  In this paper, we concentrate on the backward error and condition number of
the indefinite least squares problem. For the normwise backward error of the
indefinite least square problem, we adopt the linearization method to derive
the tight estimations for the exact normwise backward errors. Using the dual
techniques of condition number theory \cite{22.0}, we derive the explicit
expressions of the mixed and componentwise condition numbers for the linear
function of the solution for the indefinite least squares problem. 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 \cite[Chapter 15]{Higham2002Book} during
using the QR-Cholesky method \cite{1.0} for solving the indefinite 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.
