Wilkinson迭代的渐近性：三次收敛性的丧失
The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence
AI总结：
本文研究Wilkinson位移QR迭代的收敛速度，发现存在矩阵使得收敛速度严格为二次而非长期猜想的三次，并刻画了二次收敛的初始矩阵集具有Hausdorff维数1且结构复杂。
摘要：
最广泛使用的特征值计算方法之一是带有Wilkinson位移的$QR$迭代：这里位移$s$是底部$2\times 2$主子矩阵最接近角落条目的特征值。长期以来一直猜想算法的收敛速度是三次的。相反，我们证明存在矩阵使得收敛速度严格为二次。更精确地说，令$T_X$为只有两个非零条目$(T_X)_{12} = (T_X)_{21} = 1$的$3 \times 3$矩阵，并令$T_L$为与$T_X$具有相同谱的实对称三对角矩阵集合。存在$T_X$的一个邻域$U \subset T_L$，它在Wilkinson位移策略下不变，并具有以下性质。对于$T_0 \in U$，迭代序列$(T_k)$中条目$(T_k)_{23}$收敛到零的速度要么严格二次，要么严格三次。事实上，二次收敛恰好发生在$\lim T_k = T_X$时。令$X$为这样的二次收敛序列$(T_k)$的并集：集合$X$的Hausdorff维数为1，并且是由在$T_X$处相交的不相交弧$X^\sigma$的并集，其中$\sigma$遍历一个Cantor集。
英文摘要：
One of the most widely used methods for eigenvalue computation is the $QR$ iteration with Wilkinson's shift: here the shift $s$ is the eigenvalue of the bottom $2\times 2$ principal minor closest to the corner entry. It has been a long-standing conjecture that the rate of convergence of the algorithm is cubic. In contrast, we show that there exist matrices for which the rate of convergence is strictly quadratic. More precisely, let $T_X$ be the $3 \times 3$ matrix having only two nonzero entries $(T_X)_{12} = (T_X)_{21} = 1$ and let $T_L$ be the set of real, symmetric tridiagonal matrices with the same spectrum as $T_X$. There exists a neighborhood $U \subset T_L$ of $T_X$ which is invariant under Wilkinson's shift strategy with the following properties. For $T_0 \in U$, the sequence of iterates $(T_k)$ exhibits either strictly quadratic or strictly cubic convergence to zero of the entry $(T_k)_{23}$. In fact, quadratic convergence occurs exactly when $\lim T_k = T_X$. Let $X$ be the union of such quadratically convergent sequences $(T_k)$: the set $X$ has Hausdorff dimension 1 and is a union of disjoint arcs $X^σ$ meeting at $T_X$, where $σ$ ranges over a Cantor set.
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