mirror of
https://github.com/eddyem/lectures.git
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227 lines
6.9 KiB
TeX
227 lines
6.9 KiB
TeX
\documentclass[10pt,pdf,hyperref={unicode}]{beamer}
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\hypersetup{pdfpagemode=FullScreen}
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\usepackage{ed}
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\usepackage{lect}
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\title[ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ. ìÅËÃÉÑ 5.]{ëÏÍÐØÀÔÅÒÎÁÑ ÏÂÒÁÂÏÔËÁ ÒÅÚÕÌØÔÁÔÏ×
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ÉÚÍÅÒÅÎÉÊ}
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\subtitle{ìÅËÃÉÑ 5. óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ}
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\date{29 ÓÅÎÔÑÂÒÑ 2016 ÇÏÄÁ}
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\begin{document}
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% ôÉÔÕÌ
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\begin{frame}
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\maketitle
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\end{frame}
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% óÏÄÅÒÖÁÎÉÅ
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\begin{frame}
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\tableofcontents
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\end{frame}
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\section{óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ}
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\begin{frame}{óÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ}
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\begin{defin}
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\Ö óÉÓÔÅÍÁ ÌÉÎÅÊÎÙÈ ÕÒÁ×ÎÅÎÉÊ\Î ÄÌÑ $n$ ÎÅÉÚ×ÅÓÔÎÙÈ ÉÍÅÅÔ ×ÉÄ:
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$$
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\left\{
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\begin{aligned}
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a_{11}x_1+a_{12}x_2&+\cdots+a_{1n}x_n&=b_1;\\
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a_{21}x_1+a_{22}x_2&+\cdots+a_{2n}x_n&=b_2;\\
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\cdots\\
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a_{n1}x_1+a_{n2}x_2&+\cdots+a_{nn}x_n&=b_n.
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\end{aligned}
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\right.
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$$
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\end{defin}
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\begin{defin}
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åÓÌÉ ÓÕÝÅÓÔ×ÕÅÔ ×ÅËÔÏÒ--ÓÔÏÌÂÅÃ~$\B x$, ÏÂÒÁÝÁÀÝÉÊ ×ÙÒÁÖÅÎÉÅ~$\B{Ax=b}$ × ÔÏÖÄÅÓÔ×Ï, ÇÏ×ÏÒÑÔ,
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ÞÔÏ~$\B x$ Ñ×ÌÑÅÔÓÑ\Ö ÒÅÛÅÎÉÅÍ\Î ÄÁÎÎÏÊ ÓÉÓÔÅÍÙ ÕÒÁ×ÎÅÎÉÊ.
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$|\B A|\ne0$.
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\end{defin}
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\end{frame}
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\begin{frame}{íÅÔÏÄÙ ÒÅÛÅÎÉÊ}
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\only<1>{\begin{block}{}
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$\delta=\B{Ax-b}$.
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ðÒÉÂÌÉÖÅÎÎÙÅ ÍÅÔÏÄÙ: $\mathrm{min}(\delta)$. ôÏÞÎÙÅ ÍÅÔÏÄÙ: $\delta=0$.\\
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\end{block}
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\begin{block}{íÅÔÏÄ ÐÒÏÓÔÏÊ ÉÔÅÒÁÃÉÉ}
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$\B{x=Bx+c}$, $\B x_{n+1}=\B B\B x_n+\B c$.\\
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óÌÏÖÎÏÓÔØÀ ÍÅÔÏÄÁ ÐÒÏÓÔÏÊ ÉÔÅÒÁÃÉÉ ÐÒÉ ÒÅÛÅÎÉÉ ÍÁÔÒÉà ÂÏÌØÛÉÈ ÒÁÚÍÅÒÎÏÓÔÅÊ Ñ×ÌÑÅÔÓÑ ÏÓÏÂÏÅ Ó×ÏÊÓÔ×Ï
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ÔÁËÉÈ ÍÁÔÒÉÃ~--- ÓÕÝÅÓÔ×Ï×ÁÎÉÅ\Ë ÐÏÞÔÉ ÓÏÂÓÔ×ÅÎÎÙÈ ÚÎÁÞÅÎÉÊ\Î, $\lambda$:
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$||\B{Ax}-\lambda\B x||\le\epsilon||\B x||$ ÐÒÉ $||\B x||\ne0$.\\
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\end{block}
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\begin{block}{íÁÔÒÉÞÎÙÊ ÍÅÔÏÄ}
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$\B x = \B A^{-1}\B b$
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\end{block}
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}
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\only<2>{
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\begin{block}{íÅÔÏÄ çÁÕÓÓÁ}
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$$
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\B A_d\B{x} = \pmb\beta,\quad
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\B A_d=\begin{pmatrix}
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\alpha_{11}&\alpha_{12}&\alpha_{13}&\cdots&\alpha_{1m}\\
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0&\alpha_{22}&\alpha_{23}&\cdots&\alpha_{2m}\\
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\cdot&\cdot&\cdot&\cdots&\cdot\\
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0&0&0&\cdots&\alpha_{mm}
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\end{pmatrix}.
