approx_fprime — SciPy v1.15.2 Manual (original) (raw)

scipy.optimize.

scipy.optimize.approx_fprime(xk, f, epsilon=1.4901161193847656e-08, *args)[source]#

Finite difference approximation of the derivatives of a scalar or vector-valued function.

If a function maps from \(R^n\) to \(R^m\), its derivatives form an m-by-n matrix called the Jacobian, where an element \((i, j)\) is a partial derivative of f[i] with respect to xk[j].

Parameters:

xkarray_like

The coordinate vector at which to determine the gradient of f.

fcallable

Function of which to estimate the derivatives of. Has the signaturef(xk, *args) where xk is the argument in the form of a 1-D array and args is a tuple of any additional fixed parameters needed to completely specify the function. The argument xk passed to this function is an ndarray of shape (n,) (never a scalar even if n=1). It must return a 1-D array_like of shape (m,) or a scalar.

Suppose the callable has signature f0(x, *my_args, **my_kwargs), wheremy_args and my_kwargs are required positional and keyword arguments. Rather than passing f0 as the callable, wrap it to accept only x; e.g., pass fun=lambda x: f0(x, *my_args, **my_kwargs) as the callable, where my_args (tuple) and my_kwargs (dict) have been gathered before invoking this function.

Changed in version 1.9.0: f is now able to return a 1-D array-like, with the \((m, n)\)Jacobian being estimated.

epsilon{float, array_like}, optional

Increment to xk to use for determining the function gradient. If a scalar, uses the same finite difference delta for all partial derivatives. If an array, should contain one value per element of_xk_. Defaults to sqrt(np.finfo(float).eps), which is approximately 1.49e-08.

*argsargs, optional

Any other arguments that are to be passed to f.

Returns:

jacndarray

The partial derivatives of f to xk.

See also

check_grad

Check correctness of gradient function against approx_fprime.

Notes

The function gradient is determined by the forward finite difference formula:

     f(xk[i] + epsilon[i]) - f(xk[i])

f'[i] = --------------------------------- epsilon[i]

Examples

import numpy as np from scipy import optimize def func(x, c0, c1): ... "Coordinate vector x should be an array of size two." ... return c0 * x[0]*2 + c1x[1]**2

x = np.ones(2) c0, c1 = (1, 200) eps = np.sqrt(np.finfo(float).eps) optimize.approx_fprime(x, func, [eps, np.sqrt(200) * eps], c0, c1) array([ 2. , 400.00004208])