On the convergence domains of the p-cyclic SOR

Abstract

For the solution of the linear system Ax = b, where A is block p- cyclic, the block SOR iterative method is to be considered. Suppose that the block Jacobi iteration matrix B, associated with A, has eigenvalues whose pth powers are all real of the same sign. The problem of the determination of the precise convergence domains of the SOR method in case A is also consistently ordered was solved by Hadjidimos, Li and Varga by using the Schur-Cohn algorithm. The same convergence domains were later recovered by other approaches too; specifically, Wild and Niethammer and also Noutsos, independently, used hypocycloidal curves. In this manuscript it is assumed that A is not consistently ordered but A T is. By using the Schur-Cohn algorithm we successfully determine, not only: (i) the precise SOR convergence domains, but also (ii) intervals for ϱ( B), the spectral radius of B, that directly imply that the optimal value of the SOR relaxation factor ω is equal to 1. In this work new results are obtained, some well-known ones are recovered or confirmed and a number of theoretical examples are investigated further. It is worth noting that among the new results, we derived something not quite expected; specifically, in many cases there exist pairs ( ϱ( B), ω) for which the SOR method associated with the matrix A we consider converges while the corresponding SOR for the p-cyclic consistently ordered matrix A T does not!