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 ismax_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://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])