Find a root of a function, using Krylov approximation for inverse Jacobian.
This method is suitable for solving large-scale problems.
Parameters ---------- %(params_basic)s 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_maxiter : int, optional Parameter to pass to the 'inner' Krylov solver: maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. 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, with ``x`` giving the current point, and ``f`` the current function value. outer_k : int, optional Size of the subspace kept across LGMRES nonlinear iterations. See `scipy.sparse.linalg.lgmres` for details. inner_kwargs : kwargs Keyword parameters for the 'inner' Krylov solver (defined with `method`). Parameter names must start with the `inner_` prefix which will be stripped before passing on the inner method. See, e.g., `scipy.sparse.linalg.gmres` for details. %(params_extra)s
See Also -------- root : Interface to root finding algorithms for multivariate functions. See ``method=='krylov'`` in particular. scipy.sparse.linalg.gmres scipy.sparse.linalg.lgmres
Notes ----- This function implements a Newton-Krylov solver. The basic idea is to compute the inverse of the Jacobian with an iterative Krylov method. These methods require only evaluating the Jacobian-vector products, which are conveniently approximated by a finite difference:
.. math:: 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 Newton-Krylov methods, see for example 1_, and for the LGMRES sparse inverse method, see 2_.
References ---------- .. 1 D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2004). :doi:`10.1016/j.jcp.2003.08.010` .. 2 A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). :doi:`10.1137/S0895479803422014`