Fourier Analysis of the Aggregation Based Algebraic Multigrid for Stochastic Matrices

Abstract

We introduce new theoretical results on the convergence of the algebraic multigrid methods for obtaining stationary probability distribution vectors of stochastic matrices. We focus on sparse and nonsymmetric stochastic matrices. Our approach is based on the Fourier transform of the error propagation operator. For some special classes of stochastic matrices it allows one to find the optimal parameters of the algorithm and to estimate the rate of convergence. Examples related to the computation of stable gene profiles are presented.