lectures:台形公式の誤差
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lectures:台形公式の誤差 [2022/11/19 17:47] – [3] kimi | lectures:台形公式の誤差 [2022/11/21 14:37] (現在) – [3] kimi | ||
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行 32: | 行 32: | ||
\end{align} | \end{align} | ||
- | 二式の平均をとると、$\frac{1}{2}\left(\right)$ | + | 二式の平均をとると、 |
\begin{align} | \begin{align} | ||
I_0& | I_0& | ||
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 108: | 行 109: | ||
+\frac{h^{4}}{48}I_{4} | +\frac{h^{4}}{48}I_{4} | ||
-\frac{7h^{4}}{360}T_{4} | -\frac{7h^{4}}{360}T_{4} | ||
- | \\ | + | +\frac{h^{6}}{360}I_{6} |
- | & | + | -\frac{6h^{6}}{7!}T_{6} |
- | +\frac{h^{6}}{3!}\frac{1}{2(4!)}I_{6} | + | |
- | -\frac{h^{6}}{2(6!)}I_{6} | + | |
- | -\frac{h^{6}}{3!}\frac{1}{5!}T_{6} | + | |
- | +\frac{h^{6}}{7!}T_{6} | + | |
-\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} | ||
- | |||
- | $$ | ||
- | T_{2\ell}=I_{2\ell} | ||
- | +\sum_{m=1}^{\infty}\frac{h^{2m}}{2(2m)!}I_{2(m+\ell)}-\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2(m+\ell)} | ||
- | $$ | ||
- | ==== 2 ==== | ||
- | |||
- | $$ | ||
- | f(x)=f(x_{k+1})+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_{k+1})(x-x_{k+1})^n | ||
- | $$ | ||
- | |||
- | $$ | ||
- | \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x=f(x_{k+1})h-\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_{k+1})(-h)^{n+1} | ||
- | $$ | ||
\begin{align} | \begin{align} | ||
- | I(f)&=h\sum_{k=0}^{N-1}f(x_{k+1})+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}h\sum_{k=0}^{N-1}f^{(n)}(x_{k+1})\\ | + | h\left(\frac{f(a)}{2}+\sum_{k=1}^{N-1}f(x_{k})+\frac{f(b)}{2}\right)&=\int_a^b f(x)\mathrm{d}x |
- | &=h\sum_{k=1}^{N}f(x_{k})+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}h\sum_{k=1}^{N}f^{(n)}(x_{k})\\ | + | +\frac{h^{2}}{12}\left(f'(b)-f'(a)\right)\\ |
- | &=h\sum_{k=0}^{N-1}f(x_{k})-hf(x_0)+hf(x_N)+\sum_{n=1}^{\infty}\frac{(-h)^{n}}{(n+1)!}\left\{h\sum_{k=0}^{N-1}f^{(n)}(x_{k})-hf^{(n)}(x_0)+hf^{(n)}(x_N)\right\} | + | &-\frac{h^{4}}{720}\left(f^{(3)}(b)-f^{(3)}(a)\right) |
+ | +\frac{h^{6}}{30240}\left(f^{(5)}(b)-f^{(5)}(a)\right)+o(h^8) | ||
\end{align} | \end{align} | ||
- | |||
- | \begin{align} | ||
- | I(f)& | ||
- | I(f)& | ||
- | \end{align} | ||
- | |||
- | |||
- | \begin{align} | ||
- | I(f)& | ||
- | I(f)& | ||
- | -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{R_h(f^{(2m-1)})+h\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ | ||
- | & | ||
- | \end{align} | ||
- | |||
- | \begin{align} | ||
- | I(f)& | ||
- | -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{\frac{h}{2}\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ | ||
- | & | ||
- | \end{align} | ||
- | |||
- | |||
- | ==== 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.1668847648.txt.gz · 最終更新: 2022/11/19 17:47 by kimi