  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.
