Unattainability of A Perturbation Bound for Indefinite Linear Least Squares Problems

TL;DR

This paper demonstrates that a known perturbation bound for indefinite least squares problems can significantly overestimate the true error, challenging previous assumptions about the bound's reliability and the stability of related algorithms.

Contribution

It reveals the limitations of existing perturbation bounds for indefinite least squares problems and questions their applicability in assessing algorithm stability.

Findings

Perturbation bounds can overestimate errors arbitrarily.

Hyperbolic QR factorization is not proven to be forward stable.

Existing bounds may not reliably measure solution accuracy.

Abstract

Contrary to an assumption made by Bojanczyk, Higham, and Patel [SIAM J. Matrix Anal. Appl., 24(4):914-931, 2003], a perturbation bound for indefinite least square problems is capable of arbitrarily large overestimates for all perturbations of some problems. For these problems, the hyperbolic QR factorization algorithm is not proved to be forward stable because the error bound systematically overestimates the solution error of backward stable methods.