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.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 887-888, 1972.

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 103, 1990.

Arfken, G. "Appendix 2: Gaussian Quadrature." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 968-974, 1985.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 461, 1987.

Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, 1967.

Gauss, C. F. "Methodus nova integralium valores per approximationem inveniendi." Commentationes Societatis regiae scientarium Gottingensis recentiores 3, 39-76, 1814. Reprinted in Werke, Vol. 3. New York: George Olms, p. 163, 1981.

Golub, G. H. and Welsh, J. H. "Calculation of Gauss Quadrature Rules." Math. Comput. 23, 221-230, 1969.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gaussian Quadratures and Orthogonal Polynomials." §4.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 140-155, 1992.

Stroud, A. H. and Secrest, D. Gaussian Quadrature Formulas. Englewood Cliffs, NJ: Prentice-Hall, 1966.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 37-48 and 340-349, 1975.

Whittaker, E. T. and Robinson, G. "Gauss's Formula of Numerical Integration." §80 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152-163, 1967.