Mixed and componentwise condition numbers for a linear function of the solution of the linear least squares problem with equality constrains

TL;DR

This paper derives explicit mixed and componentwise condition numbers for linear functions of solutions to the constrained least squares problem, providing sharp bounds and efficient estimation methods.

Contribution

It introduces explicit formulas for mixed and componentwise condition numbers in LSE problems and offers efficient estimation techniques using existing algorithms.

Findings

Derived sharp upper bounds for condition numbers

Efficient estimation via Hager-Higham algorithm

Normwise condition numbers may overestimate errors

Abstract

In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the linear least squares problem with equality constrains (LSE). We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques. The sharp upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical Hager-Higham algorithm for estimating matrix one-norm during using the generalized QR factorization method for solving LSE. The numerical examples show that the derived condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the relative errors because normwise condition numbers ignore the data sparsity and scaling.