非Hermitian正定线性方程组的外推的广义HSS方法

计算机工程与科学 ›› 2022, Vol. 44 ›› Issue (10): 1885-1892.

非Hermitian正定线性方程组的外推的广义HSS方法

吴思婷，鲍亮

(华东理工大学数学学院,上海 200237)

收稿日期:2021-03-21 修回日期:2021-05-20 接受日期:2022-10-25 出版日期:2022-10-25 发布日期:2022-10-28

An extrapolated and generalized HSS method for non-Hermitian positive definite linear systems

WU Si-ting,BAO Liang

（School of Mathematics,East China University of Science and Technology,Shanghai 200237,China）

Received:2021-03-21 Revised:2021-05-20 Accepted:2022-10-25 Online:2022-10-25 Published:2022-10-28

摘要/Abstract

摘要： 探讨了如何高效求解非Hermitian正定线性方程组，提出了一种外推的广义Hermitian和反Hermitian (EGHSS) 迭代方法。首先，根据矩阵的广义Hermitian和反Hermitian分裂，构造出了一种新的非对称的二步迭代格式。接着，理论分析了新方法的收敛性，并给出了新方法收敛的充要条件。数值实验结果表明，在处理某些问题时，EGHSS迭代方法比GHSS迭代方法和EHSS迭代方法更有效。

关键词: 非Hermitian正定线性方程组, GHSS迭代方法, EHSS迭代方法, 收敛性

Abstract: How to efficiently solve non-Hermitian positive definite linear systems is discussed, and an extrapolated and generalized Hermitian and skew-Hermitian splitting iteration method (EGHSS) is proposed. Firstly, the method constructs a new asymmetric two-step iterative scheme based on the generalized Hermitian and skew-Hermitian splitting iteration. Secondly, the convergence of the new method is theoretically analyzed, and the necessary and sufficient conditions for the convergence are given. Finally, numerical experiments show that the EGHSS iteration method is more effective than the GHSS iteration method and the EHSS iteration method in dealing with some problems.

Key words: non-Hermitian positive definite linear systems；GHSS iteration method；EHSS iteration method；convergence ,

吴思婷, 鲍亮. 非Hermitian正定线性方程组的外推的广义HSS方法[J]. 计算机工程与科学, 2022, 44(10): 1885-1892.

WU Si-ting, BAO Liang. An extrapolated and generalized HSS method for non-Hermitian positive definite linear systems[J]. Computer Engineering & Science, 2022, 44(10): 1885-1892.

参考文献［13］

［1］	Adams M F.Algebraic multigrid methods for constrained linear systems with applications to contact problems in solid mechanics［J］.Numerical Linear Algebra with Applications,2004,11(2-3): 275-300.
［2］	Arrow Z K,Hurwica L,Uzawa H.Studies in linear and non-linear programming ［M］.Palo Alto: Stanford University Press,1958.
［3］	Bai Z Z.Splitting iteration methods for non-Hermitian positive definite systems of linear equations［J］.Hokkaido Mathematical Journal,2007,36(4): 801-814.
［4］	Young D M.Iterative solutions of large linear systems［M］.New York: Academic Press,1971.
［5］	Chu Lu,Bao Liang,Dong Bei-bei.Generalized LHSS method for non-Hermitian positive definite linear systems［J］.Journal of Zhejiang University(Science Edition),2018,45(6): 694-706.(in Chinese)
［6］	Dong Bei-bei,Bao Liang.A generalized positive-definite and skew-Hermitian splitting iteration method for saddle points problem ［J］.Computer Engineering & Science,2019,41(9): 1567-1573.(in Chinese)
［7］	Golub G H, van Loan C F.Matrix computations［M］.Baltimore: The Johns Hopkins University Press,1996.
［8］	Bai Z Z,Golub G H,Ng M K.Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems［J］.SIAM Journal on Matrix Analysis & Applications,2003,98(1): 1-32.
［9］	Guo X X,Wang S.Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems［J］.Applied Mathematics and Computation,2012,218(20):10122-10128.
［10］	Yang A L,An J, Wu Y J.A generalized preconditioned HSS method for non-Hermitian positive definite linear systems［J］.Applied Mathematics and Computation,2010,216(6): 1715-1722.
［11］	Cao Y, Tan W W, Jiang M Q. A generalization of the positive-definite and skew-Hermitian splitting iteration［J］.Numerical Algebra Control & Optimization,2013,2(4):811-821.
［12］	Benzi M. A generalization of the Hermitian and skew- Hermitian splitting iteration［J］.SIAM Journal on Matrix Analysis & Applications,2009,31(2): 360-374.
［13］	Pan Chun-ping, Wang Yu-hong, Cao Wen-fang. On extrapolated Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems ［J］.Mathematica Numerica Sinica,2019,41(1):52-64.(in Chinese)
［14］	Li C X, Wu S L. A modified GHSS method for non- Hermitian positive definite linear systems［J］.Japan Journal of Industrial and Applied Mathematics,2012,29(2): 253-268.
［15］	Bai Z Z,Golub G H,Li C K.Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices［J］.Mathematics of Computation,2007,76(257):287-298.
［16］	Bai Z Z,Golub G H. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations［J］.Numerical Linear Algebra with Applications,2007,19(4): 319-335.
	附中文参考文献：
［5］	初鲁,鲍亮,董贝贝.求解非埃尔米特正定方程组的广义LHSS迭代法［J］.浙江大学学报(理学版),2018,45(6): 694-706.
［6］	董贝贝,鲍亮.求解鞍点问题的广义正定和反Hermitian分裂方法［J］.计算机工程与科学,2019,41(9): 1567-1573.
［13］	潘春萍,王玉红,曹文方.非Hermitian正定线性方程组的外推的HSS迭代方法［J］.计算数学,2019,41(1): 52-64.