The optimal parameter of SOR-k method for p-cyclic matrices

Consider a SOR-k method for solving a p-cyclic system Ax = b (p > 2) if the p-cyclic matrix A is repartitioned as a k-cyclic matrix for 2 ⩽ k ⩽ p. Suppose that the block Jacobi matrix B associated with A is convergent and all the eigenvalues of Bp are nonnegative. A comparison of the optimal spectral radius of the SOR-k iteration matrix Lω(k) for 2 ⩽ k ⩽ p was given by Evan and Li under an assumption of the existence and differentiability of an implicit function. In this paper, the assumption is deleted. A comparison of the optimal parameter of SOR-k method, as k varies from 2 to p, is given. We will also compare the spectral radius of Lω(k) for a fixed ω and different values of k.