Abstract
We present an elegant algorithm for stably and quickly generating the weights of Fejér’s quadrature rules and of the Clenshaw–Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numer., 13 (2004), pp. 1–123.
P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd edn., Academic Press, San Diego, 612 pp.
S. Elhay and J. Kautsky, Algorithm 655 – IQPACK: FORTRAN subroutines for the weights of interpolatory quadratures, ACM Trans. Math. Softw., 13 (1987), pp. 399–415.
L. Fejér, Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Z., 37 (1933), pp. 287–309.
W. Gautschi, Numerical quadrature in the presence of a singularity, SIAM J. Numer. Anal., 4 (1967), pp. 357–362.
W. M. Gentleman, Implementing Clenshaw–Curtis quadrature, Commun. ACM, 15 (1972), pp. 337–346. Algorithm 424 (Fortran code), ibid., pp. 353–355.
J. Kautsky and S. Elhay, Calculation of the weights of interpolatory quadratures, Numer. Math., 40 (1982), pp. 407–422.
A. S. Kronrod, Nodes and Weights of Quadrature Formulas, Consultants Bureau, New York, 1965.
T. N. L. Patterson, The optimum addition of points to quadrature formulae, Math. Comput., 22 (1968), pp. 847–856. Errata, Math. Comput., 23 (1969), p. 892.
K. Petras, On the Smolyak cubature error for analytic functions, Adv. Comput. Math., 12 (2000), pp. 71–93.
K. Petras, Smolyak cubature of given polynomial degree with few nodes for increasing dimension, Numer. Math., 93 (2003), pp. 729–753.
S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov. Math. Dokl., 4 (1963), pp. 240–243.
G. von Winckel, Fast Clenshaw–Curtis Quadrature, The Mathworks Central File Exchange, Feb. 2005. URL http://www.mathworks.com/matlabcentral/files/6911/clencurt.m
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS subject classification (2000)
65D32, 65T20, 65Y20
Rights and permissions
About this article
Cite this article
Waldvogel, J. Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules. Bit Numer Math 46, 195–202 (2006). https://doi.org/10.1007/s10543-006-0045-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-006-0045-4