scipy.optimize.anderson

scipy.optimize.anderson(F, xin, iter=None, alpha=None, w0=0.01, M=5, 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 (extended) Anderson mixing.

The Jacobian is formed by for a ‘best’ solution in the space spanned by last M vectors. As a result, only a MxM matrix inversions and MxN multiplications are required. [Ey]

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).

Mfloat, optional

Number of previous vectors to retain. Defaults to 5.

w0float, optional

Regularization parameter for numerical stability. Compared to unity, good values of the order of 0.01.

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=='anderson' in particular.

References

Ey

22. Eyert, J. Comp. Phys., 124, 271 (1996).

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.anderson(fun, [0, 0])
>>> sol
array([0.84116588, 0.15883789])