640 lines
23 KiB
Python
640 lines
23 KiB
Python
"""Routines for numerical differentiation."""
|
|
|
|
from __future__ import division
|
|
|
|
import numpy as np
|
|
from numpy.linalg import norm
|
|
|
|
from scipy.sparse.linalg import LinearOperator
|
|
from ..sparse import issparse, csc_matrix, csr_matrix, coo_matrix, find
|
|
from ._group_columns import group_dense, group_sparse
|
|
|
|
EPS = np.finfo(np.float64).eps
|
|
|
|
|
|
def _adjust_scheme_to_bounds(x0, h, num_steps, scheme, lb, ub):
|
|
"""Adjust final difference scheme to the presence of bounds.
|
|
|
|
Parameters
|
|
----------
|
|
x0 : ndarray, shape (n,)
|
|
Point at which we wish to estimate derivative.
|
|
h : ndarray, shape (n,)
|
|
Desired finite difference steps.
|
|
num_steps : int
|
|
Number of `h` steps in one direction required to implement finite
|
|
difference scheme. For example, 2 means that we need to evaluate
|
|
f(x0 + 2 * h) or f(x0 - 2 * h)
|
|
scheme : {'1-sided', '2-sided'}
|
|
Whether steps in one or both directions are required. In other
|
|
words '1-sided' applies to forward and backward schemes, '2-sided'
|
|
applies to center schemes.
|
|
lb : ndarray, shape (n,)
|
|
Lower bounds on independent variables.
|
|
ub : ndarray, shape (n,)
|
|
Upper bounds on independent variables.
|
|
|
|
Returns
|
|
-------
|
|
h_adjusted : ndarray, shape (n,)
|
|
Adjusted step sizes. Step size decreases only if a sign flip or
|
|
switching to one-sided scheme doesn't allow to take a full step.
|
|
use_one_sided : ndarray of bool, shape (n,)
|
|
Whether to switch to one-sided scheme. Informative only for
|
|
``scheme='2-sided'``.
|
|
"""
|
|
if scheme == '1-sided':
|
|
use_one_sided = np.ones_like(h, dtype=bool)
|
|
elif scheme == '2-sided':
|
|
h = np.abs(h)
|
|
use_one_sided = np.zeros_like(h, dtype=bool)
|
|
else:
|
|
raise ValueError("`scheme` must be '1-sided' or '2-sided'.")
|
|
|
|
if np.all((lb == -np.inf) & (ub == np.inf)):
|
|
return h, use_one_sided
|
|
|
|
h_total = h * num_steps
|
|
h_adjusted = h.copy()
|
|
|
|
lower_dist = x0 - lb
|
|
upper_dist = ub - x0
|
|
|
|
if scheme == '1-sided':
|
|
x = x0 + h_total
|
|
violated = (x < lb) | (x > ub)
|
|
fitting = np.abs(h_total) <= np.maximum(lower_dist, upper_dist)
|
|
h_adjusted[violated & fitting] *= -1
|
|
|
|
forward = (upper_dist >= lower_dist) & ~fitting
|
|
h_adjusted[forward] = upper_dist[forward] / num_steps
|
|
backward = (upper_dist < lower_dist) & ~fitting
|
|
h_adjusted[backward] = -lower_dist[backward] / num_steps
|
|
elif scheme == '2-sided':
|
|
central = (lower_dist >= h_total) & (upper_dist >= h_total)
|
|
|
|
forward = (upper_dist >= lower_dist) & ~central
|
|
h_adjusted[forward] = np.minimum(
|
|
h[forward], 0.5 * upper_dist[forward] / num_steps)
|
|
use_one_sided[forward] = True
|
|
|
|
backward = (upper_dist < lower_dist) & ~central
|
|
h_adjusted[backward] = -np.minimum(
|
|
h[backward], 0.5 * lower_dist[backward] / num_steps)
|
|
use_one_sided[backward] = True
|
|
|
|
min_dist = np.minimum(upper_dist, lower_dist) / num_steps
|
|
adjusted_central = (~central & (np.abs(h_adjusted) <= min_dist))
