On the convergence of splittings for semidefinite linear systems

• Published in: Linear algebra and its applications 2008-11, Vol.429 (10), p.2555-2566
• Main Authors: Lin, Lijing, Wei, Yimin, Woo, Ching-Wah, Zhou, Jieyong
• Format: Article
• Language: English
• Subjects: Algebra; Exact sciences and technology; Hermitian positive semidefinite matrix; Iterative method; Linear and multilinear algebra, matrix theory; Linear system; Mathematics; Rectangular system; Regularity; Sciences and techniques of general use
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2007.12.019

Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634–641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281–290]. In view of their results, we consider here stipulations on a splitting A=M-N, which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore–Penrose solution to the system Ax=b. Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme.