| 摘要 | 计算“闭形式”的不定积分, 即符号积分, 是计算机代数的基础研究内容。在对Risch算法进行部分实现后, 人们希望实现并行积分的高效算法, 其中最著名的算法之一是Risch-Norman算法。然而, 这种方法依赖于多项式次数的估计。Norman的基于完备化的方法提供了一种避免次数估计的替代方法。然而, 根据微分域和所选的项序, 这种算法可能会不终止或产生无限数量的约简规则的情况。本文中我们将Norman方法应用于在物理学中有重要应用的Airy函数所生成的微分域。通过确定适当的相序, 我们用有限个公式表达出了两个Airy函数的具有无穷多个约简规则的完全约简系统。 |
| Abstract | The computation of indefinite integrals in some kinds of “closed form”, the so-called symbolic integration, is an important and basic research subarea of computer algebra. After implementing Risch’s algorithm partly, it was realized that efficient algorithms can be achieved in parallel integration. One of the most famous algorithms is the Risch-Norman algorithm. However, this approach relies on an analytic 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 fields 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 denoting 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, 无限约简系统 |
| Keywords | Symbolic integration, Risch-Norman algorithm, Infinite reduction systems |
| 作者 | 杜昊1,2,*, Clemens G.Raab3 |
| Author | Hao Du1,2,*, 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. |
| 浏览量 | 117 |
| 下载量 | 28 |
| 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. 艾里函数的完备约简系统[J]. 中国应用数学通报, 2023, 1(1): 10-21. |
| Citation | Hao Du, Clemens Raab. Complete reduction systems for Airy functions[J]. Bulletin of Chinese Applied Mathematics, 2023, 1(1): 10-21. |