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)

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

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

Parameters

Ffunction(x) -> f

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

xinarray_like

Initial guess for the solution

alphafloat, optional

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

reduction_methodstr 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_rankint, optional

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

iterint, optional

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

verbosebool, optional

Print status to stdout on every iteration.

maxiterint, optional

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

f_tolfloat, optional

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

f_rtolfloat, optional

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

x_tolfloat, 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_rtolfloat, optional

Relative minimum step size. If omitted, not used.

tol_normfunction(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’.

callbackfunction, 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

solndarray

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

Raises

NoConvergence

When a solution was not found.

See also

root

Interface to root finding algorithms for multivariate functions. See method=='broyden1' in particular.

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

1

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).

https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf

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])