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## Core latex/pdflatex auxiliary files:
*.aux
*.lof
*.log
*.lot
*.fls
*.out
*.toc
*.fmt
*.fot
*.cb
*.cb2
.*.lb
## Intermediate documents:
*.dvi
*.xdv
*-converted-to.*
# these rules might exclude image files for figures etc.
*.ps
*.eps
*.pdf
## Generated if empty string is given at "Please type another file name for output:"
.pdf
## Bibliography auxiliary files (bibtex/biblatex/biber):
*.bbl
*.bcf
*.blg
*-blx.aux
*-blx.bib
*.run.xml
## Build tool auxiliary files:
*.fdb_latexmk
*.synctex
*.synctex(busy)
*.synctex.gz
*.synctex.gz(busy)
*.pdfsync
## Build tool directories for auxiliary files
# latexrun
latex.out/
## Auxiliary and intermediate files from other packages:
# algorithms
*.alg
*.loa
# achemso
acs-*.bib
# amsthm
*.thm
# beamer
*.nav
*.pre
*.snm
*.vrb
# changes
*.soc
# comment
*.cut
# cprotect
*.cpt
# elsarticle (documentclass of Elsevier journals)
*.spl
# endnotes
*.ent
# fixme
*.lox
# feynmf/feynmp
*.mf
*.mp
*.t[1-9]
*.t[1-9][0-9]
*.tfm
#(r)(e)ledmac/(r)(e)ledpar
*.end
*.?end
*.[1-9]
*.[1-9][0-9]
*.[1-9][0-9][0-9]
*.[1-9]R
*.[1-9][0-9]R
*.[1-9][0-9][0-9]R
*.eledsec[1-9]
*.eledsec[1-9]R
*.eledsec[1-9][0-9]
*.eledsec[1-9][0-9]R
*.eledsec[1-9][0-9][0-9]
*.eledsec[1-9][0-9][0-9]R
# glossaries
*.acn
*.acr
*.glg
*.glo
*.gls
*.glsdefs
*.lzo
*.lzs
*.slg
*.slo
*.sls
# uncomment this for glossaries-extra (will ignore makeindex's style files!)
# *.ist
# gnuplot
*.gnuplot
*.table
# gnuplottex
*-gnuplottex-*
# gregoriotex
*.gaux
*.glog
*.gtex
# htlatex
*.4ct
*.4tc
*.idv
*.lg
*.trc
*.xref
# hyperref
*.brf
# knitr
*-concordance.tex
# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
# *.tikz
*-tikzDictionary
# listings
*.lol
# luatexja-ruby
*.ltjruby
# makeidx
*.idx
*.ilg
*.ind
# minitoc
*.maf
*.mlf
*.mlt
*.mtc[0-9]*
*.slf[0-9]*
*.slt[0-9]*
*.stc[0-9]*
# minted
_minted*
*.pyg
# morewrites
*.mw
# newpax
*.newpax
# nomencl
*.nlg
*.nlo
*.nls
# pax
*.pax
# pdfpcnotes
*.pdfpc
# sagetex
*.sagetex.sage
*.sagetex.py
*.sagetex.scmd
# scrwfile
*.wrt
# svg
svg-inkscape/
# sympy
*.sout
*.sympy
sympy-plots-for-*.tex/
# pdfcomment
*.upa
*.upb
# pythontex
*.pytxcode
pythontex-files-*/
# tcolorbox
*.listing
# thmtools
*.loe
# TikZ & PGF
*.dpth
*.md5
*.auxlock
# titletoc
*.ptc
# todonotes
*.tdo
# vhistory
*.hst
*.ver
# easy-todo
*.lod
# xcolor
*.xcp
# xmpincl
*.xmpi
# xindy
*.xdy
# xypic precompiled matrices and outlines
*.xyc
*.xyd
# endfloat
*.ttt
*.fff
# Latexian
TSWLatexianTemp*
## Editors:
# WinEdt
*.bak
*.sav
# Texpad
.texpadtmp
# LyX
*.lyx~
# Kile
*.backup
# gummi
.*.swp
# KBibTeX
*~[0-9]*
# TeXnicCenter
*.tps
# auto folder when using emacs and auctex
./auto/*
*.el
# expex forward references with \gathertags
*-tags.tex
# standalone packages
*.sta
# Makeindex log files
*.lpz
# xwatermark package
*.xwm
# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
# Uncomment the next line to have this generated file ignored.
#*Notes.bib

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DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
Version 2, December 2004
Copyright (C) 2004 Sam Hocevar <sam@hocevar.net>
Everyone is permitted to copy and distribute verbatim or modified
copies of this license document, and changing it is allowed as long
as the name is changed.
