SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 753-774, January 2021.
In the article “A Rational QZ Method” by D. Camps, K. Meerbergen, and R. Vandebril [SIAM J. Matrix Anal. Appl., 40 (2019), pp. 943--972], we introduced rational QZ (RQZ) methods. Our theoretical examinations revealed that the convergence of the RQZ method is governed by rational subspace iteration, thereby generalizing the classical QZ method, whose convergence relies on polynomial subspace iteration. Moreover the RQZ method operates on a pencil more general than Hessenberg---upper triangular, namely, a Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. However, the RQZ method can only be made competitive to advanced QZ implementations by using crucial add-ons such as small bulge multishift sweeps, aggressive early deflation, and optimal packing. In this paper we develop these techniques for the RQZ method. In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that the presented method is competitive in terms of speed and accuracy.

中文翻译：

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具有积极早期通货紧缩的多位移、多极有理 QZ 方法

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 753-774 页，2021 年 1 月。
在 D. Camps、K. Meerbergen 和 R. Vandebril 的文章“A Rational QZ Method”中 [SIAM J. Matrix Anal。Appl., 40 (2019), pp. 943--972]，我们介绍了有理 QZ (RQZ) 方法。我们的理论检验表明，RQZ 方法的收敛性受有理子空间迭代控制，从而推广了经典 QZ 方法，其收敛性依赖于多项式子空间迭代。此外，RQZ 方法对比 Hessenberg 更通用的铅笔进行操作——上三角，即 Hessenberg pen，它是由两个 Hessenberg 矩阵组成的 Pen。然而，RQZ 方法只能通过使用关键的附加组件（例如小凸起多移位扫描、积极的早期放气和最佳包装）才能与高级 QZ 实现竞争。在本文中，我们为 RQZ 方法开发了这些技术。