The Convergence Of Multi-splitting TOR Iterative Method

In this paper, we give a new parallel multi-splitting iterative method(TOR method) when the coefficient matrix is a non-singular matrix, and study the convergences of the algorithm when the coefficient matrix is an H-matrix or M-matrix. This article is arranged as follows:In the first chapter, the development of the multi-splitting iterative method for solving the large scale linear system in the past few years is simply introduced.In the second chapter, we give the definition of the multi-splitting TOR method and some preliminaries. We present two methods, which are the multi-splitting TOR algorithm and the relaxed multi-splitting TOR algorithm.The third chapter is one of the main part of this article, it gives the convergence of the multi-splitting TOR iterative algorithm and relaxed multi-splitting TOR iterative algorithm when the coefficient matrix is an H-matrix or M-matrix. We also verify the superiority of the multi-splitting iterative method by the numerical examples.The fourth chapter is the other main part of this article, it gives the convergence of the two-stage multi-splitting TOR iterative method and relaxed multi-splitting TOR iterative method, when the coefficient matrix is an H-matrix or M-matrix. We also verify the superiority of the two-stage multi-splitting iterative method by the numerical examples.The fifth chapter is a summary and outlook, it summarizes the paper and makes the outlook on the multi-splitting iterative method for the future.