多维三次样条函数及其应用

本文将一维三次样条函数推广到多维问题，它保留了三对角矩阵方程便于求解的特性，并能得到满意的插值、一阶和二阶偏导数。关于边界条件问题，若能给出边界节点处的一阶偏导数边界条件，则精度最高，但在大多数应用问题中，往往给不出此条件。为此，本文提出一种改进方法，即用拉格朗日三点插值法由域边附近的节点数据计算出边界节点处的一阶偏导数。算例表明，此法可改善精度。

THE MULTI-DIMENSIONAL CUBIC SPLINE FUNCTIONS AND THEIR APPLICATIONS

The one-dimensional cubic spline function was generalized to multi-dimensional proble-m. It retains the feature of convenient solution to the tri-diagonal matrix equation and can be obtain satisfactory interpolation values, the first and second partial derivatives. Besides, if the first derivatives at the boundary node points can be given as the boundary condition, then the accruacy must be the highest. However, in most application problems, this condition can not be given frequently. Hence, an improvement method was presented, that is, the first derivatives at the boundary node points are computed by the Laglange 3-point interpolation method with the node point data by the domain bounds. Sample cases show this method was valid for improving the accuracy.