Computing the Conditioning of the Components of a Linear Least Squares Solution
Abstract
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 Arioli, Baboulin and Gratton, SIMAX 29(2):413433, 2007, 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. 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 righthand side is exactly the condition number of this solution component when perturbations on the righthand side are considered. We also provide fragment codes using LAPACK routines to compute the variancecovariance matrix and the least squares conditioning and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace in Theorie Analytique des Probabilites, 1820, for computing the mass of Jupiter and experiments from the space industry with real physical data.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.0829
 Bibcode:
 2007arXiv0710.0829B
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Statistics