基于Taylor算子的二元向量切触有理插值

Bivariate Vector-Valued Osculatory Rational Interpolation Based on the Taylor Operator

提出了一种基于Taylor算子的二元向量切触有理插值的新方法.首先应用已知的节点定义各阶有理插值基函数,再用相应的向量值和各阶偏导数值建立一种类似二元函数Taylor公式的新型插值算子,最后进行组合运算,得出二元向量一阶、二阶切触有理插值函数的显式表达式,并自然推广到k阶情形,还给出了误差估计.算例表明,该方法计算简单,过程公式化,有应用价值.

A new approach based on the Taylor operator was proposed for the bivariate vectorvalued osculatory rational interpolation. First, the rational interpolation basis functions of each order were defined by means of the known nodes. Second, a new type of interpolation operator similar to the Taylor formula for bivariate functions was established with the corresponding vector values and partial derivative values of each order. At last, the combined operations were carried out, and the explicit expressions of the bivariate vector-valued osculatory rational interpolation functions of the 1st and 2nd orders were obtained. Naturally this approach was generalized to the kth order, and the error estimates were also made. The results of an example show that, this new approach is simple and formularized in calculation, and therefore potential for application.