|
|
h_adjusted[adjusted_central] = min_dist[adjusted_central]
|
|
use_one_sided[adjusted_central] = False
|
|
|
|
return h_adjusted, use_one_sided
|
|
|
|
|
|
relative_step = {"2-point": EPS**0.5,
|
|
"3-point": EPS**(1/3),
|
|
"cs": EPS**0.5}
|
|
|
|
|
|
def _compute_absolute_step(rel_step, x0, method):
|
|
if rel_step is None:
|
|
rel_step = relative_step[method]
|
|
sign_x0 = (x0 >= 0).astype(float) * 2 - 1
|
|
return rel_step * sign_x0 * np.maximum(1.0, np.abs(x0))
|
|
|
|
|
|
def _prepare_bounds(bounds, x0):
|
|
lb, ub = [np.asarray(b, dtype=float) for b in bounds]
|
|
if lb.ndim == 0:
|
|
lb = np.resize(lb, x0.shape)
|
|
|
|
if ub.ndim == 0:
|
|
ub = np.resize(ub, x0.shape)
|
|
|
|
return lb, ub
|
|
|
|
|
|
def group_columns(A, order=0):
|
|
"""Group columns of a 2-d matrix for sparse finite differencing [1]_.
|
|
|
|
Two columns are in the same group if in each row at least one of them
|
|
has zero. A greedy sequential algorithm is used to construct groups.
|
|
|
|
Parameters
|
|
----------
|
|
A : array_like or sparse matrix, shape (m, n)
|
|
Matrix of which to group columns.
|
|
order : int, iterable of int with shape (n,) or None
|
|
Permutation array which defines the order of columns enumeration.
|
|
If int or None, a random permutation is used with `order` used as
|
|
a random seed. Default is 0, that is use a random permutation but
|
|
guarantee repeatability.
|
|
|
|
Returns
|
|
-------
|
|
groups : ndarray of int, shape (n,)
|
|
Contains values from 0 to n_groups-1, where n_groups is the number
|
|
of found groups. Each value ``groups[i]`` is an index of a group to
|
|
which i-th column assigned. The procedure was helpful only if
|
|
n_groups is significantly less than n.
|
|
|
|
References
|
|
----------
|
|
.. [1] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
|
|
sparse Jacobian matrices", Journal of the Institute of Mathematics
|
|
and its Applications, 13 (1974), pp. 117-120.
|
|
"""
|
|
if issparse(A):
|
|
A = csc_matrix(A)
|
|
else:
|
|
A = np.atleast_2d(A)
|
|
A = (A != 0).astype(np.int32)
|
|
|
|
if A.ndim != 2:
|
|
raise ValueError("`A` must be 2-dimensional.")
|
|
|
|
m, n = A.shape
|
|
|
|
if order is None or np.isscalar(order):
|
|
rng = np.random.RandomState(order)
|
|
order = rng.permutation(n)
|
|
else:
|
|
order = np.asarray(order)
|
|
if order.shape != (n,):
|
|
raise ValueError("`order` has incorrect shape.")
|
|
|
|
A = A[:, order]
|
|
|
|
if issparse(A):
|
|
groups = group_sparse(m, n, A.indices, A.indptr)
|
|
else:
|
|
groups = group_dense(m, n, A)
|
|
|
|
groups[order] = groups.copy()
|
|
|
|
return groups
|
|
|
|
|
|
def approx_derivative(fun, x0, method='3-point', rel_step=None, f0=None,
|
|
bounds=(-np.inf, np.inf), sparsity=None,
|
|
as_linear_operator=False, args=(), kwargs={}):
|
|
"""Compute finite difference approximation of the derivatives of a
|
|
vector-valued function.
|
|
|
|
If a function maps from R^n to R^m, its derivatives form m-by-n matrix
|
|
called the Jacobian, where an element (i, j) is a partial derivative of
|
|
f[i] with respect to x[j].
