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Numerical Algorithms
Published in: Numerical Algorithms 3/2020

06-08-2019 | Original Paper

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

Authors: Lingsheng Meng, Bing Zheng, Yimin Wei

Published in: Numerical Algorithms | Issue 3/2020

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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.
previous article An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems
next article Reconstruction algorithms of an inverse source problem for the Helmholtz equation

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Literature
1.
Baboulin, M., Gratton, S.: A contribution to the conditioning of the total least-squares problem. SIAM J. Matrix Anal. Appl. 32, 685–699 (2011)CrossRefMathSciNetMATH
2.
Cline, R. E.: Representations for the generalized inverse of a partitioned matrix. J. Soc. Indust. Appl. Math. 12, 588–600 (1964)CrossRefMathSciNetMATH
3.
4.
Gohberg, I., Koltracht, I.: Mixed, componentwise, and structured condition numbers. SIAM J. Matrix Anal. Appl. 14, 688–704 (1993)CrossRefMathSciNetMATH
5.
Golub, G.H., Van Loan, C. F.: Matrix computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)MATH
6.
Graham, A.: Kroneckor products and matrix calculas with application. Wiley, New York (1981)
7.
Gratton, S., Titley-Peloquin, D., Ilunga, J. T.: Sensitivity and conditioning of the truncated total least squares solution. SIAM J. Matrix Anal. Appl. 34, 1257–1276 (2013)CrossRefMathSciNetMATH
8.
Hnětynková, I., Plešinger, M., Maria Sima, D., Strakoš, Z., Van Huffel, S.: The total least squares problem in AX ≈ B: a new classification with the relationship to the classical works. SIAM J. Matrix Anal. Appl. 32, 748–770 (2011)CrossRefMathSciNetMATH
9.
10.
Levin, M. J.: Estimation of a system pulse transfer function in the presence of noise. IEEE Trans. Automat. Control 9, 229–235 (1964)CrossRef
11.
Li, B., Jia, Z.: Some results on condition numbers of the scaled total least squares problem. Linear Algebra Appl. 435, 674–686 (2011)CrossRefMathSciNetMATH
12.
Liu, X.: On the solvability and perturbation analysis of the TLS problem (in Chinese). Acta Math. Appl. Sin. 19, 255–262 (1996)
13.
14.
Stewart, G.W., Sun, J. G.: Matrix perturbation theory. Academic Press, Boston (1990)MATH
15.
Van Huffel, S., Vandewalle, J.: The total least squares problem: Computational aspects and analysis. SIAM, Philadelphia (1991)CrossRefMATH
16.
Wei, M.: The analysis for the total least squares problem with more than one solution. SIAM J. Matrix Anal. Appl. 13, 746–763 (1992)CrossRefMathSciNetMATH
17.
Wei, M.: Algebraic relations between the total least squares and least squares problems with more than one solution. Numer. Math. 62, 123–148 (1992)CrossRefMathSciNetMATH
18.
19.
Zheng, B., Meng, L., Wei, Y.: Condition numbers of the multidimensional total least squares problem. SIAM J. Matrix Anal. Appl. 38, 924–948 (2017)CrossRefMathSciNetMATH
20.
Zhou, L., Lin, L., Wei, Y., Qiao, S.: Perturbation analysis and condition numbers of scaled total least squares problems. Numer. Algorithms 51, 381–399 (2009)CrossRefMathSciNetMATH
Metadata
Title
Condition numbers of the multidimensional total least squares problems having more than one solution
Authors
Lingsheng Meng
Bing Zheng
Yimin Wei
Publication date
06-08-2019
Publisher
Springer US
Published in
Numerical Algorithms / Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-019-00785-9

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