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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.
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