Research On Preconditioned Iteration Method For M-Matrix Linear Systems

The important problem of large scale science and engineering computation is solving large linear equation.With the development of the computer,the iterative methods have replaced by direct method to become the most important method for solving large linear equations.The criterion of judging the good or bad of the iterative method is portrayed by the convergence rate of the iteration method.In this way,we can find the new iterative method with fast convergence speed,and it has the practical significance.There is no practical value for the iterative method of non convergence or slow convergence.In many cases,the convergence rate of the iterative method is characterized by the spectral radius of the iterative matrix.In order to solve the large linear equations faster and better,this paper is proposed a new preconditioned matrix I+Sαβ on the basis of previous studies.When the coefficient matrix is a nonsingular M-matrix and nonsingular irreducible M-matrix,we discuss the convergence the preconditioned Gauss-Seidel method and classical Gauss-Seidel method,the preconditioned Jacobi method and classical Jacobi method,the preconditioned SOR iterative method and classical SOR method,preconditioned AOR iterative method and classical AOR method,the preconditioned JOR iterative method and classical JOR method.Firstly,the classic AOR general iterative method,the classical SOR iterative method,classical Gauss-Seidel iterative method,JOR iterative method and classical Jacobi iterative method for solving the large scale linear equations are given.we also give some the preconditioned matrix,and propose a new preconditioned matrix in this paper.Secondly,we give some important definitions,lemmas and theorems.Next,we consider that the coefficient matrix of linear equations is non singular and irreducible matrix,and prove the convergence analysis of the new preconditioned Gauss-Seidel iterativemethod,a new preconditioned SOR iterative method,a new preconditioned AOR iterative method and a new preconditioned JOR iteration method.Finally,the numerical experiments show that the new preconditioned methods perform much faster than the classical corresponding iterative methods,which explainthat our proposed methods are feasible and efficient.