`scipy.optimize._nonlin:BroydenFirst`

source: build-install/usr/lib/python3.14/site-packages/scipy/optimize/_nonlin.py:866

Signature

class BroydenFirst(alpha = None, reduction_method = restart, max_rank = None)

Members

Summary

Find a root of a function, using Broyden's first Jacobian approximation.

Extended Summary

This method is also known as "Broyden's good method".

Parameters

alpha : float, optional

Initial guess for the Jacobian is (-1/alpha).

reduction_method : str or tuple, optional

Method used in ensuring that the rank of the Broyden matrix stays low. Can either be a string giving the name of the method, or a tuple of the form (method, param1, param2, ...) that gives the name of the method and values for additional parameters.

Methods available:

restart: drop all matrix columns. Has no extra parameters.
simple: drop oldest matrix column. Has no extra parameters.
svd: keep only the most significant SVD components. Takes an extra parameter, to_retain, which determines the number of SVD components to retain when rank reduction is done. Default is max_rank - 2.

max_rank : int, optional

Maximum rank for the Broyden matrix. Default is infinity (i.e., no rank reduction).

Notes

This algorithm implements the inverse Jacobian Quasi-Newton update

H_{+} = H + (d x - H df) d x^{†} H / (d x^{†} H df)

which corresponds to Broyden's first Jacobian update

J_{+} = J + (df - J d x) d x^{†} / d x^{†} d x

Examples

The following functions define a system of nonlinear equations

def fun(x):
    return [x[0]  + 0.5 * (x[0] - x[1])**3 - 1.0,
            0.5 * (x[1] - x[0])**3 + x[1]]

✓

A solution can be obtained as follows.

from scipy import optimize
sol = optimize.broyden1(fun, [0, 0])

✓

sol

✗

array([0.841, 0.159])

Aliases

scipy.optimize.BroydenFirst