Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T17:07:01.780Z Has data issue: false hasContentIssue false
  • English
  • Français

On a New SSOR-Like Method with Four Parameters for the Augmented Systems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Hui-Di Wang  and
Zheng-Da Huang
Hui-Di Wang*
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China
Zheng-Da Huang*
Affiliation:
School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China
*
*Corresponding author. Email addresses:hdwang@zju.edu.cn (H.-D. Wang), zdhuang@zju.edu.cn (Z.- D. Huang)
*Corresponding author. Email addresses:hdwang@zju.edu.cn (H.-D. Wang), zdhuang@zju.edu.cn (Z.- D. Huang)
Get access

Abstract

In this paper, we propose a new SSOR-like method with four parameters to solve the augmented system. And we analyze the convergence of the method and get the optimal convergence factor under suitable conditions. It is proved that the optimal convergence factor is the same as the GMPSD method [M.A. Louka and N.M. Missirlis, A comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented systems, Numer. Math. 131(2015) 517-540] with five parameters under the same assumption.

Keywords

ConvergenceSSOR-like methodconvergence factor

MSC classification

Secondary: 65F10: Iterative methods for linear systems
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 82 - 100
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.Google Scholar
[2] Betts, J. T., Practical Methods for Optimal Control Using Nonlinear Programming, Advances in Design and Control, 3, SIAM, Philadelphia, 2001.Google Scholar
[3] Bramble, J. H., Pasciak, J. E., and Vassilev, A. T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072–1092.Google Scholar
[4] Benzi, M., Golub, G. H., and Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1–137.Google Scholar
[5] Benzi, M. and Golub, G. H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 20–41.Google Scholar
[6] Bai, Z.-Z. and Ng, M. K., On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput., 26 (2005), pp. 1710–1724.Google Scholar
[7] Bai, Z.-Z., Golub, G. H., and Li, C.-K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comp., 76 (2007), pp. 287–298.Google Scholar
[8] Bai, Z.-Z. and Wang, Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), pp. 2900–2932.CrossRefGoogle Scholar
[9] Bai, Z.-Z., Ng, M. K., and Wang, Z.-Q., Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 410–433.CrossRefGoogle Scholar
[10] Bai, Z.-Z., Parlett, B. N., and Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), pp. 1–38.Google Scholar
[11] Bai, Z.-Z., Golub, G. H., and Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603–626.Google Scholar
[12] Bai, Z.-Z., Golub, G. H., and Pan, J.-Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), pp. 1–32.Google Scholar
[13] Bai, Z.-Z. and Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle point problems, IMA J. Numer. Anal., 27 (2007), pp. 1–23.Google Scholar
[14] Bai, Z.-Z., Golub, G. H., and Ng, M. K., On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008), pp. 413–440.CrossRefGoogle Scholar
[15] Benzi, M., A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 360–374.Google Scholar
[16] Chen, Y.-L. and Tan, X.-Y., Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices, Appl. Math. Comput., 167 (2005), pp. 930–956.Google Scholar
[17] Darvishi, M. T. and Hessari, P., Symmetric SOR method for augmented systems, Appl. Math. Comput., 183 (2006), pp. 409–415.Google Scholar
[18] Darvishi, M. T. and Hessari, P., A modified symmetric successive overrelaxation method for augmented systems, Comput. Math. Appl., 61 (2011), pp. 3128–3135.Google Scholar
[19] Dyn, N. and Ferguson, W. E., The numerical solution of equality constrained quadratic programming problems, Math. Comp., 41 (1983), pp. 165–170.Google Scholar
[20] Elman, H. C. and Golub, G. H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 1645–1661.CrossRefGoogle Scholar
[21] Elman, H. C., Silvester, D. J., and Wathen, A. J., Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations, Numer. Math., 90 (2002), pp. 665–688.CrossRefGoogle Scholar
[22] Fortin, M. and Glowinski, R., Augmented Lagrangian Method: Applications to the numerical solution of boundary value problems, Studies in Mathematics and its Applications, 15, North-Holland Publishing Co., Amsterdam, 1983.Google Scholar
[23] Golub, G. H., Wu, X., and Yuan, J.-Y., SOR-like methods for augmented systems, BIT Numer. Math., 41 (2001), pp. 71–85.CrossRefGoogle Scholar
[24] Gould, N. I. M., Hribar, M. E., and Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. Sci. Comput., 23 (2001), pp. 1376–1395.Google Scholar
[25] Gill, P. E., Murray, W., and Wright, M. H., Practical Optimization, Academic Press, London-New York, 1981.Google Scholar
