broyden2(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 bad method".
This algorithm implements the inverse Jacobian Quasi-Newton update
$$H_+ = H + (dx - H df) df^\dagger / ( df^\dagger df)$$corresponding to Broyden's second method.
Function whose root to find; should take and return an array-like object.
Initial guess for the solution
Initial guess for the Jacobian is (-1/alpha)
.
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
.
Maximum rank for the Broyden matrix. Default is infinity (i.e., no rank reduction).
Number of iterations to make. If omitted (default), make as many as required to meet tolerances.
Print status to stdout on every iteration.
Maximum number of iterations to make. If more are needed to meet convergence, NoConvergence
is raised.
Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6.
Relative tolerance for the residual. If omitted, not used.
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.
Relative minimum step size. If omitted, not used.
Norm to use in convergence check. Default is the maximum norm.
Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to 'armijo'.
Optional callback function. It is called on every iteration as callback(x, f)
where x
is the current solution and f
the corresponding residual.
When a solution was not found.
An array (of similar array type as :None:None:`x0`
) containing the final solution.
Find a root of a function, using Broyden's second Jacobian approximation.
root
Interface to root finding algorithms for multivariate functions. See method=='broyden2'
in particular.
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 optimizeSee :
... sol = optimize.broyden2(fun, [0, 0])
... sol array([0.84116365, 0.15883529])
Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.
Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)
SVG is more flexible but power hungry; and does not scale well to 50 + nodes.
All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them