Multiple-Precision Correctly rounded Newton-Cotes quadrature

Laurent Fousse

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 41, Issue: 1, page 103-121
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Fousse, Laurent. "Multiple-Precision Correctly rounded Newton-Cotes quadrature." RAIRO - Theoretical Informatics and Applications 41.1 (2007): 103-121. <http://eudml.org/doc/250041>.

@article{Fousse2007,
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. },
author = {Fousse, Laurent},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {numerical integration; correct rounding; multiple-precision; Newton-Cotes; multiple-precision arithmetic; Newton-Cotes quadrature; roundoff error; error bounds; numerical examples; worst case error analysis},
language = {eng},
month = {4},
number = {1},
pages = {103-121},
publisher = {EDP Sciences},
title = {Multiple-Precision Correctly rounded Newton-Cotes quadrature},
url = {http://eudml.org/doc/250041},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Fousse, Laurent
TI - Multiple-Precision Correctly rounded Newton-Cotes quadrature
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/4//
PB - EDP Sciences
VL - 41
IS - 1
SP - 103
EP - 121
AB - 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.
LA - eng
KW - numerical integration; correct rounding; multiple-precision; Newton-Cotes; multiple-precision arithmetic; Newton-Cotes quadrature; roundoff error; error bounds; numerical examples; worst case error analysis
UR - http://eudml.org/doc/250041
ER -

References

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  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. .  URIhttp://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).  Zbl0537.65020
  4. J. Demmel and Y. Hida, Accurate floating point summation. http://www.cs.berkeley.edu/ demmel/AccurateSummation.ps (May 2002).  Zbl1061.65039
  5. W.J. Ellison and M. Mendès-France, Les nombres premiers.Actualités Scientifiques et Industrielles1366 (1975).  
  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. mathPAD8 (1998) 58–67.  
  9. The Spaces project. The MPFR library, version 2.0.1. (2002).  URIhttp://www.mpfr.org/

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