|
|
|
|
Parameters
|
|
----------
|
|
fun : callable
|
|
Function of which to estimate the derivatives. The argument x
|
|
passed to this function is ndarray of shape (n,) (never a scalar
|
|
even if n=1). It must return 1-d array_like of shape (m,) or a scalar.
|
|
x0 : array_like of shape (n,) or float
|
|
Point at which to estimate the derivatives. Float will be converted
|
|
to a 1-d array.
|
|
method : {'3-point', '2-point', 'cs'}, optional
|
|
Finite difference method to use:
|
|
- '2-point' - use the first order accuracy forward or backward
|
|
difference.
|
|
- '3-point' - use central difference in interior points and the
|
|
second order accuracy forward or backward difference
|
|
near the boundary.
|
|
- 'cs' - use a complex-step finite difference scheme. This assumes
|
|
that the user function is real-valued and can be
|
|
analytically continued to the complex plane. Otherwise,
|
|
produces bogus results.
|
|
rel_step : None or array_like, optional
|
|
Relative step size to use. The absolute step size is computed as
|
|
``h = rel_step * sign(x0) * max(1, abs(x0))``, possibly adjusted to
|
|
fit into the bounds. For ``method='3-point'`` the sign of `h` is
|
|
ignored. If None (default) then step is selected automatically,
|
|
see Notes.
|
|
f0 : None or array_like, optional
|
|
If not None it is assumed to be equal to ``fun(x0)``, in this case
|
|
the ``fun(x0)`` is not called. Default is None.
|
|
bounds : tuple of array_like, optional
|
|
Lower and upper bounds on independent variables. Defaults to no bounds.
|
|
Each bound must match the size of `x0` or be a scalar, in the latter
|
|
case the bound will be the same for all variables. Use it to limit the
|
|
range of function evaluation. Bounds checking is not implemented
|
|
when `as_linear_operator` is True.
|
|
sparsity : {None, array_like, sparse matrix, 2-tuple}, optional
|
|
Defines a sparsity structure of the Jacobian matrix. If the Jacobian
|
|
matrix is known to have only few non-zero elements in each row, then
|
|
it's possible to estimate its several columns by a single function
|
|
evaluation [3]_. To perform such economic computations two ingredients
|
|
are required:
|
|
|
|
* structure : array_like or sparse matrix of shape (m, n). A zero
|
|
element means that a corresponding element of the Jacobian
|
|
identically equals to zero.
|
|
* groups : array_like of shape (n,). A column grouping for a given
|
|
sparsity structure, use `group_columns` to obtain it.
|
|
|
|
A single array or a sparse matrix is interpreted as a sparsity
|
|
structure, and groups are computed inside the function. A tuple is
|
|
interpreted as (structure, groups). If None (default), a standard
|
|
dense differencing will be used.
|
|
|
|
Note, that sparse differencing makes sense only for large Jacobian
|
|
matrices where each row contains few non-zero elements.
|
|
as_linear_operator : bool, optional
|
|
When True the function returns an `scipy.sparse.linalg.LinearOperator`.
|
|
Otherwise it returns a dense array or a sparse matrix depending on
|
|
`sparsity`. The linear operator provides an efficient way of computing
|
|
``J.dot(p)`` for any vector ``p`` of shape (n,), but does not allow
|
|
direct access to individual elements of the matrix. By default
|
|
`as_linear_operator` is False.
|
|
args, kwargs : tuple and dict, optional
|
|
Additional arguments passed to `fun`. Both empty by default.
|
|
The calling signature is ``fun(x, *args, **kwargs)``.
|
|
|
|
Returns
|
|
-------
|
|
J : {ndarray, sparse matrix, LinearOperator}
|
|
Finite difference approximation of the Jacobian matrix.
|
|
If `as_linear_operator` is True returns a LinearOperator
|
|
with shape (m, n). Otherwise it returns a dense array or sparse
|
|
matrix depending on how `sparsity` is defined. If `sparsity`
|
|
is None then a ndarray with shape (m, n) is returned. If
|
|
`sparsity` is not None returns a csr_matrix with shape (m, n).
