Lobatto Quadrature

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  and Chandrasekhar (1960) up to  (although Chandrasekhar's  for  is incorrect).

3	0	0.00000		1.333333
				0.333333
4				0.833333
				0.166667
5	0	0.000000		0.711111
				0.544444
				0.100000
6				0.554858
				0.378475
				0.066667

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Hunter, D. and Nikolov, G. "On the Error Term of Symmetric Gauss-Lobatto Quadrature Formulae for Analytic Functions." Math. Comput. 69, 269-282, 2000.

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