# Radau quadrature formula

From Encyclopedia of Mathematics

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