Advertisement

Abstract

Product integration is a well established technique for evaluating the definite integral
$$\int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t)f(t)dt$$
(1.1)
The function f(t) is approximated by fN(t) and the resulting integral
$$\int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t){f_N}(t)dt$$
(1.2)
is taken as our approximation to the integral in (1.1). fN(t) is usually constructed by piecewise-polynomial interpolation. A simple example is the product trapezoidal rule which uses piecewise-linear interpolation, namely
$${f_N}(t) = \sum\limits_{i = 0}^N {{\phi _i}(t)f(i/N)} $$
(1.3)
. In (1.3), the ϕi(t) I = 0,...,N are the usual “hat” functions. Then we have
$$\int \begin{gathered}1 \hfill \\0 \hfill \\ \end{gathered} g(t)f(t)dt \simeq \sum\limits_{i = 0}^N {{w_i}f(i/N)} $$
(1.4)
where
$${w_i} = \int \begin{gathered}1 \hfill \\0 \hfill \\\end{gathered} g(t){\phi _i}(t)dt$$
(1.5)
.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    ATKINSON, K., SIAM J. Num. Anal., 4, 1967, pp337–348.CrossRefGoogle Scholar
  2. 2.
    CHANDLER G., Ph.D. Thesis, 1979, Australian National University.Google Scholar
  3. 3.
    FILON, L., Proc. Roy.Soc Edingburgh, 1928,49,pp38–47.Google Scholar
  4. 4.
    DE HOOG, F., & WEISS, R., Math.Comp. 27, 1973, pp295–306.CrossRefGoogle Scholar
  5. 5.
    RICHTER, G., Numer.Math., 1978, 31, pp63–70.CrossRefGoogle Scholar
  6. 6.
    SPENCE, A., in Hämmerlin G. (ed), Numerische Integration, ISNM 45 Birkhäuser Verlag Basel, 1979.Google Scholar
  7. 7.
    THOMAS, K., D. Phil. Thesis, Oxford University, 1973.Google Scholar
  8. 8.
    YOUNG, A., Proc. Roy. Soc. London Ser A ,224, 1954, pp552–561.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • K. S. Thomas

There are no affiliations available

Personalised recommendations