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On successive‐overrelaxation acceleration of the Hermitian and skew‐Hermitian splitting iterations
Author(s) -
Bai ZhongZhi,
Golub Gene H.,
Ng Michael K.
Publication year - 2007
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.517
Subject(s) - mathematics , hermitian matrix , eigenvalues and eigenvectors , convergence (economics) , power iteration , positive definite matrix , matrix (chemical analysis) , iterative method , mathematical analysis , rate of convergence , preconditioner , matrix splitting , linear system , symmetric matrix , pure mathematics , algorithm , square matrix , physics , channel (broadcasting) , materials science , engineering , quantum mechanics , electrical engineering , economics , composite material , economic growth
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.