Tanh-Sinh数值积分的积分点和权重
计算Tanh-Sinh数值积分的积分点（nodes）和权重（weights）。
| \(\normalsize {\large\int_{\small -1}^{\small 1}}f(x)dx\simeq{\large\int_{-t_a}^{t_a}}f(x(t))x'(t)dt\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ \ nodes\\ \hspace{20px} x_i=\tanh(\frac{\pi}{2}\sinh(t_i)),\hspace{10px}t_i=-t_a+(i-1)h\\ \ weights\\ \hspace{20px} w_i={\large\frac{\frac{\pi}{2}cosh(t_i)}{cosh^2(\frac{\pi}{2}\sinh(t_i))}}h,\hspace{10px}h={\large\frac{2t_a}{n-1}}\\ \) |
| \(\normalsize Tanh-Sinh\ integration\\\hspace{5px} (1)\hspace{1px} {\large\int_{\small -1}^{\small 1}}f(x)dx={\large\int_{\small -\infty}^{\small \infty}}f(x(t))x'(t)dt\\ \hspace{90px}\simeq{\large\int_{-t_a}^{t_a}}f(x(t))x'(t)dt\\ \hspace{140px}x(t)=\tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{140px}x'(t)={\large\frac{\frac{\pi}{2}\cosh(t)}{\cosh^2(\frac{\pi}{2}\sinh(t))}}\\ (2)\ Trapezoid\\ \hspace{20px} {\large\int_{\small -1}^{\small 1}}f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ \hspace{10px}nodes\\ \hspace{20px} x_i=\tanh(\frac{\pi}{2}\sinh(t_i)),\hspace{10px}t_i=-t_a+(i-1)h\\ \hspace{10px}weights\\ \hspace{20px} w_i={\large\frac{\frac{\pi}{2}\cosh(t_i)}{\cosh^2(\frac{\pi}{2}\sinh(t_i))}}h,\hspace{10px}h={\large\frac{2t_a}{n-1}}\\ \) |
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