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$$
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ðÒÑÍÏÊ ÈÏÄ~--- ÐÒÅÏÂÒÁÚÏ×ÁÎÉÅ Ë ÄÉÁÇÏÎÁÌØÎÏÊ ÆÏÒÍÅ:
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$$
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\left(\begin{matrix}
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\alpha_{11}&\alpha_{12}&\alpha_{13}&\cdots&\alpha_{1m}\\
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0&\alpha_{22}&\alpha_{23}&\cdots&\alpha_{2m}\\
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\cdot&\cdot&\cdot&\cdots&\cdot\\
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0&0&0&\cdots&\alpha_{mm}
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\end{matrix}\middle|
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\begin{matrix}\beta_1\\\beta_2\\\cdot\\\beta_m\end{matrix}\right).
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$$
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ïÂÒÁÔÎÙÊ ÈÏÄ~--- ÐÏÓÌÅÄÏ×ÁÔÅÌØÎÏÅ ÎÁÈÏÖÄÅÎÉÅ $x_m$, $x_{m-1}$, \dots, $x_1$.
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$N\propto n^3$~--- ÐÒÑÍÏÊ, $N\propto n^2$~--- ÏÂÒÁÔÎÙÊ ÈÏÄ.
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\end{block}
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}
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\only<3>{
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\begin{block}{}
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\Ö íÅÔÏÄ úÅÊÄÅÌÑ\Î: \\
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$$\B{Bx}_{n+1}+\B{Cx}_n=\B b,$$
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ÇÄÅ
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$$\B B=\begin{pmatrix}
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a_{11}&0&0&\cdots&0\\
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a_{21}&a_{22}&0&\cdots&0\\
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\cdot&\cdot&\cdot&\cdots&\cdot\\
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a_{m1}&a_{m2}&a_{m3}&\cdots&a_{mm}
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\end{pmatrix},\qquad
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\B C=\begin{pmatrix}
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0&a_{12}&a_{13}&\cdots&a_{1m}\\
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0&0&a_{23}&\cdots&a_{2m}\\
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\cdot&\cdot&\cdot&\cdots&\cdot\\
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0&0&0&\cdots&0
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\end{pmatrix}.
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$$
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ïÔÓÀÄÁ ÐÏÌÕÞÁÅÍ
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$$\B x_{n+1}=-\B B^{-1}\B{Cx}_n +\B B^{-1}\B b.$$
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\end{block}
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}
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\only<4>{
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\begin{block}{}
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åÓÌÉ $\B A$ ÓÏÄÅÒÖÉÔ~$m$ ÓÔÒÏË É~$n$ ÓÔÏÌÂÃÏ×, ÔÏ:
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\begin{description}
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\item[$m=n$] Ë×ÁÄÒÁÔÎÁÑ ÍÁÔÒÉÃÁ, ×ÏÚÍÏÖÎÏ ÓÕÝÅÓÔ×Ï×ÁÎÉÅ ÔÏÞÎÏÇÏ ÒÅÛÅÎÉÑ;
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\item[$m<n$] ÎÅÄÏÏÐÒÅÄÅÌÅÎÎÁÑ ÓÉÓÔÅÍÁ, ÒÅÛÅÎÉÅ ×ÏÚÍÏÖÎÏ ÌÉÛØ × ÏÂÝÅÍ ×ÉÄÅ
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Ó ÐÏ ËÒÁÊÎÅÊ ÍÅÒÅ~$n-m$ Ó×ÏÂÏÄÎÙÈ ËÏÜÆÆÉÃÉÅÎÔÏ×;
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\item[$m>n$] ÐÅÒÅÏÐÒÅÄÅÌÅÎÎÁÑ ÓÉÓÔÅÍÁ, ÐÒÉÂÌÉÖÅÎÎÏÅ ÒÅÛÅÎÉÅ ËÏÔÏÒÏÊ ÎÁÈÏÄÉÔÓÑ
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ÐÒÉ ÐÏÍÏÝÉ ÍÅÔÏÄÁ ÎÁÉÍÅÎØÛÉÈ Ë×ÁÄÒÁÔÏ× (× ÓÌÕÞÁÅ ÌÉÎÅÊÎÏÊ ÚÁ×ÉÓÉÍÏÓÔÉ ÓÔÒÏË
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ÄÁÎÎÏÊ ÓÉÓÔÅÍÙ ÍÏÖÅÔ ÓÕÝÅÓÔ×Ï×ÁÔØ É ÔÏÞÎÏÅ ÒÅÛÅÎÉÅ).
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\end{description}
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\end{block}
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\begin{block}{ðÒÉÂÌÉÖÅÎÎÙÅ ÒÅÛÅÎÉÑ}
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íîë ($\B{x=A\backslash b}$), ÐÓÅ×ÄÏÏÂÒÁÔÎÁÑ ÍÁÔÒÉÃÁ, \dots
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\end{block}
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}
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\end{frame}
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\section{óÔÅÐÅÎÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
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\begin{frame}{óÔÅÐÅÎÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
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\begin{defin}
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\Ö óÔÅÐÅÎÎÏÅ ÕÒÁ×ÎÅÎÉÅ\Î ÉÍÅÅÔ ×ÉÄ $p_n(x)=0$, ÇÄÅ $p_n(x)$~-- ÐÏÌÉÎÏÍ~$n$~-Ê ÓÔÅÐÅÎÉ ×ÉÄÁ
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$p_n(x)=\sum_{i=0}^n C_nx^n$.
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\end{defin}
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\begin{block}{íÅÔÏÄÙ ÒÅÛÅÎÉÑ}
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ôÏÞÎÙÅ~--- ÄÏ ÔÒÅÔØÅÊ ÓÔÅÐÅÎÉ ×ËÌÀÞÉÔÅÌØÎÏ (× ÏÂÝÅÍ ÓÌÕÞÁÅ) É ÉÔÅÒÁÃÉÏÎÎÙÅ:
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\begin{description}
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\item[ÂÉÓÅËÃÉÑ] ÄÅÌÅÎÉÅ ÐÏÐÏÌÁÍ ÏÔÒÅÚËÁ, ÇÄÅ ÎÁÈÏÄÉÔÓÑ ËÏÒÅÎØ;
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\item[ÍÅÔÏÄ ÈÏÒÄ] ÚÁÍÅÎÁ ÐÏÌÉÎÏÍÁ ÈÏÒÄÏÊ, ÐÒÏÈÏÄÑÝÅÊ ÞÅÒÅÚ ÔÏÞËÉ $(x_1, p_n(x_1)$ É $(x_2,
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p_n(x_2)$;
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\item[ÍÅÔÏÄ îØÀÔÏÎÁ] ÉÍÅÅÔ ÂÙÓÔÒÕÀ ÓÈÏÄÉÍÏÓÔØ, ÎÏ ÔÒÅÂÕÅÔ ÚÎÁËÏÐÏÓÔÏÑÎÓÔ×Á $f'(x)$ É $f''(x)$ ÎÁ
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×ÙÂÒÁÎÎÏÍ ÉÎÔÅÒ×ÁÌÅ $(x_1, x_2)$.
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\end{description}
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\end{block}
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\end{frame}
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||
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\begin{blueframe}{íÅÔÏÄ îØÀÔÏÎÁ}
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\img[0.7]{Newton_iteration}
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\end{blueframe}
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\section{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ É ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
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\begin{frame}{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ É ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
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\only<1>{
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\begin{block}{þÉÓÌÅÎÎÏÅ ÉÎÔÅÇÒÉÒÏ×ÁÎÉÅ}
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äÌÑ ÞÉÓÌÅÎÎÏÇÏ ÒÅÛÅÎÉÑ ÕÒÁ×ÎÅÎÉÑ $\displaystyle I=\Int_a^b f(x)\,dx$ ÎÁÉÂÏÌÅÅ ÐÏÐÕÌÑÒÎÙ:
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\begin{description}
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\item[ÍÅÔÏÄ ÐÒÑÍÏÕÇÏÌØÎÉËÏ×] $I\approx\sum_{i=1}^n f(x_i)[x_i-x_{i-1}]$;
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\item[ÍÅÔÏÄ ÔÒÁÐÅÃÉÊ] $I\approx\sum_{i=1}^n \frac{f(x_{i-1})+f(x_i)}{2}[x_i-x_{i-1}]$;
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\item[ÍÅÔÏÄ óÉÍÐÓÏÎÁ] $\Int_{-1}^1 f(x)\,dx\approx\frac13\bigl(f(-1)+4f(0)+f(1)\bigr)$ \so
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$I\approx\frac{b-a}{6n}\Bigl(f(x_0)+f(x_n)+2\sum_{i=1}^{n/2-1}
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f(x_{2i}) + 4\sum_{i=1}^{n/2}f(x_{2i-1})\Bigr)$.
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\end{description}
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É ÍÎÏÇÉÅ ÄÒÕÇÉÅ.
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\end{block}}
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\only<2>{
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\begin{block}{þÉÓÌÅÎÎÏÅ ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÅ}
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áÐÐÒÏËÓÉÍÁÃÉÑ ÆÕÎËÃÉÉ ÉÎÔÅÒÐÏÌÑÃÉÏÎÎÙÍ ÍÎÏÇÏÞÌÅÎÏÍ (îØÀÔÏÎÁ, óÔÉÒÌÉÎÇÁ É Ô.Ð.), ÒÁÚÄÅÌÅÎÎÙÅ
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ÒÁÚÎÏÓÔÉ.
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÷ ÎÕÌÅ×ÏÍ ÐÒÉÂÌÉÖÅÎÉÉ ÍÏÖÎÏ ÚÁÍÅÎÉÔØ ÐÒÏÉÚ×ÏÄÎÕÀ $f^{(n)}$ ÒÁÚÄÅÌÅÎÎÏÊ ÒÁÚÎÏÓÔØÀ $n$-ÇÏ ÐÏÒÑÄËÁ:
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$$f(x_0; x_1; \ldots; x_n) = \sum_{i=0}^n \frac{f(x_i)}{\displaystyle
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\prod_{j=0, j\ne i}^n\!\!(x_i - x_j)}.$$
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\end{block}}
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||
\end{frame}
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\section{äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
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\begin{frame}{äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ}
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\only<1>{
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||
\begin{defin}
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\Ö ïÂÙËÎÏ×ÅÎÎÙÅ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ\Î~(ïäõ) ÐÏÒÑÄËÁ~$n$ ÚÁÄÁÀÔÓÑ × ×ÉÄÅ
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ÆÕÎËÃÉÉ $f(x,y,y',\ldots,y^{(n)})=0$.
