The author considers the iterative solution of large sparse $n\times n$ linear systems of the form $Ax=b$ with a Krylow subspace method. He shows theoretically and experimentally that an inexact generalized minimal residual (GMRES) method can be applied to singular systems, in particular, to those obtained from Markov chain modeling. Inexact and exact GMRES are compared for consistent singular systems obtained from Markov chain matrices. It is seen that one can save the computational effort using inexact matrix-vector products. These savings come with no deterioration of the overall convergence. In cases where the index $k_0$ of the matrix is larger than one, the right-hand side needs to lie on (or be close to) the range of $A^{k_0}$ for the methods to work well.
Reviewer: Rémi Vaillancourt (Ottawa)