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\title[ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ. ìÅËÃÉÑ 5.]{ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ ÒÅÚÕÌØÔÁÔÏ× 
ÉÚÍÅÒÅÎÉÊ}
\subtitle{ìÅËÃÉÑ 5. óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ}
\date{29 ÓÅÎÔÑÂÒÑ 2016 ÇÏÄÁ}

\begin{document}
% ôÉÔÕÌ
\begin{frame}
\maketitle
\end{frame}
% óÏÄÅÒÖÁÎÉÅ
\begin{frame}
\tableofcontents
\end{frame}

\section{óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ}
\begin{frame}{óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ}
\begin{defin}
\Ö óÉÓÔÅÍÁ ÌÉÎÅÊÎÙÈ ÕÒÁ×ÎÅÎÉÊ\Î ÄÌÑ $n$ ÎÅÉÚ×ÅÓÔÎÙÈ ÉÍÅÅÔ ×ÉÄ:
$$
\left\{
\begin{aligned}
a_{11}x_1+a_{12}x_2&+\cdots+a_{1n}x_n&=b_1;\\
a_{21}x_1+a_{22}x_2&+\cdots+a_{2n}x_n&=b_2;\\
\cdots\\
a_{n1}x_1+a_{n2}x_2&+\cdots+a_{nn}x_n&=b_n.
\end{aligned}
\right.
$$
\end{defin}
\begin{defin}
åÓÌÉ ÓÕÝÅÓÔ×ÕÅÔ ×ÅËÔÏÒ--ÓÔÏÌÂÅÃ~$\B x$, ÏÂÒÁÝÁÀÝÉÊ ×ÙÒÁÖÅÎÉÅ~$\B{Ax=b}$ × ÔÏÖÄÅÓÔ×Ï, ÇÏ×ÏÒÑÔ, 
ÞÔÏ~$\B x$ Ñ×ÌÑÅÔÓÑ\Ö ÒÅÛÅÎÉÅÍ\Î ÄÁÎÎÏÊ ÓÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ.
$|\B A|\ne0$. 
\end{defin}
\end{frame}

\begin{frame}{íÅÔÏÄÙ ÒÅÛÅÎÉÊ}
\only<1>{\begin{block}{}
$\delta=\B{Ax-b}$.
ðÒÉÂÌÉÖÅÎÎÙÅ ÍÅÔÏÄÙ: $\mathrm{min}(\delta)$. ôÏÞÎÙÅ ÍÅÔÏÄÙ: $\delta=0$.\\
\end{block}
\begin{block}{íÅÔÏÄ ÐÒÏÓÔÏÊ ÉÔÅÒÁÃÉÉ}
$\B{x=Bx+c}$, $\B x_{n+1}=\B B\B x_n+\B c$.\\
óÌÏÖÎÏÓÔØÀ ÍÅÔÏÄÁ ÐÒÏÓÔÏÊ ÉÔÅÒÁÃÉÉ ÐÒÉ ÒÅÛÅÎÉÉ ÍÁÔÒÉÃ ÂÏÌØÛÉÈ ÒÁÚÍÅÒÎÏÓÔÅÊ Ñ×ÌÑÅÔÓÑ ÏÓÏÂÏÅ Ó×ÏÊÓÔ×Ï 
ÔÁËÉÈ ÍÁÔÒÉÃ~--- ÓÕÝÅÓÔ×Ï×ÁÎÉÅ\Ë ÐÏÞÔÉ ÓÏÂÓÔ×ÅÎÎÙÈ ÚÎÁÞÅÎÉÊ\Î, $\lambda$: 
$||\B{Ax}-\lambda\B x||\le\epsilon||\B x||$ ÐÒÉ $||\B x||\ne0$.\\
\end{block}
\begin{block}{íÁÔÒÉÞÎÙÊ ÍÅÔÏÄ}
$\B x = \B A^{-1}\B b$
\end{block}
}
\only<2>{
\begin{block}{íÅÔÏÄ çÁÕÓÓÁ}
$$
\B A_d\B{x} = \pmb\beta,\quad
\B A_d=\begin{pmatrix}
\alpha_{11}&\alpha_{12}&\alpha_{13}&\cdots&\alpha_{1m}\\
0&\alpha_{22}&\alpha_{23}&\cdots&\alpha_{2m}\\
\cdot&\cdot&\cdot&\cdots&\cdot\\
0&0&0&\cdots&\alpha_{mm}
\end{pmatrix}.
$$
ðÒÑÍÏÊ ÈÏÄ~--- ÐÒÅÏÂÒÁÚÏ×ÁÎÉÅ Ë ÄÉÁÇÏÎÁÌØÎÏÊ ÆÏÒÍÅ:
$$
\left(\begin{matrix}
\alpha_{11}&\alpha_{12}&\alpha_{13}&\cdots&\alpha_{1m}\\
0&\alpha_{22}&\alpha_{23}&\cdots&\alpha_{2m}\\
\cdot&\cdot&\cdot&\cdots&\cdot\\
0&0&0&\cdots&\alpha_{mm}
\end{matrix}\middle|
\begin{matrix}\beta_1\\\beta_2\\\cdot\\\beta_m\end{matrix}\right).
$$
ïÂÒÁÔÎÙÊ ÈÏÄ~--- ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏÅ ÎÁÈÏÖÄÅÎÉÅ $x_m$, $x_{m-1}$, \dots, $x_1$.

