Application of threshold partitioning of sparse matrices to Markov chains

Three algorithms which permute and partition a sparse matrix are presented as tools for the improved solution of Markov chain problems. One is the algorithm PABLO [SIAM J. Sci. Stat. Computing, vol.11, pp.811-823, 1990] while the other two are modifications of it. In the new algorithms, in addition to the location of the nonzeros, the values of the entries are taken into account. The permuted matrices are well suited for block iterative methods that find the corresponding probability distribution, as well as for block diagonal preconditioners of Krylov-based methods. Also, if the partition obtained from the ordering algorithm is used as an aggregation scheme, an iterative aggregation method performs better with this partition than with others found in the literature. Numerical experiments illustrate the performance of the iterative methods with the new orderings.