Caliari, Marco; Vianello, Marco; De Marchi, Stefano; Montagna, Roberto Hyper2d: A numerical code for hyperinterpolation on rectangles. (English) Zbl 1105.65307 Appl. Math. Comput. 183, No. 2, 1138-1147 (2006). Summary: Hyperinterpolation at Morrow-Patterson-Xu cubature points for the product Chebyshev measure provides a simple and powerful polynomial approximation method on rectangles. Here, we present an accurate and efficient Matlab/Octave implementation of the hyperinterpolation formula, accompanied by several numerical tests. Cited in 7 Documents MSC: 65D05 Numerical interpolation 41A05 Interpolation in approximation theory 41A63 Multidimensional problems Keywords:bivariate hyperinterpolation; minimal cubature; Morrow-Patterson-Xu points; Lebesgue constant; numerical examples; polynomial approximation Software:XuPad2D; Octave; Chebfun; Matlab; Hyper2d; Algorithm 792 PDFBibTeX XMLCite \textit{M. Caliari} et al., Appl. Math. Comput. 183, No. 2, 1138--1147 (2006; Zbl 1105.65307) Full Text: DOI References: [1] 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 [2] 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 [3] 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 [4] L. Bos, S. De Marchi, M. Vianello, On the Lebesgue constant for the Xu interpolation formula, J. Approx. Theory (in press) (available online 17 April 2006).; L. Bos, S. De Marchi, M. Vianello, On the Lebesgue constant for the Xu interpolation formula, J. Approx. Theory (in press) (available online 17 April 2006). · Zbl 1099.41002 [5] 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 [6] M. Caliari, S. De Marchi, M. Vianello, Hyperinterpolation on the square, J. Comput. Appl. Math. (in press) (preprint available at www.math.unipd.it/ marcov/publications.html; M. Caliari, S. De Marchi, M. Vianello, Hyperinterpolation on the square, J. Comput. Appl. Math. (in press) (preprint available at www.math.unipd.it/ marcov/publications.html · Zbl 1151.65014 [7] M. Caliari, S. De Marchi, R. Montagna, M. Vianello, Hyper2d: a Matlab/Octave interface for hyperinterpolation on rectangles, downloadable from www.math.unipd.it/ marcov/software.html; M. Caliari, S. De Marchi, R. Montagna, M. Vianello, Hyper2d: a Matlab/Octave interface for hyperinterpolation on rectangles, downloadable from www.math.unipd.it/ marcov/software.html · Zbl 1105.65307 [8] 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 [9] Franke, R., Scattered data interpolation: tests of some methods, Math. Comput., 38, 181-200 (1982) · Zbl 0476.65005 [10] K. Hesse, I.H. Sloan, Hyperinterpolation on the sphere, UNSW School of Mathematics, preprint AMR05/23, 2005.; K. Hesse, I.H. Sloan, Hyperinterpolation on the sphere, UNSW School of Mathematics, preprint AMR05/23, 2005. · Zbl 1194.41044 [11] C. Moler, MATLAB incorporates LAPACK. Increasing the speed and capabilities of matrix computation, MATLAB News & Notes - Winter 2000.; C. Moler, MATLAB incorporates LAPACK. Increasing the speed and capabilities of matrix computation, MATLAB News & Notes - Winter 2000. [12] Morrow, C. R.; Patterson, T. N.L., Construction of algebraic cubature rules using polynomial ideal theory, SIAM J. Numer. Anal., 15, 953-976 (1978) · Zbl 0402.65013 [13] Reimer, M., Multivariate Polynomial Approximation, International Series of Numerical Mathematics, vol. 144 (2003), Birkhäuser: Birkhäuser Basel · Zbl 1038.41002 [14] Renka, R. J., 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 [15] Sloan, I. H., Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83, 238-254 (1995) · Zbl 0839.41006 [16] 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.