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from __future__ import division
import math
from itertools import product
import numpy as np
from numpy.testing import (assert_raises, assert_allclose, assert_equal,
assert_, run_module_suite)
from scipy.sparse import csr_matrix, csc_matrix, lil_matrix
from scipy.optimize._numdiff import (
_adjust_scheme_to_bounds, approx_derivative, check_derivative,
group_columns)
def test_group_columns():
structure = [
[1, 1, 0, 0, 0, 0],
[1, 1, 1, 0, 0, 0],
[0, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 0],
[0, 0, 0, 1, 1, 1],
[0, 0, 0, 0, 1, 1],
[0, 0, 0, 0, 0, 0]
]
for transform in [np.asarray, csr_matrix, csc_matrix, lil_matrix]:
A = transform(structure)
order = np.arange(6)
groups_true = np.array([0, 1, 2, 0, 1, 2])
groups = group_columns(A, order)
assert_equal(groups, groups_true)
order = [1, 2, 4, 3, 5, 0]
groups_true = np.array([2, 0, 1, 2, 0, 1])
groups = group_columns(A, order)
assert_equal(groups, groups_true)
# Test repeatability.
groups_1 = group_columns(A)
groups_2 = group_columns(A)
assert_equal(groups_1, groups_2)
class TestAdjustSchemeToBounds(object):
def test_no_bounds(self):
x0 = np.zeros(3)
h = np.ones(3) * 1e-2
inf_lower = np.empty_like(x0)
inf_upper = np.empty_like(x0)
inf_lower.fill(-np.inf)
inf_upper.fill(np.inf)
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 1, '1-sided', inf_lower, inf_upper)
assert_allclose(h_adjusted, h)
assert_(np.all(one_sided))
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 2, '1-sided', inf_lower, inf_upper)
assert_allclose(h_adjusted, h)
assert_(np.all(one_sided))
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 1, '2-sided', inf_lower, inf_upper)
assert_allclose(h_adjusted, h)
assert_(np.all(~one_sided))
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 2, '2-sided', inf_lower, inf_upper)
assert_allclose(h_adjusted, h)
assert_(np.all(~one_sided))
def test_with_bound(self):
x0 = np.array([0.0, 0.85, -0.85])
lb = -np.ones(3)
ub = np.ones(3)
h = np.array([1, 1, -1]) * 1e-1
h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 1, '1-sided', lb, ub)
assert_allclose(h_adjusted, h)
h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 2, '1-sided', lb, ub)
assert_allclose(h_adjusted, np.array([1, -1, 1]) * 1e-1)
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 1, '2-sided', lb, ub)
assert_allclose(h_adjusted, np.abs(h))
assert_(np.all(~one_sided))
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 2, '2-sided', lb, ub)
assert_allclose(h_adjusted, np.array([1, -1, 1]) * 1e-1)
assert_equal(one_sided, np.array([False, True, True]))
def test_tight_bounds(self):
lb = np.array([-0.03, -0.03])
ub = np.array([0.05, 0.05])
x0 = np.array([0.0, 0.03])
h = np.array([-0.1, -0.1])
h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 1, '1-sided', lb, ub)
assert_allclose(h_adjusted, np.array([0.05, -0.06]))
h_adjusted, _ = _adjust_scheme_to_bounds(x0, h, 2, '1-sided', lb, ub)
assert_allclose(h_adjusted, np.array([0.025, -0.03]))
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 1, '2-sided', lb, ub)
assert_allclose(h_adjusted, np.array([0.03, -0.03]))
assert_equal(one_sided, np.array([False, True]))
h_adjusted, one_sided = _adjust_scheme_to_bounds(
x0, h, 2, '2-sided', lb, ub)
assert_allclose(h_adjusted, np.array([0.015, -0.015]))
assert_equal(one_sided, np.array([False, True]))
class TestApproxDerivativesDense(object):
def fun_scalar_scalar(self, x):
return np.sinh(x)
def jac_scalar_scalar(self, x):
return np.cosh(x)
def fun_scalar_vector(self, x):
return np.array([x[0]**2, np.tan(x[0]), np.exp(x[0])])
def jac_scalar_vector(self, x):
return np.array(
[2 * x[0], np.cos(x[0]) ** -2, np.exp(x[0])]).reshape(-1, 1)
