The Convergence Of Quasi-Newton Methods

Numerical methods for nonlinear equations are very important in many areas. In-exact Newton method, which is based on Newton method, is one of the main methods for solving large sparse systems of nonlinear equations. Inexact Newton method is an inner-outer iterative procedure, with Newton iteration as its outer iteration and a linear iteration as its inner iteration. There are lots of research results about inexact Newton-like methods for solving nonlinear equations and unconstrained optimization, however, no result on inexact rank one or rank two updated quasi-Newton methods has been seen, maybe because rank one and rank two updated quasi-Newton equations are easier to solve than Newton equations. When quasi-Newton methods is extended to operator equation in Banach space or minimax problems, it is difficult or impossible to solve subproblems exactly, so it is necessary to investigate inexact quasi-Newton methods. In this paper, convergence of inexact quasi-Newton methods for nonlinear equations in R~n is proven under commonly used conditions on updated matrics. The results serve as preparation for convergence theory of inexact quasi-Newton methods for operator equations in Banach space and minimax problems.