Title of Journal :Journal of Computational and Applied Mathematics

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 $\mathcal{O}\left({n}^{2}\right)$ 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 $\mathcal{O}\left(n\right)$, 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.