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Felix' Änderungen übernommen

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Martin Thoma 2013-09-18 22:58:26 +02:00
parent a9a73319a3
commit 8502acf0d7
12 changed files with 151 additions and 11 deletions
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documents/Numerik/Klausur6/Aufgabe1.tex Normal file
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\section*{Aufgabe 1}

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documents/Numerik/Klausur6/Aufgabe2.tex Normal file
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\section*{Aufgabe 2}

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documents/Numerik/Klausur6/Aufgabe3.tex Normal file
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\section*{Aufgabe 3}
\begin{table}[H]
\begin{tabular}{l|l|l|l|l}
$f_i$ & 8 & 3 & 4 & 8 \\ \hline
$x_i$ & -1 & 0 & 1 & 3 \\
\end{tabular}
\end{table}
\subsection*{Teilaufgabe i}
\begin{align}
p(x) = \sum_{i=0}^3 f_i \cdot L_i(x)
\end{align}
mit
\begin{align}
L_0(x) &= \frac{(x-x_1)(x-x_2)(x-x_3)}{(x_0 - x_1)(x_0-x_2)(x_0-x_3)}
= \ldots = \frac{x^3 - 4x^2 + 3x}{-8} \\
L_1(x) &= \frac{x^3 - 3x^2 - x + 3}{3} \\
L_2(x) &= \frac{x^3 - 2x^2 - 3x}{-4} \\
L_3(x) &= \frac{x^3 - x}{24}
\end{align}
\subsection*{Teilaufgabe ii}
Anordnung der dividierten Differenzen im so genannten Differenzenschema:
\begin{table}[H]
\begin{tabular}{llll}
$f[x_0]=f_0=8$ & ~ & ~ & ~ \\
$f[x_1]= 3$ & $f[x_0,x_1] = \frac{f[x_0] - f[x_1]}{x_0-x_1} = -5$ & ~ & ~ \\
$f[x_2] = 4$ & $1$ & $3$ & ~ \\
$f[x_3] = 8$ & $2$ & $\frac{1}{3}$ & $- \frac{2}{3} $ \\
\end{tabular}
\end{table}
Also:
\begin{align}
p(x) &= f[x_0] + f[x_0,x_1] \cdot (x-x_0) + f[x_0, x_1, x_2] \cdot (x-x_0) \cdot (x-x_1) \\ & + f[x_0, x_1, x_2, x_3] \cdot (x-x_0) \cdot (x-x_1) \cdot (x-x_2) \\
&= 8 - 5 \cdot (x-x_0) + 3 \cdot (x-x_0) \cdot (x-x_1) \\ & - \frac{2}{3} \cdot (x-x_0) \cdot (x-x_1) \cdot (x-x_2)
\end{align}

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documents/Numerik/Klausur6/Aufgabe4.tex Normal file
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\section*{Aufgabe 4}

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documents/Numerik/Klausur6/Aufgabe5.tex Normal file
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\section*{Aufgabe 5}

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documents/Numerik/Klausur6/Klausur6.tex Normal file
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\documentclass[a4paper]{scrartcl}
\usepackage{amssymb, amsmath} % needed for math
\usepackage[utf8]{inputenc} % this is needed for umlauts
\usepackage[ngerman]{babel} % this is needed for umlauts
\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf
\usepackage{pdfpages} % Signatureinbingung und includepdf
\usepackage{geometry} % [margin=2.5cm]layout
\usepackage[pdftex]{hyperref} % links im text
\usepackage{color}
\usepackage{framed}
\usepackage{enumerate} % for advanced numbering of lists
\usepackage{marvosym} % checkedbox
\usepackage{wasysym}
\usepackage{braket} % for \Set{}
\usepackage{pifont}% http://ctan.org/pkg/pifont
\usepackage{gauss}
\usepackage{algorithm,algpseudocode}
\usepackage{parskip}
\usepackage{lastpage}
\allowdisplaybreaks
\newcommand{\cmark}{\ding{51}}%
\newcommand{\xmark}{\ding{55}}%
\title{Numerik Klausur 6 - Musterlösung}
\makeatletter
\AtBeginDocument{
\hypersetup{
pdfauthor = {Martin Thoma, Peter, Felix},
pdfkeywords = {Numerik, KIT, Klausur},
pdftitle = {\@title}
}
\pagestyle{fancy}
\lhead{\@title}
\rhead{Seite \thepage von \pageref{LastPage}}
}
\makeatother
\usepackage{fancyhdr}
\fancyfoot[C]{}
\begin{document}
\input{Aufgabe1}
\input{Aufgabe2}
\input{Aufgabe3}
\input{Aufgabe4}
\input{Aufgabe5}
\end{document}

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SOURCE = Klausur6
make:
pdflatex -shell-escape $(SOURCE).tex -output-format=pdf
pdflatex -shell-escape $(SOURCE).tex -output-format=pdf
make clean
clean:
rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg