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Laguerre-Gauss Quadrature

Also called Gauss-Laguerre Quadrature or Laguerre Quadrature. A Gaussian Quadrature over the interval $[0, \infty)$ with Weighting Function $W(x)=e^{-x}$. The Abscissas for quadrature order $n$ are given by the Roots of the Laguerre Polynomials $L_n(x)$. The weights are

w_i=-{A_{n+1}\gamma_n\over A_nL_n'(x_i)L_{n+1}(x_i)}
={A_n\over A_{n-1}}{\gamma_{n-1}\over L_{n-1}(x_i)L_n'(x_i)},
\end{displaymath} (1)

where $A_n$ is the Coefficient of $x^n$ in $L_n(x)$. For Laguerre Polynomials,
A_n=(-1)^n n!,
\end{displaymath} (2)

where $n!$ is a Factorial, so
{A_{n+1}\over A_n}=-(n+1).
\end{displaymath} (3)

\end{displaymath} (4)

w_i={n+1\over L_{n+1}(x_i)L_n'(x_i)}=-{n\over L_{n-1}(x_i)L_n'(x_i)}.
\end{displaymath} (5)

(Note that the normalization used here is different than that in Hildebrand 1956.) Using the recurrence relation

xL_n'(x)=nL_n(x)-nL_{n-1}(x) =(x-n-1)L_n(x)+(n+1)L_{n+1}(x)
\end{displaymath} (6)

which implies
\end{displaymath} (7)

w_i={1\over x_i[L_n'(x_i)]^2}={x_i\over (n+1)^2[L_{n+1}(x_i)]^2}.
\end{displaymath} (8)

The error term is
\end{displaymath} (9)

Beyer (1987) gives a table of Abscissas and weights up to $n=6$.

$n$ $x_i$ $w_i$
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 $n$.

$n$ $x_i$ $w_i$
2 $2-\sqrt{2}$ ${\textstyle{1\over 4}}(2+\sqrt{2})$
  $2+\sqrt{2}$ ${\textstyle{1\over 4}}(2-\sqrt{2})$

For the associated Laguerre polynomial $L_n^\beta(x)$ with Weighting Function $w(x)=x^\beta e^{-x}$,

\end{displaymath} (10)

\gamma_n=n!\int_0^\infty x^{\beta+n}e^{-x}\,dx=n!\Gamma(n+\beta+1).
\end{displaymath} (11)

The weights are
w_i={n!\Gamma(n+\beta+1)\over x_i[{L_m^\beta}'(x_i)]^2}={n!\Gamma(n+\beta+1)x_i\over [L_{n+1}^\beta(x_i)]^2},
\end{displaymath} (12)

where $\Gamma(z)$ is the Gamma Function, and the error term is
\end{displaymath} (13)


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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© 1996-9 Eric W. Weisstein