| 释义 |
Laguerre-Gauss QuadratureAlso called Gauss-Laguerre Quadrature or Laguerre Quadrature. A Gaussian Quadrature overthe interval  with Weighting Function  . The Abscissas for quadrature order  aregiven by the Roots of the Laguerre Polynomials  . The weights are
 | (1) |
 is the Coefficient of  in . For Laguerre Polynomials,
 | (2) |
 is a Factorial, so
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 | (7) |
 | (8) |
 | (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) |
 | (11) |
 | (12) |
 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.
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