numpy.polynomial.legendre.leggauss

polynomial.legendre.leggauss(deg) [source]

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

New in version 1.7.0.

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.