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]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])