The Potential of Invariant Subspaces to Attract Krylov Iterates
Christopher Beattie
Virginia Polytechnic Institute and State University
Abstract: In earlier work (presented in part at the second IWASEP), comparisons among various Ritz-type eigenvalue estimates constructed from a fixed subspace were made - with (roughly) the conclusion that garden variety Ritz values were hard to beat asymptotically although other contenders can provide at times bounding information that Ritz values cannot recover. Ultimately, it is the quality of the subspace whence Ritz-type estimates are drawn, as an approximation of an invariant subspace that will determine the quality of the associated eigenvalue estimates.
Attention turns now in this talk to the question of how well a particular class of subspaces, Krylov subspaces and their generalizations, can do in capturing various invariant subspaces - and moreover, why various shifting and restart strategies sometimes work and sometimes don't work in the pursuit of specially desired invariant subspaces. We obtain convergence rates that depend ultimately on a generalized condensor capacity of subsets of the complex plane as introduced by Zolotarjov. However, this topic here will be stripped of much of its technical burdens with the hope of bringing to the fore the organic relationships between potential theory and the subspace approximation problem.
Although many related convergence issues have been well addressed by others from other perspectives, our approach may have particular appeal since it directly assesses the quality of the subspace approximations; it incurs little penalty for large nonnormality; and as it draws on classical potential theory (hence the pun of the title), our approach offers a pleasantly intuitive view of how polynomial restart strategies work or not, as the case may be.
This talk describes in part results that have emerged in collaboration with Mark Embree and John Rossi.