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Ein paar Kleinigkeiten geändert. Außerdem ist das ein "Test-PullRequest"
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FelixBenzBaldas
parent
493b7bf49f
commit
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13 changed files with 310 additions and 11 deletions
46
documents/Numerik/Klausur3/Aufgabe1.tex
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46
documents/Numerik/Klausur3/Aufgabe1.tex
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\section*{Aufgabe 1}
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\subsection*{Teilaufgabe a)}
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Erste Spalte:
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\begin{align}
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l_{11} &= \sqrt{a_{11}} \\
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l_{21} &= \frac{a_{21}}{l_{11}}\\
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l_{31} &= \frac{a_{31}}{l_{11}}\\
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\end{align}
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Zweite Spalte:
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\begin{align}
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l_{22} &= \sqrt{a_{22} - {l_{21}}^2}\\
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l_{32} &= (a_{32} -l_{21} \cdot l_{31}) \div l_{22} \\
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\end{align}
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Dritte Spalte:
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\begin{align}
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l_{33} &= \sqrt{a_{33}-{l_{32}^2}-{l_{31}}^2}
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\end{align}
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\subsection*{Teilaufgabe b)}
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\begin{align}
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l_{11} &= 2 \\
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l_{21} &= 1 \\
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l_{31} &= -2 \\
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l_{22} &= 3 \\
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l_{32} &= 1 \\
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l_{33} &= 1 \\
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\end{align}
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Die restlichen Einträge sind $0$. ($L$ ist immer eine untere Dreiecksmatrix)
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\subsection*{Teilaufgabe c)}
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\begin{align}
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A \cdot x = b \Leftrightarrow L \cdot L^T \cdot x = b \\
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L \cdot c = b \label{a1}
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\end{align}
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Löse \ref{a1} mit Vorwärtssubstitution.
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\begin{align}
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L^T \cdot x = c \label{a2}
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\end{align}
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Löse \ref{a2} mit Rückwärtssubstitution.
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\begin{align}
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x_3 &= 3 \\
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x_2 &= 1 \\
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x_1 &= 2
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\end{align}
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46
documents/Numerik/Klausur3/Aufgabe1.tex~
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46
documents/Numerik/Klausur3/Aufgabe1.tex~
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@ -0,0 +1,46 @@
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\section*{Aufgabe 1}
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\subsection*{Teilaufgabe a)}
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Erste Spalte:
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\begin{align}
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l_{11} &= \sqrt{a_{11}} \\
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l_{21} &= \frac{a_{21}}{l_{11}}\\
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l_{31} &= \frac{a_{31}}{l_{11}}\\
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\end{align}
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Zweite Spalte:
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\begin{align}
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l_{22} &= \sqrt{a_{22} - {l_{21}}^2}\\
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l_{32} &= (a_{32} -l_{21} \cdot l_{31}) \div l_{22} \\
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\end{align}
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Dritte Spalte:
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\begin{align}
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l_{33} &= \sqrt{a_{33}-{l_{32}^2}-{l_{31}}^2}
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\end{align}
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\subsection*{Teilaufgabe b)}
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\begin{align}
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l_{11} &= 2 \\
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l_{21} &= 1 \\
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l_{31} &= -2 \\
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l_{22} &= 3 \\
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l_{32} &= 1 \\
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l_{33} &= 1 \\
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\end{align}
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Die restlichen Einträge sind $0$. ($L$ ist immer eine untere Dreiecksmatrix)
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\subsection*{Teilaufgabe c)}
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\begin{align}
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A \cdot x = b \Leftrightarrow L \cdot L^T \cdot x = b \\
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L \cdot c = b \label{a1}
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\end{align}
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Löse \ref{a1} mit Vorwärtssubstitution.
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\begin{align}
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L^T \cdot x = c \label{a2}
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\end{align}
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Löse \ref{a2} mit Rückwärtssubstitution.
