Towards the determination of the optimal p-cyclic SSOR

Suppose that A ∈ Cn,n is a block p-cyclic consistently ordered matrix and let B and Sω denote the block Jacobi and the block symmetric successive overrelaxation (SSOR) iteration matrices associated with A, respectively. Extending previous work by Hadjidimos and Neumann, the present authors have determined the exact regions of convergence of the SSOR method in the (ϱ(B),ω)-plane, for any p ⩾ 3, under the further assumption that the eigenvalues of Bp are real of the same sign. In the present work the investigation goes on further, several questions are raised and among others the problem of the determination of the optimal regions of convergence in the spirit of Niethammer and Varga as well as that of the optimal relaxation factor are examined.