On convergence rate of the randomized Gauss-Seidel method
Bai, Zhong-Zhi1,2; Wang, Lu1,2; Wu, Wen-Ting3
AbstractThe Gauss-Seidel and Kaczmarz methods are two classic iteration methods for solving systems of linear equations, which operate in column and row spaces, respectively. Utilizing the connections between these two methods and imitating the exact analysis of the mean-squared error for the randomized Kaczmarz method, we conduct an exact closed-form formula for the mean-squared residual of the iterate generated by the randomized Gauss-Seidel method. Based on this new formula, we further estimate an upper bound for the convergence rate of the randomized Gauss-Seidel method. Theoretical analysis and numerical experiments show that this bound measurably improves the existing ones. Moreover, these theoretical results are also extended to the more general extrapolated randomized Gauss-Seidel method. (C) 2020 Elsevier Inc. All rights reserved.
KeywordSystem of linear equations Randomized Gauss-Seidel method Randomized Kaczmarz method Convergence rate
Indexed BySCI
Funding ProjectNational Natural Science Foundation, P.R. China[12071472] ; National Natural Science Foundation, P.R. China[11671393] ; National Natural Science Foundation, P.R. China[12001043] ; Grant of the Government of the Russian Federation, Russia[075-15-2019-1928] ; China-Russia (NSFC-RFBR) International Cooperative Research Project[11911530082] ; China-Russia (NSFC-RFBR) International Cooperative Research Project[19-51-53013] ; Beijing Institute of Technology Research Fund Program for Young Scholars ; Science and Technology Commission of Shanghai Municipality, P.R. China[18dz2271000]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000600065400013
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Document Type期刊论文
Corresponding AuthorBai, Zhong-Zhi
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Computat Math & Sci Engn Comp, State Key Lab Sci Engn Comp, POB 2719, Beijing 100190, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
3.Beijing Inst Technol, Sch Math & Stat, Beijing 100081, Peoples R China
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GB/T 7714
Bai, Zhong-Zhi,Wang, Lu,Wu, Wen-Ting. On convergence rate of the randomized Gauss-Seidel method[J]. LINEAR ALGEBRA AND ITS APPLICATIONS,2021,611:237-252.
APA Bai, Zhong-Zhi,Wang, Lu,&Wu, Wen-Ting.(2021).On convergence rate of the randomized Gauss-Seidel method.LINEAR ALGEBRA AND ITS APPLICATIONS,611,237-252.
MLA Bai, Zhong-Zhi,et al."On convergence rate of the randomized Gauss-Seidel method".LINEAR ALGEBRA AND ITS APPLICATIONS 611(2021):237-252.
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