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\end{defin}
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\begin{block}{}
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òÁÚÄÅÌÅÎÉÅ ÐÅÒÅÍÅÎÎÙÈ:\vspace{-2em}
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$$y'=f(x,y) \so \phi(y)\,dy=\psi(x)\,dx \so y=y_0+\Int_0^{x}\psi(x)\,dx.$$
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ïäõ ×ÔÏÒÏÇÏ ÐÏÒÑÄËÁ:
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$$Ay''+By'+Cy+Dx=0.$$
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åÓÌÉ $D\equiv0$, Á ÍÎÏÖÉÔÅÌÉ $A$, $B$ É~$C$~--- ËÏÎÓÔÁÎÔÙ, ÉÍÅÅÍ ÏÄÎÏÒÏÄÎÏÅ ïäõ.
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$y=\C_1\exp(k_1x)+\C_2\exp(k_2x)$, ÇÄÅ~$k_1$ É~$k_2$~-- ËÏÒÎÉ\Ë
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ÈÁÒÁËÔÅÒÉÓÔÉÞÅÓËÏÇÏ ÕÒÁ×ÎÅÎÉÑ\Î $Ak^2+Bk+C=0$.
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\end{block}}
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||
\only<2>{
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||
\begin{defin}
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||
\Ö äÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ × ÞÁÓÔÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ\Î~(þäõ) ÄÌÑ ÆÕÎËÃÉÉ
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$y=y(x_1,x_2,\cdots,x_n)$ ÉÍÅÀÔ ×ÉÄ
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$$f(y,x_1,\ldots,x_n;\partder{y}{x_1},\ldots;\dpartder{y}{x_1},\ldots;\cdots;
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\frac{\partial^m y}{\partial x_1^m},\ldots)=0.$$
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\end{defin}
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||
\begin{block}{}
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||
ïÄÎÁËÏ, ÎÁÉÂÏÌÅÅ ÞÁÓÔÏ ×ÓÔÒÅÞÁÀÔÓÑ þäõ ÐÅÒ×ÏÇÏ ÐÏÒÑÄËÁ ÄÌÑ ÆÕÎËÃÉÉ Ä×ÕÈ
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ÐÅÒÅÍÅÎÎÙÈ $z=z(x,y)$ ×ÉÄÁ
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$$f(z,x,y,\partder{z}{x},\partder{z}{y})=0.$$
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\end{block}}
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\only<3>{
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||
\begin{block}
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||
\Ö îÅÌÉÎÅÊÎÙÅ\Î ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÕÒÁ×ÎÅÎÉÑ ÓÏÄÅÒÖÁÔ ÎÅËÏÔÏÒÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ
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||
ÆÕÎËÃÉÉ~$y$ ÎÅ ËÁË ÐÒÏÓÔÙÅ ÍÎÏÖÉÔÅÌÉ, Á ËÁË ÁÒÇÕÍÅÎÔÙ ÆÕÎËÃÉÊ (ÞÁÝÅ ×ÓÅÇÏ~---
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||
ÓÔÅÐÅÎÎÙÈ), ÎÁÐÒÉÍÅÒ: $(y'')^3-\sin y'=\tg(xy)$. ïÂÙÞÎÙÅ ÆÉÚÉÞÅÓËÉÅ ÚÁÄÁÞÉ
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||
ÎÉËÏÇÄÁ ÎÅ ÐÒÉ×ÏÄÑÔ Ë ÔÁËÉÍ ÕÒÁ×ÎÅÎÉÑÍ, ÏÄÎÁËÏ, É ÉÈ ÒÅÛÅÎÉÑ ×ÐÏÌÎÅ ÍÏÖÎÏ
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||
ÎÁÊÔÉ ÐÒÉ ÐÏÍÏÝÉ ÞÉÓÌÅÎÎÙÈ ÍÅÔÏÄÏ×.
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||
\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
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||
ÐÏÒÑÄËÁ.
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||
\end{block}
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||
}
|
||
\end{frame}
|
||
|
||
|
||
\begin{frame}{óÐÁÓÉÂÏ ÚÁ ×ÎÉÍÁÎÉÅ!}
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||
\centering
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||
\begin{minipage}{5cm}
|
||
\begin{block}{mailto}
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||
eddy@sao.ru\\
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||
edward.emelianoff@gmail.com
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||
\end{block}\end{minipage}
|
||
\end{frame}
|
||
\end{document}
|