Gauss-Hermite Quadrature. More...
#include <FgGaussHermiteQuadrature.h>
Static Public Member Functions | |
static void | Compute (int n, af::array &x, af::array &w) |
Data direct. More... | |
Static Private Member Functions | |
static af::array | HermCompanion (const af::array &c) |
Return the scaled companion matrix of c. More... | |
static af::array | NormedHermite (const af::array &x, int n) |
Evaluate a normalized Hermite polynomial.. More... | |
Gauss-Hermite Quadrature.
Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:
$$\int_{-\inf}^{\inf} \exp(-x^2)f(x)dx.$$
In this case
$$\int_{-\inf}^{\inf} \exp(-x^2)f(x)dx\approx\sum_{i=1}^{n}w_i f(x_i),$$
where \(n\) is the number of sample points used. The \(x_i\) are the roots of the physicists' version of the Hermite polynomial \(H_n(x) (i = 1,2,...,n)\), and the associated weights \(w_i\) are given by
$$w_i=\frac{2^{n-1}n!\sqrt{\pi}}{n^2 [H_{n-1}(x_i)]^2}.$$
, 25.09.2019.
Definition at line 47 of file FgGaussHermiteQuadrature.h.
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static |
Data direct.
Computes the sample points and weights for Gauss-Hermite quadrature.
, 25.09.2019.
n | Approximation order. |
x | [in,out] The sample points. |
w | [in,out] The weights. |
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staticprivate |
Return the scaled companion matrix of c.
, 25.09.2019.
c | The af::array to process. |
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staticprivate |
Evaluate a normalized Hermite polynomial..
Compute the value of the normalized Hermite polynomial of degree $$n$$ at the points $$\mathbf{x}$$.
HmetalT, 12/11/2019.
x | Points at which to evaluate the function. |
n | Degree of the normalized Hermite function to be evaluated. |