Gauss quadrature formula

The quadrature formula

$$\int\limits_a^bp(x)f(x)\,dx\approx\sum_{i=1}^nc_if(x_i),$$

in which the nodes (cf. Node) $x_i$ and the weights $c_i$ are so selected that the formula is exact for the functions

$$\sum_{k=0}^{2n-1}a_k\omega_k(x),$$

where $\omega_k(x)$ are given linearly independent functions (the integration limits may well be infinite). The formula was introduced by C.F. Gauss [1] for $a=-1$, $b=1$, $p(x)\equiv1$. He obtained the following formula, which is exact for an arbitrary polynomial of degree not exceeding $2n-1$:

$$\int\limits_{-1}^1f(x)\,dx=A_1^{(n)}f(x_1)+\dots+A_n^{(n)}f(x_n)+R_n,$$

where the $x_k$ are the roots of the Legendre polynomial (cf. Legendre polynomials) $P_n(x)$, while $A_k^{(n)}$ and $R_n$ are defined by the formulas

$$A_k^{(n)}=\frac{2}{(1-x_k^2)[P_n'(x_k)]^2};$$

$$R_n=\frac{2^{2n+1}[n!]^4}{(2n+1)[(2n)!]^3}f^{(2n)}(c),\qquad-1<c<1.$$

The formula is used whenever the integrand is sufficiently smooth, and the gain in the number of nodes is substantial; for instance, if $f(x)$ is determined from expensive experiments or during the computation of multiple integrals as repeated integrals. In such practical applications a suitable choice of the weight function and of the functions $\omega_j(x)$ is very important.

Tables of nodes in Gauss' quadrature formula are available for wide classes of $p(x)$ and $\omega_j(x)$ [5]; in particular for $p(x)\equiv1$, $\omega_j(x)=x^j$ up to $n=512$.

If $p(x)\equiv1$, $\omega_j(x)=x^j$, Gauss' quadrature formula is employed in standard integration programs with an automatic step selection as a method of computing integrals by subdivision of subsegments [6].

References

[1]	C.F. Gauss, "Methodus nova integralium valores per approximationem inveniendi" , Werke , 3 , K. Gesellschaft Wissenschaft. Göttingen (1886) pp. 163–196
[2]	N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian)
[3]	V.I. Krylov, L.T. Shul'gina, "Handbook on numerical integration" , Moscow (1966) (In Russian)
[4]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[5]	A.H. Stroud, "Gaussian quadrature formulas" , Prentice-Hall (1966)
[6]	, A standard program for the computation of single integrals of quadratures of Gauss' type : 26 , Moscow (1967) (In Russian)

Comments

A detailed investigation of the general Gauss formulas $(w\not\equiv1)$ was carried out by E.B. Christoffel [a3] and the quadrature coefficients are therefore also called Christoffel coefficients or Christoffel numbers (see also [a1]). Tables of these coefficients may be found in [a2].

References

[a1]	F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)
[a2]	M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , 25 , Dover, reprint (1970)
[a3]	E.B. Christoffel, "Ueber die Gausssche Quadratur und eine Verallgemeinerung derselben" J. Reine Angew. Math. , 55 (1858) pp. 81–82
[a4]	P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)
[a5]	R. Piessens, et al., "Quadpack" , Springer (1983)