, Volume 87, Issue 3–4, pp 93–111 | Cite as

Modified HSS iteration methods for a class of complex symmetric linear systems

  • Zhong-Zhi BaiEmail author
  • Michele Benzi
  • Fang Chen


In this paper, we introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving a broad class of complex symmetric linear systems. We show that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. We consider acceleration of the MHSS iteration by Krylov subspace methods. Numerical experiments on a few model problems are used to illustrate the performance of the new method.


Complex symmetric matrix Hermitian and skew-Hermitian splitting Iteration method Krylov subspace method Convergence analysis Preconditioning 

Mathematics Subject Classification (2000)

65F10 65F50 65N22 CR: G1.3 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arridge SR (1999) Optical tomography in medical imaging. Inverse Probl 15: R41–R93zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Axelsson O, Kucherov A (2000) Real valued iterative methods for solving complex symmetric linear systems. Numer Linear Algebra Appl 7: 197–218zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24: 603–626zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bai Z-Z, Golub GH, Ng MK (2008) On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl 428: 413–440zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benzi M, Bertaccini D (2008) Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J Numer Anal 28: 598–618zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bertaccini D (2004) Efficient solvers for sequences of complex symmetric linear systems. Electr Trans Numer Anal 18: 49–64zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chan RH, Ng MK (1996) Conjugate gradient methods for Toeplitz systems. SIAM Rev 38: 427–482zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feriani A, Perotti F, Simoncini V (2000) Iterative system solvers for the frequency analysis of linear mechanical systems. Comput Methods Appl Mech Eng 190: 1719–1739zbMATHCrossRefGoogle Scholar
  9. 9.
    Frommer A, Lippert T, Medeke B, Schilling K (eds) (2000) Numerical challenges in lattice quantum chromodynamics. Lecture notes in computational science and engineering, vol 15. Springer, HeidelbergGoogle Scholar
  10. 10.
    Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  11. 11.
    Poirier B (2000) Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer Linear Algebra Appl 7: 715–726zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Saad Y (1993) A flexible inner–outer preconditioned GMRES algorithm. SIAM J Sci Comput 14: 461–469zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7: 856–869zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    van der Vorst HA (2003) Iterative Krylov methods for large linear systems. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  15. 15.
    van Dijk W, Toyama FM (2007) Accurate numerical solutions of the time-dependent Schrödinger equation. Phys Rev E 75: 036707-1–036707-10Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceGuizhou Normal UniversityGuiyangPeople’s Republic of China
  2. 2.State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  4. 4.Department of Mathematical SciencesXi’an Jiaotong UniversityXi’anPeople’s Republic of China
  5. 5.Department of Mathematics and PhysicsXi’an University of Post and TelecommunicationXi’anPeople’s Republic of China

Personalised recommendations