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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2022, Vol. 39 ›› Issue (5): 826-834.doi: 10.3969/j.issn.1005-3085.2022.05.011

• • 上一篇    下一篇

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

崔艳星1,   王川龙2,   江文胜3   

  1. 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
  • 出版日期:2022-10-15 发布日期:2022-12-15
  • 通讯作者: 王川龙 E-mail: clwang218@126.com
  • 基金资助:
    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).

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

CUI Yanxing1,  WANG Chuanlong2,   JIANG Wensheng3   

  1. 1. Department of Mathematics, Changzhi University, Changzhi 046000
    2. Department of Mathematics, Taiyuan Normal University, Taiyuan 030619
    3. Key Laboratory of Physics and Oceanography of Ministry of Education, Ocean University of China, Qingdao 266003
  • Online:2022-10-15 Published:2022-12-15
  • Contact: C. Wang. E-mail address: clwang218@126.com
  • Supported by:
    The National Natural Science Foundation of China (11371275); the Collaborative Education Project of the Ministry of Education (201902213010); the Higher Education Reform and Innovation Project of Shanxi Province (J2021685); the Basic Education Project of Changzhi University (2020J016).

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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。

关键词: 半范数收敛, 商收敛, 二阶段, 多分裂

Abstract:

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.

Key words: semi-norm convergence, quotient convergence, two-stage, multi-splitting

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