Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-07T23:27:23.915Z Has data issue: false hasContentIssue false
  • English
  • Français

On Preconditioned MHSS Real-Valued Iteration Methods for a Class of Complex Symmetric Indefinite Linear Systems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  12 May 2016

Zhi-Ru Ren ,
Yang Cao  and
Li-Li Zhang
Zhi-Ru Ren
Affiliation:
School of Statistics and Mathematics, Central University of Finance and Economics, Beijing 100081, P.R. China
Yang Cao*
Affiliation:
School of Transportation, Nantong University, Nantong 226019, P.R. China
Li-Li Zhang
Affiliation:
School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450046, P.R. China
*
*Corresponding author. Email addresses:caoyangnt@ntu.edu.cn (Y. Cao)
Get access

Abstract

A generalized preconditioned modified Hermitian and skew-Hermitian splitting (GPMHSS) real-valued iteration method is proposed for a class of complex symmetric indefinite linear systems. Convergence theory is established and the spectral properties of an associated preconditioned matrix are analyzed. We also give several variants of the GPMHSS preconditioner and consider the spectral properties of the preconditioned matrices. Numerical examples illustrate the effectiveness of our proposed method.

Keywords

Complex linear systemsPMHSS iterationreal-valued formconvergencepreconditioning

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 192 - 210
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Arridge, S.R., Egger, E. and Schlottbom, M., Preconditioning of complex symmetric linear systems with applications in optical tomography, Appl. Numer. Math. 74, 35–48 (2013).Google Scholar
[2]Axelsson, O. and Kucherov, A., Real valued iterativemethods for solving complex symmetric linear systems, Numer. Linear Algebra Appl. 7, 197–218 (2000).Google Scholar
[3]Axelsson, O., Neytcheva, M. and Ahmad, B., A comparison of iterative methods to solve complex valued linear algebraic systems, Numer. Algor. 66, 811–841 (2014).Google Scholar
[4]Bai, Z.-Z., Rotated block triangular preconditioning based on PMHSS, Sci. China Math. 56, 2523–2538 (2013).Google Scholar
[5]Bai, Z.-Z., Benzi, M. and Chen, F., Modified HSS iterationmethods for a class of complex symmetric linear systems, Comput. 87, 93–111 (2010).Google Scholar
[6]Bai, Z.-Z., Benzi, M. and Chen, F., On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor. 56, 297–317 (2011).Google Scholar
[7]Bai, Z.-Z., Benzi, M., Chen, F. and Wang, Z.-Q., Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal. 33, 343–369 (2013).Google Scholar
[8]Bai, Z.-Z., Chen, F. and Wang, Z.-Q., Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices, Numer. Algor. 62, 655–675 (2013).Google Scholar
[9]Bai, Z.-Z., Golub, G.H. and Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24, 603–626 (2003).Google Scholar
[10]Benzi, M. and Bertaccini, D., Block preconditioning of real-valued iterative algorithms for complex linear systems, IMA J. Numer. Anal. 28, 598–618 (2008).Google Scholar
[11]Benzi, M. and Golub, G.H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 26, 20–41 (2004).Google Scholar
[12]Bertaccini, D., Efficient preconditioning for sequences of parametric complex symmetric linear systems, Electron. Trans. Numer. Anal. 18, 49–64 (2004).Google Scholar
[13]Cao, Y. and Ren, Z.-R., Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems, Appl. Math. Comput. 264, 61–71 (2015).Google Scholar
[14]Day, D. and Heroux, M., Solving complex-valued linear systems via equivalent real formulations, SIAM J. Sci. Comput. 23, 480–498 (2001).CrossRefGoogle Scholar
[15]Feriani, A., Perotti, F. and Simoncini, V., Iterative system solvers for the frequency analysis of linear mechanical systems, Comput. Methods. Appl. Mech. Eng. 190, 1719–1739 (2000).Google Scholar
[16]Frommer, A., Lippert, T., Medeke, B. and Schilling, K. (eds), Numerical Challenges in Lattice QuantumChromodynamics, Lecture Notes in Computational Science and Engineering, vol. 15, Springer, Heidelberg (2000).Google Scholar
[17]Howle, V.E. and Vavasis, S.A., An iterative method for solving complex-symmetric systems arising in electrical power modeling, SIAM J. Matrix Anal. Appl. 26, 1150–1178 (2005).Google Scholar
[18]Ipsen, I.C.F., A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput. 23, 1050–1051 (2002).Google Scholar
[19]Lang, C. and Ren, Z.-R., Inexact rotated block triangular preconditioners for a class of block two-by-two matrices, J. Eng. Math. 93, 87–98 (2015).Google Scholar
[20]Poirier, B., Efficient preconditioning scheme for block partitioned matrices with structured sparsity, Numer. Linear Algebra Appl. 7, 715–726 (2000).Google Scholar
[21]Saad, Y., Iterative Methods for Sparse Linear Systems (2nd edn), SIAM: Philadelphia (2003).Google Scholar
[22]van Dijk, W. and Toyama, F.M., Accurate numerical solutions of the time-dependent Schrödinger equation, Phys. Rev. E 75, 036707-1–36707-10 (2007).Google Scholar
[23]Li, X., Yang, A.-L. and Wu, Y.-J., Lopsided PMHSS iterationmethod for a class of complex symmetric linear systems, Numer. Algor. 66, 555–568 (2014).Google Scholar
[24]Xu, W.-W., A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems, Appl. Math. Comput. 219, 10510–10517 (2013).Google Scholar