lectures:台形公式の誤差
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lectures:台形公式の誤差 [2022/11/19 09:54] – [2] kimi | lectures:台形公式の誤差 [2022/11/21 12:37] – [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 69: | 行 69: | ||
I_{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}}{2(2m)!}I_{2(m+\ell)} | ||
- | +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2(m+\ell)} | + | +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2(m+\ell)}\\ |
+ | T_{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)} | ||
\end{align} | \end{align} | ||
- | ==== 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})\\ | + | I_0&=T_{0} |
- | &=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}}{2(2!)}I_{2}+\frac{h^{2}}{3!}T_{2} |
- | &=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}}{2(4!)}I_{4}+\frac{h^{4}}{5!}T_{4} |
+ | -\frac{h^{6}}{2(6!)}I_{6}+\frac{h^{6}}{7!}T_{6} | ||
+ | -\cdots\\ | ||
+ | T_{2}&=I_{2} | ||
+ | +\frac{h^{2}}{2(2!)}I_{4}-\frac{h^{2}}{3!}T_{4} | ||
+ | +\frac{h^{4}}{2(4!)}I_{6}-\frac{h^{4}}{5!}T_{6} | ||
+ | +\cdots\\ | ||
+ | T_{4}&=I_{4} | ||
+ | +\frac{h^{4}}{2(2!)}I_{6}-\frac{h^{2}}{3!}T_{6} | ||
+ | +\cdots | ||
\end{align} | \end{align} | ||
+ | $\left(\right)$ | ||
\begin{align} | \begin{align} | ||
- | I(f)&=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)})\\ | + | I_0&=T_{0} |
- | I(f)&=R_h(f)+h\left(f(b)-f(a)\right)+\sum_{n=1}^{\infty}\frac{(-h)^n}{(n+1)!}\left\{R_h(f^{(n)})+h\left(f^{(n)}(b)-f^{(n)}(a)\right)\right\}\\ | + | -\frac{h^{2}}{2(2!)}I_{2} |
- | \end{align} | + | +\frac{h^{2}}{3!}\left( |
+ | I_{2} | ||
+ | +\frac{h^{2}}{2(2!)}I_{4}-\frac{h^{2}}{3!}T_{4} | ||
+ | +\frac{h^{4}}{2(4!)}I_{6}-\frac{h^{4}}{5!}T_{6} | ||
+ | +\cdots\right)\\ | ||
+ | &-\frac{h^{4}}{2(4!)}I_{4}+\frac{h^{4}}{5!}T_{4} | ||
+ | -\frac{h^{6}}{2(6!)}I_{6}+\frac{h^{6}}{7!}T_{6} | ||
+ | -\cdots\\ | ||
+ | & | ||
+ | -\frac{h^{2}}{12}I_{2} | ||
+ | +\frac{h^{4}}{48}I_{4} | ||
+ | -\frac{7h^{4}}{360}T_{4} | ||
+ | +\frac{h^{6}}{360}I_{6} | ||
+ | -\frac{6h^{6}}{7!}T_{6} | ||
+ | -\cdots\\ | ||
+ | & | ||
+ | -\frac{h^{2}}{2(2!)}I_{2} | ||
+ | \frac{h^{2}}{3!}I_{2} | ||
+ | +\frac{h^{2}}{3!}\frac{h^{2}}{2(2!)}I_{4}-\frac{h^{2}}{3!}\frac{h^{2}}{3!}T_{4} | ||
+ | +\frac{h^{2}}{3!}\frac{h^{4}}{2(4!)}I_{6}-\frac{h^{2}}{3!}\frac{h^{4}}{5!}T_{6} | ||
+ | \\ | ||
+ | & | ||
+ | -\frac{h^{6}}{2(6!)}I_{6}+\frac{h^{6}}{7!}T_{6} | ||
+ | -\cdots | ||
- | |||
- | \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} | \end{align} | ||
$$ | $$ | ||
- | I(f)=T_h(f) | + | T_{2\ell}=I_{2\ell} |
- | -\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m)!}\frac{1}{2}I(f^{(2m)}) | + | +\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)} |
- | +\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