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Journal Articles ACM Transactions on Mathematical Software Year : 2023

Efficient and Validated Numerical Evaluation of Abelian Integrals

Abstract

Abelian integrals play a key role in the infinitesimal version of Hilbert's 16th problem. Being able to evaluate such integrals - with guaranteed error bounds - is a fundamental step in computer-aided proofs aimed at this problem. Using interpolation by trigonometric polynomials and quasi-Newton-Kantorovitch validation, we develop a validated numerics method for computing Abelian integrals in a quasi-linear number of arithmetic operations. Our approach is both effective, as exemplified on two practical perturbed integrable systems, and amenable to an implementation in a formal proof assistant, which is key to provide fully reliable computer-aided proofs.
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Dates and versions

hal-03561096 , version 1 (08-02-2022)
hal-03561096 , version 2 (12-12-2023)

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Florent Bréhard, Nicolas Brisebarre, Mioara Joldeş, Warwick Tucker. Efficient and Validated Numerical Evaluation of Abelian Integrals. ACM Transactions on Mathematical Software, inPress, ⟨10.1145/3637550⟩. ⟨hal-03561096v2⟩
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