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)
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The free abscissas for , ..., are the roots of the polynomial , where is a Legendre polynomial. The weights of the free abscissas are
(2)
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(3)
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and of the endpoints are
(4)
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The error term is given by
(5)
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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 |