同伦分支点的分布与失效概率

IF 3.5 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics and Computation Pub Date : 2025-01-08 DOI:10.1016/j.amc.2024.129273

Jonathan D. Hauenstein , Caroline Hills , Andrew J. Sommese, Charles W. Wampler

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引用次数: 0

摘要

同伦延拓是数值代数几何中求解多元多项式方程组的一种标准方法。诸如伽马技巧之类的技术产生概率为1的可跟踪同伦。然而，由于数值代数几何采用数值路径跟踪方法，靠近分支点可能会引起有限精度计算的问题。本文系统地研究了同伦的分支点，阐明了分支点是如何分布的，并利用这些信息研究了在有限精度情况下的失效概率。文中还举例说明了该理论分析的正确性，其中包括一个运动学系统的各种起动系统。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Branch points of homotopies: Distribution and probability of failure

Homotopy continuation is a standard method used in numerical algebraic geometry for solving multivariate systems of polynomial equations. Techniques such as the so-called gamma trick yield trackable homotopies with probability one. However, since numerical algebraic geometry employs numerical path tracking methods, being close to a branch point may cause concern with finite precision computations. This paper provides a systematic study of branch points of homotopies to elucidate how branch points are distributed and use this information to study the probability of failure when using finite precision. Several examples, including a system arising in kinematics, with various start systems are included to demonstrate the theoretical analysis.

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来源期刊

Applied Mathematics and Computation 数学-应用数学

CiteScore

7.90

自引率

10.00%

发文量

755

审稿时长

36 days

期刊介绍： Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results. In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.