Computing the conditioning of the components of a linear least-squares solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least-squares problem. We recall theoretical results obtained in (SIAM J. Matrix Anal. Appl. 2007; 29(2):413-433) on conditioning of the least-squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities when the regression matrix and the right-hand side are perturbed. In particular, we show that in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right-hand side is exactly the condition number of this solution component when only perturbations on the right-hand side are considered. We explain how to compute the variance-covariance matrix and the least-squares conditioning using the libraries LAPACK (LAPACK Users' Guide (3rd edn). SIAM: Philadelphia, 1999) and ScaLAPACK (ScaLAPACK Users' Guide. SIAM: Philadelphia, 1997) and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace (Théorie Analytique des Probabilités. Mme Ve Courcier, 1820; 497-530) for computing the mass of Jupiter and a physical application if the area of space geodesy.