Convergence Rate of the Cyclic Reduction Algorithm for Null Recurrent Quasi-Birth-Death Problems

The minimal nonnegative solution G of the matrix equation G = A 0 + A 1 G + A 2 G 2 , where the matrices A 0 , A 1 and A 2 are nonnegative and A 0 + A 1 + A 2 is stochastic, plays an important role in the study of quasi-birth-death processes (QBDs). The cyclic reduction algorithm is a very efficient iteration for finding the matrix G, under the standard assumption that the transition probability matrix of the QBD and the matrix A 0 + A 1 + A 2 are both irreducible. The convergence is known to be quadratic for positive recurrent QBDs and for transient QBDs. For the null recurrent case, the convergence of a closely related algorithm, the Latouche-Ramaswami algorithm, has been shown to be linear with rate 1/2 under two additional assumptions. In this talk, we show that the convergence of the cyclic reduction algorithm and hence of the Latouche-Ramaswami algorithm is at least linear with rate 1/2 in the null recurrent case, without those two additional assumptions, and the proof here is much simpler.