Abstract
By utilizing the minimum residual technique to the two-parameter TSCSP (TTSCSP) iteration method, we propose the two-step minimum residual TTSCSP (TMRTTSCSP) method for solving the large sparse complex symmetric linear systems in this article. Meanwhile, we explore the convergence conditions of this method. Moreover, by synthesizing the iterative scheme of the TTSCSP method into one step, we derive the single-step minimum residual TTSCSP (SMRTTSCSP) method and discuss its convergence properties as well. Finally, we verify the effectiveness of these two new methods and compare them with several existing ones by two numerical experiments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bai Z-Z (2008) Several splittings for non-Hermitian linear systems. Sci China Ser A Math 51:1339–1348. https://doi.org/10.1007/s11425-008-0106-z
Bai Z-Z (2009) Optimal parameters in the HSS-like methods for saddle-point problems. Numer Linear Algebra Appl 16:447–479. https://doi.org/10.1002/nla.626
Bai Z-Z (2013) Rotated block triangular preconditioning based on PMHSS. Sci China Math 56:2523–2538. https://doi.org/10.1007/s11425-013-4695-9
Bai Z-Z (2015) On preconditioned iteration methods for complex linear systems. J Eng Math 93:41–60. https://doi.org/10.1007/s10665-013-9670-5
Bai Z-Z (2018) Quasi-HSS iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. Numer Linear Algebra Appl 25:e2116. https://doi.org/10.1002/nla.2116
Bai Z-Z, Golub GH (2007) Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J Numer Anal 27:1–23. https://doi.org/10.1093/imanum/drl017
Bai Z-Z, Wang Z-Q (2008) On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl 428:2900–2932. https://doi.org/10.1016/j.laa.2008.01.018
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. https://doi.org/10.1137/s0895479801395458
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. https://doi.org/10.1007/s00211-004-0521-1
Bai Z-Z, Parlett BN, Wang Z-Q (2005) On generalized successive overrelaxation methods for augmented linear systems. Numer Math 102:1–38. https://doi.org/10.1007/s00211-005-0643-0
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. https://doi.org/10.1137/050623644
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. https://doi.org/10.1016/j.laa.2007.02.018
Bai Z-Z, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87:93–111. https://doi.org/10.1007/s00607-010-0077-0
Bai Z-Z, Benzi M, Chen F (2011) On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer Algorithms 56:297–317. https://doi.org/10.1007/s11075-010-9441-6
Bai Z-Z, Benzi M, Chen F, Wang Z-Q (2013) 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. https://doi.org/10.1093/imanum/drs001
Benzi M (2009) A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J Matrix Anal Appl 31:360–374. https://doi.org/10.1137/080723181
Benzi M, Gander MJ, Golub GH (2003) Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT Numer Math 43:881–900. https://doi.org/10.1023/b:bitn.0000014548.26616.65
Cao Y, Ren Z-R, Yao L-Q (2019) Improved relaxed positive-definite and skew-Hermitian splitting preconditioners for saddle point problems. J Comput Math 37:95–111. https://doi.org/10.4208/jcm.1710-m2017-0065
Chen F (2015) On choices of iteration parameter in HSS method. Appl Math Comput 271:832–837. https://doi.org/10.1016/j.amc.2015.09.003
Chen F, Jiang Y-L (2012) On contraction factors of Hermitian and skew-Hermitian splitting iteration method for generalized saddle point problems. Commun Appl Math Comput 26:28–34
Chen F, Li T-Y (2021) Fast and improved scaled HSS preconditioner for steady-state space-fractional diffusion equations. Numer Algorithms 87:651–665. https://doi.org/10.1007/s11075-020-00982-x
Chen F, Li T-Y, Muratova GV (2021) Lopsided scaled HSS preconditioner for steady-state space-fractional diffusion equations. Calcolo 58:26. https://doi.org/10.1007/s10092-021-00419-4
Edalatpour V, Hezari D, Salkuyeh DK (2015) Accelerated generalized SOR method for a class of complex systems of linear equations. Math Commun 20:37–52
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. https://doi.org/10.1016/s0045-7825(00)00187-0
