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.

翻译：QZ算法用于计算矩阵束的Schur形式。该算法是一个迭代过程，在某一时刻必须判断某个特征值是否已收敛，并转向处理另一个特征值。选择合适的判定准则并非易事：若准则过于严格，算法可能在已收敛的特征值上浪费迭代次数；若准则不够严格，则计算得到的特征值可能不精确。此外，准则的计算开销不应过大。本文提出了一种基于特征值大小及其间距的新准则，该思路与Ahues和Tissier针对QR算法的工作类似。理论论证与数值实验表明，该准则在精度上优于现有最常用的准则。同时，本文还评估了处理无穷特征值的几种常用准则。