scipy.special.roots_hermite¶

scipy.special.
roots_hermite
(n, mu=False)[source]¶ GaussHermite (physicist’s) quadrature.
Compute the sample points and weights for GaussHermite 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
quadrature order
 mubool, optional
If True, return the sum of the weights, optional.
 Returns
 xndarray
Sample points
 wndarray
Weights
 mufloat
Sum of the weights
See also
Notes
For small n up to 150 a modified version of the GolubWelsch 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 wellknown 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.olver2014
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.olver2015
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.