A Class Of Matrix Saor Methods And Saddle Point The Problem Gsor-like Method Of Convergence Analysis

There are two main methods to solve the linear equations Ax=b.One is the direct solving method,the other is the iterative method.Gauss elimination is the most important method of direct solving method.The solutions of many problems in mathematics,physics,mechanics,engineering and so on are sumed up to the solutions of one or some large sparse linear systems.With the rapid development of computer,the scale of the problem becomes larger and larger,how to solve the large linear equations is the core of the large-scale science and engineering project computation which usually are solved by iterative method.Generally speaking,the iterative methods that we experts usually use are Jacobi, Gauss-Seidel,SOR(succesive overrelaxation),AOR(accelerate overrelaxation), SSOR(symmetric successive overrelaxation),SAOR(symmetric accelerate overrelaxation). GAOR(generalize accelerate overrelaxation)and so on.The critical matter of iterative method is the convergence and the convergence rate of the iterative patterns,which is a core question.Of course,the divergent patterns will not be adopted.If the pattern has a low rate of convergence,the time of the human and machines will be wasted and the answer will not be available.Furthermore,the value of application is very small.Hence,we must look for the patterns with the high rate of convergence and try to settle some parameters of the iterative patterns,such as the overrelaxation parameter of SOR iterative method.In general,the convergence of the iterative method is closely related to the property of the coefficient matrix of linear systems,for instance,nonnegative matrices,cyclic matrices,M matrices,H matrices and so on.If the matrices are different,the research methods of iterative methods will also be different.Therefore,we often make sure the type of matrices before studying the convergence of the iterative method.At the same time,there are other methods for accelerate the convergence rate of iterative methods,such as preconditioned and semi-iterative methods etc.The study of optimum parameter is cared about by many experts,which is very important for solving the equations.In this paper,the necessary and sufficient condition and the optimum parameter for the non-zero diagonal elements(1,1) consistently ordered matrix of SAOR method and GSOR-Like method of saddle-point are discussed.The followings are the construction and main contents of this paper:In chapter 1,we outline the development process of the priorities compatibility matrix method.At the same time,we introduce a saddle point for the research method in recent years and the significance of optimal parameters.Our main research work is summaried finally.In chapter 2,the convergence of SAOR method is mainly discussed for the coefficient matrix of the linear system to the non-zero diagonal elements(1,1) consistently ordered matrix.Using the relationship between iterative matrix SAOR eigenvalueλand Jacobi iteration matrix eigenvalueμ.We seperately discuss the eigenvalueμof Jacobi iteration matrix is real number and the whole is greater than 1,purely imaginary and general complex areas.At the same time,convergence of the necessary and sufficient conditions for the optimal parameters are gotten when the parameters ofγis 2,ωis complex from the situation.Finally,we give numerical examples.In chapter 3,through equations coefficient matrix decomposition.We introduce the preconditioning matrix Q.According to research issues GSOR saddle-point for the iterative method.First of all,we give the GSOR-Like iterative matrix H(ω,Υ). Secondly,the H(ω,Υ) eigenvalueλand Q^-1 B^T A^-1B of the eigenvalueμ,as well as the relationship between the convergence of the method necessary and sufficient conditions forρ(H(ω,Υ))＜1 is derived.Finally,we get the optimal parameters and give an numerical example.