def test_gauss_newton_nonlinear_overdetermined_solve():
"""Test that gauss_newton solves nonlinear, overdetermined systems."""
# The function being optimized is c0 + c1^2 * x^2 + c2^3 * x^3
x0 = np.array([20., 2.5, 500.])
result = nonlinear_solve(f3, x0, j3, method="gaussnewton")
f_expt = np.array([0.] * 4)
x_expt = np.array([2., 3., 5.])
message_expt = "Convergence obtained."
jac_expt = j3(x_expt)
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'])
assert result['success']
assert result['message'] == message_expt
# Because of symmetry the second coefficient of -3 should work
x0 = np.array([1000., -200., 500.])
result = nonlinear_solve(f3, x0, j3, method="gaussnewton")
x_expt = np.array([2., -3., 5.])
message_expt = "Convergence obtained."
jac_expt = j3(x_expt)
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'])
assert result['success']
assert result['message'] == message_expt
def test_gauss_newton_nonlinear_solve():
"""Test that gauss_newton solves nonlinear systems."""
# The function being optimized is c0 + c1^2 * x^2 + c2^3 * x^3
# The points are evaluated on [1., 2., 3.]
x0 = np.array([20., 2.5, 500.])
result = nonlinear_solve(f1, x0, j1, maxiter=5000, method="gaussnewton")
f_expt = np.array([0.] * 3)
x_expt = np.array([2., 3., 5.])
message_expt = "Convergence obtained."
jac_expt = j1(x_expt)
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'])
assert result['success']
assert result['message'] == message_expt
# Because of symmetry the second coefficient of -3 should work
x0 = np.array([200., -200., 200.])
result = nonlinear_solve(f1, x0, j1, maxiter=5000, method="gaussnewton")
x_expt = np.array([2., -3., 5.])
jac_expt = j1(x_expt)
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'])
assert result['success']
assert result['message'] == message_expt
def test_nonlinear_singular_matrix_error():
"""Test that newton raises error when Jacobian is singular."""
x0 = np.array([5., 5.])
result = nonlinear_solve(f7, x0, j7)
message_expt = "Singular Jacobian; no solution found."
assert not result['success']
assert result["message"] == message_expt
assert result["eps"] == 1e-6
def test_nonlinear_no_solution1():
"""Test that no solution in nonlinear_solve raises error."""
x_0 = np.array([1., 1.])
result = nonlinear_solve(f6, x_0, j6, eps=1.0e-9, maxiter=100)
assert not result["success"]
assert not np.allclose(result["f"], [0., 0.], atol=1.0e-3)
assert result["message"] == "Maximum number of iterations reached."
assert result["niter"] == 100
def test_nonlinear_bfgs():
"""Test that newton solves system with bfgs inverse update."""
x_0 = np.array([0.5, 0.5])
result = nonlinear_solve(f5, x_0, j5, stepsize=0.5, eps=1.0e-5, maxiter=10000, method="bfgs")
assert result["success"]
assert result["niter"] < 10000
assert np.allclose(result["f"], [0., 0.], rtol=1.0e-5, atol=1.0e-3)
assert result["message"] == "Convergence obtained."
assert np.allclose(result["x"], [1., 0.], rtol=1.0e-5, atol=1.0e-2)
def test_nonlinear_dfp():
"""Test that newton solves system with dfp update."""
x_0 = np.array([3., 1.])
result = nonlinear_solve(f5, x_0, j5, stepsize=1, eps=1.0e-5, maxiter=1000, method="dfp")
assert result["success"]
assert result["message"] == "Convergence obtained."
assert np.allclose(result["f"], [0., 0.], atol=1.0e-3)
assert np.allclose(result["x"], [1., 0.], atol=1.0e-3)
assert result["niter"] < 1000
def test_nonlinear_badbroyden():
"""Test that newton solves system with bad broyden update."""
x_0 = np.array([1.1, 0.5])
result = nonlinear_solve(f5, x_0, stepsize=1, eps=1.0e-6, maxiter=100, method="badbroyden")
assert result["success"]
assert np.allclose(result["f"], [0., 0.], rtol=1.0e-5, atol=1.0e-5)
assert result["message"] == "Convergence obtained."
assert result["niter"] < 100
assert np.allclose(result["x"], [1., 0.], rtol=1.0e-5, atol=1.0e-5)
assert np.allclose(result["f"], [0., 0.], rtol=1.0e-5, atol=1.0e-5)
def test_gauss_newton_nonlinear_underdetermined_solve():
"""Test that gauss_newton solves nonlinear, underdetermined systems."""
