Also called Gauss-Laguerre quadrature or Laguerre quadrature. A Gaussian quadrature over the interval with weighting function (Abramowitz and Stegun 1972, p. 890). The abscissas for quadrature order are given by the roots of the Laguerre polynomials . The weights are

			(1)
			(2)

where is the coefficient of in . For Laguerre polynomials,

(3)

where is a factorial, so

			(4)
			(5)

Additionally,

(6)

so

			(7)
			(8)

Using the recurrence relation

			(9)
			(10)

which, since is a root of , gives

(11)

so (10) becomes

(12)

gives

			(13)
			(14)

The error term is

(15)

(Abramowitz and Stegun 1972, p. 890).

Beyer (1987) gives a table of abscissas and weights up to .

2	0.585786	0.853553
	3.41421	0.146447
3	0.415775	0.711093
	2.29428	0.278518
	6.28995	0.0103893
4	0.322548	0.603154
	1.74576	0.357419
	4.53662	0.0388879
	9.39507	0.000539295
5	0.26356	0.521756
	1.4134	0.398667
	3.59643	0.0759424
	7.08581	0.00361176
	12.6408	0.00002337

The abscissas and weights can be computed analytically for small .

2

For the generalized Laguerre polynomial with weighting function ,

(16)

is the coefficient of in and

			(17)
			(18)

where is the gamma function. The weights are then

			(19)
			(20)

and the error term is

(21)

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 890 and 923, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 463, 1987.

Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 64-65, 1960.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 325-327, 1956.

CITE THIS AS:

Weisstein, Eric W. "Laguerre-Gauss Quadrature." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Laguerre-GaussQuadrature.html