From db3a08596eb1f1f0886ca578f4e7049a1529c1b6 Mon Sep 17 00:00:00 2001 From: salagne Date: Fri, 29 May 2020 14:42:07 +0200 Subject: [PATCH] minor corrctions, added ref to spektralsatz --- ana10.pdf | Bin 147471 -> 145978 bytes ana10.tex | 16 +++++++-------- ana11.pdf | Bin 168120 -> 168339 bytes ana11.tex | 52 ++++++++++++++++++++++++------------------------- ana7.pdf | Bin 220501 -> 220522 bytes ana7.tex | 1 + analysisII.pdf | Bin 576206 -> 576082 bytes 7 files changed, 35 insertions(+), 34 deletions(-) diff --git a/ana10.pdf b/ana10.pdf index b57d61c341395b2c524bdc1d900f72d8d5bf882f..dbc33e0b182c7a1c12c29ac4eaa934f3329b57c4 100644 GIT binary patch delta 19257 zcmXVWWmug((=8i!-4u5%?hXYC#VPLY?rt0RVmIzq+}*W6ad&qpR*Gv+pZ7bzGPyF5 znXHvuGf8|PQpYV~0u>A=I|o3)OAlGp)%7H5#R^!i)#_C~PDi}#Tf4A6Tbr`Reral6 zixBY6pTzB!;z&tgTvwj^y9KJlGxGG1kxicKiUcF=ailr+5KIKP-%s~8+BFLotTGg> zF4IjXL;J)Q?wc`xFhYuQ3Z@gY?0DjN#y 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Fixpunktiteration: Startwert $x^{(0)}$. Iterationsschritt @@ -34,7 +34,7 @@ $x = g(x) = x - \sigma (f(x) - b) \implies f(x) = b$. Frage: Unter welchen Bedingungen konvergiert die Fixpunktiteration? \begin{definition}[Lipschitz-Stetigkeit] - Eine Funktion $f\colon D \subseteq \mathbb{K}^{n} \to \mathbb{K}^{n}$ heißt\\ + Eine Funktion $g\colon D \subseteq \mathbb{K}^{n} \to \mathbb{K}^{n}$ heißt\\ \underline{Lipschitz-stetig}, wenn eine Konstante $L < \infty$ existiert, s.d. \[ \Vert g(x) - g(y) \Vert \le L \Vert x - y \Vert, \qquad \forall x, y \in D @@ -88,7 +88,7 @@ Frage: Unter welchen Bedingungen konvergiert die Fixpunktiteration? \intertext{Seien $k, m$ beliebig. Dann gilt $\forall \epsilon > 0$} \Vert x^{(k+m)} - x^{(k)} \Vert &= \Vert x^{(k+m)} - x^{(k+m-1)} + x^{(k+m-1)} - \ldots x^{(k+1)} - x^{(k)} \Vert \\ - &\le \Vert x^{(k+m)} - x^{(k+m+1)} \Vert + &\le \Vert x^{(k+m)} - x^{(k+m-1)} \Vert + \ldots + \Vert x^{(k+1)} - x^{(k)} \Vert \\ &= L^{m-1} \Vert x^{(k+1)} - x^{(k)} \Vert + L^{m-2} \Vert x^{k+1} - x^{k} \Vert @@ -123,7 +123,7 @@ Frage: Unter welchen Bedingungen konvergiert die Fixpunktiteration? \begin{bem}[Anwendung: Lineare Gleichungssysteme] $A = \left( a_{ij} \right)_{i,j = 1}^{n} \in \mathbb{K}^{n \times n}$ regulär und $b = (b_i)_{i = 1}^{n} \in \mathbb{K}^{n}$. Da $A$ regulär, hat das LGS $Ax = b$ genau - eine Lösung $x^{*} = A^{-1}b$. Sei $g(x) := x - \sigma (Ax - b)$ mit + eine Lösung $x^{*} = A^{-1}b$. Sei $g(x) \coloneqq x - \sigma (Ax - b)$ mit $\sigma \in \mathbb{K} \setminus \{0\} $. Fixpunktiteration $x^{(k)} = x^{(k-1)} - \sigma (Ax^{(k-1}) - b)$, $k \in \N$ konvergiert, wenn @@ -139,8 +139,8 @@ Frage: Unter welchen Bedingungen konvergiert die Fixpunktiteration? $\left\Vert \mathbb{I} - \frac{A}{\Vert A \Vert_{\infty}} \right\Vert_2 < 1$. Da $A$ positiv definit und hermitesch, sind alle Eigenwerte $\lambda > 0$. Es gilt $\forall $ EW: $0 < \lambda \le \Vert A \Vert_{\infty}$. Für EW von $\mathbb{I} - \frac{A}{\Vert A\Vert_{\infty}} $ gilt $\mu = 1 - \frac{\lambda}{\Vert A \Vert_{\infty}}$, $\lambda$ Eigenwert von $A$. Also - $0 \le \underbrace{1 - \frac{\lambda}{\Vert A \Vert_{\infty}}}_{= \mu} < 1$, also - $\underbrace{\Big\Vert \underbrace{\mathbb{I} - \frac{A}{\Vert A \Vert_{\infty}}}_{\text{hermitesch}} \Big\Vert_2}_{\text{Spektralnorm}} < 1$. Falls $A$ hermitesch und positiv definit, ist also + $0 \le \underbrace{1 - \frac{\lambda}{\Vert A \Vert_{\infty}}}_{= \mu} < 1$, mit \ref{lemma:spektralnorm} folgt + $\Big\Vert \underbrace{\mathbb{I} - \frac{A}{\Vert A \Vert_{\infty}}}_{\text{hermitesch}} \Big\Vert_2 < 1$. Falls $A$ hermitesch und positiv definit, ist also die Richardson Iteration konvergent. \end{bem} @@ -154,7 +154,7 @@ Frage: Unter welchen Bedingungen konvergiert die Fixpunktiteration? \begin{bem}[Anwendung: Nichtlineare Gleichungssysteme] Sei $f\colon D \subseteq \R^{n} \to \R^{n}$ Lipschitz stetig mit $L$ und stark monoton mit $m > 0$. - Betrachte $f(x) = b$, $g(x) := x - \theta (f(x) - b)$. + Betrachte $f(x) = b$, $g(x) \coloneqq x - \theta (f(x) - b)$. Frage: Wahl von $\theta$, s.d. $\forall x ^{(0)} \in D$ die Fixpunktiteration konvergiert? Es ist @@ -175,7 +175,7 @@ Frage: Unter welchen Bedingungen konvergiert die Fixpunktiteration? 0 &= (\underbrace{f(x) - b + b -f(x')}_{= 0}, x - x')_2 \\ &= (f(x) - f(x'), x-x')_2 \\ &\stackrel{f\text{ stark monoton}}{\ge} m \Vert x - x' \Vert_2^2 \\ - &> 0 + &\ge 0 .\end{salign*} Also $x = x'$, damit ist $x^{*}$ eindeutig. \end{bem} diff --git a/ana11.pdf b/ana11.pdf index 87eaf13995f85ffe703cbfdc30f8023cd64f6af2..42d71601d358da33c36ff33cf3be97ad68d49d7d 100644 GIT binary patch delta 46310 zcmV($A_Fb5n6jwC1XMeO(zwq>z=ChG?U!*H_igs~9d%)dpewX#Go8(+9+E~yapu5SO;)VnvEU5$68TYo)2 zw`TRBeRpW?+PaP3HEo1 z3nPe8vRZ3L1eI~*H}zq+`SqOJ)m#71!3${F>Z<7;zJBwj*3{#CFoh;u4g)T_0e>=GgA;0p5to$nK})G1P&u3Eh*8Y$jIdD<=5!^9 zX1Rv4wV*(&pr>)v-%5)KqM+Cli>bn5nK=}ZTx5%psPT(l^{)=~p}Bc?&dsWB&e`g) zKVQQ#H6Gzr(*Mzh@BNd1xbvQ5`RbqEl6P&%VnPxvCI-vIa>R@Vfopo7*nhv@HNA@% zzg+N;dN+u<_7h<*LJ@}DtYHtQx?ROBSZu++D1?