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Multiple-Precision Correctly rounded Newton-Cotes quadrature

Published online by Cambridge University Press:  24 April 2007

Laurent Fousse
Laurent Fousse*
Affiliation:
Univ. Nancy I/LORIA, 615 rue du Jardin Botanique, 54602 Villers-lès-Nancy Cedex, France; laurent@komite.net
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Abstract

Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics.
We study the Newton-Cotes quadrature scheme in the context of multiple-precision arithmetic and give enough details on the algorithms and the error bounds to enable software developers to write a Newton-Cotes quadrature with bounded error.

Keywords

numerical integrationcorrect roundingmultiple-precisionNewton-Cotes
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 1: Real Numbers , January 2007 , pp. 103 - 121
Copyright
© EDP Sciences, 2007

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References

D.H. Bailey and X.S. Li, A comparison of three high-precision quadrature schemes, in Proceedings of the RNC'5 conference (Real Numbers and Computers) (September 2003) 81–95. http://www.ens-lyon.fr/LIP/Arenaire/RNC5.
C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, User's Guide to PARI/GP (2000). ftp://megrez.math.u-bordeaux.fr/pub/pari/manuals/users.pdf.
P.J. Davis and P. Rabinowitz, Methods of numerical integration. Academic Press, New York, 2nd edition (1984).
J. Demmel and Y. Hida, Accurate floating point summation. http://www.cs.berkeley.edu/ demmel/AccurateSummation.ps (May 2002).
W.J. Ellison and M. Mendès-France, Les nombres premiers. Actualités Scientifiques et Industrielles 1366 (1975).
J.-M. Chesneaux F. Jezequel and M. Charikhi, Dynamical control of computations of multiple integrals. SCAN2002 conference, Paris (France) (23–27 September 2002).
B. Fuchssteiner, K. Drescher, A. Kemper, O. Kluge, K. Morisse, H. Naundorf, G. Oevel, F. Postel, T. Schulze, G. Siek, A. Sorgatz, W. Wiwianka and P. Zimmermann, MuPAD User's Manual. Wiley Ltd. (1996).
W. Oevel, Numerical computations in MuPAD 1.4. mathPAD 8 (1998) 58–67.
The Spaces project. The MPFR library, version 2.0.1. http://www.mpfr.org/ (2002).