DO WHAT THE FUCK YOU WANT TO PUBLIC LICENSE
TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION
0. You just DO WHAT THE FUCK YOU WANT TO.

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MAIN = sheet
FLAGS = -pdf -lualatex
all:
latexmk $(FLAGS) $(MAIN)
dev:
latexmk $(FLAGS) -pvc $(MAIN)
clean:
latexmk -C

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\documentclass[11pt, a4paper, twoside]{article}
\usepackage[
a4paper,
headsep=5mm,
footskip=0mm,
top=12mm,
left=10mm,
right=10mm,
bottom=10mm
]{geometry}
\usepackage{amsmath}
\usepackage{gauss}
\usepackage{nicematrix}
\usepackage{tikz}
\usepackage{amsfonts}
\usepackage{makecell}
\usepackage{multicol}
\usepackage[noend]{algorithm2e}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{tikz}
\usetikzlibrary{arrows,automata,positioning, graphs, graphdrawing}
\usegdlibrary {trees}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
linkcolor=blue,
filecolor=magenta,
urlcolor=cyan,
pdftitle={Overleaf Example},
pdfpagemode=FullScreen,
}
\setlength{\algomargin}{0pt}
\begin{document}
\pagestyle{fancy}
\fancyhead{}
\fancyhead[L]{Numerische Mathematik für die Fachrichtungen Informatik}
\fancyhead[R]{Gero Beckmann - \url{https://github.com/Geronymos/}}
\fancyfoot{}
\fancyfoot[R]{\thepage}
\newenvironment{definition}[1]{\noindent\textbf{#1:}}{}
\section{Computergenauigkeit}
\[
FL = \{ +- B^e \Sigma_{l=1}^{l_m} a_l B^{-l} : e = e_{min} +
\Sigma_{l=0}^{L_e-1} c_l B^l, a_l, c_l \in \{0, ..., B-1 \}, a \neq 0 \} \cup
\{ 0 \} \subset \mathbb{Q} \\
\]
\begin{multicols}{2}
\section{Normen und Kondition}
\begin{align*}
\|A\|_1 &= \max_{n=0,...,N} \Sigma_{m=0}^{N} |a_{mn}| & \text{Spaltennorm} \\
\|A\|_2 &= \sqrt {\max \lambda \text{ von } A^T A} & \text{Spektralnorm} \\
\|A\|_\infty &= \max_{m=0,...,N} \Sigma_{n=0}^{N} |a_{mn}| & \text{Zeilennorm} \\
\end{align*}
\subsection{Kondition}
\begin{align*}
\kappa(A) &= \|A\|\|A^{-1}\| \\
\kappa(A) &= \frac{\max_{\|y\|=1} \|A_y\|}{\min_{\|z\|=1} \|Az\|} \\
\kappa_2(A^TA) &= \kappa_2(A)^2 = \sqrt{\frac{\max \lambda \text{ von } A^TA}{\min \lambda}}
\end{align*}
\end{multicols}
\begin{multicols}{2}
\section{Cholesky-Zerlegung}
\begin{enumerate}
\item Berechne $A=LL^T$
\item Löse durch Vorwärtssubstitution $Ly = b$
\item Löse durch rückwärtssubstitution $L^T = y$
\end{enumerate}
\begin{align*}
Ax &= b \\
A &= \begin{pmatrix}
l_{11} & & \\
l_{21} & l_{22} & \\
l_{31} & l_{32} & l_{33}
\end{pmatrix} \begin{pmatrix}
l_{11} & l_{21} & l_{31} \\
& l_{22} & l_{32} \\
& & l_{33}
\end{pmatrix}
\end{align*}
\end{multicols}
\hspace{-.6cm}
\begin{minipage}{.42\textwidth}
\section{LR-Zerlegung}
\begin{enumerate}
\item Berechne Zerlegung $A = CR$
\item Löse $Ly = b$ durch Vorwaärtssubstitution
\item Löse $Rx =y$ durch Rückwärtssubstitution
\end{enumerate}
\end{minipage}
\hspace{-2cm}
\begin{minipage}{.6\textwidth}
\hspace{-10cm}
\begin{align*}
\begin{gmatrix}[p]
1 & 4 & -1 \\
3 & 0 & 5 \\
2 & 2 & 1
\rowops
\add[-3]{0}{1}
\add[-2]{0}{2}
\end{gmatrix} \leadsto \begin{pNiceMatrix}
1 & 4 & -1 \\
3 & -12 & 8 \\
2 & -6 & 3
\CodeAfter
\tikz \draw (2-|1) -| (4-|2);
\end{pNiceMatrix} \begin{gmatrix}
\\ \\
\rowops
\add[\frac{1}{-2}]{1}{2}
\end{gmatrix} \leadsto \begin{pNiceMatrix}
1 & 4 & -1 \\
3 & -12 & 8 \\
2 & \frac 1 2 & -1
\CodeAfter
\tikz \draw (2-|1) -| (3-|2) -| (4-|3);
\end{pNiceMatrix} \\
