艾里函数的完备约化系统

杜昊; HAO Du; ClemensG.Raab; CLEMENS G.Raab

doi:10.48014/bcam.20230724002

艾里函数的完备约化系统

Complete Reduction Systems for Airy Functions

中国应用数学通报 2023年第1卷第1期页码[10-22] 下载全文[0.6MB] [HTML]

摘要

计算某种“闭形式”的不定积分,即符号积分,是计算机代数的一个重要研究领域。在部分实现递归Risch算法后,人们发现并行积分方法可以实现更高效的算法,其中最著名的算法之一是Risch-Norman算法。然而,这种方法依赖于积分中无法准确得到的多项式次数的估计。Norman基于完备化思想提供了一种避免次数估计的替代方法。然而,根据微分域的构造和项序的选择,可能会发生完备化过程不能终止并产生无限多约化法则的情况。我们将Norman方法优化并应用于在物理学中有重要应用的Airy函数生成的微分环。通过确定适当的项序,我们用有限个公式表示无限多个约化法则,并给出了Airy函数的两个完备约化系统。

Abstract

The computation of indefinite integrals in certain kind of “closed form”,which is known as symbolic integration,is an important research subarea of computer algebra.After implementing the recursive Risch algorithm partly,it was realized that efficient algorithms can be achieved by a parallel approach.This led to the famous Risch-Norman algorithm.However,this approach relies on an ansatz with heuristic degree bounds.Norman’s completion-based approach provides an alternative for finding the numerator of the integral avoiding heuristic degree bounds.However,depending on the differential field and on the selected ordering of terms,it may happen that the completion process does not terminate and yields an infinite number of reduction rules.We apply Norman’s approach to the differential field generated by Airy functions,which play an important role in physics.By fixing adapted orderings and analyzing the process in the concrete case,we present two complete reduction systems for Airy functions by finitely many formulae to denote infinitely many reduction rules.

DOI	10.48014/bcam.20230724002
文章类型	研究性论文
收稿日期	2023-07-24
接收日期	2023-08-28
出版日期	2023-12-28
关键词	符号积分, Risch-Norman算法, 无限约化系统
Keywords	Symbolic integration, Risch-Norman algorithm, Infinite reduction systems
作者	杜昊^1,2,*, Clemens G.Raab³
Author	Hao Du^1,2,*, Clemens G.Raab³
所在单位	1. 北京邮电大学理学院, 北京 102206 2. 教育部数学与信息网络重点实验室, 北京 102206 3. 约翰·开普勒 (林茨) 大学代数教研室, 林茨 4040.
Company	1. School of Science, Beijing University of Posts and Telecommunications (BUPT) , Beijing 102206, China 2. Key Laboratory of Mathematics and Information Networks (BUPT) , Ministry of Education, Beijing 102206, China 3. Institute for Algebra, Johannes Kepler University Linz (JKU) , Linz 4040, Austria
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基金项目	This research was supported by the Austrian Science Fund(FWF): P 31952,by the National Natural Science Foundation of China(NSFC): 12201065 and by the Basic Research Fund of Beijing University of Posts and Telecommunications( BUPT): 500422372,500423226.
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引用本文	杜昊,Clemens Raab. 艾里函数的完备约化系统[J]. 中国应用数学通报,2023,1(1):10-22.
Citation	Hao Du,Clemens Raab.Complete reduction systems for Airy functions[J].Bulletin of Chinese Applied Mathematics,2023,1(1):10-22.