Bivariate Lagrange interpolation at the Padua points: Computational aspects

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 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.

[1]  Péter Vértesi,et al.  Exact order of the Lebesgue constants for bivariate Lagrange interpolation at certain node-systems , 2009 .

[2]  Péter Vértesi,et al.  On multivariate projection operators , 2009, J. Approx. Theory.

[3]  Jesús M. Carnicer,et al.  Interpolation lattices in several variables , 2006, Numerische Mathematik.

[4]  Marco Vianello,et al.  Bivariate interpolation at Xu points: results, extensions and applications. , 2006 .

[5]  Yuan Xu,et al.  Lagrange Interpolation on Chebyshev Points of Two Variables , 1996 .

[6]  Marco Vianello,et al.  Bivariate Lagrange interpolation at the Padua points: The generating curve approach , 2006, J. Approx. Theory.

[7]  Marco Vianello,et al.  Algorithm xxx: Padua2D: Lagrange Interpolation at Padua Points on Bivariate Domains , 2008 .

[8]  Yuan Xu.  Christoffel functions and Fourier series for multivariate orthogonal polynomials , 1995 .

[9]  Tomas Sauer,et al.  Polynomial interpolation in several variables , 2000, Adv. Comput. Math..

[10]  N. Levenberg,et al.  Multivariate simultaneous approximation , 2002 .

[11]  Marco Vianello,et al.  Bivariate Lagrange interpolation at the Padua points: the ideal theory approach , 2007, Numerische Mathematik.

[12]  Marco Vianello,et al.  Bivariate polynomial interpolation on the square at new nodal sets , 2005, Appl. Math. Comput..

[13]  Lloyd N. Trefethen,et al.  An Extension of MATLAB to Continuous Functions and Operators , 2004, SIAM J. Sci. Comput..

[14]  Marco Vianello,et al.  On the Lebesgue constant for the Xu interpolation formula , 2006, J. Approx. Theory.

[15]  Ron Brown,et al.  Algorithm 792: accuracy test of ACM algorithms for interpolation of scattered data in the plane , 1999, TOMS.

[16]  XuYuan,et al.  Bivariate Lagrange interpolation at the Padua points: the ideal theory approach , 2007 .

[17]  Marco Vianello,et al.  A Numerical Study of the Xu Polynomial Interpolation Formula in Two Variables , 2005, Computing.

[18]  Borislav Bojanov,et al.  On polynomial interpolation of two variables , 2003, J. Approx. Theory.

[19]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .