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Greedy Motzkin–Kaczmarz methods for solving linear systems
Author(s) -
Zhang Yanjun,
Li Hanyu
Publication year - 2022
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2429
Subject(s) - mathematics , greedy algorithm , convergence (economics) , row , residual , block (permutation group theory) , selection (genetic algorithm) , mathematical optimization , algorithm , linear system , combinatorics , computer science , artificial intelligence , mathematical analysis , database , economics , economic growth
The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy randomized Motzkin–Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. The computational complexities of the proposed methods are also given. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices and some realistic practical problems, and their block versions have almost the same performance for most of cases.