The Convergence Analysis Of Many Iteration Method For Linear System

To meet the needs of the rapid development of science and technology, people are growing the scales of the problems solving. But directly iterative method on the division of matrix can not sustain the original problem. When the spectral distribution is dispersed, the convergence speed is very slow, or even not convergent. Therefore, it is an effective way to solve the convergence problem by using pretreatment technology to gather the spectra of the coefficient matrix of the linear algebra system. In this paper, we study the convergence theorems and comparison theorems of the parallel alternating two-stage method, parallel synchronous iterative methods and improved preconditioned iterative method.This paper includes five parts: In part one and part two we give the background of this paper, the basic definitions and some relation knowledge. In part three, our study concentrates on the parallel alternating two-stage method. We present the convergence conditions of these methods for solving nonsingular linear systems when the matrix is Hermite positive definite matrix. In part four, we study the parallel synchronous iterative methods. Also, we present convergence results for these methods. In Part five, we present the improving preconditioned mixed-type iterative method for solving the linear systems Ax = b, where A is a Z-matrix. And give some comparison theorems to show that the rate of convergence of the improved preconditioned mixed-type iterative method is faster than the rate of convergence of the preconditioned mixed-type iterative method. Finally, we give one numerical example to illustrate our results. In Part six, we present the improving preconditioned AOR-type iterative method for solving the linear systems Ax = b, where A is a Z-matrix. And give some comparison theorems to show that the rate of convergence of the improved preconditioned AOR-type iterative method is faster than the rate of convergence of the previous preconditioned iterative methods.Finally, we give one numerical example to illustrate our results.