Gaussian Quadrature

Gaussian Quadrature

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 x_i and weights w_i for i = 1,...,n. The domain of integration for such a rule is conventionally taken as, 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 . 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 such that

Common weighting functions include (Chebyshev–Gauss) and (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.