Nonsingular H Matrices The Iterative Solution Of A Class Of Nonlinear Equations Ax = F (x)

The paper discusses one type of discriminace for H-matrix and a kind of iterative methods for a special nonlinear equations Ax=F(x).The solutions of many problems in mathematics,mechanics and so on are sumed up as solving a large-scale sparse linear equations Ax=b.For the linear equations, there are two main methods:one is the direct method,the other is the iterative method.When the step of linear system equations is not very high,the direct method is better.Instead,we choose the iterative method in many cases.In recent years,with the emergence of computers and its rapid development,the scale of the problem become larger and larger,which enables the iterative method become the current main way for solving the linear system equations.For the iterative method, since the iterative format must be convergent,the convergence of iterative format become a key issue.As early as 1976,when people studied the iterative matrices of JOR,SOR and AOR,they found the very important relationship between the convergence of iterative matrices and H-matrix,namely,as long as the matrix we discussed is the H-matrix,then JOR,SOR and AOR are convergent.Based on this theory,Chapter 2 conducts research work on the criteria of H-matrix and obtains a class of methods for distinction H-matrix,the numerical example indicats this method is effective.Moreover,Chapter 3 introduces a type of iterative method and its convergence analysis for the nonlinear systems Ax=F(x) when A is singular and symmertic semi-positive definite.With the development of science and technology, people pay more attiation to the question of nonlinear equation.Before 70's,they had done massive research on nonlinear systems,not only in theory,but also in numerical solutions.In the book of <> and <>, there are system inductions about it.However,for the question how to obtain the equation solution,nonlinear systems is inferior to linear systems,not only in theory,but also in numerical solutions,therefore,there are many questions in root's existence of nonlinear system and the effective methods to find numerical solutions. All of these need further research and summery.This paper contains three chapters.The main results of this paper as following:Chapter 1 Preliminaries.This part makes preparation for chapter 2 and chapter 3.Mainly introduced the background knowledge of the nonsingular H- matrix and nonlinear systems,and our main research work is summaried finally.Chapter 2 The criteria of nonsingular H-matrices.This chapter contains four parts:Part one introduced the preliminaries about H-matrix;Part two,Part three,Part four are the matrix determination theorem from three different angles, The last part is a summary of this chapter.Chapter 3 A type of iterative methods for nonlinear equations Ax=F(x). Firstly,we introduce a few of background knowledge about nonlinear systems Ax= F(x),and construct iterative scheme for equations Ax=F(x),when A is singular and symmetric semi-positive definite,then we obtain the convergence theorem. Secondly,the theorem which we have obtained is put onto the two-stage iterative methods and block two-stage iterative methods.