scipy.special.roots_hermitenorm¶

scipy.special.roots_hermitenorm(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, $$He_n(x)$$. 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

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 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.

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