| Title:
|
Extrapolated positive definite and positive semi-definite splitting methods for solving non-Hermitian positive definite linear systems (English) |
| Author:
|
Shokrpour, Raheleh |
| Author:
|
Ebadi, Ghodrat |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
67 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
319-340 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Recently, Na Huang and Changfeng Ma in (2016) proposed two kinds of typical practical choices of the PPS method. In this paper, we extrapolate two versions of the PPS iterative method, and we introduce the extrapolated Hermitian and skew-Hermitian positive definite and positive semi-definite splitting (EHPPS) iterative method and extrapolated triangular positive definite and positive semi-definite splitting (ETPPS) iterative method. We also investigate convergence analysis and consistency of the proposed iterative methods. Then, we study upper bounds for the spectral radius of iteration matrices and give upper bounds for the extrapolation parameter of the methods. Moreover, the optimal parameters which minimize upper bounds of the spectral radius are obtained. Finally, several numerical examples are given to show the efficiency of the presented method. (English) |
| Keyword:
|
extrapolated |
| Keyword:
|
non-Hermitian |
| Keyword:
|
positive definite |
| Keyword:
|
skew-Hermitian |
| Keyword:
|
splitting |
| Keyword:
|
HSS iteration method |
| MSC:
|
15A06 |
| MSC:
|
15B48 |
| MSC:
|
65B05 |
| MSC:
|
65F10 |
| idZBL:
|
Zbl 07547198 |
| idMR:
|
MR4409309 |
| DOI:
|
10.21136/AM.2021.0256-20 |
| . |
| Date available:
|
2022-04-14T13:36:22Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150318 |
| . |
| Reference:
|
[1] Albrecht, P., Klein, M. P.: Extrapolated iterative methods for linear systems.SIAM J. Numer. Anal. 21 (1984), 192-201. Zbl 0531.65015, MR 0731223, 10.1137/0721014 |
| Reference:
|
[2] Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems.Numer. Linear Algebra Appl. 7 (2000), 197-218. Zbl 1051.65025, MR 1762967, 10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S |
| Reference:
|
[3] Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems.Computing 87 (2010), 93-111. Zbl 1210.65074, MR 2640009, 10.1007/s00607-010-0077-0 |
| Reference:
|
[4] Bai, Z.-Z., Golub, G. H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems.SIAM J. Sci. Comput. 26 (2005), 844-863. Zbl 1079.65028, MR 2126115, 10.1137/S1064827503428114 |
| Reference:
|
[5] 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:
|
[6] 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:
|
[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] Cao, Z.: A convergence theorem on an extrapolated iterative method and its applications.Appl. Numer. Math. 27 (1998), 203-209. Zbl 0927.65052, MR 1634345, 10.1016/S0168-9274(98)00013-0 |
| Reference:
|
[9] Ebadi, G., Alipour, N., Vuik, C.: Deflated and augmented global Krylov subspace methods for the matrix equations.Appl. Numer. Math. 99 (2016), 137-150. Zbl 1329.65087, MR 3413898, 10.1016/j.apnum.2015.08.010 |
| Reference:
|
[10] Ebadi, G., Rashedi, S.: New variants of global Krylov type methods for linear systems with multiple right-hand sides arising in elliptic PDEs.Comput. Methods Differ. Equ. 6 (2018), 111-127. Zbl 1424.65028, MR 3778524 |
| Reference:
|
[11] Evans, D. J., Martins, M. M.: On the convergence of the extrapolated AOR method.Int. J. Comput. Math. 43 (1992), 161-171. Zbl 0754.65032, 10.1080/00207169208804083 |
