Gauss-Hermite Numerical Integration
Compute the integral over the infinite interval (-∞,∞) using Gauss–Hermite quadrature.
| \(\normalsize Gauss-Hermite\ quadrature\\ {\large\int_{\small -\infty}^{\small \infty}}e^{-x^2}f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ {\large\int_{\small -\infty}^{\small \infty}}g(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}e^{x_i^2}g(x_i)\\\) |
The integrand f(x) is assumed to be analytic and not a periodic function.
| \(\normalsize Gaussian\ quadrature\\ \hspace{10px} {\large\int_{\small a}^{\small b}}w(x)f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i), \ {\large\int_{\small a}^{\small b}}g(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}{\large\frac{w_{i}}{w(x_i)}}g(x_i)\\ Gauss-Hermite\ quadrature\\ \hspace{30px} interval(a,b):\hspace{20px} (-\infty,\infty)\\ \hspace{30px} w(x):\hspace{80px} e^{-x^2}\\ \hspace{30px} polynomialsl:\hspace{10px} H_n (x) \\ \) |
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