Global Convergence of Hessenberg Shifted QR II: Numerical Stability

TL;DR

This paper presents a framework proving rapid convergence and numerical stability of shifted QR algorithms with Ritz shifts, addressing forward-instability issues and ensuring efficient computation in finite arithmetic.

Contribution

It introduces a dichotomy framework that guarantees either stable Ritz value approximation or early matrix decoupling, enhancing the understanding of QR algorithm convergence.

Findings

Proves rapid convergence of shifted QR with Ritz shifts in finite arithmetic.

Establishes a stability framework addressing forward-instability issues.

Shows polylogarithmic bit complexity for convergence on well-conditioned matrices.

Abstract

We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap.