The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence
Resolves a conjecture about the convergence rate of a fundamental eigenvalue algorithm, revealing that cubic convergence is not universal.
Wilkinson's iteration for eigenvalue computation is shown to have strictly quadratic convergence for certain matrices, contradicting the long-standing conjecture of cubic convergence. The set of quadratically convergent sequences has Hausdorff dimension 1 and is structured as a union of disjoint arcs meeting at a specific matrix.
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.