scipy.special.roots_hermitenorm¶
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scipy.special.roots_hermitenorm(n, mu=False)[source]¶ 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
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.