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. Copyright © 2010 John Wiley & Sons, Ltd.