Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than fifty years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of the shifted QR algorithm on nonsymmetric matrices has remained elusive. We introduce a new family of shifting strategies for the Hessenberg shifted QR algorithm. We prove that when the input is a diagonalizable Hessenberg matrix $H$ of bounded eigenvector condition number $\kappa_V(H)$ -- defined as the minimum condition number of $V$ over all diagonalizations $VDV^{-1}$ of $H$ -- then the shifted QR algorithm with a certain strategy from our family is guaranteed to converge rapidly to a Hessenberg matrix with a zero subdiagonal entry, in exact arithmetic. Our convergence result is nonasymptotic, showing that the geometric mean of certain subdiagonal entries of $H$ decays by a fixed constant in every $QR$ iteration. The arithmetic cost of implementing each iteration of our strategy scales roughly logarithmically in the eigenvector condition number $\kappa_V(H)$, which is a measure of the nonnormality of $H$. The key ideas in the design and analysis of our strategy are: (1) We are able to precisely characterize when a certain shifting strategy based on Ritz values stagnates. We use this information to design certain ``exceptional shifts'' which are guaranteed to escape stagnation whenever it occurs. (2) We use higher degree shifts (of degree roughly $\log \kappa_V(H)$) to dampen transient effects due to nonnormality, allowing us to treat nonnormal matrices in a manner similar to normal matrices.

翻译：五十多年前，对称矩阵上移位QR算法的快速收敛性已被证明。然而，尽管该算法具有显著的理论意义和实际应用价值，但非对称矩阵上移位QR算法的动态特性与收敛性质至今仍难以理解。本文提出了一类新的海森伯格移位QR算法移位策略。我们证明：当输入为可对角化海森伯格矩阵$H$，且其特征向量条件数$\kappa_V(H)$（定义为所有对角化分解$VDV^{-1}$中$V$的最小条件数）有界时，采用本族中特定策略的移位QR算法在精确算术条件下能快速收敛至次对角线元素为零的海森伯格矩阵。我们的收敛结果为非渐近形式，表明$H$的某些次对角线元素的几何平均值在每个QR迭代中按固定常数衰减。实现该策略每次迭代的算术代价与特征向量条件数$\kappa_V(H)$（衡量$H$非正规性的度量）呈对数级增长。该策略设计与分析的核心思想包括：(1) 精确定义了基于Ritz值的特定移位策略停滞的条件，并据此设计可确保脱离停滞的"异常移位"；(2) 采用高次移位（次数约$\log \kappa_V(H)$）抑制非正规性导致的暂态效应，使非正规矩阵的处理方式可类同于正规矩阵。