Also called ``the'' Gaussian Quadrature or Legendre Quadrature. A Gaussian Quadrature over the
interval with Weighting Function . The Abscissas for quadrature order are
given by the roots of the Legendre Polynomials , which occur symmetrically about 0.
The weights are
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
2 | ± 0.57735 | 1.000000 |
3 | 0 | 0.888889 |
± 0.774597 | 0.555556 | |
4 | ± 0.339981 | 0.652145 |
± 0.861136 | 0.347855 | |
5 | 0 | 0.568889 |
± 0.538469 | 0.478629 | |
± 0.90618 | 0.236927 |
The Abscissas and weights can be computed analytically for small .
2 | 1 | |
3 | 0 | |
4 | ||
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 462-463, 1987.
Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 56-62, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 323-325, 1956.
© 1996-9 Eric W. Weisstein