Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

SSOR-LIKE METHOD FOR AUGMENTED SYSTEMy

  • Liang, Mao-Lin (School of Mathematics and Statistics, Tianshui Normal University) ;
  • Dai, Li-Fang (School of Mathematics and Statistics, Tianshui Normal University) ;
  • Wang, San-Fu (School of Mathematics and Statistics, Tianshui Normal University)
  • Received : 2010.07.10
  • Accepted : 2010.11.01
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

This paper proposes a new generalized iterative method (SSOR-like method) for solving augmented system. A functional equation relating two involved parameters is obtained, and some convergence conditions for this method are derived. This paper generalizes some foregone results. Numerical examples show that, this method is efficient by suitable choices of the involved parameters.

Keywords

Acknowledgement

Supported by : Education Department of Gansu Province, Tianshui Normal University

References

  1. H. Elman, D.J. Silvester, A. Wathen, Iterative methods for problems in computational fluid dynmics. in : Iterative Methods in Scientifilc Computing, Springer-Verlag, Singapore, 1997, 271-327.
  2. J.T. Betts, Practical Methods for Opitmal Control Using Nonlinear Programming, SIAM, Philadelphia, PA, 2001.
  3. H. Elman and D.J. Silvester, Fast nonsymmetric iteraton and preconditoning for Navier- Stokes equations, SIAM J. Sci. Comput. 17 (1996), 33-46. https://doi.org/10.1137/0917004
  4. J.Y. Yuan, Iterative methods for generalized least squares problems, Ph.D. Thesis, IMPA, Rio de Janeiro, Brazil, 1993.
  5. P.E. Gill, W. Murray and M.H.Wright, Practical Optimization, Academic Press, New York, NY, 1981.
  6. Z.Z. Bai and G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007), 1-23.
  7. N. Dyn and W.E. Ferguson Jr., The numerical solution of equality constrained quadratic programming problems, Math. Comput. 41 (1983), 165-170.
  8. G.H. Golub, X. Wu, J.Y. Yuan, SOR-like method for augmented system, BIT 41(2001), 71-85. https://doi.org/10.1023/A:1021965717530
  9. Z.Z. Bai, B.N. Parlett and Z.-Q.Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005), 1-38. https://doi.org/10.1007/s00211-005-0643-0
  10. K. Arrow, L. Hurwicz and H. Uzawa, Studies in Nonlinear Programming, Stanford University Press, Stanford, 1958.
  11. H.C. Elman and G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994), 1645-1661. https://doi.org/10.1137/0731085
  12. S.L. Wu, T.Z. Huang, X.L. Zhao, A modifiled SSOR iterative method for augmented systems, J. Comput. Appl. Math. 228 (2009), 424-433. https://doi.org/10.1016/j.cam.2008.10.006
  13. M.T. Darvishi and P. Hessari, Symmetric SOR method for augmented systems, Appl. Math. Comput. 183 (2006), 409-415. https://doi.org/10.1016/j.amc.2006.05.094
  14. J.C. Li, B.J. Li and D. J. Evans, A generalized successive overrelaxation method for least squares problems, BIT 38 (1998), 347-355. https://doi.org/10.1007/BF02512371
  15. J.C. Li, X. Kong, Optimal parameters of GSOR-like method for solving the augmented linear system, Appl. Math. Comput. 204 (2008), 150-161. https://doi.org/10.1016/j.amc.2008.06.010
  16. M.T. Darvishi, F. Khani, S. Hamedi-Nezhad and B. Zheng, Symmetric block-SOR methods for rank-defilcient least squares problems, J. Comput. Appl. Math. 215 (2008), 14-27. https://doi.org/10.1016/j.cam.2007.03.023
  17. G.F. Zhang and Q.H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math. 219 (2008), 51-58. https://doi.org/10.1016/j.cam.2007.07.001
  18. D.M. Young, Iterative solutions of large linear systems, Academic Press, NY, 1971.
  19. R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962, 123-125.
  20. Z.Z. Bai, G.H.Golub and J.Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefilnite linear systems, Numer.Math. 98 (2004) 1-32. https://doi.org/10.1007/s00211-004-0521-1