Reusing Chebyshev points for polynomial interpolation
Abstract
Let XlC${X_{l}^{C}}$ 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 Xn0C XnC$X_{n_{0}}^{C} \subset {X_{n}^{C}}$. 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.
References
[1]
Baglama, J., Calvetti, D., Reichel, L.: Fast Leja points. Electron. Trans. Numer. Anal. 7, 124---140 (1998)
[2]
Bernstein, S.: Sur la limitation des valeurs d'un polynôme. Bull. Acad. Sci. de l'URSS 8, 1025---1050 (1931)
[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)
[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)
[5]
Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197---205 (1960)
[6]
Erdös, P.: Problems and results on the theory of interpolation. II. Acta Math. Acad. Sci. Hung. 12, 235---244 (1961)
[7]
Gil, A., Segura, J., Temme, N.M.: Numerical Methods for Special Functions. SIAM. 978-0-898716-34-4 (2007)
[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)
[9]
Leja, F.: Sur certaines suites liées aux ensembles plans et leur application à la représentation conforme. Ann. Polon. Math. 3, 8---13 (1957)
[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)
[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)
[12]
Salzer, H.E.: Lagrangian interpolation at the Chebyshev points xn,=cos(¿¿/n),¿=0(1)n$x_{n,¿} = \cos (\nu \pi /n), \nu = 0(1)n$; some unnoted advantages. Comput. J. 15, 156---159 (1972)
[13]
Taylor, R.: Lagrange interpolation on Leja points, Graduate School Theses and Dissertations. http://scholarcommons.usf.edu/etd/530 (2008)
[14]
Trefethen, L.N.: Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev. 50, 67---87 (2008)
[15]
Trefethen, L.N., et al.: Chebfun Version 4.2. The Chebfun Development Team. http://www.chebfun.org (2011)
[16]
Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2012)
- Reusing Chebyshev points for polynomial interpolation
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Published: 01 October 2015
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