The Asymptotics of Wilkinson's Iteration: Loss of Cubic Convergence

TL;DR

This paper investigates the convergence rates of Wilkinson's QR iteration for eigenvalues, revealing that contrary to the long-standing belief of cubic convergence, some matrices exhibit only quadratic convergence.

Contribution

The authors demonstrate the existence of matrices where Wilkinson's shift strategy converges quadratically, challenging the prior assumption of universal cubic convergence.

Findings

Existence of matrices with quadratic convergence under Wilkinson's shift.

The set of such matrices forms a Hausdorff dimension 1 set.

Quadratic convergence occurs precisely when the sequence approaches a specific matrix T_X.

Abstract

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.