关键词: 切触有理插值, 牛顿插值, 分段组合, 承袭性, 高阶导数

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