Recursive Schemes for Scattered Data Interpolation via Bivariate Continued Fractions

DOI：10.3770/j.issn:2095-2651.2016.05.010

 作者 单位 钱江 河海大学理学院,江苏 南京 211100 王凡 南京农业大学工学院基础课部, 江苏 南京 210031 傅卓佳 河海大学力学与材料学院, 江苏 南京 211100 吴云标 河海大学文天学院基础课部, 安徽 马鞍山 243031

本文首先基于新的非张量积型偏逆差商递推算法,分别构造奇数与偶数个插值节点上的二元连分式散乱数据插值格式,进而得到被插函数与二元连分式间的恒等式.接着,利用连分式三项递推关系式,提出特征定理来研究插值连分式的分子分母次数.然后,数值算例表明新的递推格式可行有效,同时,通过比较二元Thiele型插值连分式的分子分母次数,发现新的二元插值连分式的分子分母次数较低,这主要归功于节省了冗余的插值节点. 最后,计算此有理函数插值所需要的四则运算次数少于计算径向基函数插值.

In the paper, firstly, based on new non-tensor-product-typed partially inverse divided differences algorithms in a recursive form, scattered data interpolating schemes are constructed via bivariate continued fractions with odd and even nodes, respectively. And equivalent identities are also obtained between interpolated functions and bivariate continued fractions. Secondly, by means of three-term recurrence relations for continued fractions, the characterization theorem is presented to study on the degrees of the numerators and denominators of the interpolating continued fractions. Thirdly, some numerical examples show it feasible for the novel recursive schemes. Meanwhile, compared with the degrees of the numerators and denominators of bivariate Thiele-typed interpolating continued fractions, those of the new bivariate interpolating continued fractions are much low, respectively, due to the reduction of redundant interpolating nodes. Finally, the operation count for the rational function interpolation is smaller than that for radial basis function interpolation.