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_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_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)
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=1e-8, atol=1e-9)
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)
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=1e-8, atol=1e-9)