Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


On node distributions for interpolation and spectral methods

Author: N. S. Hoang
Journal: Math. Comp. 85 (2016), 667-692
MSC (2010): Primary 65D05; Secondary 41A05, 41A10
Published electronically: September 23, 2015
MathSciNet review: 3434875
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $ C_M^{s+1}[-1,1]$, the set of $ (s+1)$-time differentiable functions whose $ (s+1)$-th derivatives are bounded in absolute value by a constant $ M>0$. Node distributions for computing spectral differentiation matrices and integration matrices are proposed and studied. Numerical experiments have shown that the proposed node distributions can yield results of higher accuracy than those obtained by the most commonly used Chebyshev-Gauss-Lobatto node distribution.

References [Enhancements On Off] (What's this?)

  • [1] L. Brutman, Lebesgue functions for polynomial interpolation--a survey, Ann. Numer. Math. 4 (1997), no. 1-4, 111-127. MR 1422674 (97m:41003)
  • [2] Bengt Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR 1386891 (97g:65001)
  • [3] R. Günttner, On asymptotics for the uniform norms of the Lagrange interpolation polynomials corresponding to extended Chebyshev nodes, SIAM J. Numer. Anal. 25 (1988), no. 2, 461-469. MR 933735 (89e:41005),
  • [4] Myron S. Henry, Approximation by polynomials: interpolation and optimal nodes, Amer. Math. Monthly 91 (1984), no. 8, 497-499. MR 761404,
  • [5] J. S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal. 35 (1998), no. 2, 655-676. MR 1618874 (99b:65009),
  • [6] Rainer Kress, Numerical Analysis, Graduate Texts in Mathematics, vol. 181, Springer-Verlag, New York, 1998. MR 1621952 (99c:65001)
  • [7] Heinz-Joachim Rack, An example of optimal nodes for interpolation, Internat. J. Math. Ed. Sci. Tech. 15 (1984), no. 3, 355-357. MR 750010 (86b:65010),
  • [8] Simon J. Smith, Lebesgue constants in polynomial interpolation, Ann. Math. Inform. 33 (2006), 109-123. MR 2385471 (2009e:41013)
  • [9] Lloyd N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1776072 (2001c:65001)
  • [10] P. Vértesi, Optimal Lebesgue constant for Lagrange interpolation, SIAM J. Numer. Anal. 27 (1990), no. 5, 1322-1331. MR 1061132 (91k:41010),
  • [11] Zhimin Zhang, Superconvergence of a Chebyshev spectral collocation method, J. Sci. Comput. 34 (2008), no. 3, 237-246. MR 2377618 (2008k:65156),

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65D05, 41A05, 41A10

Retrieve articles in all journals with MSC (2010): 65D05, 41A05, 41A10

Additional Information

N. S. Hoang
Affiliation: Department of Mathematics, University of West Georgia, Carrollton, Georgia 30118

Keywords: Interpolation, pseudospectral methods, node distributions, differentiation matrices, integration matrices, scaled Chebyshev nodes, Chebyshev nodes, Chebyshev-Gauss-Lobatto nodes.
Received by editor(s): August 5, 2013
Received by editor(s) in revised form: July 18, 2014
Published electronically: September 23, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society