From 24f205189d9fc4e2f308c0d2c734d61dc0613a78 Mon Sep 17 00:00:00 2001 From: NeevoLima Date: Thu, 19 Sep 2013 15:32:23 +0200 Subject: [PATCH 1/2] =?UTF-8?q?Aufgabe=202=20f=C3=BCr=20Klausur=202=20Must?= =?UTF-8?q?erl=C3=B6sung=20geschrieben?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Habe die Musterlösung zu Aufgabe 2 Klausur 2 verfasst. Bisher war dort nur ein Copy&Paste Inhalt der Aufgabe 2 der Klausur 1. --- documents/Numerik/Klausur2/Aufgabe2.tex | 59 +++++++++--------------- documents/Numerik/Klausur2/Klausur2.pdf | Bin 205975 -> 155540 bytes 2 files changed, 23 insertions(+), 36 deletions(-) diff --git a/documents/Numerik/Klausur2/Aufgabe2.tex b/documents/Numerik/Klausur2/Aufgabe2.tex index 3e0825b..15e5bf4 100644 --- a/documents/Numerik/Klausur2/Aufgabe2.tex +++ b/documents/Numerik/Klausur2/Aufgabe2.tex @@ -1,43 +1,30 @@ \section*{Aufgabe 2} \subsection*{Teilaufgabe a)} -Formel: $y_i = \frac{b_i - \sum_{j=1}^{i-1} y_j \cdot l_{ij}}{l_{ii}} $ -Anmerkung: $l_{ii}$ kann nicht $0$ sein, da L dann nicht mehr invertierbar wäre. +\textbf{Behauptung:} Für $x \in \mathbb{R}$ gilt, dass $cos(x_k) = x_{k+1}$ gegen den einzigen Fixpunkt $x^{*} = cos(x^{*})$ konvergiert. -Algorithmus: +\textbf{Beweis:} +Sei $ D := [-1, 1]$.\\ +Trivial: $D$ ist abgeschlossen. -\begin{algorithm}[H] - \begin{algorithmic} - \For{$i=1$ to $i=n$} - \State $sum \gets 0$ - \For{$j = 1$ to $j = i-1$} - \State $sum \gets sum + y_i \cdot l_{ij}$ - \EndFor - \State $y_i \gets \frac{b_i - sum}{l_{ii}}$ - \EndFor - \end{algorithmic} -\caption{TODO} -\end{algorithm} +Sei $ x \in D$, so gilt: +\begin{align*} + 0 < cos(x) \leq 1 +\end{align*} +Also: $cos(x) \in D$.\\ Wenn $x \not\in D$, so gilt $y := cos(x)$ und $cos(y) \in D$. D.h. bereits nach einem Iterationschritt wäre $cos(x) \in D$ für $x \in \mathbb{R}$! Dies ist wichtig, da damit gezeigt ist, dass $cos(x_k) = x_{k+1}$ für jedes $x \in \mathbb{R}$ konvergiert! Es kommt nur dieser einzige Iteratationsschritt für $x \not\in \mathbb{R}$ hinzu. -\subsubsection*{(b)} -\begin{algorithm}[H] - \begin{algorithmic} - \Require Matrix $A$, Vektor $b$ - \Procedure{LoeseLGS}{$A$, $b$} - \State $P, L, R \gets \Call{LRZer}{A}$ - \State $b^* \gets P \cdot b$ - \State $c \gets \Call{VorSub}{L, b^*}$ - \State $x \gets \Call{RueckSub}{R, c}$ - \State \Return $x$ - \EndProcedure - \end{algorithmic} -\caption{Löse ein LGS $Ax = b$} -\end{algorithm} +Nun gilt mit $ x, y \in [0,1], x < y, \xi \in (x,y) $ und dem Mittelwert der Differentialrechnung: +\begin{align*} + \frac{cos(x) - cos(y)}{x - y} = cos'(\xi) \\ + \Leftrightarrow cos(x) - cos(y) = cos'(\xi) * (x - y) \\ + \Leftrightarrow | cos(x) - cos(y) | = | cos'(\xi) * (x - y) | \leq | cos'(\xi) | * | (x - y) | +\end{align*} +Da $ \xi \in (0, 1) $ gilt: +\begin{align*} + 0 \leq cos'(\xi) < 1 +\end{align*} +Damit ist gezeigt, dass $cos(x) : D \rightarrow D$ Kontraktion auf $D$. -\subsection*{Teilaufgabe c)} -Aufwand: -\begin{itemize} -\item Vorwärts-/Rückwärtssubstitution: jeweils $\frac{1}{2} \cdot n^2$ -\item LR-Zerlegung: $\frac{1}{3}n^3$ -\item gesamt: $\frac{1}{3}n^3+n^2$ -\end{itemize} +Damit sind alle Voraussetzung des Banachschen Fixpunktsatzes erfüllt. + +Nach dem Banachschen Fixpunktsatz folgt die Aussage. \ No newline at end of file diff --git a/documents/Numerik/Klausur2/Klausur2.pdf b/documents/Numerik/Klausur2/Klausur2.pdf index f4f5d2e60e612beab3867d446e018c526544e65a..88a66d04c4b1bc3c0e3cb0e9a87d8863486767b6 100644 GIT binary patch delta 108571 zcmZU(Q+J>Z(=-~}wr$(CZBA@WcxB>=ZQIGj6Wg}UiS1^m5CFMa&~cO6^=6j1p|zmH>nSwCP4`aK*%Gr$(JGIfsLmMJ;xCooEB|4 zHSE)KsFGHDS@C+Al}fbnGJGIYJ9N(7nT+RIqu}xG@p1d>^B8*w+b3hLm^jfPVI#q* zXu*VnS+7Q=edaVU-L&+{XC7VLvph75So2vx`)HT({m$XuWpi$qdSaL8Dc-1SC4sj3 zYG1hx{M5U1P_%4!S~g}D?