$N\propto n^3$~--- ÐÒÑÍÏÊ, $N\propto n^2$~--- ÏÂÒÁÔÎÙÊ ÈÏÄ.
\end{block}
}
\only<3>{
\begin{block}{}
\Ö íÅÔÏÄ úÅÊÄÅÌÑ\Î: \\
$$\B{Bx}_{n+1}+\B{Cx}_n=\B b,$$
ÇÄÅ
$$\B B=\begin{pmatrix}
a_{11}&0&0&\cdots&0\\
a_{21}&a_{22}&0&\cdots&0\\
\cdot&\cdot&\cdot&\cdots&\cdot\\
a_{m1}&a_{m2}&a_{m3}&\cdots&a_{mm}
\end{pmatrix},\qquad
\B C=\begin{pmatrix}
0&a_{12}&a_{13}&\cdots&a_{1m}\\
0&0&a_{23}&\cdots&a_{2m}\\
\cdot&\cdot&\cdot&\cdots&\cdot\\
0&0&0&\cdots&0
\end{pmatrix}.
$$
ïÔÓÀÄÁ ÐÏÌÕÞÁÅÍ
$$\B x_{n+1}=-\B B^{-1}\B{Cx}_n +\B B^{-1}\B b.$$
\end{block}
}
\only<4>{
\begin{block}{}
åÓÌÉ $\B A$ ÓÏÄÅÒÖÉÔ~$m$ ÓÔÒÏË É~$n$ ÓÔÏÌÂÃÏ×, ÔÏ:
\begin{description}
\item[$m=n$] Ë×ÁÄÒÁÔÎÁÑ ÍÁÔÒÉÃÁ, ×ÏÚÍÏÖÎÏ ÓÕÝÅÓÔ×Ï×ÁÎÉÅ ÔÏÞÎÏÇÏ ÒÅÛÅÎÉÑ;
\item[$m<n$] ÎÅÄÏÏÐÒÅÄÅÌÅÎÎÁÑ ÓÉÓÔÅÍÁ, ÒÅÛÅÎÉÅ ×ÏÚÍÏÖÎÏ ÌÉÛØ × ÏÂÝÅÍ ×ÉÄÅ
Ó ÐÏ ËÒÁÊÎÅÊ ÍÅÒÅ~$n-m$ Ó×ÏÂÏÄÎÙÈ ËÏÜÆÆÉÃÉÅÎÔÏ×;
\item[$m>n$] ÐÅÒÅÏÐÒÅÄÅÌÅÎÎÁÑ ÓÉÓÔÅÍÁ, ÐÒÉÂÌÉÖÅÎÎÏÅ ÒÅÛÅÎÉÅ ËÏÔÏÒÏÊ ÎÁÈÏÄÉÔÓÑ
ÐÒÉ ÐÏÍÏÝÉ ÍÅÔÏÄÁ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ× (× ÓÌÕÞÁÅ ÌÉÎÅÊÎÏÊ ÚÁ×ÉÓÉÍÏÓÔÉ ÓÔÒÏË
ÄÁÎÎÏÊ ÓÉÓÔÅÍÙ ÍÏÖÅÔ ÓÕÝÅÓÔ×Ï×ÁÔØ É ÔÏÞÎÏÅ ÒÅÛÅÎÉÅ).
\end{description}
\end{block}
\begin{block}{ðÒÉÂÌÉÖÅÎÎÙÅ ÒÅÛÅÎÉÑ}
íîë ($\B{x=A\backslash b}$), ÐÓÅ×ÄÏÏÂÒÁÔÎÁÑ ÍÁÔÒÉÃÁ, \dots
\end{block}
}
\end{frame}

