On condition numbers of the total least squares problem with linear equality constraint

A contribution to condition numbers of the multidimensional total least squares problem with linear equality constraint

“…Since there expressions involve matrix Kronecker product operations which may make the computation more expensive, we provide compact upper bounds to enhance the computation efficiency. All expressions and upper bounds of these condition numbers generalize those for the single-dimensional TLSE problem [22] and multidimensional TLS problem [26].…”

“…Further perturbation results were given in [21], which provides perturbation analysis and tighter bounds for the forward error of the solution, when the perturbation in input data are of different magnitude. The condition numbers and perturbation results in [21,22] unify those for standard TLS problem [16,20,42,44]. When C, D are nonzero matrices and d = 1, under some condition (see (2.12) with σ n−p+1 = 0), the TLSE solution reduces to a solution to the least squares problem with equality constraint (LSE), whose perturbation results were studied in [4,5,40], that are also unified by the ones [21] for the TLSE problem.…”

“…Notice that φ ′ (c) is vital for above condition numbers, while a simple and Fréchet differentiable expression of φ (c) is not easy to derive. To get φ ′ (c), as did in [22], we start from the differentiability of the weighted TLS solution X t(ε) by defining the mapping for the multidimensional WTLS problem…”

“…Remark 4.1 When p > 0, k = n − p and d = 1, for the single dimensional TLSE problem, Liu and Jia [22] derived the first order perturbation estimate as…”

“…The condition number of the truncated TLS solution of an ill-conditioned TLS problem was studied by Gratton, Titley-Peloquin, and Ilunga [11], Meng, Diao and Bai [24]. By making use of the perturbation results in [1,16,20], and the close relation of the single dimensional TLSE to an unconstrained weighted TLS problems, Liu and Jia [22] derived closed formulae for condition numbers of the single dimensional TLSE problem. Further perturbation results were given in [21], which provides perturbation analysis and tighter bounds for the forward error of the solution, when the perturbation in input data are of different magnitude.…”