The asymptotics of Wilkinson's shift iteration

TL;DR

This paper investigates the convergence rates of Wilkinson's shift iteration on Jacobi matrices, revealing cubic convergence for AP-free spectra and detailed structure of initial conditions near specific limit points.

Contribution

It establishes the cubic convergence for AP-free spectra and characterizes the complex structure of initial conditions leading to different convergence behaviors.

Findings

Convergence is cubic for AP-free spectra.

Existence of a limit matrix P_0 with quadratic convergence.

The set of initial conditions converging to P_0 has a Cantor set structure.

Abstract

We study the rate of convergence of Wilkinson's shift iteration acting on Jacobi matrices with simple spectrum. We show that for AP-free spectra (i.e., simple spectra containing no arithmetic progression with 3 terms), convergence is cubic. In order 3, there exists a tridiagonal symmetric matrix P_0 which is the limit of a sequence of a Wilkinson iteration, with the additional property that all iterations converging to P_0 are strictly quadratic. Among tridiagonal matrices near P_0, the set X of initial conditions with convergence to P_0 is rather thin: it is a union of disjoint arcs X_s meeting at P_0, where s ranges over the Cantor set of sign sequences s: N -> {1,-1}. Wilkinson's step takes X_s to X_{s'}, where s' is the left shift of s. Among tridiagonal matrices conjugate to P_0, initial conditions near P_0 but not in X converge at a cubic rate.