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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. Advance Publication 1-27 (2022). DOI: 10.11650/tjm/220305


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).


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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. Advance Publication 1 - 27, 2022.


Published: 2022
First available in Project Euclid: 30 March 2022

Digital Object Identifier: 10.11650/tjm/220305

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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