A new deflation criterion for the QZ algorithm

TL;DR

This paper proposes a new eigenvalue convergence criterion for the QZ algorithm, improving accuracy and efficiency by balancing convergence strictness with computational cost, based on eigenvalue size and gaps.

Contribution

It introduces a novel deflation criterion for the QZ algorithm, inspired by QR algorithm methods, with theoretical and experimental validation showing superior accuracy.

Findings

Outperforms existing criteria in accuracy

Reduces unnecessary iterations on converged eigenvalues

Effective for infinite eigenvalues detection

Abstract

The QZ algorithm computes the Schur form of a matrix pencil. It is an iterative algorithm and at some point, it must decide that an eigenvalue has converged and move on with another one. Choosing a criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might be inaccurate. Additionally, the criterion should not be computationally expensive to evaluate. This paper introduces a new criterion based on the size of and the gap between the eigenvalues. This is similar to the work of Ahues and Tissuer for the QR algorithm. Theoretical arguments and numerical experiments suggest that it outperforms the most popular criteria in terms of accuracy. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.