Bivariate Lagrange interpolation at the Padua points: Computational aspects

Abstract

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 (nonpolynomial) Padua-like interpolation to other domains, such as triangles and ellipses; we give complexity and error estimates, and several numerical tests

Nontensorial Clenshaw–Curtis cubature

“…A further improvement can be obtained by integrating the interpolation (instead of an hyperinterpolation) polynomial at the Padua points (24). This procedure is well located in the present framework, since it has been shown in [12] that the interpolation polynomial can be written as…”

“…The first family is given by the "MorrowPatterson-Xu" points, which have been studied in the contexts of minimal cubature [27,17,44,1], interpolation [44,2] and hyperinterpolation [11,9,13]. The second family, termed the "Padua points", has been recently studied in the interpolation context [10,3,5,12,14]. In what follows we use the following notation for the set of ν + 1 one-dimensional Chebyshev-Gauss-Lobatto nodes…”

“…They have been introduced experimentally in [10], and then studied from the theoretical [3,5] and the computational [12,14] point of view. In fact, the Padua points are the first known example of non tensor-product optimal interpolation in two variables, since they are unisolvent and their Lebesgue constant has optimal order of growth O (log n) 2 .…”

“…It is also worth recalling that there are other three families of Padua points, obtained from (4) by successive 90-degree rotations of the square. We refer to [12] for a full description.…”

“…[27,44,2,9,11]) and the "Padua points" (cf. [10,3,5,12,14]). Finally, we compare the performance of the resulting formulas (implemented in Matlab) with tensor-product Clenshaw-Curtis and Gauss-Legendre formulas on several test functions.…”