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\begin{document}
\section{不等間隔格子の二次精度差分}
本ドキュメントでは, NumRu::Derivative で定義される threepoints\_O2nd\_deriv で用いる
不等間隔格子の二次精度差分についてまとめる. この差分は極端に不等間隔ではないデータに
対して二次精度の差分を与えるものである.

今, 関数$f(x)$を, 数列 $x_n (x_0, x_1, ..., x_{i}, ..., x_{n})$ 上に離散化する.
%
\begin{align}
  f_i &\equiv f(x_i)\\
  t   &\equiv (x_{i+1} - x_{i})\\
  s   &\equiv (x_{i}   - x_{i-1})
\end{align}
%
ここで, $s$と$t$はほぼ同じオーダーの値である場合を想定して議論を進める.

ここで, $f(x)$を各格子点近傍にてテイラー展開する.
%
\begin{align}
  f(x_{i+1}) - f(x_{i}) &= tf'(x_i) + \frac{t^2}{2}f''(x_i) + O(t^3)\\
  f(x_{i-1}) - f(x_{i}) &= -sf'(x_i) + \frac{s^2}{2}f''(x_i) + O(s^3)
\end{align}
%
ここで, $f'(x_i), f''(x_i)$はそれぞれ$x_i$における$f$の$x$に関する一階および二階の微分項, $O(t^3)$は$t^3$のオーダーの値を表す. 両式から$f''$の項を消去するために, $s^2\times$(1.4) - $t^2\times$(1.5) を計算すると,
%
\begin{align}
  s^2f_{i+1} + (t^2 -s^2)f_i - t^2f_{i-1} &= (s^2 + st^2)f'(x_i) + s^2O(t^3) + t^2O(s^3)
\end{align}
%
となる. 上式を変形して
%
\begin{align}
  \frac{s^2f_{i+1} + (t^2 -s^2)f_i - t^2f_{i-1}}{st(s + t)} 
         &= f'(x_i) + \frac{O(s^2t^3) + O(t^2s^3)}{st(s + t)}\\
         &= O(t^2).
\end{align}
%
これより, 2次精度差分の公式は
\begin{align}
  f'(x_i) &= \frac{s^2f_{i+1} + (t^2 -s^2)f_i - t^2f_{i-1}}{st(s + t)} 
\end{align}
%
と書くことができる.

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