A generalization of Saad's bound on harmonic Ritz vectors of Hermitian matrices

TL;DR

This paper extends Saad's bound to harmonic Ritz vectors of Hermitian matrices, highlighting the influence of the condition number and suggesting preconditioning to improve the harmonic Rayleigh--Ritz method.

Contribution

It generalizes Saad's bound for harmonic Ritz vectors, revealing the role of the condition number and proposing preconditioning strategies.

Findings

The new bound depends on the condition number of a shifted operator.

Preconditioning can enhance the harmonic Rayleigh--Ritz procedure.

Practical implications for iterative eigenvalue algorithms.

Abstract

We prove a Saad's type bound for harmonic Ritz vectors of a Hermitian matrix. The new bound reveals a dependence of the harmonic Rayleigh--Ritz procedure on the condition number of a shifted problem operator. Several practical implications are discussed. In particular, the bound motivates incorporation of preconditioning into the harmonic Rayleigh--Ritz scheme.