Approximate inverse-free rational Krylov methods and the link with FOM and GMRES

Abstract

In this presentation we revisit the approximate rational Krylov method. We present two alternative but mathematically equivalent formulations of the same algorithm. The first reformulation uses a pole swapping technique and is an implicit method, just like the original algorithm. The second reformulation explicitly solves shifted linear systems using the Arnoldi Hessenberg matrix. This reformulation leads us to a connection between the approximate rational Krylov method and the full orthogonalization method (FOM). Finally, we show how the approximate rational Krylov method can be modified to obtain a similar connection with GMRES.