The following code is required to get ready before we proceed:
scipy.optimize.newton_krylov(F, xin, iter=None, rdiff=None, method='lgmres', inner_maxiter=20, inner_M=None, outer_k=10, 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)
In the following table can see the parameters and their description of the Anderson method
| Parameters |
F: function(x) -> f. Function whose root to find; should take and return an array-like object. xin: array_like. Initial guess for the solution. rdiff: float, optional. Relative step size to use in numerical differentiation. method: {'lgmres', 'gmres', 'bicgstab', 'cgs', 'minres'} or function. Krylov method to use to approximate the Jacobian. Can be a string, or a function implementing the same interface as the iterative solvers in scipy.sparse.linalg. The default is scipy.sparse.linalg.lgmres. inner_M: LinearOperator or InverseJacobian Preconditioner for the inner Krylov iteration. Note that you can also use inverse Jacobians as (adaptive) preconditioners. For example: from scipy.optimize.nonlin import BroydenFirst, KrylovJacobian
from scipy.optimize.nonlin import InverseJacobian
jac = BroydenFirst()
kjac = KrylovJacobian(inner_M=InverseJacobian(jac))If the preconditioner has a method named update, it will be called as update(x, f) after each non-linear step, with x giving the current point, and f the current function value. inner_tol, inner_maxiter, ... |
|
Parameters to pass on to the inner Krylov solver. See scipy.sparse.linalg.gmres for details. outer_k: int, optional. Size of the subspace kept across LGMRES non-linear iterations. See scipy.sparse.linalg.lgmres documentation for details. iter: int, optional. Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose: bool, optional. Print status to stdout on every iteration. maxiter: int, optional. Maximum number of iterations to make. If more are needed to meet convergence, NoConvergence is raised. f_tol: float, optional. Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol: float, optional. Relative tolerance for the residual. If omitted, not used. x_tol: float, 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_rtol: float, optional. Relative minimum step size. If omitted, not used. tol_norm: function(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. callback: Function, 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 |
sol: ndarray. An array (of similar array type as x0) containing the final solution. |
| Raises |
NoConvergence. When a solution was not found. |