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Bivariate Polynomial Interpolation over Nonrectangular Meshes

  • Jiang Qian (a1), Sujuan Zheng (a1), Fan Wang (a2) and Zhuojia Fu (a3)
Abstract
Abstract

In this paper, bymeans 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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Copyright
Corresponding author
*Corresponding author. Email addresses:qianjianghhu@sina.com (J. Qian), zsj@hhu.edu.cn (S.-J. Zheng), wangfan@njau.edu.cn (F. Wang), paul212063@hhu.edu.cn (Z.-J. Fu)
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[2] B. A.Bailey , Multivariate polynomial interpolation and sampling in Paley-Wiener spaces, J. Approx. Theory, vol. 164 (2012), pp. 460487.

[3] J.Chai , N.Lei , Y.Li , and P.Xia , The proper interpolation space for multivariate Birkhoff interpolation, J. Comput. Appl. Math., vol. 235 (2011), pp. 32073214.

[5] N.Dyn , and M. S.Floater , Multivariate polynomial interpolation on lower sets, J. Approx. Theory, vol. 177 (2014), pp. 3442.

[6] M.Gasca , T.Sauer , On the history of multivariate polynomial interpolation, J. Comput. and Appl. Math., vol. 122 (2000), pp. 2335.

[7] M. J.Lai , Convex preserving scattered data interpolation using bivariate C1 cubic splines, J. Comput. Appl. Math., vol. 119 (2000), pp. 249258.

[9] W. R.Madych , An estimate for multivariate interpolation II, J. Approx. Theory, vol. 142 (2006), pp. 116128.

[10] A.Mazroui , D.Sbibih , and A.Tijini , Recursive computation of bivariate Hermite spline interpolants, Appl. Numer. Math., vol. 57 (2007), pp. 962973.

[12] J.Qian , R. H.Wang , C. J.Li , The bases of the Non-uniform cubic spline space , Numer. Math. Theor. Meth. Appl. vol. 5(4) (2012), pp. 635652.

[18] R. H.Wang , J.Qian , On branched continued fractions rational interpolation over pyramid-typed grids, Numer. Algor., vol. 54 (2010), pp. 4772.

[19] R. H.Wang , J.Qian , Bivariate polynomial and continued fraction interpolation over orthotriples, Applied Mathematics and Computation, vol. 217 (2011), pp. 76207635.

[22] T. H.Zhou , and M. J.Lai , Scattered data interpolation by bivariate splines with higher approximation order, J. Comput. Appl. Math., vol. 242 (2013), pp. 125140.

[23] C. G.Zhu , and R. H.Wang , Lagrange interpolation by bivariate splines on cross-cut partitions, J. Comp. Appl. Math., vol. 195 (2006), pp. 326340.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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