2nd International Workshop on the Numerical Solution of Markov Chains 1 Preconditioned Krylov Subspace Methods for the Numerical Solution of Markov Chains

In a general projection technique the original matrix problem of size N is approximated by one of dimension m, typically much smaller than N. A particularly successful class of techniques based on this principle is that of Krylov subspace methods which utilize subspaces of the form spanfv; Av; ::::;A m?1 vg. This general principle can be used to solve linear systems and eigenvalue problems which arise when computing stationary probability distributions of Markov chains. It can also be used to approximate the product of the exponential of a matrix by a vector as occurs when following the solutions of transient models. In this paper we give an overview of these ideas and discuss preconditioning techniques which constitute an essential ingredient in the success of Krylov subspace methods.