## A fast algorithm for computing the mock-Chebyshev nodes

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

• 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

#### 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 $\left[-1,1\right]$ 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 $\left(n+1\right)$ points from an equispaced grid with $\mathcal{O}\left({n}^{2}\right)$ points chosen to mimic the non-uniform $\left(n+1\right)$-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 $\left[-1,1\right]$. A discussion of bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points in ${\left[-1,1\right]}^{2}$is given and numerical results are also provided.