  Multistep matrix splitting iterations serve as preconditioning for Krylov
subspace methods for solving singular linear systems. The preconditioner is
applied to the generalized minimal residual (GMRES) method and the flexible
GMRES (FGMRES) method. We present theoretical and practical justifications for
using this approach. Numerical experiments show that the multistep generalized
shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS)
iteration preconditioning are more robust and efficient compared to standard
preconditioners for some test problems of large sparse singular linear systems.
