"""Routines for numerical differentiation."""
import functools
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
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 absolute 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 absolute 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
@functools.lru_cache()
def _eps_for_method(x0_dtype, f0_dtype, method):
"""
Calculates relative EPS step to use for a given data type
and numdiff step method.
Progressively smaller steps are used for larger floating point types.
Parameters
----------
f0_dtype: np.dtype
dtype of function evaluation
x0_dtype: np.dtype
dtype of parameter vector
method: {'2-point', '3-point', 'cs'}
Returns
-------
EPS: float
relative step size. May be np.float16, np.float32, np.float64
Notes
-----
The default relative step will be np.float64. However, if x0 or f0 are
smaller floating point types (np.float16, np.float32), then the smallest
floating point type is chosen.
"""
# the default EPS value
EPS = np.finfo(np.float64).eps
x0_is_fp = False
if np.issubdtype(x0_dtype, np.inexact):
# if you're a floating point type then over-ride the default EPS
EPS = np.finfo(x0_dtype).eps
x0_itemsize = np.dtype(x0_dtype).itemsize
x0_is_fp = True
if np.issubdtype(f0_dtype, np.inexact):
f0_itemsize = np.dtype(f0_dtype).itemsize
# choose the smallest itemsize between x0 and f0
if x0_is_fp and f0_itemsize < x0_itemsize:
EPS = np.finfo(f0_dtype).eps
if method in ["2-point", "cs"]:
return EPS**0.5
elif method in ["3-point"]:
return EPS**(1/3)
else:
raise RuntimeError("Unknown step method, should be one of "
"{'2-point', '3-point', 'cs'}")
def _compute_absolute_step(rel_step, x0, f0, method):
"""
Computes an absolute step from a relative step for finite difference
calculation.
Parameters
----------
rel_step: None or array-like
Relative step for the finite difference calculation
x0 : np.ndarray
Parameter vector
f0 : np.ndarray or scalar
method : {'2-point', '3-point', 'cs'}
Returns
-------
h : float
The absolute step size
Notes
-----
`h` will always be np.float64. However, if `x0` or `f0` are
smaller floating point dtypes (e.g. np.float32), then the absolute
step size will be calculated from the smallest floating point size.
"""
# this is used instead of np.sign(x0) because we need
# sign_x0 to be 1 when x0 == 0.
sign_x0 = (x0 >= 0).astype(float) * 2 - 1
rstep = _eps_for_method(x0.dtype, f0.dtype, method)
if rel_step is None:
abs_step = rstep * sign_x0 * np.maximum(1.0, np.abs(x0))
else:
# User has requested specific relative steps.
# Don't multiply by max(1, abs(x0) because if x0 < 1 then their
# requested step is not used.
abs_step = rel_step * sign_x0 * np.abs(x0)
# however we don't want an abs_step of 0, which can happen if
# rel_step is 0, or x0 is 0. Instead, substitute a realistic step
dx = ((x0 + abs_step) - x0)
abs_step = np.where(dx == 0,
rstep * sign_x0 * np.maximum(1.0, np.abs(x0)),
abs_step)
return abs_step
def _prepare_bounds(bounds, x0):
"""
Prepares new-style bounds from a two-tuple specifying the lower and upper
limits for values in x0. If a value is not bound then the lower/upper bound
will be expected to be -np.inf/np.inf.
Examples
--------
>>> _prepare_bounds([(0, 1, 2), (1, 2, np.inf)], [0.5, 1.5, 2.5])
(array([0., 1., 2.]), array([ 1., 2., inf]))
"""
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 ith 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, abs_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. If None (default) the absolute step size is
computed as ``h = rel_step * sign(x0) * max(1, abs(x0))``, with
`rel_step` being selected automatically, see Notes. Otherwise
``h = rel_step * sign(x0) * abs(x0)``. For ``method='3-point'`` the
sign of `h` is ignored. The calculated step size is possibly adjusted
to fit into the bounds.
abs_step : array_like, optional
Absolute step size to use, possibly adjusted to fit into the bounds.
For ``method='3-point'`` the sign of `abs_step` is ignored. By default
relative steps are used, only if ``abs_step is not None`` are absolute
steps used.
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-D structure, for ndarrays, if m=1 it is returned
as a 1-D 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 as ``EPS**(1/s)``, where EPS is
determined from the smallest floating point dtype of `x0` or `fun(x0)`,
``np.finfo(x0.dtype).eps``, 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]_. Relative steps are used by
default. However, absolute steps are used when ``abs_step is not None``.
If any of the absolute or relative steps produces an indistinguishable
difference from the original `x0`, ``(x0 + dx) - x0 == 0``, then a
automatic step size is substituted for that particular entry.
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._numdiff 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 = _eps_for_method(x0.dtype, f0.dtype, method)
return _linear_operator_difference(fun_wrapped, x0,
f0, rel_step, method)
else:
# by default we use rel_step
if abs_step is None:
h = _compute_absolute_step(rel_step, x0, f0, method)
else:
# user specifies an absolute step
sign_x0 = (x0 >= 0).astype(float) * 2 - 1
h = abs_step
# cannot have a zero step. This might happen if x0 is very large
# or small. In which case fall back to relative step.
dx = ((x0 + h) - x0)
h = np.where(dx == 0,
_eps_for_method(x0.dtype, f0.dtype, method) *
sign_x0 * np.maximum(1.0, np.abs(x0)),
h)
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._numdiff 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)))