{Jl%Khw(rLaprw zd|z-{#6EYY;)4=3F7vW5OMiIk2Nb=Y6C)n1*k^^EteI7SHCsW~a6*^qgoz}6;6>C` 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Die Stetigkeitsdefinition von $f$ entspricht der Stetigkeit von $f$ in $\R$. Aber: Differenzierbarkeit einer Funktion $g\colon I \subseteq \R \to \R$, \[ - g'(x_0) := \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} + g'(x_0) \coloneqq \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} .\] Für $h \in \R^{n}$ \underline{nicht sinnvoll}. \section{Partielle Differenzierbarkeit} @@ -19,8 +19,8 @@ der Stetigkeit von $f$ in $\R$. Aber: Differenzierbarkeit einer Funktion $g\colo im Punkt $x \in D$ partiell differenzierbar nach $i$-ter Koordinatenrichtung, falls der Grenwert \[ - \partial_i f(x) := \lim_{h \to 0} \frac{f\left( x + h \cdot e^{(i)} \right) - f(x)}{h} - .\] existiert mit $e^{(i)} := $ $i$-te Spalte der $n \times n$ Einheitsmatrix. + \partial_i f(x) \coloneqq \lim_{h \to 0} \frac{f\left( x + h \cdot e^{(i)} \right) - f(x)}{h} + .\] existiert mit $e^{(i)} \coloneqq $ $i$-te Spalte der $n \times n$ Einheitsmatrix. Schreibweise auch $\frac{\partial f}{\partial x_i}(x)$ oder $\frac{\partial f(x)}{\partial x_i}$. \item $f$ heißt partiell differenzierbar in $x \in D$, falls $\partial_i f(x)$ für alle $1 \le i \le n$ existieren. @@ -44,7 +44,7 @@ der Stetigkeit von $f$ in $\R$. Aber: Differenzierbarkeit einer Funktion $g\colo \begin{bsp} Die Funktion \[ - r(x) := \Vert x \Vert_2 = \sqrt{\sum_{i=1}^{n} x_i^2} + r(x) \coloneqq \Vert x \Vert_2 = \sqrt{\sum_{i=1}^{n} x_i^2} .\] ist in $D = \R^{n} \setminus \{0\} $ stetig partiell differenzierbar mit den partiellen Ableitungen \begin{align*} @@ -71,7 +71,7 @@ der Stetigkeit von $f$ in $\R$. Aber: Differenzierbarkeit einer Funktion $g\colo s.d. die partiellen Ableitungen $\partial_i f(y)$, $i = 1, \ldots, n$ beschränkt sind $\forall y \in K_r(x)$, d.h. \[ - \sup_{x \in K_r(x)} | \partial_i f(y) | \le M, \quad i = 1, \ldots, n + \sup_{y \in K_r(x)} | \partial_i f(y) | \le M, \quad i = 1, \ldots, n .\] Dann gilt $f$ stetig in Punkt $x$. \end{satz} @@ -82,15 +82,15 @@ der Stetigkeit von $f$ in $\R$. Aber: Differenzierbarkeit einer Funktion $g\colo \intertext{Mit $y_2$ fest, folgt mit dem Mittelwertsatz der Differentialrechnung: $\exists \xi = \xi(y_2)$ zwischen $y_1$ und $x_1$, also} f(y_1, y_2) - f(x_1, y_2) &\stackrel{\text{MWS}}{=} \partial_1 f(\xi, y_2) (y_1 - x_1) - \intertext{Analog für $x_1$ fest und $\zeta(x_1)$ zwischen $y_2$ und $x_2$} - f(x_1, y_2) - f(x_1, x_2) &\stackrel{\text{MWS}}{=} \partial_2 f(x_1, \zeta) (y_2 - x_2) + \intertext{Analog für $x_1$ fest und $\eta(x_1)$ zwischen $y_2$ und $x_2$} + f(x_1, y_2) - f(x_1, x_2) &\stackrel{\text{MWS}}{=} \partial_2 f(x_1, \eta) (y_2 - x_2) \intertext{Dann folgt} - |f(y) - f(y)| &\le \underbrace{|\partial_1 f(\xi, y_2)|}_{\le M} |y_1 - x_1| - + \underbrace{|\partial_2 f(x_2, \zeta)|}_{\le M} |y_2 - x_2| \\ + |f(y) - f(x)| &\le \underbrace{|\partial_1 f(\xi, y_2)|}_{\le M} |y_1 - x_1| + + \underbrace{|\partial_2 f(x_2, \eta)|}_{\le M} |y_2 - x_2| \\ &\le M (|y_1 - x_1| + |y_2 - x_2|) \\ &= M \Vert y - x \Vert_1 .