Abstract
Let \({X_{l}^{C}}\) be the set of l Chebyshev points in the interval [−1,1]. If n and n 0 are such that n=2m n 0−1 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 n 0 and n, n>n 0, where n≠2m n 0−1, we give a simple procedure to build a set of n points in the interval [−1,1] that include the set of n 0 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baglama, J., Calvetti, D., Reichel, L.: Fast Leja points. Electron. Trans. Numer. Anal. 7, 124–140 (1998)
Bernstein, S.: Sur la limitation des valeurs d’un polynôme. Bull. Acad. Sci. de l’URSS 8, 1025–1050 (1931)
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)
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)
Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)
Erdös, P.: Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hung. 12, 235–244 (1961)
Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. SIAM. 978-0-898716-34-4 (2007)
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)
Leja, F.: Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme. Ann. Polon. Math. 3, 8–13 (1957)
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)
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)
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)
Taylor, R.: Lagrange interpolation on Leja points, Graduate School Theses and Dissertations. http://scholarcommons.usf.edu/etd/530 (2008)
Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67–87 (2008)
Trefethen, L.N., et al.: Chebfun Version 4.2. The Chebfun Development Team. http://www.chebfun.org (2011)
Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the PECASE Award, sponsored by the Lawrence Livermore National Laboratory under grant B597952 and the Office of Science of the U.S. Department of Energy under grant DE-SC0005384.
Rights and permissions
About this article
Cite this article
Ghili, S., Iaccarino, G. Reusing Chebyshev points for polynomial interpolation. Numer Algor 70, 249–267 (2015). https://doi.org/10.1007/s11075-014-9945-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-014-9945-6