Also called Gauss-Laguerre Quadrature or Laguerre Quadrature. A Gaussian Quadrature over
the interval with Weighting Function . The Abscissas for quadrature order are
given by the Roots of the Laguerre Polynomials . The weights are
|
(1) |
where is the Coefficient of in . For Laguerre Polynomials,
|
(2) |
where is a Factorial, so
|
(3) |
Additionally,
|
(4) |
so
|
(5) |
(Note that the normalization used here is different than that in Hildebrand 1956.) Using the recurrence relation
|
(6) |
which implies
|
(7) |
gives
|
(8) |
The error term is
|
(9) |
Beyer (1987) gives a table of Abscissas and weights up to .
|
|
|
2 |
0.585786 |
0.853553 |
|
3.41421 |
0.146447 |
3 |
0.415775 |
0.711093 |
|
2.29428 |
0.278518 |
|
6.28995 |
0.0103893 |
4 |
0.322548 |
0.603154 |
|
1.74576 |
0.357419 |
|
4.53662 |
0.0388879 |
|
9.39507 |
0.000539295 |
5 |
0.26356 |
0.521756 |
|
1.4134 |
0.398667 |
|
3.59643 |
0.0759424 |
|
7.08581 |
0.00361176 |
|
12.6408 |
0.00002337 |
The Abscissas and weights can be computed analytically for small .
For the associated Laguerre polynomial with Weighting Function
,
|
(10) |
and
|
(11) |
The weights are
|
(12) |
where is the Gamma Function, and the error term is
|
(13) |
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 463, 1987.
Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 64-65, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 325-327, 1956.
© 1996-9 Eric W. Weisstein
1999-05-26