Angle-free cluster robust Ritz value bounds for restarted block eigensolvers

TL;DR

This paper introduces angle-free, robust bounds for Ritz value errors in restarted block eigensolvers, extending previous single-value bounds to all Ritz values and improving convergence analysis for clustered eigenvalues.

Contribution

It extends existing angle-free Ritz value bounds to all Ritz values and clustered eigenvalues, applicable to various restarted block eigensolvers.

Findings

Bounds are applicable to all Ritz values, not just extreme ones.

The new bounds are numerically advantageous and robust for clustered eigenvalues.

Applicable to shifted-invert and deflation variants of block Lanczos.

Abstract

Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or iterative subspace and the invariant subspace associated with the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed for restarted block eigensolvers, since the left- and right-hand sides of the bounds use different terms. They must be combined with additional bounds which could cause an overestimation. Alternative repeatable bounds that are angle-free and depend only on the errors of the Ritz values have been pioneered for Hermitian eigenvalue problems in doi:10.1515/rnam.1987.2.5.371 but only for a single extreme Ritz value. We extend this result to all Ritz values and achieve robustness for clustered eigenvalues by utilizing nonconsecutive eigenvalues. Our new bounds cover the restarted block Lanczos method and its modifications with shift-and-invert and deflation, and are numerically advantageous.