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\Dauthor{���� �O��}
\Ddate{2010 �N 7 �� 29 ��}
\Dfile{2010-0729-ogihara.tex}
\Dtitle{���l�v�Z�̏펯}

\markright{Newton �@ }

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\begin{document}
\chapter{Newton �@}

����`�������������Ƃ��ɂ͌v�Z���ԂȂǂ̃R�X�g���l����Ɓu�j���[�g���@�v 
\footnote{�j���[�g���E���t�g��(Newton-Raphson)�@�Ƃ�����}��p����̂��ǂ�. 
�j���[�g���@�́u����`�̐��E
\footnote{�f���^�֐���s�A���Ȋ֐��ȊO�̐����̗ǂ��֐��̏ꍇ. �Ⴆ�΂ׂ������Ȃ�. }���ɂ������ȗ̈悾������΂قڐ��`�̐��E�ł���
\footnote{����`�̊֐�$f(x + h)$������̎�$x = a$�̂܂��ł̃e�C���[�W�J���s��. 
\begin{equation}
	f(a + h) = f(a) + f^\prime(a)h + \frac{f^{\prime \prime}(a)h^2}{2!} + �c \nonumber\
\end{equation}
������$h\ll 1$�̂Ƃ���h��2���ȏ�̍��͖����ł���. �����
\begin{equation}
	f(a + h) \sim f(a) + f^\prime(a)h. \nonumber\
\end{equation}
�E�ӂ͐��`�Ȃ̂ŋɂ������ȗ̈悾�Ɛ��`�Ƃ݂Ȃ���. }�v�Ƃ����u��ʌ����v �Ɋ�Â��Ă���̂�
, ���_�I�ɂ͓K�p�\�͈͂���ύL��, ���p�I�ł���. 
\section{����}

