Laguerre-Gauss Quadrature
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
5-12-2021
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Laguerre-Gauss Quadrature
Laguerre-Gauss quadrature, also called Gauss-Laguerre quadrature or Laguerre quadrature, is a Gaussian quadrature over the interval 
with weighting function 
(Abramowitz and Stegun 1972, p. 890). It fits all polynomials of degree 
exactly (Chandrasekhar 1960, p. 61).
The abscissas for quadrature order 
are given by the roots of the Laguerre polynomials 
. The weights are
where 
is the coefficient of 
in 
. For Laguerre polynomials,
 |
(3)
|
where 
is a factorial, so
Additionally,
![gamma_n=int_0^inftyW(x)[L_n(x)]^2dx=1,](https://mathworld.wolfram.com/images/equations/Laguerre-GaussQuadrature/NumberedEquation2.gif) |
(6)
|
so
Using the recurrence relation
which, since 
is a root of 
, gives
 |
(11)
|
so (10) becomes
 |
(12)
|
gives
The error term is
 |
(15)
|
(Abramowitz and Stegun 1972, p. 890).
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 generalized Laguerre polynomial 
with weighting function 
,
 |
(16)
|
is the coefficient of 
in 
and
where 
is the gamma function. The weights are then
and the error term is
 |
(21)
|
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 890 and 923, 1972.
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. 61 and 64-65, 1960.
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 325-327, 1956.
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