def fun_vector_scalar(self, x):
return np.sin(x[0] * x[1]) * np.log(x[0])
def wrong_dimensions_fun(self, x):
return np.array([x**2, np.tan(x), np.exp(x)])
def jac_vector_scalar(self, x):
return np.array([
x[1] * np.cos(x[0] * x[1]) * np.log(x[0]) +
np.sin(x[0] * x[1]) / x[0],
x[0] * np.cos(x[0] * x[1]) * np.log(x[0])
])
def fun_vector_vector(self, x):
return np.array([
x[0] * np.sin(x[1]),
x[1] * np.cos(x[0]),
x[0] ** 3 * x[1] ** -0.5
])
def jac_vector_vector(self, x):
return np.array([
[np.sin(x[1]), x[0] * np.cos(x[1])],
[-x[1] * np.sin(x[0]), np.cos(x[0])],
[3 * x[0] ** 2 * x[1] ** -0.5, -0.5 * x[0] ** 3 * x[1] ** -1.5]
])
def fun_parametrized(self, x, c0, c1=1.0):
return np.array([np.exp(c0 * x[0]), np.exp(c1 * x[1])])
def jac_parametrized(self, x, c0, c1=0.1):
return np.array([
[c0 * np.exp(c0 * x[0]), 0],
[0, c1 * np.exp(c1 * x[1])]
])
def fun_with_nan(self, x):
return x if np.abs(x) <= 1e-8 else np.nan
def jac_with_nan(self, x):
return 1.0 if np.abs(x) <= 1e-8 else np.nan
def fun_zero_jacobian(self, x):
return np.array([x[0] * x[1], np.cos(x[0] * x[1])])
def jac_zero_jacobian(self, x):
return np.array([
[x[1], x[0]],
[-x[1] * np.sin(x[0] * x[1]), -x[0] * np.sin(x[0] * x[1])]
])
def fun_non_numpy(self, x):
return math.exp(x)
def jac_non_numpy(self, x):
return math.exp(x)
def test_scalar_scalar(self):
x0 = 1.0
jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0,
method='2-point')
jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0)
jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0,
method='cs')
jac_true = self.jac_scalar_scalar(x0)
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
assert_allclose(jac_diff_4, jac_true, rtol=1e-12)
def test_scalar_vector(self):
x0 = 0.5
jac_diff_2 = approx_derivative(self.fun_scalar_vector, x0,
method='2-point')
jac_diff_3 = approx_derivative(self.fun_scalar_vector, x0)
jac_diff_4 = approx_derivative(self.fun_scalar_vector, x0,
method='cs')
jac_true = self.jac_scalar_vector(np.atleast_1d(x0))
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
assert_allclose(jac_diff_4, jac_true, rtol=1e-12)
def test_vector_scalar(self):
x0 = np.array([100.0, -0.5])
jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0,
method='2-point')
jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0)
jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0,
method='cs')
jac_true = self.jac_vector_scalar(x0)
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-7)
assert_allclose(jac_diff_4, jac_true, rtol=1e-12)
def test_vector_vector(self):
x0 = np.array([-100.0, 0.2])
jac_diff_2 = approx_derivative(self.fun_vector_vector, x0,
method='2-point')
jac_diff_3 = approx_derivative(self.fun_vector_vector, x0)
jac_diff_4 = approx_derivative(self.fun_vector_vector, x0,
method='cs')
jac_true = self.jac_vector_vector(x0)
assert_allclose(jac_diff_2, jac_true, rtol=1e-5)
assert_allclose(jac_diff_3, jac_true, rtol=1e-6)
assert_allclose(jac_diff_4, jac_true, rtol=1e-12)
def test_wrong_dimensions(self):
x0 = 1.0
assert_raises(RuntimeError, approx_derivative,
self.wrong_dimensions_fun, x0)
f0 = self.wrong_dimensions_fun(np.atleast_1d(x0))
assert_raises(ValueError, approx_derivative,
self.wrong_dimensions_fun, x0, f0=f0)
def test_custom_rel_step(self):
x0 = np.array([-0.1, 0.1])
jac_diff_2 = approx_derivative(self.fun_vector_vector, x0,
method='2-point', rel_step=1e-4)
jac_diff_3 = approx_derivative(self.fun_vector_vector, x0,
rel_step=1e-4)
jac_true = self.jac_vector_vector(x0)
assert_allclose(jac_diff_2, jac_true, rtol=1e-2)
assert_allclose(jac_diff_3, jac_true, rtol=1e-4)
def test_options(self):
x0 = np.array([1.0, 1.0])
c0 = -1.0
c1 = 1.0
lb = 0.0
ub = 2.0
f0 = self.fun_parametrized(x0, c0, c1=c1)
rel_step = np.array([-1e-6, 1e-7])
