<正> 一、引 言 在实际中,往往需要把从实验而来的列表函数一点列(X_i、Y_i),i=0,1,…,n,用较简单的分析式近似表达,求其近似导函数或积分,这实际上是一个平面点列的插值或拟合问题。流行的方法常不能保证对曲线的凸性要求,产生多余拐点或波动,尤其当点子很多,或点列蕴含曲率剧变、平直微曲、曲直相接时更是难以适应。 从处理方式讲,流行方法大都可归于“整体性”方法,在作插值或拟合时着眼于点列整体。例如多项式插值,一般是寻求一个通过全体点子的多项式,因而有n+1个点就得到n次多项式。当n很大时该多项式一般拐点很多就很难保凸。另外,再以三次样条函数插值为例:连续性方程(如所谓三弯距方程)加上适当的边界条件构成一个n+1阶线性代数方程组,从中解出各点的一阶或二阶导数,由此确定一条曲线。由于如此而来的各点的导数值显然同全体点子有关,故任一点的改变必然会引起远离该点处曲线的改变,这对于实际计算是不利的。

In this paper a series of approximate formulas for the expression, derivation and integration of functions given by way of n data points is suggested, and in addition, some computational test results are also given. These formulas are derived from so-called "part ideas", so they have many advantages, one of which is the guarantee of convexity. With the aid of a digital computer, they can be effectively used in experimental data processing or fitting, contour designing, approximate derivation, numerical integration, etc.