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\title[ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ. ìÅËÃÉÑ 4.]{ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ ÒÅÚÕÌØÔÁÔÏ×
ÉÚÍÅÒÅÎÉÊ}
\subtitle{ìÅËÃÉÑ 4. óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ. éÎÔÅÇÒÉÒÏ×ÁÎÉÅ. äÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ.}
\date{}

\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.
$\mathrm{det}\,A\equiv |\B A|\ne0$.
\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}{íÁÔÒÉÞÎÙÊ ÍÅÔÏÄ}
$\B x = \B A^{-1}\B b$\\
$\B A \cdot \B A^{-1} = \B A^{-1} \cdot \B A = \B E$.
îÁÈÏÖÄÅÎÉÅ ÏÂÒÁÔÎÏÊ ÍÁÔÒÉÃÙ:
\begin{itemize}
\item Ó ÐÏÍÏÝØÀ ÐÒÉÓÏÅÄÉÎÅÎÎÏÊ: $(\B A | \B E )$ \so $(\B E | \B A^{-1})$;
\item $\B A^{-1} = \dfrac{\mathrm{adj\,}\B A}{|\B A|}$, ÐÒÉÓÏÅÄÉÎÅÎÎÁÑ ÍÁÔÒÉÃÁ $\mathrm{adj\,}\B A$
Ñ×ÌÑÅÔÓÑ ÔÒÁÎÓÐÏÎÉÒÏ×ÁÎÎÏÊ ÍÁÔÒÉÃÅÊ ÁÌÇÅÂÒÁÉÞÅÓËÉÈ ÄÏÐÏÌÎÅÎÉÊ ($(-1)^{i+j}M_{ij}$, $M_ij$~--
ÓÏÏÔ×ÅÔÓÔ×ÕÀÝÉÊ ÄÏÐÏÌÎÉÔÅÌØÎÙÊ ÍÉÎÏÒ~--- ÏÐÒÅÄÅÌÉÔÅÌØ ÍÁÔÒÉÃÙ Ó ×ÙÞÅÒËÎÕÔÙÍÉ $i$-Ê ÓÔÒÏËÏÊ É $j$-Í
ÓÔÏÌÂÃÏÍ).
\item É Ô.Ä., É Ô.Ð.
\end{itemize}
æÏÒÍÕÌÙ ëÒÁÍÅÒÁ: $x_j = |A_j|/|A|$, $A_j$ ÐÏÌÕÞÁÅÔÓÑ ÉÚ $A$ ÚÁÍÅÎÏÊ $j$-ÇÏ ÓÔÏÌÂÃÁ ÎÁ $\B b$.
\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}{íÅÔÏÄ úÅÊÄÅÌÑ}
$$\B{Bx}_{n+1}+\B{Cx}_n=\B b,$$
ÇÄÅ
$$\B B=\begin{pmatrix}
a_{11}&0&0&\cdots&0\\
a_{21}&a_{22}&0&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
a_{m1}&a_{m2}&a_{m3}&\cdots&a_{mm}
\end{pmatrix},\qquad
\B C=\begin{pmatrix}
0&a_{12}&a_{13}&\cdots&a_{1m}\\
0&0&a_{23}&\cdots&a_{2m}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\cdots&0
\end{pmatrix}.
$$
ïÔÓÀÄÁ ÐÏÌÕÞÁÅÍ
$$\B x_{n+1}=-\B B^{-1}\B{Cx}_n +\B B^{-1}\B b.$$
\end{block}
}
\only<4>{
\begin{block}{LU-ÍÅÔÏÄ}
$$\B A=\B L\cdot \B U,$$
ÇÄÅ
$$\B L=\begin{pmatrix}
l_{11}&0&0&\cdots&0\\
l_{21}&l_{22}&0&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
l_{m1}&l_{m2}&l_{m3}&\cdots&l_{mm}
\end{pmatrix},\qquad
\B U=\begin{pmatrix}
1&u_{12}&u_{13}&\cdots&u_{1m}\\
0&1&u_{23}&\cdots&u_{2m}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\cdots&1
\end{pmatrix}.
$$
ðÒÑÍÏÊ ÈÏÄ:  $\B L\cdot \B U\cdot \B x \equiv \B L\cdot\B y= \B b$, ÎÁÈÏÄÉÍ $\B y$,
ÉÚ $\B U\cdot \B x =\B y$ ÎÁÈÏÄÉÍ $\B x$.
$$\begin{cases}
l_{ij}=a_{ij}-\Sum_{s=1}^{j-1}l_{is}u_{sj},& i\ge j;\\
u_{ij}=\frac1{l_{ii}}\Bigl(a_{ij}-\Sum_{s=1}^{j-1}l_{is}u_{sj}\Bigr), & i < j.
\end{cases}
$$
LU-ÒÁÚÌÏÖÅÎÉÅ ×ÏÚÍÏÖÎÏ ÄÌÑ ÍÁÔÒÉÃ Ó ÐÒÅÏÂÌÁÄÁÎÉÅÍ ÄÉÁÇÏÎÁÌØÎÙÈ ÜÌÅÍÅÎÔÏ×
\end{block}
}
\only<5>{
\begin{block}{òÁÚÌÏÖÅÎÉÅ èÏÌÅÃËÏÇÏ}
$\B A=\B L\cdot \B L^T$, ÌÉÂÏ $\B A=\B U^T\cdot \B U$, ÇÄÅ $\B L$~-- ÎÉÖÎÑÑ ÔÒÅÕÇÏÌØÎÁÑ ÍÁÔÒÉÃÁ
ÓÏ ÓÔÒÏÇÏ ÐÏÌÏÖÉÔÅÌØÎÙÍÉ ÜÌÅÍÅÎÔÁÍÉ ÎÁ ÄÉÁÇÏÎÁÌÉ, $\B U$~-- ×ÅÒÈÎÑÑ ÔÒÅÕÇÏÌØÎÁÑ ÍÁÔÒÉÃÁ.
òÁÚÌÏÖÅÎÉÅ èÏÌÅÃËÏÇÏ ×ÓÅÇÄÁ ÓÕÝÅÓÔ×ÕÅÔ É ÅÄÉÎÓÔ×ÅÎÎÏ ÄÌÑ ÌÀÂÏÊ ÓÉÍÍÅÔÒÉÞÎÏÊ (ÏÔÎÏÓÉÔÅÌØÎÏ ÇÌÁ×ÎÏÊ
ÄÉÁÇÏÎÁÌÉ) ÐÏÌÏÖÉÔÅÌØÎÏ-ÏÐÒÅÄÅÌÅÎÎÏÊ ÍÁÔÒÉÃÙ (×ÓÅ ÄÉÁÇÏÎÁÌØÎÙÅ ÍÉÎÏÒÙ ÐÏÌÏÖÉÔÅÌØÎÙ).

$$\begin{cases}
l_{ii}=\sqrt{a_{ii} - \Sum_{s=1}^{i-1}l^2_{is}}; \\
l_{ij} = \frac1{l_{ii}}\Bigl( a_{ij} - \Sum_{s=1}^{j-1}l_{is}l_{js}\Bigr), & j < i.
\end{cases}
$$

ðÒÑÍÏÊ É ÏÂÒÁÔÎÙÊ ÈÏÄÙ ÁÎÁÌÏÇÉÞÎÙ LU-ÒÁÚÌÏÖÅÎÉÀ.
\end{block}
}
\only<6>{
\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}
}
\only<7>{
\begin{block}{íÅÔÏÄ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ× ($m > n$)}
$\B{x=A\backslash b}$, %), ÐÓÅ×ÄÏÏÂÒÁÔÎÁÑ ÍÁÔÒÉÃÁ, \dots
$S=\sum_i(\sum_j a_{ij}x_j - b_i)^2$, $\partder{S}{x_j}=0$ \so
$\B C\B x = \B d$, ÇÄÅ
$c_{kj} = \sum_i a_{ik}a_{ij}$, $k,j=\overline{1,n}$, $d_k = \sum_i a_{ik}b_i$.
ô.Ï. $\B C = \B A^T\cdot \B A$, $\B d=\B A^T\B b$.
\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}
}
\end{frame}

\begin{frame}{þÉÓÌÏ ÏÂÕÓÌÏ×ÌÅÎÎÏÓÔÉ ÍÁÔÒÉÃÙ}
\begin{block}{ïÃÅÎËÁ ÏÛÉÂËÉ ÒÅÛÅÎÉÑ}
ðÕÓÔØ $\B x'$~-- ÐÒÉÂÌÉÖÅÎÎÏÅ ÒÅÛÅÎÉÅ. áÂÓÏÌÀÔÎÁÑ É ÏÔÎÏÓÉÔÅÌØÎÁÑ ÏÛÉÂËÉ: $||\B x-\B x'||$
É $\frc{||\B x-\B x'||}{||\B x||}$. îÁÍ ÉÚ×ÅÓÔÎÁ ÎÅ×ÑÚËÁ $\B r=\B b-A\B x'$:
$$\B r=A\B x-A\B x'=A(\B x-\B x')\so ||\B x-\B x'||=||A^{-1}\B r||\le ||A^{-1}||\,||\B r||,$$
Á Ô.Ë. $||\B b||\le||A||\,||\B x||$, $\frc{1}{||\B x||}\le\frc{||A||}{||\B b||}$:
$$\frac{||\B x-\B x'||}{||\B x||}\le||A^{-1}||\,||\B r||\,\frac{||A||}{||\B b||}=k(A)\frac{||\B
r||}{||\B b||}.$$
\Ö þÉÓÌÏ ÏÂÕÓÌÏ×ÌÅÎÎÏÓÔÉ\Î: $k(A)=||A||\,||A^{-1}||$. þÅÍ ÏÎÏ ÂÏÌØÛÅ, ÔÅÍ ÂÏÌØÛÅ ÆÌÕËÔÕÁÃÉÉ $\B x$
×ÌÉÑÀÔ ÎÁ ÏÂÝÅÅ ÒÅÛÅÎÉÅ. õ ÈÏÒÏÛÏ ÏÂÕÓÌÏ×ÌÅÎÎÙÈ ÍÁÔÒÉÃ $K(A)\equiv1$ (ÎÁÐÒ., ÏÒÔÏÇÏÎÁÌØÎÙÅ ÍÁÔÒÉÃÙ,
Õ ËÏÔÏÒÙÈ $A^T=A^{-1}$).
\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{frame}{âÉÓÅËÃÉÑ (ÄÉÈÏÔÏÍÉÑ)}
\img{bisect}
ïÔÒÅÚÏË ÄÅÌÉÔÓÑ ×ÐÌÏÔØ ÄÏ ÚÁÄÁÎÎÏÊ ÔÏÞÎÏÓÔÉ $b_n-a_n\le\epsilon$, ËÏÒÅÎØ $x\approx(b_n+a_n)/2$.

ðÒÉÍÅÎÑÅÔÓÑ É ÄÌÑ ÐÏÉÓËÁ ÚÎÁÞÅÎÉÊ × ÕÐÏÒÑÄÏÞÅÎÎÏÍ ÒÑÄÕ.
\end{frame}

\begin{frame}{íÅÔÏÄ ÈÏÒÄ (ÓÅËÕÝÉÈ)}
\begin{block}{}
$$x_{i+1}=x_{i-1}+\frac{y_{i-1}\cdot(x_i-x_{i-1})}{y_i-y_{i-1}}.$$
\end{block}
\begin{pict}\smimg[0.5]{chords1}\,\smimg[0.5]{chords2}\end{pict}
\end{frame}

\begin{blueframe}{íÅÔÏÄ îØÀÔÏÎÁ}
\begin{block}{}
$$x_{i+1}=x_i+\frac{y_i}{y'_i}.$$
\end{block}
\begin{pict}\smimg[0.5]{newton1}\,\smimg[0.5]{newton2}\end{pict}
\end{blueframe}

\section{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ É ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
\begin{frame}{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ}
\begin{block}{}
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}
\end{frame}

\begin{frame}{íÅÔÏÄ ÐÒÑÍÏÕÇÏÌØÎÉËÏ×}
\begin{columns}
\column{0.45\textwidth}
\begin{block}{}
$$\int_a^b f(x)\,dx\approx$$
\begin{list}{}{}
\item $\displaystyle\sum_{i=0}^{n-1}f(x_{i})(x_{i+1}-x_{i})$;
\item $\displaystyle\sum_{i=1}^{n}f(x_{i})(x_i-x_{i-1})$;
\item $\displaystyle\sum_{i=0}^{n-1}f\bigl(\frac{x_{i}+x_{i+1}}{2}\bigr)(x_{i+1}-x_{i})$.
\end{list}
äÌÑ ÒÁ×ÎÏÍÅÒÎÙÈ ÓÅÔÏË:

$\displaystyle h\sum_{i=0}^{n-1} f_i$;
$\displaystyle h\sum_{i=1}^{n} f_i$;
$\displaystyle h\bigl(\sum_{i=1}^{n-1} f_i + \frac{f_0+f_n}{2}\bigr)$.

\end{block}
\column{0.45\textwidth}
\img{rectangmeth}
\end{columns}
\end{frame}

\begin{lightframe}{íÅÔÏÄ ÔÒÁÐÅÃÉÊ}
\begin{columns}
\column{0.6\textwidth}
\begin{block}{}
$$\int_a^b f(x)\,dx\approx
\sum_{i=0}^{n-1}\frac{f(x_i)+f(x_{i+1})}{2}(x_{i+1}-x_i).$$

äÌÑ ÒÁ×ÎÏÍÅÒÎÙÈ ÓÅÔÏË~--- ÆÏÒÍÕÌÁ ëÏÔÅÓÁ:
$$\int_a^b f(x)\,dx =
h\left(\frac{f_0+f_n}{2}+\sum_{i=1}^{n-1}f_i\right) + E_n(f),$$
$$E_n(f)=-\frac{f''(\xi)}{12}(b-a)h^2, \xi\in[a,b].$$
\end{block}
\column{0.4\textwidth}
\img{trapezmeth}
\end{columns}
\end{lightframe}

\begin{blueframe}{íÅÔÏÄ óÉÍÐÓÏÎÁ}
\begin{columns}
\column{0.6\textwidth}
\begin{block}{}
$$\int_a^b f(x)\,dx\approx \frac{b-a}{6}\left(f(a)+4f\bigl(\frac{a+b}{2}\bigr)\right)$$
\end{block}
\column{0.4\textwidth}
\img{Simpsons_method_illustration}
\end{columns}
\begin{block}{}
æÏÒÍÕÌÁ ëÏÔÅÓÁ:
$$I\approx \frac{h}{3}\Bigl(f(x_0)+
2\sum_{i=1}^{N/2-1}f(x_{2i}) + 4\sum_{i=1}^{N/2}f(x_{2i-1} + f(x_N)\Bigr).$$

\end{block}
\end{blueframe}


\begin{frame}{þÉÓÌÅÎÎÏÅ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
\begin{block}{}
áÐÐÒÏËÓÉÍÁÃÉÑ ÆÕÎËÃÉÉ ÉÎÔÅÒÐÏÌÑÃÉÏÎÎÙÍ ÍÎÏÇÏÞÌÅÎÏÍ (îØÀÔÏÎÁ, óÔÉÒÌÉÎÇÁ É Ô.Ð.), ÒÁÚÄÅÌÅÎÎÙÅ
ÒÁÚÎÏÓÔÉ.
\end{block}
\begin{block}{áÐÐÒÏËÓÉÍÁÃÉÑ ÍÎÏÇÏÞÌÅÎÏÍ}
$$f(x)\approx P_N(x)\quad\Arr\quad f^{(r)}(x)\approx P_N^{(r)}(x).$$
ðÏÌÉÎÏÍ îØÀÔÏÎÁ:
$$P_N(x)=\sum_{m=0}^{N}C_x^m\sum _{k=0}^{m}(-1)^{m-k}\,C_m^k\,f(k).$$
ðÏÌÉÎÏÍ ìÁÇÒÁÎÖÁ:
$$P_N(x) = \sum_{k=0}^N y_k \frac {(x-x_0)(x-x_1) \ldots (x-x_{k-1})(x-x_{k+1}) \ldots  (x-x_n)} {(x_k-x_0)(x_k-x_1) \ldots
(x_k-x_{k-1})(x_k-x_{k+1}) \ldots  (x_k-x_n)}.$$
á ÔÁËÖÅ: ÉÎÔÅÒÐÏÌÑÃÉÑ ËÕÂÉÞÅÓËÉÍÉ ÓÐÌÁÊÎÁÍÉ, ÒÁÚÌÏÖÅÎÉÅ ÐÏ ÂÁÚÉÓÕ ÔÒÉÇÏÎÏÍÅÔÒÉÞÅÓËÉÈ ÆÕÎËÃÉÊ É Ô.Ð.
\end{block}
\end{frame}

\begin{frame}{}
\begin{block}{òÁÚÄÅÌÅÎÎÙÅ ÒÁÚÎÏÓÔÉ}
 ÷ ÎÕÌÅ×ÏÍ ÐÒÉÂÌÉÖÅÎÉÉ ÍÏÖÎÏ ÚÁÍÅÎÉÔØ ÐÒÏÉÚ×ÏÄÎÕÀ $f^{(n)}$ ÒÁÚÄÅÌÅÎÎÏÊ ÒÁÚÎÏÓÔØÀ $n$-ÇÏ ÐÏÒÑÄËÁ.
$$f(x_{0};\;x_{1};\;\ldots ;\;x_{n})=\sum _{j=0}^{n}{\frac {f(x_j)}{\prod \limits _{i=0 \atop i\neq j}^{n}(x_{j}-x_{i})}}.$$
÷ ÞÁÓÔÎÏÓÔÉ:
$$f(x_0;\;x_1)={\frac {f(x_1)}{x_1-x_0}}+{\frac {f(x_0)}{x_0-x_1}},$$
$$f(x_0;\;x_1;\;x_2)={\frac {f(x_2)}{(x_2-x_1)(x_2-x_0)}}+{\frac {f(x_1)}{(x_1-x_2)(x_1-x_0)}}+{\frac
{f(x_0)}{(x_0-x_2)(x_0-x_1)}}\ldots$$
\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{blueframe}{}
\begin{block}{íÅÔÏÄ üÊÌÅÒÁ}
áÐÐÒÏËÓÉÍÁÃÉÑ ÉÎÔÅÇÒÁÌØÎÏÊ ËÒÉ×ÏÊ ËÕÓÏÞÎÏ-ÌÉÎÅÊÎÏÊ ÆÕÎËÃÉÅÊ. úÁÄÁÞÁ ëÏÛÉ × ÐÒÏÓÔÅÊÛÅÍ ×ÉÄÅ:
$\frac{dy}{dx}=f(x,y)$, $y|_{x=x_0}=y_0$. òÅÛÅÎÉÅ ÉÝÅÔÓÑ ÎÁ ÉÎÔÅÒ×ÁÌÅ $(x_0, b]$.
$$y_i=y_{i-1}+(x_i-x_{i-1})f(x_{i-1},y_{i-1}),\qquad i=\overline{1,n}.$$
\end{block}
\img[0.5]{Euler_method}
\end{blueframe}

\begin{frame}{}
\begin{block}{íÅÔÏÄ òÕÎÇÅ-ëÕÔÔÙ}
$$y_{n+1}=y_n+\frac{h}{6}\bigl(k_1+2k_2+2k_3+k_4\bigr),\qquad \text{ÇÄÅ}$$
$k_1=f(x_n,y_n)$, $k_2=f\bigl(x_n+\frc{h}2, y_n+\frc{h}2 k_1\bigr)$,
$k_3=f\bigl(x_n+\frc{h}2, y_n+\frc{h}2 k_2\bigr)$,
$k_4=f(x_n+h,y_n+hk_3)$ ($h$~-- ÛÁÇ ÓÅÔËÉ ÐÏ $x$).
\end{block}
\img[0.5]{Runge-Kutta}
\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}
