An Operator Relation of the Ussor and the Jacobi Iteration Matrices of a p-Cyclic Matrix

Abstract

Let the Jacobi matrix B associated with the linear system $Ax = b$ be a weakly cyclic matrix, generated by the cyclic permutation $\sigma = ( \sigma _1 ,\sigma _2 , \cdots ,\sigma _p )$ as this is defined by Li and Varga. The same authors derived the corresponding functional equation connecting the eigenvalues $\lambda $ of the unsymmetric successive overrelaxation (USSOR) iteration matrix $T_{\omega \hat \omega } $ and the eigenvalues $\mu $ of the Jacobi matrix B extending previous results by Gong and Cal. In this paper, the validity of an analogous matrix relationship connecting the operators $T_{\omega \hat \omega } $ and B is proved. Moreover, the “equivalence” of the USSOR method and a certain two-parametric p-step method for the solution of the initial system is established. The tool for the proof of our main result is elementary graph theory.