基于块的 New ton-Hermite 混合切触有理插值

作为New ton多项式插值在重节点情形时的推广，New ton-Hermite多项式插值是很常用的切触线性插值，它建立在广义差商基础之上，广义差商能被递归地计算并产生有用的中间结果。New ton-Hermite插值实际上是基于点的插值，可以通过增加新的节点来获得一个新的插值多项式。这里将基于点的插值推广到基于块的插值。受现代建筑设计的启发，将插值点集划分为一些子集（块），然后将在每个子集上选择切触插值，线性或有理插值，最后用类似于New ton-Hermite插值的格式进行装配。显然，在切触有理插值上提供了灵活的选择，这里也包括它的特殊情形New ton-Hermite多项式插值。本文介绍了所谓的基于块的广义差商并给出递归算法，给出的数值例子说明了方法的有效性。

Block Based Newton-Hermite-like Blending Osculatory Rational Interpolation

As the generalization of New ton＇s polynomial interpolation so as to accommodate repeated abscissae ,New ton-Hermite polynomial interpolation may be the favourite osculatory linear interpolation in the sense that is built up by means of the generalized divided differences which can be calculated recursively and produce useful intermediate results .However Newton-Hermite interpolation is in fact point based interpolation since a new interpolating polynomial is obtained by adding a new support point into the current set of support points once at a time .In this paper we extend the point based interpolation to the block based interpolation .Inspired by the idea of the modern architectural design ,we first divide the original set of support points into some subsets （blocks） ,then construct each block by using whatever osculatory interpolation means ,linear or rational and finally assemble these blocks by New ton-Hermite＇s method to shape the w hole interpolation scheme .Clearly our method offers many flexible osculatory interpolation schemes for choices which include the classical Newton-Hermite＇s polynomial interpolation as its special case .We introduce so-called block based generalized divided differences and give a recursive algorithm accompanied with a numerical examples to show the effectiveness of our method .