关于用节点的一、二阶导数或各段三阶导数确定三次样条的问题

关于用节点的一、二阶导数或各段三阶导数确定三次样条的问题 ON THE PROBLEMS TO DETERMINE THE SPLINES OF DEGREE 3 BY THE FIRST AND SECOND DERIVATIVES AT THE KNOT POINTS OR A THIRD DERIVATIVE FOR EACH SEGMENT 杨文茂 Yang Wenmao first-author < 正 > 一引言与引理确定三次样条的一般提法是对于区间[a,b]上的一个分法已知型值y j （j=0,1,…,n）和一定的边界条件（如y o 和y n ）,通过解三转角方程或三弯矩方程分别获得节点的一阶导数mj（y’ j ）或二阶导数M j （y’’ j ）（j=0,1,…,n）,然后通过样条的表达式就可以计算[a,b]上任意x所对应的样条函数值y,即确定了样条[1] 。本文探讨与上述提法相反的问题,即从已知各节点的一阶或二阶导数或各段三阶导数出发,在通常所见各边界条件下,求样条的型值,即确定样条。此问题是有意义的,在处理曲线光顺时,有一种弯势设计法[2]。曲线的二阶导数的符号与大小分别决定曲线的弯向（凹凸性）与 Let a division of the interval [a,b] be α=x 0 < x 1 < … < x n-1 < x n =b. In Order to determine a spline of degree 3 in this interval, in general case, we give the spline funetion values y j （j=0, 1,…, n） at the knot points and some boundary conditions （for example y 0 ′ and y n ′ ）. Fiest of all we find the first or the second derivatives y j ′ （≡m j ） or y-j ″ （≡M j ） at the knot points; further using the formulas for interpolation we obtain a spline, i.e: In this paper we would consider the problems converse to the above, i. e. suppose to give the first or the second derivatives （m j or M j ） at the knot points, or a third derivative （σ j ） for every segment of a spline, under some boundary conditions, first of all we find the spline function values y j at the knot points, further using the formulas for interpolation, we obtain a spline. i.e. Main results obtained in this paper are theorems 1, 2 and 3, corrensponding to the given values m j , M j and σ j respectively. For some boundary conditions, we obtain the spline function values y j at the knot points. In all cases, m j , M j and σ j must satisfy one or two equations, and spline function values y j contain one or two arbitrary parameters. 1981-04-01 2021-04-01 4