Square & 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 the Exact Interpolation Scheme multigrid approach of Brandt and Ron, whereby the prolongation is adapted to yield a better and better coarse representation of the sought eigenvector. 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 novel bottom-up procedure for defining coarse-grid aggregates. Copyright c © 2000 John Wiley & Sons, Ltd.