OPTIMAL P-CYCLIC SOR FOR COMPLEX SPECTRA

We consider the successive overrelaxation (SOR) method for the solutio n of a linear system Ax = b, when the matrix A has a block p x p parti tioned p-cyclic form and its associated block Jacobi matrix J(p) is we akly cyclic of index p. Following the pioneering work by Young and Var ga in the 1950s, many researchers have considered various cases for th e spectrum sigma(J(p)) and have determined (optimal) values for the re laxation factor omega is an element of (0, 2) so that the SOR method c onverges as fast as possible. After the most recent work on the best b lock p-cyclic repartitioning and that on the solution of large-scale s ystems arising in queueing network problems in Markov analysis, the op timization of the convergence of the p-cyclic SOR for more complex spe ctra sigma(J(p)) has become more demanding. Here we state the one-poin t problem for the general p-cyclic complex SOR case. The existence and the uniqueness of its solution are established by analyzing and devel oping further the theory of the associated hypocycloidal curves. For t he determination of the optimal parameter(s) an algorithm is presented and a number of illustrative numerical examples are given. (C) 1997 P ublished by Elsevier Science Inc.