numpy.polynomial.legendre.leggauss

polynomial.legendre.leggauss(deg)

Gauss-Legendre quadrature.

Computes the sample points and weights for Gauss-Legendre quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([-1, 1]\) with the weight function \(f(x) = 1\).

Parameters:

degint

Number of sample points and weights. It must be >= 1.

Returns:

xndarray

1-D ndarray containing the sample points.

yndarray

1-D ndarray containing the weights.

Notes

The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that

\[w_k = c / (L'_n(x_k) * L_{n-1}(x_k))\]

where \(c\) is a constant independent of \(k\) and \(x_k\) is the k’th root of \(L_n\), and then scaling the results to get the right value when integrating 1.