严格对角占优矩阵与SOR迭代法的收敛性定理

第34卷第1期 长春理工大学学报(自然科学版) Vol.34 No. 1

20 11年3月 Journal of Changchun University of Science and Technology (Natural Science Edition) Mar. 2011

严格对角占优矩阵与SOR 迭代法的收敛性定理

宋岱才，敬长红，陈德艳

(辽宁石油化工大学 理学院，抚顺 113001)

摘 要：针对线性方程组的系数矩阵为 -链严格对角占优矩阵和双严格对角占优矩阵的情况，讨论了线性方程组求

解时常用的SOR 迭代方法的收敛性，给出了迭代法收敛性定理，解决了以往估计迭代矩阵谱半径的问题。结果不

仅适用于这两类矩阵，还适用于广义严格对角占优矩阵类，最后举例说明了所给结果的优越性。

关键词： -链严格对角占优矩阵；双严格对角占优矩阵；迭代法；收敛性

中图分类号：O24 1.6；O151.2 文献标识码：A 文章编号：1672-9870 (20 11)01-0 170-03

Diagonal Strictly Dominance Matrix and Convergence

Theorem of SOR Iteration Method

SONG Daicai ，JING Changhong，CHEN Deyan

(School of Sciences，Liaoning University of Petroleum & Chemical Technology ，Fushun 11300 1)

Abstract ：In this paper Convergence theorem of SOR iteration method for solving linear system is studied，when coefficient ma-

trix is -chain diagonal strictly dominance or doubly diagonal strictly dominance，and some convergence theorems are given ，

which solves the problem of spectral radius of iterative matrices. Results obtained are applicable for - chain diagonal strictly domi-

nance matrix or doubly diagonal strictly dominance matrix ，and improve the known results and are applicable for generalized di-

agonal strictly dominance matrices.Finally ，a numerical example is given for illustrating advantage of the results in this paper.

Key words ： -chain diagonal strictly dominance matrix ；doubly diagonal strictly dominance matrix ；iteration method ；

convergence theorem

界限问题。最后举例说明这一结果的适用性。

1 基本概念及引理

设方程组的系数矩阵A 分解为 = ，其

n×n 中D=diag ( ， ，…，