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Title: A new optimized iterative method for solving $M$-matrix linear systems (English)
Author: Fakharzadeh Jahromi, Alireza
Author: Nasseri Shams, Nafiseh
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 67
Issue: 3
Year: 2022
Pages: 251-272
Summary lang: English
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Category: math
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Summary: In this paper, we present a new iterative method for solving a linear system, whose coefficient matrix is an $M$-matrix. This method includes four parameters that are obtained by the accelerated overrelaxation (AOR) splitting and using the Taylor approximation. First, under some standard assumptions, we establish the convergence properties of the new method. Then, by minimizing the Frobenius norm of the iteration matrix, we find the optimal parameters. Meanwhile, numerical results on test examples show the efficiency of the new proposed method in contrast with the Hermitian and skew-Hermitian splitting (HSS), AOR methods and a modified version of the AOR (QAOR) iteration. (English)
Keyword: linear system
Keyword: $M$-matrix
Keyword: optimal parameter
Keyword: Taylor approximation
Keyword: optimization
MSC: 65F10
MSC: 90C99
idZBL: Zbl 07547195
idMR: MR4409306
DOI: 10.21136/AM.2021.0246-20
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Date available: 2022-04-14T13:34:14Z
Last updated: 2024-07-01
Stable URL: http://hdl.handle.net/10338.dmlcz/150314
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Reference: [1] Avdelas, G., Hadjidimos, A., Yeyios, A.: Some theoretical and computational results concerning the accelerated overrelaxation (AOR) method.Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 9 (1980), 5-10. Zbl 0445.65018, MR 0617249
Reference: [2] Bai, Z., Chi, X.: Asymptotically optimal successive overrelaxation methods for systems of linear equations.J. Comput. Math. 21 (2003), 503-612. Zbl 1031.65050, MR 1999971
Reference: [3] Bai, Z.-Z., Golub, G. H., Ng, M. K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems.SIAM J. Matrix Anal. Appl. 24 (2003), 603-626. Zbl 1036.65032, MR 1972670, 10.1137/S0895479801395458
Reference: [4] Bai, Z.-Z., Golub, G. H., Ng, M. K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations.Numer. Linear Algebra Appl. 14 (2007), 319-335. Zbl 1199.65097, MR 2310394, 10.1002/nla.517
Reference: [5] Bai, Z.-Z., Golub, G. H., Pan, J.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems.Numer. Math. 98 (2004), 1-32. Zbl 1056.65025, MR 2076052, 10.1007/s00211-004-0521-1
Reference: [6] Beik, F. P. A., Shams, N. N.: Preconditioned generalized mixed-type splitting iterative method for solving weighted least-squares problems.Int. J. Comput. Math. 91 (2014), 944-963. Zbl 1304.65134, MR 3230032, 10.1080/00207160.2013.810215
Reference: [7] Benzi, M.: A generalization of the Hermitian and skew-Hermitian splitting iteration.SIAM. J. Matrix Anal. Appl. 31 (2009), 360-374. Zbl 1191.65025, MR 2530254, 10.1137/080723181
Reference: [8] Benzi, M., Golub, G. H.: A preconditioner for generalized saddle point problems.SIAM J. Matrix Anal. Appl. 26 (2004), 20-41. Zbl 1082.65034, MR 2112850, 10.1137/S0895479802417106
Reference: [9] Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences.Computer Science and Applied Mathematics. Academic Press, New York (1979). Zbl 0484.15016, MR 0544666, 10.1016/c2013-0-10361-3
Reference: [10] Demmel, J. W.: Applied Numerical Linear Algebra.SIAM, Philadelphia (1997). Zbl 0879.65017, MR 1463942, 10.1137/1.9781611971446
Reference: [11] Golub, G. H., Vanderstraeten, D.: On the preconditioning of matrices with skew-symmetric splittings.Numer. Algorithms 25 (2000), 223-239. Zbl 0983.65041, MR 1827156, 10.1023/A:1016637813615
Reference: [12] Guo, P., Wu, S.-L., Li, C.-X.: On the SOR-like iteration method for solving absolute value equations.Appl. Math. Lett. 97 (2019), 107-113. Zbl 1437.65044, MR 3957497, 10.1016/j.aml.2019.03.033
Reference: [13] Hadjidimos, A.: Accelerated overrelaxation method.Math. Comput. 32 (1978), 149-157. Zbl 0382.65015, MR 0483340, 10.1090/S0025-5718-1978-0483340-6
Reference: [14] Ke, Y.: The new iteration algorithm for absolute value equation.Appl. Math. Lett. 99 (2020), Article ID 105990, 7 pages. Zbl 07112056, MR 3989672, 10.1016/j.aml.2019.07.021
Reference: [15] Li, L., Huang, T.-Z., Liu, X.-P.: Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems.Numer. Linear Algebra Appl. 14 (2007), 217-235. Zbl 1199.65109, MR 2301913, 10.1002/nla.528
Reference: [16] Meng, G.-Y.: A practical asymptotical optimal SOR method.Appl. Math. Comput. 242 (2014), 707-715. Zbl 1336.65044, MR 3239699, 10.1016/j.amc.2014.06.034
Reference: [17] Ren, L., Ren, F., Wen, R.: A selected method for the optimal parameters of the AOR iteration.J. Inequal. Appl. 2016 (2016), Article ID 279, 14 pages. Zbl 1353.65025, MR 3571336, 10.1186/s13660-016-1196-8
Reference: [18] Saad, Y.: Iterative Methods for Sparse Linear Systems.SIAM, Philadelphia (2003). Zbl 1031.65046, MR 1990645, 10.1137/1.9780898718003
Reference: [19] Salkuyeh, D. K.: The Picard-HSS iteration method for absolute value equations.Optim. Lett. 8 (2014), 2191-2202. Zbl 1335.90102, MR 3279597, 10.1007/s11590-014-0727-9
Reference: [20] Varga, R. S.: Matrix Iterative Analysis.Prentice-Hall Series in Automatic Computation. Prentice-Hall, Englewood Cliffs (1962). Zbl 0133.08602, MR 0158502, 10.1007/978-3-642-05156-2
Reference: [21] Woźnicki, Z. I.: Basic comparison theorems for weak and weaker matrix splitting.Electron. J. Linear Algebra 8 (2001), 53-59. Zbl 0981.65041, MR 1836055, 10.13001/1081-3810.1060
Reference: [22] Wu, S.-L., Liu, Y.-J.: A new version of the accelerated overrelaxation iterative method.J. Appl. Math. 2014 (2014), Article ID 725360, 6 pages. Zbl 1442.65050, MR 3256322, 10.1155/2014/725360
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