A Gaussian Quadrature-like formula for numerical estimation of integrals. It requires points and fits all
Polynomials to degree , so it effectively fits exactly all Polynomials of
degree . It uses a Weighting Function in which the endpoint in the interval is included
in a total of Abscissas, giving free abscissas. The general formula is
(1) |
(2) |
(3) |
(4) |
(5) |
2 | 0.5 | |
0.333333 | 1.5 | |
3 | 0.222222 | |
1.02497 | ||
0.689898 | 0.752806 | |
4 | 0.125 | |
0.657689 | ||
0.181066 | 0.776387 | |
0.822824 | 0.440924 | |
5 | 0.08 | |
0.446208 | ||
0.623653 | ||
0.446314 | 0.562712 | |
0.885792 | 0.287427 |
The Abscissas and weights can be computed analytically for small .
2 | ||
3 | ||
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.
© 1996-9 Eric W. Weisstein