5 蝗ｺ螳夂せ縺ｮ霑ｽ霍｡

縺�∪�後ヱ繝ｩ繝｡繝ｼ繧ｿ縺ｯ荳螳壼､縺ｫ螳壹ａ縺ｦ縺翫￥�弱＠縺溘′縺｣縺ｦ�御ｻ･荳九�蠑上〒縺ｯ螟画焚 $\lambda$ 繧堤怐逡･縺吶ｋ �� 蝗ｺ螳夂せ縺ｮ縺�縺�◆縺��菴咲ｽｮ縺ｯ�梧焚蛟､繧ｷ繝･繝溘Ξ繝ｼ繧ｷ繝ｧ繝ｳ縺ｫ繧医▲縺ｦ螳ｹ譏薙↓蠕励ｉ繧後ｋ�� 縺薙�繝��繧ｿ縺九ｉ莉ｻ諢上�邊ｾ蠎ｦ縺ｮ豁｣遒ｺ縺ｪ菴咲ｽｮ縺後＞縺ｾ�檎衍繧翫◆縺�→縺励ｈ縺�ｼ� 縺昴ｌ縺ｫ縺ｯNewton豕輔ｒ逕ｨ縺�ｋ��

Newton譁ｹ縺ｮ繧｢繝ｫ繧ｴ繝ｪ繧ｺ繝�繧堤ｰ｡蜊倥↓蠕ｩ鄙偵☆繧具ｼ� $F(\mbox{\boldmath$\ x $})=0,\, \mbox{\boldmath$\ x $} \in \mbox{\boldmath$R$}^n$ 縺ｨ縺吶ｋ縺ｨ�碁未謨ｰ $F(\mbox{\boldmath$\ x $})$ 縺ｮ�� 隨ｬ霑台ｼｼ $\mbox{\boldmath$\ x $}^{(k)}$ 縺ｾ繧上ｊ縺ｮTaylor螻暮幕縺ｯ��

$\begin{displaymath} F(\mbox{\boldmath$ x $}) = F(\mbox{\boldmath$ x $}^{(k)}) + ... ...(\mbox{\boldmath$ x $} - \mbox{\boldmath$ x $}^{(k)}) + \cdots \end{displaymath}$

(5.10)

縺ｧ陦ｨ縺輔ｌ繧具ｼ弱％縺薙〒 $F'(\mbox{\boldmath$\ x $}^{(k)}) = (\partial f_i/ \partial x_j)$ 縺ｯJacobi陦悟�縺ｧ縺ゅｋ�手ｿ台ｼｼ蛟､縺ｮ菫ｮ豁｣驥� $\mbox{\boldmath$\ h $} = \mbox{\boldmath$\ x $}^{(k+1)} - \mbox{\boldmath$\ x $}^{(k)}$ 縺ｯ谺｡縺ｮ蠑上〒荳弱∴繧峨ｌ繧具ｼ�

$\begin{displaymath} F'(\mbox{\boldmath$ x $}^{(k)})\mbox{\boldmath$ h $} = -F(\mbox{\boldmath$ x $}^{(k)}) \end{displaymath}$

(5.11)

Newton豕輔�莠梧ｬ｡蜿取據縺ｧ縺ゅｊ�後°縺ｪ繧翫ｈ縺�ｬｬ

霑台ｼｼ $\mbox{\boldmath$\ x $}^{(k)}$ 繧剃ｸ弱∴縺溘ｉ�� 謨ｰ蝗槭�蜿榊ｾｩ縺ｧ邊ｾ蠎ｦ縺ｮ濶ｯ縺�ｿ台ｼｼ隗｣縺悟ｾ励ｉ繧後ｋ莉慕ｵ�∩縺ｫ縺ｪ縺｣縺ｦ縺�ｋ��

縺輔※�訓oincaré 蜀吝ワ縺ｮ譁ｹ遞句ｼ�(8)縺ｮ蝗ｺ螳夂せ縺ｮ霑台ｼｼ蛟､繧� $\mbox{\boldmath$\ x $}_0^{(k)}$ 縺ｨ縺吶ｋ�弱＞縺ｾ��

$\begin{displaymath} F(\mbox{\boldmath$ x $}_0) = \mbox{\boldmath$ x $}_0 - T(\mbox{\boldmath$ x $}_0) = 0 \end{displaymath}$

(5.12)

縺ｨ縺励◆縺ｨ縺搾ｼ檎ｬｬ

谺｡縺ｮ霑台ｼｼ隗｣繧貞ｾ励ｋ繧｢繝ｫ繧ｴ繝ｪ繧ｺ繝�縺ｯ谺｡縺ｮ縺ｨ縺翫ｊ��

$\begin{displaymath} \begin{array}{rcl} \mbox{\boldmath$ x $}^{(k+1)}_0 & = & \mb... ...math$ h $} & = & -F(\mbox{\boldmath$ x $}^{(k)}_0) \end{array}\end{displaymath}$

(5.13)

縺薙％縺ｧ�� $F'(\mbox{\boldmath$\ x $}_0^{(k)})$ 縺ｯ蛻晄悄蛟､ $\mbox{\boldmath$\ x $}_0$ 縺ｫ髢｢縺吶ｋ蠕ｮ蛻�↓繧医ｋ隨ｬ

谺｡縺ｮ Jacobi陦悟� $(2 \times 2)$ 縺ｧ��

$\begin{displaymath} F'(\mbox{\boldmath$ x $}^{(k)}_0) = I - \frac{\partial T}{\partial \mbox{\boldmath$ x $}}(\mbox{\boldmath$ x $}^{(k)}_0) \end{displaymath}$

(5.14)

$\begin{displaymath} \frac{\partial T}{\partial \mbox{\boldmath$ x $}}(\mbox{\bol... ...\partial y_0} (2\pi, x_0^{(k)}, y_0^{(k)}) \end{array}\right] \end{displaymath}$

(5.15)

縺ｨ縺ｪ繧具ｼ弱◆縺�縺暦ｼ�

縺ｯ $(2 \times 2)$ 縺ｮ蜊倅ｽ崎｡悟�縺ｧ縺ゅｋ�� 蠑�(13)縺ｮ隨ｬ2蠑上�� $\mbox{\boldmath$\ h $}$ 縺ｫ縺､縺�※�� Gauss縺ｮ謗�″蜃ｺ縺玲ｳ輔↑縺ｩ縺ｧ隗｣縺九↑縺上※縺ｯ縺ｪ繧峨↑縺�ｼ� $F(\mbox{\boldmath$\ x $}_0^{(k)})$ 縺ｯ髢｢謨ｰ蛟､縺昴�繧ゅ�繧剃ｸ弱∴繧九→繧医＞縺鯉ｼ� 蝠城｡後��悟ｼ�(15)縺ｮ Jacobi陦悟�縺ｮ隕∫ｴ�(蛛丞ｾｮ蛻��蛟､��)繧偵←縺�ｄ縺｣縺ｦ豎ゅａ繧九°縺ｧ縺ゅｋ��

縺｡繧�▲縺ｨ蜈�↓繧ゅ←縺｣縺ｦ閠�∴縺ｦ縺ｿ繧医≧�� 蠑�(2)繧呈嶌縺咲峩縺�:

$\begin{displaymath} \frac{d \mbox{\boldmath$ x $}}{dt} = \mbox{\boldmath$ f $}(t,\mbox{\boldmath$ x $}). \end{displaymath}$

(5.16)

縺溘□縺暦ｼ� $\mbox{\boldmath$\ x $} = (x,y)^t, \, \mbox{\boldmath$\ f $}(t, \mbox{\boldmath$\ x $}) = (f_1(t, \mbox{\boldmath$\ x $}),f_2(t, \mbox{\boldmath$\ x $}))^t$ 縺ｨ縺吶ｋ�� $(\varphi_1, \varphi_2)^t = \mbox{\boldmath$\ \varphi $}$ 縺ｨ縺励◆縺ｨ縺搾ｼ� 蛻晄悄蛟､ $\mbox{\boldmath$\ x $}_0$ 縺九ｉ蜃ｺ逋ｺ縺吶ｋ蠑�(16)縺ｮ隗｣繧�

