Numerische Mathematik

, Volume 2, Issue 1, pp 197–205 | Cite as

A method for numerical integration on an automatic computer

  • C. W. Clenshaw
  • A. R. Curtis


A new method for the numerical integration of a “well-behaved” function over a finite range of argument is described. It consists essentially of expanding the integrand in a series of Chebyshev polynomials, and integrating this series term by term. Illustrative examples are given, and the method is compared with the most commonly-used alternatives, namelySimpson's rule and the method ofGauss.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Hildebrand, F. B.: Introduction to numerical analysis. New York: McGraw-Hill 1956.Google Scholar
  2. [2]
    Kopal, Z.: Numerical analysis. London: Chapman & Hall 1955.Google Scholar
  3. [3]
    Goodwin, E. T.: Evaluation of integrals of the form\(\int\limits_{ - \infty }^{ + \infty } {f\left( \chi \right)e^{ - \chi ^2 } } d\chi \). Proc. Cambridge Phil. Soc.45, 241–245 (1949).Google Scholar
  4. [4]
    Longman, I. M.: Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Phil. Soc.52, 764–768 (1956).Google Scholar
  5. [5]
    National Bureau of Standards Appl. Math. Series No. 9. Tables of Chebyshev Polynomials. Washington: Government Printing Office 1952.Google Scholar
  6. [6]
    Clenshaw, C. W.: The numerical solution of linear differential equations in Chebyshev series. Proc. Cambridge Phil. Soc.53, 134–149 (1957).Google Scholar
  7. [7]
    Clenshaw, C. W.: A note on the summation of Chebyshev series. Math. Tab., Wash.9, 118 (1955).Google Scholar

Copyright information

© Springer-Verlag 1960

Authors and Affiliations

  • C. W. Clenshaw
    • 1
  • A. R. Curtis
    • 1
  1. 1.Mathematics DivisionNational Physical LaboratoryTeddington

Personalised recommendations