BIT Numerical Mathematics

, Volume 40, Issue 1, pp 84–101 | Cite as

Adaptive Quadrature—Revisited

  • Walter Gander
  • Walter Gautschi


First, the basic principles of adaptive quadrature are reviewed. Adaptive quadrature programs being recursive by nature, the choice of a good termination criterion is given particular attention. Two Matlab quadrature programs are presented. The first is an implementation of the well-known adaptive recursive Simpson rule; the second is new and is based on a four-point Gauss-Lobatto formula and two successive Kronrod extensions. Comparative test results are described and attention is drawn to serious deficiencies in the adaptive routines quad and quad8 provided by Matlab.

Adaptive quadrature Gauss quadrature Kronrod rules 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carl de Boor, On writing an automatic integration algorithm, in Mathematical Software, John R. Rice ed., Academic Press, New York, 1971, pp. 201–209.Google Scholar
  2. 2.
    Philip J. Davis and Philip Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, Orlando, 1984.Google Scholar
  3. 3.
    Walter Gander, A simple adaptive quadrature algorithm, Research Report No. 83–03, Seminar für Angewandte Mathematik, ETH, Zürich, 1993.Google Scholar
  4. 4.
    Walter Gander, Computermathematik, Birkhäuser, Basel, 1992.Google Scholar
  5. 5.
    Walter Gander and Walter Gautschi, Adaptive quadrature-revisited, Research Report #306, Institut für Wissenschaftliches Rechnen, ETH, Zürich, 1998.Google Scholar
  6. 6.
    S. Garribba, L. Quartapelle, and G. Reina, Algorithm 36-SNIFF: Efficient selftuning algorithm for numerical integration, Computing, 20 (1978), pp. 363–375.Google Scholar
  7. 7.
    Walter Gautschi, Gauss-Kronrod quadrature-a survey, in Numerical Methods and Approximation Theory III, G. V. Milovanović, ed., Faculty of Electronic Engineering, University of Niš, Niš, 1988, pp. 39–66.Google Scholar
  8. 8.
    Walter Gautschi, Numerical Analysis: An Introduction, Birkhäuser, Boston, 1997.Google Scholar
  9. 9.
    Michael T. Heath, Scientific Computing, McGraw-Hill, New York, 1997.Google Scholar
  10. 10.
    William M. Kahan, Handheld calculator evaluates integrals, Hewlett-Packard Journal 31:8 (1980), pp. 23–32.Google Scholar
  11. 11.
    David K. Kahaner, Comparison of numerical quadrature formulas, in Mathematical Software, John R. Rice ed., Academic Press, New York, 1971, pp. 229–259.Google Scholar
  12. 12.
    J. N. Lyness, Notes on the adaptive Simpson quadrature routine, J. Assoc. Comput. Mach., 16 (1969), pp. 483–495.Google Scholar

Copyright information

© Swets & Zeitlinger 2000

Authors and Affiliations

  • Walter Gander
    • 1
  • Walter Gautschi
    • 2
  1. 1.Institut für Wissenschaftliches RechnenETHZürichSwitzerland
  2. 2.Institut für Wissenschaftliches RechnenETHZürichSwitzerland

Personalised recommendations