{"title":"Lauricella precision:根据参数对Lauricella函数进行数值计算的软件包","authors":"M.A. Bezuglov , B.A. Kniehl , A.I. Onishchenko , O.L. Veretin","doi":"10.1016/j.cpc.2025.109812","DOIUrl":null,"url":null,"abstract":"<div><div>We introduce the <span>PrecisionLauricella</span> package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of the Laurent expansion coefficients of Lauricella functions whose parameters depend linearly on a small regulator, <em>ε</em>. In practical multi-loop calculations, Lauricella functions are required only as series around <span><math><mi>ε</mi><mo>=</mo><mn>0</mn></math></span>, and <span>PrecisionLauricella</span> is designed specifically to deliver such coefficients with arbitrary precision. The package leverages a method based on analytic continuation via Frobenius generalized power series, providing an efficient and accurate alternative to conventional approaches relying on multi-dimensional series expansions or Mellin–Barnes representations. This one-dimensional approach is particularly advantageous for high-precision calculations and facilitates further optimization through <em>ε</em>-dependent reconstruction from evaluations at specific numerical values, enabling efficient parallelization. The underlying mathematical framework for this method has been detailed in our previous work, while the current paper focuses on the design, implementation, and practical applications of the <span>PrecisionLauricella</span> package.</div></div><div><h3>Program summary</h3><div><em>Program Title:</em> PrecisionLauricella</div><div><em>CPC Library link to program files:</em> <span><span>https://doi.org/10.17632/6f958yz2dr.1</span><svg><path></path></svg></span></div><div><em>Developer's repository link:</em> <span><span>https://bitbucket.org/BezuglovMaxim/precisionlauricella-package/src/main/</span><svg><path></path></svg></span></div><div><em>Licensing provisions:</em> GPLv3</div><div><em>Programming language:</em> Wolfram Mathematica</div><div><em>Supplementary material:</em> PrecisionLauricella_Examples.nb</div><div><em>Nature of problem:</em> Lauricella functions, generalizations of hypergeometric functions, appearing in physics and mathematics, including Feynman integrals and string theory. When their indices depend linearly on a small parameter <em>ε</em>, their numerical evaluation becomes challenging due to the complexity of high-dimensional series and singularities. Traditional methods, like hypergeometric re-expansion or Mellin–Barnes integrals, often lack efficiency and precision.</div><div>Managing multi-dimensional sums exacerbates computational costs, especially for high-precision requirements, making these approaches unsuitable for many practical applications. Thus, there is a pressing need for efficient, scalable methods capable of maintaining numerical accuracy and effectively handling parameter dependencies.</div><div><em>Solution method:</em> Our method uses the Frobenius approach to achieve analytic continuations of Lauricella functions through generalized power series. Representing the functions as one-dimensional series simplifies high-precision numerical evaluations compared to traditional methods relying on multi-dimensional expansions or Mellin–Barnes integrals.</div><div>We further optimize calculations by reconstructing <em>ε</em> dependencies from evaluations at specific values, enabling efficient parallelization and reducing computational costs.</div><div>A comprehensive mathematical exposition of the method is provided in our previous work [1].</div></div><div><h3>References</h3><div><ul><li><span>[1]</span><span><div>M. Bezuglov, B. Kniehl, A. Onishchenko, O. Veretin, High-precision numerical evaluation of Lauricella functions, <span><span>arXiv:2502.03276</span><svg><path></path></svg></span>, 2 2025.</div></span></li></ul></div></div>","PeriodicalId":285,"journal":{"name":"Computer Physics Communications","volume":"316 ","pages":"Article 109812"},"PeriodicalIF":3.4000,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"PrecisionLauricella: Package for numerical computation of Lauricella functions depending on a parameter\",\"authors\":\"M.A. Bezuglov , B.A. Kniehl , A.I. Onishchenko , O.L. Veretin\",\"doi\":\"10.1016/j.cpc.2025.109812\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We introduce the <span>PrecisionLauricella</span> package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of the Laurent expansion coefficients of Lauricella functions whose parameters depend linearly on a small regulator, <em>ε</em>. In practical multi-loop calculations, Lauricella functions are required only as series around <span><math><mi>ε</mi><mo>=</mo><mn>0</mn></math></span>, and <span>PrecisionLauricella</span> is designed specifically to deliver such coefficients with arbitrary precision. The package leverages a method based on analytic continuation via Frobenius generalized power series, providing an efficient and accurate alternative to conventional approaches relying on multi-dimensional series expansions or Mellin–Barnes representations. This one-dimensional approach is particularly advantageous for high-precision calculations and facilitates further optimization through <em>ε</em>-dependent reconstruction from evaluations at specific numerical values, enabling efficient parallelization. The underlying mathematical framework for this method has been detailed in our previous work, while the current paper focuses on the design, implementation, and practical applications of the <span>PrecisionLauricella</span> package.</div></div><div><h3>Program summary</h3><div><em>Program Title:</em> PrecisionLauricella</div><div><em>CPC Library link to program files:</em> <span><span>https://doi.org/10.17632/6f958yz2dr.1</span><svg><path></path></svg></span></div><div><em>Developer's repository link:</em> <span><span>https://bitbucket.org/BezuglovMaxim/precisionlauricella-package/src/main/</span><svg><path></path></svg></span></div><div><em>Licensing provisions:</em> GPLv3</div><div><em>Programming language:</em> Wolfram Mathematica</div><div><em>Supplementary material:</em> PrecisionLauricella_Examples.nb</div><div><em>Nature of problem:</em> Lauricella functions, generalizations of hypergeometric functions, appearing in physics and mathematics, including Feynman integrals and string theory. When their indices depend linearly on a small parameter <em>ε</em>, their numerical evaluation becomes challenging due to the complexity of high-dimensional series and singularities. Traditional methods, like hypergeometric re-expansion or Mellin–Barnes integrals, often lack efficiency and precision.