Inexact Newton-type methods (Q609737)

Inexact Newton-type methods are discussed for approximating a locally unique solution of the nonlinear equation \(A(x)^{\#}(F(x)+G(x))=0\) in Banach space. Here \(F\) is a Fréchet-differentiable operator, \(G\) is a continuous operator and \(A(x)^{\#}\) is an analog of the Moore-Penrose generalized inverse of \(A(x)\) which is an approximation of the derivative of \(F(x)\). Based on outer inverses, the authors consider recurrent functions and establish a semi-local convergence for the proposed inexact Newton-type method. They show that the proposed method includes many other classical methods as special cases and its sufficient convergence conditions are weaker than those in the earlier studies.