scipy.special.

roots_hermitenorm

scipy.special.roots_hermitenorm(n, mu=False)

Gauss-Hermite (statistician’s) quadrature.

Compute the sample points and weights for Gauss-Hermite quadrature. The sample points are the roots of the nth degree Hermite polynomial, \(He_n(x)\). These sample points and weights correctly integrate polynomials of degree \(2n - 1\) or less over the interval \([-\infty, \infty]\) with weight function \(w(x) = e^{-x^2/2}\). See 22.2.15 in [AS] for more 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

scipy.integrate.fixed_quad

numpy.polynomial.hermite_e.hermegauss

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 used which computes nodes and weights in a numerical stable manner. The algorithm has linear runtime making computation for very large n (several thousand or more) feasible.

References

[AS]

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