## 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
• Journal Name: Journal of Computational and Applied Mathematics
• Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, Metadex, zbMATH, DIALNET, Civil Engineering Abstracts
• 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 [&#x2212;1,1]" role="presentation" >$\left[-1,1\right]$ 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" >$\left(n+1\right)$ points from an equispaced grid with O(n2)" role="presentation" >$\mathcal{O}\left({n}^{2}\right)$ points chosen to mimic the non-uniform (n+1)" role="presentation" >$\left(n+1\right)$-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 O(n)" role="presentation" >$\mathcal{O}\left(n\right)$, where n+1" role="presentation" >$n+1$ is the number of the Chebyshev–Lobatto nodes on the interval [&#x2212;1,1]" role="presentation" >$\left[-1,1\right]$. A discussion of bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points in [&#x2212;1,1]2" role="presentation" >${\left[-1,1\right]}^{2}$is given and numerical results are also provided.