Gauss quadrature formula
The quadrature formula
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in which the nodes (cf. Node) and the weights
are so selected that the formula is exact for the functions
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where are given linearly independent functions (the integration limits may well be infinite). The formula was introduced by C.F. Gauss [1] for
,
,
. He obtained the following formula, which is exact for an arbitrary polynomial of degree not exceeding
:
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where the are the roots of the Legendre polynomial (cf. Legendre polynomials)
, while
and
are defined by the formulas
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The formula is used whenever the integrand is sufficiently smooth, and the gain in the number of nodes is substantial; for instance, if 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
is very important.
Tables of nodes in Gauss' quadrature formula are available for wide classes of and
[5]; in particular for
,
up to
.
If ,
, 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 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) |
Gauss quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_quadrature_formula&oldid=34105