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(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, by a suitable choice of the n points x_i and n weights w_i. The domain of integration for such a rule is conventionally taken as [-1, 1], so the rule is stated as

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.

1 Rules for the basic problem

For the integration problem stated above, the associated polynomials are Legendre polynomials. Some low-order rules for solving the integration problem are listed below.

2 Change of interval for Gaussian quadrature

An integral over [a, b] must be changed into an integral over [-1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way:

After applying the Gaussian quadrature rule, the following approximation is obtained:

3 Other forms of Gaussian quadrature

The integration problem can be expressed in a slightly more general way by introducing a weight function ω into the integrand, and allowing an interval other than [-1, 1]. That is, the problem is to calculate

for some choices of a, b, and ω. For a = -1, b = 1, and ω(x) = 1, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for Abramowitz and Stegun (A&S).

3.1 Error estimates

The error of a Gaussian quadrature rule can be stated as follows (theorem 3.6.24 in Stoer and Bulirsch). For an integrand which has 2n continuous derivatives,

for some ξ in (a, b), where p_n is the orthogonal polynomial of order n.

Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the 2n'th derivative, and furthermore the actual error may be much less than a bound established by the derivative. Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss-Kronrod rules can be useful.