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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)

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

Notes

This algorithm implements the inverse Jacobian Quasi-Newton update

$$H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df)$$

which corresponds to Broyden's first Jacobian update

$$J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx$$

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.

Raises

NoConvergence: When a solution was not found.

Returns

sol : ndarray: An array (of similar array type as :None:None:`x0`) containing the final solution.

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

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([0.84116396, 0.15883641])

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