Computing the Conditioning of the Components of a Linear Least Squares Solution

Baboulin, Marc and Dongarra, Jack and Gratton, Serge and Langou, Julien (2007) Computing the Conditioning of the Components of a Linear Least Squares Solution. [MIMS Preprint]

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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 [2] 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 right-hand side is exactly the condition number of this solution component when perturbations on the right-hand side are considered. We also provide fragment codes using LAPACK [1] routines to compute the variance-covariance 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 [19] for computing the mass of Jupiter and experiments from the space industry with real physical data.

Item Type:	MIMS Preprint
Additional Information:	Appears also as Technical Report ut-cs-07-604, Department of Computer Science, University of Tennessee, Knoxville, TN, USA, September 2007.
Uncontrolled Keywords:	Linear least squares, statistical linear least squares, parameter estimation, condition number, variance-covariance matrix, LAPACK, ScaLAPACK.
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis MSC 2010, the AMS's Mathematics Subject Classification > 68 Computer science
Depositing User:	Ms Lucy van Russelt
Date Deposited:	10 Oct 2007
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/855

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