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

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

where $A_n$ is the Coefficient of $x^n$ in $P_n(x)$. For Legendre Polynomials,

$$A_n={(2n)!\over 2^n(n!)^2},$$ (2)

so

$${A_{n+1}\over A_n} = {[2(n+1)]!\over 2^{n+1}[(n+1)!]^2} {2^n(n!)^2\over (2n)!}$$

$$= {(2n+1)(2n+2)\over 2(n+1)^2} = {2n+1\over n+1}.$$ (3)

Additionally,

$$\gamma_n={2\over 2n+1},$$ (4)

so

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

Using the Recurrence Relation

$$(1-x^2)P_n'(x)=nxP_n(x)+nP_{n-1}(x) = (n+1)xP_n(x)-(n+1)P_{n+1}(x)$$ (6)

gives

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

The error term is

$$E={2^{2n+1}(n!)^4\over(2n+1)[(2n)!]^3}f^{(2n)}(\xi).$$ (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.