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Dina analisis numeris, quadrature rule mangrupa salah sahiji "pendekatan" definite integral tina function, biasana "dinyatakan" salaku jumlah beurat tina nilai fungsi dina titik husus di jero domain  integration.

(Tempo numerical integration keur "quadrature rules" lianna.)

Dina n-titikGaussian quadrature rule, dingaranan ieu sanggeus Carl Friedrich Gauss, mangrupa quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1,

by a suitable choice of the n points xi and n weights wi.

The domain of integration for such a rule is conventionally taken as [-1, 1],

so the rule is stated as

It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belong to a class of orthogonal polynomials.

Rules for the basic problem

For the integration problem stated above,

the associated polynomials are Legendre polynomials.

Some low-order rules for solving the integration problem are listed below.

Change of interval for Gaussian quadrature

An integral over [a, b] must be changed into an integral over [-1, 1] before applying the Gaussian quadrature rule.  This change of interval can be done in the following way:

After applying the Gaussian quadrature rule, the following approximation is obtained:

Other forms of Gaussian quadrature

The integration problem can be expressed in a slightly more general way by introducing a weighting function ω into the integrand,

and allowing an interval other than [-1, 1].

That is, the problem is to calculate

for some choices of a, b, and ω.

For a = -1, b = 1, and ω(x) = 1,

the problem is the same as that considered above.

Other choices léad to other integration rules.

Some of these are tabulated below.

Equation numbers are given for Abramowitz and Stegun (A&S).