The asymptotics of Wilkinson’s shift iteration

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 P0 which is the limit of a sequence of a Wilkinson iteration, with the additional property that all iterations converging to P0 are strictly quadratic. Among tridiagonal matrices near P0, the set X of initial conditions with convergence to P0 is rather thin: it is a union of disjoint arcs Xs meeting at P0, where s ranges over the Cantor set of sign sequences s : N → {1,−1}. Wilkinson’s step takes Xs to Xs′ , where s ′ is the left shift of s. Among tridiagonal matrices conjugate to P0, initial conditions near P0 but not in X converge at a cubic rate.