Analyses Of The Convergence Of Aor Method For A Type Of Matrices And Convergence Theorems For Two Preconditioned Iterative Methods

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.Therefore,the convergence and the convergence rate of the iteration patterns become focus of attention of many people (see[l]-[6]).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 are not surely attainable.So,we must look for the patterns with the high rate of convergence or try to settle some parameters of the iteration patterns (for instance the overrelaxation parameter of SOR iterative method).In chapter 2,when the coefficient matrix of a linear system is (1,1) consistently ordered matrix and the eigenvalues of its Jacobi matrix are all pure imaginaries or zeroes ,the convergence and the optimum parameters of its AOR iterative method are given.Puthermore,we make a comparison between its optimum spectral radius and that of SOR method and draw a conclusion that these two methods have their own advantages respectively under different conditions.Therefore the method to how to choose the better method between AOR and SOR is given successfully.At the same time,we also gain an ideal result,that is to say,when the eigenvalues of Jacobi matrix of the consistently ordered matrix A are all zeros or a pair of conjugate pure imaginaries of multiplicity n/2(assuming their modular is a),for its AOR iterative method, the range of convergence and the optimum parameters are given,and it is shown that this method has a better characteristic than other iterative methods,that is,for the optimum parametersγb = 2/(1 + (1+α²)¹/2 ,w_b - l/(1 + α²)^1/2, ρ{L_γb,wb = 0 is obtained.This means,for this type of equation systems ,that we can get the exact solution only after n iterations by using the optimum AOR method,which is just the best answer we want.Except for the research and improvement for iterative method,one of the effective methods to improve the convergence and enhance the convergence rate is making some changes in equation systems themselves,for example preconditioning the system.After studying lots of preconditioned iterative methods (see[7]-[12]),on the one hand, in chapter 3,the preconditioner (I + S_α) is considered to be applied...