The matrix analogue of the scalar AOR iterative method

Abstract

The Accelerated Overrelaxation (AOR) and the Generalized AOR (GAOR) iterative methods for the solution of linear systems of algebraic equations (Ax=b,A∈Cn×n,det(A)≠0,b∈Cn) have been around for about four decades and a plethora of variations of them have been proposed. In this work a novel algorithm is introduced, the Matrix Analogue of the AOR (MAAOR) iterative method, which is analysed and studied. The MAAOR method generalizes both the AOR and the GAOR. Sufficient convergence conditions for the GAOR method are determined when the coefficient matrix A of the linear system to be solved is a Hermitian matrix with positive diagonal elements. Similarly, sufficient convergence conditions for the MAAOR method are determined when A is a nonsingular H-matrix. The new convergence conditions are the most general ones so far. Numerical examples are presented in support of the theory developed.