Condition numbers of the mixed least squares-total least squares problem: revisited

TL;DR

This paper derives a new closed-form formula for the first-order perturbation estimate of the MTLS solution, unifying the condition numbers of TLS and MTLS problems, and providing efficient computational bounds.

Contribution

It introduces a new closed formula for perturbation estimates of MTLS solutions, unifies TLS and MTLS condition numbers, and offers computationally efficient bounds.

Findings

Derived a new closed-form perturbation estimate formula.

Unified the condition numbers of TLS and MTLS problems.

Provided efficient bounds for condition numbers and perturbation estimates.

Abstract

A new closed formula for the first order perturbation estimate of the mixed least squares-total least squares (MTLS) solution is presented. It is mathematically equivalent to the one by Zheng and Yang(Numer. Linear Algebra Appl. 2019; 26(4):e2239). With this formula, general and structured normwise, mixed and componentwise condition numbers of the MTLS problem are derived. Perturbation bounds based on the normwise condition number, and compact forms for the upper bounds of mixed and componentwise condition numbers are also given in order for economic storage and efficient computation. It is shown that the condition numbers and perturbation bound of the TLS problem are unified in the ones of the MTLS problem.