Preconditioned AOR iterative method for linear systems (Q881487)

This paper presents the preconditioned accelerated overrelaxation (AOR) iterative method for solving linear systems. In detail, two preconditioners \(P_\alpha\) and \(P_\beta\) are adopted, where \(P_\alpha = I+S_\alpha\) with \[ S_\alpha = \begin{pmatrix} 0 & - \alpha_1 a_{12} & 0 & \cdots & 0 \\ 0 & 0 & - \alpha_2 a_{23} & \cdots & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & \cdots & - \alpha_{n-1} a_{n-1 , n} \\ 0 & 0 & 0 & \cdots & 0 \end{pmatrix} \] and \(P_\beta = I+ \beta U.\) Note that \(\alpha_i s\) and \(\beta \) are positive real numbers. They are introduced by \textit{H. Kotakemori, H. Niki} and \textit{N. Okamoto} [J. Comput. Appl. Math. 75, No.~1, 87--97 (1996; Zbl 0872.65027)] and used as preconitioners for the Gauss-Seidel method. In this paper, this preconditioner is used for the AOR iterative method. In addition, the theory shows that the preconditioned system converges. The numerical results show that the spectral radius of the preconditioned matrix is smaller than the matrix of the basic AOR method and the preconditioned systems converge faster.