descriptionPublicationkeyboard_double_arrow_right Article 16 Mar 2024 English Publisher:Association for Computing Machinery (ACM)Journal:ACM Transactions on Mathematical Software, volume 50, pages 1-38 (issn: 0098-3500, eissn: 1557-7295,

Authors: Florent Bréhard; Nicolas Brisebarre; Mioara Joldes; Warwick Tucker;

doi: 10.1145/3637550

Efficient and Validated Numerical Evaluation of Abelian Integrals

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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.

Related Organizations

UNIVERSITE FEDERALE DE TOULOUSE MIDI-PYRENEES
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Paul Sabatier University
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Laboratory for Analysis and Architecture of Systems
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Monash University
Australia
UNIVERSITE DE TOULOUSE
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Keywords

[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Newton-like operator, [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS], rigorous numerics, Hilbert's 16th problem, Secondary: 37G15, 510, 42A10, 42A15, [INFO.INFO-SC]Computer Science [cs]/Symbolic Computation [cs.SC], limit cycles, [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, [INFO.INFO-SC] Computer Science [cs]/Symbolic Computation [cs.SC], Abelian integral, 37M20, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], 620, 2020 AMS classification: Primary: 34C07, [INFO.INFO-NA] Computer Science [cs]/Numerical Analysis [cs.NA], [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic, abelian integral, trigonometric polynomial interpolation, 65G20, Numerical analysis

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TypedPolynomials.jl software on GitHub
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