Modified HSS iteration methods for a class of complex symmetric linear systems. (English) Zbl 1210.65074

The authors present a modification of the Hermitian and skew-Hermitian splitting (HSS) iteration, which consists in the fact that solution of linear system with coefficient matrix \(\alpha I +\) i \(T\) is avoided and only two linear sub-systems with real symmetric and positive definite matrices \(\alpha I + W\) and \(\alpha I + T\) are solved at each step. They prove that this modified HSS iteration is unconditionally convergent.


65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65F08 Preconditioners for iterative methods
Full Text: DOI


[1] Arridge SR (1999) Optical tomography in medical imaging. Inverse Probl 15: R41–R93 · Zbl 0926.35155
[2] Axelsson O, Kucherov A (2000) Real valued iterative methods for solving complex symmetric linear systems. Numer Linear Algebra Appl 7: 197–218 · Zbl 1051.65025
[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: 603–626 · Zbl 1036.65032
[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: 413–440 · Zbl 1135.65016
[5] Benzi M, Bertaccini D (2008) Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J Numer Anal 28: 598–618 · Zbl 1145.65022
[6] Bertaccini D (2004) Efficient solvers for sequences of complex symmetric linear systems. Electr Trans Numer Anal 18: 49–64 · Zbl 1066.65048
[7] Chan RH, Ng MK (1996) Conjugate gradient methods for Toeplitz systems. SIAM Rev 38: 427–482 · Zbl 0863.65013
[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: 1719–1739 · Zbl 0981.70005
[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, Heidelberg · Zbl 0957.00052
[10] Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, Baltimore · Zbl 0865.65009
[11] Poirier B (2000) Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer Linear Algebra Appl 7: 715–726 · Zbl 1051.65059
[12] Saad Y (1993) A flexible inner–outer preconditioned GMRES algorithm. SIAM J Sci Comput 14: 461–469 · Zbl 0780.65022
[13] Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7: 856–869 · Zbl 0599.65018
[14] van der Vorst HA (2003) Iterative Krylov methods for large linear systems. Cambridge University Press, Cambridge · Zbl 1023.65027
[15] van Dijk W, Toyama FM (2007) Accurate numerical solutions of the time-dependent Schrödinger equation. Phys Rev E 75: 036707-1–036707-10
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.