Gaussian quadrature

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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 n-point 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 xi and weights wi 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 f(x) = W(x) g(x)\,, where g(x) is approximately polynomial, and W(x) is known, then there are alternative weights wi such that

Common weighting functions include W(x)=(1-x^2)^{-1/2}\, (Gauss-Chebyshev) and W(x)=e^{-x^2} (Gauss-Hermite).

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, Pn(x). With the nth polynomial normalized to give Pn(1) = 1, the ith Gauss node, xi, is the ith root of Pn; its weight is given by (Abramowitz & Stegun 1972, p. 887)

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