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\begin{align}
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x_3 &= 3 \\
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x_2 &= 1 \\
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x_1 &= 2 \\
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\end{align}
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17
documents/Numerik/Klausur3/Aufgabe2.tex
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17
documents/Numerik/Klausur3/Aufgabe2.tex
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\section*{Aufgabe 2}
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\subsection*{Teilaufgabe a)}
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\begin{align}
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r_{ij} = a_{ij} - \sum_{k=1}^{i-1} l_{ik} \cdot r_{kj} \\
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l_{ij} = \frac{a_{ij} - \sum_{k=1}^{j-1} l_{ik} \cdot r_{kj}}{r_{jj}}
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\end{align}
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\begin{algorithm}
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\begin{algorithmic}
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\For{$d \in \Set{1, \dots n}$}
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\State berechne d-te Zeile von R
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\State berechne d-te Spalte von L
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\EndFor
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\end{algorithmic}
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\end{algorithm}
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17
documents/Numerik/Klausur3/Aufgabe2.tex~
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17
documents/Numerik/Klausur3/Aufgabe2.tex~
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\section*{Aufgabe 2}
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\subsection*{Teilaufgabe a)}
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\begin{align}
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r_{ij} = a_{ij} - \sum_{k=1}^{i-1} l_{ik} \cdot r_{kj} \\
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l_{ij} = \frac{a_{ij} - \sum_{k=1}^{j-1} l_{ik} \cdot r_{kj}}{r_{jj}}
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\end{align}
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\begin{algorithm}
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\begin{algorithmic}
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\For{$d \in \Set{1, \dots n}$}
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\State berechne d-te Zeile von R \\
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\State berechne d-te Spalte von L
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\EndFor
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\end{algorithmic}
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\end{algorithm}
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38
documents/Numerik/Klausur3/Klausur3.tex
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38
documents/Numerik/Klausur3/Klausur3.tex
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\documentclass[a4paper]{scrartcl}
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\usepackage{amssymb, amsmath} % needed for math
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\usepackage[utf8]{inputenc} % this is needed for umlauts
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\usepackage[ngerman]{babel} % this is needed for umlauts
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\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
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\usepackage{pdfpages} % Signatureinbingung und includepdf
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\usepackage{geometry} % [margin=2.5cm]layout
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\usepackage[pdftex]{hyperref} % links im text
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\usepackage{color}
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\usepackage{framed}
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\usepackage{enumerate} % for advanced numbering of lists
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\usepackage{marvosym} % checkedbox
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\usepackage{wasysym}
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\usepackage{braket} % for \Set{}
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\usepackage{pifont}% http://ctan.org/pkg/pifont
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\usepackage{gauss}
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\usepackage{algorithm,algpseudocode}
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\usepackage{parskip}
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\usepackage{lastpage}
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\allowdisplaybreaks
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\newcommand{\cmark}{\ding{51}}%
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\newcommand{\xmark}{\ding{55}}%
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\title{Numerik Klausur 3 - Musterlösung}
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\makeatletter
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\makeatother
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\usepackage{fancyhdr}
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\fancyfoot[C]{}
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\begin{document}
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\include{Aufgabe1}
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\include{Aufgabe2}
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\include{Aufgabe3}
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\include{Aufgabe4}
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\include{Aufgabe5}
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\end{document}
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48
documents/Numerik/Klausur3/Klausur3.tex~
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48
documents/Numerik/Klausur3/Klausur3.tex~
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\documentclass[a4paper]{scrartcl}
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\usepackage{amssymb, amsmath} % needed for math
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\usepackage[utf8]{inputenc} % this is needed for umlauts
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\usepackage[ngerman]{babel} % this is needed for umlauts
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\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
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\usepackage{pdfpages} % Signatureinbingung und includepdf
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\usepackage{geometry} % [margin=2.5cm]layout
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\usepackage[pdftex]{hyperref} % links im text
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\usepackage{color}
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\usepackage{framed}
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\usepackage{enumerate} % for advanced numbering of lists
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\usepackage{marvosym} % checkedbox
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\usepackage{wasysym}
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\usepackage{braket} % for \Set{}
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\usepackage{pifont}% http://ctan.org/pkg/pifont
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\usepackage{gauss}
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\usepackage{algorithm,algpseudocode}
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\usepackage{parskip}
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\usepackage{lastpage}
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\allowdisplaybreaks
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\newcommand{\cmark}{\ding{51}}%
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\newcommand{\xmark}{\ding{55}}%
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\title{Numerik Klausur 2 - Musterlösung}
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\makeatletter
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\AtBeginDocument{
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\hypersetup{
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pdfauthor = {Felix Benz-Baldas, Martin Thoma, Peter},
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pdfkeywords = {Numerik, KIT, Klausur},
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pdftitle = {\@title}
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}
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\pagestyle{fancy}
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\lhead{\@title}
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\rhead{Seite \thepage von \pageref{LastPage}}
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}
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\makeatother
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\usepackage{fancyhdr}
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\fancyfoot[C]{}
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\begin{document}
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\include{Aufgabe1}
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\include{Aufgabe2}
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\include{Aufgabe3}
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\include{Aufgabe4}
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\include{Aufgabe5}
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\end{document}
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8
documents/Numerik/Klausur3/Makefile
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8
documents/Numerik/Klausur3/Makefile
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SOURCE = Klausur3
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make:
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pdflatex -shell-escape $(SOURCE).tex -output-format=pdf
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pdflatex -shell-escape $(SOURCE).tex -output-format=pdf
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make clean
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clean:
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rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg
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8
documents/Numerik/Klausur3/Makefile~
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8
documents/Numerik/Klausur3/Makefile~
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SOURCE = Klausur2
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make:
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pdflatex -shell-escape $(SOURCE).tex -output-format=pdf
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pdflatex -shell-escape $(SOURCE).tex -output-format=pdf
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make clean
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clean:
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rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg
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