New inverse data for tridiagonal matrices and the asymptotics of Wilkinson’s shift iteration

We introduce bidiagonal coordinates, a new set of spectral coordinates on open dense charts covering the space of real symmetric tridiagonal matrices. In contrast to the standard Jacobi inverse variables, reduced tridiagonal matrices now lie in the interior of some chart. Bidiagonal coordinates are thus convenient for the study of asymptotics of isospectral dynamics, both for continuous and discrete time. In particular, we study the rate of convergence of Wilkinson’s shift iteration. For AP-free spectra (i.e., simple spectra containing no arithmetic progression with 3 terms), convergence is cubic. In order 3, for AP-spectra, however, there exists a matrix P0 such that if Wilkinson’s iteration converges to P0 then convergence is strictly quadratic. Near p0 ∈ R , the bidiagonal coordinates of P0, the set X of initial conditions with convergence to p0 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 X s ′ , where s′ is the left shift of s. The set X is rather thin and for initial conditions near p0 but not in X , cubic convergence still applies.