# Non-Exact Initial Guess. Here a really Good Initial guess is needed
x_expt = np.array([2., 3., 5., 8.])
f_expt = np.array([0.] * 3)
x0 = np.array([2.001, 3.001, 5.001, 10.01])
result = nonlinear_solve(f4, x0, j4, maxiter=12, method="gaussnewton")
message_expt = "Maximum number of iterations reached."
assert result['message'] == message_expt
# Repeat with a higher number of iterations and a really good initial guess
# Still it only converges to the first decimal place.
x0 = np.array([2.00001, 3.00001, 5.00001, 8.00001])
result = nonlinear_solve(f4, x0, j4, maxiter=10000, method="gaussnewton")
jac_expt = j4(result["x"])
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'], rtol=1e-1, atol=1e-1)
assert result['message'] == "Convergence obtained."
assert result['success']
def test_nonlinear_linear_solve():
"""Test that newton solves linear systems in 1 step."""
x_0 = np.array([1.0, 1.0])
result = nonlinear_solve(f4, x_0, j1, stepsize=np.array([1., 1.]), eps=1.0e-9, maxiter=1)
assert result["success"]
assert result["message"] == "Convergence obtained."
assert result["niter"] == 1
assert np.allclose(result["f"], [0., 0.], atol=1.0e-6)
assert np.allclose(result["x"], [2., 1.], atol=1.0e-9)
assert np.allclose(result["f"], [0., 0.], atol=1.0e-6)
assert np.allclose(result["J"], [[1., 1.], [-1., 1]], atol=1.0e-6)
def test_nonlinear_nonlinear_solve():
"""Test that newton solves nonlinear systems."""
x_0 = np.array([0.5, 0.5])
result = nonlinear_solve(f5, x_0, j5, eps=1.0e-9, maxiter=100)
assert result["success"]
assert result["niter"] < 101
assert np.allclose(result["f"], [0., 0.], atol=1.0e-6)
assert result["message"] == "Convergence obtained."
assert result["niter"] < 100
assert np.allclose(result["x"], [1., 0.], atol=1.0e-6)
assert np.allclose(result["f"], [0., 0.], atol=1.0e-6)
assert np.allclose(result["J"], [[3., 1.], [-1., 0]], atol=1.0e-6)
def test_gauss_newton_approximation_overdetermined():
"""Test gauss newton using an approximation finite diff Jacobian."""
x0 = np.array([20., 2.5, 500.])
result = nonlinear_solve(f3, x0, method="gaussnewton")
f_expt = np.array([0.] * 4)
x_expt = np.array([2., 3., 5.])
message_expt = "Convergence obtained."
jac_expt = j3(x_expt)
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'])
assert result['success']
assert result['message'] == message_expt
def test_gauss_newton_linear_solve():
"""Test that gauss_newton solves linear systems in 1 step."""
# Obtained from pg 489 of Numerical Mathematics and Computing Sixth Edition.
x0 = np.array([100., -200.])
result = nonlinear_solve(f_lin, x0, j_lin, eps=1e-20, method="gaussnewton")
x_expt = np.array([0.4864, -1.6589])
f_expt = np.array([0., 0.])
jac_expt = j_lin(1.)
message_expt = "Convergence obtained."
assert np.allclose(f_expt, result['f'], rtol=1e-5, atol=1e-5)
assert np.allclose(jac_expt, result['J'], rtol=1e-5, atol=1e-5)
assert np.allclose(x_expt, result['x'], rtol=1e-4, atol=1e-4)
assert result['success']
assert result['message'] == message_expt