×

zbMATH — the first resource for mathematics

The revisited total least squares problems with linear equality constraint. (English) Zbl 1441.65044
Summary: The total least squares problem with linear equality constraint is proved to be approximated by an unconstrained total least squares problem with a large weight on the constraint. A criterion for choosing the weighting factor is given, and a QR-based inverse (QR-INV) iteration method is presented. Numerical results show that the QR-INV method is more efficient than the standard QR-SVD procedure and Schaffrin’s inverse iteration method, especially for large and sparse matrices.

MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F50 Computational methods for sparse matrices
Software:
VanHuffel
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Björck, Å.; Heggernes, P.; Matstoms, P., Methods for large scale total least squares problems, SIAM J. Matrix Anal. Appl., 22, 413-429 (2000) · Zbl 0974.65037
[2] Dowling, E. M.; Degroat, R. D.; Linebarger, D. A., Total least squares with linear constraints, IEEE Int. Conf. Acoust., 5, 5, 341-344 (1992)
[3] Dutta, A.; Li, X., On a problem of weighted low-rank approximation of matrices, SIAM J. Matrix Anal. Appl., 38, 2, 530-553 (2017) · Zbl 1365.65110
[4] Golub, G. H.; Van Loan, C. F., An analysis of total least squares problem, SIAM J. Matrix Anal. Appl., 17, 6, 883-893 (1980) · Zbl 0468.65011
[5] Golub, G. H.; Van Loan, C. F., Matrix Computations (2013), Johns Hopkins University Press: Johns Hopkins University Press Baltimore · Zbl 1268.65037
[6] Hermus, K.; Verhelst, W.; Lemmerling, P.; Wambacq, P.; Van Huffel, S., Perceptual audio modeling with exponentially damped sinusoids, Signal Process., 85, 163-176 (2005) · Zbl 1148.94332
[7] Lemmerling, P.; De Moor, B., Misfit versus latency, Automatica, 37, 2057-2067 (2001) · Zbl 1010.93007
[8] Lemmerling, P.; Mastronardi, N.; Van Huffel, S., Efficient implementation of a structured total least squares based speech compression method, Linear Algebra Appl., 366, 295-315 (2003) · Zbl 1020.94506
[9] Levin, M., Estimation of a system pulse transfer function in the presence of noise, IEEE Trans. Autom. Control, 9, 229-235 (1964)
[10] Liu, Q.; Wang, M., On the weighting method for mixed least squares-total least squares problems, Numer. Linear Algebra Appl., 24, 5, Article e2094 pp. (2017) · Zbl 1438.65072
[11] Markovsky, I.; Van Huffel, S., Overview of total least squares methods, Signal Process., 87, 2283-2302 (2007) · Zbl 1186.94229
[12] Ng, M.; Plemmons, R.; Pimentel, F., A new approach to constrained total least squares image restoration, Linear Algebra Appl., 316, 237-258 (2000) · Zbl 0960.65044
[13] Ng, M.; Bose, N.; Koo, J., Constrained total least squares for color image reconstruction, (Total Least Squares and Errors-in-Variables Modelling III: Analysis, Algorithms and Applications (2002), Kluer Academic Publishers), 365-374 · Zbl 1008.68151
[14] Nocedal, J.; Wright, S. J., Numerical Optimization (2006), Springer: Springer New York · Zbl 1104.65059
[15] Pearson, K., On lines and planes of closest fit to points in space, Philos. Mag., 2, 559-572 (1901) · JFM 32.0710.04
[16] Rhode, S.; Usevich, K.; Markovsky, I.; Gauterin, F., A recursive restricted total least-squares algorithm, IEEE Trans. Signal Process., 62, 21, 5652-5662 (2014) · Zbl 1394.94485
[17] Schaffrin, B., A note on constrained total least squares estimation, Linear Algebra Appl., 417, 245-258 (2006) · Zbl 1117.62061
[18] Schaffrin, B.; Felus, Y., An algorithmic approach to the total least-squares problem with linear and quadratic constraints, Stud. Geophys. Geod., 53, 1, 1-16 (2009)
[19] Stewart, G., On the weighting method for least squares problems with linear equality constraints, BIT Numer. Math., 37, 4, 961-967 (1997) · Zbl 0893.65024
[20] Stewart, G.; Sun, J., Matrix Perturbation Theory (1990), Academic Press
[21] (Van Huffel, S., Recent Advances in Total Least Squares Techniques and Errors-in Variables Modeling. Recent Advances in Total Least Squares Techniques and Errors-in Variables Modeling, Proceedings of the Second International Workshop on Total Least Squares Techniques and Errors-in-Variables Modeling (1997), SIAM: SIAM Philadelphia) · Zbl 0861.00018
[22] (Van Huffel, S.; Lemmerling, P., Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications (2002), Kluwer: Kluwer Dordrecht, Boston, London) · Zbl 0984.00011
[23] Van Huffel, S.; Vandevalle, J., The Total Least Squares Problems: Computational Aspects and Analysis, Frontiers in Applied Mathematics, vol. 9 (1991), SIAM: SIAM Philadelphia
[24] Van Huffel, S.; Zha, H., Restricted total least squares problem: formulation, algorithm, and properties, SIAM J. Matrix Anal. Appl., 12, 292-309 (1991) · Zbl 0732.65041
[25] Van Loan, C. F., On the method of weighting for equality-constrained least-squares problems, SIAM J. Numer. Anal., 22, 5, 851-864 (1985) · Zbl 0584.65015
[26] Yan, S.; Fan, S., The solution set of the mixed LS-TLS problem, Int. J. Comput. Math., 77, 4, 545-561 (2001) · Zbl 0986.65039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.