Formally Verified Approximations of Definite Integrals
- 26 Downloads
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.
KeywordsFormal proof Numeric computations Definite integrals Improper integrals Decision procedure Interval arithmetic Polynomial approximations Real analysis
We would like to thank Érik Martin-Dorel for his improvements to the Coq framework for computing rigorous polynomial approximations and Philippe Dumas for stimulating discussions and suggestions.
- 1.Ahmed, Z.: Ahmed’s integral: the maiden solution. Math. Spectr. 48(1), 11–12 (2015)Google Scholar
- 5.Eaton, J.W., Bateman, D., Hauberg, S., Wehbring, R.: GNU Octave version 3.8.1 manual: a high-level interactive language for numerical computations (2014). http://www.gnu.org/software/octave/doc/interpreter
- 7.Helfgott, H.A.: Major arcs for Goldbach’s problem (2014). arXiv:1305.2897
- 9.Mahboubi, A., Melquiond, G., Sibut-Pinote, T.: Formally verified approximations of definite integrals. In: Blanchette, J.C., Merz, S. (eds.) 7th Conference on Interactive Theorem Proving, LNCS, vol. 9807, pp. 274–289, Nancy (2016). https://doi.org/10.1007/978-3-319-43144-4_17
- 10.Makarov, E., Spitters, B.: The Picard algorithm for ordinary differential equations in Coq. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) 4th International Conference on Interactive Theorem Proving, LNCS, vol. 7998, pp. 463–468. Springer, Rennes (2013). https://doi.org/10.1007/978-3-642-39634-2_34 CrossRefGoogle Scholar
- 12.Mayero, M.: Formalisation et automatisation de preuves en analyses réelle et numérique. Ph.D. Thesis, Université Paris VI (2001)Google Scholar