钱江,王凡,傅卓佳,吴云标.基于二元连分式的散乱数据递推插值格式[J].数学研究及应用,2016,36(5):583~607
基于二元连分式的散乱数据递推插值格式
Recursive Schemes for Scattered Data Interpolation via Bivariate Continued Fractions
投稿时间:2016-01-10  最后修改时间:2016-03-18
DOI:10.3770/j.issn:2095-2651.2016.05.010
中文关键词:  散乱数据插值  二元连分式  三项递推关系式  特征定理  径向基函数
英文关键词:Scattered data interpolation  bivariate continued fraction  three-term recurrence relation  characterization theorem  radial basis function
基金项目:国家自然科学基金天元专项基金(Grant No.11426086), 中央高校基本科研业务费项目 (Grant No.2016B08714), 江苏省自然科学基金青年基金 (Grant No.BK20160853).
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 
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中文摘要:
      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.
英文摘要:
      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.
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