On the partial condition numbers for the indefinite least squares problem
For numerical analysts, this work provides a unified theoretical framework for condition numbers of indefinite least squares, but the contribution is incremental as it generalizes existing results.
The paper derives expressions for partial condition numbers of the indefinite least squares problem under general weighted norms, and extends them to structured problems and total least squares. Numerical experiments demonstrate the effectiveness of proposed estimation algorithms.
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, the expression of the partial unified condition number is first presented when the data space is measured by the 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, whose expressions are given. 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.