Volume 9, Issue 4
Bivariate Polynomial Interpolation over Nonrectangular Meshes

Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 549-578.

Published online: 2016-09

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• Abstract

In this paper, by means of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

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@Article{NMTMA-9-549, author = {}, title = {Bivariate Polynomial Interpolation over Nonrectangular Meshes}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2016}, volume = {9}, number = {4}, pages = {549--578}, abstract = {

In this paper, by means of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2016.y15027}, url = {http://global-sci.org/intro/article_detail/nmtma/12389.html} }
TY - JOUR T1 - Bivariate Polynomial Interpolation over Nonrectangular Meshes JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 549 EP - 578 PY - 2016 DA - 2016/09 SN - 9 DO - http://doi.org/10.4208/nmtma.2016.y15027 UR - https://global-sci.org/intro/article_detail/nmtma/12389.html KW - AB -

In this paper, by means of a new recursive algorithm of non-tensor-product-typed divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimation is worked out, and an equivalence is obtained between high-order non-tensor-product-typed divided differences and high-order partial derivatives in the case of odd and even interpolating nodes, respectively. Thirdly, several numerical examples illustrate the recursive algorithms valid for the non-tensor-product-typed interpolating polynomials, and disclose that these polynomials change as the order of the interpolating nodes, although the node collection is invariant. Finally, from the aspect of computational complexity, the operation count with the bivariate polynomials presented is smaller than that with radial basis functions.

Jiang Qian, Sujuan Zheng, Fan Wang & Zhuojia Fu. (2019). Bivariate Polynomial Interpolation over Nonrectangular Meshes. Numerical Mathematics: Theory, Methods and Applications. 9 (4). 549-578. doi:10.4208/nmtma.2016.y15027
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