一类有理保形插值

SHAPE PRESERVING INTERPOLATION BY PIECEWISE RATIONAL POLYNOMIALS

给出了一个适用于一般数据集的有理保形插值函数。它不仅具有插值函数的形式简单、参数易于选取等特点,而且其C^2保形插值的参数可以很方便地求得,而不必求解非线性方程组。在给出C^k类函数(k=1,…,4)的Hermite插值的最佳误差估计的基础上,本文得到了一般数据集的保形插值的误差估计;作为其推论,又得到了对严格凸函数的保凸插值的误差估计。本文中C^1和C^2保形插值的参数仍可在一定范围内自由选取。故可利用其适当调整曲线的形状,使之更符合设计要求;或利用参数的适当选取以获得较好的逼近阶。提出的有理保形插值已用于正在开发的微机CAD并行系统。

A rational scheme for shape preserving interpolation to general data sets is given, which possesses the characteristics of the rational interpolation function such as simple form and parameters to be easily choosen. The parameters of the C2 interpolation to a convex data set can be easily found, with no need to resolve a nonlinear system. Error estimation is given, by using the best error estimation of Hermite interpolation. As the corollary, the error estimation for the convexity (and / or monotonicity ) preserving interpolation to convex (and / or monotonic) data is obtained. Furthermore, the parameters for C1 and C2 interpolation can be freely choosen within certain extent to adjust the shape of the interpolating curve or to obtain a better approximation to the original function that the data comes from.The rational shape preserving interpolation method in this paper has been used in the microcomputer CAD parallel system (based on Transputer) which is being developed.