Numerical Integration

  • Vilmos Komornik
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


For his celestial mechanics Newton had to evaluate complicated integrals. He and his followers generalized the trapezoidal rule \(\int _{a}^{b}f(x)\ dx \approx \frac{f(a)+f(b)} {2} (b - a)\) and Simpson’s rule \(\int _{a}^{b}f(x)\ dx \approx \frac{f(a)+4f(\frac{a+b} {2} )+f(b)} {6} (b - a)\) by seeking approximations of the form \(\int _{I}f(x)\ dx \approx A_{1}f(x_{1}) + \cdots + A_{n}\,f(x_{n})\) where the points x k and coefficients A k do not depend on the particular choice of the function f.




  1. 3.
    N.I. Achieser, Theory of Approximation, Dover, New York, 1992.Google Scholar
  2. 70.
    J.C. Burkill, The Theory of Ordinary Differential Equations, Oliver & Boyd, Edinburgh, 1956.Google Scholar
  3. 140.
    A. Erdélyi, Asymptotic Expansions, Dover, New York, 1956.Google Scholar
  4. 217.
    E. Hairer, G. Wanner, Analysis by Its History, Springer, New York, 1996.Google Scholar
  5. 359.
    I.P. Natanson, Constructive Function Theory I-III, Frederick Ungar, New York, 1961–1965.Google Scholar
  6. 404.
    A. Ralston, P. Rabinowitz, A First Course in Numerical Analysis, McGraw-Hill, New York, 1978.Google Scholar
  7. 491.
    A.F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press Ltd., Oxford, 1963.Google Scholar
  8. 506.
    G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, 1995.Google Scholar
  9. 511.
    E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 1996.Google Scholar

Copyright information

© Springer-Verlag London Ltd. 2017

Authors and Affiliations

  • Vilmos Komornik
    • 1
  1. 1.Department of MathematicsUniversity of StrasbourgStrasbourgFrance

Personalised recommendations