Radau 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
2054
Radau Quadrature
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 ![[-1,1]](https://mathworld.wolfram.com/images/equations/RadauQuadrature/Inline6.gif)
is included in a total of 
abscissas, giving 
free abscissas. The general formula is
 |
(1)
|
The free abscissas 
for 
, ..., 
are the roots of the polynomial
 |
(2)
|
where 
is a Legendre polynomial. The weights of the free abscissas are
and of the endpoint
 |
(5)
|
The error term is given by
![E=(2^(2n-1)n[(n-1)!]^4)/([(2n-1)!]^3)f^((2n-1))(xi),](https://mathworld.wolfram.com/images/equations/RadauQuadrature/NumberedEquation4.gif) |
(6)
|
for 
.
 |
 |
 |
| 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 
.
REFERENCES:
Abramowitz, M. and Stegun, I. 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.
Ueberhuber, C. W. Numerical Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag, p. 105, 1997.
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