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)

and of the endpoint
 (4)

The error term is given by
 (5)

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, C. 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.