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.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs and mathematical tables. U.S. Government Printing Office, NBS, Washington (1964)
Cipra, B.A.: A new testament for special functions. SIAM News 31(2) (March 1998)
Lozier, D.: NIST Digital Library of Mathematical Functions. Annals of Mathematics and Artificial Intelligence 38(1–3) (May 2003)
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)
Floating-Point Working Group: IEEE standard for binary floating-point arithmetic. SIGPLAN 22, 9–25 (1987)
Muller, J.M.: Elementary functions: Algorithms and implementation. Birkhäuser, Basel (1997)
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/
Higham, N.: Accuracy and stability of numerical algorithms. SIAM, Philadelphia (1996)
Cuyt, A., Verdonk, B., Waadeland, H.: Efficient and reliable multiprecision implementation of elementary and special functions. SIAM J. Sci. Comput. 28, 1437–1462 (2006)
Cuyt, A., Brevik Petersen, V., Verdonk, B., Waadeland, H., Jones, W.: Handbook of Continued Fractions for Special Functions. Springer, Heidelberg (2008)
Jones, W., Thron, W.: Numerical stability in evaluating continued fractions. Math. Comp. 28, 795–810 (1974)
Author information
Authors and Affiliations
Editor information
Rights 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)