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

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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 number-based estimates are much tighter.

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Acknowledgements

The authors would like to thank the handling editor Prof. Claude Brezinski and two anonymous referees for their constructive comments and suggestions, which greatly improve the presentation of this paper.

Funding

This paper is supported in part by the National Natural Science Foundation of China under grants 12090011, 11771188, 12171210; the Priority Academic Program Development Project (PAPD); and the Top-notch Academic Programs Project (No. PPZY2015A013) of Jiangsu Higher Education Institution.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Qiaohua Liu

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, People’s Republic of China

    Zhigang Jia

  3. Research Institute of Mathematical Science, Jiangsu Normal University, Xuzhou, 221116, People’s Republic of China

    Zhigang Jia

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  1. Qiaohua Liu
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  2. Zhigang Jia
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Correspondence to Qiaohua Liu.

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Liu, Q., Jia, Z. On condition numbers of the total least squares problem with linear equality constraint. Numer Algor 90, 363–385 (2022). https://doi.org/10.1007/s11075-021-01191-w

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