Multiple-precision correctly rounded Newton-Cotes quadrature
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 103-121.

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.

DOI : 10.1051/ita:2007004
Classification : 65D30, 65D32, 65G50
Mots-clés : numerical integration, correct rounding, multiple-precision, Newton-Cotes 
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     author = {Fousse, Laurent},
     title = {Multiple-precision correctly rounded {Newton-Cotes} quadrature},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {103--121},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     doi = {10.1051/ita:2007004},
     mrnumber = {2330046},
     zbl = {1136.65032},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2007004/}
}
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Fousse, Laurent. Multiple-precision correctly rounded Newton-Cotes quadrature. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 1, pp. 103-121. doi : 10.1051/ita:2007004. http://www.numdam.org/articles/10.1051/ita:2007004/

[1] 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.

[2] 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.

[3] P.J. Davis and P. Rabinowitz, Methods of numerical integration. Academic Press, New York, 2nd edition (1984). | MR | Zbl

[4] J. Demmel and Y. Hida, Accurate floating point summation. http://www.cs.berkeley.edu/ demmel/AccurateSummation.ps (May 2002).

[5] W.J. Ellison and M. Mendès-France, Les nombres premiers. Actualités Scientifiques et Industrielles 1366 (1975). | MR | Zbl

[6] J.-M. Chesneaux F. Jezequel and M. Charikhi, Dynamical control of computations of multiple integrals. SCAN2002 conference, Paris (France) (23-27 September 2002).

[7] 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).

[8] W. Oevel, Numerical computations in MuPAD 1.4. mathPAD 8 (1998) 58-67.

[9] The Spaces project. The MPFR library, version 2.0.1. http://www.mpfr.org/ (2002).

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