Condition numbers of the multidimensional total least squares problems having more than one solution
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Condition numbers of the multidimensional total least squares problems having more than one solution Lingsheng Meng1,2 · Bing Zheng2 · Yimin Wei3 Received: 1 September 2018 / Accepted: 22 July 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
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. Keywords Multidimensional total least squares · Condition numbers · Singular value decomposition · Upper bound estimates Mathematics Subject Classification (2010) 65F35 · 65F20
1 Introduction The total least squares (TLS) problem min E F F , subject to (A + E)X = B + F, E,F
(1.1)
where A ∈ Rm×n , B ∈ Rm×d (d ≥ 1), and X ∈ Rn×d are the parameters of interest, proved to be a realistic parameter estimation technique. It has been intensively Bing Zheng
[email protected] 1
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730070, China
3
School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai, 200433, China
Numerical Algorithms
studied and widely used in various applications since the early eighties due to the work of Golub and Van Loan on the reliable solution methods based on the singular value decomposition (SVD). The same technique is known in the system identification literature as the Koopmans-Levin method [10] and in the statistical literature as orthogonal regression [3]. We refer the readers to the monograph [15] for a good overview of theory, computational methods, and applications of the TLS problem. The TLS problem (1.1) with multiple right-hand sides (i.e., d > 1) is sometimes called the (classic) multidimensional TLS problem, and the case with a single righthand side (i.e., d = 1) being the basic TLS problem. However, unlike the ordinary least squares problems, the TLS problem (1.1) may have no solution, and even if there is a solution, it may not be unique [8, 15]. The solvability conditions of the TLS problem (1.1) have been investigated by many researchers [5, 8, 12, 15, 18]. According to their works, the solvability can be classified in three conditions (see C 1, C 2, and C 3 in Section 2 ). Most research works on the theory, computations, and applications of the TLS problem in the known literature, including its perturba
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