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

Abstract

A recent study on the condition numbers of the mixed least squares-total least squares (MTLS) problem is due to Zheng and Yang (Numer Linear Algebra Appl. 2019;26(4):e2239). However, the associated expressions are not compact and the Kronecker-product operations make the computation costly. In this paper, we first present new and alternative closed formula for the first order perturbation estimate and condition numbers of the MTLS solution. Then we reveal the relationship between the new formula and Zheng and Yang's result. Several new computable formulae and perturbation bounds for the normwise condition number of the MTLS solution are also provided. Finally, mixed and componentwise condition numbers, structured condition numbers are investigated. Through a number of tests, they are shown to be tighter than the normwise condition numbers for sparse and structured problems.

Keywords:

• Mixed least squares-total least squares problem
• condition number
• perturbation bound
• linear structure

Acknowledgments

The authors would like to thank the handling editor Ren-Cang Li and two anonymous referees for their valuable comments and suggestions which led to great improvements to this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is partially supported by the National Natural Science Foundation of China (11001167).