Abstract
In this paper, we consider the iterative solution of a class of complex symmetric linear systems and construct a preconditioned variant of the modified block product (PMBP) iteration method. The proposed method is established by applying the preconditioning technique to the MBP method presented recently. The convergence conditions, optimal iteration parameters, and corresponding optimal convergence factor of the PMBP iteration method are determined. Meanwhile, a practical way to choose the iteration parameters for the PMBP iteration method is proposed. Furthermore, comparisons between the PMBP iteration method and some existing ones verify that the PMBP iteration method owns faster convergence rate than some known ones. Finally, some numerical experiments are provided to demonstrate the effectiveness and the superiority of the proposed PMBP method.
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References
Arridge SR (1999) Optical tomography in medical imaging. Inverse Probl 15:R41. https://doi.org/10.1088/0266-5611/15/2/022
Axelsson O, Kucherov A (2000) Real valued iterative methods for solving complex symmetric linear systems. Numer Linear Algebra with Appl 7:197–218. https://doi.org/10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S
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 (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, Pan J-Y (2021) Matrix analysis and computations. SIAM, Philadelphia. https://doi.org/10.1137/1.9781611976632.bm
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, Parlett BN, Wang Z-Q (2005) On generalized successive overrelaxation methods for augmented linear systems. Numerische Mathematik 102:1–38. https://doi.org/10.1007/s00211-005-0643-0
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
Balani FB, Hajarian M (2022) Modified block product preconditioner for a class of complex symmetric linear systems. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2022.2065231
Benzi M, Bertaccini D (2008) Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J Numer Anal 28:598–618. https://doi.org/10.1093/imanum/drm039
Bertaccini D (2004) Efficient solvers for sequences of complex symmetric linear systems. Electron Trans Numer Anal 18:49–64. http://eudml.org/doc/124865
Dijk WV, Toyama FM (2007) Accurate numerical solutions of the time-dependent Schrödinger equation. Phys Rev E 75:1–10. https://doi.org/10.1103/PhysRevE.75.036707
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. https://hrcak.srce.hr/140386
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
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
Huang Z-G (2020) A new double-step splitting iteration method for certain block two-by-two linear systems. Comput Appl Math 39:193. https://doi.org/10.1007/s40314-020-01220-9
Huang Z-G (2021) Efficient block splitting iteration methods for solving a class of complex symmetric linear systems. J Comput Appl Math 395:113574. https://doi.org/10.1016/j.cam.2021.113574
Huang Y-Y, Chen G-L (2018) A relaxed block splitting preconditioner for complex symmetric indefinite linear systems. Open Math 16:561–573. https://doi.org/10.1515/math-2018-0051
Huang Z-G, Wang L-G, Xu Z, Cui J-J (2018) An efficient two-step iterative method for solving a class of complex symmetric linear systems. Comput Math Appl 75:2473–2498. https://doi.org/10.1016/j.camwa.2017.12.026
Huang Z-G, Wang L-G, Xu Z, Cui J-J (2019a) 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
Huang Z-G, Xu Z, Cui J-J (2019b) Preconditioned triangular splitting iteration method for a class of complex symmetric linear systems. Calcolo 56:1–39. https://doi.org/10.1007/s10092-019-0318-3
Li X-A, Zhang W-H, Wu Y-J (2018) On symmetric block triangular splitting iteration method for a class of complex symmetric system of linear equations. Appl Math Lett 79:131–137. https://doi.org/10.1016/j.aml.2017.12.008
Liang Z-Z, Zhang G-F (2019) Robust additive block triangular preconditioners for block two-by-two linear systems. Numer Algorithms 82:503–537. https://doi.org/10.1007/s11075-018-0611-2
Poirier B (2000) Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer Linear Algebra Appl 7:715–726. https://doi.org/10.1002/1099-1506(200010/12)7:7/8<715::AID-NLA220>3.0.CO;2-R
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
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
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. https://doi.org/10.1002/nla.1952
Zeng M-L, Zhang G-F (2015) Parameterized rotated block preconditioning techniques for block two-by-two systems with application to complex linear systems. Comput Math Appl 70:2946–2957. https://doi.org/10.1016/j.camwa.2015.10.011
Zhang J-H, Dai H (2017) A new block preconditioner for complex symmetric indefinite linear systems. Numer Algorithms 74:889–903. https://doi.org/10.1007/s11075-016-0175-y
Zhang J-H, Wang Z-W, Zhao J (2018a) Preconditioned symmetric block triangular splitting iteration method for a class of complex symmetric linear systems. Appl Math Lett 86:95–102. https://doi.org/10.1016/j.aml.2018.06.024
Zhang J-L, Fan H-T, Gu C-Q (2018b) An improved block splitting preconditioner for complex symmetric indefinite linear systems. Numer Algorithms 77:451–478. https://doi.org/10.1007/s11075-017-0323-z
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
Zheng Q, Lu L (2017) A shift-splitting preconditioner for a class of block two-by-two linear systems. Appl Math Lett 66:54–60. https://doi.org/10.1016/j.aml.2016.11.009
Acknowledgements
We would like to express our sincere thanks to editor and anonymous reviewers for their valuable suggestions and constructive comments which greatly improved the presentation of this paper.
Funding
This work was supported by the National Science Foundation of China (no. 12361078), the Guangxi Natural Science Foundations (nos. 2023JJA110017, 2021JJB110006, 2021AC19147) and the Graduate Innovation Program of Guangxi University for Nationalities (no. gxun-chxs 2021056).
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Xie, X., Huang, Z., Cui, J. et al. A preconditioned version of the MBP iteration method for a class of complex symmetric linear systems. Comp. Appl. Math. 43, 123 (2024). https://doi.org/10.1007/s40314-024-02643-4
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DOI: https://doi.org/10.1007/s40314-024-02643-4