AN OPERATOR RELATION OF THE USSOR AND THE JACOBI ITERATION MATRICES OF A P-CYCLIC MATRIX

Let the Jacobi matrix B associated with the linear system Aa = b be a weakly cyclic matrix, generated by the cyclic permutation sigma = (sig ma(1),sigma(2),...,sigma(p)) as this is defined by Li and Varga. The s ame authors derived the corresponding functional equation connecting t he eigenvalues lambda of the unsymmetric successive overrelaxation (US SOR) iteration matrix T(<omega(omega)over cap>) and the eigenvalues mu of the Jacobi matrix B extending previous results by Gong and Cai. In this paper, the validity of an analogous matrix relationship connecti ng the operators T(<omega(omega)over cap>) 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. Th e tool for the proof of our main result is elementary graph theory.