\section{óÔÅÐÅÎÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
\begin{frame}{óÔÅÐÅÎÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
\begin{defin}
\Ö óÔÅÐÅÎÎÏÅ ÕÒÁ×ÎÅÎÉÅ\Î ÉÍÅÅÔ ×ÉÄ $p_n(x)=0$, ÇÄÅ $p_n(x)$~-- ÐÏÌÉÎÏÍ~$n$~-Ê ÓÔÅÐÅÎÉ ×ÉÄÁ 
$p_n(x)=\sum_{i=0}^n C_nx^n$.
\end{defin}
\begin{block}{íÅÔÏÄÙ ÒÅÛÅÎÉÑ}
ôÏÞÎÙÅ~--- ÄÏ ÔÒÅÔØÅÊ ÓÔÅÐÅÎÉ ×ËÌÀÞÉÔÅÌØÎÏ (× ÏÂÝÅÍ ÓÌÕÞÁÅ) É ÉÔÅÒÁÃÉÏÎÎÙÅ:
\begin{description}
\item[ÂÉÓÅËÃÉÑ] ÄÅÌÅÎÉÅ ÐÏÐÏÌÁÍ ÏÔÒÅÚËÁ, ÇÄÅ ÎÁÈÏÄÉÔÓÑ ËÏÒÅÎØ;
\item[ÍÅÔÏÄ ÈÏÒÄ] ÚÁÍÅÎÁ ÐÏÌÉÎÏÍÁ ÈÏÒÄÏÊ, ÐÒÏÈÏÄÑÝÅÊ ÞÅÒÅÚ ÔÏÞËÉ $(x_1, p_n(x_1)$ É $(x_2, 
p_n(x_2)$;
\item[ÍÅÔÏÄ îØÀÔÏÎÁ] ÉÍÅÅÔ ÂÙÓÔÒÕÀ ÓÈÏÄÉÍÏÓÔØ, ÎÏ ÔÒÅÂÕÅÔ ÚÎÁËÏÐÏÓÔÏÑÎÓÔ×Á $f'(x)$ É $f''(x)$ ÎÁ 
×ÙÂÒÁÎÎÏÍ ÉÎÔÅÒ×ÁÌÅ $(x_1, x_2)$.
\end{description}
\end{block}
\end{frame}

