  In this paper, the normwise condition number of a linear function of the
equality constrained linear least squares solution called the partial condition
number is considered. Its expression and closed formulae are first presented
when the data space and the solution space are measured by the weighted
Frobenius norm and the Euclidean norm, respectively. Then, we investigate the
corresponding structured partial condition number when the problem is
structured. To estimate these condition numbers with high reliability, the
probabilistic spectral norm estimator and the small-sample statistical
condition estimation method are applied and two algorithms are devised. The
obtained results are illustrated by numerical examples.
