Hans Georg Schaathun
Oktober 2016
$$S_{2n} = \frac{h}{3}\cdot\bigg[ \sum_{i=0,n} y_i +4\sum_{\mathrm{odd}\;i}y_i +2\mathop{\sum_{\mathrm{even}\;i}}\limits_{i\neq0,n}y_i \bigg]$$
$$T_n = 2h\big(\frac{y_0}2+y_2+y_4+\ldots+y_{n-1}+ \frac{y_n}2\big)$$
$$M_n = 2h(y_1+y_3+\ldots+y_{n-1})$$
$$S_{2n} = \frac{T_n+2M_n}{3}$$
$$S_{2n} = \frac{2T_{2n}+M_n}{3}$$
$$S_{2n} = \frac{4T_{2n}-M_n}{3}$$
$$I = \int_a^b f(x)dx$$
| $$S_{2n} = \frac{h}{3}\cdot\bigg[ \sum_{i=0,n} y_i +4\sum_{\mathrm{odd}\;i}y_i +2\mathop{\sum_{\mathrm{even}\;i}}\limits_{i\neq0,n}y_i \bigg]$$ | $$ \begin{align} \begin{split} \bigg|I-S_{2n}\bigg| & \le \frac{K(b-a)}{180}h^4 \\& = \frac{K(b-a)^5}{180n^4} \end{split} \end{align} $$ |
| $$T_{2n} = h\big(\frac{y_0}2+y_1+y_2+\ldots+y_{n-1}+ \frac{y_{2n}}2\big)$$ | $$\bigg|I-T_{2n}\bigg| \le \frac K{48} \cdot\frac{(b-a)^3}{n^2}$$ |
| $$M_{2n} = 2h(y'_1+y'_2+\ldots+y'_{2n})$$ | $$\bigg|I-M_{2n}\bigg| \le \frac K{96} \cdot\frac{(b-a)^3}{n^2}$$ |