Partial condition number for the equality constrained linear least squares problem

TL;DR

This paper introduces a new partial condition number for the equality constrained linear least squares problem, providing formulas, structured variants, and reliable estimation algorithms, with numerical validation.

Contribution

It presents the first explicit formulas for the partial condition number, including structured cases, and develops algorithms for their reliable estimation.

Findings

Formulas for the partial condition number in different norms

Algorithms for high-reliability estimation of condition numbers

Numerical examples demonstrating effectiveness

Abstract

In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the data space and the solution space are measured by the weighted Frobenius norm and the Euclidean norm, respectively. Then, we investigate the corresponding structured partial condition number when the problem is structured. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical examples.