|
|
For sparse matrices and linear operators it is always returned as
|
|
a 2-dimensional structure, for ndarrays, if m=1 it is returned
|
|
as a 1-dimensional gradient array with shape (n,).
|
|
|
|
See Also
|
|
--------
|
|
check_derivative : Check correctness of a function computing derivatives.
|
|
|
|
Notes
|
|
-----
|
|
If `rel_step` is not provided, it assigned to ``EPS**(1/s)``, where EPS is
|
|
machine epsilon for float64 numbers, s=2 for '2-point' method and s=3 for
|
|
'3-point' method. Such relative step approximately minimizes a sum of
|
|
truncation and round-off errors, see [1]_.
|
|
|
|
A finite difference scheme for '3-point' method is selected automatically.
|
|
The well-known central difference scheme is used for points sufficiently
|
|
far from the boundary, and 3-point forward or backward scheme is used for
|
|
points near the boundary. Both schemes have the second-order accuracy in
|
|
terms of Taylor expansion. Refer to [2]_ for the formulas of 3-point
|
|
forward and backward difference schemes.
|
|
|
|
For dense differencing when m=1 Jacobian is returned with a shape (n,),
|
|
on the other hand when n=1 Jacobian is returned with a shape (m, 1).
|
|
Our motivation is the following: a) It handles a case of gradient
|
|
computation (m=1) in a conventional way. b) It clearly separates these two
|
|
different cases. b) In all cases np.atleast_2d can be called to get 2-d
|
|
Jacobian with correct dimensions.
|
|
|
|
References
|
|
----------
|
|
.. [1] W. H. Press et. al. "Numerical Recipes. The Art of Scientific
|
|
Computing. 3rd edition", sec. 5.7.
|
|
|
|
.. [2] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of
|
|
sparse Jacobian matrices", Journal of the Institute of Mathematics
|
|
and its Applications, 13 (1974), pp. 117-120.
|
|
|
|
.. [3] B. Fornberg, "Generation of Finite Difference Formulas on
|
|
Arbitrarily Spaced Grids", Mathematics of Computation 51, 1988.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.optimize import approx_derivative
|
|
>>>
|
|
>>> def f(x, c1, c2):
|
|
... return np.array([x[0] * np.sin(c1 * x[1]),
|
|
... x[0] * np.cos(c2 * x[1])])
|
|
...
|
|
>>> x0 = np.array([1.0, 0.5 * np.pi])
|
|
>>> approx_derivative(f, x0, args=(1, 2))
|
|
array([[ 1., 0.],
|
|
[-1., 0.]])
|
|
|
|
Bounds can be used to limit the region of function evaluation.
|
|
In the example below we compute left and right derivative at point 1.0.
|
|
|
|
>>> def g(x):
|
|
... return x**2 if x >= 1 else x
|
|
...
|
|
>>> x0 = 1.0
|
|
>>> approx_derivative(g, x0, bounds=(-np.inf, 1.0))
|
|
array([ 1.])
|
|
>>> approx_derivative(g, x0, bounds=(1.0, np.inf))
|
|
array([ 2.])
|
|
"""
|
|
if method not in ['2-point', '3-point', 'cs']:
|
|
raise ValueError("Unknown method '%s'. " % method)
|
|
|
|
x0 = np.atleast_1d(x0)
|
|
if x0.ndim > 1:
|
|
raise ValueError("`x0` must have at most 1 dimension.")
|
|
|
|
lb, ub = _prepare_bounds(bounds, x0)
|
|
|
|
if lb.shape != x0.shape or ub.shape != x0.shape:
|
|
raise ValueError("Inconsistent shapes between bounds and `x0`.")
|
|
|
|
if as_linear_operator and not (np.all(np.isinf(lb))
|
|
and np.all(np.isinf(ub))):
|
|
raise ValueError("Bounds not supported when "
|
|
"`as_linear_operator` is True.")
