Inexact Newton Methods for Singular Problems

this paper we describe the effects of an inexact implementation of Newton's method on the behavior of the iteration for certain nonlinear equations in Banach space for which the Fr'echet derivative is singular at the solution. We give a termination criterion for the inner iteration that preserves not only the q-linear convergence of the Newton iterates but also the fine structure required for an acceleration method. KEY WORDS: inexact Newton method, singular nonlinear equation, simple fold, acceleration of convergence 1 INTRODUCTION In this paper we describe the effects of an inexact [12] implementation of Newton's method on the behavior of the iteration for certain nonlinear equations in Banach space for which the Fr'echet derivative is singular at the solution. As a particular example we consider the Newton-GMRES iteration [2]. We give a termination criterion for the inner iteration that preserves not only the q-linear convergence of the Newton iterates but also the fine structure required for a generalization of the acceleration method given in [22]. Let F be a map from a Banach space E into itself. Assume that F (