Gauss–Hermite quadrature

In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind:

${\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx.}$

Weights versus x_i for four choices of n

In this case

${\displaystyle \int _{-\infty }^{+\infty }e^{-x^{2}}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i})}$

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 ^[1]

${\displaystyle w_{i}={\frac {2^{n-1}n!{\sqrt {\pi }}}{n^{2}[H_{n-1}(x_{i})]^{2}}}.}$

Example with change of variable

Consider a function h(y), where the variable y is Normally distributed: ${\displaystyle y\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$. The expectation of h corresponds to the following integral:

${\displaystyle E[h(y)]=\int _{-\infty }^{+\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(y-\mu )^{2}}{2\sigma ^{2}}}\right)h(y)dy}$

As this does not exactly correspond to the Hermite polynomial, we need to change variables:

${\displaystyle x={\frac {y-\mu }{{\sqrt {2}}\sigma }}\Leftrightarrow y={\sqrt {2}}\sigma x+\mu }$

Coupled with the integration by substitution, we obtain:

${\displaystyle E[h(y)]=\int _{-\infty }^{+\infty }{\frac {1}{\sqrt {\pi }}}\exp(-x^{2})h({\sqrt {2}}\sigma x+\mu )dx}$

leading to:

${\displaystyle E[h(y)]\approx {\frac {1}{\sqrt {\pi }}}\sum _{i=1}^{n}w_{i}h({\sqrt {2}}\sigma x_{i}+\mu )}$

As an illustration, in the simplest non-trivial case, with ${\displaystyle n=2}$, we have ${\displaystyle x_{1}=-{\frac {1}{\sqrt {2}}},x_{2}={\frac {1}{\sqrt {2}}}}$ and ${\displaystyle w_{1}=w_{2}={\frac {\sqrt {\pi }}{2}}}$, so the estimate reduces to:

${\displaystyle E[h(y)]\approx {\frac {1}{2}}(h(\mu -\sigma )+h(\mu +\sigma ))}$

– i.e. the average of the function's values one standard deviation below and above the mean.