, 3e7 * y[1] ** 2 ]) def jac(t, y): return np.array([ [-0.04, 1e4 * y[2], 1e4 * y[1]], [0.04, -1e4 * y[2] - 6e7 * y[1], -1e4 * y[1]], [0, 6e7 * y[1], 0] ]) t = 1 y = np.array([1, 0, 0]) J_true = jac(t, y) threshold = 1e-5 f = fun(t, y).ravel() J_num, factor = num_jac(fun, t, y, f, threshold, None) assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5) J_num, factor = num_jac(fun, t, y, f, threshold, factor) assert_allclose(J_num, J_true, rtol=1e-5, atol=1e-5) def test_num_jac_sparse(): def fun(t, y): e = y[1:]**3 - y[:-1]**2 z = np.zeros(y.shape[1]) return np.vstack((z, 3 * e)) + np.vstack((2 * e, z)) def structure(n): A = np.zeros((n, n), dtype=int) A[0, 0] = 1 A[0, 1] = 1 for i in range(1, n - 1): A[i, i - 1: i + 2] = 1 A[-1, -1] = 1 A[-1, -2] = 1 return A np.random.seed(0) n = 20 y = np.random.randn(n) A = structure(n) groups = group_columns(A) f = fun(0, y[:, None]).ravel() # Compare dense and sparse results, assuming that dense implementation # is correct (as it is straightforward). J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, None, sparsity=(A, groups)) J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, None) assert_allclose(J_num_dense, J_num_sparse.toarray(), rtol=1e-12, atol=1e-14) assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14) # Take small factors to trigger their recomputing inside. factor = np.random.uniform(0, 1e-12, size=n) J_num_sparse, factor_sparse = num_jac(fun, 0, y.ravel(), f, 1e-8, factor, sparsity=(A, groups)) J_num_dense, factor_dense = num_jac(fun, 0, y.ravel(), f, 1e-8, factor) assert_allclose(J_num_dense, J_num_sparse.toarray(), rtol=1e-12, atol=1e-14) assert_allclose(factor_dense, factor_sparse, rtol=1e-12, atol=1e-14)