A Note on the Convergence of the Generalized AOR Iterative Method for Linear Systems

Authors: Zhong-xi Gao, Hou-biao Li

Abstract:

Recently, some convergent results of the generalized AOR iterative (GAOR) method for solving linear systems with strictly diagonally dominant matrices are presented in [Darvishi, M.T., Hessari, P.: On convergence of the generalized AOR method for linear systems with diagonally dominant cofficient matrices. Appl. Math. Comput. 176, 128-133 (2006)] and [Tian, G.X., Huang, T.Z., Cui, S.Y.: Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant cofficient matrices. J. Comp. Appl. Math. 213, 240-247 (2008)]. In this paper, we give the convergence of the GAOR method for linear systems with strictly doubly diagonally dominant matrix, which improves these corresponding results.

Keywords: Diagonally dominant matrix, GAOR method, Linear system, Convergence

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1330847

References:

[1] Darvishi, M.T., Hessari, P.: On convergence of the generalized AOR method for linear systems with diagonally dominant cofficient matrices. Appl. Math. Comput. 176(2006), 128-133.
[2] Tian, G.X., Huang, T.Z., Cui, S.Y.: Convergence of generalized AOR iterative method for linear systems with strictly diagonally dominant cofficient matrices. J. Comp. Appl. Math. 213(2008), 240-247.
[3] Yuan, J.Y.: Numerical methods for generalized least squares problems. J. Comp. Appl. Math. 66, 571-584 (1996).
[4] Yuan, J.Y., Jin, X.Q.: Convergence of the generalized AOR method. Appl. Math. Comput. 99(1999), 35-46.
[5] Searle, S., Casella, G., McCulloch, C.: Variance Components. Wiley Interscience, New York, 1992.
[6] Hadjidimos, A.: Accelerated overralation method. Math. Comput. 32(141)(1978), 149-157.
[7] Hadjidimos, A., Yeyios, A.: The principle of extrapolation in connection with the accelerated overralation method. Linear Algebra Appl. 30(1980), 115-128.
[8] Li, Hou-Biao, Huang, T. Z. and Li H., An improvement on a new upper bound for moduli of eigenvalues of iterative matrices, Appl. Math. Comp. 173(2006), 977-984.
[9] Huang, T. Z., Gao Zh. X., A new upper bound for moduli of eigenvalues of iterative matrices, Intern. J. Computer Math., 80(6)(2003), 799-803.