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 [−1, 1], so the rule is stated as

Gaussian quadrature as above will only produce good 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 ${\displaystyle f(x)=\omega (x)g(x)}$, where g(x) is approximately polynomial and ω(x) is known, then alternative weights ${\displaystyle w_{i}'}$ and points ${\displaystyle x_{i}'}$ that depend on the weighting function ω(x) may give better results, where

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