Also called Radau Quadrature (Chandrasekhar 1960). A Gaussian Quadrature with Weighting Function in which the endpoints of the interval are included in a total of Abscissas, giving free abscissas. Abscissas are symmetrical about the origin, and the general Formula is

 (1)

The free Abscissas for , ..., are the roots of the Polynomial , where is a Legendre Polynomial. The weights of the free abscissas are
 (2) (3)

and of the endpoints are
 (4)

The error term is given by
 (5)

for . Beyer (1987) gives a table of parameters up to =11 and Chandrasekhar (1960) up to =9 (although Chandrasekhar's for is incorrect).

 3 0 1.33333 ± 1 0.333333 4 ± 0.447214 0.833333 ± 1 0.166667 5 0 0.711111 ± 0.654654 0.544444 ± 1 0.100000 6 ± 0.285232 0.554858 ± 0.765055 0.378475 ± 1 0.0666667

The Abscissas and weights can be computed analytically for small .

 3 0 4 5 0

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 888-890, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 465, 1987.

Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 63-64, 1960.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 343-345, 1956.