Deflation in Krylov Subspace Methods and Distance to Uncontrollability

The task of extracting from a Krylov decomposition the approximation to an eigenpair that yields the smallest backward error can be phrased as finding the smallest perturbation which makes an associated matrix pair uncontrollable. Exploiting this relationship, we propose a new deflation criterion, which potentially admits earlier deflations than standard deflation criteria. Along these lines, a new deflation procedure for shift-and-invert Krylov methods is developed. Numerical experiments demonstrate the merits and limitations of this approach. AMS subject classifications: 65F15, 65F50, 93B05.