具有承袭性的高阶导数有理插值算法

High Order Derivative Rational Interpolation Algorithm With Heredity

切触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Newton(牛顿)多项式插值的承袭性和分段组合的方法,构造出了一种无极点且满足高阶导数插值条件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.最后给出误差估计,并通过数值实例说明该算法具有承袭性、计算量低、便于编程等特点.

Osculatory rational interpolation was an important theme of function approximation,meanwhile,reducing the degree and solving the existence of the osculatory rational interpolation function made a crucial problem for rational interpolation. The previous algorithms of osculatory rational interpolation functions mostly depended on the continued fraction with conditional feasibility and high computation complexity. Based on heredity of the Newton interpolation and the method of piecewise combination,an osculatory rational interpolation function without real poles was constructed to meet the condition of high order derivative interpolation,and was in turn extended to the vector-valued cases. It not only solved the existence problem for the osculatory rational interpolation function,but reduced the degree of the rational function. Furthermore,the error estimates of the new algorithm was given. Results of the numerical examples illustrate the new algorithm＇s heredity,low computation complexity and easy programmability.