Simpson’s Rule
Calculate the integral over the finite interval (a, b) using Simpson's rule.
The number of subdivisions is doubled at each step: 2, 4, 8, 16, ..., N. The integrand f(x) is assumed to be analytic and nonperiodic.
| \(\normalsize\ Simpson:\\ \hspace{20px}S={\large\int_a^{b}}f(x)dx=\hspace{5px}{\large\frac{h}{3}}\{f(a)+2{\large \displaystyle \sum_{j=1}^{\frac{n}{2}-1}}f(a+2jh)+4{\large \displaystyle \sum_{j=1}^{\frac{n}{2}}}f(a+(2j-1)h)+f(b)\}\\ \hspace{60px}h={\large\frac{b-a}{n}}\\ \) |
| \(\normalsize \hspace{10px}S={\large\int_a^{b}}f(x)dx\\ (1)\ Simpson\\ \hspace{20px}S_{\normalsize s}={\large\frac{S_{\normalsize t}+2S_m}{3}}\\ \hspace{45px}={\large\frac{h}{3}}\{f(a)+2{\large \displaystyle \sum_{j=1}^{\frac{n}{2}-1}}f(a+2jh)+4{\large \displaystyle \sum_{j=1}^{\frac{n}{2}}}f(a+(2j-1)h)+f(b)\},\hspace{10px} h={\large\frac{b-a}{n}}\\ \) |
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