Florent Bréhard, Nicolas Brisebarre, Mioara Joldes, Warwick Tucker
{"title":"Efficient and Validated Numerical Evaluation of Abelian Integrals","authors":"Florent Bréhard, Nicolas Brisebarre, Mioara Joldes, Warwick Tucker","doi":"10.1145/3637550","DOIUrl":null,"url":null,"abstract":"<p>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.</p>","PeriodicalId":50935,"journal":{"name":"ACM Transactions on Mathematical Software","volume":"80 1","pages":""},"PeriodicalIF":2.7000,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACM Transactions on Mathematical Software","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3637550","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
As a scientific journal, ACM Transactions on Mathematical Software (TOMS) documents the theoretical underpinnings of numeric, symbolic, algebraic, and geometric computing applications. It focuses on analysis and construction of algorithms and programs, and the interaction of programs and architecture. Algorithms documented in TOMS are available as the Collected Algorithms of the ACM at calgo.acm.org.