lectures:台形公式の誤差
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lectures:台形公式の誤差 [2022/11/18 23:15] – [3] kimi | lectures:台形公式の誤差 [2022/11/21 13:25] – [3] kimi | ||
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====== 台形公式の誤差 ====== | ====== 台形公式の誤差 ====== | ||
===== 1 ===== | ===== 1 ===== | ||
- | $$x_k = a + hk$$ | + | 区間$[a, |
- | + | $h=\frac{b-a}{N}$。ここで、$x_0=a$、$x_N=b$であることに注意。 | |
- | $$h=\frac{b-a}{N}$$ | + | |
- | + | ||
- | $x_0=a$、$x_N=b$ | + | |
- | + | ||
- | + | ||
- | + | ||
- | $$ | + | |
- | f(x)=f(x_k)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)(x-x_k)^n | + | |
- | $$ | + | |
+ | また、 | ||
+ | \begin{align} | ||
+ | I_0& | ||
+ | I_n& | ||
+ | \end{align} | ||
+ | と定義する。 | ||
+ | $f(x)$を$x=x_k$の周りでテーラー展開したものを$[{x_{k}}, | ||
\begin{align} | \begin{align} | ||
+ | f(x)& | ||
\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x& | \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x& | ||
- | &=f(x_k)h+\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_k)h^{n+1} | + | &=hf(x_k)+\sum_{n=1}^{\infty}\frac{h^{n+1}}{(n+1)!}f^{(n)}(x_k)\\ |
+ | I_0& | ||
\end{align} | \end{align} | ||
+ | 同様に$f(x)$を$x=x_{k+1}$の周りでテーラー展開したものを$[{x_{k}}, | ||
\begin{align} | \begin{align} | ||
- | I(f)\equiv \int_a^b f(x)\mathrm{d}x& | + | I_0&=h\sum_{k=0}^{N-1}f(x_{k+1})-\sum_{n=1}^{\infty}\frac{(-h)^{n+1}}{(n+1)!}\sum_{k=0}^{N-1}f^{(n)}(x_{k+1}) |
- | & | + | |
\end{align} | \end{align} | ||
+ | ===== 2 ===== | ||
- | $$ | + | 和をとる変数をそろえると、 |
- | R_h(f)\equiv h\sum_{k=0}^{N-1}f(x_k) | + | |
- | $$ | + | |
- | $$ | + | \begin{align} |
- | I(f)=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)}) | + | I_0& |
- | $$ | + | I_0& |
+ | \end{align} | ||
- | ==== 2 ==== | + | 二式の平均をとると、 |
- | + | \begin{align} | |
- | $$ | + | I_0&=\frac{h}{2}\left(\sum_{k=0}^{N-1}f(x_k)+\sum_{k=1}^{N}f(x_{k})\right)\\ |
- | f(x)=f(x_{k+1})+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_{k+1})(x-x_{k+1})^n | + | &+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}\frac{h}{2}\left(\sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k)\right) |
- | $$ | + | \end{align} |
- | + | ||
- | $$ | + | |
- | \int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x=f(x_{k+1})h-\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_{k+1})(-h)^{n+1} | + | |
- | $$ | + | |
+ | $n=2m-1$のとき、 | ||
\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})\\ | + | \sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k) |
- | &=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})\\ | + | & |
- | &=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\} | + | &=f^{(2m-1)}(x_0)-f^{(2m-1)}(x_N) |
+ | =-\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\\ | ||
+ | & | ||
\end{align} | \end{align} | ||
- | + | $n=2m$のとき、 | |
\begin{align} | \begin{align} | ||
- | I(f)&=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)})\\ | + | \frac{h}{2}\left(\sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k)\right) |
