Optimal p-cyclic SOR for complex spectra

We consider the successive overrelaxation (SOR) method for the solution of a linear system Ax = b, when the matrix A has a block p × p partitioned p-cyclic form and its associated block Jacobi matrix Jp is weakly cyclic of index p. Following the pioneering work by Young and Varga in the 1950s, many researchers have considered various cases for the spectrum σ(Jp) and have determined (optimal) values for the relaxation factor ω (0,2) so that the SOR method converges as fast as possible. After the most recent work on the best block p-cyclic repartitioning and that on the solution of large-scale systems arising in queueing network problems in Markov analysis, the optimization of the convergence of the p-cyclic SOR for more complex spectra σ(Jp) has become more demanding. Here we state the one-point problem for the general p-cyclic complex SOR case. The existence and the uniqueness of its solution are established by analyzing and developing further the theory of the associated hypocycloidal curves. For the determination of the optimal parameter(s) an algorithm is presented and a number of illustrative numerical examples are given.