Lobatto Quadrature

Also called Radau Quadrature (Chandrasekhar 1960). A Gaussian Quadrature with Weighting Function $W(x)=1$ in which the endpoints of the interval $[-1,1]$ are included in a total of $n$ Abscissas, giving $r=n-2$ free abscissas. Abscissas are symmetrical about the origin, and the general Formula is

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

The free Abscissas $x_i$ for $i=2$, ..., $n-1$ are the roots of the Polynomial $P_{n-1}'(x)$, where $P(x)$ is a Legendre Polynomial. The weights of the free abscissas are

$$w_i = -{2n\over(1-{x_i}^2)P_{n-1}''(x_i)P_m'(x_i)}$$ (2)

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

and of the endpoints are

$$w_{1,n}={2\over n(n-1)}.$$ (4)

The error term is given by

$$E=-{n(n-1)^32^{2n-1}[(n-2)!]^4\over (2n-1)[(2n-2)!]^3} f^{(2n-2)}(\xi),$$ (5)

for $\xi\in(-1,1)$. Beyer (1987) gives a table of parameters up to $n$=11 and Chandrasekhar (1960) up to $n$=9 (although Chandrasekhar's $\mu_{3,4}$ for $m=5$ is incorrect).

$n$	$x_i$	$w_i$
3	0	1.33333
	± 1	0.333333
4	± 0.447214	0.833333
	± 1	0.166667
5	0	0.711111
	± 0.654654	0.544444
	± 1	0.100000
6	± 0.285232	0.554858
	± 0.765055	0.378475
	± 1	0.0666667

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

$n$	$x_i$	$w_i$
3	0	${\textstyle{4\over 3}}$
	$\pm 1$	${\textstyle{1\over 3}}$
4	$\pm{\textstyle{1\over 5}}\sqrt{5}$	${\textstyle{1\over 6}}$
	$\pm 1$	${\textstyle{5\over 6}}$
5	0	${\textstyle{32\over 45}}$
	$\pm{\textstyle{1\over 7}}\sqrt{21}$	${\textstyle{49\over 90}}$
	$\pm 1$	${\textstyle{1\over 10}}$

See also Chebyshev Quadrature, Radau Quadrature

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 888-890, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 465, 1987.

Chandrasekhar, S. Radiative Transfer. New York: Dover, pp. 63-64, 1960.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 343-345, 1956.