Numerical Algorithms

, Volume 70, Issue 2, pp 249–267 | Cite as

Reusing Chebyshev points for polynomial interpolation

Original Paper

Abstract

Let \({X_{l}^{C}}\) be the set of l Chebyshev points in the interval [−1,1]. If n and n0 are such that n=2mn0−1 for some positive integer m, then \(X_{n_{0}}^{C} \subset {X_{n}^{C}}\). This property can be utilized in order to reuse previous function values when one wants to increase the degree of the polynomial interpolation. For given n0 and n, n>n0, where n≠2mn0−1, we give a simple procedure to build a set of n points in the interval [−1,1] that include the set of n0 Chebyshev points and have favorable interpolation properties. We show that the nodal polynomial for these points has a maximum norm that is at most O(n) times larger than that of the Chebyshev points of the same size. We also present numerical evidence suggesting that the Lebesgue constant for these points grows at most linearly in n.

Keywords

Polynomial interpolation Chebyshev points Lebesgue constant 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baglama, J., Calvetti, D., Reichel, L.: Fast Leja points. Electron. Trans. Numer. Anal. 7, 124–140 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Bernstein, S.: Sur la limitation des valeurs d’un polynôme. Bull. Acad. Sci. de l’URSS 8, 1025–1050 (1931)MATHGoogle Scholar
  3. 3.
    Boyd, J.P., Xu, F.: Divergence (Runge phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock-Chebyshev subset interpolation. Appl. Math. Comput. 210, 158–168 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boyd, J.P., Gildersleeve, K.W.: Numerical experiments on the condition number of the interpolation matrices for radial basis functions. Appl. Numer. Math. 61, 443–459 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Erdös, P.: Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hung. 12, 235–244 (1961)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. SIAM. 978-0-898716-34-4 (2007)Google Scholar
  8. 8.
    Jung, J., Stefan, W.: A simple regularization of the polynomial interpolation for the resolution of the Runge phenomenon. J. Sci. Comput. 46, 225–242 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Leja, F.: Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme. Ann. Polon. Math. 3, 8–13 (1957)MathSciNetMATHGoogle Scholar
  10. 10.
    Narcowich, F.J., Ward, J.D.: Norms of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64, 69–94 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Salzer, H.E.: Lagrangian interpolation at the Chebyshev points \(x_{n,ν} = \cos (\nu \pi /n), \nu = 0(1)n\); some unnoted advantages. Comput. J. 15, 156–159 (1972)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Taylor, R.: Lagrange interpolation on Leja points, Graduate School Theses and Dissertations. http://scholarcommons.usf.edu/etd/530 (2008)
  14. 14.
    Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Trefethen, L.N., et al.: Chebfun Version 4.2. The Chebfun Development Team. http://www.chebfun.org (2011)
  16. 16.
    Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of Mechanical Engineering and Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA

Personalised recommendations