×

Bivariate Lagrange interpolation at the Padua points: Computational aspects. (English) Zbl 1152.65018

Summary: The so-called “Padua points” give a simple, geometric and explicit construction of bivariate polynomial interpolation in the square. Moreover, the associated Lebesgue constant has minimal order of growth \(\mathcal O(\log ^2(n))\). Here we show four families of Padua points for interpolation at any even or odd degree \(n\), and we present a stable and efficient implementation of the corresponding Lagrange interpolation formula, based on the representation in a suitable orthogonal basis. We also discuss extension of (non-polynomial) Padua-like interpolation to other domains, such as triangles and ellipses; we give complexity and error estimates, and several numerical tests.

MSC:

65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
41A63 Multidimensional problems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bagby, T.; Bos, L.; Levenberg, N., Multivariate simultaneous approximation, Constr. Approx., 18, 569-577 (2002) · Zbl 1025.41002
[2] Battles, Z.; Trefethen, L. N., An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25, 1743-1770 (2004) · Zbl 1057.65003
[3] Bojanov, B.; Xu, Y., On polynomial interpolation of two variables, J. Approx. Theory, 120, 267-282 (2003) · Zbl 1029.41001
[4] Bos, L.; Caliari, M.; De Marchi, S.; Vianello, M.; Xu, Y., Bivariate Lagrange interpolation at the Padua points: The generating curve approach, J. Approx. Theory, 143, 1, 15-25 (2006) · Zbl 1113.41001
[5] Bos, L.; Caliari, M.; De Marchi, S.; Vianello, M., Bivariate interpolation at Xu points: Results, extensions and applications, Electron. Trans. Numer. Anal., 25, 1-16 (2006) · Zbl 1115.65304
[6] Bos, L.; Caliari, M.; De Marchi, S.; Vianello, M., A numerical study of the Xu polynomial interpolation formula, Computing, 76, 311-324 (2005) · Zbl 1087.65009
[7] L. Bos, S. De Marchi, M. Vianello, Y. Xu, Bivariate Lagrange interpolation at the Padua points: The ideal theory approach, Numer. Math. (2007). Available on line, DOI 10.1007/s00211-007-0112-z; L. Bos, S. De Marchi, M. Vianello, Y. Xu, Bivariate Lagrange interpolation at the Padua points: The ideal theory approach, Numer. Math. (2007). Available on line, DOI 10.1007/s00211-007-0112-z · Zbl 1126.41002
[8] Caliari, M.; De Marchi, S.; Montagna, R.; Vianello, M., XuPad2D: A Matlab code for hyperinterpolation/interpolation at Xu/Padua points on rectangles, available at
[9] Caliari, M.; De Marchi, S.; Vianello, M., Padua2D: A Fortran code for bivariate Lagrange interpolation at Padua points, available at
[10] Caliari, M.; De Marchi, S.; Vianello, M., Bivariate polynomial interpolation on the square at new nodal sets, Appl. Math. Comput., 165, 261-274 (2005) · Zbl 1081.41001
[11] Carnicer, J. M.; Gasca, M.; Sauer, T., Interpolation lattices in several variables, Numer. Math., 102, 559-581 (2006) · Zbl 1087.41030
[12] B. Della Vecchia, G. Mastroianni, P. Vertesi, Exact order of the Lebesgue constants for bivariate Lagrange interpolation at certain node systems, Studia Sci. Math. Hungar. (2007) (in press); B. Della Vecchia, G. Mastroianni, P. Vertesi, Exact order of the Lebesgue constants for bivariate Lagrange interpolation at certain node systems, Studia Sci. Math. Hungar. (2007) (in press) · Zbl 1240.41007
[13] Dunkl, C. F.; Xu, Y., (Orthogonal Polynomials of Several Variables. Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 81 (2001), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0964.33001
[14] Gasca, M.; Sauer, T., Polynomial interpolation in several variables, Adv. Comput. Math., 12, 377-410 (2000) · Zbl 0943.41001
[15] Renka, R. J.; Brown, R., Algorithm 792: Accuracy tests of ACM algorithms for interpolation of scattered data in the plane, ACM Trans. Math. Software, 25, 79-93 (1999) · Zbl 0963.65014
[16] L. Szili, P. Vertesi, On multivariate projection operators (2007) (in press); L. Szili, P. Vertesi, On multivariate projection operators (2007) (in press) · Zbl 1171.42001
[17] Xu, Y., Christoffel functions and Fourier series for multivariate orthogonal polynomials, J. Approx. Theory, 82, 205-239 (1995) · Zbl 0874.42018
[18] Xu, Y., Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory, 87, 220-238 (1996) · Zbl 0864.41002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.