TOPICS
Search

Radau Quadrature


A Gaussian quadrature-like formula for numerical estimation of integrals. It requires m+1 points and fits all polynomials to degree 2m, so it effectively fits exactly all polynomials of degree 2m-1. It uses a weighting function W(x)=1 in which the endpoint -1 in the interval [-1,1] is included in a total of n abscissas, giving r=n-1 free abscissas. The general formula is

 int_(-1)^1f(x)dx=w_1f(-1)+sum_(i=2)^nw_if(x_i).
(1)

The free abscissas x_i for i=2, ..., n are the roots of the polynomial

 (P_(n-1)(x)+P_n(x))/(1+x),
(2)

where P(x) is a Legendre polynomial. The weights of the free abscissas are

w_i=(1-x_i)/(n^2[P_(n-1)(x_i)]^2)
(3)
=1/((1-x_i)[P_(n-1)^'(x_i)]^2),
(4)

and of the endpoint

 w_1=2/(n^2).
(5)

The error term is given by

 E=(2^(2n-1)n[(n-1)!]^4)/([(2n-1)!]^3)f^((2n-1))(xi),
(6)

for xi in (-1,1).

nx_iw_i
2-10.5
0.3333331.5
3-10.222222
-0.2898981.02497
0.6898980.752806
4-10.125
-0.5753190.657689
0.1810660.776387
0.8228240.440924
5-10.08
-0.720480.446208
-0.1671810.623653
0.4463140.562712
0.8857920.287427

The abscissas and weights can be computed analytically for small n.

nx_iw_i
2-11/2
1/33/2
3-12/9
1/5(1-sqrt(6))1/(18)(16+sqrt(6))
1/5(1+sqrt(6))1/(18)(16-sqrt(6))

See also

Chebyshev Quadrature, Lobatto Quadrature

Explore with Wolfram|Alpha

References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 888, 1972.Chandrasekhar, S. Radiative Transfer. New York: Dover, p. 61, 1960.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 338-343, 1956.Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 105, 1997.

Referenced on Wolfram|Alpha

Radau Quadrature

Cite this as:

Weisstein, Eric W. "Radau Quadrature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RadauQuadrature.html

Subject classifications