(`X1)8H8)u#pqtrhb03lW+3z{rhHTiR~g5a0*?Ptl;QX zRaM7BYkDmngt|cYdPU;}OU*mpET~PLr`=QM%;Lk#M;)lrk9xqH!}rTpCC1xQDZE3` z1)OVkQLj)(F56Syvp0$=goH8VAT%i&mBQKtaQnL~z=_GjT2CIEgJI(mGn7qPkEZ?% 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zvXgO=vHx?bIykugbF!1Mk*OlE%Gg^t{IfXzV@Z(d3W>A-Q&D2ve3CruJe=a9e4-K@ zY@(u)JUpD-66~T9JRq|Fe+crwi~ggMxHvh|(RCo{0qR*nr8yO};cc&}`kz}hbO?-) zkQ8tXCp*Y!k~L!tWRM1;or797Rn_4k3;knDnHkdwTB}#gOfiLpJaq-){^GZL!afH8 zi{Qw(9d9Oe@*4xbOhO>ESLXb%#1M&pPD-e37;`a$D(9RSF;>Q}#5hr5qR;_(b#Y{( ztrpB;0G$sGSvkS{(8Lf&Aw0k#cTciliT;6dyxxJr3zLh(?Dyb7q-pUV<$trc!(%EK zygSi@*s5!ED02wlX%yN*C`D0PS5eeE@s=uk;cp-)%Wa~&ey>&t+(KwSM|hS2>{f(D zBQ*QofU0AkDqrHTEgYl)uCskDYH^NSn*kNI`!d09e3ig`LP zdsi&hRN}B;tOvm**M@cHAJ5Wj#ogS+z4cy9NAHWfwd1x&1~uOvkI)D33o2Bl95~(9 W5RwR)gN>Vmj~9WOT2e_0;eP<*G&E@d From 2b98fb8039f032a89d9b59f994c0fb4ce3c83c90 Mon Sep 17 00:00:00 2001 From: NeevoLima Date: Thu, 19 Sep 2013 15:37:51 +0200 Subject: [PATCH 2/2] Fehler in Aufgabe 2 Klausur 2 korrigiere MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Die Kontraktionskonstante war falsch und die Variablen für den MWS sind nicht aus dem Intervall D gekommen. --- documents/Numerik/Klausur2/Aufgabe2.tex | 4 ++-- documents/Numerik/Klausur2/Klausur2.pdf | Bin 155540 -> 155804 bytes 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/documents/Numerik/Klausur2/Aufgabe2.tex b/documents/Numerik/Klausur2/Aufgabe2.tex index 15e5bf4..49389b0 100644 --- a/documents/Numerik/Klausur2/Aufgabe2.tex +++ b/documents/Numerik/Klausur2/Aufgabe2.tex @@ -13,7 +13,7 @@ Sei $ x \in D$, so gilt: \end{align*} Also: $cos(x) \in D$.\\ Wenn $x \not\in D$, so gilt $y := cos(x)$ und $cos(y) \in D$. D.h. bereits nach einem Iterationschritt wäre $cos(x) \in D$ für $x \in \mathbb{R}$! Dies ist wichtig, da damit gezeigt ist, dass $cos(x_k) = x_{k+1}$ für jedes $x \in \mathbb{R}$ konvergiert! Es kommt nur dieser einzige Iteratationsschritt für $x \not\in \mathbb{R}$ hinzu. -Nun gilt mit $ x, y \in [0,1], x < y, \xi \in (x,y) $ und dem Mittelwert der Differentialrechnung: +Nun gilt mit $ x, y \in D, x < y, \xi \in (x,y) $ und dem Mittelwert der Differentialrechnung: \begin{align*} \frac{cos(x) - cos(y)}{x - y} = cos'(\xi) \\ \Leftrightarrow cos(x) - cos(y) = cos'(\xi) * (x - y) \\ @@ -21,7 +21,7 @@ Nun gilt mit $ x, y \in [0,1], x < y, \xi \in (x,y) $ und dem Mittelwert der Dif \end{align*} Da $ \xi \in (0, 1) $ gilt: \begin{align*} - 0 \leq cos'(\xi) < 1 + 0 \leq | cos'(\xi) | = | sin(\xi) | < 1 \end{align*} Damit ist gezeigt, dass $cos(x) : D \rightarrow D$ Kontraktion auf $D$. diff --git a/documents/Numerik/Klausur2/Klausur2.pdf b/documents/Numerik/Klausur2/Klausur2.pdf index 88a66d04c4b1bc3c0e3cb0e9a87d8863486767b6..c22c76d1fe302c860808925ed43642a76192a076 100644 GIT binary patch delta 46641 zcmZU%Q*fY7+l3q3wmr$jw(W^++k9dt6WhkbnAo;$V`6*1|K7EazJspngFd+HuIhEI 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