Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem

TL;DR

This paper derives explicit mixed and componentwise condition numbers for the total least squares problem, providing sharp bounds and analyzing structured perturbations, which improve understanding of problem sensitivity and data sparsity effects.

Contribution

It introduces new explicit formulas for mixed and componentwise condition numbers in TLS, including structured cases, and compares their effectiveness to normwise measures.

Findings

Structured condition numbers are smaller than unstructured ones.

Derived condition numbers provide sharper perturbation bounds.

Normwise condition numbers can overestimate errors.

Abstract

In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the total least squares (TLS) problem. 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. For the structured TLS problem, we consider the structured perturbation analysis and obtain the corresponding expressions of the mixed and componentwise condition numbers. We prove that the structured ones are smaller than their corresponding unstructured ones based on the derived expressions. Moreover, we point out that the new derived expressions can recover the previous results on the condition analysis for the TLS problem. 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. Meanwhile, from the observations of numerical examples, it is more suitable to adopt structured condition numbers to measure the conditioning for the structured TLS problem.