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Numerical Integration
Numerical integration is the approximate computation of anintegral using numerical techniques. The numerical computation of anintegral is sometimes calledquadrature. Ueberhuber (1997, p. 71) uses the word "quadrature" to mean numerical computation of a univariateintegral, and "cubature" to mean numerical computation of amultiple integral.
There are a wide range of methods available for numerical integration. A good source for such techniques is Presset al.(1992). Numerical integration is implemented in theWolfram Language asNIntegrate[f,
x,xmin,xmax
].
The most straightforward numerical integration technique uses theNewton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spacedintervals by various degreepolynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called thetrapezoidal rule andSimpson's rule, respectively. The 5-point formula is calledBoole's rule. A generalization of thetrapezoidal rule isRomberg 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 calledGaussian 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 theGaussian 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 forGaussian quadrature is Hildebrand (1956).
Modern numerical integrations methods based oninformation 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 onapproximation theory) are not as efficient (Smith 1974).
See also
Cubature,Double Exponential Integration,Filon's Integration Formula,Gauss-Kronrod Quadrature,Gregory's Formula,Integral,Integration,Monte Carlo Integration,Numerical Differentiation,Quadrature,Quasi-Monte Carlo Integration,T-IntegrationExplore with Wolfram|Alpha
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 Control96, 61-70, Mar. 1974.Ueberhuber, C. W. "Numerical Integration." Ch. 12 inNumerical 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 inThe Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 132-163, 1967.Referenced on Wolfram|Alpha
Numerical IntegrationCite this as:
Weisstein, Eric W. "Numerical Integration."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NumericalIntegration.html