On the condition number theory of the equality constrained indefinite least squares problem

TL;DR

This paper develops a unified condition number theory for the equality constrained indefinite least squares problem, providing explicit formulas, special cases, and efficient computation methods, with applications to related least squares problems.

Contribution

It introduces a comprehensive framework for the condition number analysis of equality constrained indefinite least squares problems, including explicit formulas and computationally efficient bounds.

Findings

Derived explicit projected condition number expressions.

Unified framework encompassing various condition numbers.

Numerical experiments validate theoretical results.

Abstract

In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares problem. By setting specific norms and parameters, some widely used condition numbers, like the normwise, mixed and componentwise condition numbers follow as its special cases. Considering practical applications and computation, some new compact forms or upper bounds of the projected condition numbers are given to improve the computational efficiency. The new compact forms are of particular interest in calculating the exact value of the 2-norm projected condition numbers. When the equality constrained indefinite least squares problem degenerates into some specific least squares problems, our results give some new findings on the condition number theory of these specific least squares problems. Numerical experiments are given to illustrate our theoretical results.