Bounds on the convergence of Ritz values from Krylov subspaces to interior eigenvalues of Hermitean matrices

TL;DR

This paper derives bounds on how quickly Ritz values from Krylov subspaces converge to interior eigenvalues of Hermitian matrices, especially in regions with low spectral density, using the Kaniel-Paige-Saad formalism applied to shifted and squared matrices.

Contribution

It introduces new bounds for Ritz value convergence near spectrum voids by applying the Kaniel-Paige-Saad formalism to shifted and squared matrices.

Findings

Bounds are effective in low spectral density regions.

The formalism improves understanding of Ritz value convergence.

Applicable to Hermitian matrices in spectral analysis.

Abstract

We consider bounds on the convergence of Ritz values from a sequence of Krylov subspaces to interior eigenvalues of Hermitean matrices. These bounds are useful in regions of low spectral density, for example near voids in the spectrum, as is required in many applications. Our bounds are obtained by considering the usual Kaniel-Paige-Saad formalism applied to the shifted and squared matrix.