$\begin{displaymath} \mbox{\boldmath$ x $}(t) = \mbox{\boldmath$ \varphi $}(t, \mbox{\boldmath$ x $}_0) \end{displaymath}$

(5.17)

縺ｨ縺励ｈ縺�ｼ� 縺薙�隗｣繧貞ｼ�(16)縺ｫ莉｣蜈･縺吶ｋ縺ｨ��

$\begin{displaymath} \frac{d \mbox{\boldmath$ \varphi $}}{dt}(t, \mbox{\boldmath$... ...}(t, \mbox{\boldmath$ \varphi $}(t, \mbox{\boldmath$ x $}_0)). \end{displaymath}$

(5.18)

縺薙�蠑上ｒ $\mbox{\boldmath$\ x $}_0$ 縺ｧ蠕ｮ蛻�☆繧具ｼ弱☆繧九→��

$\begin{displaymath} \frac{\partial}{\partial \mbox{\boldmath$ x $}_0}\left( \fr... ...box{\boldmath$ \varphi $}(t, \mbox{\boldmath$ x $}_0)) \right) \end{displaymath}$

(5.19)

蟾ｦ霎ｺ縺ｮ蠕ｮ蛻��鬆�ｺ上�螟画峩縺ｧ縺阪ｋ�主承霎ｺ繧りｨ育ｮ励☆繧九→��

$\begin{displaymath} \frac{d}{dt}\left( \frac{\partial \mbox{\boldmath$ \varphi $... ...{\partial \mbox{\boldmath$ x $}_0}(t,\mbox{\boldmath$ x $}_0). \end{displaymath}$

(5.20)

縺薙�蠑上��後ｏ縺九ｊ繧�☆縺乗嶌縺肴鋤縺医ｋ縺ｨ��

$\begin{displaymath} \frac{dX}{dt} = \frac{d \mbox{\boldmath$ f $}}{d \mbox{\boldmath$ x $}}X \end{displaymath}$

(5.21)

縺ｨ縺ｪ繧具ｼ弱▽縺ｾ繧� $X = \partial \mbox{\boldmath$\ \varphi $} / \partial \mbox{\boldmath$\ x $}_0$ 縺ｯ邱壼ｽ｢陦悟�蠕ｮ蛻�婿遞句ｼ�縺ｮ蝓ｺ譛ｬ陦悟�隗｣縺ｨ縺ｪ縺｣縺ｦ縺�ｋ�� 縺薙�蠑�(20)縺ｯ蠑�(16)縺ｮ螟牙�譁ｹ遞句ｼ上→縺�ｏ繧後ｋ��

縺ｨ縺薙ｍ縺ｧ

$\begin{displaymath} \frac{\partial \mbox{\boldmath$ \varphi $}} {\partial \mbox{\boldmath$ x $}_0}(0,\mbox{\boldmath$ x $}_0) = I \end{displaymath}$

(5.22)

縺ｧ縺ゅｋ��

縺ｯ $(n \times n)$ 縺ｮ蜊倅ｽ崎｡悟�縺ｧ縺ゅｋ�� 蠑�(22)繧貞�譛溷､縺ｨ縺励※�悟ｼ�(20)繧呈凾蛻ｻ

縺九ｉ $2\pi$ 縺ｾ縺ｧ謨ｰ蛟､遨榊�縺吶ｌ縺ｰ�悟ｼ�(15)繧呈ｱゅａ繧� 縺薙→縺後〒縺搾ｼ君ewton豕輔↓繧医▲縺ｦ邊ｾ蟇�↑蝗ｺ螳夂せ縺ｮ菴咲ｽｮ $\mbox{\boldmath$\ x $}_0$ 縺� 豎ゅａ繧峨ｌ繧具ｼ弱∪縺�

-蜻ｨ譛溽せ縺梧ｱゅａ縺溘＞蝣ｴ蜷医�遨榊�繧� $2m\pi$ 縺ｾ縺ｧ螳溯｡後☆繧後�繧医＞��

User & 2017-09-07