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Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

Authors: Zhong-Zhi Bai, Gene H. Golub and Chi-Kwong Li
Journal: Math. Comp. 76 (2007), 287-298
MSC (2000): Primary 65F10, 65F50
Published electronically: August 31, 2006
MathSciNet review: 2261022
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Abstract: For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

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Zhong-Zhi Bai
Affiliation: Department of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China, and State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, People’s Republic of China

Gene H. Golub
Affiliation: Scientific Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, Stanford, California 94305-9025

Chi-Kwong Li
Affiliation: Department of Mathematics, The College of William & Mary, P.O. Box 8795, Williamsburg, Virginia 23187-8795

Keywords: Non-Hermitian matrix, positive semidefinite matrix, Hermitian and skew-Hermitian splitting, splitting iteration method, convergence.
Received by editor(s): February 11, 2005
Published electronically: August 31, 2006
Additional Notes: The work of the first author was supported by The Special Funds For Major State Basic Research Projects (No. G1999032803), The National Basic Research Program (No. 2005CB321702), The China NNSF Outstanding Young Scientist Foundation (No. 10525102) and The National Natural Science Foundation (No. 10471146), P.R. China, and The 2004 Ky and Yu-Fen Fan Fund Travel Grant of American Mathematical Society
The work of the second author was in part supported by the Department of Energy: DE-FC02-01ER41177
The research of the third author was partially supported by an NSF grant.
Article copyright: © Copyright 2006 American Mathematical Society