TITLE: On the Effects of Using the Grassmann-Taksar-Heyman Method in 
Iterative Aggregation-Disaggregation

AUTHORS: Tugrul Dayar and William J. Stewart

ABSTRACT: Iterative aggregation-disaggregation (IAD) is an effective method 
for solving finite nearly completely decomposable (NCD) Markov chains. Small 
perturbations in the transition probabilities of these chains may lead to 
considerable changes in the stationary probabilities; NCD Markov chains are 
known to be ill-conditioned. During an IAD step, this undesirable condition 
is inherited by the coupling matrix and one confronts the problem of finding 
the stationary probabilities of a stochastic matrix whose diagonal elements 
are close to 1. In this paper, the effects of using the 
Grassmann-Taksar-Heyman (GTH) method to solve the coupling matrix formed in 
the aggregation step is investigated. Then, the idea is extended in such a 
way that the same direct method can be incorporated into the disaggregation 
step. Finally, the effects of using the GTH method in the IAD algorithm on 
various examples are demonstrated, and the conditions under which it should 
be employed are explained.

KEY WORDS: Markov chains, decomposability, stationary probability, 
aggregation-disaggregation, Gaussian elimination, sparsity schemes