In this talk we shall consider inexact Newton methods for finding a zero of F:Rn→Rn. We will give an overview on these procedures and we shall focus on Newton-Krylov methods. These latter methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimension. the specific Newton-Krylov method considered here is Newton-GMRES, where the Generalized Minimum RESidual iterative solver is used to solve the linear system arising at each iteration of an inexact Newton method
In order to enlarge convergence domain, globally convergent modifications of the basic Newton-Krylov methods have been considered. Here we will focus on a hybrid Newton-GMRES method where a global strategy restricted to a low-dimensional subspace generated by GMRES is performed. This method is an extension of the classical linesearch Newton-GMRES method and computational results indicate that it enhances the classical linesearch Newton-GMRES approach.
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