| Reference:
|
[12] Hadjidimos, A.: The optimal solution to the problem of complex extrapolation of a first-order scheme.Linear Algebra Appl. 62 (1984), 241-261. Zbl 0567.65015, MR 0761072, 10.1016/0024-3795(84)90100-9 |
| Reference:
|
[13] Hadjidimos, A., Psimarni, A., Yeyios, A.: On the convergence of some generalized iterative methods.Linear Algebra Appl. 75 (1986), 117-132. Zbl 0589.65027, MR 0825402, 10.1016/0024-3795(86)90184-9 |
| Reference:
|
[14] Hadjidimos, A., Yeyios, A.: The principle of extrapolation in connection with the accelerated overrelaxation method.Linear Algebra Appl. 30 (1980), 115-128. Zbl 0428.65015, MR 0568784, 10.1016/0024-3795(80)90187-1 |
| Reference:
|
[15] Huang, N., Ma, C.: Positive definite and semi-definite splitting methods for non-Hermitian positive definite linear systems.J. Comput. Math. 34 (2016), 300-316. Zbl 1363.65050, MR 3504482, 10.4208/jcm.1511-m2015-0299 |
| Reference:
|
[16] Krukier, L. A., Chikina, L. G., Belokon, T. V.: Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations.Appl. Numer. Math. 41 (2002), 89-105. Zbl 1004.65042, MR 1908751, 10.1016/S0168-9274(01)00112-X |
| Reference:
|
[17] Krukier, L. A., Krukier, B. L., Ren, Z.-R.: Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems.Numer. Linear Algebra Appl. 21 (2014), 152-170. Zbl 1324.65053, MR 3150615, 10.1002/nla.1870 |
| Reference:
|
[18] Li, C.-X., Wu, S.-L.: A modified GHSS method for non-Hermitian positive definite linear systems.Japan J. Ind. Appl. Math. 29 (2012), 253-268. Zbl 1267.65038, MR 2931411, 10.1007/s13160-012-0059-z |
| Reference:
|
[19] Missirlis, N. M., Evans, D. J.: On the convergence of some generalized preconditioned iterative methods.SIAM J. Numer. Anal. 18 (1981), 591-596. Zbl 0464.65018, MR 0622695, 10.1137/0718037 |
| Reference:
|
[20] Salkuyeh, D. K., Siahkalaei, T. S.: Two-parameter TSCSP method for solving complex symmetric system of linear equations.Calcolo 55 (2018), Article ID 8, 22 pages. Zbl 1392.65071, MR 3761177, 10.1007/s10092-018-0252-9 |
| Reference:
|
[21] Song, Y.: Semiconvergence of extrapolated iterative methods for singular linear systems.J. Comput. Appl. Math. 106 (1999), 117-129. Zbl 0930.65033, MR 1696806, 10.1016/S0377-0427(99)00060-6 |
| Reference:
|
[22] Song, Y., Wang, L.: On the semiconvergence of extrapolated iterative methods for singular linear systems.Appl. Numer. Math. 44 (2003), 401-413. Zbl 1027.65040, MR 1954432, 10.1016/S0168-9274(02)00168-X |
| Reference:
|
[23] Wang, L., Song, Y.: On the optimization of extrapolation methods for singular linear systems.J. Comput. Math. 26 (2008), 227-239. Zbl 1174.65012, MR 2395592 |
| Reference:
|
[24] Yeyios, A.: On an accelerated procedure of extrapolation.Int. J. Math. Math. Sci. 4 (1981), 753-762. Zbl 0473.65013, MR 0663659, 10.1155/S0161171281000586 |
| Reference:
|
[25] Yeyios, A.: On the optimization of an extrapolation method.Linear Algebra Appl. 57 (1984), 191-203. Zbl 0527.65027, MR 0729272, 10.1016/0024-3795(84)90187-3 |
| Reference:
|
[26] Young, D. M.: Iterative Solution of Large Linear Systems.Computer Science and Applied Mathematics. Academic Press, New York (1971). Zbl 0231.65034, MR 0305568, 10.1016/c2013-0-11733-3 |
| Reference:
|
[27] Zeng, M.-L., Zhang, G.-F.: Complex-extrapolated MHSS iteration method for singular complex symmetric linear systems.Numer. Algorithms 76 (2017), 1021-1037. Zbl 1383.65029, MR 3736226, 10.1007/s11075-017-0295-z |
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