新预条件下矩阵不同分裂的收敛性分析

The Convergence Analysis of the Different Matrix Splitting under the New Pre-condition

近年来对于求解线性方程组的技术有了很大的发展,特别是预条件技术的出现使得解线性方程组的速度有了很大的提高,在预条件技术中最主要的是怎样去找一个合适的预条件子,本文提出了一个新的预条件子,不但证明了当线性方程组的Ax=b系数矩阵为非奇异的M-矩阵和H-矩阵时,在新预条件子作用下它们的收敛性,还得到了在新预条件下PSOR、PJOR等的收敛速度明显快于以往经典SOR、JOR迭代法,从而证明了本文提出的新预条件子的优越性.

In recent years there is a great development in solving the linear equations, especially af- ter the introduction of the pre-condition matrix, and the linear method＇s convergence rate accelerates greatly. What is the most important in the pre-condition technology is how to find a suitable pre-condi- tion. In this paper we propose a new pre-condition matrix, and when the Ax=b coefficient matrix is nonsingular irreducible M-matrix and H-matrix respectively we discuss the convergence analysis of not only the pre-conditioned ,but also we prove their convergence results under the new pre-condition, and demonstrate the convergence rate of PSOR and PJOR is clearly much faster than that of the classical SOR and JOR method, which show the superiority of our new per-condition iterative method in this paper.