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Difference between revisions of "Chebyshev quadrature formula"

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An interpolation quadrature formula with equal coefficients:
 
An interpolation quadrature formula with equal coefficients:
  
<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/c/c021/c021950/c0219501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\int\limits_{-1}^1f(x)dx\cong C\sum_{k=1}^Nf(x_k).\tag{*}$$
  
The weight function is equal to one, and the integration interval is finite and is taken to coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219502.png" />. The number of parameters defining the quadrature formula (*) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219503.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219504.png" /> nodes and the value of the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219505.png" />). The parameters are determined by the requirement that (*) is exact for all polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219506.png" /> or less, or equivalently, for the monomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219507.png" />. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219508.png" /> is obtained from the condition that the quadrature formula is exact for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c0219509.png" />, and is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195010.png" />. The nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195011.png" /> turn out to be real only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195013.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195014.png" /> the nodes were calculated by P.L. Chebyshev. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195015.png" /> among the nodes of the Chebyshev quadrature formula there always are complex ones (cf. [[#References|[1]]]). The algebraic degree of precision of the Chebyshev quadrature formula is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195016.png" /> for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195018.png" /> for even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021950/c02195019.png" />. Formula (*) was proposed by Chebyshev in 1873.
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The weight function is equal to one, and the integration interval is finite and is taken to coincide with $[-1,1]$. The number of parameters defining the quadrature formula \ref{*} is $N+1$ ($N$ nodes and the value of the coefficient $C$). The parameters are determined by the requirement that \ref{*} is exact for all polynomials of degree $N$ or less, or equivalently, for the monomials $1,x,\ldots,x^N$. The parameter $C$ is obtained from the condition that the quadrature formula is exact for $f(x)=1$, and is equal to $2/N$. The nodes $x_1,\ldots,x_N$ turn out to be real only for $N=1,\ldots,7$ and $N=9$. For $N=1,\ldots,7$ the nodes were calculated by P.L. Chebyshev. For $N\geq10$ among the nodes of the Chebyshev quadrature formula there always are complex ones (cf. [[#References|[1]]]). The algebraic degree of precision of the Chebyshev quadrature formula is $N$ for odd $N$ and $N+1$ for even $N$. Formula \ref{*} was proposed by Chebyshev in 1873.
  
 
====References====
 
====References====

Revision as of 17:02, 3 August 2014

An interpolation quadrature formula with equal coefficients:

$$\int\limits_{-1}^1f(x)dx\cong C\sum_{k=1}^Nf(x_k).\tag{*}$$

The weight function is equal to one, and the integration interval is finite and is taken to coincide with $[-1,1]$. The number of parameters defining the quadrature formula \ref{*} is $N+1$ ($N$ nodes and the value of the coefficient $C$). The parameters are determined by the requirement that \ref{*} is exact for all polynomials of degree $N$ or less, or equivalently, for the monomials $1,x,\ldots,x^N$. The parameter $C$ is obtained from the condition that the quadrature formula is exact for $f(x)=1$, and is equal to $2/N$. The nodes $x_1,\ldots,x_N$ turn out to be real only for $N=1,\ldots,7$ and $N=9$. For $N=1,\ldots,7$ the nodes were calculated by P.L. Chebyshev. For $N\geq10$ among the nodes of the Chebyshev quadrature formula there always are complex ones (cf. [1]). The algebraic degree of precision of the Chebyshev quadrature formula is $N$ for odd $N$ and $N+1$ for even $N$. Formula \ref{*} was proposed by Chebyshev in 1873.

References

[1] N.M. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian)


Comments

This formula is to be distinguished from the Gauss–Chebyshev quadrature formula (cf. Gauss quadrature formula), which is defined using a weight function .

The original reference for Chebyshev's quadrature formula is [a3]. S.N. Bernshtein [a2] has shown that the nodes are real only if or . A detailed discussion of the formula can be found in [a4]. Tables of quadrature nodes are given in [a1].

References

[a1] A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[a2] S.N. Bernshtein, "Sur les formules quadratures de Cotes et Chebyshev" C.R. Acad. Sci. USSR , 14 pp. 323–326
[a3] P.L. Chebyshev, "Sur les quadratures" J. Math. Pures Appl. , 19 : 2 (1874) pp. 19–34 (Oeuvres, Vol. 2, pp. 165–180)
[a4] F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)
[a5] P.J. Davis, P. Rabinowitz, "Methods of numerical integration" , Acad. Press (1984)
How to Cite This Entry:
Chebyshev quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebyshev_quadrature_formula&oldid=16279
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article