From 33ba7ff9819368ce9aa4c3d2d8c0ef832607df25 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 10 Sep 2013 05:50:43 +0200 Subject: [PATCH] =?UTF-8?q?Musterl=C3=B6sung=20angefangen?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Numerik/Klausur1/Aufgabe1.tex | 195 ++++++++++++++++++++++++ documents/Numerik/Klausur1/Aufgabe2.tex | 49 ++++++ documents/Numerik/Klausur1/Aufgabe3.tex | 1 + documents/Numerik/Klausur1/Aufgabe4.tex | 1 + documents/Numerik/Klausur1/Aufgabe5.tex | 1 + documents/Numerik/Klausur1/Klausur1.pdf | Bin 0 -> 171383 bytes documents/Numerik/Klausur1/Klausur1.tex | 36 +++++ documents/Numerik/Klausur1/Makefile | 7 + documents/Numerik/Klausur1/Readme.md | 1 + 9 files changed, 291 insertions(+) create mode 100644 documents/Numerik/Klausur1/Aufgabe1.tex create mode 100644 documents/Numerik/Klausur1/Aufgabe2.tex create mode 100644 documents/Numerik/Klausur1/Aufgabe3.tex create mode 100644 documents/Numerik/Klausur1/Aufgabe4.tex create mode 100644 documents/Numerik/Klausur1/Aufgabe5.tex create mode 100644 documents/Numerik/Klausur1/Klausur1.pdf create mode 100644 documents/Numerik/Klausur1/Klausur1.tex create mode 100644 documents/Numerik/Klausur1/Makefile create mode 100644 documents/Numerik/Klausur1/Readme.md diff --git a/documents/Numerik/Klausur1/Aufgabe1.tex b/documents/Numerik/Klausur1/Aufgabe1.tex new file mode 100644 index 0000000..25e5706 --- /dev/null +++ b/documents/Numerik/Klausur1/Aufgabe1.tex @@ -0,0 +1,195 @@ +\section*{Aufgabe 1} +\subsection*{Teilaufgabe a} +\textbf{Gegeben:} + +\[A = +\begin{pmatrix} + 3 & 15 & 13 \\ + 6 & 6 & 6 \\ + 2 & 8 & 19 +\end{pmatrix}\] + +\textbf{Aufgabe:} LR-Zerlegung von $A$ mit Spaltenpivotwahl + +\textbf{Lösung:} + +\[P = +\begin{pmatrix} + 0 & 1 & 0 \\ + 1 & 0 & 0 \\ + 0 & 0 & 1 +\end{pmatrix}\] + +durch scharfes hinsehen. + +Nun $L, R$ berechnen: + +\begin{align} + &\begin{gmatrix}[p] + 6 & 6 & 6 \\ + 3 & 15 & 13 \\ + 2 & 8 & 19 + \rowops + \add[\cdot (-\frac{1}{2})]{0}{1} + \add[\cdot (-\frac{1}{3})]{0}{2} + \end{gmatrix} + \\ + = \begin{pmatrix} + 1 & 0 & 0 \\ + -\frac{1}{2} & 1 & 0 \\ + -\frac{1}{3} & 0 & 1 + \end{pmatrix} \cdot + &\begin{gmatrix}[p] + 6 & 6 & 6 \\ + 0 & 12 & 10 \\ + 0 & 6 & 17 + \rowops + \add[\cdot (-\frac{1}{2})]{1}{2} + \end{gmatrix} + \\ + = \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & -\frac{1}{2} & 1 + \end{pmatrix} \cdot + \begin{pmatrix} + 1 & 0 & 0 \\ + -\frac{1}{2} & 1 & 0 \\ + -\frac{1}{3} & 0 & 1 + \end{pmatrix} \cdot + &\begin{gmatrix}[p] + 6 & 6 & 6 \\ + 0 & 12 & 10 \\ + 0 & 0 & 12 + \colops + \add[\cdot (-1)]{0}{1} + \add[\cdot (-1)]{0}{2} + \end{gmatrix} + \\ + = \begin{pmatrix} + 1 & 0 & 0 \\ + -\frac{1}{2} & 1 & 0 \\ + -\frac{1}{12} & - \frac{1}{2} & 1 + \end{pmatrix} \cdot + &\begin{gmatrix}[p] + 6 & 0 & 0 \\ + 0 & 12 & 10 \\ + 0 & 0 & 12 + \colops + \add[\cdot (-\frac{10}{12})]{1}{2} + \end{gmatrix} + \cdot + \begin{pmatrix} + 1 & -1 & -1 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} + \\ + = \begin{pmatrix} + 1 & 0 & 0 \\ + -\frac{1}{2} & 1 & 0 \\ + -\frac{1}{12} & - \frac{1}{2} & 1 + \end{pmatrix} \cdot + &\begin{gmatrix}[p] + 6 & 0 & 0 \\ + 0 & 12 & 0 \\ + 0 & 0 & 12 + \colops + \mult{0}{\cdot \frac{1}{6}} + \mult{1}{\cdot \frac{1}{12}} + \mult{2}{\cdot \frac{1}{12}} + \end{gmatrix} + \cdot + \begin{pmatrix} + 1 & -1 & -1 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & 1 & -\frac{10}{12} \\ + 0 & 0 & 1 + \end{pmatrix} + \\ + = \begin{pmatrix} + 1 & 0 & 0 \\ + -\frac{1}{2} & 1 & 0 \\ + -\frac{1}{12} & - \frac{1}{2} & 1 + \end{pmatrix} \cdot + &\begin{gmatrix}[p] + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{gmatrix} + \cdot + \begin{pmatrix} + 1 & -1 & -\frac{1}{6} \\ + 0 & 1 & -\frac{5}{6} \\ + 0 & 0 & 1 + \end{pmatrix} + \cdot + \begin{pmatrix} + \frac{1}{6} & 0 & 0 \\ + 0 & \frac{1}{12} & 0 \\ + 0 & 0 & \frac{1}{12} + \end{pmatrix} + \\ + = \underbrace{\begin{pmatrix} + 1 & 0 & 0 \\ + -\frac{1}{2} & 1 & 0 \\ + -\frac{1}{12} & - \frac{1}{2} & 1 + \end{pmatrix}}_L \cdot + &\begin{gmatrix}[p] + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{gmatrix} + \cdot \underbrace{\frac{1}{72} + \begin{pmatrix} + 12 & -6 & -1 \\ + 0 & 6 & -5 \\ + 0 & 0 & 6 + \end{pmatrix}}_R +\end{align} + +ACHTUNG: Ich habe mich irgendwo verrechnet! +Siehe \href{http://www.wolframalpha.com/input/?i=%7B%7B1%2C0%2C0%7D%2C%7B-1%2F2%2C1%2C0%7D%2C%7B-1%2F12%2C-1%2F2%2C1%7D%7D*%7B%7B12%2C-6%2C-1%7D%2C%7B0%2C6%2C-5%7D%2C%7B0%2C0%2C6%7D%7D}{WolframAlpha} + +\subsection*{Teilaufgabe b} + +\textbf{Gegeben:} + +\[A = +\begin{pmatrix} + 9 & 4 & 12 \\ + 4 & 1 & 4 \\ + 12 & 4 & 17 +\end{pmatrix}\] + +\textbf{Aufgabe:} $A$ auf positive Definitheit untersuchen, ohne Eigenwerte zu berechnen. + +\textbf{Lösung:} +Eine Matrix $A \in \mathbb{R}^{n \times n}$ heißt positiv Definit $\dots$ +\begin{align*} + \dots & \Leftrightarrow \forall x \in \mathbb{R}^n: x^T A x > 0\\ + & \Leftrightarrow \text{Alle Eigenwerte sind größer als 0} +\end{align*} + +Falls $A$ symmetrisch ist, gilt: +\begin{align*} + \text{$A$ ist pos. Definit} & \Leftrightarrow \text{alle führenden Hauptminore von $A$ sind positiv}\\ + & \Leftrightarrow \text{es gibt eine Cholesky-Zerlegung $A=GG^T$ mit $G$ ist reguläre untere Dreiecksmatrix}\\ +\end{align*} + +Mit dem Hauptminor-Kriterium gilt: + +\begin{align} + \det(A_1) &= 9 > 0\\ + \det(A_2) &= + \begin{vmatrix} + 9 & 4 \\ + 4 & 1 \\ + \end{vmatrix} = 