GMRES on singular systems revisited

被引：0

作者：

Hayami K. ^{[1
]}

Sugihara K. ^{[2
]}

机构：

[1] National Institute of Informatics, The Graduate University for Advanced Studies (SOKENDAI), 2-1-2 Hitotsubashi Chiyoda-ku, Tokyo

[2] National Institute of Informatics, 2-1-2 Hitotsubashi Chiyoda-ku, Tokyo

来源：

| 1600年 / National Institute of Informatics卷 / 2020期

关键词：

GMRES method; Krylov subspace method; least squares problem; singular system;

D O I：

10.48550/arXiv.2009.00371

中图分类号：

学科分类号：

摘要：

In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449-469], the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem minxϵRn ∥b - Ax∥2 2, where A ϵ Rn×n may be singular and b ϵ Rn, by decomposing the algorithm into the range R(A) and its orthogonal complement R(A)⊥ components. However, we found that the proof of the fact that GMRES gives a least squares solution if R(A) = R(AT) was not complete. In this paper, we will give a complete proof. © 2020 National Institute of Informatics. All rights reserved.

引用

共 4 条

[1] Hayami K, Sugihara M., A geometric view of Krylov subspace methods on singular systems, Numer Linear Algebra Appl, 18, pp. 449-469, (2011)
[2] Saad Y, Schultz MH., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J Sci Statist Comput, 7, pp. 856-869, (1986)
[3] Brown P, Walker HF., GMRES on (nearly) singular systems, SIAM J Matrix Anal Appl, 18, pp. 37-51, (1997)
[4] Sugihara K, Hayami K, Zheng N., Right preconditioned MINRES for singular systems, Numer Linear Algebra Appl, 27, (2020)

← 1 →