Han D-R, Yuan X-M (2013) Local linear convergence of the alternating direction method of multipliers for quadratic programs. SIAM J Numer Anal 51:3446–3457. https://doi.org/10.1137/120886753
Hezari D, Edalatpour V, Salkuyeh DK (2015) Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer Linear Algebra Appl 22:761–776. https://doi.org/10.1002/nla.1987
Hezari D, Salkuyeh DK, Edalatpour V (2016) A new iterative method for solving a class of complex symmetric system of linear equations. Numer Algorithms 73:927–955. https://doi.org/10.1007/s11075-016-0123-x
Huang Y-M (2014) A practical formula for computing optimal parameters in the HSS iteration methods. J Comput Appl Math 255:142–149. https://doi.org/10.1016/j.cam.2013.01.023
Huang Z-G, Wang L-G, Xu Z, Cui J-J (2019) Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems. Comput Math Appl 77:1902–1916. https://doi.org/10.1016/j.camwa.2018.11.024
Li X, Wu Y-J, Yang A-L, Yuan J-Y (2014) Modified accelerated parameterized inexact Uzawa method for singular and nonsingular saddle point problems. Appl Math Comput 244:552–560. https://doi.org/10.1016/j.amc.2014.07.031
Liang Z-Z, Zhang G-F (2016) On SSOR iteration method for a class of block two-by-two linear systems. Numer Algorithms 71:655–671. https://doi.org/10.1007/s11075-015-0015-5
Liao L-D, Zhang G-F (2019) A generalized variant of simplified HSS preconditioner for generalized saddle point problems. Appl Math Comput 346:790–799. https://doi.org/10.1016/j.amc.2018.10.073
Salkuyeh DK (2017) Two-step scale-splitting method for solving complex symmetric system of linear equations. arXiv:1705.02468
Salkuyeh DK, Siahkolaei TS (2018) Two-parameter TSCSP method for solving complex symmetric system of linear equations. Calcolo 55:1–22. https://doi.org/10.1007/s10092-018-0252-9
Salkuyeh DK, Hezari D, Edalatpour V (2015) Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. Int J Comput Math 92:802–815. https://doi.org/10.1080/00207160.2014.912753
Simoncini V, Benzi M (2004) Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems. SIAM J Matrix Anal Appl 26:377–389. https://doi.org/10.1137/s0895479803434926
van Dijk W, Toyama FM (2007) Accurate numerical solutions of the time-dependent Schrüdinger equation. Phys Rev E 75:036707. https://doi.org/10.1103/physreve.75.036707
Wang T, Zheng Q-Q, Lu L-Z (2017) A new iteration method for a class of complex symmetric linear systems. J Comput Appl Math 325:188–197. https://doi.org/10.1016/j.cam.2017.05.002
Yang A-L, Cao Y, Wu Y-J (2019) Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems. BIT Numer Math 59:299–319. https://doi.org/10.1007/s10543-018-0729-6
Zhang J-L (2018) An efficient variant of HSS preconditioner for generalized saddle point problems. Numer Linear Algebra Appl 25:e2166. https://doi.org/10.1002/nla.2166
Zhang G-F, Lu Q-H (2008) On generalized symmetric SOR method for augmented systems. J Comput Appl Math 219:51–58. https://doi.org/10.1016/j.cam.2007.07.001
Zhang J-H, Wang Z-W, Zhao J (2019) Double-step scale splitting real-valued iteration method for a class of complex symmetric linear systems. Appl Math Comput 353:338–346. https://doi.org/10.1016/j.amc.2019.02.020
Zhang W-H, Yang A-L, Wu Y-J (2021) Minimum residual modified HSS iteration method for a class of complex symmetric linear systems. Numer Algorithms 86:1543–1559. https://doi.org/10.1007/s11075-020-00944-3
Acknowledgements
We would like to express our sincere thanks to the editor and the anonymous reviewers for their valuable suggestions and constructive comments which greatly improved the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Science Foundation of China (No. 11901123), the Guangxi Natural Science Foundations (Nos. 2018JJB110062, 2019AC20062, 2021JJB110006, 2021AC19147), the Natural Science Foundation of Guangxi University for Nationalities (No. 2019KJQN001) and the Graduate Innovation Program of Guangxi University for Nationalities (No. gxun-chxs 2021056).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xie, X., Huang, Z., Cui, J. et al. Minimum residual two-parameter TSCSP method for solving complex symmetric linear systems. Comp. Appl. Math. 42, 52 (2023). https://doi.org/10.1007/s40314-023-02195-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-023-02195-z