In this paper, the preconditioned accelerated overrelaxation (AOR) method for solving a class of two-by-two linear systems is presented. A new preconditioner is proposed according to the idea of [1] by Wu and Huang. The spectral radii of the iteration matrix of the preconditioned and the original methods are compared. The comparison results show that the convergence rate of the preconditioned AOR methods is indeed better than that of the original AOR methods, whenever the original AOR methods are convergent under certain conditions. Finally, a numerical example is presented to confirm our results.
References
[1]
S.-L. Wu and T.-Z. Huang, “A Modified AOR-Type Iterative Method for L-Matrix Linear Systems,” Australian & New Zealand Industrial and Applied Mathematics Journal, Vol. 49, 2007, pp. 281-292.
[2]
J.-Y. Yuan, “Iterative Methods for Generalized Least Squares Problems,” Ph.D. Thesis, IMPA, Rio de Janeiro, Brazil, 1993.
[3]
J.-Y. Yuan, “Numerical Methods for Generalized Least Squares Problems,” Journal of Computational and Applied Mathematic, Vol. 66, No. 1-2, 1996, pp. 571-584. doi:10.1016/0377-0427(95)00167-0
[4]
J.-Y. Yuan and A. N. Iusem, “SOR-Type Methods for Generalized Least Squares Problems,” Acta Mathematicae Applicatae Sinica, Vol. 16, 2000, pp. 130-139. doi:10.1007/BF02677673
[5]
J.-Y. Yuan and X.-Q. Jin, “Convergence of the Generalized AOR Method,” Applied Mathematics and Computation, Vol. 99, No. 1, 1999, pp. 35-46. doi:10.1016/S0096-3003(97)10175-8
[6]
R. S. Varga, “Matrix Iterative Analysis,” Springer Series in Computational Mathematics: 27, Springer-Verlag, Berlin, 2000.
[7]
D. M. Young, “Iterative Solution of Large Linear Systems,” Academic Press, New York, 1971.
[8]
A. Hadjidimos, “Accelerated over Relaxtion Method,” Mathematics of Computation, Vol. 32, No. 141, 1978, pp. 149-157. doi:10.1090/S0025-5718-1978-0483340-6
[9]
X.-X. Zhou, Y.-Z. Song, L. Wang and Q.-S. Liu, “Preconditioned GAOR Methods for Solving Weighted Linear Least Squares Problems,” Journal of Computational and Applied Mathematics, Vol. 224, No. 1, 2009, pp. 242-249. doi:10.1016/j.cam.2008.04.034
[10]
A. Berman and R.J. Plemmons, “Nonnegative Matrices in the Mathematics Sciences,” SIAM, Philadelphia, 1994.
