On inexact hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

156 Citations (Scopus)

Abstract

We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by considering both global convergence speed and overall computation workload, and computational efficiencies of both inexact iterations are analyzed and compared deliberately.

Original languageEnglish
Pages (from-to)413-440
Number of pages28
JournalLinear Algebra and Its Applications
Volume428
Issue number2-3
DOIs
Publication statusPublished - 15 Jan 2008

Scopus Subject Areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

User-Defined Keywords

  • Conjugate gradient (CG) method
  • Conjugate gradient for normal equations (CGNE) method
  • Hermitian matrix
  • Inexact iterations
  • Lanczos method
  • Skew-Hermitian matrix

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