\end{salign*} - Sei $\epsilon \in (0, r]$ beliebig, dann gilt für $\delta := \frac{\epsilon}{M}$ + Sei $\epsilon \in (0, r]$ beliebig, dann gilt für $\delta \coloneqq \frac{\epsilon}{M}$ \[ \Vert y -x \Vert_1 \le \delta \implies | f(y) - f(y) | \le \epsilon .\] Also $f$ stetig in $x$ und wegen $|f(y) - f(x)| \le M \Vert y - x \Vert_1$ sogar Lipschitz-stetig. @@ -103,8 +103,8 @@ Für $n > 2$ analog. \item Falls $\partial_i f$ partiell differenzierbar sind, dann heißt $f$ zweimal partiell differenzierbar mit den partiellen Ableitungen zweiter Ordnung \[ - \partial_i \partial_j f(x) := \frac{\partial^2 f(x)}{\partial x_i \partial x_j} - := \frac{\partial}{\partial x_i} \left( \frac{\partial f(x)}{\partial x_j} \right) + \partial_i \partial_j f(x) \coloneqq \frac{\partial^2 f(x)}{\partial x_i \partial x_j} + \coloneqq \frac{\partial}{\partial x_i} \left( \frac{\partial f(x)}{\partial x_j} \right) .\] \item $f$ heißt $k$-mal stetig partiell differenzierbar, falls alle partiellen Ableitungen $k$-ter Ordnung von $f$ existieren und stetig sind. @@ -128,9 +128,9 @@ Für $n > 2$ analog. \begin{enumerate}[1)] \item Sei $n = 2$ und \[ - A := \underbrace{f(x_1 + h_1, x_2 + h_2) - f(x_1 + h_1, x_2)}_{= \varphi(x_1 + h_1)} + A \coloneqq \underbrace{f(x_1 + h_1, x_2 + h_2) - f(x_1 + h_1, x_2)}_{= \varphi(x_1 + h_1)} - \underbrace{f(x_1, x_2 + h_2) + f(x_1, x_2)}_{= \varphi(x_1)} - .\] Definiere $\varphi(x) := f(x, x_2 + h_2) - f(x, x_2)$. Dann ist + .\] Definiere $\varphi(x) \coloneqq f(x, x_2 + h_2) - f(x, x_2)$. Dann ist $A = \varphi(x_1 + h_1) - \varphi(x_1)$. Mit dem MWS bezügl. $x_1$ folgt \begin{salign*} \varphi(x_1 + h_1) - \varphi(x_1) &= \varphi'(x_1 + \theta_1) \cdot h_1, \quad \theta_1 \in (0, h_1) @@ -145,11 +145,11 @@ Für $n > 2$ analog. \begin{salign*} A &= \varphi'(x_1 + \theta_1) h_1 = \partial_2 (\partial_1 f(x_1 + \theta_1, x_2 + \theta_1')) \cdot h_1 \cdot h_2 - \intertext{Analog definiere $\psi(x) := f(x_1 + h_1, x) - f(x_1, x)$, dann} + \intertext{Analog definiere $\psi(x) \coloneqq f(x_1 + h_1, x) - f(x_1, x)$, dann} A &= \psi(x_2 + h_2) - \psi(x_2) \\ &\stackrel{\text{wie oben}}{=} h_2 \psi'(x_2 + \theta_2) \\ - &= h_1 \cdot h_2 \partial_ - (\partial_2 f(x_1 + \partial_2, x_2 + \partial_2')), \quad \theta_2 \in (0, h_1), + &= h_1 \cdot h_2 \partial_1 + (\partial_2 f(x_1 + \theta_2, x_2 + \theta_2')), \quad \theta_2 \in (0, h_1), \theta_2' \in (0, h_2) .