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Fast interpolation and cubature at the Padua points ∗

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Fast interpolation and cubature at the Padua points ∗

Marco Caliari, Stefano De Marchi

Dept. of Computer Science, University of Verona (Italy) Alvise Sommariva, Marco Vianello

†

Dept. of Pure and Applied Mathematics, University of Padova (Italy)

Abstract

The Padua points are the first known example of optimal polynomial interpolation points in two variables, with a Lebesgue constant increas- ing like log square of the degree. Moreover, they generate a nontenso- rial Clenshaw-Curtis-like cubature formula, which is competitive with the tensor-product Gauss-Legendre formula and even with the few known min- imal formulas, on integrands that are not “too regular”. Here we com- pare in Matlab two fast algorithms for interpolation and cubature at the Padua points, based on representation in the bivariate Chebyshev orthog- onal basis via matrix products or via the FFT. The former turns out to be superior up to degree n =?, and thus an hybrid algorithm comes out as the most reasonable choice.

2000 AMS subject classification: ???.

Keywords: bivariate Lagrange interpolation, Padua points, bivariate Chebyshev or- thogonal basis, nontensorial Clenshaw-Curtis-like cubature, matrix product formula- tion, FFT.

∗Work supported by the “ex-60%” funds of the University of Padova, and by the INdAM GNCS.

†Corresponding author: e-mail: marcov@math.unipd.it

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