A fast algorithm for computing the mock-Chebyshev nodes


İBRAHİMOĞLU B. A.

Journal of Computational and Applied Mathematics, vol.373, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 373
  • Publication Date: 2020
  • Doi Number: 10.1016/j.cam.2019.07.001
  • Title of Journal : Journal of Computational and Applied Mathematics
  • Keywords: Interpolation, Runge phenomenon, Mock-Chebyshev interpolation

Abstract

Runge Phenomenon which is a very well-known example and published by C. Runge in 1901 is as follows: polynomial interpolation of a function f" role="presentation" >f, using equidistant interpolation points on [−1,1]" role="presentation" >[1,1] could diverge on certain parts of this interval even if f" role="presentation" >f is analytic anywhere on the interval. Among all the techniques that have been proposed to defeat this phenomenon in the literature of approximation theory, there is the mock-Chebyshev interpolation on a grid: a subset of (n+1)" role="presentation" >(n+1) points from an equispaced grid with O(n2)" role="presentation" >O(n2) points chosen to mimic the non-uniform (n+1)" role="presentation" >(n+1)-point Chebyshev–Lobatto grid [1]. This study suggests a fast algorithm for computing the mock-Chebyshev nodes using the distance between each pair of consecutive points. The complexity of the algorithm is O(n)" role="presentation" >O(n), where n+1" role="presentation" >n+1 is the number of the Chebyshev–Lobatto nodes on the interval [−1,1]" role="presentation" >[1,1]. A discussion of bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points in [−1,1]2" role="presentation" >[1,1]2is given and numerical results are also provided.