## Abstract

The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems *Ax* ≈ *b* were analyzed by Fierro, Golub, Hansen, and O’Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of *A* applied to *b*. This paper focuses on the situation when multiple observations *b*
_{1},..., *b*
_{
d
} are available, i.e., the T-TLS method is applied to the problem *AX* ≈ *B*, where *B* = [*b*
_{1},..., *b*
_{
d
}] is a matrix. It is proved that the filtering representation of the T-TLS solution can be generalized to this case. The corresponding filter factors are explicitly derived.

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This work has been partially supported by the GĂCR grant No. GA13-06684S. Iveta Hnětynková is a member of the University Center for Mathematical Modeling, Applied Analysis and Computational Mathematics (Math MAC). The research of Martin Plěsinger and Jana Žáková has been supported by the SGS grant of Technical University of Liberec No. 21161/2016.

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Hnětynková, I., Plešinger, M. & Žáková, J. Filter factors of truncated tls regularization with multiple observations.
*Appl Math* **62, **105–120 (2017). https://doi.org/10.21136/AM.2017.0228-16

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### Keywords

- truncated total least squares
- multiple right-hand sides
- eigenvalues of rank-
*d*update - ill-posed problem
- regularization
- filter factors

### MSC 2010

- 15A18
- 65F20
- 65F22
- 65F30