|
|
|
|
def fun_wrapped(x):
|
|
f = np.atleast_1d(fun(x, *args, **kwargs))
|
|
if f.ndim > 1:
|
|
raise RuntimeError("`fun` return value has "
|
|
"more than 1 dimension.")
|
|
return f
|
|
|
|
if f0 is None:
|
|
f0 = fun_wrapped(x0)
|
|
else:
|
|
f0 = np.atleast_1d(f0)
|
|
if f0.ndim > 1:
|
|
raise ValueError("`f0` passed has more than 1 dimension.")
|
|
|
|
if np.any((x0 < lb) | (x0 > ub)):
|
|
raise ValueError("`x0` violates bound constraints.")
|
|
|
|
if as_linear_operator:
|
|
if rel_step is None:
|
|
rel_step = relative_step[method]
|
|
|
|
return _linear_operator_difference(fun_wrapped, x0,
|
|
f0, rel_step, method)
|
|
else:
|
|
h = _compute_absolute_step(rel_step, x0, method)
|
|
|
|
if method == '2-point':
|
|
h, use_one_sided = _adjust_scheme_to_bounds(
|
|
x0, h, 1, '1-sided', lb, ub)
|
|
elif method == '3-point':
|
|
h, use_one_sided = _adjust_scheme_to_bounds(
|
|
x0, h, 1, '2-sided', lb, ub)
|
|
elif method == 'cs':
|
|
use_one_sided = False
|
|
|
|
if sparsity is None:
|
|
return _dense_difference(fun_wrapped, x0, f0, h,
|
|
use_one_sided, method)
|
|
else:
|
|
if not issparse(sparsity) and len(sparsity) == 2:
|
|
structure, groups = sparsity
|
|
else:
|
|
structure = sparsity
|
|
groups = group_columns(sparsity)
|
|
|
|
if issparse(structure):
|
|
structure = csc_matrix(structure)
|
|
else:
|
|
structure = np.atleast_2d(structure)
|
|
|
|
groups = np.atleast_1d(groups)
|
|
return _sparse_difference(fun_wrapped, x0, f0, h,
|
|
use_one_sided, structure,
|
|
groups, method)
|
|
|
|
|
|
def _linear_operator_difference(fun, x0, f0, h, method):
|
|
m = f0.size
|
|
n = x0.size
|
|
|
|
if method == '2-point':
|
|
def matvec(p):
|
|
if np.array_equal(p, np.zeros_like(p)):
|
|
return np.zeros(m)
|
|
dx = h / norm(p)
|
|
x = x0 + dx*p
|
|
df = fun(x) - f0
|
|
return df / dx
|
|
|
|
elif method == '3-point':
|
|
def matvec(p):
|
|
if np.array_equal(p, np.zeros_like(p)):
|
|
return np.zeros(m)
|
|
dx = 2*h / norm(p)
|
|
x1 = x0 - (dx/2)*p
|
|
x2 = x0 + (dx/2)*p
|
|
f1 = fun(x1)
|
|
f2 = fun(x2)
|
|
df = f2 - f1
|
|
return df / dx
|
|
|
|
elif method == 'cs':
|
|
def matvec(p):
|
|
if np.array_equal(p, np.zeros_like(p)):
|
|
return np.zeros(m)
|
|
dx = h / norm(p)
|
|
x = x0 + dx*p*1.j
|
|
f1 = fun(x)
|
|
df = f1.imag
|
|
return df / dx
|
|
|
|
else:
|
|
raise RuntimeError("Never be here.")
|
|
|
|
return LinearOperator((m, n), matvec)
|
|
|
|
|
|
def _dense_difference(fun, x0, f0, h, use_one_sided, method):
|
|
m = f0.size
|
|
n = x0.size
|
|
J_transposed = np.empty((n, m))
|
|
h_vecs = np.diag(h)
|
|
|
|
for i in range(h.size):
|
|
if method == '2-point':
|
|
x = x0 + h_vecs[i]
|
|
dx = x[i] - x0[i] # Recompute dx as exactly representable number.
