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

Zhong Zhi Bai*, Gene H. Golub, Kwok Po NG

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

171 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)319-335
Number of pages17
JournalNumerical Linear Algebra with Applications
Issue number4
Publication statusPublished - May 2007

Scopus Subject Areas

  • Algebra and Number Theory
  • Applied Mathematics

User-Defined Keywords

  • Hermitian matrix
  • Non-hermitian matrix
  • Normal matrix
  • Skew-Hermitian matrix
  • Splitting iteration method
  • Successive overrelaxation


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