Inexact GMRES for singular linear systems

Inexact Krylov subspace methods have been shown to be practical alternatives for the solution of certain linear systems of equations. In this paper, the solution of singular systems with inexact matrix-vector products is explored. Criteria are developed to prescribe how inexact the matrix-vector products can be, so that the computed residual remains close to the true residual, thus making the inexact method of practical applicability. Cases are identified for which the methods work well, and this is the case in particular for systems representing certain Markov chains. Numerical experiments illustrate the effectiveness of the inexact approach.