Gaussian Quadrature

Seeks to obtain the best numerical estimate of an integral by picking optimal abscissas at which to evaluate the function . The fundamental theorem of Gaussian quadrature states that the optimal abscissas of the -point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomial for the same interval and weighting function. Gaussian quadrature is optimal because it fits all polynomials up to degree exactly. Slightly less optimal fits are obtained from Radau quadrature and Laguerre-Gauss quadrature.

	interval	are roots of
1

To determine the weights corresponding to the Gaussian abscissas , compute a Lagrange interpolating polynomial for by letting

(1)

(where Chandrasekhar 1967 uses instead of ), so

(2)

Then fitting a Lagrange interpolating polynomial through the points gives

(3)

for arbitrary points . We are therefore looking for a set of points and weights such that for a weighting function ,

			(4)
			(5)

with weight

(6)

The weights are sometimes also called the Christoffel numbers (Chandrasekhar 1967). For orthogonal polynomials with , ..., ,

(7)

(Hildebrand 1956, p. 322), where is the coefficient of in , then

			(8)
			(9)

where

(10)

Using the relationship

(11)

(Hildebrand 1956, p. 323) gives

(12)

(Note that Press et al. 1992 omit the factor .) In Gaussian quadrature, the weights are all positive. The error is given by

			(13)
			(14)

where (Hildebrand 1956, pp. 320-321).

Other curious identities are

(15)

and

			(16)
			(17)

(Hildebrand 1956, p. 323).

In the notation of Szegö (1975), let be an ordered set of points in , and let , ..., be a set of real numbers. If is an arbitrary function on the closed interval , write the Gaussian quadrature as

(18)

Here are the abscissas and are the Cotes numbers.