Radau quadrature formula

From Encyclopedia of Mathematics
Jump to: navigation, search

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].


[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:
This article was adapted from an original article by I.P. Mysovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article