Backward error and condition number analysis for the indefinite linear least squares problem

Abstract

In this paper, we concentrate on the backward error and condition number of the indefinite least squares problem. For the normwise backward error of the indefinite least square problem, we adopt the linearization method to derive the tight estimations for the exact normwise backward errors. Using the dual techniques of condition number theory [1], we derive the explicit expressions of the mixed and componentwise condition numbers for the linear function of the solution for the indefinite least squares problem. The tight upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical power method for estimating matrix 1-norm [18, Chapter 15] during using the QR-Cholesky method [4] for solving the indefinite least squares problem. The numerical examples show that the derived condition numbers can give sharp perturbation bound with respect to the interested component of the solution. And the linearization estimations are effective for the normwise backward errors.

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

“…Note that the condition number theory of ILS problem has been studied [29,27,12]. In this paper we only present a new result on its 2-norm projected condition number.…”

“…Diao and Zhou [12] used the dual techniques to recover the explicit expressions of mixed and componentwise condition numbers of the ILS problem. 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].…”

“…Based on the results on the projected condition numbers of the EILS problem, we present some interesting new findings on the condition number theory of some specific LS problems. Note that the condition number theory of ILS problem has been studied [29,27,12]. In this paper we only present a new result on its 2-norm projected condition number.…”

“…and ) is still more compact than (3.8) with the assumption that m > n. The explicit expressions of the projected mixed and componentwise condition numbers and its upper bounds have been given in [27]. Diao and Zhou [12] used the dual techniques to recover the explicit expressions of mixed and componentwise condition numbers of the ILS problem. 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.…”

Splitting-based randomized iterative methods for solving indefinite least squares problem

“…The ILS problem has found many applications in some fields, such as the total least squares problems (see, e.g., Golub & Van Loan, 1980;Van Huffel & Vandewalle, 1991) and H ∞ smoothing (Hassibi et al, 1993). Extensive works on computations, perturbation analysis, and applications of this problem have been published (see, e.g., Chandrasekaran et al, 1998;Bojanczyk et al, 2003a;Xu, 2004;Liu & Li, 2011;Liu & Zhang, 2013;Liu & Liu, 2014;Li et al, 2014;Li & Wang, 2018;Diao & Zhou, 2019;Song, 2020;Bojanczyk, 2021). In this paper, we mainly focus on its numerical methods.…”