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_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_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_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_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_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_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_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_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_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)
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)
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_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_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_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_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)