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\begin{document}
\section{�s���Ԋu�i�q�̓񎟐��x����}
�{�h�L�������g�ł�, NumRu::Derivative �Œ�`����� threepoints\_O2nd\_deriv �ŗp����
�s���Ԋu�i�q�̓񎟐��x�����ɂ��Ă܂Ƃ߂�. ���̍����͋ɒ[�ɕs���Ԋu�ł͂Ȃ��f�[�^��
�΂��ē񎟐��x�̍�����^������̂ł���.

��, �֐�$f(x)$��, ���� $x_n (x_0, x_1, ..., x_{i}, ..., x_{n})$ ��ɗ��U������.
%
\begin{align}
  f_i &\equiv f(x_i)\\
  t   &\equiv f(x_{i+1} - x_{i})\\
  s   &\equiv f(x_{i}   - x_{i-1})
\end{align}
%
������, $s$��$t$�͂قړ����I�[�_�[�̒l�ł���ꍇ��z�肵�ċc�_��i�߂�.

������, $f(x)$���e�i�q�_�ߖT�ɂăe�C���[�W�J����.
%
\begin{align}
  f(x_{i+1} - x_{i}) &= tf'(x_i) + \frac{t^2}{2}(x_i) + O(t^3)\\
  f(x_{i-1} - x_{i}) &= -sf'(x_i) + \frac{s^2}{2}(x_i) + O(s^3)
\end{align}
%
������, $f'(x_i), F''(x_i)$�͂��ꂼ��$x_i$�ɂ�����$f$��$x$�Ɋւ����K����ѓ�K�̔�����, $O(t^3)$��$t^3$�̃I�[�_�[�̒l��\��. ��������$f''$�̍����������邽�߂�, $s^2\times$(1.4) - $t^2\times$(1.5) ���v�Z�����,
%
\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}
%
�ƂȂ�. �㎮��ό`����
%
\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}
%
������, 2�����x�����̌�����
\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}
%
�Ə������Ƃ��ł���.

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%%%%%%%%            ���� (���x���t��)           %%%%%%%%
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%\begin{eqnarray}
% \Deqlab{1.1} % ���ȏ��ł̎��ԍ�������.
%  \DP{\rho}{t} + \Ddiv (\rho \Dvect{V}) = 0.
%\end{eqnarray}
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%%%%%%%%  ���� (���ԍ���Ɨ����ď��������ꍇ)   %%%%%%%%
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%$$
% \DP{p}{z} = \rho g.
% \eqno \textrm{(1.11)}
%$$
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%%%%%%%%         �Q�l���� (�{���ɏ����ꍇ)      %%%%%%%%
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%{\bfseries �Q�l����}
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% \item	���Җ�, 2000:
%	���Ж�, (��, ��).
%	�o�Ŏ�, 319pp.
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