A Class Of Preconditioned Iterative 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.Iterative method is one of the most important method.The rule whether the iterative method is good or no is usually described by convergence rate.Thus,we should look for an iterative method which has fast convergence rate.In order to solve linear systems more faster and more better,we quote nonsingular preconditioned matrices.By the preconditioned matrices,we accelerate the convergence rate of the iterative method.From[1]to[10],some diffrent interative methods have been proposed according to diffrent preconditioned matrices.In general,the convergence of the interative methods is closely related to the property of the coefficient matrix of linear systems.If the coefficient matrices of linear systemes are diffrent,then the research methods of iterative methods will also be diffrent.In this paper,the coefficient matrices of the linear systems are H-matrix and M-matrix.The followings are the construction of this paper and main contents of the evry chapter:In chapter 2,preliminaries.This part mainly makes preparations for chapter 3,chapter 4 and chapter 5.Firstly,we introduce some definitions and theorems which will be used in the later chapters,such as M-matrix,H-matrix,regular splitting'definitions and famous Perron-Frobenins theorem etc.Secondly,we point out several stipulations as a matter of convenience.In chapter 3,preconditioned Gauss-Seidel iterative method.Based on the preconditioned Gauss-Seidel iterative method had been proposed in[1],in this paper,the author propose the preconditioned Gauss-Seidel iterative method under the preconditioner I+C_α.When the coefficient matrix of linear system is anH-matrix,we obtain several convergence results.In chapter 4,preconditioned AOR iterative method.Firstly,when the coefficient matrix of linear system is an H-matrix,this paper proposes the preconditioned AOR iterative method under the preconditioner I+C_α,and we obtain convergence theorem;Secondly,when the coefficient matrixof linear system is a nonsingular M-matrix,a we obtain preconditioned comparison theorem.At the last,several numerical examples are given for verifying the results in this chaper.In chapter 5,preconditioned USSOR iterative method.Firstly,when the coefficient matrix of linear system is an H-matrix,the author propose the preconditioned USSOR iterative method under the preconditioner I+C_α,and we obtain convergence theorem;Secondly,when the coefficient matrix of linear system is a nonsingular M-matrix,we obtain preconditioned comparison theorem.At the last,several numerical examples are given for verifying the results in this chaper.