# A Condition Analysis of the Weighted Linear Least Squares Problem Using Dual Norms

## TL;DR

This paper introduces a new approach to analyze the sensitivity of solutions in weighted linear least squares problems using dual norms, providing explicit condition numbers and efficient estimators.

## Contribution

It develops explicit normwise and componentwise condition numbers based on dual operator theory and proposes efficient estimators for these condition numbers.

## Key findings

- Condition numbers accurately bound perturbations in solutions.
- Componentwise analysis reveals individual component sensitivities.
- Proposed estimators are both accurate and computationally efficient.

## Abstract

In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise condition numbers derived in this paper can be computed at low cost when the dimension of the linear function is low due to dual operator theory. Moreover, we use the augmented system to perform a componentwise perturbation analysis of the solution and residual of the weighted linear least squares problems. We also propose two efficient condition number estimators. Our numerical experiments demonstrate that our condition numbers give accurate perturbation bounds and can reveal the conditioning of individual components of the solution. Our condition number estimators are accurate as well as efficient.