Abstract: Rational interpolation is an important element of function approximation, and reducing the degree of osculatory rational interpolation function and solving the existence of osculatory rational interpolation function is an important problem of rational interpolation. The algorithm of osculatory rational interpolation function mostly depends on the continued fraction. But the algorithm is conditional and the computation is large. Based on heredity of Newton interpolation and the method of piecewise combination, this paper constructs the function  of osculatory rational interpolation and extends it to vector-valued case, which has no real poles. It not only solves the existence of such osculatory rational interpolation function, but also reduces the degree of rational function. Furthermore, it gives the error estimation of rational function, and by use of the numerical example, illustrates the algorithm has heredity, needs less computation, and it facilitates the practical application.

Key words: osculatory rational interpolation, Newton interpolation, piecewise combination, heredity, high order derivatives

摘要： 有理插值是函数逼近的一个重要内容，而降低切触有理插值的次数和解决切触有理插值函数的存在性是有理插值的一个重要问题。切触有理插值函数的算法大都是基于连分式进行的，其算法可行性是有条件的，且计算量较大。利用牛顿多项式插值承袭性的思想和分段组合的方法，构造出了一种无极点的切触有理插值函数，并推广到向量值切触有理插值情形；既解决了此类切触有理插值函数存在性问题，又降低了切触有理插值函数的次数。给出误差估计，并通过数值实例说明该算法具有承袭性、计算量低、便于实际应用等特点。

关键词: 切触有理插值, 牛顿插值, 分段组合, 承袭性, 高阶导数