Square and stretch multigrid for stochastic matrix eigenproblems

A novel multigrid algorithm for computing the principal eigenvector of column-stochastic matrices is developed. The method is based on an approach originally introduced by Horton and Leutenegger (Perform. Eval. Rev. 1994; 22:191-200) whereby the coarse-grid problem is adapted to yield a better and better coarse representation of the original problem. A special feature of the present approach is the squaring of the stochastic matrix-followed by a stretching of its spectrum-just prior to the coarse-grid correction process. This procedure is shown to yield good convergence properties, even though a cheap and simple aggregation is used for the restriction and prolongation matrices, which is important for maintaining competitive computational costs. A second special feature is a bottom-up procedure for defining coarse-grid aggregates.