Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than 50 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.

对称矩阵上的移位 QR 算法的快速收敛早在 50 多年前就已被证明。从那时起，尽管人们对非对称矩阵上的移位 QR 算法的动力学和收敛特性产生了浓厚的兴趣及其实际相关性，但人们仍然难以理解。我们为 Hessenberg 移位 QR 算法引入了一系列新的移位策略。我们证明，当输入是有界特征向量条件数\(\kappa _V(H)\)的可对角化 Hessenberg 矩阵H时，定义为V在所有对角化\(VDV^{-1}\)上的最小条件数H——那么采用我们家族的某种策略的移位 QR 算法保证在精确算术中快速收敛到下对角项为零的 Hessenberg 矩阵。我们的收敛结果是非渐近的，表明H的某些下对角项的几何平均值在每次QR迭代中衰减固定常数。实现策略每次迭代的算术成本在特征向量条件数\(\kappa _V(H)\)中大致呈对数缩放，这是H的非正态性的度量。我们的策略设计和分析的关键思想是：（1）我们能够精确地描述基于 Ritz 价值观的某种转移策略何时停滞。我们利用这些信息来设计某些“特殊转变”，保证在停滞发生时避免停滞。 (2) 我们使用更高的阶次移位（阶数大致为\(\log \kappa _V(H)\) ）来抑制非正态性造成的瞬态效应，使我们能够以与正态矩阵类似的方式处理非正态矩阵。