A note on GMRES preconditioned by a perturbed LDL(T) decomposition with static pivoting

A strict adherence to threshold pivoting in the direct solution of symmetric indefinite problems can result in substantially more work and storage than forecast by a sparse analysis of the symmetric problem. One way of avoiding this is to use static pivoting where the data structures and pivoting sequence generated by the analysis are respected and pivots that would otherwise be very small are replaced by a user defined quantity. This can give a stable factorization but of a perturbed matrix. The conventional way of solving the sparse linear system is then to use iterative refinement, but there are cases where this fails to converge. In this paper, we discuss the use of more robust iterative methods, namely GMRES and flexible GMRES (FGMRES). We show both theoretically and experimentally that both approaches are more robust than iterative refinement and furthermore that FGMRES is far more robust than GMRES and that, under reasonable hypotheses, FGMRES is backward stable. We also show how restarted variants can be beneficial, although again the GMRES variant is not as robust as FGMRES.