ai-content-maker/.venv/Lib/site-packages/scipy/optimize/tests/test__numdiff.py

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2024-05-03 04:18:51 +03:00
import math
from itertools import product
import numpy as np
from numpy.testing import assert_allclose, assert_equal, assert_
from pytest import raises as assert_raises
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, _eps_for_method, _compute_absolute_step)
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)
def test_correct_fp_eps():
# check that relative step size is correct for FP size
EPS = np.finfo(np.float64).eps
relative_step = {"2-point": EPS**0.5,
"3-point": EPS**(1/3),
"cs": EPS**0.5}
for method in ['2-point', '3-point', 'cs']:
assert_allclose(
_eps_for_method(np.float64, np.float64, method),
relative_step[method])
assert_allclose(
_eps_for_method(np.complex128, np.complex128, method),
relative_step[method]
)
# check another FP size
EPS = np.finfo(np.float32).eps
relative_step = {"2-point": EPS**0.5,
"3-point": EPS**(1/3),
"cs": EPS**0.5}
for method in ['2-point', '3-point', 'cs']:
assert_allclose(
_eps_for_method(np.float64, np.float32, method),
relative_step[method]
)
assert_allclose(
_eps_for_method(np.float32, np.float64, method),
relative_step[method]
)
assert_allclose(
_eps_for_method(np.float32, np.float32, method),
relative_step[method]
)
class TestAdjustSchemeToBounds:
def test_no_bounds(self):
x0 = np.zeros(3)
h = np.full(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:
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 jac_non_numpy(self, x):
# x can be a scalar or an array [val].
# Cast to true scalar before handing over to math.exp
xp = np.asarray(x).item()
return math.exp(xp)
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_scalar_abs_step(self):
# can approx_derivative use abs_step?
x0 = 1.0
jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0,
method='2-point', abs_step=1.49e-8)
jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0,
abs_step=1.49e-8)
jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0,
method='cs', abs_step=1.49e-8)
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_scalar_abs_step(self):
# can approx_derivative use abs_step?
x0 = np.array([100.0, -0.5])
jac_diff_2 = approx_derivative(self.fun_vector_scalar, x0,
method='2-point', abs_step=1.49e-8)
jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0,
abs_step=1.49e-8, rel_step=np.inf)
jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0,
method='cs', abs_step=1.49e-8)
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=3e-9)
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_fp(self):
# checks that approx_derivative works for FP size other than 64.
# Example is derived from the minimal working example in gh12991.
np.random.seed(1)
def func(p, x):
return p[0] + p[1] * x
def err(p, x, y):
return func(p, x) - y
x = np.linspace(0, 1, 100, dtype=np.float64)
y = np.random.random(100).astype(np.float64)
p0 = np.array([-1.0, -1.0])
jac_fp64 = approx_derivative(err, p0, method='2-point', args=(x, y))
# parameter vector is float32, func output is float64
jac_fp = approx_derivative(err, p0.astype(np.float32),
method='2-point', args=(x, y))
assert err(p0, x, y).dtype == np.float64
assert_allclose(jac_fp, jac_fp64, atol=1e-3)
# parameter vector is float64, func output is float32
def err_fp32(p):
assert p.dtype == np.float32
return err(p, x, y).astype(np.float32)
jac_fp = approx_derivative(err_fp32, p0.astype(np.float32),
method='2-point')
assert_allclose(jac_fp, jac_fp64, atol=1e-3)
# check upper bound of error on the derivative for 2-point
def f(x):
return np.sin(x)
def g(x):
return np.cos(x)
def hess(x):
return -np.sin(x)
def calc_atol(h, x0, f, hess, EPS):
# truncation error
t0 = h / 2 * max(np.abs(hess(x0)), np.abs(hess(x0 + h)))
# roundoff error. There may be a divisor (>1) missing from
# the following line, so this contribution is possibly
# overestimated
t1 = EPS / h * max(np.abs(f(x0)), np.abs(f(x0 + h)))
return t0 + t1
for dtype in [np.float16, np.float32, np.float64]:
EPS = np.finfo(dtype).eps
x0 = np.array(1.0).astype(dtype)
h = _compute_absolute_step(None, x0, f(x0), '2-point')
atol = calc_atol(h, x0, f, hess, EPS)
err = approx_derivative(f, x0, method='2-point',
abs_step=h) - g(x0)
assert abs(err) < atol
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:
