A Flexible Inner-Outer Preconditioned GMRES Algorithm

We present a variant of the GMRES algorithm which allows changes in the precon-ditioning at every step. There are many possible applications of the new algorithm some of which are brieey discussed. In particular, a result of the exibility of the new variant is that any iterative method can be used as a preconditioner. For example, the standard GMRES algorithm itself can be used as a preconditioner, as can CGNR (or CGNE) the conjugate gradient method applied to the normal equations. However, the more appealing utilization of the method is in conjunction with relaxation techniques, possibly multi-level techniques. The possibility of changing preconditioners may be exploited to develop eecient iterative methods and to enhance robustness. A few numerical experiments are reported to illustrate this fact.