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Difference between revisions of "Radau quadrature formula"

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A [[Quadrature formula of highest algebraic accuracy|quadrature formula of highest algebraic accuracy]] for the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770201.png" /> and with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770202.png" />, with one fixed node, namely an end point of the interval, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770203.png" />. The Radau quadrature formula has the form
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A [[Quadrature formula of highest algebraic accuracy|quadrature formula of highest algebraic accuracy]] for the interval $[a,b]=[-1,1]$ and with weight $p(x)=1$, with one fixed node, namely an end point of the interval, say $-1$. The Radau quadrature formula has the form
  
<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/r/r077/r077020/r0770204.png" /></td> </tr></table>
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$$\int\limits_{-1}^1f(x)dx\cong Af(-1)+\sum_{j=1}^nC_jf(x_j).$$
  
The nodes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770205.png" /> are the roots of the Jacobi polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770206.png" /> (the [[Jacobi polynomials|Jacobi polynomials]] form an orthogonal system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770207.png" /> with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770208.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r0770209.png" />. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r07702010.png" /> are positive. The algebraic degree of accuracy is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077020/r07702011.png" />. There exist tables of the nodes and coefficients for the Radau quadrature formula, see, for example, [[#References|[2]]].
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The nodes $x_j$ are the roots of the Jacobi polynomial $P_n^{(0,1)}(x)$ (the [[Jacobi polynomials|Jacobi polynomials]] form an orthogonal system on $[-1,1]$ with weight $1+x$), and $A=2/(n+1)^2$. The coefficients $C_j$ are positive. The algebraic degree of accuracy is $2n$. There exist tables of the nodes and coefficients for the Radau quadrature formula, see, for example, [[#References|[2]]].
  
 
The formula was found by R. Radau [[#References|[1]]].
 
The formula was found by R. Radau [[#References|[1]]].

Latest revision as of 12:26, 27 October 2014

A quadrature formula of highest algebraic accuracy for the interval $[a,b]=[-1,1]$ and with weight $p(x)=1$, with one fixed node, namely an end point of the interval, say $-1$. The Radau quadrature formula has the form

$$\int\limits_{-1}^1f(x)dx\cong Af(-1)+\sum_{j=1}^nC_jf(x_j).$$

The nodes $x_j$ are the roots of the Jacobi polynomial $P_n^{(0,1)}(x)$ (the Jacobi polynomials form an orthogonal system on $[-1,1]$ with weight $1+x$), and $A=2/(n+1)^2$. The coefficients $C_j$ are positive. The algebraic degree of accuracy is $2n$. There exist tables of the nodes and coefficients for the Radau quadrature formula, see, for example, [2].

The formula was found by R. Radau [1].

References

[1] R. Radau, "Etude sur les formules d'approximation qui servent à calculer la valeur numérique d'une intégrale définie" J. Math. Pures et Appl. , 6 (1880) pp. 283–336
[2] V.I. Krylov, "Approximate calculation of integrals" , Macmillan (1962) (Translated from Russian)
[3] A.M. Stroud, D. Secrest, "Gaussian quadrature formulas" , Prentice-Hall (1966)
How to Cite This Entry:
Radau quadrature formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radau_quadrature_formula&oldid=34104
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article