|
|
df = fun(x) - f0
|
|
elif method == '3-point' and use_one_sided[i]:
|
|
x1 = x0 + h_vecs[i]
|
|
x2 = x0 + 2 * h_vecs[i]
|
|
dx = x2[i] - x0[i]
|
|
f1 = fun(x1)
|
|
f2 = fun(x2)
|
|
df = -3.0 * f0 + 4 * f1 - f2
|
|
elif method == '3-point' and not use_one_sided[i]:
|
|
x1 = x0 - h_vecs[i]
|
|
x2 = x0 + h_vecs[i]
|
|
dx = x2[i] - x1[i]
|
|
f1 = fun(x1)
|
|
f2 = fun(x2)
|
|
df = f2 - f1
|
|
elif method == 'cs':
|
|
f1 = fun(x0 + h_vecs[i]*1.j)
|
|
df = f1.imag
|
|
dx = h_vecs[i, i]
|
|
else:
|
|
raise RuntimeError("Never be here.")
|
|
|
|
J_transposed[i] = df / dx
|
|
|
|
if m == 1:
|
|
J_transposed = np.ravel(J_transposed)
|
|
|
|
return J_transposed.T
|
|
|
|
|
|
def _sparse_difference(fun, x0, f0, h, use_one_sided,
|
|
structure, groups, method):
|
|
m = f0.size
|
|
n = x0.size
|
|
row_indices = []
|
|
col_indices = []
|
|
fractions = []
|
|
|
|
n_groups = np.max(groups) + 1
|
|
for group in range(n_groups):
|
|
# Perturb variables which are in the same group simultaneously.
|
|
e = np.equal(group, groups)
|
|
h_vec = h * e
|
|
if method == '2-point':
|
|
x = x0 + h_vec
|
|
dx = x - x0
|
|
df = fun(x) - f0
|
|
# The result is written to columns which correspond to perturbed
|
|
# variables.
|
|
cols, = np.nonzero(e)
|
|
# Find all non-zero elements in selected columns of Jacobian.
|
|
i, j, _ = find(structure[:, cols])
|
|
# Restore column indices in the full array.
|
|
j = cols[j]
|
|
elif method == '3-point':
|
|
# Here we do conceptually the same but separate one-sided
|
|
# and two-sided schemes.
|
|
x1 = x0.copy()
|
|
x2 = x0.copy()
|
|
|
|
mask_1 = use_one_sided & e
|
|
x1[mask_1] += h_vec[mask_1]
|
|
x2[mask_1] += 2 * h_vec[mask_1]
|
|
|
|
mask_2 = ~use_one_sided & e
|
|
x1[mask_2] -= h_vec[mask_2]
|
|
x2[mask_2] += h_vec[mask_2]
|
|
|
|
dx = np.zeros(n)
|
|
dx[mask_1] = x2[mask_1] - x0[mask_1]
|
|
dx[mask_2] = x2[mask_2] - x1[mask_2]
|
|
|
|
f1 = fun(x1)
|
|
f2 = fun(x2)
|
|
|
|
cols, = np.nonzero(e)
|
|
i, j, _ = find(structure[:, cols])
|
|
j = cols[j]
|
|
|
|
mask = use_one_sided[j]
|
|
df = np.empty(m)
|
|
|
|
rows = i[mask]
|
|
df[rows] = -3 * f0[rows] + 4 * f1[rows] - f2[rows]
|
|
|
|
rows = i[~mask]
|
|
df[rows] = f2[rows] - f1[rows]
|
|
elif method == 'cs':
|
|
f1 = fun(x0 + h_vec*1.j)
|
|
df = f1.imag
|
|
dx = h_vec
|
|
cols, = np.nonzero(e)
|
|
i, j, _ = find(structure[:, cols])
|
|
j = cols[j]
|
|
else:
|
|
raise ValueError("Never be here.")
