Convergence Rate of the Cyclic Reduction Algorithm for Null Recurrent Quasi-Birth-Death Problems
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چکیده
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.
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A Shifted Cyclic Reduction Algorithm for Quasi-Birth-Death Problems
A shifted cyclic reduction algorithm has been proposed by He, Meini, and Rhee [SIAM J. Matrix Anal. Appl., 23 (2001), pp. 673–691] for finding the stochastic matrix G associated with discrete-time quasi-birth-death (QBD) processes. We point out that the algorithm has quadratic convergence even for null recurrent QBDs. We also note that the approximations (to the matrix G) obtained by their algo...
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