scipy.optimize.newton_krylov¶

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)¶ Find a root of a function, using Krylov approximation for inverse Jacobian.
This method is suitable for solving largescale problems.
Parameters:  F : function(x) > f
Function whose root to find; should take and return an arraylike 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 use also 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 nonlinear step, withx
giving the current point, andf
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 nonlinear iterations. See
scipy.sparse.linalg.lgmres
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 maxnorm) for the residual. If omitted, default is 6e6.
 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.
Notes
This function implements a NewtonKrylov solver. The basic idea is to compute the inverse of the Jacobian with an iterative Krylov method. These methods require only evaluating the Jacobianvector products, which are conveniently approximated by a finite difference:
\[J v \approx (f(x + \omega*v/v)  f(x)) / \omega\]Due to the use of iterative matrix inverses, these methods can deal with large nonlinear problems.
Scipy’s
scipy.sparse.linalg
module offers a selection of Krylov solvers to choose from. The default here is lgmres, which is a variant of restarted GMRES iteration that reuses some of the information obtained in the previous Newton steps to invert Jacobians in subsequent steps.For a review on NewtonKrylov methods, see for example [1], and for the LGMRES sparse inverse method, see [2].
References
[1] (1, 2) D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). DOI:10.1016/j.jcp.2003.08.010 [2] (1, 2) A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). DOI:10.1137/S0895479803422014