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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Å. Björck: Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics, Philadelphia, 1996.
J. R. Bunch, C. P. Nielsen: Updating the singular value decomposition. Numer. Math. 31 (1978), 111–129.
J. R. Bunch, C. P. Nielsen, D. C. Sorensen: Rank-one modification of the symmetric eigenproblem. Numer. Math. 31 (1978), 31–48.
R. D. Fierro, G. H. Golub, P. C. Hansen, D. P. O’Leary: Regularization by truncated total least squares. SIAM J. Sci. Comput. 18 (1997), 1223–1241.
G. H. Golub: Some modified matrix eigenvalue problems. SIAM Rev. 15 (1973), 318–334.
G. H. Golub, C. F. Van Loan: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17 (1980), 883–893.
G. H. Golub, C. F. Van Loan: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, 2013.
J. Hadamard: Le problème de Cauchy et les équations aux dérivées partielles linéaires hyperboliques. Hermann, Paris, 1932.
P. C. Hansen: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM Monographs on Mathematical Modeling and Computation 4, Society for Industrial and Applied Mathematics, Philadelphia, 1998.
P. C. Hansen: Discrete Inverse Problems. Insight and Algorithms. Fundamentals of Algorithms 7, Society for Industrial and Applied Mathematics, Philadelphia, 2010.
P. C. Hansen, J. G. Nagy, D. P. O’Leary: Deblurring Images. Matrices, Spectra, and Filtering. Fundamentals of Algorithms 3, Society for Industrial and Applied Mathematics, Philadelphia, 2006.
P. C. Hansen, V. Pereyra, G. Scherer: Least Squares Data Fitting with Applications, Johns Hopkins University Press, Baltimore. 2013.
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.
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.
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.
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.
F. Natterer: The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Chichester. 1986.
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.
X.-F. Wang: Total least squares problem with the arbitrary unitarily invariant norms. Linear Multilinear Algebra 65 (2017), 438–456.
M. Wei: Algebraic relations between the total least squares and least squares problems with more than one solution. Numer. Math. 62 (1992), 123–148.
M. Wei: The analysis for the total least squares problem with more than one solution. SIAM J. Matrix Anal. Appl. 13 (1992), 746–763.
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
- truncated total least squares
- multiple right-hand sides
- eigenvalues of rank-d update
- ill-posed problem
- filter factors