Convergence rates to deflation of simple shift strategies

TL;DR

This paper analyzes the convergence behavior of simple shift strategies in QR algorithms for eigenvalue computation, showing conditions for quadratic and cubic convergence rates and implications for Wilkinson's shift.

Contribution

It introduces the concept of simple shift strategies, establishes conditions for convergence rates, and demonstrates cubic convergence for Wilkinson's shift under generic spectra.

Findings

Continuous shift strategies may fail to induce deflation.

Convergence to zero of subdiagonal entries is at least quadratic.

Under certain conditions, convergence is cubic, including for Wilkinson's shift with generic spectra.

Abstract

The computation of eigenvalues of real symmetric tridiagonal matrices frequently proceeds by a sequence of QR steps with shifts. We introduce simple shift strategies, functions sigma satisfying natural conditions, taking each n x n matrix T to a real number sigma(T). The strategy specifies the shift to be applied by the QR step at T. Rayleigh and Wilkinson's are examples of simple shift strategies. We show that if sigma is continuous then there exist initial conditions for which deflation does not occur, i.e., subdiagonal entries do not tend to zero. In case of deflation, we consider the rate of convergence to zero of the (n, n-1) entry: for simple shift strategies this is always at least quadratic. If the function sigma is smooth in a suitable region and the spectrum of T does not include three consecutive eigenvalues in arithmetic progression then convergence is cubic. This implies cubic convergence to deflation of Wilkinson's shift for generic spectra. The study of the algorithm near deflation uses tubular coordinates, under which QR steps with shifts are given by a simple formula.