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Condition numbers of the multidimensional total least squares problems having more than one solution

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Abstract

Recently, the condition numbers of the total least squares (TLS) problems having a unique solution have been studied at length in Zheng et al. (SIAM J. Matrix Anal. Appl. 38: 924–948, 2017). However, it is known that the TLS problem may have no solution, and even if an existing solution, it may not be unique. As a continuation of their work, in this paper, we investigate the condition numbers of the minimum Frobenius norm solution of the (multidimensional) TLS problem when having more than one solution. The tight and computable upper bound estimates of the normwise, mixed, and componentwise condition numbers are respectively derived. Some numerical experiments are performed to illustrate the tightness of these upper bounds.

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Acknowledgments

We would like to express our sincere thanks to the anonymous reviewers for their valuable suggestions which greatly improved the presentation of this paper.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11571004, 11701458 and 11771099).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China

    Lingsheng Meng

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730070, China

    Lingsheng Meng & Bing Zheng

  3. School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai, 200433, China

    Yimin Wei

Authors
  1. Lingsheng Meng
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  2. Bing Zheng
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  3. Yimin Wei
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Correspondence to Bing Zheng.

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Meng, L., Zheng, B. & Wei, Y. Condition numbers of the multidimensional total least squares problems having more than one solution. Numer Algor 84, 887–908 (2020). https://doi.org/10.1007/s11075-019-00785-9

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  • DOI: https://doi.org/10.1007/s11075-019-00785-9

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