• Non ci sono risultati.

Near-optimal interpolation and quadrature in two variables: the Padua points

N/A
N/A
Info
Scarica
Protected

Academic year: 2021

Condividi "Near-optimal interpolation and quadrature in two variables: the Padua points"

Copied!
1
0
0

Caricamento.... (Visualizza il testo completo ora)

Scarica adesso ( 1 pagine )

Testo completo

(1)

Near-optimal interpolation and quadrature in two variables:

the Padua points ∗

Marco Vianello

Department of Pure and Applied Mathematics University of Padova

via Trieste 63, 35121 - Padova (Italy)

Poster subject: Numerical Analysis and Scientific Computing

Abstract

The Padua points are the first known example of optimal points for total degree polynomial interpolation in two variables, with a Lebesgue constant increasing like log

2

of the degree; cf. [1, 2, 3]. Moreover, they gen- erate a nontensorial Clenshaw-Curtis-like cubature formula, which turns out to be competitive with the tensorial Gauss-Legendre formula and even with the few known minimal formulas in the square, on integrands that are not “too regular”; cf. [4]. Such a behavior is analogous to that of the univariate Clenshaw-Curtis formula; cf. [5]. We present a survey about properties, software implementations and applications of interpolation and numerical cubature at the Padua points.

References

[1] L. Bos, M. Caliari, S. De Marchi, M. Vianello and Y. Xu, Bivariate Lagrange interpolation at the Padua points: the generating curve approach, J. Approx. Theory 143 (2006), 15–

25.

[2] L. Bos, S. De Marchi, M. Vianello and Y. Xu, Bivariate Lagrange interpolation at the Padua points: the ideal theory approach, Numer. Math. 108 (2007), 43–57.

[3] M. Caliari, S. De Marchi and M. Vianello, Bivariate Lagrange interpolation at the Padua points: computational aspects, J. Comput. Appl. Math., published online 23 October 2007.

[4] A. Sommariva, M. Vianello and R. Zanovello, Nontensorial Clenshaw-Curtis cubature, submitted (downloadable at: http://www.math.unipd.it/∼marcov/publications.html).

[5] L.N. Trefethen, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev., to ap- pear.

∗

Work in collaboration with M. Caliari and S. De Marchi (University of Verona), A. Som-

mariva (University of Padova); supported by the “ex-60%” funds of the University of Padova

and by the GNCS-INdAM.

Riferimenti

Scarica adesso ( PDF - 1 pagine - 37.23 KB )

Documenti correlati

Bivariate Lagrange interpolation at the Padua points: Computational aspects ?

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

Bivariate interpolation at Xu points

We have recently proved that the Lebesgue constant of these points grows like log 2 (n), n being the de- gree, as with the best known points for the square, and we have imple- mented

Efficient implementation of bivariate interpolation and cubature at Padua points

In this talk we present a stable and efficient Fortran BLAS-based implemen- tation and a new Matlab FFT-based implementation of the Lagrange interpo- lation formula and cubature at

We present a stable and efficient Fortran implementation of polynomial interpolation at the “Padua points” on the square [−1, 1]

In this paper we present a stable and efficient Fortran implementation of the Lagrange interpolation formula at the Padua points, with cost O(n 3 ) flops for the evaluation (once

Near optimal interpolation and quadrature in two variables: the Padua points

The Padua points, recently studied during an international collaboration at the University of Padua, are the first known example of near-optimal point set for bivariate polynomial

ON THE COMPUTATION OF SETS OF POINTS WITH LOW LEBESGUE CONSTANT ON THE UNIT DISK

In section 3, we evaluate the Lebesgue constants on some sets that have a different structure, showing that there exist at least a set with at most quadratic growth of the

Polynomial interpolation and quadrature on subregions of the sphere

SIAM Conference on Mathematical and Computational Issues in the Geosciences (SIAM GS13) Padova (Italy) June 17-20 2013M. Polynomial interpolation and quadrature on subregions of

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

Abstract We have implemented in Matlab/Octave two fast algorithms for bivariate Lagrange interpolation at the so-called Padua points on rectangles, and the corresponding versions