## Abstract

Linear matrix approximation problems *AX* ≈ *B* are often solved by the total least squares minimization (TLS). Unfortunately, the TLS solution may not exist in general. The so-called core problem theory brought an insight into this effect. Moreover, it simplified the solvability analysis if *B* is of column rank one by extracting a core problem having always a unique TLS solution. However, if the rank of *B* is larger, the core problem may stay unsolvable in the TLS sense, as shown for the first time by Hnětynková, Plešinger, and Sima (2016). Full classification of core problems with respect to their solvability is still missing. Here we fill this gap. Then we concentrate on the so-called composed (or reducible) core problems that can be represented by a composition of several smaller core problems. We analyze how the solvability class of the components influences the solvability class of the composed problem. We also show on an example that the TLS solvability class of a core problem may be in some sense improved by its composition with a suitably chosen component. The existence of irreducible problems in various solvability classes is discussed.

This is a preview of subscription content, access via your institution.

## References

- [1]
*G. H. Golub, C. F. Van Loan*: An analysis of the total least squares problem. SIAM J. Numer. Anal.*17*(1980), 883–893. - [2]
*I. Hnětynková, M. Plešinger, D. M. Sima*: Solvability of the core problem with multiple right-hand sides in the TLS sense. SIAM J. Matrix Anal. Appl.*37*(2016), 861–876. - [3]
*I. Hnětynková, M. Plešinger, D. M. Sima, Z. Strakoš, S. Van Huffel*: The total least squares problem in*AX*≈*B*: a new classification with the relationship to the classical works. SIAM J. Matrix Anal. Appl.*32*(2011), 748–770. - [4]
*I. Hnětynková, M. Plešinger, Z. Strakoš*: The core problem within a linear approximation problem*AX*≈*B*with multiple right-hand sides. SIAM J. Matrix Anal. Appl.*34*(2013), 917–931. - [5]
*I. Hnětynková, M. Plešinger, Z. Strakoš*: Band generalization of the Golub-Kahan bidiagonalization, generalized Jacobi matrices, and the core problem. SIAM J. Matrix Anal. Appl.*36*(2015), 417–434. - [6]
*I. Hnětynková, M. Plešinger, J. Žáková*: Modification of TLS algorithm for solving*ℱ*_{2}linear data fitting problems. PAMM, Proc. Appl. Math. Mech.*17*(2017), 749–750. - [7]
*I. Markovsky, S. Van Huffel*: Overview of total least-squares methods. Signal Process.*87*(2007), 2283–2302. - [8]
*C. C. Paige, Z. Strakoš*: Core problems in linear algebraic systems. SIAM J. Matrix Anal. Appl.*27*(2005), 861–875. - [9]
*D. A. Turkington*: Generalized Vectorization, Cross-Products, and Matrix Calculus. Cambridge University Press, Cambridge, 2013. - [10]
*S. Van Huffel, J. Vandewalle*: The Total Least Squares Problem: Computational Aspects and Analysis. Frontiers in Applied Mathematics 9, Society for Industrial and Applied Mathematics, Philadelphia, 1991. - [11]
*X.-F. Wang*: Total least squares problem with the arbitrary unitarily invariant norms. Linear Multilinear Algebra*65*(2017), 438–456. - [12]
*M. S. Wei*: Algebraic relations between the total least squares and least squares problems with more than one solution. Numer. Math.*62*(1992), 123–148. - [13]
*M. Wei*: The analysis for the total least squares problem with more than one solution. SIAM J. Matrix Anal. Appl.*13*(1992), 746–763. - [14]
*S. Yan, K. Huang*: The original TLS solution sets of the multidimensional TLS problem. Int. J. Comput. Math.*73*(2000), 349–359.

## Acknowledgements

We wish to thank the anonymous referee for her or his careful reading the paper and useful comments which led to improvements of our manuscript.

## Author information

### Affiliations

### Corresponding author

## Additional information

The research of Iveta Hnětynková has been supported by the GAČR grant No. GA17-04150J. The research of Martin Plešinger and Jana Žáková has been supported by the SGS grant of Technical University of Liberec No. 21254/2018.

## Rights and permissions

## About this article

### Cite this article

Hnětynková, I., Plešinger, M. & Žáková, J. Solvability classes for core problems in matrix total least squares minimization.
*Appl Math* **64, **103–128 (2019). https://doi.org/10.21136/AM.2019.0252-18

Received:

Published:

Issue Date:

### Keywords

- linear approximation problem
- core problem theory
- total least squares
- classification
- (ir)reducible problem

### MSC 2010

- 15A06
- 15A09
- 15A18
- 15A23
- 65F20