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

Abstract

ABSTRACTIn 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.