Linear Equations Of The Preconditioned Iterative Solution Of The Theorem

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 which usually are solved by iterative method.The critical matter of the study of iterative methods is the convergence and the convergence rate of the iterative patterns. Of course,the disconvergent 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 are not surely attainable.So,we must look for the patterns with the high rate of convergence and the convergence rate of the iterative method has become a very important problem.Thus,we should find an iterative method which has fast convergence rate.This owns practical value.In order to solve linear system more faster and more better,we quote nonsingular preconditioned matrices.By preconditioned matrices,we improve the convergence rate of the iterative method.Based on preconditioned matrix P_R,this paper proposes the preconditioned AOR,2PPJ, USSOR iterative methods which have the generalization in practice,the comparison theorems are also obtained,this generalizes the results given by former researchers.The construction and main results of this paper are as follows:Chapter 1 Firstly,we mainly explain iterative solving method in solving linear system of equations.Secondly,we review some proposed preconditioned matrices, also different iterative methods under this different preconditioned matrices and some preconditioned comparison theorems which has been discussed under this proposed preconditioned matrices.Chapter 2 Preliminaries.We provide the basic knowledge and the basic lemma required in this paper,such as Z-matrix,M-matrix,regular splittings and weak regular splittings,on which the theorems and methods are presented.Chapter 3 Iterative methods under the preconditioned matrix I+S+R.This part is the main part of this paper.Firstly,under preconditioner P_R=I+S+R, AOR,2PPJ iterative methods are presented,about which comparison theorems are obtained when the coefficient matrix A is a Z-matrix;Secondly,if the coefficient matrix A is an H-matrix,then(I+S+ R)A is also an H-matrix,that is,Gauss-Seidel iterative method under the preconditioner P_R is convergent;Lastly,numerical examples verifies the results in this chapter.Chapter 4 USSOR iterative method under the preconditioner I + R.The preconditioned matrix P_R is matrix I+R when S equals to zero matrix.The USSOR iterative method under the preconditioner I+R is proposed in this chapter,on which comparison theorem is also obtained when the coefficient matrix A is Z- matrix.