On the optimum relaxation factor associated with p-cyclic matrices, Linear Algebra Appl . 162/164

Assume that the matrix coefficient of the nonsingular linear system Ax = b belongs to the class of the generalized consistently ordered ( p- 9, 9) matrices, where p and 9 are relatively prime. It is well known that under the additional assumption that the pth powers of the eigenvalues of the Jacobi matrix T associated with A are nonnegative (nonpositive), the problem of determining the optimum relaxation factor that maximizes the asymptotic convergence rate of the successive overrelaxation method for the solution of Ax = b has been solved in many cases. Thus, in the works by Young, by Varga, and by Nichols and Fox, the problem has been solved in the nonnegative case for any ( p, 9). In the nonpositive case, in view of the work by Kredell, by Niethammer, de Pillis, and Varga, by Galanis, Hadjidimos, and Noutsos, and by Wild and Niethammer, the corresponding problem seems to be more difficult; it has been solved only for 9 = p- 1. The present work is a contribution towards the solution of the problem in the latter case. In particular, we study the case 9 = 1, p> 3, with detailed results for p = 3,4