Abstract
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.
Similar content being viewed by others
References
Arridge SR (1999) Optical tomography in medical imaging. Inverse Probl 15: R41–R93
Axelsson O, Kucherov A (2000) Real valued iterative methods for solving complex symmetric linear systems. Numer Linear Algebra Appl 7: 197–218
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: 603–626
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: 413–440
Benzi M, Bertaccini D (2008) Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J Numer Anal 28: 598–618
Bertaccini D (2004) Efficient solvers for sequences of complex symmetric linear systems. Electr Trans Numer Anal 18: 49–64
Chan RH, Ng MK (1996) Conjugate gradient methods for Toeplitz systems. SIAM Rev 38: 427–482
Feriani A, Perotti F, Simoncini V (2000) Iterative system solvers for the frequency analysis of linear mechanical systems. Comput Methods Appl Mech Eng 190: 1719–1739
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, Heidelberg
Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore
Poirier B (2000) Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer Linear Algebra Appl 7: 715–726
Saad Y (1993) A flexible inner–outer preconditioned GMRES algorithm. SIAM J Sci Comput 14: 461–469
Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7: 856–869
van der Vorst HA (2003) Iterative Krylov methods for large linear systems. Cambridge University Press, Cambridge
van Dijk W, Toyama FM (2007) Accurate numerical solutions of the time-dependent Schrödinger equation. Phys Rev E 75: 036707-1–036707-10
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C.C. Douglas.
This work was supported by The National Basic Research Program (No. 2005CB321702) and The National Outstanding Young Scientist Foundation (No. 10525102), P. R. China, and by The US National Science Foundation grants DMS-0511336 and DMS-0810862.
Rights and permissions
About this article
Cite this article
Bai, ZZ., Benzi, M. & Chen, F. Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010). https://doi.org/10.1007/s00607-010-0077-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-010-0077-0
Keywords
- Complex symmetric matrix
- Hermitian and skew-Hermitian splitting
- Iteration method
- Krylov subspace method
- Convergence analysis
- Preconditioning