`scipy.special._orthogonal:roots_hermite`

source: /scipy/special/_orthogonal.py :827

Signature

def roots_hermite ( n , mu = False )

Summary

Gauss-Hermite (physicist's) quadrature.

Extended Summary

Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth degree Hermite polynomial, $H_{n} (x)$ . These sample points and weights correctly integrate polynomials of degree $2 n - 1$ or less over the interval $[- \infty, \infty]$ with weight function $w (x) = e^{- x^{2}}$ . See 22.2.14 in [AS] for details.

Parameters

n : int: quadrature order
mu : bool, optional: If True, return the sum of the weights, optional.

Returns

x : ndarray: Sample points
w : ndarray: Weights
mu : float: Sum of the weights

Notes

For small n up to 150 a modified version of the Golub-Welsch algorithm is used. Nodes are computed from the eigenvalue problem and improved by one step of a Newton iteration. The weights are computed from the well-known analytical formula.

For n larger than 150 an optimal asymptotic algorithm is applied which computes nodes and weights in a numerically stable manner. The algorithm has linear runtime making computation for very large n (several thousand or more) feasible.

Aliases

scipy.special.h_roots