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]](/images/equations/RadauQuadrature/Inline6.svg)
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
 |  | ![(1-x_i)/(n^2[P_(n-1)(x_i)]^2)](/images/equations/RadauQuadrature/Inline15.svg) |
(3)
|
 |  | ![1/((1-x_i)[P_(n-1)^'(x_i)]^2),](/images/equations/RadauQuadrature/Inline18.svg) |
(4)
|
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),](/images/equations/RadauQuadrature/NumberedEquation4.svg) |
(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 
.
 |  |  |
| 2 |  |  |
 |  | |
| 3 |  |  |
 |  | |
 |  |
