We discuss and compare two greedy algorithms, that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called "Approximate Fekete Points" by QR factorization with column pivoting of Vandermonde-like matrices. The second computes Discrete Leja Points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted from Weakly Admissible Meshes.2000 AMS subject classification: 41A10, 41A63, 65D05.
Abstract. Using the concept of Geometric Weakly Admissible Meshes (see §2 below) together with an algorithm based on the classical QR factorization of matrices, we compute efficient points for discrete multivariate least squares approximation and Lagrange interpolation.
We propose a numerical method (implemented in Matlab) for computing
approximate Fekete points on compact multivariate domains. It relies
on the search of maximum volume submatrices of Vandermonde matrices
computed on suitable discretization meshes, and uses a simple greedy algorithm
based on QR factorization with column pivoting. The method
gives also automatically an algebraic cubature formula, provided that the
moments of the underlying polynomial basis are known. Numerical tests
are presented for the interval and the square, which show that approximate
Fekete points are well-suited for polynomial interpolation and cubature
We have implemented in Matlab a Gauss-like cubature formula over convex, nonconvex or even multiply connected polygons. The formula is exact for polynomials of degree at most 2n − 1 using N ∼ mn 2 nodes, m being the number of sides that are not orthogonal to a given line, and not lying on it. It does not need any preprocessing like triangulation of the domain, but relies directly on univariate Gauss-Legendre quadrature via Green's integral formula. Several numerical tests are presented.2000 AMS subject classification: 65D32.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.