BIT Numerical Mathematics

, Volume 57, Issue 3, pp 927–941 | Cite as

Error estimates for the summation of real numbers with application to floating-point summation

  • Marko Lange
  • Siegfried M. Rump


Standard Wilkinson-type error estimates of floating-point algorithms involve a factor \(\gamma _k:=k\mathbf {u}/(1-k\mathbf {u})\) for \(\mathbf {u}\) denoting the relative rounding error unit of a floating-point number system. Recently, it was shown that, for many standard algorithms such as matrix multiplication, LU- or Cholesky decomposition, \(\gamma _k\) can be replaced by \(k\mathbf {u}\), and the restriction on k can be removed. However, the arguments make heavy use of specific properties of both the underlying set of floating-point numbers and the corresponding arithmetic. In this paper, we derive error estimates for the summation of real numbers where each sum is afflicted with some perturbation. Recent results on floating-point summation follow as a corollary, in particular error estimates for rounding to nearest and for directed rounding. Our new estimates are sharp and unveil the necessary properties of floating-point schemes to allow for a priori estimates of summation with a factor omitting higher order terms.


Floating-point Summation Wilkinson-type error estimates Error analysis Real numbers 

Mathematics Subject Classification

65G50 65F05 



The authors would like to thank the anonymous referees for their valuable suggestions which helped us improving this paper.


  1. 1.
    ANSI/IEEE 754-1985: IEEE Standard for Binary Floating-Point Arithmetic. New York (1985)Google Scholar
  2. 2.
    ANSI/IEEE 754-2008: IEEE Standard for Floating-Point Arithmetic. New York (2008)Google Scholar
  3. 3.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Jeannerod, C.-P., Rump, S.M.: Improved error bounds for inner products in floating-point arithmetic. SIAM J. Matrix Anal. Appl. (SIMAX) 34(2), 338–344 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jeannerod, C.-P., Rump, S.M.: On relative errors of floating-point operations: optimal bounds and applications. Preprint (2014)Google Scholar
  6. 6.
    Knuth, D.E.: The Art of Computer Programming: Seminumerical Algorithms, vol. 2, 3rd edn. Addison Wesley, Reading (1998)zbMATHGoogle Scholar
  7. 7.
    Ozaki, K., Ogita, T., Bünger, F., Oishi, S.: Accelerating interval matrix multiplication by mixed precision arithmetic. Nonlinear Theory Appl. IEICE 6(3), 364–376 (2015)CrossRefGoogle Scholar
  8. 8.
    Rump, S.M.: Error estimation of floating-point summation and dot product. BIT Numer. Math. 52(1), 201–220 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rump, S.M., Jeannerod, C.-P.: Improved backward error bounds for LU and Cholesky factorizations. SIAM J. Matrix Anal. Appl. (SIMAX) 35(2), 684–698 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rump, S.M., Lange, M.: On the definition of unit roundoff. BIT Numer. Math. 56(1), 309–317 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Faculty of Science and EngineeringWaseda UniversityTokyoJapan
  2. 2.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany

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