`numpy.polynomial.legendre:leggauss`

source: /numpy/polynomial/legendre.py :1470

Signature

def leggauss ( deg )

Summary

Gauss-Legendre quadrature.

Extended Summary

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

Parameters

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

Returns

x : ndarray: 1-D ndarray containing the sample points.
y : ndarray: 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.

Aliases

numpy.polynomial.legendre.leggauss