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(n2) 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.