基于符号-数值计算的区间代数曲线插值
Algebraic Curve Interpolation for Intervals via Symbolic-Numeric Computation
AI总结：
本文针对代数曲线插值问题，提出了基于符号-数值计算的新方法，可求解最小次数插值多项式，还能重构最小次数、最少单项式的整系数插值代数曲线，将问题转化为优化问题后用拉格朗日乘子法与符号计算求解，并通过示例验证了方法有效性。
摘要：
代数曲线插值通过指定平面上N个点的位置，构造一条应经过这些点的函数f的代数曲线。本文提出了一种构建可对一组数据（点或邻域）进行插值的代数曲线的新方法，该方法旨在搜索能对给定数据进行插值的最小次数多项式。此外，本文还提出了一种高效方法，用于重构具有最小次数和最少单项式、且系数为整数的代数曲线，以对给定数据进行插值。研究将相关问题转化为优化问题，通过拉格朗日乘子法和符号计算求解，并给出了多个示例以说明所提方法的有效性。
英文摘要：
Algebraic curve interpolation is described by specifying the location of N points in the plane and constructing an algebraic curve of a function f that should pass 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, the paper also presents 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