�j���[�g���@�̌����͔��������ł���. 
������$f(x) = 0$�̈�̉�$\alpha$��``�߂���" ���鏉���ߎ��l$x^{(0)}$����o�������Ƃ�, 
\begin{equation}
	f(x^{(0)}) = f^{\prime}(x^{(0)})\times (x^{(0)} - x^{(1)})\nonumber
\end{equation}
�Ȃ̂�
\begin{equation}
	\frac{f(x^{(0)})}{f^{\prime}(x^{(0)})} = x^{(0)} - x^{(1)}. \nonumber
\end{equation}
�����, 
\begin{equation}
	x^{(1)} = x^{(0)} - \frac{f(x^{(0)})}{f^{\prime} (x^{(0)})}. \nonumber
\end{equation}
�������ʉ������
\begin{equation}
	x^{(\nu + 1)} = x^{(\nu )} - \frac{f(x^{(\nu )})}{f^\prime(x^{(\nu )})}. 
	\Deqlab{1}
\end{equation}
�����p����, ���X�Ɩ]�ނ炭
\footnote{$x_o$��$\alpha $�ƂƂĂ�����Ă���Ɗ��҂�����$\alpha$�̋ߎ��l�����߂��Ȃ��ꍇ������̂�, $\alpha $���������t���Ă��Ȃ��Ǝg���Ȃ�.}�͉��ǂ��ꂽ�ߎ���$x^{(1)}, x^{(2)}$�c���v�Z����. 
����ɑ��ϐ��̔���`�A��������
\begin{equation}
	f_i(x_1, �c, x_n) = 0 \hspace{4em}(i = 1, �c, n)\nonumber
\end{equation}
�̏ꍇ�ɂ�\Deqref{1}�����R�Ɋg�����邱�Ƃ��ł���. $\alpha _i$($i = 1,�cn$)��$f_i(x_i)(i = 1, �c, n)$�̂Ƃ�����Ƃ���. 
$x_i^{(0)}(i = 1, �c, n)$�����̏����ߎ����Ƃ����Ƃ��X�ɐ��x�̗ǂ��ߎ���$x_i^{(1)}(i = 1, �c, n)$������Ƃ����, 
$f_i(x_i^{(1)})(i = 1, �c, n)$��
\begin{equation}
  f_i(x_1^{(1)}, x_2^{(1)}, �c, x_n^{(1)}) \sim f_i(x_1^{(0)}, x_2^{(0)}, �c, x_n^{(0)}) + \sum^n_{j = 1}\DP{f_i(x_1^{(0)}, x_2^{(0)}, �c, x_n^{(0)})}{x_j}(x_j^{(1)} - x_j^{(0)})\nonumber
\end{equation}
�ƕ\�����. �������ʉ������
\begin{equation}
  f_i(x_i^{(\mu + 1)}) \sim f_i(x_i^{(\mu)}) + \sum^n_{j = 1}\DP{f_i(x_i^{(\mu )})}{x_j}(x_j^{(\mu )} - x_j^{(\mu )}). \tag{6.3'}
  \Deqlab{2}
\end{equation}
������$x_i^{(\mu + 1)}(i = 1, �c, n)$��$\alpha_i(i = 1, �c, n)$�ɏ\���߂��Ƃ����$f_i(x_i^{(\mu + 1)}) \sim f_i(\alpha_i) = 0$
�ƌ��Ȃ���. ���̂Ƃ�\Deqref{2}��
\begin{equation}
0 \sim f_i(x_i^{(\mu)}) + \sum^n_{j = 1}\DP{f_i(x_i^{(\mu )})}{x_j}(x_j^{(\mu )} - x_j^{(\mu )}). \nonumber
\end{equation}
�Ȃ̂�
\begin{equation}
f_i(x_i^{(\mu)}) \sim \sum^n_{j = 1}\DP{f_i(x_i^{(\mu )})}{x_j}(x_j^{(\mu )} - x_j^{(\mu )}).\nonumber
\end{equation}
������$\Delta x_j \equiv x_j^{(\mu )} - x_i^{(\mu + 1)}$�Ƃ����
\begin{equation}
x_i^{(\mu + 1)} = x_j^{(\mu )} - \Delta x_j^{(\mu)}.
\Deqlab{3}
\end{equation}
\begin{equation}
f_i(x_i^{(\mu)}) \sim \sum^n_{j = 1}\DP{f_i(x_i^{(\mu )})}{x_j}\Delta x_j^{(\mu)}.
\Deqlab{4}
\end{equation}
�ƂȂ���\Deqref{1}�𑽕ϐ��Ɋg���������ƂȂ�. \Deqref{4}��$\Delta x^{(\nu)}_1$, �c, $\Delta x^{(\nu)}_n$�𖢒m���Ƃ���
�A��1���������Ƃ��ĉ�����$\Delta x^{(\nu)}_j$����, �����\Deqref{3}�ɓ���邱�Ƃŉ������߂���. 
$f_i(x)$�����f�ϐ�$x$�̕��f���֐��ł����Ă���\Deqref{4}�͌`���I�ɂ��̂܂܎g����
\footnote{���f��$z$��$z \equiv x + iy$($i$�͋����P��)�Ƃ����$f(z) = u + iv$�ƒu����. 
������$u = u(x,y), v = v(x,y)$�Ȃ̂�$f(z) = f_2(u, v)$�ƕ\����2�ϐ��A���������Ə�����.
�����\Deqref{4}��$n = 2$�Ƃ��ĉ����΂悢}. 

