lectures:台形公式の誤差
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| 両方とも前のリビジョン前のリビジョン次のリビジョン | 前のリビジョン次のリビジョン両方とも次のリビジョン | ||
| lectures:台形公式の誤差 [2022/11/21 12:36] – [2] kimi | lectures:台形公式の誤差 [2022/11/21 13:25] – [3] kimi | ||
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| 行 87: | 行 87: | ||
| +\cdots\\ | +\cdots\\ | ||
| T_{4}& | T_{4}& | ||
| - | +\frac{h^{4}}{2(2!)}I_{6}-\frac{h^{2}}{3!}T_{6} | + | +\frac{h^{2}}{2(2!)}I_{6}-\frac{h^{2}}{3!}T_{6} |
| + | +\cdots\\ | ||
| + | T_{6}& | ||
| +\cdots | +\cdots | ||
| \end{align} | \end{align} | ||
| 行 112: | 行 114: | ||
| -\cdots\\ | -\cdots\\ | ||
| &=T_{0} | &=T_{0} | ||
| - | -\frac{h^{2}}{2(2!)}I_{2} | + | -\frac{h^{2}}{12}I_{2} |
| - | \frac{h^{2}}{3!}I_{2} | + | +\frac{h^{4}}{48}I_{4} |
| - | +\frac{h^{2}}{3!}\frac{h^{2}}{2(2!)}I_{4}-\frac{h^{2}}{3!}\frac{h^{2}}{3!}T_{4} | + | -\frac{7h^{4}}{360}\left(I_{4} |
| - | +\frac{h^{2}}{3!}\frac{h^{4}}{2(4!)}I_{6}-\frac{h^{2}}{3!}\frac{h^{4}}{5!}T_{6} | + | +\frac{h^{2}}{2(2!)}I_{6}-\frac{h^{2}}{3!}T_{6} |
| - | \\ | + | +\cdots\right)\\ |
| - | & | + | &+\frac{h^{6}}{360}I_{6} |
| - | -\frac{h^{6}}{2(6!)}I_{6}+\frac{h^{6}}{7!}T_{6} | + | -\frac{6h^{6}}{7!}T_{6} |
| - | -\cdots | + | -\cdots\\ |
| + | & | ||
| + | -\frac{h^{2}}{12}I_{2} | ||
| + | +\frac{h^{4}}{720}I_{4} | ||
| + | -\frac{h^{6}}{480}I_{6}+\frac{31h^{6}}{15120}T_{6} | ||
| + | +\cdots\\ | ||
| + | &=T_{0} | ||
| + | -\frac{h^{2}}{12}I_{2} | ||
| + | +\frac{h^{4}}{720}I_{4} | ||
| + | -\frac{h^{6}}{480}I_{6}+\frac{31h^{6}}{15120}\left(I_{6}+\cdots\right) | ||
| + | +\cdots\\ | ||
| + | & | ||
| + | -\frac{h^{2}}{12}I_{2} | ||
| + | +\frac{h^{4}}{720}I_{4} | ||
| + | -\frac{h^{6}}{30240}I_{6}+o(h^8) | ||
| \end{align} | \end{align} | ||
| 行 128: | 行 143: | ||
| $$ | $$ | ||
| - | ==== 3 ==== | ||
| - | \begin{align} | ||
| - | T_h(f)& | ||
| - | I(f^{(n)})& | ||
| - | \end{align} | ||
| - | |||
| - | $$ | ||
| - | I(f)=T_h(f) | ||
| - | -\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) | ||
| - | +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_h(f^{(2m)})\\ | ||
| - | $$ | ||
| - | |||
| - | \begin{align} | ||
| - | I(f)=T_h(f) | ||
| - | -\frac{h^{2}}{2!}\frac{1}{2}I(f^{(2)})+\frac{h^{2}}{3!}T_h(f^{(2)}) | ||
| - | -\frac{h^{4}}{4!}\frac{1}{2}I(f^{(4)})+\frac{h^{4}}{5!}T_h(f^{(4)}) | ||
| - | -\frac{h^{6}}{6!}\frac{1}{2}I(f^{(6)})+\frac{h^{4}}{7!}T_h(f^{(6)})\\ | ||
| - | +\cdots | ||
| - | -\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) | ||
| - | +\frac{h^{2m}}{(2m+1)!}T_h(f^{(2m)})+\cdots\\ | ||
| - | 0=\frac{h^{2}}{3!}I(f^{(2)})-\frac{h^{2}}{3!}T_h(f^{(2)}) | ||
| - | +\frac{h^{2}}{3!}\frac{h^{2}}{2!}\frac{1}{2}I(f^{(4)})-\frac{h^{2}}{3!}\frac{h^{2}}{3!}T_h(f^{(4)}) | ||
| - | +\frac{h^{2}}{3!}\frac{h^{4}}{4!}\frac{1}{2}I(f^{(6)})-\frac{h^{2}}{3!}\frac{h^{4}}{5!}T_h(f^{(6)})+\cdots& | ||
| - | 0=\frac{h^{2}}{3!}I(f^{(2)})-\frac{h^{2}}{3!}T_h(f^{(2)}) | ||
| - | +\frac{h^{4}}{4!}I(f^{(4)})-\frac{h^{4}}{3!^2}T_h(f^{(4)}) | ||
| - | +\frac{h^{6}}{3!4!2}I(f^{(6)})-\frac{h^{6}}{3!5!}T_h(f^{(6)})+\cdots& | ||
| - | |||
| - | \end{align} | ||
| - | \begin{align} | ||
| - | I(f)=T_h(f) | ||
| - | -\frac{h^{2}}{2!}\frac{1}{6}I(f^{(2)}) | ||
| - | +\frac{h^{4}}{4!}\frac{1}{2}I(f^{(4)}) | ||
| - | +\frac{h^{4}}{3!}(\frac{1}{20}-\frac{1}{6})T_h(f^{(4)}) | ||
| - | +\frac{h^{6}}{3!4!2}I(f^{(6)})-\frac{h^{6}}{6!}\frac{1}{2}I(f^{(6)}) | ||
| - | +\frac{h^{4}}{7!}T_h(f^{(6)})-\frac{h^{6}}{3!5!}T_h(f^{(6)})+\cdots\\ | ||
| - | +\cdots | ||
| - | -\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) | ||
| - | +\frac{h^{2m}}{(2m+1)!}T_h(f^{(2m)})+\cdots\\ | ||
| - | \end{align} | ||
lectures/台形公式の誤差.txt · 最終更新: 2022/11/21 14:37 by kimi