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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 12:36] – [2] kimi
@@ 行 1: / 行 1: @@
 ====== 台形公式の誤差 ======
 ===== 1 =====
-$$x_k = a + hk$$
+区間$[a,\,b]$を$N$分割した標本点を$x_k = a + hk$とおく、ただし、$h$は分割幅で
+$h=\frac{b-a}{N}$。ここで、$x_0=a$、$x_N=b$であることに注意。
-$$h=\frac{b-a}{N}$$
+また、
+\begin{align}
+I_0&\equiv\int_a^b f(x)\mathrm{d}x=\sum_{k=0}^{N-1}\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x\\
+I_n&\equiv\int_a^b f^{(n)}(x)\mathrm{d}x=f^{(n-1)}(b)-f^{(n-1)}(a)
+\end{align}
+と定義する。
-$x_0=a$、$x_N=b$
+$f(x)$を$x=x_k$の周りでテーラー展開したものを$[{x_{k}},\,{x_{k+1}}]$で積分し、和をとると、
+\begin{align}
+f(x)&=f(x_k)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)(x-x_k)^n\\
+\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x&=f(x_k)\left({x_{k}}-{x_{k+1}}\right)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)\int_{x_{k}}^{x_{k+1}}(x-x_k)^n\mathrm{d}x\\
+&=hf(x_k)+\sum_{n=1}^{\infty}\frac{h^{n+1}}{(n+1)!}f^{(n)}(x_k)\\
+I_0&=h\sum_{k=0}^{N-1}f(x_k)+\sum_{n=1}^{\infty}\frac{h^{n+1}}{(n+1)!}\sum_{k=0}^{N-1}f^{(n)}(x_k)
+\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$[{x_{k}},\,{x_{k+1}}]$で積分し、和をとると、
+\begin{align}
+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}
+===== 2 =====
+和をとる変数をそろえると、
-$$
+\begin{align}
-f(x)=f(x_k)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)(x-x_k)^n
+I_0&=h\sum_{k=0}^{N-1}f(x_k)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}\left(h\sum_{k=0}^{N-1}f^{(n)}(x_k)\right)\\
-$$
+I_0&=h\sum_{k=1}^{N}f(x_{k})+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}\left((-1)^{n}h\sum_{k=1}^{N}f^{(n)}(x_{k})\right)
+\end{align}
+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)\\
+&+\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}
+$n=2m-1$のとき、
 \begin{align}
-\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x&=f(x_k)\left({x_{k}}-{x_{k+1}}\right)+\sum_{n=1}^{\infty}\frac{1}{n!}f^{(n)}(x_k)\int_{x_{k}}^{x_{k+1}}(x-x_k)^n\mathrm{d}x\\
+\sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k)
-&=f(x_k)h+\sum_{n=1}^{\infty}\frac{1}{(n+1)!}f^{(n)}(x_k)h^{n+1}
+&=\sum_{k=0}^{N-1}f^{(2m-1)}(x_k)-\sum_{k=1}^{N}f^{(2m-1)}(x_k)\\
+&=f^{(2m-1)}(x_0)-f^{(2m-1)}(x_N)
+=-\left(f^{(2m-1)}(b)-f^{(2m-1)}(a)\right)\\
+&=-\int_a^b f^{(2m)}(x)\mathrm{d}x=-I_{2m}
 \end{align}
+$n=2m$のとき、
 \begin{align}
-I(f)\equiv \int_a^b f(x)\mathrm{d}x&=\sum_{k=0}^{N-1}\int_{x_{k}}^{x_{k+1}}f(x)\mathrm{d}x\\
+\frac{h}{2}\left(\sum_{k=0}^{N-1}f^{(n)}(x_k)+(-1)^{n}\sum_{k=1}^{N}f^{(n)}(x_k)\right)
-&=h\sum_{k=0}^{N-1}f(x_k)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}h\sum_{k=0}^{N-1}f^{(n)}(x_k)
+&=\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}
 $$
-R_h(f)\equiv h\sum_{k=0}^{N-1}f(x_k)
+I_0=T_{0}
+-\sum_{m=1}^{\infty}\frac{h^{2m-1}}{(2m)!}\frac{h}{2}I_{2m}
++\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2m}
 $$
+===== 3 =====
+\begin{align}
+I_0&=T_{0}
+-\sum_{m=1}^{\infty}\frac{h^{2m}}{2(2m)!}I_{2m}
++\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}
+\begin{align}
+I_0&=T_{0}
+-\frac{h^{2}}{2(2!)}I_{2}+\frac{h^{2}}{3!}T_{2}
+-\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}
+$\left(\right)$
+\begin{align}
+I_0&=T_{0}
+-\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^{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_{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\\
+&=T_{0}
+-\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^{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
+\end{align}
 $$
-I(f)=R_h(f)+\sum_{n=1}^{\infty}\frac{h^n}{(n+1)!}R_h(f^{(n)})
+T_{2\ell}=I_{2\ell}
++\sum_{m=1}^{\infty}\frac{h^{2m}}{2(2m)!}I_{2(m+\ell)}-\sum_{m=1}^{\infty}\frac{h^{2m}}{(2m+1)!}T_{2(m+\ell)}
 $$
 ==== 2 ====
@@ 行 90: / 行 185: @@
 -\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}}{3!}I(f^{(2)})-\frac{h^{2}}{3!}T_h(f^{(2)})
--\frac{h^{2}}{2!}\frac{1}{2}I(f^{(4)})+\frac{h^{2}}{3!}T_h(f^{(4)})
++\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^{4}}{4!}\frac{1}{2}I(f^{(6)})+\frac{h^{4}}{5!}T_h(f^{(6)})+\cdots&\\
++\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&\\
+=\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}