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.
(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