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Error estimates for the regularization of least squares problems

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Abstract

The a posteriori estimate of the errors in the numerical solution of ill-conditioned linear systems with contaminated data is a complicated problem. Several estimates of the norm of the error have been recently introduced and analyzed, under the assumption that the matrix is square and nonsingular. In this paper we study the same problem in the case of a rectangular and, in general, rank-deficient matrix. As a result, a class of error estimates previously introduced by the authors (Brezinski et al., Numer Algorithms, in press, 2008) are extended to the least squares solution of consistent and inconsistent linear systems. Their application to various direct and iterative regularization methods are also discussed, and the numerical effectiveness of these error estimates is pointed out by the results of an extensive experimentation.

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Authors and Affiliations

  1. Laboratoire Paul Painlevé, UMR CNRS 8524, Université des Sciences et Technologies de Lille, 59655, Villeneuve d’Ascq Cedex, France

    C. Brezinski

  2. Dipartimento di Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    G. Rodriguez & S. Seatzu

Authors
  1. C. Brezinski
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  2. G. Rodriguez
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  3. S. Seatzu
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Corresponding author

Correspondence to G. Rodriguez.

Additional information

This work was supported by MIUR under the PRIN grant no. 2006017542-003, and the University of Cagliari.

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Brezinski, C., Rodriguez, G. & Seatzu, S. Error estimates for the regularization of least squares problems. Numer Algor 51, 61–76 (2009). https://doi.org/10.1007/s11075-008-9243-2

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  • DOI: https://doi.org/10.1007/s11075-008-9243-2

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