Lobatto Quadrature
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
7-12-2021
2022
Lobatto Quadrature
Also called Radau quadrature (Chandrasekhar 1960). A Gaussian quadrature with weighting function 
in which the endpoints of the interval ![[-1,1]](https://mathworld.wolfram.com/images/equations/LobattoQuadrature/Inline2.gif)
are included in a total of 
abscissas, giving 
free abscissas. Abscissas are symmetrical about the origin, and the general formula is
 |
(1)
|
The free abscissas 
for 
, ..., 
are the roots of the polynomial 
, where 
is a Legendre polynomial. The weights of the free abscissas are
and of the endpoints are
 |
(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),](https://mathworld.wolfram.com/images/equations/LobattoQuadrature/NumberedEquation3.gif) |
(5)
|
for 
. Beyer (1987) gives a table of parameters up to 
and Chandrasekhar (1960) up to 
(although Chandrasekhar's 
for 
is incorrect).
 |
 |
 |
 |
 |
| 3 |
0 |
0.00000 |
 |
1.333333 |
| |
 |
 |
 |
0.333333 |
| 4 |
 |
 |
 |
0.833333 |
| |
 |
 |
 |
0.166667 |
| 5 |
0 |
0.000000 |
 |
0.711111 |
| |
 |
 |
 |
0.544444 |
| |
 |
 |
 |
0.100000 |
| 6 |
 |
 |
 |
0.554858 |
| |
 |
 |
 |
0.378475 |
| |
 |
 |
 |
0.066667 |
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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