Optimal parameters of the generalized symmetric SOR method for augmented systems. (English) Zbl 1293.65046
Summary: For the augmented system of linear equations, G.-F. Zhang and Q.-H. Lu [J. Comput. Appl. Math. 219, No. 1, 51–58 (2008; Zbl 1196.65071)] recently studied the generalized symmetric SOR method (GSSOR) with two parameters. In this note, the optimal parameters of the GSSOR method are obtained, and numerical examples are given to illustrate the corresponding results.
MSC:
| 65F10 | Iterative numerical methods for linear systems |
Citations:
Zbl 1196.65071References:
| [1] | Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 1-137 (2005) · Zbl 1115.65034 |
| [2] | Björck, A., Numerical Methods for Least Squares Problems (1996), SIAM: SIAM Philadelphia, PA · Zbl 0847.65023 |
| [3] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer: Springer New York · Zbl 0788.73002 |
| [4] | Elman, H. C., Preconditioners for saddle point problems arising in computational fluid dynamics, Appl. Numer. Math., 43, 75-89 (2002) · Zbl 1168.76348 |
| [5] | Girault, V.; Raviart, P. A., Finite Element Approximation of the Navier Stokes Equations (1979), Springer: Springer Berlin, New York · Zbl 0413.65081 |
| [6] | Nocedal, J.; Wright, S., Numerical Optimization (1999), Springer: Springer New York · Zbl 0930.65067 |
| [7] | Bai, Z.-Z.; Wang, Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428, 2900-2932 (2008) · Zbl 1144.65020 |
| [8] | Bramble, J. H.; Pasciak, J. E.; Vassilev, A. T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34, 1072-1092 (1997) · Zbl 0873.65031 |
| [9] | Chen, F.; Jiang, Y.-L., A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput., 206, 765-771 (2008) · Zbl 1159.65034 |
| [10] | Elman, H. C.; Golub, G. H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31, 1645-1661 (1994) · Zbl 0815.65041 |
| [11] | Bai, Z.-Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comp., 75, 791-815 (2006) · Zbl 1091.65041 |
| [12] | Bai, Z.-Z.; Li, G.-Q., Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23, 561-580 (2003) · Zbl 1046.65018 |
| [13] | Bai, Z.-Z.; Pan, J.-Y.; Ng, M. K., New preconditioners for saddle point problems, Appl. Math. Comput., 172, 762-771 (2006) · Zbl 1088.65040 |
| [14] | Bai, Z.-Z.; Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle point problems, IMA J. Numer. Anal., 27, 1-23 (2007) · Zbl 1134.65022 |
| [15] | Bai, Z.-Z.; Golub, G. H.; Li, C.-K., Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28, 583-603 (2006) · Zbl 1116.65039 |
| [16] | Bai, Z.-Z.; Golub, G. H.; Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive defitine linear systems, SIAM J. Matrix Anal. Appl., 24, 603-626 (2003) · Zbl 1036.65032 |
| [17] | Bai, Z.-Z.; Golub, G. H.; Pan, J.-Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98, 1-32 (2004) · Zbl 1056.65025 |
| [18] | Golub, G. H.; Wu, X.; Yuan, J.-Y., SOR-like methods for augmented systems, BIT, 41, 71-85 (2001) · Zbl 0991.65036 |
| [19] | Li, C.-J.; Li, Z.; Shao, X.-H.; Nie, Y.-Y.; Evans, D. J., Optimum parameter for the SOR-like method for augmented systems, Int. J. Comput. Math., 81, 749-763 (2004) · Zbl 1068.65043 |
| [20] | Shao, X.-H.; Li, Z.; Li, C.-J., Modified SOR-like method for the augment system, Int. J. Comput. Math., 84, 1653-1662 (2007) · Zbl 1141.65024 |
| [21] | Bai, Z.-Z.; Parlett, B. N.; Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102, 1-38 (2005) · Zbl 1083.65034 |
| [22] | Darvishi, M. T.; Hessari, P., Symmetric SOR method for augmented systems, Appl. Math. Comput., 173, 404-420 (2006) · Zbl 1111.65029 |
| [23] | Darvishi, M. T.; Hessari, P., A modified symmetric successive overrelaxation method for augmented systems, Comput. Math. Appl., 61, 3128-3135 (2011) · Zbl 1222.65030 |
| [24] | Wu, S.-L.; Huang, T.-Z.; Zhao, X.-L., A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math., 228, 424-433 (2009) · Zbl 1167.65016 |
| [25] | Zhang, L.-T.; Huang, T.-Z.; Cheng, S.-H.; Wang, Y.-P., Convergence of a generalized MSSOR method for augmented systems, J. Comput. Appl. Math., 236, 1841-1850 (2012) · Zbl 1244.65050 |
| [26] | Zhang, G.-F.; Lu, Q.-H., On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math., 219, 51-58 (2008) · Zbl 1196.65071 |
| [27] | Zhang, N.-M.; Lu, T.-T.; Wei, Y.-M., Semi-convergence analysis of Uzawa methods for singular saddle point problems, J. Comput. Appl. Math., 255, 334-345 (2014) · Zbl 1291.65116 |
| [28] | Elman, H. C., Preconditioning for the steady-state Navier-Stokes equations with low viscosity, SIAM J. Sci. Comput., 20, 4, 1299-1316 (1999) · Zbl 0935.76057 |
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