A numerical study of the Xu polynomial 2 interpolation formula in two variables 3
نویسندگان
چکیده
8 In his paper " Lagrange interpolation on Chebyshev points of two variables " (J. 9 220–238), Y. Xu proposed a set of Chebyshev like points for polynomial interpolation in the square 10 [−1, 1] 2 , and derived a compact form of the corresponding Lagrange interpolation formula. We inves-11 tigate computational aspects of the Xu polynomial interpolation formula like numerical stability and 12 efficiency, the behavior of the Lebesgue constant, and its application to the reconstruction of various test 13 functions.
منابع مشابه
Near-optimal interpolation and quadrature in two variables: the Padua points∗
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 of the degree; cf. [1, 2, 3]. Moreover, they generate 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 ...
متن کاملLagrange Interpolation on Chebyshev Points of Two Variables
We study interpolation polynomials based on the points in [−1, 1]× [−1, 1] that are common zeros of quasi-orthogonal Chebyshev polynomials and nodes of near minimal degree cubature formula. With the help of the cubature formula we establish the mean convergence of the interpolation polynomials. 1991 Mathematics Subject Classification: Primary 41A05, 33C50.
متن کاملOn the Sauer-Xu formula for the error in multivariate polynomial interpolation
Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree ≤ n in d variables, at a ‘correct’ point set. It is the purpose of this note to give a short proof of a remarkable formula for the error in polynomial interpolation given in [3]. In [3], Sauer and Xu consider interpolation from...
متن کاملDiscrete Fourier Analysis, Cubature, and Interpolation on a Hexagon and a Triangle
Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonome...
متن کاملOn the numerical stability of the second barycentric formula for trigonometric interpolation in shifted equispaced points
We consider the numerical stability of the second barycentric formula for evaluation at points in [0, 2π ] of trigonometric interpolants in an odd number of equispaced points in that interval. We show that, contrary to the prevailing view, which claims that this formula is always stable, it actually possesses a subtle instability that seems not to have been noticed before. This instability can ...
متن کامل