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October, 2022 On Relaxed Greedy Randomized Iterative Methods for the Solution of Factorized Linear Systems
Shi-Min Liu, Yong Liu
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Taiwanese J. Math. 26(5): 953-979 (October, 2022). DOI: 10.11650/tjm/220305

Abstract

RK-RK and REK-RK methods are the two latest while very effective randomized iteration solvers for factorized linear system $UV \beta = y$ by interlacing randomized Kaczmarz (RK) and randomized extended Kaczmarz (REK) updates. This paper considers two latest randomized iterative methods for solving large-scale linear systems and linear least-squares problems—greedy randomized Kaczmarz (GRK) method and greedy randomized Gauss–Seidel (GRGS) method. By introducing a relaxation parameter $\omega$ into the iterates of GRK and GRGS, we construct relaxed GRK and GRGS methods, respectively. In addition, by interlacing their updates, we propose relaxed GRK-GRK and GRGS-GRK methods to solve consistent and inconsistent factorized linear systems, respectively. We prove the exponential convergence of these two interlaced methods and show that relaxed GRK-GRK and GRGS-GRK can be more efficient than RK-RK and REK-RK, respectively, if the relaxation parameters are chosen appropriately.

Funding Statement

The work is supported by National Natural Science Foundation (11371243).

Citation

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Shi-Min Liu. Yong Liu. "On Relaxed Greedy Randomized Iterative Methods for the Solution of Factorized Linear Systems." Taiwanese J. Math. 26 (5) 953 - 979, October, 2022. https://doi.org/10.11650/tjm/220305

Information

Received: 26 June 2021; Revised: 18 January 2022; Accepted: 27 March 2022; Published: October, 2022
First available in Project Euclid: 20 March 2022

Digital Object Identifier: 10.11650/tjm/220305

Subjects:
Primary: 15A06 , 65F10 , 65F20 , 65K05 , 90C25

Keywords: factorized linear systems , greedy randomized Gauss–Seidel , greedy randomized Kaczmarz , relaxation parameter

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

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Vol.26 • No. 5 • October, 2022
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