In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An npoint Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points x_{i} and weights w_{i} for i = 1,...,n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as
Gaussian quadrature as above will only produce accurate results if the function f(x) is well approximated by a polynomial function within the range [1,1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as , where g(x) is approximately polynomial, and W(x) is known, then there are alternative weights w_{i} such that
Common weighting functions include (GaussChebyshev) and (GaussHermite).
It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.
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Rules for the basic problem
For the integration problem stated above, the associated polynomials are Legendre polynomials, P_{n}(x). With the n^{th} polynomial normalized to give P_{n}(1) = 1, the i^{th} Gauss node, x_{i}, is the i^{th} root of P_{n}; its weight is given by (Abramowitz & Stegun 1972, p. 887)
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