numpy.polynomial.laguerre.laggauss#

polynomial.laguerre.laggauss(deg)[source]#

Gauss-Laguerre quadrature.

Computes the sample points and weights for Gauss-Laguerre quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([0, \inf]\) with the weight function \(f(x) = \exp(-x)\).

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.

Examples

>>> from numpy.polynomial.laguerre import laggauss
>>> laggauss(2)
(array([0.58578644, 3.41421356]), array([0.85355339, 0.14644661]))