Convergence rates to deflation of simple shift strategies

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 σ satisfying natural conditions, taking each n × n matrix T to a real number σ(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 σ 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 σ 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.