info prev up next book cdrom email home

Legendre-Gauss Quadrature

Also called ``the'' Gaussian Quadrature or Legendre Quadrature. A Gaussian Quadrature over the interval $[-1, 1]$ with Weighting Function $W(x)=1$. The Abscissas for quadrature order $n$ are given by the roots of the Legendre Polynomials $P_n(x)$, which occur symmetrically about 0. The weights are

\begin{displaymath}
w_i=-{A_{n+1}\gamma_n\over A_nP_n'(x_i)P_{n+1}(x_i)}={A_n\over A_{n-1}}{\gamma_{n-1}\over P_{n-1}(x_i)P_n'(x_i)},
\end{displaymath} (1)

where $A_n$ is the Coefficient of $x^n$ in $P_n(x)$. For Legendre Polynomials,
\begin{displaymath}
A_n={(2n)!\over 2^n(n!)^2},
\end{displaymath} (2)

so
$\displaystyle {A_{n+1}\over A_n}$ $\textstyle =$ $\displaystyle {[2(n+1)]!\over 2^{n+1}[(n+1)!]^2} {2^n(n!)^2\over (2n)!}$  
  $\textstyle =$ $\displaystyle {(2n+1)(2n+2)\over 2(n+1)^2} = {2n+1\over n+1}.$ (3)

Additionally,
\begin{displaymath}
\gamma_n={2\over 2n+1},
\end{displaymath} (4)

so
\begin{displaymath}
w_i=-{2\over(n+1)P_{n+1}(x_i)P_n'(x_i)}={2\over nP_{n-1}(x_i)P_n'(x_i)}.
\end{displaymath} (5)

Using the Recurrence Relation


\begin{displaymath}
(1-x^2)P_n'(x)=nxP_n(x)+nP_{n-1}(x) = (n+1)xP_n(x)-(n+1)P_{n+1}(x)
\end{displaymath} (6)

gives
\begin{displaymath}
w_i={2\over(1-{x_i}^2)[P_n'(x_i)]^2}={2(1-{x_i}^2)\over(n+1)^2[P_{n+1}(x_i)]^2}.
\end{displaymath} (7)

The error term is
\begin{displaymath}
E={2^{2n+1}(n!)^4\over(2n+1)[(2n)!]^3}f^{(2n)}(\xi).
\end{displaymath} (8)

Beyer (1987) gives a table of Abscissas and weights up to $n=16$, and Chandrasekhar (1960) up to $n=8$ for $n$ Even.

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

$n$ $x_i$ $w_i$
2 $\pm{\textstyle{1\over 3}}\sqrt{3}$ 1
3 0 ${\textstyle{8\over 9}}$
  $\pm{\textstyle{1\over 5}}\sqrt{15}$ ${\textstyle{5\over 9}}$
4 $\pm{\textstyle{1\over 35}}\sqrt{525-70\sqrt{30}}$ ${\textstyle{1\over 36}}(18+\sqrt{30}\,)$
  $\pm{\textstyle{1\over 35}}\sqrt{525+70\sqrt{30}}$ ${\textstyle{1\over 36}}(18-\sqrt{30}\,)$


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.



info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein
1999-05-26