\begin{blueframe}{íÅÔÏÄ îØÀÔÏÎÁ}
\img[0.7]{Newton_iteration}
\end{blueframe}

\section{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ É ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
\begin{frame}{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ É ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
\only<1>{
\begin{block}{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ}
äÌÑ ÞÉÓÌÅÎÎÏÇÏ ÒÅÛÅÎÉÑ ÕÒÁ×ÎÅÎÉÑ $\displaystyle I=\Int_a^b f(x)\,dx$ ÎÁÉÂÏÌÅÅ ÐÏÐÕÌÑÒÎÙ:
\begin{description}
\item[ÍÅÔÏÄ ÐÒÑÍÏÕÇÏÌØÎÉËÏ×] $I\approx\sum_{i=1}^n f(x_i)[x_i-x_{i-1}]$;
\item[ÍÅÔÏÄ ÔÒÁÐÅÃÉÊ] $I\approx\sum_{i=1}^n \frac{f(x_{i-1})+f(x_i)}{2}[x_i-x_{i-1}]$;
\item[ÍÅÔÏÄ óÉÍÐÓÏÎÁ] $\Int_{-1}^1 f(x)\,dx\approx\frac13\bigl(f(-1)+4f(0)+f(1)\bigr)$ \so
$I\approx\frac{b-a}{6n}\Bigl(f(x_0)+f(x_n)+2\sum_{i=1}^{n/2-1}
f(x_{2i}) + 4\sum_{i=1}^{n/2}f(x_{2i-1})\Bigr)$.
\end{description}
É ÍÎÏÇÉÅ ÄÒÕÇÉÅ.
\end{block}}
\only<2>{
\begin{block}{þÉÓÌÅÎÎÏÅ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
áÐÐÒÏËÓÉÍÁÃÉÑ ÆÕÎËÃÉÉ ÉÎÔÅÒÐÏÌÑÃÉÏÎÎÙÍ ÍÎÏÇÏÞÌÅÎÏÍ (îØÀÔÏÎÁ, óÔÉÒÌÉÎÇÁ É Ô.Ð.), ÒÁÚÄÅÌÅÎÎÙÅ 
ÒÁÚÎÏÓÔÉ.

÷ ÎÕÌÅ×ÏÍ ÐÒÉÂÌÉÖÅÎÉÉ ÍÏÖÎÏ ÚÁÍÅÎÉÔØ ÐÒÏÉÚ×ÏÄÎÕÀ $f^{(n)}$ ÒÁÚÄÅÌÅÎÎÏÊ ÒÁÚÎÏÓÔØÀ $n$-ÇÏ ÐÏÒÑÄËÁ:
$$f(x_0; x_1; \ldots; x_n) = \sum_{i=0}^n \frac{f(x_i)}{\displaystyle
\prod_{j=0, j\ne i}^n\!\!(x_i - x_j)}.$$
\end{block}}
\end{frame}

\section{äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
\begin{frame}{äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
\only<1>{
\begin{defin}
\Ö ïÂÙËÎÏ×ÅÎÎÙÅ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ\Î~(ïäõ) ÐÏÒÑÄËÁ~$n$ ÚÁÄÁÀÔÓÑ × ×ÉÄÅ
ÆÕÎËÃÉÉ $f(x,y,y',\ldots,y^{(n)})=0$.
\end{defin}
\begin{block}{}
òÁÚÄÅÌÅÎÉÅ ÐÅÒÅÍÅÎÎÙÈ:\vspace{-2em}
$$y'=f(x,y) \so \phi(y)\,dy=\psi(x)\,dx \so y=y_0+\Int_0^{x}\psi(x)\,dx.$$

