Research On The Existence Criterion And Properties Of Osculatory Rational Interpolation

Spline function is a useful and powerful tool for designing a curve or surface, which is generally used in CAGD and engineering mathematics. Rational spline, as a family member of spline functions, integrates advantages of spline function and rational approximation, and is more flexible and more general. So in recent years, rational interpolation splines have received more and more attention.First, this thesis gives a necessary and sufficient condition to judge the existence of the rational interpolating function, and obtains a concrete expression form of the rational interpolating function. By using the generalized Vadermonde matrix, we obtain a necessary and sufficient condition to judge the existence of the osculatory rational interpolating function and give its representation formula. A few examples are given to judge and compute the interpolating problem.A kind of rational quadratic Hermite interpolation spline is also derived. This rational spline not only belongs to C 1 in the interpolation interval, but also has the precision of simple polynomial. Then the jump values of deviation in nodes are discussed, and the error estimations of jump values are obtained when the interpolated function has different smoothness. At last the error estimation of rational quadratic Hermite interpolation spline is given,and a necessary and sufficient condition is derived for the interpolating curves to be convex in the interpolating intervals.