lectures:台形公式の誤差
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lectures:台形公式の誤差 [2022/11/21 12:37] – [3] kimi | lectures:台形公式の誤差 [2022/11/21 14:37] (現在) – [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} | ||
- | $\left(\right)$ | ||
\begin{align} | \begin{align} | ||
行 112: | 行 113: | ||
-\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} | ||
+ | \begin{align} | ||
+ | h\left(\frac{f(a)}{2}+\sum_{k=1}^{N-1}f(x_{k})+\frac{f(b)}{2}\right)& | ||
+ | +\frac{h^{2}}{12}\left(f' | ||
+ | & | ||
+ | +\frac{h^{6}}{30240}\left(f^{(5)}(b)-f^{(5)}(a)\right)+o(h^8) | ||
\end{align} | \end{align} | ||
- | $$ | ||
- | T_{2\ell}=I_{2\ell} | ||
- | +\sum_{m=1}^{\infty}\frac{h^{2m}}{2(2m)!}I_{2(m+\ell)}-\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2(m+\ell)} | ||
- | $$ | ||
lectures/台形公式の誤差.1669001820.txt.gz · 最終更新: 2022/11/21 12:37 by kimi