# Example from Numerical Optimization 2nd edition, p. 198.
def setup_method(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 = np.full_like(self.x0, 1e-8)
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_allclose(J_dense, J_sparse.toarray(),
rtol=5e-16, atol=7e-15)
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)
class TestApproxDerivativeLinearOperator:
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 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 test_scalar_scalar(self):
x0 = 1.0
jac_diff_2 = approx_derivative(self.fun_scalar_scalar, x0,
method='2-point',
as_linear_operator=True)
jac_diff_3 = approx_derivative(self.fun_scalar_scalar, x0,
as_linear_operator=True)
jac_diff_4 = approx_derivative(self.fun_scalar_scalar, x0,
method='cs',
as_linear_operator=True)
jac_true = self.jac_scalar_scalar(x0)
np.random.seed(1)
for i in range(10):
p = np.random.uniform(-10, 10, size=(1,))
assert_allclose(jac_diff_2.dot(p), jac_true*p,
rtol=1e-5)
assert_allclose(jac_diff_3.dot(p), jac_true*p,
rtol=5e-6)
assert_allclose(jac_diff_4.dot(p), jac_true*p,
rtol=5e-6)
def test_scalar_vector(self):
x0 = 0.5
jac_diff_2 = approx_derivative(self.fun_scalar_vector, x0,
method='2-point',
as_linear_operator=True)
jac_diff_3 = approx_derivative(self.fun_scalar_vector, x0,
as_linear_operator=True)
jac_diff_4 = approx_derivative(self.fun_scalar_vector, x0,
method='cs',
as_linear_operator=True)
jac_true = self.jac_scalar_vector(np.atleast_1d(x0))
np.random.seed(1)
for i in range(10):
p = np.random.uniform(-10, 10, size=(1,))
assert_allclose(jac_diff_2.dot(p), jac_true.dot(p),
rtol=1e-5)
assert_allclose(jac_diff_3.dot(p), jac_true.dot(p),
rtol=5e-6)
assert_allclose(jac_diff_4.dot(p), jac_true.dot(p),
rtol=5e-6)
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',
as_linear_operator=True)
jac_diff_3 = approx_derivative(self.fun_vector_scalar, x0,
as_linear_operator=True)
jac_diff_4 = approx_derivative(self.fun_vector_scalar, x0,
method='cs',
as_linear_operator=True)
jac_true = self.jac_vector_scalar(x0)
np.random.seed(1)
for i in range(10):
p = np.random.uniform(-10, 10, size=x0.shape)
assert_allclose(jac_diff_2.dot(p), np.atleast_1d(jac_true.dot(p)),
rtol=1e-5)
assert_allclose(jac_diff_3.dot(p), np.atleast_1d(jac_true.dot(p)),
rtol=5e-6)
assert_allclose(jac_diff_4.dot(p), np.atleast_1d(jac_true.dot(p)),
rtol=1e-7)
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',
as_linear_operator=True)
jac_diff_3 = approx_derivative(self.fun_vector_vector, x0,
as_linear_operator=True)
jac_diff_4 = approx_derivative(self.fun_vector_vector, x0,
method='cs',
as_linear_operator=True)
jac_true = self.jac_vector_vector(x0)
np.random.seed(1)
for i in range(10):
p = np.random.uniform(-10, 10, size=x0.shape)
assert_allclose(jac_diff_2.dot(p), jac_true.dot(p), rtol=1e-5)
assert_allclose(jac_diff_3.dot(p), jac_true.dot(p), rtol=1e-6)
assert_allclose(jac_diff_4.dot(p), jac_true.dot(p), rtol=1e-7)
def test_exception(self):
x0 = np.array([-100.0, 0.2])
assert_raises(ValueError, approx_derivative,
self.fun_vector_vector, x0,
method='2-point', bounds=(1, np.inf))
def test_absolute_step_sign():
# test for gh12487
# if an absolute step is specified for 2-point differences make sure that
# the side corresponds to the step. i.e. if step is positive then forward
# differences should be used, if step is negative then backwards
# differences should be used.
# function has double discontinuity at x = [-1, -1]
# first component is \/, second component is /\
def f(x):
return -np.abs(x[0] + 1) + np.abs(x[1] + 1)
# check that the forward difference is used
grad = approx_derivative(f, [-1, -1], method='2-point', abs_step=1e-8)
assert_allclose(grad, [-1.0, 1.0])
# check that the backwards difference is used
grad = approx_derivative(f, [-1, -1], method='2-point', abs_step=-1e-8)
assert_allclose(grad, [1.0, -1.0])
# check that the forwards difference is used with a step for both
# parameters
grad = approx_derivative(
f, [-1, -1], method='2-point', abs_step=[1e-8, 1e-8]
)
assert_allclose(grad, [-1.0, 1.0])
# check that we can mix forward/backwards steps.
grad = approx_derivative(
f, [-1, -1], method='2-point', abs_step=[1e-8, -1e-8]
)
assert_allclose(grad, [-1.0, -1.0])
grad = approx_derivative(
f, [-1, -1], method='2-point', abs_step=[-1e-8, 1e-8]
)
assert_allclose(grad, [1.0, 1.0])
# the forward step should reverse to a backwards step if it runs into a
# bound
# This is kind of tested in TestAdjustSchemeToBounds, but only for a lower level
# function.
grad = approx_derivative(
f, [-1, -1], method='2-point', abs_step=1e-8,
bounds=(-np.inf, -1)
)
assert_allclose(grad, [1.0, -1.0])
grad = approx_derivative(
f, [-1, -1], method='2-point', abs_step=-1e-8, bounds=(-1, np.inf)
)
assert_allclose(grad, [-1.0, 1.0])
def test__compute_absolute_step():
# tests calculation of absolute step from rel_step
methods = ['2-point', '3-point', 'cs']
x0 = np.array([1e-5, 0, 1, 1e5])
EPS = np.finfo(np.float64).eps
relative_step = {
"2-point": EPS**0.5,
"3-point": EPS**(1/3),
"cs": EPS**0.5
}
f0 = np.array(1.0)
for method in methods:
rel_step = relative_step[method]
correct_step = np.array([rel_step,
rel_step * 1.,
rel_step * 1.,
rel_step * np.abs(x0[3])])
abs_step = _compute_absolute_step(None, x0, f0, method)
assert_allclose(abs_step, correct_step)
sign_x0 = (-x0 >= 0).astype(float) * 2 - 1
abs_step = _compute_absolute_step(None, -x0, f0, method)
assert_allclose(abs_step, sign_x0 * correct_step)
# if a relative step is provided it should be used
rel_step = np.array([0.1, 1, 10, 100])
correct_step = np.array([rel_step[0] * x0[0],
relative_step['2-point'],
rel_step[2] * 1.,
rel_step[3] * np.abs(x0[3])])
abs_step = _compute_absolute_step(rel_step, x0, f0, '2-point')
assert_allclose(abs_step, correct_step)
sign_x0 = (-x0 >= 0).astype(float) * 2 - 1
abs_step = _compute_absolute_step(rel_step, -x0, f0, '2-point')
assert_allclose(abs_step, sign_x0 * correct_step)