端点导数的误差对插值三次样条函数的影响

A Practical Application of Cubic Splines

研究了等距节点情况下端点一阶导数的误差，对插值三次样条函数的影响，得到了由端点导数的误差所引起的插值样条函数的绝对误差的上界估计式。

Fig. 1 shows a number of discrete points representing observed data such as displacements. In some trajectory problems, the effect of difference in initial velocity on later velocities need to be studied. Fig. 1 is mathematically more seneral, because for one thing it shows two curves passing through observed points but with different first derivatives at both ends.We make use of the powerful mathematical tool-cubic splines -to pass through the given points and have first derivatives at both ends as given.From Fin. 1, we can see that differences of the two curves quickly attenuate and after,say three equal intervals, the two curves are almost undistinguishable. Thus differences in first deriyatives at endpoints have almost no effect on function values and first derivatives in intervals removed from the endpoints.Eq. (13) gives mathematically the estimated upper limit of the difference in function value due to differences of first derivatives at both ends.