A Gaussian quadrature-like formula for numerical estimation of integrals. It requires  points and fits all polynomials to degree , so it effectively fits exactly all polynomials of degree . It uses a weighting function  in which the endpoint  in the interval  is included in a total of  abscissas, giving  free abscissas. The general formula is

(1)

The free abscissas  for , ...,  are the roots of the polynomial

(2)

where  is a Legendre polynomial. The weights of the free abscissas are

			(3)
			(4)

and of the endpoint

(5)

The error term is given by

(6)

for .

2		0.5
	0.333333	1.5
3		0.222222
		1.02497
	0.689898	0.752806
4		0.125
		0.657689
	0.181066	0.776387
	0.822824	0.440924
5		0.08
		0.446208
		0.623653
	0.446314	0.562712
	0.885792	0.287427

The abscissas and weights can be computed analytically for small .

2
3

REFERENCES:

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

Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 61, 1960.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 338-343, 1956.

Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 105, 1997.