bundles / scipy 1.18.0 / scipy / optimize / _elementwise / bracket_root

function

`scipy.optimize._elementwise:bracket_root`

source: /scipy/optimize/_elementwise.py:474

Signature

def bracket_root(f, xl0, xr0 = None, *, xmin = None, xmax = None, factor = None, args = (), kwargs = None, maxiter = 1000)

Summary

Bracket the root of a monotonic, real-valued function of a real variable.

Extended Summary

For each element of the output of f, bracket_root seeks the scalar bracket endpoints xl and xr such that sign(f(xl)) == -sign(f(xr)) elementwise.

The function is guaranteed to find a valid bracket if the function is monotonic, but it may find a bracket under other conditions.

This function works elementwise when xl0, xr0, xmin, xmax, factor, and the elements of args are (mutually broadcastable) arrays.

Parameters

f : callable

The function for which the root is to be bracketed. The signature must be

f(x: array, *args) -> array

where each element of x is a finite real and args is a tuple, which may contain an arbitrary number of arrays that are broadcastable with x.

f must be an elementwise function: each element f(x)[i] must equal f(x[i]) for all indices i. It must not mutate the array x or the arrays in args.

xl0, xr0 : float array_like

Starting guess of bracket, which need not contain a root. If xr0 is not provided, xr0 = xl0 + 1. Must be broadcastable with all other array inputs.

xmin, xmax : float array_like, optional

Minimum and maximum allowable endpoints of the bracket, inclusive. Must be broadcastable with all other array inputs.

factor : float array_like, default: 2

The factor used to grow the bracket. See Notes.

args : tuple of array_like, optional

Additional positional array arguments to be passed to f. If the callable for which the root is desired requires arguments that are not broadcastable with x, wrap that callable with f such that f accepts only x and broadcastable *args.

kwargs : dict of str:array_like, optional

Additional keyword arguments to be passed to f. See args.

maxiter : int, default: 1000

The maximum number of iterations of the algorithm to perform.

Returns

res : _RichResult

An object similar to an instance of scipy.optimize.OptimizeResult with the following attributes. The descriptions are written as though the values will be scalars; however, if f returns an array, the outputs will be arrays of the same shape.

success

True where the algorithm terminated successfully (status 0); False otherwise.

status

An integer representing the exit status of the algorithm.

0The algorithm produced a valid bracket.
-1The bracket expanded to the allowable limits without success.
-2The maximum number of iterations was reached.
-3A non-finite value was encountered.
-4Iteration was terminated by callback.
-5: The initial bracket does not satisfy`xmin <= xl0 < xr0 < xmax`.

bracket

The lower and upper endpoints of the bracket, if the algorithm terminated successfully.

f_bracket

The values of f evaluated at the endpoints of res.bracket, respectively.

nfev

The number of abscissae at which f was evaluated to find the root. This is distinct from the number of times f is called because the the function may evaluated at multiple points in a single call.

nit

The number of iterations of the algorithm that were performed.

Notes

This function generalizes an algorithm found in pieces throughout the scipy.stats codebase. The strategy is to iteratively grow the bracket (l, r) until f(l) < 0 < f(r) or f(r) < 0 < f(l). The bracket grows to the left as follows.

If xmin is not provided, the distance between xl0 and l is iteratively increased by factor.
If xmin is provided, the distance between xmin and l is iteratively decreased by factor. Note that this also increases the bracket size.

Growth of the bracket to the right is analogous.

Growth of the bracket in one direction stops when the endpoint is no longer finite, the function value at the endpoint is no longer finite, or the endpoint reaches its limiting value (xmin or xmax). Iteration terminates when the bracket stops growing in both directions, the bracket surrounds the root, or a root is found (by chance).

If two brackets are found - that is, a bracket is found on both sides in the same iteration, the smaller of the two is returned.

If roots of the function are found, both xl and xr are set to the leftmost root.

Array API Standard Support

bracket_root has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ✅                   
PyTorch               ✅                     ✅                   
JAX                   ⛔                     ⛔                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

Suppose we wish to find the root of the following function.

def f(x, c=5):
    return x**3 - 2*x - c

✓

First, we must find a valid bracket. The function is not monotonic, but `bracket_root` may be able to provide a bracket.

from scipy.optimize import elementwise
res_bracket = elementwise.bracket_root(f, 0)

✓

res_bracket.success
res_bracket.bracket

✗

(np.float64(2.0), np.float64(4.0))

Indeed, the values of the function at the bracket endpoints have opposite signs.

res_bracket.f_bracket

✗

(np.float64(-1.0), np.float64(51.0))

Once we have a valid bracket, `find_root` can be used to provide a precise root.

res_root = elementwise.find_root(f, res_bracket.bracket)

✓

res_root.x

✗

np.float64(2.0945514815423265)

`bracket_root` and `find_root` accept arrays for most arguments. For instance, to find the root for a few values of the parameter ``c`` at once:

import numpy as np
c = np.asarray([3, 4, 5])
res_bracket = elementwise.bracket_root(f, 0, args=(c,))
res_bracket.bracket
res_root = elementwise.find_root(f, res_bracket.bracket, args=(c,))

✓

res_root.x

✗

array([1.89329, 2.     , 2.09455])

Aliases

scipy.optimize._elementwise.bracket_root