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quadrature (Definition)

Quadrature is the computation of a univariate definite integral. It can refer to either numerical or analytic techniques; one must gather from context which is meant. The term refers to the geometric origin of integration in determining the area of a plane figure by approximating it with squares.

Cubature refers to higher-dimensional definite integral computation. Likewise, this term refers to the geometric operation of approximating the volume of a solid by means of cubes (and has since been extended to higher dimensions).

The terms “quadrature” and “cubature” are typically used in numerical analysis to denote the approximation of a definite integral, typically by a suitable weighted sum. Perhaps the simplest possibility is approximation by a sum of values at equidistant points, i.e. approximate $ \int_0^1 f(x) \, dx$ by $ \sum_{k=0}^n f(k/n) / n$. More complicated approximations involve variable weights and evaluation of the function at points which may not be spaced equidistantly. Some such numerical quadrature methods are Simpson's rule, the trapezoidal rule, and Gaussian quadrature.

"quadrature" is owned by rspuzio. [ full author list (2) | owner history (1) ]
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Also defines:  cubature

Cross-references: Gaussian, trapezoidal rule, Simpson's rule, function, weights, variable, points, sum, approximation, dimensions, cubes, volume, operation, squares, plane, area, origin, term, analytic, definite integral
There are 6 references to this entry.

This is version 9 of quadrature, born on 2002-01-05, modified 2007-02-15.
Object id is 1286, canonical name is Quadrature.
Accessed 4170 times total.

AMS MSC26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type)
 41A55 (Approximations and expansions :: Approximate quadratures)
 65D32 (Numerical analysis :: Numerical approximation and computational geometry :: Quadrature and cubature formulas)
 28-00 (Measure and integration :: General reference works )

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