Gaussian Quadrature

Gaussian quadrature is a numerical integration rule that chooses abscissas and weights to integrate polynomials of the highest possible polynomial degree exactly. The fundamental theorem of Gaussian quadrature states that the optimal abscissas of the -point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomials of polynomial degree for the same interval and weighting function. For a nonnegative weighting function with finite moments of all orders and positive integral against the square of every nonzero polynomial, Gaussian quadrature fits all polynomials up to polynomial degree exactly. On a finite interval, prescribing one endpoint gives Radau quadrature, with exactness through polynomial degree for . Prescribing both endpoints gives Lobatto quadrature, with exactness through polynomial degree for .

The weighting function specifies the weighted integral approximated by the rule,

(1)

For a chosen rule, the interval and are fixed while varies. The weighting function is distinct from the discrete weights multiplying .

	interval	are roots of
1

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

(2)

(where Chandrasekhar 1967 uses instead of ), so

(3)

Then fitting a Lagrange interpolating polynomial through the points gives

(4)

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

			(5)
			(6)

with weight

(7)

The weights are sometimes also called the Christoffel numbers (Chandrasekhar 1967). For the member of polynomial degree in a family of orthogonal polynomials, set above. Then

(8)

where is its leading coefficient. The weights are given by

			(9)
			(10)

(Hildebrand 1956, p. 322), where

(11)

Using the relationship

(12)

(Hildebrand 1956, p. 323) gives

(13)

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

			(14)
			(15)

where (Hildebrand 1956, pp. 320-321).

Other curious identities are

(16)

and

			(17)
			(18)

(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

(19)

Here are the abscissas and are the Cotes numbers.

The nodes and weights can also be obtained from the leading block of the Jacobi matrix for the normalized orthogonal polynomials. Its eigenvalues are the nodes . If is a real unit eigenvector for and , then

(20)

This is the Golub-Welsch algorithm (Golub and Welsch 1969). Equivalently, the Christoffel function gives .

For example, the three-point Legendre-Gauss quadrature rule has

(21)

Its nodes are , 0, and , with corresponding weights , , and . The rule is exact through polynomial degree 5.

The rule's higher-degree exactness is illustrated above by the quartic polynomial and its quadratic Lagrange interpolating polynomial ,

			(22)
			(23)

Their values agree at the three nodes. The blue and orange shaded regions lie where is positive and negative, respectively. Their signed integrals cancel. Consequently,

(24)

The numbers beneath the nodes are the corresponding weights.

Numerical checks of computed nodes and weights require attention to conditioning. Small errors in reproduced moments need not imply comparably small errors in the nodes and weights, because the map from moments to a quadrature rule can be severely ill-conditioned. Checks comparing the coefficients of the reconstructed three-term recurrence relation with independently determined values, or testing the sum of the nodes, can therefore be more informative than checking moments alone (Gautschi 1983).

The modified Chebyshev algorithm supplies the required recurrence coefficients from modified moments (Gautschi 1982).