Condition numbers of multidimensional mixed least squares-total least squares problems

This paper considers the multidimensional mixed least squares-total least squares (MTLS) problem which arises in the regression model, signal processing and the problem of space coordinate transformation. Firstly, the MTLS problem is proved to be equivalent to a weighted total least squares problem in the limit sense. Then, the perturbation analysis and explicit normwise, mixed and componentwise condition number formulae for the MTLS problem are presented. Tight and computable upper bounds for these condition numbers are also given. The results include the ones for the single-right-hand-side MTLS as special cases. Finally in the numerical experiments, the tightness of condition numbers and upper bounds in evaluating the forward errors is also shown. (c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

Automated summary

This paper discusses the multidimensional mixed least squares-total least squares (MTLS) problem in the context of regression models, signal processing, and space coordinate transformation. It proves that the MTLS problem is equivalent to a weighted total least squares problem in the limit sense, and provides perturbation analysis and explicit condition number formulae for the MTLS problem. Tight and computable upper bounds for the condition numbers are also given. The numerical experiments demonstrate the tightness of the condition numbers and upper bounds in evaluating the forward errors.