A new fast algorithm for computing the mock-Chebyshev nodes


İBRAHİMOĞLU B. A.

Applied Numerical Mathematics, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1016/j.apnum.2024.03.002
  • Journal Name: Applied Numerical Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, DIALNET
  • Keywords: Interpolation, Mock-Chebyshev points, Mock-Padua points
  • Yıldız Technical University Affiliated: Yes

Abstract

Interpolation by polynomials on equispaced points is not always convergent due to the Runge phenomenon, and also, the interpolation process is exponentially ill-conditioned. By taking advantage of the optimality of the interpolation processes on the Chebyshev-Lobatto nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev nodes for polynomial interpolation. Mock-Chebyshev nodes asymptotically follow the Chebyshev distribution, and they are selected from a sufficiently large set of equispaced nodes. However, there are few studies in the literature regarding the computation of these points. In a recent paper [1], we have introduced a fast algorithm for computing the mock-Chebyshev nodes for a given set of (n+1) Chebyshev-Lobatto points using the distance between each pair of consecutive points. In this study, we propose a modification of the algorithm by changing the function to compute the quotient of the distance and show that this modified algorithm is also fast and stable; and gives a more accurate grid satisfying the conditions of a mock-Chebyshev grid with the complexity being O(n). Some numerical experiments using the points obtained by this modified algorithm are given to show its effectiveness and numerical results are also provided. A bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points is discussed.