On a matrix identity connecting iteration operators associated with a p-cyclic matrix

Abstract

The successive overrelaxation (SOR) and the symmetric SOR (SSOR) iteration matrices are connected with the Jacobi iteration matrix in case these operators are associated with a ( q, p − q)-generalized consistently ordered matrix through certain matrix identities. The validity of these identities has been proved in the last couple of years. Very recently an analogous matrix identity was shown to hold for the modified SOR (MSOR) and the Jacobi iteration matrices in the particular cases ( p,q) = (2,1), (3,1), and 3,2). It is the main objective of this paper to extend the validity of this identity to cover an entire class or pairs ( p, q). The identity in question is not only of theoretical interest but of practical importance too, since it can be used to show the equivalence of the MSOR and a class of p-step iterative methods for the solution of a linear system whose matrix coefficient possesses the ( q,p − q)-generalized consistently ordered property.