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Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse

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Abstract

The singular value decomposition and spectral norm of a matrix are ubiquitous in numerical analysis. They are extensively used in proofs, but usually it is not necessary to compute them. However, there are some important applications in the realm of verified error bounds for the solution of ordinary and partial differential equations where reasonably tight error bounds for the spectral norm of a matrix are mandatory. We present various approaches to this together with some auxiliary useful estimates.

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Authors and Affiliations

  1. Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg, 21071, Germany

    Siegfried M. Rump

  2. Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, 169-8555, Tokyo, Japan

    Siegfried M. Rump

  3. Laboratoire LIP6, Département Calcul Scientifique, Université Pierre et Marie Curie (Paris 6), 4 place Jussieu, 75252, Paris cedex 05, France

    Siegfried M. Rump

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  1. Siegfried M. Rump
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Correspondence to Siegfried M. Rump.

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Communicated by Axel Ruhe.

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Rump, S.M. Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse. Bit Numer Math 51, 367–384 (2011). https://doi.org/10.1007/s10543-010-0294-0

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  • DOI: https://doi.org/10.1007/s10543-010-0294-0

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