|
|
|
|
# All that's left is to compute the fraction. We store i, j and
|
|
# fractions as separate arrays and later construct coo_matrix.
|
|
row_indices.append(i)
|
|
col_indices.append(j)
|
|
fractions.append(df[i] / dx[j])
|
|
|
|
row_indices = np.hstack(row_indices)
|
|
col_indices = np.hstack(col_indices)
|
|
fractions = np.hstack(fractions)
|
|
J = coo_matrix((fractions, (row_indices, col_indices)), shape=(m, n))
|
|
return csr_matrix(J)
|
|
|
|
|
|
def check_derivative(fun, jac, x0, bounds=(-np.inf, np.inf), args=(),
|
|
kwargs={}):
|
|
"""Check correctness of a function computing derivatives (Jacobian or
|
|
gradient) by comparison with a finite difference approximation.
|
|
|
|
Parameters
|
|
----------
|
|
fun : callable
|
|
Function of which to estimate the derivatives. The argument x
|
|
passed to this function is ndarray of shape (n,) (never a scalar
|
|
even if n=1). It must return 1-d array_like of shape (m,) or a scalar.
|
|
jac : callable
|
|
Function which computes Jacobian matrix of `fun`. It must work with
|
|
argument x the same way as `fun`. The return value must be array_like
|
|
or sparse matrix with an appropriate shape.
|
|
x0 : array_like of shape (n,) or float
|
|
Point at which to estimate the derivatives. Float will be converted
|
|
to 1-d array.
|
|
bounds : 2-tuple of array_like, optional
|
|
Lower and upper bounds on independent variables. Defaults to no bounds.
|
|
Each bound must match the size of `x0` or be a scalar, in the latter
|
|
case the bound will be the same for all variables. Use it to limit the
|
|
range of function evaluation.
|
|
args, kwargs : tuple and dict, optional
|
|
Additional arguments passed to `fun` and `jac`. Both empty by default.
|
|
The calling signature is ``fun(x, *args, **kwargs)`` and the same
|
|
for `jac`.
|
|
|
|
Returns
|
|
-------
|
|
accuracy : float
|
|
The maximum among all relative errors for elements with absolute values
|
|
higher than 1 and absolute errors for elements with absolute values
|
|
less or equal than 1. If `accuracy` is on the order of 1e-6 or lower,
|
|
then it is likely that your `jac` implementation is correct.
|
|
|
|
See Also
|
|
--------
|
|
approx_derivative : Compute finite difference approximation of derivative.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy as np
|
|
>>> from scipy.optimize import check_derivative
|
|
>>>
|
|
>>>
|
|
>>> def f(x, c1, c2):
|
|
... return np.array([x[0] * np.sin(c1 * x[1]),
|
|
... x[0] * np.cos(c2 * x[1])])
|
|
...
|
|
>>> def jac(x, c1, c2):
|
|
... return np.array([
|
|
... [np.sin(c1 * x[1]), c1 * x[0] * np.cos(c1 * x[1])],
|
|
... [np.cos(c2 * x[1]), -c2 * x[0] * np.sin(c2 * x[1])]
|
|
... ])
|
|
...
|
|
>>>
|
|
>>> x0 = np.array([1.0, 0.5 * np.pi])
|
|
>>> check_derivative(f, jac, x0, args=(1, 2))
|
|
2.4492935982947064e-16
|
|
"""
|
|
J_to_test = jac(x0, *args, **kwargs)
|
|
if issparse(J_to_test):
|
|
J_diff = approx_derivative(fun, x0, bounds=bounds, sparsity=J_to_test,
|
|
args=args, kwargs=kwargs)
|
|
J_to_test = csr_matrix(J_to_test)
|
|
abs_err = J_to_test - J_diff
|
|
i, j, abs_err_data = find(abs_err)
|
|
J_diff_data = np.asarray(J_diff[i, j]).ravel()
|
|
return np.max(np.abs(abs_err_data) /
|
|
np.maximum(1, np.abs(J_diff_data)))
|
|
else:
|
|
J_diff = approx_derivative(fun, x0, bounds=bounds,
|
|
args=args, kwargs=kwargs)
|
|
abs_err = np.abs(J_to_test - J_diff)
|
|
return np.max(abs_err / np.maximum(1, np.abs(J_diff)))
|