The incomplete factorization preconditioners applied to the GMRES(m) method for solving Markov chains

This paper is a review and a comparison of some preconditioners based on incomplete factorizations of matrices - for matrices describing Markov chains. Three preconditioners are considered: ILU(0), ILU3, IWZ(0). Two of them (ILU(0), ILU3) are based on the LU factorization, the latter (IWZ(0)) - on the WZ factorization. The preconditioners are investigated in respect of their usability for decreasing number of iterations in a projection method, namely GMRES(m). To chose the best preconditioner for such methods, authors introduce a measure called iteration speed-up (p) and some of its relatives, as well as they define a function giving an average number of restarts needed to achieve a given accuracy for matrices from a some set (Is). These measures are studied for two different cases of matrices describing Markov chains to compare influence of the examined incomplete preconditioners for GMRES(m).