Abstract
In this paper, the modifications of the Hermitian-Normal splitting iteration methods for solving a class of complex symmetric linear systems are presented. Theoretical analysis shows that the modified iteration methods of Hermitian-normal splitting are unconditionally convergent; the coefficient matrices of the two linear systems solved in each iteration of the methods are real symmetric positive definite. Inexact version of the methods employs the Krylov subspace method as an internal iteration to accelerate. Numerical examples from two model problems are given to illustrate the effectiveness of the modified iteration methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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 (2008) Several splittings for non-Hermitian linear systems. Sci China A 51:1339–1348
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, Pan J-Y (2004) Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer Math 98:1–32
Bai Z-Z, Golub GH, Li C-K (2006) Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J Sci Comput 28:583–603
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
Bai Z-Z, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87:93–111
Bertaccini D (2004) Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron Trans Numer Anal 18:49–64
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
Hageman LA, Young DM (1971) Iterative solution of large linear systems. Academic Press, New York
Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Natl Bur Stand 49:409–435
Saad Y (1993) A flexible inner-outer preconditioned GMRES algorithm. SIAM J Sci Comput 14:461–469
Saad Y (1996) Iterative methods for sparse linear systems. PWS Publishing Company, Boston
Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7:856–869
Widlund O (1978) A Lanczos method for a class of nonsymmetric systems of linear equations. SIAM J Numer Anal 15:801–812
Wu S-L (2015) Several variants of the Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems. Numer Linear Algebra Appl 22:338–356
Acknowledgements
We gratefully acknowledge the reviewers for their patience in reading the first draft of this paper and for their valuable suggestions on revision.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Zhong-Zhi Bai.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by National Natural Science Foundation of China (No. 11101282), by Shanghai Leading Academic Discipline Project (No. XTKX2012), and by Innovation Program of Shanghai Municipal Education Commission (No. 14YZ096).
Rights and permissions
About this article
Cite this article
Du, YK., Qin, M. Modified Hermitian-normal splitting iteration methods for a class of complex symmetric linear systems. Comp. Appl. Math. 39, 190 (2020). https://doi.org/10.1007/s40314-020-01219-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01219-2