A contribution to the conditioning of the total least squares problem

TL;DR

This paper derives formulas and bounds for the condition number of linear functions of the total least squares solution, aiding in understanding the sensitivity of solutions to perturbations.

Contribution

It provides closed-form formulas and bounds for the condition number of total least squares solutions, enhancing stability analysis.

Findings

Closed formulas for condition numbers using singular values and vectors.

Upper bounds requiring only extreme singular values.

Numerical comparisons with existing error estimates.

Abstract

We derive closed formulas for the condition number of a linear function of the total least squares solution. Given an over determined linear system Ax=b, we show that this condition number can be computed using the singular values and the right singular vectors of [A,b] and A. We also provide an upper bound that requires the computation of the largest and the smallest singular value of [A,b] and the smallest singular value of A. In numerical examples, we compare these values and the resulting forward error bounds with existing error estimates.