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In numerical analysis, an n-point Gaussian quadrature rule, named after Carl Friedrich Gauss,[1] is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n.
The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826.[2] The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as
which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].
The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as
where g(x) is well-approximated by a low-degree polynomial, then alternative nodes xi' and weights wi' will usually give more accurate quadrature rules. These are known as Gauss–Jacobi quadrature rules, i.e.,
Common weights include (Chebyshev–Gauss) and . One may also want to integrate over semi-infinite (Gauss–Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature).
It can be shown (see Press et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.