A note on the condition number of the scaled total least squares problem

Abstract

In this paper, we consider the explicit expressions of the normwise condition number for the scaled total least squares problem. Some techniques are introduced to simplify the expression of the condition number, and some new results are derived. Based on these new results, new expressions of the condition number for the total least squares problem can be deduced as a special case. New forms of the condition number enjoy some storage and computational advantages. We also proposed three different methods to estimate the condition number. Some numerical experiments are carried out to illustrate the effectiveness of our results.Keywords: condition number, Fréchet derivative, the scaled total least squares problem, power method, probabilistic condition estimation methodwhere λ is a positive real number, · F denotes the Frobenius norm and R(·) is the range space. Let [E S r S ] be the solution to (1.1), then the solution to the linear system (A + E S )λx = λb − r S is called the STLS solution and denoted by x S . As shown in [2], when λ = 1, λ → 0 and λ → ∞, x S becomes the TLS solution x ⊺ , OLS solution x O and DLS solution x D , respectively. The condition number gives a quantitative measurement of the maximum amplification of the resulting change in solution with respect to a perturbation in the data and has been extensively studied for too many topics to list here. For the STLS problem, Zhou et al. [3] considered its perturbation analysis and presented the normwise, mixed and componentwise condition numbers. Based on the ✩

A contribution to the conditioning of the mixed least-squares scaled total least-squares problem

“…Condition numbers measure the worst-case sensitivity of a solution of a problem with respect to small perturbations in the input data. The condition numbers of the TLS problem, the STLS problem, and the MTLS problem have been studied widely, e.g., by Zhou et al [16], Baboulin and Gratton [1], Li and Jia [5,4], Zheng et al [14], Wang et al [12], Zheng and Yang [15]. Recently, Zhang and Wang [13] studied a closed formula for a first-order perturbation estimate of the MLSSTLS solution and gave explicit expressions for the condition numbers of the MLSSTLS problem.…”

On the Condition Number Theory of the Equality Constrained Indefinite Least Squares Problem

“…15] can be used to estimate the 2-norm projected condition number and the upper bounds of the projected mixed and componentwise condition numbers. The corresponding algorithms can be easily derived similar to [37] and [11] in estimating the condition numbers of the TLS problem. These condition number estimation methods have been well developed and can be adapted to our settings without any technical difficulty.…”

“…Thus, considering the relationship between the EILS and the ILS problems, we may say that the results given in [29,27,12] can be treated as special cases of our work. Moreover, based on the relationship between the ILS and the TLS problems [8], Li and Wang [27] also established the condition number of the TLS problem but they did not give the compact forms, which were later given in [37]. More results on the condition number theory of the TLS problem can be found in [1,24,25,44].…”

Structured condition numbers for the total least squares problem with linear equality constraint and their statistical estimation

“…Condition number plays an important role in perturbation theory and error analysis for algorithms; see e.g., [8,9,10]. Recently, the condition numbers of TLS problem, the scaled TLS problem, the multidimensional TLS problem, the mixed LS-TLS problem, truncated-TLS problem and TLSE problem have been considered; see [19,20,21,22,23,24,25,26,27]. Structured TLS problems [11,12,13] had been studied extensively in the past decades.…”