Mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem

Abstract

In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques. The sharp upper bounds for the derived mixed and componentwise condition numbers are obtained. For the structured TLS problem, we consider the structured perturbation analysis and obtain the corresponding expressions of the mixed and componentwise condition numbers. We prove that the structured ones are smaller than their corresponding unstructured ones based on the derived expressions. Moreover, we point out that the new derived expressions can recover the previous results on the condition analysis for the TLS problem.The numerical examples show that the derived condition numbers can give sharp perturbation bounds, on the other hand normwise condition numbers can severely overestimate the relative errors because normwise condition numbers ignore the data sparsity and scaling. Meanwhile, from the observations of numerical examples, it is more suitable to adopt structured condition numbers to measure the conditioning for the structured TLS problem.

Condition numbers for the truncated total least squares problem and their estimations

“…Recently, Diao and Sun [13] defined and gave mixed and componentwise condition numbers for the linear function Proof. From the proof of Reference [22], lemma 2, for P given by (33) we have…”

“…Reference [11]). The Kronecker product-based formulas for the mixed and componentwise condition numbers to the TLS problem (2) were derived in References [12,13]. The corresponding componentwise perturbation analysis for the multidimensional TLS problem and mixed least squares-TLS problem can be found in References [14,15].…”

“…For structured TLS problems, it is suitable to investigate structured perturbations on the input data, because structure-preserving algorithms that preserve the underlying matrix structure can enhance the accuracy and efficiency of the TLS solution computation. Structured condition numbers for structured TLS problems can be found in References [13,21,22] and references therein. In this paper, we introduce structured perturbation analysis for the structured TTLS (STTLS) problem.…”

Condition Numbers for a Linear Function of the Solution to the Constrained and Weighted Least Squares Problem and Their Statistical Estimation

“…This technique was proposed by Baboulin and Gratton [2] for the mixed and componentwise condition numbers of least squares problem. Later, Diao et al applied this technique to weighted least squares problem [9] and the total least squares problem [10]. In addition, we consider the statistical estimation of these condition numbers by the probabilistic spectral norm estimator [25] and the small-sample statistical condition estimation (SSCE) method [29].…”

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

“…For structured TLS problems, it is suitable to investigate structured perturbations on the input data, because structure-preserving algorithms that preserve the underlying matrix structure can enhance the accuracy and efficiency of the TLS solution computation. Recently, structured condition numbers of TLS problem, the mixed LS-TLS problem, and truncated-TLS problem have been considered; see [14,15,16,17,18] However, to our best knowledge, there is no work on structured perturbation analysis and structured condition numbers of TLSE problem so far. Therefore, in this paper, we study structured condition numbers of TLSE problem, including the normwise, mixed, and componentwise condition number and their relationships corresponding to the unstructured ones.…”