ïäõ ×ÔÏÒÏÇÏ ÐÏÒÑÄËÁ:
$$Ay''+By'+Cy+Dx=0.$$
åÓÌÉ $D\equiv0$, Á ÍÎÏÖÉÔÅÌÉ $A$, $B$ É~$C$~--- ËÏÎÓÔÁÎÔÙ, ÉÍÅÅÍ ÏÄÎÏÒÏÄÎÏÅ ïäõ. 
$y=\C_1\exp(k_1x)+\C_2\exp(k_2x)$, ÇÄÅ~$k_1$ É~$k_2$~-- ËÏÒÎÉ\Ë
ÈÁÒÁËÔÅÒÉÓÔÉÞÅÓËÏÇÏ ÕÒÁ×ÎÅÎÉÑ\Î $Ak^2+Bk+C=0$.
\end{block}}
\only<2>{
\begin{defin}
\Ö äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ × ÞÁÓÔÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ\Î~(þäõ) ÄÌÑ ÆÕÎËÃÉÉ
$y=y(x_1,x_2,\cdots,x_n)$ ÉÍÅÀÔ ×ÉÄ
$$f(y,x_1,\ldots,x_n;\partder{y}{x_1},\ldots;\dpartder{y}{x_1},\ldots;\cdots;
\frac{\partial^m y}{\partial x_1^m},\ldots)=0.$$
\end{defin}
\begin{block}{}
ïÄÎÁËÏ, ÎÁÉÂÏÌÅÅ ÞÁÓÔÏ ×ÓÔÒÅÞÁÀÔÓÑ þäõ ÐÅÒ×ÏÇÏ ÐÏÒÑÄËÁ ÄÌÑ ÆÕÎËÃÉÉ Ä×ÕÈ
ÐÅÒÅÍÅÎÎÙÈ $z=z(x,y)$ ×ÉÄÁ
$$f(z,x,y,\partder{z}{x},\partder{z}{y})=0.$$
\end{block}}
\only<3>{
\begin{block}
\Ö îÅÌÉÎÅÊÎÙÅ\Î ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ ÓÏÄÅÒÖÁÔ ÎÅËÏÔÏÒÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ
ÆÕÎËÃÉÉ~$y$ ÎÅ ËÁË ÐÒÏÓÔÙÅ ÍÎÏÖÉÔÅÌÉ, Á ËÁË ÁÒÇÕÍÅÎÔÙ ÆÕÎËÃÉÊ (ÞÁÝÅ ×ÓÅÇÏ~---
ÓÔÅÐÅÎÎÙÈ), ÎÁÐÒÉÍÅÒ: $(y'')^3-\sin y'=\tg(xy)$. ïÂÙÞÎÙÅ ÆÉÚÉÞÅÓËÉÅ ÚÁÄÁÞÉ
ÎÉËÏÇÄÁ ÎÅ ÐÒÉ×ÏÄÑÔ Ë ÔÁËÉÍ ÕÒÁ×ÎÅÎÉÑÍ, ÏÄÎÁËÏ, É ÉÈ ÒÅÛÅÎÉÑ ×ÐÏÌÎÅ ÍÏÖÎÏ
ÎÁÊÔÉ ÐÒÉ ÐÏÍÏÝÉ ÞÉÓÌÅÎÎÙÈ ÍÅÔÏÄÏ×.
\end{block}
\begin{block}{íÅÔÏÄÙ ÒÅÛÅÎÉÑ}
òÕÎÇÅ--ëÕÔÔÙ, üÊÌÅÒÁ, áÄÁÍÓÁ, ËÏÎÅÞÎÙÈ ÒÁÚÎÏÓÔÅÊ É Ô.Ð.

äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ× Ó×ÏÄÑÔ ÐÕÔÅÍ ÚÁÍÅÎÙ ÐÅÒÅÍÅÎÎÙÈ Ë ÓÉÓÔÅÍÅ ïäõ ÐÅÒ×ÏÇÏ 
ÐÏÒÑÄËÁ.
\end{block}
}
\end{frame}


\begin{frame}{óÐÁÓÉÂÏ ÚÁ ×ÎÉÍÁÎÉÅ!}
\centering
\begin{minipage}{5cm}
\begin{block}{mailto}
eddy@sao.ru\\
edward.emelianoff@gmail.com
\end{block}\end{minipage}
\end{frame}
\end{document}
