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Lobatto Quadrature

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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)^1f(x)dx=w_1f(-1)+w_nf(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)/((1-x_i^2)P_(n-1)^('')(x_i)P_m^'(x_i))
(2)
=2/(n(n-1)[P_(n-1)(x_i)]^2),
(3)

and of the endpoints are

w_(1,n)=2/(n(n-1)).
(4)

The error term is given by

E=-(n(n-1)^32^(2n-1)[(n-2)!]^4)/((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).

nx_ix_iw_iw_i
300.000004/31.333333
+/-1+/-1.000001/30.333333
4+/-1/5sqrt(5)+/-0.4472145/60.833333
+/-1+/-1.0000001/60.166667
500.000000(32)/(45)0.711111
+/-1/7sqrt(21)+/-0.654654(49)/(90)0.544444
+/-1+/-1.0000001/(10)0.100000
6sqrt(1/(21)(7-2sqrt(7)))+/-0.2852321/(30)(14+sqrt(7))0.554858
sqrt(1/(21)(7+2sqrt(7)))+/-0.7650551/(30)(14-sqrt(7))0.378475
+/-1+/-1.0000001/(15)0.066667

See also

Chebyshev Quadrature, Radau Quadrature

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References

Abramowitz, M. and Stegun, I. 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.Hunter, D. and Nikolov, G. "On the Error Term of Symmetric Gauss-Lobatto Quadrature Formulae for Analytic Functions." Math. Comput. 69, 269-282, 2000.Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 105, 1997.

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Lobatto Quadrature

Cite this as:

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

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