Received June 28, 2023. Published online April 5, 2024.
Abstract: We multiply both sides of the complex symmetric linear system $Ax=b$ by $1-i\omega$ to obtain a new equivalent linear system, then a dual-parameter double-step splitting (DDSS) method is established for solving the new linear system. In addition, we present an upper bound for the spectral radius of iteration matrix of the DDSS method and obtain its quasi-optimal parameter. Theoretical analyses demonstrate that the new method is convergent when some conditions are satisfied. Some tested examples are given to illustrate the effectiveness of the proposed method.
Keywords: DDSS iteration method; linear equations; SPD matrix; SPSD matrix; convergence property
References: [1] S. R. Arridge: Optical tomography in medical imaging. Inverse Probl. 15 (1999), R41-R93. DOI 10.1088/0266-5611/15/2/022 | MR 1684463 | Zbl 0926.35155
[2] O. Axelsson, A. Kucherov: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7 (2000), 197-218. DOI 10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S | MR 1762967 | Zbl 1051.65025
[3] Z.-Z. Bai: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109 (2000), 273-285. DOI 10.1016/S0096-3003(99)00027-2 | MR 1738197 | Zbl 1026.65028
[4] Z.-Z. Bai: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283 (2015), 71-78. DOI 10.1016/j.cam.2015.01.025 | MR 3317271 | Zbl 1311.65032
[5] Z.-Z. Bai: Quasi-HSS iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. Numer. Linear Algebra Appl. 25 (2018), Article ID e2116, 19 pages. DOI 10.1002/nla.2116 | MR 3826931 | Zbl 1513.65063
[6] Z.-Z. Bai, M. Benzi, F. Chen: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87 (2010), 93-111. DOI 10.1007/s00607-010-0077-0 | MR 2640009 | Zbl 1210.65074
[7] Z.-Z. Bai, M. Benzi, F. Chen: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56 (2011), 297-317. DOI 10.1007/s11075-010-9441-6 | MR 2755673 | Zbl 1209.65037
[8] Z.-Z. Bai, G. H. Golub: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27 (2007), 1-23. DOI 10.1093/imanum/drl017 | MR 2289269 | Zbl 1134.65022
[9] Z.-Z. Bai, G. H. Golub, M. K. Ng: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24 (2003), 603-626. DOI 10.1137/S0895479801395458 | MR 1972670 | Zbl 1036.65032
[10] Z.-Z. Bai, G. H. Golub, J.-Y. Pan: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98 (2004), 1-32. DOI 10.1007/s00211-004-0521-1 | MR 2076052 | Zbl 1056.65025
[11] Z.-Z. Bai, J.-Y. Pan: Matrix Analysis and Computations. Other Titles in Applied Mathematics 173. SIAM, Philadelphia (2021). DOI 10.1137/1.9781611976632 | MR 4362581 | Zbl 07417710
[12] M. Benzi, D. Bertaccini: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28 (2008), 598-618. DOI 10.1093/imanum/drm039 | MR 2433214 | Zbl 1145.65022
[13] D. Bertaccini: Efficient preconditioning for sequences of parametric complex symmetric linear systems. ETNA, Electron. Trans. Numer. Anal. 18 (2004), 49-64. MR 2083294 | Zbl 1066.65048
[14] F. Chen, T.-Y. Li, K.-Y. Lu, G. V. Muratova: Modified QHSS iteration methods for a class of complex symmetric linear systems. Appl. Numer. Math. 164 (2021), 3-14. DOI 10.1016/j.apnum.2020.01.018 | MR 4207970 | Zbl 1460.65033
