On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations

Authors

  • Zhong-Zhi Bai,

    Corresponding author
    1. 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
    • 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
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  • Gene H. Golub,

    1. Scientific Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, Stanford, CA 94305-9025, U.S.A.
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  • Michael K. Ng

    1. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
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Errata

This article is corrected by:

  1. Errata: Erratum Volume 19, Issue 5, 891, Article first published online: 20 June 2012

Abstract

We further generalize the technique for constructing the Hermitian/skew-Hermitian splitting (HSS) iteration method for solving large sparse non-Hermitian positive definite system of linear equations to the normal/skew-Hermitian (NS) splitting obtaining a class of normal/skew-Hermitian splitting (NSS) iteration methods. Theoretical analyses show that the NSS method converges unconditionally to the exact solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the NSS iteration which is dependent solely on the spectrum of the normal splitting matrix, and is independent of the eigenvectors of the matrices involved. We present a successive-overrelaxation (SOR) acceleration scheme for the NSS iteration, which specifically results in an acceleration scheme for the HSS iteration. Convergence conditions for this SOR scheme are derived under the assumption that the eigenvalues of the corresponding block Jacobi iteration matrix lie in certain regions in the complex plane. A numerical example is used to show that the SOR technique can significantly accelerate the convergence rate of the NSS or the HSS iteration method. Copyright © 2006 John Wiley & Sons, Ltd.

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