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 arraylike 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 ismax_rank  2
.
 max_rankint, optional
Maximum rank for the Broyden matrix. Default is infinity (ie., 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 maxnorm) for the residual. If omitted, default is 6e6.
 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.
Notes
This algorithm implements the inverse Jacobian QuasiNewton 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 highdimensional 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