Tanh-Sinh Numerical Integration (a, b)
Calculate the integral over the finite interval (a, b) using the Tanh-Sinh rule (double exponential type) with the specified number of subdivisions.
This method is also suitable for integrating functions with singularities at the endpoints (±∞).
The integrand f(x) is assumed to be analytic except at the endpoints and not to be a periodic function.
The integrand f(x) is assumed to be analytic except at the endpoints and not to be a periodic function.
| \(\normalsize Tanh-Sinh\ integration\\\hspace{5px} (1)\ x\rightarrow \frac{b-a}{2}y+\frac{b+a}{2}\\ \hspace{30px} {\large\int_{\small a}^{\small b}}f(x)dx= {\large\int_{\small -1}^{\small 1}}f(\frac{b-a}{2}y+\frac{b+a}{2})\frac{b-a}{2}dy\\ \hspace{20px} y\rightarrow \tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{30px}\simeq {\large\int_{\small -t_a}^{\small t_a}}f(\frac{b-a}{2}y(t)+\frac{b+a}{2})y'(t)\frac{b-a}{2}dt\\ \hspace{140px}y(t)=\tanh(\frac{\pi}{2}\sinh(t))\\ \hspace{140px}y'(t)={\large\frac{\frac{\pi}{2}\cosh(t)}{\cosh^2(\frac{\pi}{2}\sinh(t))}}\\ (2)\ Trapezoid\\ \hspace{20px}S={\large\displaystyle \sum_{\small i=1}^{\small n}}f( \frac{b-a}{2}y_i+\frac{b+a}{2})w_i\\ \hspace{10px}nodes\\ \hspace{20px} y_i=\tanh(\frac{\pi}{2}\sinh(t_i)),\hspace{10px}t_i=-t_a+(i-1)*h\ \\ \hspace{10px}weights\\ \hspace{20px} w_i={\large\frac{\frac{\pi}{2}\cosh(t_i)}{\cosh^2(\frac{\pi}{2}\sinh(t_i))}}\frac{b-a}{2}h,\hspace{10px}h={\large\frac{2t_a}{n-1}}\\ \) |
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