Gauss-Laguerre Numerical Integration
Calculate the integral over the infinite interval (0,∞) using the Gauss-Laguerre method.
| \(\normalsize Gauss-Laguerre\ quadrature\\ {\large\int_{\small 0}^{\small \infty}}x^{\alpha}e^{-x}f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ {\large\int_{\small 0}^{\small \infty}}g(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}x_i^{-\alpha}e^{x_i}g(x_i)\\\) |
The integrand f(x) is assumed to be analytic and not a periodic function.
| \(\normalsize Gaussian\ quadrature\\ \hspace{20px} {\large\int_{\small a}^{\small b}}w(x)f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i), \hspace{20px} {\large\int_{\small a}^{\small b}}g(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}{\large\frac{w_{i}}{w(x_i)}}g(x_i)\\ Gauss-Laguerre\ quadrature\\ \hspace{30px} interval(a,b):\hspace{20px} [0,\infty)\\ \hspace{30px} w(x):\hspace{80px} x^{\alpha} e^{-x} \\ \hspace{30px} polynomialsl:\hspace{10px} L_n^\alpha (x) \\ \) |
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