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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Structured preconditioners for nonsingular matrices of block two-by-two structures


Author: Zhong-Zhi Bai
Journal: Math. Comp. 75 (2006), 791-815
MSC (2000): Primary 65F10, 65F50
Published electronically: November 30, 2005
MathSciNet review: 2196992
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Abstract: For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.


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Additional Information

Zhong-Zhi Bai
Affiliation: State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, People’s Republic of China
Email: bzz@lsec.cc.ac.cn

DOI: http://dx.doi.org/10.1090/S0025-5718-05-01801-6
PII: S 0025-5718(05)01801-6
Keywords: Block two-by-two matrix, preconditioner, modified block relaxation iteration, eigenvalue distribution, positive definiteness
Received by editor(s): October 16, 2003
Received by editor(s) in revised form: March 16, 2005
Published electronically: November 30, 2005
Additional Notes: Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803), The National Basic Research Program (No. 2005CB321702), and The National Natural Science Foundation (No. 10471146), P.R. China
Article copyright: © Copyright 2005 American Mathematical Society