9 - 16 < 0\\ + &\Rightarrow \text{$A$ ist nicht positiv definit} +\end{align} diff --git a/documents/Numerik/Klausur1/Aufgabe2.tex b/documents/Numerik/Klausur1/Aufgabe2.tex new file mode 100644 index 0000000..bf71571 --- /dev/null +++ b/documents/Numerik/Klausur1/Aufgabe2.tex @@ -0,0 +1,49 @@ +\section*{Aufgabe 2} +\subsection*{Teilaufgabe a} +\textbf{Aufgabe} +Formulieren Sie einen Algorithmus in Pseudocode zum Lösen des Gleichungssystems +\[Ly = b,\] +wobei $L$ eine invertierbare, untere Dreiecksmatrix ist. + +Geben Sie die Formel zur Berechnung von $y_i$ an. + +\textbf{Lösung:} TODO! %TODO! + + \begin{algorithm}[H] + \begin{algorithmic} + \Require $p \in \mathbb{P}, a \in \mathbb{Z}, p \geq 3$ + \Procedure{CalculateLegendre}{$a$, $p$} + \If{$a \geq p$ or $a < 0$}\Comment{rule (III)} + \State \Return $\Call{CalculateLegendre}{a \mod p, p}$ \Comment{now: $a \in [0, \dots, p-1]$} + \ElsIf{$a == 0$ or $a == 1$} + \State \Return $a$ \Comment{now: $a \in [2, \dots, p-1]$} + \ElsIf{$a == 2$} \Comment{rule (VII)} + \If{$p \equiv \pm 1 \mod 8$} + \State \Return 1 + \Else + \State \Return -1 + \EndIf \Comment{now: $a \in [3, \dots, p-1]$} + \ElsIf{$a == p-1$} \Comment{rule (VI)} + \If{$p \equiv 1 \mod 4$} + \State \Return 1 + \Else + \State \Return -1 + \EndIf \Comment{now: $a \in [3, \dots, p-2]$} + \ElsIf{!$\Call{isPrime}{a}$} \Comment{rule (II)} + \State $p_1, p_2, \dots, p_n \gets \Call{Factorize}{a}$ + \State \Return $\prod_{i=1}^n \Call{CalculateLegendre}{p_i, p}$ + \Else \Comment{now: $a \in \mathbb{P}, \sqrt{p-2} \geq a \geq 3$} + \If{$\frac{p-1}{2} \equiv 0 \mod 2$ or $\frac{a-1}{2} \equiv 0 \mod 2$} + \State \Return $\Call{CalculateLegendre}{p, a}$ + \Else + \State \Return $(-1) \cdot \Call{CalculateLegendre}{p, a}$ + \EndIf + \EndIf + \EndProcedure + \end{algorithmic} + \caption{Calculate Legendre symbol} + \label{alg:calculateLegendreSymbol} + \end{algorithm} + +\subsection*{Teilaufgabe b} +\subsection*{Teilaufgabe c} diff --git a/documents/Numerik/Klausur1/Aufgabe3.tex b/documents/Numerik/Klausur1/Aufgabe3.tex new file mode 100644 index 0000000..3d0108e --- /dev/null +++ b/documents/Numerik/Klausur1/Aufgabe3.tex @@ -0,0 +1 @@ +\section*{Aufgabe 3} diff --git a/documents/Numerik/Klausur1/Aufgabe4.tex b/documents/Numerik/Klausur1/Aufgabe4.tex new file mode 100644 index 0000000..4afac90 --- /dev/null +++ b/documents/Numerik/Klausur1/Aufgabe4.tex @@ -0,0 +1 @@ +\section*{Aufgabe 4} diff --git a/documents/Numerik/Klausur1/Aufgabe5.tex b/documents/Numerik/Klausur1/Aufgabe5.tex new file mode 100644 index 0000000..701ee58 --- /dev/null +++ b/documents/Numerik/Klausur1/Aufgabe5.tex 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