On partially randomized extended Kaczmarz method for solving large sparse overdetermined inconsistent linear systems

• Published in: Linear algebra and its applications Vol. 578; pp. 225 - 250
• Main Authors: Bai, Zhong-Zhi, Wu, Wen-Ting
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.10.2019; American Elsevier Company, Inc
• Subjects: Convergence; Convergence property; Inconsistency; Iterative methods; Kaczmarz method; Linear algebra; Linear equations; Linear systems; Matrices (mathematics); Randomization; Randomized iteration; System of linear equations; Upper bounds; System of linear equations; Randomized iteration; 65K05; 65F10; Inconsistency; 90C25; Convergence property; 65F20; Kaczmarz method; 15A06
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2019.05.005

For solving large sparse, overdetermined, and inconsistent system of linear equations by iteration methods, by further reconstructing the randomized extended Kaczmarz method proposed by Zouzias and Freris in 2013 (SIAM J. Matrix Anal. Appl. 34 (2013), 773–793), we propose a partially randomized extended Kaczmarz method. When the coefficient matrix is assumed to be of full column rank, we prove the convergence and derive an upper bound for the expected convergence rate of the partially randomized extended Kaczmarz method. This bound could be smaller than that of the randomized extended Kaczmarz method under certain conditions. Moreover, with numerical results we show that the partially randomized extended Kaczmarz method can be much more effective than the randomized extended Kaczmarz method.