AUTHORS: Chunping Pan

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ABSTRACT: In this paper, we study the iterative algorithms for saddle point problems(SPP). We present a new symmetric successive over-relaxation method with three parameters, which is the extension of the SSOR iteration method. Under some suitable conditions, we give the convergence results. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.

KEYWORDS: iterative method, saddle point problems, SOR-like, SSOR-like; symmetric and positive definite matrix

REFERENCES:

[1] M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numerica, vol.14, 2005, pp. 1-137.

[2] J. H. Bramble, J. E. Pasciak and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problem, SIAM J. Numer. Anal., vol.34, 1997, No.3, pp. 1072-1092.

[3] G. H. Golub, X. Wu, J.-Y. Yuan, SOR-like methods for augmented systems, BIT, vol.41, 2001, pp. 71-85.

[4] Z. Z. Bai, B. N. Parlett., Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., vol.102, 2005, pp. 1-38.

[5] J. C. Li, X. Kong, Optimum parameters of GSOR-1ike methods for the augmented systems, Applied Mathematics and Computation, vol.204, 2008, No.1, pp. 150-161.

[6] C. P. Pan, H. Y. Wang, The polynomial Acceleration of GSOR Iterative Algorithms for Large Sparse Saddle point problems, Chinese Journal of Engineering Mathematics, vol.28, 2011, No.3, pp. 307-314.

[7] C. P. Pan, H. Y.Wang, W. L. Zhao, On generalized SSOR iterative method for Saddle point problems, Journal of Mathematics, vol.31, 2011, No.3, pp. 569-574.

[8] Z. Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM journal of Matrix Analysis and Applications, vol.24, 2003, pp. 603-626.

[9] Z. Z. Bai, G. H. Golub and J. Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Numer. Math., vol.98, 2004, pp. 1- 32.

[10] C. P. Pan, H. Y. Wang, On generalized Preconditioned Hermitian and skew-Hermitian splitting methods for saddle point problems, Journal On Numerical Methods and Computer Applications, vol.32, 2011, No.3, pp. 174-182.

[11] D. M. Young, Iterative Solutions of Large Linear Systems, Academic Press, New York, 1971.

[12] Q. Hu and J. Zou, An iterative method with variable relaxation parameters for saddle-point problems, SIAM Journal of Matrix Analysis and Applications, vol.23, 2001, pp. 317-338.

[13] M. T. Darvishi and P. Hessari, Symmetric SOR method for augmented systems, Appl. Math. Comput., vol.183, 2006, pp. 409-415.

[14] S. L. Wu, T. Z. Huang, X. L. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math., vol.228, 2009, pp. 424- 433.

[15] L. T. Zhang, T. Z. Huang, S. H. Cheng, Y. P. Wang, Convergence of a generalized MSSOR method for augmented systems, J. Comput. Appl. Math., vol.236, 2012, pp. 1841-1850.

[16] C. P. Pan, On generalized preconditioned Hermitian and skew-Hermitian splitting methods for saddle point problems, WSEAS Transactions on Mathematics, vol.11,2012, pp.1147-1156.

[17] Z. Li, C. Li and D. J. Evans, Chebyshev acceleration for SOR-like method,Int. J. Comput. Math., vol.82, 2005, pp.583-593.

[18] M. M. Martins, W. Yousif and J. L. Santos, A variant of the AOR method for augmented systems, Math. Comput, vol.81, 2012, pp.399-417.