Spectral properties of the preconditioned AHSS iteration method for generalized saddle point problems

• Published in: Computational & applied mathematics Vol. 29; no. 2; pp. 269 - 295
• Main Authors: Huang, Zhuo-Hong, Huang, Ting-Zhu
• Format: Journal Article
• Language: English; Portuguese
• Published: Sociedade Brasileira de Matemática Aplicada e Computacional 01.01.2010
• Subjects: MATHEMATICS, APPLIED; PAHSS; positive stable; generalized saddle point problem; splitting iteration method
• ISSN: 1807-0302
• DOI: 10.1590/S1807-03022010000200009

In this paper, we study the distribution on the eigenvalues of the preconditioned matrices that arise in solving two-by-two block non-Hermitian positive semidefinite linear systems by use of the accelerated Hermitian and skew-Hermitian splitting iteration methods. According to theoretical analysis, we prove that all eigenvalues of the preconditioned matrices are very clustered with any positive iteration parameters α and β; especially, when the iteration parameters α and β approximate to 1, all eigenvalues approach 1. We also prove that the real parts of all eigenvalues of the preconditioned matrices are positive, i.e., the preconditioned matrix is positive stable. Numerical experiments show the correctness and feasibility of the theoretical analysis. Mathematical subject classification: 65F10, 65N22, 65F50.