- | 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}\left(\sum_{k=0}^{N-1}f^{(2m)}(x_k)+\sum_{k=1}^{N}f^{(2m)}(x_k)\right)\\ |
+ | &=\frac{h}{2}\left(f^{(2m)}(x_0)+2\sum_{k=1}^{N-1}f^{(2m)}(x_k)+f^{(2m)}(x_N)\right)\\ | ||
+ | &=h\left(\frac{f^{(2m)}(a)}{2}+\sum_{k=1}^{N-1}f^{(2m)}(x_k)+\frac{f^{(2m)}(b)}{2}\right)\\ | ||
+ | &\equiv T_{2m} | ||
\end{align} | \end{align} | ||
+ | $$ | ||
+ | I_0=T_{0} | ||
+ | -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\frac{h}{2}I_{2m} | ||
+ | +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2m} | ||
+ | $$ | ||
+ | ===== 3 ===== | ||
\begin{align} | \begin{align} | ||
- | I(f)&=R_h(f)+\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}R_h(f^{(2m-1)})+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}R_h(f^{(2m)})\\ | + | I_0&=T_{0} |
- | I(f)&=R_h(f)+h\left(f(b)-f(a)\right) | + | -\sum_{m=1}^{\infty}\frac{h^{2m}}{2(2m)!}I_{2m} |
- | -\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\}\\ | + | +\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2m}\\ |
- | & | + | I_{2\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)}\\ | ||
+ | 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)} | ||
\end{align} | \end{align} | ||
+ | |||
\begin{align} | \begin{align} | ||
- | I(f)&=R_h(f)+\frac{h}{2}\left(f(b)-f(a)\right) | + | I_0&=T_{0} |
- | -\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\left\{\frac{h}{2}\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\right\}\\ | + | -\frac{h^{2}}{2(2!)}I_{2}+\frac{h^{2}}{3!}T_{2} |
- | &+\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}\left\{R_h(f^{(2m)})+\frac{h}{2}\left(f^{(2m)}(b)-f^{(2m)}(a)\right)\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^{2}}{2(2!)}I_{6}-\frac{h^{2}}{3!}T_{6} | ||
+ | +\cdots\\ | ||
+ | T_{6}& | ||
+ | +\cdots | ||
\end{align} | \end{align} | ||
+ | $\left(\right)$ | ||
- | ==== 3 ==== | ||
\begin{align} | \begin{align} | ||
- | T_h(f)&\equiv R_h(f)+\frac{h}{2}\left(f(b)-f(a)\right)\\ | + | I_0&=T_{0} |
- | I(f^{(n)})&=f^{(n-1)}(b)-f^{(n-1)}(a) | + | -\frac{h^{2}}{2(2!)}I_{2} |
+ | +\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^{6}}{2(6!)}I_{6}+\frac{h^{6}}{7!}T_{6} | ||
+ | -\cdots\\ | ||
+ | &=T_{0} | ||
+ | -\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}}{12}I_{2} | ||
+ | +\frac{h^{4}}{48}I_{4} | ||
+ | -\frac{7h^{4}}{360}\left(I_{4} | ||
+ | +\frac{h^{2}}{2(2!)}I_{6}-\frac{h^{2}}{3!}T_{6} | ||
+ | +\cdots\right)\\ | ||
+ | & | ||
+ | -\frac{6h^{6}}{7!}T_{6} | ||
+ | -\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\\ | ||
+ | & | ||
+ | -\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} | ||
$$ | $$ | ||
- | 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\\ | ||
- | I(f^{(2)})=T_h(f^{(2)}) | ||
- | -\frac{h^{2}}{2!}\frac{1}{2}I(f^{(4)})+\frac{h^{2}}{3!}T_h(f^{(4)}) | ||
- | -\frac{h^{4}}{4!}\frac{1}{2}I(f^{(6)})+\frac{h^{4}}{5!}T_h(f^{(6)})+\cdots& | ||
- | |||
- | \end{align} | ||
lectures/台形公式の誤差.txt · 最終更新: 2022/11/21 14:37 by kimi