\end{salign*} Also folgt @@ -176,22 +176,22 @@ Für $n > 2$ analog. \begin{definition}[Begriffe der Vektoranalysis] \begin{itemize} - \item \underline{Gradient}: Sei $D \subseteq \R^{n}$ offen und $f\colon D \to \R$ eine partiell - differenzierbare Funktion. Der Vektor + \item \underline{Gradient}: Sei $D \subseteq \R^{n}$ offen und + $f \colon D \to \R$ eine partiell differenzierbare Funktion. Der Vektor \[ - \text{grad} f(x) := \nabla f(x) := \begin{pmatrix} \partial_1 f(x) \\ \vdots \\ \partial_n f(x) \end{pmatrix} \in \R^{n} + \text{grad} f(x) \coloneqq \nabla f(x) \coloneqq \begin{pmatrix} \partial_1 f(x) \\ \vdots \\ \partial_n f(x) \end{pmatrix} \in \R^{n} \] heißt der Gradient von $f$ in $x \in D$. \item \underline{Hesse-Matrix}: Sei $D \subseteq \R^{n}$ offen, $f\colon D \to \R$ eine zweimal partiell differenzierbare Funktion. Die Matrix \[ - H_f(x) := \nabla^2 f(x) := (\partial_i \partial_j f(x))_{i,j=1}^{n} \in \R^{n \times n} - .\] heißt Hesse-Matrix von $f$ im Punkt $x \in D$. + H_f(x) \coloneqq \nabla^2 f(x) \coloneqq (\partial_i \partial_j f(x))_{i,j=1}^{n} \in \R^{n \times n} + \] heißt Hesse-Matrix von $f$ im Punkt $x \in D$. \item \underline{Jacobi-Matrix}: Sei $D \subseteq \R^{n}$ offen, $f\colon D \to \R^{m}$ eine partiell differenzierbare Vektorfunktion. Die Matrix \[ - J_f(x) := \begin{pmatrix} \partial_1 f_1 & \ldots &\partial_n f_1 \\ - \vdots & & \vdots \\ - \partial_1 f_m & \ldots & \partial_n f_m + J_f(x) \coloneqq \begin{pmatrix} \partial_1 f_1 & \cdots &\partial_n f_1 \\ + \vdots & \ddots & \vdots \\ + \partial_1 f_m & \cdots & \partial_n f_m \end{pmatrix} \in \R^{m \times n} .\] heißt Funktionalmatrix oder auch Jacobimatrix von $f$ in $x \in D$. @@ -208,7 +208,7 @@ Für $n > 2$ analog. \[ \partial_j \left( \frac{x_i}{r(x)}\right) = \begin{cases} - \frac{r(x) - x_i \frac{x_i}{r(x)}}{r(x)^2} = \frac{1}{r(x)} - \frac{x_i^2}{r(x)^2} & i = j \\ + \frac{r(x) - x_i \frac{x_i}{r(x)}}{r(x)^2} = \frac{1}{r(x)} - \frac{x_i^2}{r(x)^3} & i = j \\ - \frac{x_i \frac{x_j}{r(x)}}{r(x)^2} = - \frac{x_i x_j}{r(x)^{3}} & i \neq j \end{cases} .\] Diese Hesse-Matrix ist die Jacobi-Matrix der Vektorfunktion $v(x) = \frac{x}{r(x)} = \begin{pmatrix} \vdots \\ \frac{x_i}{r(x)} \\ \vdots \end{pmatrix} $. diff --git a/ana7.pdf b/ana7.pdf index 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Dann ist $\bar{A}^TA \in \K^{n\times n}$ hermitesch und positiv semidefinit. 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