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Numerische Mathematik

, Volume 34, Issue 4, pp 387–401 | Cite as

Product integration with the Clenshaw-Curtis points: Implementation and error estimates

  • Ian H. Sloan
  • William E. Smith
Article

Summary

This paper is concerned with the practical implementation of a product-integration rule for approximating\(\int\limits_{ - 1}^1 {k(x)f(x)dx} \), wherek is integrable andf is continuous. The approximation is\(\sum\limits_{i = 0}^n {w_{ni} } f(\cos i {\pi \mathord{\left/ {\vphantom {\pi n}} \right. \kern-\nulldelimiterspace} n})\), where the weightsw ni are such as to make the rule exact iff is any polynomial of degree ≦n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |λx|α with α>−1 and |λ|≦1, or of the form cosαx or sinαx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)≡1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.

Subject Classifications

AMS(MOS): 65D30 CR: 5.16 

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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Ian H. Sloan
    • 1
  • William E. Smith
    • 1
  1. 1.School of MathematicsUniversity of New South WalesKensingtonAustralia

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