\section{������}

��, �ȒP�̂��߂ɘA���łȂ������l����. ��$\alpha $���P���Ǘ���������, $\alpha$�̋ߖT��$f$���f���Ȋ֐��ł���
\begin{equation}
	f^{\prime}(\alpha) \neq 0 \hspace{4em}{\left(�A���������̂Ƃ���, �s��\left[\DP{f_i(x^{(\nu )})}{x^{(\nu )}_j}\right]������\right)}. \nonumber
\end{equation}
�����ďo���l$x^{(0)}$��$\alpha $�Ɂu�\���v �߂����,�܂�$\nu$��$\infty$�܂ł����Ă�����
\begin{equation}
	x^{(\nu)}\rightarrow  \alpha \hspace{2em}{(\nu \rightarrow \infty )}\nonumber
\end{equation}
�ł���, ������$N$�����$\nu$ ($< N$)�ɑ΂���
\begin{equation}
	x^{(\nu  + 1)} = x^{(\nu )} - \frac{f(x^{(\nu )})}{f^{\prime}(x^{(\nu )})}. \tag{6.4'}\Deqlab{4'}
\end{equation}
���̂Ƃ�$\epsilon _\nu  = x^{(\nu )} - \alpha $�Ƃ���. $f(x^{(\nu )})$���e�C���[�W�J����
\begin{equation}
	f(x^{(\nu )}) = f(\alpha ) + f^{\prime}(\alpha )\epsilon _\nu  + \frac{1}{2!}f^{\prime \prime}(\alpha )\epsilon _\nu ^2 + �c. \nonumber
\end{equation}
$f^{\prime}(x^{(\nu )})$���e�C���[�W�J����
\begin{equation}
	f^{\prime}(x^{(\nu )}) = f^{\prime}(\alpha ) + f^{\prime \prime}(\alpha )\epsilon _\nu  + \frac{1}{2!}f^{\prime \prime \prime}(\alpha )\epsilon ^2_\nu  + �c. \nonumber
\end{equation}
�����\Deqref{4'}�̉E�ӑ�2���ɑ������. 
\begin{eqnarray*}
	\frac{f(x^{(\nu )})}{f^{\prime}(x^{(\nu )})}
	&=&	\frac{f(\alpha ) + f^{\prime}(\alpha )\epsilon _\nu  + f^{\prime \prime}(\alpha )\frac{\epsilon ^2}{2}}{f^{\prime}(\alpha ) + f^{\prime \prime}(\alpha )\epsilon _\nu } + O(\epsilon ^3_\nu )\\
	&=& \frac{1}{f^{\prime}(\alpha )}\frac{f(\alpha ) + f^{\prime}(\alpha )\epsilon _\nu  + f^{\prime \prime}(\alpha )\frac{{\epsilon _\nu }^2}{2}}{1 + \frac{f^{\prime \prime}(\alpha )}{f^{\prime}(\alpha )}\epsilon _\nu }  + O(\epsilon ^3_\nu ). 
\end{eqnarray*}
$\displaystyle \left|\frac{f^{\prime \prime}(\alpha )}{f^{\prime}(\alpha )}\epsilon _\nu  \right| \ll 1$���
\footnote{$|\delta| \ll  1$�̂Ƃ�$\frac{1}{1 + \delta}$���}�N���[�����W�J����
\begin{eqnarray*}
  \frac{1}{1 + \delta }
  &=& \frac{1}{1 - 0} + \frac{d}{d\delta}\left(\frac{1}{1 + \delta} \right)|_{\delta = 0} \cdot (\delta)  + \frac{1}{2!}\frac{d^2}{d\delta ^2}\left(\frac{1}{1 + \delta} \right)|_\delta = 0 \cdot (\delta)^2  + �c\\
  &=& 1 + (-\frac{1}{(1 + 0)^2}\cdot \delta ) +  \frac{1}{(1 + 0)^3}\cdot (\delta)^2 + �c\\
  &\sim& 1 - \delta
\end{eqnarray*}
�Ƌߎ��ł��邱�Ƃ�p����}
\begin{eqnarray*}
	\frac{f(x^{(\nu )})}{f^{\prime}(x^{(\nu )})}
	&\approx &	\frac{1}{f^{\prime}(\alpha )}\left( f(\alpha ) + f^{\prime}(\alpha )\epsilon _\nu  + f^{\prime \prime}(\alpha )\frac{\epsilon ^2_\nu }{2}\right)\left( 1 - \frac{f^{\prime \prime}(\alpha )}{f^{\prime}(\alpha )}\epsilon _\nu  \right)�@+ O(\epsilon ^3_\nu )\\
