Rich Lehoucq
Sandia National Laboratories

Abstract: We present an isomorphism between discrete dynamical systems and a class of iterations for the eigenvalue problem. This isomorphism provides a framework to investigate the convergence and stability of existing eigen-iterations and propose new techniques. The isomorphism with discrete dynamical systems allows us to consider geometric properties. In particular, we demonstrate that there are first integrals, or invariants, that must be satisfied by the eigen-iteration. Satisfaction of these invariants is demonstrated to be crucial to the numerical stability of the eigen-iteration. Moreover, these invariants provide a mechanism for monitoring the stability of the eigen-iteration.