[15] M. Dehghan, M. Dehghani-Madiseh, M. Hajarian: A generalized preconditioned MHSS method for a class of complex symmetric linear systems. Math. Model. Anal. 18 (2013), 561-576. DOI 10.3846/13926292.2013.839964 | MR 3175665 | Zbl 1281.65058
[16] A. Feriani, F. Perotti, V. Simoncini: Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Eng. 190 (2000), 1719-1739. DOI 10.1016/S0045-7825(00)00187-0 | Zbl 0981.70005
[17] A. Frommer, T. Lippert, B. Medeke, K. Schilling (Eds.): Numerical Challenges in Lattice Quantum Chromodynamics. Lecture Notes in Computational Science and Engineering 15. Springer, Berlin (2000). DOI 10.1007/978-3-642-58333-9 | MR 1861777 | Zbl 0957.00052
[18] Y. Huang, G. Chen: A relaxed block splitting preconditioner for complex symmetric indefinite linear systems. Open Math. 16 (2018), 561-573. DOI 10.1515/math-2018-0051 | MR 3812172 | Zbl 1388.65033
[19] Z.-G. Huang: A new double-step splitting iteration method for certain block two-by-two linear systems. Comput. Appl. Math. 39 (2020), Article ID 193, 42 pages. DOI 10.1007/s40314-020-01220-9 | MR 4116897 | Zbl 1463.65047
[20] Z.-G. Huang: Efficient block splitting iteration methods for solving a class of complex symmetric linear systems. J. Comput. Appl. Math. 395 (2021), Article ID 113574, 21 pages. DOI 10.1016/j.cam.2021.113574 | MR 4246150 | Zbl 1470.65054
[21] Z.-G. Huang: Modified two-step scale-splitting iteration method for solving complex symmetric linear systems. Comput. Appl. Math. 40 (2021), Article ID 122, 35 pages. DOI 10.1007/s40314-021-01514-6 | MR 4248586 | Zbl 1476.65041
[22] Z.-G. Huang, L.-G. Wang, Z. Xu, J.-J. Cui: Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems. Comput. Math. Appl. 77 (2019), 1902-1916. DOI 10.1016/j.camwa.2018.11.024 | MR 3926852 | Zbl 1442.65041
[23] B. Li, J. Cui, Z. Huang, X. Xie: On preconditioned MQHSS iterative method for solving a class of complex symmetric linear systems. Comput. Appl. Math. 41 (2022), Article ID 250, 23 pages. DOI 10.1007/s40314-022-01942-y | MR 4455168 | Zbl 1513.65076
[24] C.-X. Li, S.-L. Wu: A single-step HSS method for non-Hermitian positive definite linear systems. Appl. Math. Lett. 44 (2015), 26-29. DOI 10.1016/j.aml.2014.12.013 | MR 3311417 | Zbl 1315.65032
[25] L. Li, T.-Z. Huang, X.-P. Liu: Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Linear Algebra Appl. 14 (2007), 217-235. DOI 10.1002/nla.528 | MR 2301913 | Zbl 1199.65109
[26] X. Li, A.-L. Yang, Y.-J. Wu: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms 66 (2014), 555-568. DOI 10.1007/s11075-013-9748-1 | MR 3225002 | Zbl 1298.65058
[27] H. Noormohammadi Pour, H. Sadeghi Goughery: New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems. Numer. Algorithms 69 (2015), 207-225. DOI 10.1007/s11075-014-9890-4 | MR 3339219 | Zbl 1317.65091
[28] B. Poirier: Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer. Linear Algebra Appl. 7 (2000), 715-726. DOI 10.1002/1099-1506(200010/12)7:7/8<715::AID-NLA220>3.0.CO;2-R | MR 1802367 | Zbl 1051.65059
[29] A. Shirilord, M. Dehghan: Single step iterative method for linear system of equations with complex symmetric positive semi-definite coefficient matrices. Appl. Math. Comput. 426 (2022), Article ID 127111, 17 pages. DOI 10.1016/j.amc.2022.127111 | MR 4408297 | Zbl 1511.65055
