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

ABSTRACT

In this paper, we consider the backward error and condition number of the indefinite linear least squares (ILS) problem. For the normwise backward error of ILS, we adopt the linearization method to derive the tight estimations for the exact normwise backward errors. We derive the explicit expressions of the normwise, mixed and componentwise condition numbers for the linear function of the solution for ILS. 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 [N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, PA, 2002, Chapter 15] during using the QR-Cholesky method [S. Chandrasekaran, M. Gu, and A.H. Sayed, A stable and efficient algorithm for the indefinite linear least-squares problem, SIAM J. Matrix Anal. Appl. 20(2) (1999), pp. 354–362] for solving ILS. Moreover, we revisit the previous results on condition numbers for ILS and linear 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.

KEYWORDS:

• Indefinite least squares
• normwise backward error
• linearization estimate
• componentwise perturbation
• condition number
• power method

2010 AMS SUBJECT CLASSIFICATIONS:

• 15A09
• 15A12
• 65F35

Acknowledgements

The author would like to thank the referees for their constructive comments, which led to improvements of our manuscript. One referee reminds us to pay attention to the following papers [1,3,6,8,9,41].

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Fundamental Research Funds for the Central Universities under the grant 2412017FZ007.