jac_true = self.jac_parametrized(x0, c0, c1)
jac_diff_2 = approx_derivative(
self.fun_parametrized, x0, method='2-point', rel_step=rel_step,
f0=f0, args=(c0,), kwargs=dict(c1=c1), bounds=(lb, ub))
jac_diff_3 = approx_derivative(
self.fun_parametrized, x0, rel_step=rel_step,
f0=f0, args=(c0,), kwargs=dict(c1=c1), bounds=(lb, ub))
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
def test_with_bounds_2_point(self):
lb = -np.ones(2)
ub = np.ones(2)
x0 = np.array([-2.0, 0.2])
assert_raises(ValueError, approx_derivative,
self.fun_vector_vector, x0, bounds=(lb, ub))
x0 = np.array([-1.0, 1.0])
jac_diff = approx_derivative(self.fun_vector_vector, x0,
method='2-point', bounds=(lb, ub))
jac_true = self.jac_vector_vector(x0)
assert_allclose(jac_diff, jac_true, rtol=1e-6)
def test_with_bounds_3_point(self):
lb = np.array([1.0, 1.0])
ub = np.array([2.0, 2.0])
x0 = np.array([1.0, 2.0])
jac_true = self.jac_vector_vector(x0)
jac_diff = approx_derivative(self.fun_vector_vector, x0)
assert_allclose(jac_diff, jac_true, rtol=1e-9)
jac_diff = approx_derivative(self.fun_vector_vector, x0,
bounds=(lb, np.inf))
assert_allclose(jac_diff, jac_true, rtol=1e-9)
jac_diff = approx_derivative(self.fun_vector_vector, x0,
bounds=(-np.inf, ub))
assert_allclose(jac_diff, jac_true, rtol=1e-9)
jac_diff = approx_derivative(self.fun_vector_vector, x0,
bounds=(lb, ub))
assert_allclose(jac_diff, jac_true, rtol=1e-9)
def test_tight_bounds(self):
x0 = np.array([10.0, 10.0])
lb = x0 - 3e-9
ub = x0 + 2e-9
jac_true = self.jac_vector_vector(x0)
jac_diff = approx_derivative(
self.fun_vector_vector, x0, method='2-point', bounds=(lb, ub))
assert_allclose(jac_diff, jac_true, rtol=1e-6)
jac_diff = approx_derivative(
self.fun_vector_vector, x0, method='2-point',
rel_step=1e-6, bounds=(lb, ub))
assert_allclose(jac_diff, jac_true, rtol=1e-6)
jac_diff = approx_derivative(
self.fun_vector_vector, x0, bounds=(lb, ub))
assert_allclose(jac_diff, jac_true, rtol=1e-6)
jac_diff = approx_derivative(
self.fun_vector_vector, x0, rel_step=1e-6, bounds=(lb, ub))
assert_allclose(jac_true, jac_diff, rtol=1e-6)
def test_bound_switches(self):
lb = -1e-8
ub = 1e-8
x0 = 0.0
jac_true = self.jac_with_nan(x0)
jac_diff_2 = approx_derivative(
self.fun_with_nan, x0, method='2-point', rel_step=1e-6,
bounds=(lb, ub))
jac_diff_3 = approx_derivative(
self.fun_with_nan, x0, rel_step=1e-6, bounds=(lb, ub))
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
x0 = 1e-8
jac_true = self.jac_with_nan(x0)
jac_diff_2 = approx_derivative(
self.fun_with_nan, x0, method='2-point', rel_step=1e-6,
bounds=(lb, ub))
jac_diff_3 = approx_derivative(
self.fun_with_nan, x0, rel_step=1e-6, bounds=(lb, ub))
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-9)
def test_non_numpy(self):
x0 = 1.0
jac_true = self.jac_non_numpy(x0)
jac_diff_2 = approx_derivative(self.jac_non_numpy, x0,
method='2-point')
jac_diff_3 = approx_derivative(self.jac_non_numpy, x0)
assert_allclose(jac_diff_2, jac_true, rtol=1e-6)
assert_allclose(jac_diff_3, jac_true, rtol=1e-8)
# math.exp cannot handle complex arguments, hence this raises
assert_raises(TypeError, approx_derivative, self.jac_non_numpy, x0,
**dict(method='cs'))
def test_check_derivative(self):
x0 = np.array([-10.0, 10])
accuracy = check_derivative(self.fun_vector_vector,
self.jac_vector_vector, x0)
assert_(accuracy < 1e-9)
accuracy = check_derivative(self.fun_vector_vector,
self.jac_vector_vector, x0)
assert_(accuracy < 1e-6)
x0 = np.array([0.0, 0.0])
accuracy = check_derivative(self.fun_zero_jacobian,
self.jac_zero_jacobian, x0)
assert_(accuracy == 0)
accuracy = check_derivative(self.fun_zero_jacobian,
self.jac_zero_jacobian, x0)
assert_(accuracy == 0)
class TestApproxDerivativeSparse(object):
