Optimal successive overrelaxation iterative methods for P-cyclic matrices

Summary We consider linear systems whose associated block Jacobi matricesJ_p are weakly cyclic of indexp. In a recent paper, Pierce, Hadjidimos and Plemmons [13] proved that the block two-cyclic successive overrelaxation (SOR) iterative method is numerically more effective than the blockq-cyclic SOR-method, 2<qp, if the eigenvalues ofJ_p^p are either all non-negative or all non-positive. Based on the theory of stationaryp-step methods, we give an alternative proof of their theorem. We further determine the optimal relaxation parameter of thep-cyclic SOR method under the assumption that the eigenvalues ofJ_p^p are contained in a real interval, thereby extending results due to Young [19] (for the casep=2) and Varga [15] (forp>2). Finally, as a counterpart to the result of Pierce, Hadjidimos and Plemmons, we show that, under this more general assumption, the two-cyclic SOR method is not always superior to theq-cyclic SOR method, 2<qp.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch supported in part by the Deutsche Forschungsgemeinschaft