基于符号数值计算的代数曲线区间插值  

Algebraic curve interpolation for intervals via symbolic-numeric computation

作　　者：Lydia Dehbi 杨争峰[1] 彭超[1] 徐姚晨 曾振柄[2] Lydia Dehbi;Zhengfeng Yang;Chao Peng;Yaochen Xu;Zhenbing Zeng

机构地区：[1]华东师范大学软件工程学院,上海200062 [2]上海大学数学系,上海200444

出　　处：《中国科学：数学》2024年第5期699-730,共32页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12171159)资助项目。

摘　　要：本文研究的代数曲线区间插值问题,是针对预先给定平面上的若干矩形小邻域,构造经过它们的次数最低的代数曲线、项数最少的代数曲线以及系数是整数的代数曲线.本文将上述问题转化为优化问题,给出基于符号数值计算和Lagrange乘子法的求解方法,应用这一方法解决了几个具体的有趣问题,包括基于太阳系行星、小行星和矮行星的轨道数据重新发现Kepler第三定律.Algebraic curve interpolation is described by specifying the location of N points in the plane and constructing an algebraic curve of function f that passes through them.In this paper,we propose a novel approach to construct the algebraic curve that interpolates a set of data(points or neighborhoods).This approach aims to search the polynomial with the smallest degree interpolating the given data.Moreover,we also present an efficient method to reconstruct the algebraic curve of integer coefficients with the smallest degree and the least monomials that interpolates the provided data.The problems are converted into optimization problems and are solved via Lagrange multipliers methods and symbolic computation.Various examples are presented to illustrate the proposed approaches,including an example to recover Kepler's third law of planet motion by analyzing the orbital data of 38 planets,asteroids,and dwarfs.

关 键 词：代数曲线 数学机械化 符号数值计算 稀疏插值 

分 类 号：O241.3[理学—计算数学]