钱江,王凡,郭庆杰.二元非张量积型连分式插值[J].计算机科学,2018,45(3):83-91
二元非张量积型连分式插值
Bivariate Non-tensor-product-typed Continued Fraction Interpolation
投稿时间:2017-07-18  修订日期:2017-09-15
DOI:10.11896/j.issn.1002-137X.2018.03.014
中文关键词:  散乱数据插值,二元连分式,径向基函数,非张量积型,特征定理
英文关键词:Scattered data interpolation,Bivariate continued fraction,Radial basis function,Non-tensor product type,Characterization theorem
基金项目:本文受国家自然科学基金天元专项基金(11426086),江苏省自然科学基金青年基金项目(BK20160853),河海大学中央高校业务费基金项目(2016B08714)资助
作者单位
钱江 河海大学理学院 南京211100 
王凡 南京农业大学工学院基础课部 南京210031 
郭庆杰 大连理工大学盘锦校区基础教学部 辽宁 盘锦124221 
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中文摘要:
      首先,基于新的二元非张量积型逆差商递推算法,分别建立奇数与偶数个插值节点上的二元连分式插值格式,并得到被插函数的两类恒等式。接着,利用连分式三项递推关系式,分别确定渐近式的分子和分母的次数,即特征定理,并给出推导分子、分母的递推算法。同时,研究表明所提连分式的分子、分母次数分别小于相应的二元Thiele型插值连分式分子、分母次数,这主要是因为所提连分式插值减少了对冗余的插值节点的采用。然后,从计算复杂性的角度出发,所提二元有理函数插值的计算量小于相同插值节点上的径向基函数插值的计算量。最后,数值算例表明所提二元连分式插值方法有效且可行,同时也揭示了即使插值节点集合不变,所提插值连分式的表达式也会随着插值节点顺序的改变而改变。
英文摘要:
      Based on the new recursive algorithms of bivariate non-tensor-product-typed inverse divided differences,the scattered data interpolating schemes via bivariate continued fractions were established in the case of odd and even interpolating nodes,respectively.Then two equivalent identities of the interpolated function were obtained.Moreover,by means of the three-term recurrence relations,the degrees of the numerators and denominators were determined,i.e.,the characterization theorem,so do the corresponding recursive algorithms.Meanwhile,compared with the degrees of the numerators and denominators of the well-known bivariate Thiele-typed interpolating continued fractions,those of the presented bivariate rational interpolating functions are much lower respectively,due to the reduction of redundant interpolating nodes.Furthermore,the operation count for the rational function interpolation is smaller than that of radial basis function interpolation from the aspect of complexity of the operations.Finally,some numerical examples show that it’s valid for the recursive continued fraction interpolation,and imply that these interpolating continued fractions change as the order of the interpolating nodes change,although the node collection is invariant.
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