Advertisement

Reliable Computing

, Volume 7, Issue 4, pp 321–340 | Cite as

Automatic Forward Error Analysis for Floating Point Algorithms

  • Walter Krämer
  • Armin Bantle
Article

Abstract

We investigate absolute and relative error bounds for floating point calculations determined by means of sequences of instructions (as, for example, given by a computer program). We get rigorous error bounds on the round-off or generated error due to the actual machine floating point operations, as well as the propagated error from one sequence to the next in a very convenient way by the computer itself. The results stated in the theorems can be used to implement software tools for the automatic computation of a priori worst case error bounds for floating point computations. These automatically computed bounds are valid simultaneously for all data vectors varying in the domain specified and their corresponding machine vectors fulfilling a maximum prescribed error bound.

With great success we have used our method in the past to implement a fast interval library for elementary functions called FI_LIB [12]. Further numerical examples often show a high quality of the computed a priori bounds.

Keywords

Software Tool Elementary Function Error Bound Great Success Case Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alefeld, G. and Herzberger, J.: Einführung in die Intervallrechnung, Bibliographisches Institut, Mannheim, 1974.Google Scholar
  2. 2.
    Bantle, A. and Krämer, W.: Ein Kalkül f ür verläßliche absolute und relative Fehlerabsch ätzungen, Preprint 98/5 des IWRMM, Universität Karlsruhe, 1998.Google Scholar
  3. 3.
    Bauer, F. L.: Computational Graphs and Rounding Error, SIAM J. Numer. Anal. 11 (1) (1974), pp.87-96.Google Scholar
  4. 4.
    Blomquist, F. and Krämer, W.: Algorithmen mit garantierten Fehlerschranken für die Fehlerund die komplementäre Fehlerfunktion, Preprint 97/3 desIWRMM, Universität Karlsruhe, 1997.Google Scholar
  5. 5.
    Carr, J.W. III: Error Analysis in Floating Point Arithmetic, Comm. ACM 2 (5) (1959), pp.10-15.Google Scholar
  6. 6.
    Demmel, J. W.: Underflow and the Reliability of Numerical Software, SIAM J. Sci. Statist. Comput. 5 (4) (1984), pp.887-919.Google Scholar
  7. 7.
    Ferguson, Jr, W. E.: Exact Computation of a Sum or Difference with Applications to Argument Reduction, in: Knowles, S. and McAllister, W. H. (eds), Proc. 12th IEEE Symposium on Computer Arithmetic, Bath, England, IEEE Computer Society Press, Los Alamitos, CA, USA, 1995, pp. 216-221.Google Scholar
  8. 8.
    Fischer, H.-Ch.: Schnelle automatische Differentiation, Einschließungsmethoden und Anwendungen, Dissertation, Universität Karlsruhe, 1990.Google Scholar
  9. 9.
    Higham, N. J.: Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1996.Google Scholar
  10. 10.
    Hofschuster, W. and Krämer, W.: Ein rechnergestützter Fehlerkalkül mit Anwendung auf ein genaues Tabellenverfahren, Preprint 96/5 des IWRMM, Universität Karlsruhe, 1996.Google Scholar
  11. 11.
    Hofschuster, W. and Krämer, W.: A Computer Oriented Approach to Get Sharp Reliable Error Bounds, Reliable Computing 3 (3) (1997), pp. 239–248.Google Scholar
  12. 12.
    Hofschuster, W. and Krämer, W.: FI LIB, eine schnelle und portable Funktionsbibliothek für reelle Argumente und reelle Intervalle im IEEE-Double-Format, Preprint 98/5 des IWRMM, Universität Karlsruhe, 1998.Google Scholar
  13. 13.
    IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard 754–1985, Institute of Electrical and Electronics Engineers, New York, 1985. Reprinted in SIGPLAN Notices 22 (2), (1987), pp. pp 9–25.Google Scholar
  14. 14.
    Iri, M.: Simultaneous Computation of Functions, Partial Derivatives and Error Estimates of Rounding Errors, Japan J. Appl. Math. 1 (1984), pp 223–252.Google Scholar
  15. 15.
    Klatte, R., Kulisch, U., Lawo, C., Rauch, M., and Wiethoff, A.: C-XSC: A C++ Class Library for Extended Scientific Computing, Springer-Verlag, Heidelberg, 1993.Google Scholar
  16. 16.
    Krämer, W.:A Priori Worst Case Error Bounds for Floating-Point Computations, in: Proceedings of the 13th IEEE Symp. on Computer Arithmetic, Asilomar, California, 1997, pp.64–71. Also in: IEEE Transactions on Computers 47 (7) (1998).Google Scholar
  17. 17.
    Krämer, W.: Constructive Error Analysis, Journal of Universal Computer Science (JUCS) 4 (2) (1998), pp.147–163.Google Scholar
  18. 18.
    Kulisch, U.: Grundlagen des Numerischen Rechnens, Bibliographisches Institut, Mannheim,1976.Google Scholar
  19. 19.
    Miller, W.: Software for Roundoff Analysis, ACM Trans. Math. Software 1 (2) (1975), pp.108–128.Google Scholar
  20. 20.
    Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, 1990.Google Scholar
  21. 21.
    Priest, D. M.: On Properties of Floating Point Arithmetics: Numerical Stability and the Cost of Accurate Computations, Ph.D. thesis, Mathematics Department, University of California, Berkeley, CA, USA, 1992, ftp://ftp.icsi.berkeley.edu/pub/theory/priest-thesis.ps.Z.Google Scholar
  22. 22.
    Richman, P. L.: Automatic Error Analysis for Determining Precision, Comm. ACM 15 (9) (1972), pp.813–817.Google Scholar
  23. 23.
    Scherer, R. and Zeller, K.: Shorthand Notation for Rounding Errors, Computing Suppl. 2 (1980), pp.165–168.Google Scholar
  24. 24.
    Schumacher, G.: Genauigkeitsfragen bei algebraisch-numerischen Algorithmen auf Skalar-und Vektorrechnern, Dissertation, Universität Karlsruhe, 1989.Google Scholar
  25. 25.
    Sterbenz, P. H.: Floating-Point Computation, Prentice-Hall, Englewood Cliffs, NJ, 1974.Google Scholar
  26. 26.
    Stummel, F.: Rounding Error Analysis of Elementary Numerical Algorithms, Computing. Suppl. 2 (1980), pp.169–195.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Walter Krämer
    • 1
  • Armin Bantle
    • 1
  1. 1.Wessenschaftliches Rechnen / Softwaretechnologie, Bergische Universität GH WuppertalWuppertalGermany

Personalised recommendations