Backward error and condition number analysis for the indefinite linear least squares problem

TL;DR

This paper analyzes the backward error and condition numbers for the indefinite least squares problem, providing explicit formulas, bounds, and efficient estimation methods, with numerical validation demonstrating sharpness and effectiveness.

Contribution

It introduces explicit expressions for condition numbers and tight bounds, along with efficient estimation techniques for the indefinite least squares problem.

Findings

Derived explicit formulas for condition numbers.

Provided tight upper bounds and estimation methods.

Numerical examples confirm the sharpness and effectiveness.

Abstract

In this paper, we concentrate on the backward error and condition number of the indefinite least squares problem. For the normwise backward error of the indefinite least square problem, we adopt the linearization method to derive the tight estimations for the exact normwise backward errors. Using the dual techniques of condition number theory \cite{22.0}, we derive the explicit expressions of the mixed and componentwise condition numbers for the linear function of the solution for the indefinite least squares problem. The tight upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical power method for estimating matrix 1-norm \cite[Chapter 15]{Higham2002Book} during using the QR-Cholesky method \cite{1.0} for solving the indefinite least squares problem. The numerical examples show that the derived condition numbers can give sharp perturbation bound with respect to the interested component of the solution. And the linearization estimations are effective for the normwise backward errors.