[26] Hu, Q.-Y. and Zou, J., An iterative method with variable relaxation parameters for saddle point problems, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 317–338.Google Scholar
[27] Hu, Q.-Y. and Zou, J., Two new variants of nonlinear inexact Uzawa algorithms for saddle point problems, Numer. Math., 93 (2002), pp. 333–359.Google Scholar
[28] Huang, Z.-D. and Zhou, X.-Y., On the minimum convergence factor of a class of GSOR-like methods for augmented systems, Numer. Algor., 70 (2014), pp. 113–132.CrossRefGoogle Scholar
[29] Jiang, M.-Q. and Cao, Y., On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), pp. 973–982.Google Scholar
[30] Keller, C., Gould, N. I. M., and Wathen, A. J., Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300–1317.Google Scholar
[31] Li, C.-J., Li, Z., Nie, Y.-Y., and Evans, D. J., Generalized AOR method for the augmented system, Int. J. Comput. Math., 81 (2004), pp. 495–504.Google Scholar
[32] Li, Z., Li, C.-J., Evans, D. J., and Zhang, T., Two parameter GSOR method for the augmented systems, Int. J. Comput. Math., 82 (2005), pp. 1033–1042.Google Scholar
[33] Lin, Y.-Q. and Wei, Y.-M., Fast corrected Uzawa methods for solving symmetric saddle point problems, Calcolo, 43 (2006), pp. 65–82.Google Scholar
[34] Lin, Y.-Q. and Wei, Y.-M., A note on constraint preconditioners for nonsymmetric saddle point problems, Numer. Linear Algebra Appl., 14 (2007), pp. 659–664.Google Scholar
[35] Li, J.-C. and Kong, X., Optimal parameters of GSOR-like methods for solving the augmented linear systems, Appl. Math. Comput., 204 (2008), pp. 150–161.Google Scholar
[36] Lu, J.-F. and Zhang, Z.-Y., A modified nonlinear inexact Uzawa algorithm with a variable relaxation parameter for the stabilized saddle point problem, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1934–1957.CrossRefGoogle Scholar
[37] Li, X., Yang, A.-L., and Wu, Y.-J., Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for saddle-point problems, Int. J. Comput. Math., 91 (2014), pp. 1224–1283.Google Scholar
[38] Louka, M. A. and Missirlis, N. M., A comparison of the extrapolated successive overrelaxation and the preconditioned simultaneous displacement methods for augmented linear systems, Numer. Math., 131 (2015), pp. 517–540.Google Scholar
[39] Martins, M. M., Yousif, W., and Santos, J. L., A variant of the AOR method for augmented systems, Math. Comput., 81 (2012), pp. 399–417.Google Scholar
[40] Najafi, H. S. and Edalatpanah, S. A., A new modified SSOR iteration method for solving augmented linear systems, Int. J. Comput. Math., 91 (2014), pp. 539–552.Google Scholar
[41] Najafi, H. S. and Edalatpanah, S. A., On the modified symmetric successive over-relaxation method for augmented systems, Comput. Appl. Math., 34 (2015), pp. 607–617.Google Scholar
[42] Pour, H. N. and Goughery, H. S., New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, Numer. Algor., 69 (2015), pp. 207–225.Google Scholar
[43] Sturier, E. D. and Liesen, J., Block-diagonal preconditioners for indefinite linear algebraic systems, SIAM J. Sci. Comput., 26 (2005), pp. 1598–1619.Google Scholar
[44] Shao, X.-H., Li, Z., and Li, C.-J., Modified SOR-like method for the augmented system, Int. J. Comput. Math., 84 (2007), pp. 1653–1662.CrossRefGoogle Scholar
[45] Wen, C. and Huang, T.-Z., Modified SSOR-like method for augmented system, Math. Model. Anal., 16 (2011), pp. 475–487.Google Scholar
[46] Wu, S.-L., Huang, T.-Z., and Zhao, X.-L., A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math., 228 (2009), pp. 424–433.Google Scholar
[47] Wang, K., Di, J.-J., and Liu, D., Improved PHSS iterative methods for solving saddle point problems, Numer. Algor., 71 (2016), pp. 753–773.Google Scholar
[48] Young, D. M., Iterative Solution of Large Linear Systems, Academic Press, New York-London, 1971.Google Scholar
[49] Zheng, B., Wang, K., and Wu, Y.-J., SSOR-like methods for saddle point problems, Int. J. Comput. Math., 86 (2009), pp. 1405–1423.Google Scholar
[50] Zheng, B., Bai, Z.-Z., and Yang, X., On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 431 (2009), pp. 808–817.Google Scholar
[51] Zhang, G.-F. and Lu, Q.-H., On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math., 219 (2008), pp. 51–58.Google Scholar
[52] Zhou, Y.-Y., Zhang, G.-F., A generalization of parameterized inexact Uzawa method for generalized saddle point problems, Appl. Math. Comput., 215 (2009), pp. 599–607.Google Scholar
[53] Zhang, L.-T., Huang, T.-Z., Cheng, S.-H., and Wang, Y.-P., Convergence of a generalized MSSOR method for augmented systems, J. Comput. Appl. Math., 236 (2012), pp. 1841–1850.Google Scholar
[54] Zhu, M.-Z., A generalization of the local Hermitian and skew-Hermitian splitting iteration methods for the non-Hermitian saddle point problems, Appl. Math. Comput., 218 (2012), pp. 8816–8824.Google Scholar