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In this paper we introduce the Modified Accelerated Overrelaxation (MAOR) method, a generalization of the AOR one, for the iterative solution of the nonsingular linear system Ax = b. We assume that A is in a p x p partitioned fonn and belongs to a subclass of the p-cyclic consistently ordered matrices. It is pointed out that for specific choices of the "acceleration" and "relaxation" matrices the MAOR method reduces to an extrapolation of the Jacobi or the Modified (M)SOR method with different parameters corresponding to the row blocks of A. First, an eigenvalue relationship connecting the spectra of the block Jacobi and MAOR matrices associated with A is derived from which many wellRknown eigenvalue relationships can be recovered. Then, by considering the particular case p = 2 it is shown that a matrix analogue of the aforementioned eigenvalue relationship holds and the MAOR method is equivalent to a cenain 2-step one. Finally, the precise domains of convergence of the MAOR method are derived, when the spectrum of the Jacobi matrix is real or pure imaginary and a brief discussion follows which concludes the present study.

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