  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 $X^\sigma$ meeting at
$T_X$, where $\sigma$ ranges over a Cantor set.
