scipy.optimize.

BroydenFirst#

class scipy.optimize.BroydenFirst(alpha=None, reduction_method='restart', max_rank=None)[source]#

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

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

Parameters:

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).

Methods

aspreconditioner
matvec
rmatvec
rsolve
setup
solve
todense
update

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://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])