On the equivalence of the k-step iterative euler methods and successive overrelaxation (SOR) methods for k-cyclic matrices

For the solution of the linear system x = Tx + c (1), where T is weakly cyclic of index k ≥ 2, the block SOR method together with two classes of monoparametric k-step iterative Euler methods, whose (optimum) convergence properties were studied in earlier papers, are considered. By establishing the existence of the matrix analog of the Varga's relation, connecting the eigenvalues of the SOR and the Jacobi matrices associated with (1), it is proved that the aforementioned SOR method is equivalent to a certain monoparametric k-step iterative Euler method derived from (1). By suitably modifying the existing theory, one can then determine (optimum) relaxation factors for which the SOR method in question converges, (optimum) regions of convergence etc., so that one can obtain, what is known, several new results. Finally, a number of theoretical applications of practical importance is also presented.