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
x0 : array_like
rdiff : float, optional
method : {‘lgmres’, ‘gmres’, ‘bicgstab’, ‘cgs’, ‘minres’} or function
inner_M : LinearOperator or InverseJacobian
inner_tol, inner_maxiter, ... :
outer_k : int, optional
iter : int, optional
verbose : bool, optional
maxiter : int, optional
f_tol : float, optional
f_rtol : float, optional
x_tol : float, optional
x_rtol : float, optional
tol_norm : function(vector) > scalar, optional
line_search : {None, ‘armijo’ (default), ‘wolfe’}, optional
callback : function, optional


Returns :  sol : ndarray

Raises :  NoConvergence :

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 numerical differentiation:
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 [KK], and for the LGMRES sparse inverse method, see [BJM].
References
[KK]  (1, 2) D.A. Knoll and D.E. Keyes, J. Comp. Phys. 193, 357 (2003). 
[BJM]  (1, 2) A.H. Baker and E.R. Jessup and T. Manteuffel, SIAM J. Matrix Anal. Appl. 26, 962 (2005). 