{"title":"Certifying Zeros of Polynomial Systems Using Interval Arithmetic","authors":"Paul Breiding, Kemal Rose, Sascha Timme","doi":"https://dl.acm.org/doi/10.1145/3580277","DOIUrl":null,"url":null,"abstract":"<p>We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software <monospace>HomotopyContinuation.jl</monospace> now has a built-in function <monospace>certify</monospace>, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk’s method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as a powerful new tool in numerical algebraic geometry, which can make certification the default and not just an option.</p>","PeriodicalId":50935,"journal":{"name":"ACM Transactions on Mathematical Software","volume":"75 ","pages":""},"PeriodicalIF":2.7000,"publicationDate":"2023-03-21","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/https://dl.acm.org/doi/10.1145/3580277","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
We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk’s method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as a powerful new tool in numerical algebraic geometry, which can make certification the default and not just an option.
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.