On condition numbers of the total least squares problem with linear equality constraint

TL;DR

This paper derives new, computable formulas for condition numbers of the total least squares problem with linear equality constraints, improving error estimation especially for sparse or badly scaled data.

Contribution

It introduces novel limit techniques to obtain closed-form expressions and bounds for various condition numbers of TLSE, unifying and extending previous results.

Findings

Normwise condition number estimates are sharp for equilibratory data.

Mixed and componentwise estimates are tighter for sparse, badly scaled matrices.

New formulas avoid costly Kronecker product operations.

Abstract

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers-based estimates are much tighter.