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Error estimation of floating-point summation and dot product

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Abstract

We improve the well-known Wilkinson-type estimates for the error of standard floating-point recursive summation and dot product by up to a factor 2. The bounds are valid when computed in rounding to nearest, no higher order terms are necessary, and they are best possible. For summation there is no restriction on the number of summands. The proofs are short by using a new tool for the estimation of errors in floating-point computations which cures drawbacks of the “unit in the last place (ulp)”. The presented estimates are nice and simple, and closer to what one may expect.

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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, Tokyo, 169-8555, Japan

    Siegfried M. Rump

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

Additional information

Communicated by Axel Ruhe.

Part of this research was done while being visiting Professor at Université Pierre et Marie Curie (Paris 6), Laboratoire LIP6, Département Calcul Scientifique, 4 place Jussieu, 75252 Paris cedex 05, France.

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Rump, S.M. Error estimation of floating-point summation and dot product. Bit Numer Math 52, 201–220 (2012). https://doi.org/10.1007/s10543-011-0342-4

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  • DOI: https://doi.org/10.1007/s10543-011-0342-4

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