[30] T. S. Siahkoalaei, D. K. Salkuyeh: A new double-step method for solving complex Helmholtz equation. Hacet. J. Math. Stat. 49 (2020), 1245-1260. DOI 10.15672/hujms.494876 | MR 4199075 | Zbl 1478.65102
[31] W. van Dijk, F. M. Toyama: Accurate numerical solutions of the time-dependent Schrödinger equation. Phys. Rev. E 75 (2007), Article ID 036707, 10 pages. DOI 10.1103/PhysRevE.75.036707 | MR 2358574
[32] T. Wang, Q. Zheng, L. Lu: A new iteration method for a class of complex symmetric linear systems. J. Comput. Appl. Math. 325 (2017), 188-197. DOI 10.1016/j.cam.2017.05.002 | MR 3658905 | Zbl 1365.65087
[33] S.-L. Wu: Several variants of the Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems. Numer. Linear Algebra Appl. 22 (2015), 338-356. DOI 10.1002/nla.1952 | MR 3313262 | Zbl 1363.65055
[34] X.-Y. Xiao, X. Wang, H.-W. Yin: Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems. Comput. Math. Appl. 74 (2017), 2269-2280. DOI 10.1016/j.camwa.2017.07.007 | MR 3718115 | Zbl 1398.65053
[35] X.-Y. Xiao, X. Wang, H.-W. Yin: Efficient preconditioned NHSS iteration methods for solving complex symmetric linear systems. Comput. Math. Appl. 75 (2018), 235-247. DOI 10.1016/j.camwa.2017.09.004 | MR 3758701 | Zbl 1478.65023
[36] A.-L. Yang: On the convergence of the minimum residual HSS iteration method. Appl. Math. Lett. 94 (2019), 210-216. DOI 10.1016/j.aml.2019.02.031 | MR 3924568 | Zbl 1411.65055
[37] A.-L. Yang, Y. Cao, Y.-J. Wu: Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems. BIT 59 (2019), 299-319. DOI 10.1007/s10543-018-0729-6 | MR 3921381 | Zbl 1432.65033
[38] M.-L. Zeng: Inexact modified QHSS iteration methods for complex symmetric linear systems of strong skew-Hermitian parts. IAENG, Int. J. Appl. Math. 51 (2021), 109-115.
[39] J. Zhang, H. Dai: A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems. Appl. Math. Lett. 49 (2015), 100-106. DOI 10.1016/j.aml.2015.05.006 | MR 3361702 | Zbl 1382.65083
[40] J.-H. Zhang, H. Dai: A new block preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms 74 (2017), 889-903. DOI 10.1007/s11075-016-0175-y | MR 3611559 | Zbl 1366.65049
[41] J. Zhang, Z. Wang, J. Zhao: Double-step scale splitting real-valued iteration method for a class of complex symmetric linear systems. Appl. Math. Comput. 353 (2019), 338-346. DOI 10.1016/j.amc.2019.02.020 | MR 3916000 | Zbl 1429.65073
[42] J.-L. Zhang, H.-T. Fan, C.-Q. Gu: An improved block splitting preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms 77 (2018), 451-478. DOI 10.1007/s11075-017-0323-z | MR 3748379 | Zbl 1388.65031
[43] W.-H. Zhang, A.-L. Yang, Y.-J. Wu: Minimum residual modified HSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms 86 (2021), 1543-1559. DOI 10.1007/s11075-020-00944-3 | MR 4229637 | Zbl 1470.65057
[44] Z. Zheng, F.-L. Huang, Y.-C. Peng: Double-step scale splitting iteration method for a class of complex symmetric linear systems. Appl. Math. Lett. 73 (2017), 91-97. DOI 10.1016/j.aml.2017.04.017 | MR 3659913 | Zbl 1375.65056
Affiliations: Beibei Li, Jingjing Cui (corresponding author), Zhengge Huang, Xiaofeng Xie, College of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University, No. 188, East Daxue Road, Xixiangtang District, 530006, Nanning, P. R. China, e-mail: beibeili1997@163.com, jingjingcui1990@163.com, ZhenggeHuang@163.com, xiexiaofeng2022@163.com
