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)
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The free abscissas for , ..., are the roots of the polynomial
(2)
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where is a Legendre polynomial. The weights of the free abscissas are
(3)
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(4)
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and of the endpoint
(5)
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The error term is given by
(6)
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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 | ||