BIT Numerical Mathematics

, Volume 56, Issue 1, pp 293–307 | Cite as

Improved error bounds for floating-point products and Horner’s scheme

  • Siegfried M. RumpEmail author
  • Florian Bünger
  • Claude-Pierre Jeannerod


Let \(\mathbf{u}\) denote the relative rounding error of some floating-point format. Recently it has been shown that for a number of standard Wilkinson-type bounds the typical factors \(\gamma _k:=k\mathbf{u}/(1-k\mathbf{u})\) can be improved into \(k\mathbf{u}\), and that the bounds are valid without restriction on \(k\). Problems include summation, dot products and thus matrix multiplication, residual bounds for \(LU\)- and Cholesky-decomposition, and triangular system solving by substitution. In this note we show a similar result for the product \(\prod _{i=0}^k{x_i}\) of real and/or floating-point numbers \(x_i\), for computation in any order, and for any base \(\beta \geqslant 2\). The derived error bounds are valid under a mandatory restriction of \(k\). Moreover, we prove a similar bound for Horner’s polynomial evaluation scheme.


Floating-point product IEEE 754 standard  Wilkinson type error estimates Horner scheme 

Mathematics Subject Classification

65G50 65F05 



The authors wish to thank Marko Lange, Vincent Lefèvre, and Jean-Michel Muller for their fruitful comments on a preliminary version of this note. Moreover, our thanks to two anonymous referees for their thoughtful suggestions.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Siegfried M. Rump
    • 1
    • 2
    Email author
  • Florian Bünger
    • 1
  • Claude-Pierre Jeannerod
    • 3
  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany
  2. 2.Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  3. 3.Inria, Laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL)Université de LyonLyon Cedex 07France

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