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Numerische Mathematik

, Volume 134, Issue 3, pp 473–512 | Cite as

Structured eigenvalue condition numbers for parameterized quasiseparable matrices

  • Froilán M. DopicoEmail author
  • Kenet Pomés
Article

Abstract

The development of fast algorithms for performing computations with \(n\times n\) low-rank structured matrices has been a very active area of research during the last two decades, as a consequence of the numerous applications where these matrices arise. The key ideas behind these fast algorithms are that low-rank structured matrices can be described in terms of O(n) parameters and that these algorithms operate on the parameters instead on the matrix entries. Therefore, the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer, since this determines the maximum accuracy of a given fast computation. In other words, it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low-rank structured matrices, but, as far as we know, this has not yet been accomplished in any case. In this paper, we derive structured relative eigenvalue condition numbers for the important class of low-rank structured matrices known as \(\{1;1\}\)-quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the Givens-vector representations of these matrices, and we provide fast algorithms for computing them. Comparisons among the new structured condition numbers and the unstructured one are also presented, as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge. In addition, the approach presented in this paper is general and may be extended to other problems and classes of low-rank structured matrices.

Mathematics Subject Classification

65F15 65F35 15A12 15A18 

Notes

Acknowledgments

The authors sincerely thank two anonymous referees and the Associate Editor handling this manuscript (Prof. Ilse Ipsen) for their extremely detailed readings of this long and technical paper, for correcting many grammatical typos, and for providing constructive and insightful comments that have contributed to a significant improvement of the original manuscript.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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