Hessenberg移位QR的全局收敛性 I：精确算术
Global Convergence of Hessenberg Shifted QR I: Exact Arithmetic
AI总结：
针对非对称Hessenberg矩阵的移位QR算法，提出新移位策略族，证明在精确算术下快速收敛到可分解形式，并给出非渐近收敛速率与对数级算术成本。
摘要：
移位QR算法在对称矩阵上的快速收敛性在五十多年前就已得到证明。此后，尽管该算法具有重要的实际意义并引起了广泛关注，但对其在非对称矩阵上的动力学和收敛性质的理解仍然难以捉摸。\n我们为Hessenberg移位QR算法引入了一个新的移位策略族。我们证明，当输入是一个可对角化的Hessenberg矩阵$H$，其特征向量条件数$κ_V(H)$有界——定义为在所有对角化$VDV^{-1}$中$V$的最小条件数——那么采用我们策略族中某一特定策略的移位QR算法，在精确算术下，保证快速收敛到一个具有零次对角线元素的Hessenberg矩阵。我们的收敛结果是非渐近的，表明在每次QR迭代中，$H$的某些次对角线元素的几何平均值以固定常数衰减。实现我们策略每次迭代的算术成本大致随特征向量条件数$κ_V(H)$的对数增长，而$κ_V(H)$是衡量$H$非正规性的指标。\n我们策略设计和分析的关键思想是：（1）我们能够精确刻画基于Ritz值的某种移位策略何时停滞。我们利用这一信息设计某些“异常移位”，这些移位在停滞发生时保证能够逃脱停滞。（2）我们使用高次移位（次数约为$\log κ_V(H)$）来抑制由非正规性引起的瞬态效应，从而能够以类似于正规矩阵的方式处理非正规矩阵。
英文摘要：
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 $κ_V(H)$ -- 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 $κ_V(H)$, 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 $\log κ_V(H)$) to dampen transient effects due to nonnormality, allowing us to treat nonnormal matrices in a manner similar to normal matrices.
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