numpy.polynomial.legendre.leggauss¶

numpy.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:

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.

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.