Block cyclic SOR for Markov chains with p-cyclic infinitesimal generator

Abstract

We introduce a new application of p-cyclic iterations, for arbitrary p⩾2. The block SOR method for the computation of the steady state-distribution of finite Markov chains that possess p-cyclic infinitesimal generators is considered. It is shown that convergence, in a sense more general than the usual, may be obtained even if the SOR iteration violates the usual conditions for semiconvergence. Necessary and sufficient conditions for convergence in this extended sense are derived. They are then applied in the case where the pth power of the associated Jacobi matrix of the system to be solved possesses only nonnegative eigenvalues. Exact convergence intervals and the optimal ω-values are derived for this case. In addition to the “usual” optimal ω in the interval (1, p (p−1) ) , other ω-values that yield convergence in the extended sense are found to achieve the same optimal convergence rate. Numerical tests indicate that small perturbations of ω around the optimal value affect the convergence factor much less if these newly introduced optimal ω-values are used.