Gaussian Quadrature

Seeks to obtain the best numerical estimate of an integral by picking optimal Abscissas $x_i$ at which to evaluate the function $f(x)$. The Fundamental Theorem of Gaussian Quadrature states that the optimal Abscissas of the $m$-point Gaussian Quadrature Formulas are precisely the roots of the orthogonal Polynomial for the same interval and Weighting Function. Gaussian quadrature is optimal because it fits all Polynomials up to degree $2m$ exactly. Slightly less optimal fits are obtained from Radau Quadrature and Laguerre Quadrature.

$W(x)$	Interval	$x_i$ Are Roots Of
1	$(-1,1)$	$P_n(x)$
$e^{-t}$	$(0,\infty)$	$L_n(x)$
$e^{-t^2}$	$(-\infty,\infty)$	$H_n(x)$
$(1-t^2)^{-1/2}$	$(-1,1)$	$T_n(x)$
$(1-t^2)^{1/2}$	$(-1,1)$	$U_n(x)$
$x^{1/2}$	$(0,1)$	$x^{-1/2}P_{2n+1}(\sqrt{x}\,)$
$x^{-1/2}$	$(0,1)$	$P_n(\sqrt{x}\,)$

To determine the weights corresponding to the Gaussian Abscissas, compute a Lagrange Interpolating Polynomial for $f(x)$ by letting

$$\pi(x)\equiv \prod_{j=1}^m (x-x_j)$$ (1)

(where Chandrasekhar 1967 uses $F$ instead of $\pi$), so

$$\pi'(x_j)=\left[{d\pi\over dx}\right]_{x=x_j} = \prod_{\scriptstyle i=1\atop\scriptstyle i\not= j}^m (x_j-x_i).$$ (2)

Then fitting a Lagrange Interpolating Polynomial through the $m$ points gives

$$\phi(x)=\sum_{j=1}^m {\pi(x)\over (x-x_j)\pi'(x_j)} f(x_j)$$ (3)

for arbitrary points $x$. We are therefore looking for a set of points $x_j$ and weights $w_j$ such that for a Weighting Function $W(x)$,

$$\int_a^b \phi(x)W(x)\,dx = \int_a^b \sum_{j=1}^m {\pi(x)W(x)\over(x-x_j)\pi'(x_j)}\,dx\,f(x_j)$$

$$\equiv \sum_{j=1}^m w_jf(x_j),$$ (4)

with Weight

$$w_j ={1\over \pi'(x_j)} \int_a^b {\pi(x)W(x)\over x-x_j}\,dx.$$ (5)

The weights $w_j$ are sometimes also called the Christoffel Number (Chandrasekhar 1967). For orthogonal Polynomials $\phi_j(x)$ with $j$=1, ..., $n$,

$$\phi_j(x)=A_j\pi(x)$$ (6)

(Hildebrand 1956, p. 322), where $A_n$ is the Coefficient of $x^n$ in $\phi_n(x)$, then

$$w_j = {1\over\phi_n'(x_j)} \int_a^b W(x){\phi(x)\over x-x_j}\,dx$$

$$= -{A_{n+1}\gamma_n\over A_n\phi_n'(x_j)\phi_{n+1}(x)},$$ (7)

where

$$\gamma_m\equiv \int [\phi_m(x)]^2W(x)\,dx.$$ (8)

Using the relationship

$$\phi_{n+1}(x_i)=-{A_{n+1}A_{n-1}\over{A_n}^2} {\gamma_n\over\gamma_{n-1}}\phi_{n-1}(x_i)$$ (9)

(Hildebrand 1956, p. 323) gives

$$w_j={A_n\over A_{n-1}}{\gamma_{n-1}\over \phi_n'(x_j)\phi_{n-1}(x_j)}.$$ (10)

(Note that Press et al. 1992 omit the factor $A_n/A_{n-1}$.) In Gaussian quadrature, the weights are all Positive. The error is given by

$$E_n={f^{(2n)}(\xi)\over(2n)!}\int_a^b W(x)[\pi(x)]^2\,dx={\gamma_n\over{A_n}^2} {f^{(2n)}(\xi)\over (2n)!},$$ (11)

where $a<\xi<b$ (Hildebrand 1956, pp. 320-321).

Other curious identities are

$$\sum_{k=0}^m {[\phi_k(x)]^2\over\gamma_k} = {A_m\over A_{m+1}\gamma_m} [\phi_{m+1}'(x)\phi_m(x)-\phi_m'(x)\phi_{m+1}(x)]$$ (12)

and

$$\sum_{k=0}^m {[\phi_k(x)]^2\over\gamma_k}=-{A_m\phi_m'(x_i)\phi_{m+1}(x_i)\over A_{m+1}\gamma_m}={1\over w_i}$$ (13)

(Hildebrand 1956, p. 323).

In the Notation of Szegö (1975), let $x_{1n}<\ldots<x_{nn}$ be an ordered set of points in $[a, b]$, and let $\lambda_{1n}$, ..., $\lambda_{nn}$ be a set of Real Numbers. If $f(x)$ is an arbitrary function on the Closed Interval $[a, b]$, write the Mechanical Quadrature as

$$Q_n(f)=\sum_{\nu=1}^n \lambda_{\nu n}f(x_{\nu n}).$$ (14)

Here $x_{\nu n}$ are the Abscissas and $\lambda_{\nu n}$ are the Cotes Numbers.

See also Chebyshev Quadrature, Chebyshev-Gauss Quadrature, Chebyshev-Radau Quadrature, Fundamental Theorem of Gaussian Quadrature, Hermite-Gauss Quadrature, Jacobi-Gauss Quadrature, Laguerre-Gauss Quadrature, Legendre-Gauss Quadrature, Lobatto Quadrature, Mehler Quadrature, Radau Quadrature

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 887-888, 1972.

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 103, 1990.

Arfken, G. ``Appendix 2: Gaussian Quadrature.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 968-974, 1985.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 461, 1987.

Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, 1967.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Gaussian Quadratures and Orthogonal Polynomials.'' §4.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 140-155, 1992.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 37-48 and 340-349, 1975.

Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152-163, 1967.