Finite difference (FD) formulas approximate derivatives by weighted sums of function values. Given arbitrarily distributed node locations in one-dimension, a previous algorithm by the present author (1988, Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput., 51, 699–706) provides FD weights of optimal order of accuracy for approximating any order derivative at a specified location. This algorithm is extended here to the case of finding weights to apply not only to function values but also to first derivative values in the case that these also are available. The MATLAB code for the algorithm is provided, and two examples are given to illustrate how this type of FD stencil can be applied to solving partial differential equations.

中文翻译：

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一种基于Hermite的有限差分权重的计算算法

有限差分（FD）公式通过函数值的加权和来近似导数。鉴于一维任意分布的节点位置，前一算法由本作者（1988年，一代上任意空间格栅有限差分公式。数学。COMPUT。，51，699-706）提供了精度的最优顺序的权重FD逼近指定位置的任何阶数导数。该算法在此扩展到寻找权重的情况，该权重不仅适用于函数值，而且在这些值也可用的情况下也适用于一阶导数。提供了该算法的MATLAB代码，并给出了两个示例来说明如何将此类型的FD模板应用于求解偏微分方程。