# Example from Numerical Optimization 2nd edition, p. 198.
def __init__(self):
np.random.seed(0)
self.n = 50
self.lb = -0.1 * (1 + np.arange(self.n))
self.ub = 0.1 * (1 + np.arange(self.n))
self.x0 = np.empty(self.n)
self.x0[::2] = (1 - 1e-7) * self.lb[::2]
self.x0[1::2] = (1 - 1e-7) * self.ub[1::2]
self.J_true = self.jac(self.x0)
def fun(self, x):
e = x[1:]**3 - x[:-1]**2
return np.hstack((0, 3 * e)) + np.hstack((2 * e, 0))
def jac(self, x):
n = x.size
J = np.zeros((n, n))
J[0, 0] = -4 * x[0]
J[0, 1] = 6 * x[1]**2
for i in range(1, n - 1):
J[i, i - 1] = -6 * x[i-1]
J[i, i] = 9 * x[i]**2 - 4 * x[i]
J[i, i + 1] = 6 * x[i+1]**2
J[-1, -1] = 9 * x[-1]**2
J[-1, -2] = -6 * x[-2]
return J
def structure(self, 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
def test_all(self):
A = self.structure(self.n)
order = np.arange(self.n)
groups_1 = group_columns(A, order)
np.random.shuffle(order)
groups_2 = group_columns(A, order)
for method, groups, l, u in product(
['2-point', '3-point', 'cs'], [groups_1, groups_2],
[-np.inf, self.lb], [np.inf, self.ub]):
J = approx_derivative(self.fun, self.x0, method=method,
bounds=(l, u), sparsity=(A, groups))
assert_(isinstance(J, csr_matrix))
assert_allclose(J.toarray(), self.J_true, rtol=1e-6)
rel_step = 1e-8 * np.ones_like(self.x0)
rel_step[::2] *= -1
J = approx_derivative(self.fun, self.x0, method=method,
rel_step=rel_step, sparsity=(A, groups))
assert_allclose(J.toarray(), self.J_true, rtol=1e-5)
def test_no_precomputed_groups(self):
A = self.structure(self.n)
J = approx_derivative(self.fun, self.x0, sparsity=A)
assert_allclose(J.toarray(), self.J_true, rtol=1e-6)
def test_equivalence(self):
structure = np.ones((self.n, self.n), dtype=int)
groups = np.arange(self.n)
for method in ['2-point', '3-point', 'cs']:
J_dense = approx_derivative(self.fun, self.x0, method=method)
J_sparse = approx_derivative(
self.fun, self.x0, sparsity=(structure, groups), method=method)
assert_equal(J_dense, J_sparse.toarray())
def test_check_derivative(self):
def jac(x):
return csr_matrix(self.jac(x))
accuracy = check_derivative(self.fun, jac, self.x0,
bounds=(self.lb, self.ub))
assert_(accuracy < 1e-9)
accuracy = check_derivative(self.fun, jac, self.x0,
bounds=(self.lb, self.ub))
assert_(accuracy < 1e-9)
if __name__ == '__main__':
run_module_suite()
|
