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

$$\int_{-1}^1 f(x)\,dx=w_1 f(-1)+\sum_{i=2}^{n} w_if(x_i).$$ (1)

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

$${P_{n-1}(x)+P_n(x)\over 1+x},$$ (2)

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

$$w_i={1-x_i\over n^2[P_{n-1}(x_i)]^2}={1\over(1-x_i)[P_{n-1}'(x_i)]^2},$$ (3)

and of the endpoint

$$w_1={2\over n^2}.$$ (4)

The error term is given by

$$E={2^{2n-1}n[(n-1)!]^4\over[(2n-1)!]^3} f^{(2n-1)}(\xi),$$ (5)

for $\xi\in(-1,1)$.

$n$	$x_i$	$w_i$
2	$-1$	0.5
	0.333333	1.5
3	$-1$	0.222222
	$-0.289898$	1.02497
	0.689898	0.752806
4	$-1$	0.125
	$-0.575319$	0.657689
	0.181066	0.776387
	0.822824	0.440924
5	$-1$	0.08
	$-0.72048$	0.446208
	$-0.167181$	0.623653
	0.446314	0.562712
	0.885792	0.287427

The Abscissas and weights can be computed analytically for small $n$.

$n$	$x_i$	$w_i$
2	$-1$	${\textstyle{1\over 2}}$
	${\textstyle{1\over 3}}$	${\textstyle{3\over 2}}$
3	$-1$	${\textstyle{2\over 9}}$
	${\textstyle{1\over 5}}(1-\sqrt{6}\,)$	${\textstyle{1\over 18}}(16+\sqrt{6}\,)$
	${\textstyle{1\over 5}}(1+\sqrt{6}\,)$	${\textstyle{1\over 18}}(16-\sqrt{6}\,)$

See also Chebyshev Quadrature, Lobatto Quadrature

References

Abramowitz, M. and Stegun, C. 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.