Gauss-Lobatto Numerical Integration
Calculate the integral over the finite interval (a, b) using the Gauss-Lobatto integration method.
| \(\normalsize Gauss-Lobatto\ quadrature\\ {\large\int_{\small -1}^{\small 1}}f(x)dx\simeq{\large\frac{2}{n(n-1)}}(f(-1)+f(1))+{\large\displaystyle \sum_{\small i=2}^{n-1}}w_{i}f(x_i)\\{\large\int_a^{b}}f(x)dx\simeq{\large\frac{b-a}{2}[{\large\frac{2(f(a)+f(b))}{n(n-1)}}+\displaystyle \sum_{\small i=1}^{n}}w_{i}f({\large\frac{b-a}{2}}x_i+{\large\frac{b+a}{2}})]\\\) |
The integrand f(x) is assumed to be analytic and not a periodic function.
| \(\normalsize Gaussian\ quadrature\\ \hspace{20px} {\large\int_{\small a}^{\small b}}w(x)f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ Gauss-Lobatto\ quadrature\\ \hspace{30px} interval(a,b):\hspace{20px} [-1,\ 1]\\ \hspace{30px} w(x):\hspace{80px} 1\\ \hspace{30px} polynomials:\hspace{10px} P'_{n-1} (x)\\ \) |
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