Formally Verified Approximations of Definite Integrals (original) (raw)
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Article Dans Une Revue Journal of Automated Reasoning Année : 2019
Assia Mahboubi (1, 2) , Guillaume Melquiond (3) , Thomas Sibut-Pinote (2)
1 LS2N - équipe GALLINETTE - Gallinette : vers une nouvelle génération d'assistant à la preuve
2 SPECFUN - Symbolic Special Functions : Fast and Certified
3 TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
- Fonction : Auteur
LS2N - équipe GALLINETTE - Gallinette : vers une nouvelle génération d'assistant à la preuve
SPECFUN - Symbolic Special Functions : Fast and Certified
- Fonction : Auteur
- PersonId : 1146
- IdHAL : guillaume-melquiond
- ORCID : 0000-0002-6697-1809
- IdRef : 117280836
TOCCATA - Formally Verified Programs, Certified Tools and Numerical Computations
- Fonction : Auteur
- PersonId : 993935
SPECFUN - Symbolic Special Functions : Fast and Certified
Résumé
Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis. This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. Our approach also handles improper integrals, provided that a factor of the integrand belongs to a catalog of identified integrable functions. This work has been integrated to the CoqInterval library.
Mots clés
- numeric computations
- polynomial approximations
- decision procedure
- real analysis
- interval arithmetic
- formal proof
- definite integrals
- Improper integrals
Domaines
Fichier principal
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| Origine | Fichiers produits par l'(les) auteur(s) |
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| Licence |  Autorisation HAL |
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https://inria.hal.science/hal-01630143
Soumis le : jeudi 15 mars 2018-14:39:17
Dernière modification le : lundi 19 janvier 2026-16:46:18
Archivage à long terme le : lundi 10 septembre 2018-22:09:57
Dates et versions
hal-01630143 , version 1 (07-11-2017)
hal-01630143 , version 2 (15-03-2018)
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Identifiants
- HAL Id : hal-01630143 , version 2
- DOI : 10.1007/s10817-018-9463-7
Citer
Assia Mahboubi, Guillaume Melquiond, Thomas Sibut-Pinote. Formally Verified Approximations of Definite Integrals. Journal of Automated Reasoning, 2019, 62 (2), pp.281-300. ⟨10.1007/s10817-018-9463-7⟩. ⟨hal-01630143v2⟩
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