A contribution to condition numbers of the multidimensional total least squares problem with linear equality constraint

TL;DR

This paper investigates the condition numbers of the multidimensional total least squares problem with linear equality constraints, providing formulas, bounds, and numerical validation to enhance understanding and computation of problem sensitivity.

Contribution

It introduces a unified approach to derive condition numbers for multidimensional TLSE problems using perturbation theory and limit techniques, including compact bounds for efficient computation.

Findings

Derived Kronecker-product-based formulas for condition numbers.

Provided compact upper bounds to reduce computational cost.

Numerical experiments validate the theoretical results.

Abstract

This paper is devoted to condition numbers of the multidimensional total least squares problem with linear equality constraint (TLSE). Based on the perturbation theory of invariant subspace, the TLSE problem is proved to be equivalent to a multidimensional unconstrained weighed total least squares problem in the limit sense. With a limit technique, Kronecker-product-based formulae for normwise, mixed and componentwise condition numbers of the minimum Frobenius norm TLSE solution are given. Compact upper bounds of these condition numbers are provided to reduce the storage and computation cost. All expressions and upper bounds of these condition numbers unify the ones for the single-dimensional TLSE problem and multidimensional total least squares problem. Some numerical experiments are performed to illustrate our results.