scipy.optimize.broyden1¶

scipy.optimize.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)[source]¶

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

This method is also known as “Broyden’s good method”.

Parameters:

Parameters:	F : function(x) -> f Function whose root to find; should take and return an array-like object. x0 : 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 (ie., 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.

F : function(x) -> f

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

x0 : 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 (ie., 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

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

which corresponds to Broyden’s first Jacobian update

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

References

[R138]

B.A. van der Rotten, PhD thesis, “A limited memory Broyden method to solve high-dimensional systems of nonlinear equations”. Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003).

http://www.math.leidenuniv.nl/scripties/Rotten.pdf