On the partial condition numbers for the indefinite least squares problem

TL;DR

This paper introduces a unified framework for partial condition numbers in indefinite least squares problems, deriving explicit formulas, considering structure, and proposing effective estimation algorithms with numerical validation.

Contribution

It develops a comprehensive unified condition number framework for indefinite least squares, including structured variants and estimation methods, advancing theoretical understanding and practical computation.

Findings

Derived explicit formulas for partial condition numbers

Proposed algorithms for efficient condition number estimation

Validated methods through numerical experiments

Abstract

The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be called the unified condition number, the expression of the partial unified condition number is first presented when the data space is measured by the general weighted product norm. Then, by setting the specific norms and weight parameters, we obtain the expressions of the partial normwise, mixed and componentwise condition numbers. Moreover, the corresponding structured partial condition numbers are also taken into consideration when the problem is structured, whose expressions are given. Considering the connections between the indefinite and total least squares problems, we derive the (structured) partial condition numbers for the latter, which generalize the ones in the literature. To estimate these condition numbers effectively and reliably, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and three related algorithms are devised. Finally, the obtained results are illustrated by numerical experiments.