Numerical Integration

Numerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature. Ueberhuber (1997, p. 71) uses the word "quadrature" to mean numerical computation of a univariate integral, and "cubature" to mean numerical computation of a multiple integral.

There are a wide range of methods available for numerical integration. A good source for such techniques is Press et al. (1992). Numerical integration is implemented in the Wolfram Language as NIntegrate[f, x, xmin, xmax].

The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spaced intervals by various degree polynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called the trapezoidal rule and Simpson's rule, respectively. The 5-point formula is called Boole's rule. A generalization of the trapezoidal rule is Romberg integration, which can yield accurate results for many fewer function evaluations.

If the functions are known analytically instead of being tabulated at equally spaced intervals, the best numerical method of integration is called Gaussian quadrature. By picking the abscissas at which to evaluate the function, Gaussian quadrature produces the most accurate approximations possible. However, given the speed of modern computers, the additional complication of the Gaussian quadrature formalism often makes it less desirable than simply brute-force calculating twice as many points on a regular grid (which also permits the already computed values of the function to be re-used). An excellent reference for Gaussian quadrature is Hildebrand (1956).

Modern numerical integrations methods based on information theory have been developed to simulate information systems such as computer controlled systems, communication systems, and control systems since in these cases, the classical methods (which are based on approximation theory) are not as efficient (Smith 1974).

REFERENCES:

Corbit, D. "Numerical Integration: From Trapezoids to RMS: Object-Oriented Numerical Integration." Dr. Dobb's J., No. 252, 117-120, Oct. 1996.

Davis, P. J. and Rabinowitz, P. Methods of Numerical Integration, 2nd ed. New York: Academic Press, 1984.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.

Krommer, A. R. and Ueberhuber, C. W. Numerical Integration on Advanced Computer Systems. Berlin: Springer-Verlag, 1994.

Milne, W. E. Numerical Calculus: Approximations, Interpolation, Finite Differences, Numerical Integration and Curve Fitting. Princeton, NJ: Princeton University Press, 1949.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, 1992.

Smith, J. M. "Recent Developments in Numerical Integration." J. Dynam. Sys., Measurement and Control 96, 61-70, Mar. 1974.

Ueberhuber, C. W. "Numerical Integration." Ch. 12 in Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, pp. 65-169, 1997.

Weisstein, E. W. "Books about Numerical Methods." http://www.ericweisstein.com/encyclopedias/books/NumericalMethods.html.

Whittaker, E. T. and Robinson, G. "Numerical Integration and Summation." Ch. 7 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 132-163, 1967.