BIT Numerical Mathematics

, Volume 46, Issue 1, pp 195–202 | Cite as

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules



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.

Key words

numerical quadrature Fejér’s quadrature rules Clenshaw–Curtis quadrature discrete Fourier transform 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H.-J. Bungartz and M. Griebel, Sparse grids, Acta Numer., 13 (2004), pp. 1–123.Google Scholar
  2. 2.
    P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd edn., Academic Press, San Diego, 612 pp.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    L. Fejér, Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Z., 37 (1933), pp. 287–309.Google Scholar
  5. 5.
    W. Gautschi, Numerical quadrature in the presence of a singularity, SIAM J. Numer. Anal., 4 (1967), pp. 357–362.Google Scholar
  6. 6.
    W. M. Gentleman, Implementing Clenshaw–Curtis quadrature, Commun. ACM, 15 (1972), pp. 337–346. Algorithm 424 (Fortran code), ibid., pp. 353–355.Google Scholar
  7. 7.
    J. Kautsky and S. Elhay, Calculation of the weights of interpolatory quadratures, Numer. Math., 40 (1982), pp. 407–422.Google Scholar
  8. 8.
    A. S. Kronrod, Nodes and Weights of Quadrature Formulas, Consultants Bureau, New York, 1965.Google Scholar
  9. 9.
    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.Google Scholar
  10. 10.
    K. Petras, On the Smolyak cubature error for analytic functions, Adv. Comput. Math., 12 (2000), pp. 71–93.Google Scholar
  11. 11.
    K. Petras, Smolyak cubature of given polynomial degree with few nodes for increasing dimension, Numer. Math., 93 (2003), pp. 729–753.Google Scholar
  12. 12.
    S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov. Math. Dokl., 4 (1963), pp. 240–243.Google Scholar
  13. 13.
    G. von Winckel, Fast Clenshaw–Curtis Quadrature, The Mathworks Central File Exchange, Feb. 2005. URL Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Seminar for Applied MathematicsSwiss Federal Institute of Technology ETHZurichSwitzerland

Personalised recommendations