On the partial condition numbers for the indefinite least squares problem

Abstract

The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be called the unified condition number, we first present an expression of the partial unified condition number when the data space is measured by a general weighted product norm. Then, by setting the specific norms and weight parameters, we obtain the expressions of the partial normwise, mixed and componentwise condition numbers. Moreover, the corresponding structured partial condition numbers are also taken into consideration when the problem is structured. Considering the connections between the indefinite and total least squares problems, we derive the (structured) partial condition numbers for the latter, which generalize the ones in the literature. To estimate these condition numbers effectively and reliably, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and three related algorithms are devised. Finally, the obtained results are illustrated by numerical experiments.

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

“…Yang [13] presented a unified definition of condition number to cope with the conditioning of equality constrained indefinite least squares problem, which include the normwise, mixed and componentwise condition numbers as its special cases, for further discussions see [13,14].…”

“…With a little algebra, we can check that κ OLSF (A, b) can be rewritten as follows [16,17] [14] in investigating the condition number for indefinite least squares problem.…”

“…This method is based on the probabilistic spectral norm estimator proposed by Hochstenbach in [23], which gives an interval containing the spectral norm of a matrix with high probability. The PCE method has been used to estimate the condition number of various problems (see [14,24]). For the convenience of presentation, we summarize the method given in [23] as the following lemma.…”

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

“…Also the linearization estimate for the normwise backward error was obtained [36]. In 2016, the condition numbers for a linear function of the solution for ILS also named as "partial condition numbers for ILS " [20] is studied. The explicit expressions for these condition numbers are derived.…”

“…The explicit expressions for these condition numbers are derived. Also the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are proposed to estimate condition numbers [20], but the authors do not take account of the numerical method for ILS to reduce the computational complexity of condition estimations. Usually, in the field of condition estimations [18,Chapter 15] in numerical linear algebra, how to incorporate condition estimations to the numerical method by utilizing the already computed matrix decompositions is crucial, thus the computational complexity during condition estimations can be reduced when the already computed matrix decompositions are used to devise the algorithms for condition estimations.…”