bundles / scipy 1.17.1 / scipy / optimize / _nonlin / broyden1

function

`scipy.optimize._nonlin:broyden1`

Signature

   def     broyden1 (    F    ,    xin    ,    iter    =  None   ,    alpha    =  None   ,    reduction_method    =  restart   ,    max_rank    =  None   ,    verbose    =  False   ,    maxiter    =  None   ,    f_tol    =  None   ,    f_rtol    =  None   ,    x_tol    =  None   ,    x_rtol    =  None   ,    tol_norm    =  None   ,    line_search    =  armijo   ,    callback    =  None   ,   ** kw      )

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

F : function(x) -> f

Function whose root to find; should take and return an array-like object.

xin : array_like

Initial guess for the solution

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).

iter : int, optional

Number of iterations to make. If omitted (default), make as many as required to meet tolerances.

verbose : bool, optional

Print status to stdout on every iteration.

maxiter : int, optional

Maximum number of iterations to make. If more are needed to meet convergence, NoConvergence is raised.

f_tol : float, optional

Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6.

f_rtol : float, optional

Relative tolerance for the residual. If omitted, not used.

x_tol : float, optional

Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used.

x_rtol : float, optional

Relative minimum step size. If omitted, not used.

tol_norm : function(vector) -> scalar, optional

Norm to use in convergence check. Default is the maximum norm.

line_search : {None, 'armijo' (default), 'wolfe'}, optional

Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'.

callback : function, optional

Optional callback function. It is called on every iteration as callback(x, f) where x is the current solution and f the corresponding residual.

Returns

sol : ndarray: An array (of similar array type as x0) containing the final solution.

Raises

: NoConvergence: When a solution was not found.

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

✗

Aliases

scipy.optimize.broyden1