# scipy.special.roots_hermite#

scipy.special.roots_hermite(n, mu=False)[source]#

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 $$2n - 1$$ or less over the interval $$[-\infty, \infty]$$ with weight function $$w(x) = e^{-x^2}$$. See 22.2.14 in [AS] for details.

Parameters
nint

mubool, optional

If True, return the sum of the weights, optional.

Returns
xndarray

Sample points

wndarray

Weights

mufloat

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.

References

townsend.trogdon.olver-2014

Townsend, A. and Trogdon, T. and Olver, S. (2014) Fast computation of Gauss quadrature nodes and weights on the whole real line. arXiv:1410.5286.

townsend.trogdon.olver-2015

Townsend, A. and Trogdon, T. and Olver, S. (2015) Fast computation of Gauss quadrature nodes and weights on the whole real line. IMA Journal of Numerical Analysis DOI:10.1093/imanum/drv002.

AS

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.