</div><div>Managing multi-dimensional sums exacerbates computational costs, especially for high-precision requirements, making these approaches unsuitable for many practical applications. Thus, there is a pressing need for efficient, scalable methods capable of maintaining numerical accuracy and effectively handling parameter dependencies.</div><div><em>Solution method:</em> Our method uses the Frobenius approach to achieve analytic continuations of Lauricella functions through generalized power series. Representing the functions as one-dimensional series simplifies high-precision numerical evaluations compared to traditional methods relying on multi-dimensional expansions or Mellin–Barnes integrals.</div><div>We further optimize calculations by reconstructing <em>ε</em> dependencies from evaluations at specific values, enabling efficient parallelization and reducing computational costs.</div><div>A comprehensive mathematical exposition of the method is provided in our previous work [1].</div></div><div><h3>References</h3><div><ul><li><span>[1]</span><span><div>M. Bezuglov, B. Kniehl, A. Onishchenko, O. Veretin, High-precision numerical evaluation of Lauricella functions, <span><span>arXiv:2502.03276</span><svg><path></path></svg></span>, 2 2025.</div></span></li></ul></div></div>\",\"PeriodicalId\":285,\"journal\":{\"name\":\"Computer Physics Communications\",\"volume\":\"316 \",\"pages\":\"Article 109812\"},\"PeriodicalIF\":3.4000,\"publicationDate\":\"2025-08-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computer Physics Communications\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0010465525003145\",\"RegionNum\":2,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Physics Communications","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0010465525003145","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
PrecisionLauricella: Package for numerical computation of Lauricella functions depending on a parameter
We introduce the PrecisionLauricella package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of the Laurent expansion coefficients of Lauricella functions whose parameters depend linearly on a small regulator, ε. In practical multi-loop calculations, Lauricella functions are required only as series around , and PrecisionLauricella is designed specifically to deliver such coefficients with arbitrary precision. The package leverages a method based on analytic continuation via Frobenius generalized power series, providing an efficient and accurate alternative to conventional approaches relying on multi-dimensional series expansions or Mellin–Barnes representations. This one-dimensional approach is particularly advantageous for high-precision calculations and facilitates further optimization through ε-dependent reconstruction from evaluations at specific numerical values, enabling efficient parallelization. The underlying mathematical framework for this method has been detailed in our previous work, while the current paper focuses on the design, implementation, and practical applications of the PrecisionLauricella package.
Program summary
Program Title: PrecisionLauricella
CPC Library link to program files:https://doi.org/10.17632/6f958yz2dr.1
Nature of problem: Lauricella functions, generalizations of hypergeometric functions, appearing in physics and mathematics, including Feynman integrals and string theory. When their indices depend linearly on a small parameter ε, their numerical evaluation becomes challenging due to the complexity of high-dimensional series and singularities. Traditional methods, like hypergeometric re-expansion or Mellin–Barnes integrals, often lack efficiency and precision.
Managing multi-dimensional sums exacerbates computational costs, especially for high-precision requirements, making these approaches unsuitable for many practical applications. Thus, there is a pressing need for efficient, scalable methods capable of maintaining numerical accuracy and effectively handling parameter dependencies.
Solution method: Our method uses the Frobenius approach to achieve analytic continuations of Lauricella functions through generalized power series. Representing the functions as one-dimensional series simplifies high-precision numerical evaluations compared to traditional methods relying on multi-dimensional expansions or Mellin–Barnes integrals.
We further optimize calculations by reconstructing ε dependencies from evaluations at specific values, enabling efficient parallelization and reducing computational costs.
A comprehensive mathematical exposition of the method is provided in our previous work [1].
References
[1]
M. Bezuglov, B. Kniehl, A. Onishchenko, O. Veretin, High-precision numerical evaluation of Lauricella functions, arXiv:2502.03276, 2 2025.
期刊介绍:
The focus of CPC is on contemporary computational methods and techniques and their implementation, the effectiveness of which will normally be evidenced by the author(s) within the context of a substantive problem in physics. Within this setting CPC publishes two types of paper.
Computer Programs in Physics (CPiP)
These papers describe significant computer programs to be archived in the CPC Program Library which is held in the Mendeley Data repository. The submitted software must be covered by an approved open source licence. Papers and associated computer programs that address a problem of contemporary interest in physics that cannot be solved by current software are particularly encouraged.
Computational Physics Papers (CP)
These are research papers in, but are not limited to, the following themes across computational physics and related disciplines.
mathematical and numerical methods and algorithms;
computational models including those associated with the design, control and analysis of experiments; and
algebraic computation.
Each will normally include software implementation and performance details. The software implementation should, ideally, be available via GitHub, Zenodo or an institutional repository.In addition, research papers on the impact of advanced computer architecture and special purpose computers on computing in the physical sciences and software topics related to, and of importance in, the physical sciences may be considered.