Skip to main content

Validated Evaluation of Special Mathematical Functions

  • Conference paper
Intelligent Computer Mathematics (CICM 2008)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5144))

Included in the following conference series:

Abstract

Because of the importance of special functions, several books and a large collection of papers have been devoted to the numerical computation of these functions, the most well-known being the Abramowitz and Stegun handbook [1]. But up to this date, no environment offers routines for the provable correct evaluation of these special functions.

We point out how series and limit-periodic continued fraction representations of the functions can be helpful in this respect. Our scalable precision technique is mainly based on the use of sharpened a priori truncation and round-off error upper bounds, in case of real arguments. The implementation is validated in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs and mathematical tables. U.S. Government Printing Office, NBS, Washington (1964)

    MATH  Google Scholar 

  2. Cipra, B.A.: A new testament for special functions. SIAM News 31(2) (March 1998)

    Google Scholar 

  3. Lozier, D.: NIST Digital Library of Mathematical Functions. Annals of Mathematics and Artificial Intelligence 38(1–3) (May 2003)

    Google Scholar 

  4. Moler, C.B.: Cleve’s corner: The tetragamma function and numerical craftsmanship: MATLAB’s special mathematical functions rely on skills from another era. Technical note, The MathWorks, Inc (2002)

    Google Scholar 

  5. Floating-Point Working Group: IEEE standard for binary floating-point arithmetic. SIGPLAN 22, 9–25 (1987)

    Google Scholar 

  6. Muller, J.M.: Elementary functions: Algorithms and implementation. Birkhäuser, Basel (1997)

    MATH  Google Scholar 

  7. Lozier, D., Olver, F.: Numerical Evaluation of Special Functions. In: Gautschi, W. (ed.) AMS Proceedings of Symposia in Applied Mathematics, vol. 48, pp. 79–125 (1994); Updated version (December 2000), http://math.nist.gov/nesf/

  8. Higham, N.: Accuracy and stability of numerical algorithms. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  9. Cuyt, A., Verdonk, B., Waadeland, H.: Efficient and reliable multiprecision implementation of elementary and special functions. SIAM J. Sci. Comput. 28, 1437–1462 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cuyt, A., Brevik Petersen, V., Verdonk, B., Waadeland, H., Jones, W.: Handbook of Continued Fractions for Special Functions. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  11. Jones, W., Thron, W.: Numerical stability in evaluating continued fractions. Math. Comp. 28, 795–810 (1974)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Serge Autexier John Campbell Julio Rubio Volker Sorge Masakazu Suzuki Freek Wiedijk

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Backeljauw, F., Becuwe, S., Cuyt, A. (2008). Validated Evaluation of Special Mathematical Functions. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds) Intelligent Computer Mathematics. CICM 2008. Lecture Notes in Computer Science(), vol 5144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85110-3_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-85110-3_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85109-7

  • Online ISBN: 978-3-540-85110-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics