The Convergence Analysis Of Preconditioned Iteration Method And Parallel Alternating Two-stage Method

Stationary iterative method is a class of very important ways for the solution of large linear algebra systems. However, to meet the needs of the rapid development of science and technology, people are growing the scales of the problems solving. And for the stationary iterative method based on the division of matrix, the convergence speed is very slow, or even not convergent, when the spectral distribution is dispersed. 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. And it is becoming a hot issue in the study of iterative method. In this paper, we mainly study the convergence theorems and comparison theorems of the preconditioned iterative method and parallel alternating two-stage method. We present several new preconditioners and obtain some new algorithms.This paper mainly 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, we talk about the research of solving the linear equations for Z-matrix by using the preconditioned AOR method. We present the convergence proving and the convergence speed comparisons between the classical methods and the preconditioned methods. We also give the convergence speed comparisons among different preconditioned methods with different preconditioners. In part four, we mainly study the preconditioned Mixed-type splitting method. We introduce the pretreatment into the Mixed-type splitting method. And we analyze the convergence of the method and get some comparison theorems. In Part five, we mainly discuss the parallel alternating two-stage method and some new relaxation algorithms for the solution of equations whose coefficient matrices are monotone or H-matrices. Moreover, we give the convergence theorems.