Dynamics of the symmetric eigenvalue problem with shift strategies

TL;DR

This paper analyzes the dynamics of shifted QR algorithms for symmetric eigenvalue problems, focusing on convergence rates and the impact of spectrum structure and strategy discontinuities.

Contribution

It provides a detailed theoretical analysis of the convergence behavior of shift strategies in symmetric eigenvalue computations, including the effects of spectrum structure and discontinuities.

Findings

Most shift strategies lead to cubic convergence to zero of off-diagonal entries.

Discontinuities in shift strategies are necessary for fast deflation.

Spectrum arithmetic progressions can cause quadratic convergence.

Abstract

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_\sigma$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal T}_\Lambda$, the manifold of real symmetric tridiagonal matrices conjugate to the diagonal matrix $\Lambda$. More precisely, a (generic) shift $s \in \RR$ defines a map $F_s: {\cal T}_\Lambda \to {\cal T}_\Lambda$. A strategy $\sigma: {\cal T}_\Lambda \to \RR$ specifies the shift to be applied at $T$ so that $F_\sigma(T) = F_{\sigma(T)}(T)$. Good shift strategies should lead to fast deflation: some off-diagonal coordinate tends to zero, allowing for reducing of the problem to submatrices. For topological reasons, continuous shift strategies do not obtain fast deflation; many standard strategies are indeed discontinuous. Practical implementation only gives rise systematically to bottom deflation, convergence to zero of the lowest off-diagonal entry $b(T)$. For most shift strategies, convergence to zero of $b(T)$ is cubic, $|b(F_\sigma(T))| = \Theta(|b(T)|^k)$ for $k = 3$. The existence of arithmetic progressions in the spectrum of $T$ sometimes implies instead quadratic convergence, $k = 2$. The complete integrability of the Toda lattice and the dynamics at non-smooth points are central to our discussion. The text does not assume knowledge of numerical linear algebra.