半正定线性系统广义非定常多分裂二阶段迭代方法的收敛性  

The Convergence of Generalized Non-stationary Multi-splitting Two-stage Iterative Methods for Semi-definite Linear Systems

作　　者：崔艳星[1] 王川龙[2] 江文胜[3] CUI Yanxing;WANG Chuanlong;JIANG Wensheng(Department of Mathematics,Changzhi University,Changzhi 046000;Department of Mathematics,Taiyuan Normal University,Taiyuan 030619;Key Laboratory of Physics and Oceanography of Ministry of Education,Ocean University of China,Qingdao 266003)

机构地区：[1]长治学院数学系,长治046000 [2]太原师范学院数学系,太原030619 [3]中国海洋大学物理海洋教育部重点实验室,青岛266003

出　　处：《工程数学学报》2022年第5期826-834,共9页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11371275);教育部协同育人项目(201902213010);山西省高等学校教学改革创新项目(J2021685);长治学院基础教育项目(2020J016)。

摘　　要：为了高效求解正定或半正定的大型稀疏线性方程组,在第一阶段采用经典矩阵分裂的基础上,广义非定常多分裂二阶段迭代方法的第二阶段分裂融合了多分裂和矩阵预处理技术,对非定常多分裂二阶段迭代方法进行了推广。为了研究收敛性,将该迭代方法的算法形式和逻辑语言表达形式改写为紧凑的迭代格式。由此得到,广义非定常多分裂二阶段迭代算法在一个充分条件下收敛。最后,具有五对角系数矩阵的大型稀疏线性系统的数值算例验证了广义非定常多分裂二阶段迭代算法的普适性,并且从迭代次数和CPU时间上体现了算法的高效性。To effectively solve the large sparse linear equations of positive definite or positive semidefinite,the second stage splitting of generalized non-stationary multi-splitting two-stage iterative method is proposed,which combines the techniques of multi-splitting and matrix preprocessing generalizes the non-stationary multi-splitting two-stage iterative method based on the classical matrix splitting in the first stage.The algorithm and logical expression of the generalized non-stationary multi-stage iterative method are rewritten into a compact iterative scheme to consider the convergence.According to the iterative scheme,the generalized non-stationary multi-splitting two-stage iterative algorithm is convergent under a sufficient condition.Finally,a numerical example of a large sparse linear system with a five-diagonal coefficient matrix shows the feasibility of the generalized non-stationary multi-split two-stage iterative algorithm,and the efficiency of the algorithm is verified in terms of iteration steps and CPU time.

关 键 词：半范数收敛 商收敛 二阶段 多分裂 

分 类 号：O241.6[理学—计算数学]