A fast algorithm for computing the mock-Chebyshev nodes

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, using equidistant interpolation points on [-1, 1] could diverge on certain parts of this interval even if 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) points from an equispaced grid with O(n(2)) points chosen to mimic the non-uniform (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), where n + 1 is the number of the Chebyshev-Lobatto nodes on the interval [-1, 1]. A discussion of bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points in [-1, 1](2) is given and numerical results are also provided. (C) 2019 Elsevier B.V. All rights reserved.