	&=&	\frac{1}{f^{\prime}(\alpha )}\left( f(\alpha ) + f^{\prime}(\alpha )\epsilon _\nu  + f^{\prime \prime}(\alpha )\frac{\epsilon ^2_\nu }{2} - \frac{f(\alpha )f^{\prime \prime}(\alpha )}{f^{\prime}(\alpha )}\epsilon _\nu  - f^{\prime \prime}\epsilon ^2_\nu  + \frac{(f^{\prime \prime}(\alpha ))^2}{f^{\prime}(\alpha )\frac{\epsilon ^3_\nu }{2}} \right) + O(\epsilon ^3_\nu )\\
	&=&	\frac{f(\alpha )}{f^{\prime}(\alpha )} + \frac{f^{\prime}(\alpha )\epsilon _\nu }{f^{\prime}(\alpha )} + \frac{f^{\prime \prime}(\alpha )}{f^{\prime}(\alpha )}\frac{\epsilon ^2_\nu }{2} - \frac{f(\alpha )f^{\prime \prime}(\alpha )}{(f^{\prime}(\alpha ))^2}\epsilon _\nu  - \frac{f^{\prime \prime}(\alpha )}{f^{\prime}(\alpha )}\epsilon ^2_\nu   + O(\epsilon ^3_\nu ). 
\end{eqnarray*}
������$f(\alpha) = 0$�Ȃ̂�
\begin{equation}
	\frac{f(x^k)}{f^{\prime}(x^k)} \approx   - \frac{f^{\prime \prime}(\alpha)}{2f^{\prime}(\alpha)}\epsilon ^2_\nu  + \epsilon _\nu  + O(\epsilon _\nu ^3). \tag{6.4''}
    \Deqlab{4''}
\end{equation}
��\Deqref{4'}�ɖ߂�, ���ӂ���$\alpha$������
\begin{equation}
	x^{(\nu  + 1)} - \alpha = x^{(\nu )} - \alpha - \frac{f(x^{(\nu )})}{f^{\prime}(x^{(\nu )})}. \nonumber
\end{equation}
������\Deqref{4''}���������
\begin{eqnarray*}
	\epsilon _{(\nu  + 1)} 
	&=& \epsilon _\nu  - \epsilon _\nu  + \frac{f^{\prime \prime}(\alpha)}{2f^{\prime}(\alpha)}{\epsilon _\nu }^2 + O(\epsilon _\nu ^3)\\
	&=&	\frac{f^{\prime \prime}(\alpha)}{2f^{\prime}(\alpha)}{\epsilon _\nu }^2 + O(\epsilon _\nu ^3). 
\end{eqnarray*}
������$O(\epsilon _\nu ^3)$�𖳎�����$\epsilon$�̒�`���v���o����
\begin{equation}
	\parallel x^{(\nu  + 1)} - \alpha \parallel   \approx \frac{f^{\prime \prime}(\alpha)}{2f^{\prime}(\alpha)}\parallel x^{(\nu )} - \alpha \parallel ^2.  \Deqlab{5}
\end{equation}
\Deqref{5}�̌`�̎����󋵂�2�������ƌĂ΂��. ��G�c�ɂ����āu�덷������2��Ō���v���邢�́u����������������1�񂲂Ƃɔ{�ɑ�����v. ���̏�������, ��������Ă��邩�ǂ�����O�����ċ�̓I�ɒ��ׂ��i���Ȃ��悤�Ȃ��̂ł���.
\Deqref{5}�̂悤�Ȏ����󋵂ɓ��B�������ǂ�����, $x^{(\nu)} - x^{(\nu + 1)}$���ώ@���Ă���΂킩��. 
$x^{(\mu)} - x^{(\mu + 1)}$��\Deqref{5}�̗l�ɐU����������悤�ɂȂ����ꍇ��
\begin{equation}
	x^{(\nu)} - x^{(\nu + 1)} \approx x^{(\nu)} - \alpha \nonumber
\end{equation}
�Ƃ݂Ȃ��Ă悢
\footnote{
	\begin{equation}
		x^{(\nu  + 1)} = x^{(\nu )} - \frac{f(x^{(\nu )})}{f^{\prime}(x^{(\nu )})}\nonumber
	\end{equation}
	�Ȃ̂�\Deqref{4''}���
	\begin{equation}
		x^{(\nu )} - x^{(\nu + 1)} \approx \frac{f^{\prime \prime}(\alpha)}{2f^{\prime}(\alpha)}\epsilon ^2_\nu  + \epsilon _\nu  + O(\epsilon _\nu ^3). \nonumber
	\end{equation}
	$\epsilon _\nu \ll 1$���
	\begin{eqnarray*}
		x^{(\nu )} - x^{(\nu + 1)} 
		&\approx &	\epsilon _\nu  \\
		&=&	x^{(\nu )} - \alpha . 
	\end{eqnarray*}}. 

\section{�v�Z��}

���ۂɈȉ��̓�̕��������v�Z���Ă݂�.

(1)$f(x) = \tanh(x) + 0.2x + 0.3 = 0$

(2)$f(x) = x^5 - 7x^4 + 16x^3 - 8x^2 -16x + 16 =0$

{\bf ��:}

(1)�̃v���O�����ڍׂ͕t�^1���Q�Ƃ��ꂽ��. ���ʂ͕t�^1$^\prime$���Q�Ƃ̂���. 

(2)�̃v���O�����ڍׂ͕t�^2���Q�Ƃ��ꂽ��. ���ʂ͕t�^2$^\prime$���Q�Ƃ̂���. 
\end{document}
