Namespaces
Variants
Actions

Difference between revisions of "Gauss quadrature formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
 +
{{TEX|done}}
 
The quadrature formula
 
The quadrature formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435101.png" /></td> </tr></table>
+
$$\int\limits_a^bp(x)f(x)dx\approx\sum_{i=1}^nc_if(x_i),$$
  
in which the nodes (cf. [[Node|Node]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435102.png" /> and the weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435103.png" /> are so selected that the formula is exact for the functions
+
in which the nodes (cf. [[Node|Node]]) $x_i$ and the weights $c_i$ are so selected that the formula is exact for the functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435104.png" /></td> </tr></table>
+
$$\sum_{k=0}^{2n-1}a_k\omega_k(x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435105.png" /> are given linearly independent functions (the integration limits may well be infinite). The formula was introduced by C.F. Gauss [[#References|[1]]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435108.png" />. He obtained the following formula, which is exact for an arbitrary polynomial of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g0435109.png" />:
+
where $\omega_k(x)$ are given linearly independent functions (the integration limits may well be infinite). The formula was introduced by C.F. Gauss [[#References|[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 $2n-1$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351010.png" /></td> </tr></table>
+
$$\int\limits_{-1}^1f(x)dx=A_1^{(n)}f(x_1)+\ldots+A_n^{(n)}f(x_n)+R_n,$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351011.png" /> are the roots of the Legendre polynomial (cf. [[Legendre polynomials|Legendre polynomials]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351012.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351014.png" /> are defined by the formulas
+
where the $x_k$ are the roots of the Legendre polynomial (cf. [[Legendre polynomials|Legendre polynomials]]) $P_n(x)$, while $A_k^{(n)}$ and $R_n$ are defined by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351015.png" /></td> </tr></table>
+
$$A_k^{(n)}=\frac{2}{(1-x_k^2)[P_n'(x_k)]^2};$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351016.png" /></td> </tr></table>
+
$$R_n=\frac{2^{2n+1}[n!]^4}{(2n+1)[(2n)!]^3}f^{(2n)}(c),\quad-1<c<1.$$
  
The formula is used whenever the integrand is sufficiently smooth, and the gain in the number of nodes is substantial; for instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351017.png" /> 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|weight function]] and of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351018.png" /> is very important.
+
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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351020.png" /> [[#References|[5]]]; in particular for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351022.png" /> up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351023.png" />.
+
Tables of nodes in Gauss' quadrature formula are available for wide classes of $p(x)$ and $\omega_j(x)$ [[#References|[5]]]; in particular for $p(x)\equiv1$, $\omega_j(x)=x^j$ up to $n=512$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351025.png" />, Gauss' quadrature formula is employed in standard integration programs with an automatic step selection as a method of computing integrals by subdivision of subsegments [[#References|[6]]].
+
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 [[#References|[6]]].
  
 
====References====
 
====References====
Line 29: Line 30:
  
 
====Comments====
 
====Comments====
A detailed investigation of the general Gauss formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043510/g04351026.png" /> was carried out by E.B. Christoffel [[#References|[a3]]] and the quadrature coefficients are therefore also called Christoffel coefficients or [[Christoffel numbers|Christoffel numbers]] (see also [[#References|[a1]]]). Tables of these coefficients may be found in [[#References|[a2]]].
+
A detailed investigation of the general Gauss formulas $(w\not\equiv1)$ was carried out by E.B. Christoffel [[#References|[a3]]] and the quadrature coefficients are therefore also called Christoffel coefficients or [[Christoffel numbers|Christoffel numbers]] (see also [[#References|[a1]]]). Tables of these coefficients may be found in [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , '''25''' , Dover, reprint  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.B. Christoffel,  "Ueber die Gausssche Quadratur und eine Verallgemeinerung derselben"  ''J. Reine Angew. Math.'' , '''55'''  (1858)  pp. 81–82</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Piessens,  et al.,  "Quadpack" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Hildebrand,  "Introduction to numerical analysis" , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abramowitz,  I.A. Stegun,  "Handbook of mathematical functions" , '''25''' , Dover, reprint  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.B. Christoffel,  "Ueber die Gausssche Quadratur und eine Verallgemeinerung derselben"  ''J. Reine Angew. Math.'' , '''55'''  (1858)  pp. 81–82</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.J. Davis,  P. Rabinowitz,  "Methods of numerical integration" , Acad. Press  (1984)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Piessens,  et al.,  "Quadpack" , Springer  (1983)</TD></TR></table>

Revision as of 12:40, 27 October 2014

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 $2n-1$:

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

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

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

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

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)
How to Cite This Entry:
Gauss quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_quadrature_formula&oldid=34105
This article was adapted from an original article by N.S. BakhvalovV.P. Motornyi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article