Convergence of Restarted Krylov Subspaces to Invariant Subspaces

Abstract

The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired set of eigenvalues is a maximum "reachable invariant subspace" that can be developed from the given starting vector. Convergence for this distinguished subspace is bounded in terms involving a polynomial approximation problem. Elementary results from potential theory lead to convergence rate estimates and suggest restarting strategies based on optimal approximation points (e.g., Leja or Chebyshev points); exact shifts are evaluated within this framework. Computational examples illustrate the utility of these results. Origins of superlinear effects are also described.

On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

“…be interpreted as the opening between R(Q) and the "closest" subspace of R(Y ) of dimension k. This one-sided gap was used in [4], [5].…”

“…We mention that in [4], [5], invariant subspaces have been used in the convergence analysis of Krylov subspace methods in the context of eigenvalue problems. We also point out that using invariant subspaces and not the spectrum of A directly allows us to treat diagonalizable and nondiagonalizable matrices in the same manner.…”

Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computations

“…The resulting analysis incorporates a different polynomial approximation problem. In typical situations the new bounds are weaker at early iterations, though the asymptotic convergence rate we establish is never worse than that obtained in [3]. In certain situations where the desired eigenvalues are ill-conditioned, these new bounds improve the earlier analysis.…”

“…In this section we characterize those good invariant subspaces (within X g ) that can be captured with Krylov subspaces, adapting the discussion from [3]. Throughout we assume that the starting vector v 1 is fixed.…”

“…Recently, new bounds have been derived for single-vector Krylov subspace methods that impose no restriction on the dimension of the desired invariant subspace or diagonalizability of A, yet still result in a conventional polynomial approximation problem [3]. While examples demonstrate that these bounds can be descriptive, their derivation involves fairly intricate arguments.…”