Skip to main content
SpringerLink
Log in

A Computer Oriented Approach to Get Sharp Reliable Error Bounds

  • Published:
Reliable Computing

Abstract

We present interval methods to get reliable a priori error bounds for the machine evaluation of algorithms implementing some mathematical expression. The term expression not only means simple arithmetical expressions but also more complex program parts including loops or recursive structures (e.g. a complete elementary function routine).

We sketch a method that can be used to get an upper bound for the approximation error of a polynomial or a rational approximation. We also discuss a method to compute worst case a priori error estimates for arbitrary IEEE double floating-point computations. Our theoretical results lead to reliable and easy to use public domain software tools. The application of these tools to an accurate table method shows that error bounds of high quality can be derived.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. American National Standards Institute / Institute of Electrical and Electronics Engineers: A Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Std. 754-1985, New York, 1985 (reprinted in SIGPLAN 22 (2) (1987), pp 9–25).

    Google Scholar 

  2. Hofschuster, W. and Krämer, W.: Ein rechnergest ützter Fehlerkalkül mit Anwendung auf ein genaues Tabellenverfahren, Preprint 96/5 des Instituts für Wissenschaftliches Rechnen und Mathematische Modellbildung, 1996, ftp://iamk4515.mathematik.uni-karlsruhe.de, directory: /pub/iwrmm/preprints.

  3. Krämer, W.: Sichere und genaue Abschätzung des Approximationsfehlers bei rationalen Approximationen, Bericht des Instituts für Angewandte Mathematik, Universität Karlsruhe, 1996, ftp://iamk4515.mathematik.uni-karlsruhe.de, directory: /pub/documents/reports.

  4. Tang, P. T. P.: Table-Driven Implementation of the Expm1 Function in IEEE Floating-Point Arithmetic, ACM Trans. on Math. Software 18 (2) (1992), pp. 211–222.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Karlsruhe (TH), D-76128, Karlsruhe, Germany

    Werner Hofschuster

  2. Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung, Universität Karlsruhe (TH), D-76128, Karlsruhe, Germany

    Walter Krämer

Authors
  1. Werner Hofschuster
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Krämer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hofschuster, W., Krämer, W. A Computer Oriented Approach to Get Sharp Reliable Error Bounds. Reliable Computing 3, 239–248 (1997). https://doi.org/10.1023/A:1009966622475

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009966622475

Keywords

Search

Navigation