def test_secant_condition_dfp():
"""Test that dfp update satisfies secant condition."""
x = (np.random.rand(1, 3) * 50. - 10.)[0]
dx = np.array([0.5, 0.5, 0.5])
df = f2(x + dx) - f2(x)
dfp = np.eye(3)
Jacobian.update_dfp(dfp, x + dx, dx, f2(x + dx) - f2(x))
assert np.allclose(df, dfp.dot(dx))
def test_jacobian_inputs():
"""Test invalid inputs to Jacobian class."""
# Test adding a non-callable jacobian to the class.
npt.assert_raises(TypeError, Jacobian, jac=5.)
# Test Jacobian calls the correct function.
jac_obj = Jacobian(jac=j1)
x_0 = np.random.rand(2)
assert np.allclose(jac_obj(x_0), j1(x_0))
jac_obj_obj = Jacobian(jac_obj)
assert np.allclose(jac_obj_obj(x_0), j1(x_0))
def test_bfgs_secant_condition():
"""Test the secant condition for BFGS."""
x = (np.random.rand(1, 3) * 50. - 10.)[0]
dx = np.array([0.5, 0.5, 0.5])
df = f2(x + dx) - f2(x)
# Compute the updated bfgs Jacobian
bfgs = np.eye(3)
Jacobian.update_bfgs(bfgs, x + dx, dx, f2(x + dx) - f2(x))
bfgs = np.linalg.inv(bfgs)
assert np.allclose(df, bfgs.dot(dx), atol=1e-1)
def test_positive_definiteness_dfp():
"""Test that DFP is positive definite."""
# It's crucial that the x values are greater than 0.5
# or else positive definiteness is not required.
random_vecs = np.random.rand(10, 3) * 50. + 0.5
dx = np.array([0.5, 0.5, 0.5])
x = random_vecs[0]
dfp = np.eye(3)
Jacobian.update_dfp(dfp, x + dx, dx, f2(x + dx) - f2(x))
assert np.all(
np.asarray([np.dot(x, dfp.dot(x)) for x in random_vecs]) > 0.)
def test_update_dfp():
"""Test the definition of DFP."""
# Test wikipedia definition/format.
random_vecs = np.random.rand(10, 3) * 50. + 0.5
x = random_vecs[0]
dx = np.array([0.0001, 0.0001, 0.0001])
df = f2(x + dx) - f2(x)
b0 = j2(x)
gamma = 1. / np.dot(df, dx)
expected_answer = (np.eye(3) - gamma * np.outer(df, dx)
).dot(b0).dot(np.eye(3) - gamma * np.outer(dx, df))
expected_answer += gamma * np.outer(df, df)
Jacobian.update_dfp(b0, x + dx, dx, df)
assert np.allclose(b0, expected_answer)
def test_update_goodbroyden_j1():
"""Test Jacobian.update_goodbroyden using analytical Jacobian and function."""
# Initial point x_k
x_0 = np.random.rand(2)
# Function at initial x_k
f1_0 = f1(x_0)
# Jacobian at initial x_k
j1_0 = j1(x_0)
# Define arbitrary step
dx = np.asarray([0.01, 0.01])
# Take step dx
x_1 = x_0 + dx
# Function at x_{k+1}
f1_1 = f1(x_1)
# Compute analytical Jacobian at x_{k+1}`
j1_1 = j1(x_1)
delta_f = f1_1 - f1_0
expected_ans = j1_0 + \
np.outer((delta_f - j1_0.dot(dx)) / np.dot(dx, dx), dx)
# Approximate Jacobian at (x_0 + dx) with Good Broyden method
# from Jacobian and vector function at initial x_0
Jacobian.update_goodbroyden(j1_0, x_1, x_1 - x_0, f1_1 - f1_0)
assert np.allclose(j1_0, expected_ans)
assert np.allclose(j1_1, expected_ans, atol=1e-1)
def test_update_sr1_inv_j1():
"""Test Jacobian.update_sr1_inv against analytical Jacobian and function."""
# Initial point x_0
x_0 = np.random.rand(2)
# Function at initial x_k
f1_0 = f1(x_0)
# Inverse Jacobian at initial x_0
j1_inv_0 = np.linalg.inv(j1(x_0))
# Define arbitrary step and take step
dx = np.asarray([0.01, 0.01])
x_1 = x_0 + dx
# Function at x_{k+1}
f1_1 = f1(x_1)
delta_f = f1_1 - f1_0
# Compute analytical inverse Jacobian at `x_0 + dx`
tmp = dx - j1_inv_0.dot(delta_f)
expected_ans = j1_inv_0 + (np.outer(tmp, tmp.T)) / np.dot(tmp.T, dx)
# Approximate inverse Jacobian at (x_0 + dx) with Bad Broyden method
# from inverse Jacobian and vector function at initial x_0
Jacobian.update_sr1inv(j1_inv_0, x_1, x_1 - x_0, f1_1 - f1_0)
assert np.allclose(j1_inv_0, expected_ans, atol=1e-1)
# assert that the approximation jacobian matches the analytic jacobian.
j1_1 = np.linalg.inv(j1(x_1))
assert np.allclose(j1_1, expected_ans, atol=1e-1)