\Rightarrow L = \begin{pmatrix}
1 & 0& 0 \\
3 & 1 & 0 \\
2 & \frac 1 2 & 1
\end{pmatrix}, R = \begin{pmatrix}
1 & 4 & -1 \\
0 & -12 & 8 \\
0 & 0 & -1
\end{pmatrix}
\end{align*}
\end{minipage}
\subsection{Mit Pivotwahl / Permutationsmatrix $PA = LR$}
\begin{enumerate}
\item Berechne Zerlegung $PA = LR$ durch Gauß-Elimitation
\item Löse $Ly = Pb$ durch Vorwärtssubstition
\item Löse $Rx = y$ durch Rückwärtssubstitution
\end{enumerate}
\def\rowswapfromlabel#1{#1}
\def\rowswaptolabel#1{#1}
\def\colswapfromlabel#1{#1}
\def\colswaptolabel#1{#1}
\begin{align*}
\begin{pmatrix}
1 \\ 2 \\ 3
\end{pmatrix}
\begin{gmatrix}[p]
1 & 2 & 2 \\
-2 & -2 & 4 \\
2 & 4 & 2
\rowops
\swap[|-2| > |1|][]01
\end{gmatrix} \leadsto
\begin{pmatrix}
2 \\ 1 \\ 3
\end{pmatrix}
\begin{gmatrix}[p]
-2 & -2 & 4 \\
1 & 2 & 2 \\
2 & 4 & 2
\rowops
\add[\frac 1 2 ]01
\add[1]02
\end{gmatrix} \leadsto
\begin{pmatrix}
2 \\ 1 \\ 3
\end{pmatrix}
\begin{pNiceMatrix}
-2 & -2 & 4 \\
-\frac 1 2 & 1 & 4 \\
-1 & 2 & 6
\CodeAfter
\tikz \draw (2-|1) -| (4-|2);
\end{pNiceMatrix}
\begin{gmatrix}
\\ \\
\rowops
\swap[|2| > |1|]12
\end{gmatrix} \\ \leadsto
\begin{pmatrix}
2 \\ 3 \\ 1
\end{pmatrix}
\begin{pNiceMatrix}
-2 & -2 & 4 \\
-1 & 2 & 6 \\
-\frac 1 2 & 1 & 4
\CodeAfter
\tikz \draw (2-|1) -| (4-|2);
\end{pNiceMatrix}
\begin{gmatrix}
\\ \\
\rowops
\add[-\frac 1 2]12
\end{gmatrix} \leadsto
\begin{pmatrix}
2 \\ 3 \\ 1
\end{pmatrix}
\begin{pNiceMatrix}
-2 & -2 & 4 \\
-1 & 2 & 6 \\
-\frac 1 2 & \frac 1 2 & 1
\CodeAfter
\tikz \draw (2-|1) -| (3-|2) -| (4-|3);
\end{pNiceMatrix} \Rightarrow
L = \begin{pmatrix}
1 & 0 & 0 \\
-1 & 1 & 0 \\
-\frac12 & \frac12 &1
\end{pmatrix},
R = \begin{pmatrix}
-2 & -2 & 4 \\
0 & 2 & 6 \\
0 & 0 & 1
\end{pmatrix},
P = \begin{pmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 0 & 0
\end{pmatrix}
\end{align*}
Für Eliminierung in Spalte n werden Zeilen so getauscht, dass in der n-ten
Spaten ab dre n-ten Zeile, sodass das Betraglich größte Element in Zeile n
steht.
\newpage
\begin{multicols}{2}
\section{QR-Zerlegung $A = QR$}
\begin{enumerate}
\item Bestimme Matrizen Q und R durch Householder-Transformationen
\item Löse $Qx = b$ ($Q^{-1} = Q^T$, also $c = Q^Tb$)
\item Löse $Rx = c$ durch Rückwärtssubstitution
\end{enumerate}
\begin{enumerate}
\item Bestimme Teilmatrix $A'^{(j-1)}$
\item Berechne $v^{(j)} = {a'}_{I}^{(j-1)} + sign({a'}_{II}^{(j-1)}) \cdot \| {a'}_I^{(j-1)} \| e_I$
\item Berechne $H'^{(j-1)} = I - \frac {2v^{(j)}v^{(j)T}} {v^{(j)T}v^{(j)}}$
\item Bestime $H^{(j)} = \begin{pmatrix} 1 & 0 \\ 0 & H'^{(j-1)}\end{pmatrix}$
\item Berechne $A^{(j)} = H^{(j)}A^{(j-1)}$ bis $A^{(j)} = R$
\end{enumerate}
\begin{align*}
j = 1 \rightarrow j = k = min(m-1, n) \\
Q^T = H^{(k)} \cdot ... \cdot H^{(2)} H^{(1)}
\end{align*}
\subsection{Minimale Fehlerquote}
\[
|y_i - f(x_i)|_2^2 = \Sigma_{i=1}^{N} (y_i - f(x_i))^2
\]
\subsection{Ausgleichssystem}
Der Vektor $x \in \mathbb{R}^N$ löst genau dann $\|Ax -b \|_2 = min!$, falls er
$A^TAx = A^Tb$ (Normalgleichung) löst.
\columnbreak
\section{Singilärwertzerlegung}
\begin{enumerate}
\item Rechne $S = A^TA$
\item Berechne EW und EV von S
\item Bilde ONB $u_1, u_2, ..., u_N$ aus EV von S
\item Berechne $\sigma_k = \sqrt{\lambda_k}$
\item $U = \begin{pNiceArray}{c|c|c} U_1 & ... & U_k \end{pNiceArray} =
diag(\sqrt{\lambda_1}, ..., \sqrt{\lambda_k}) =
diag(\sigma_1, ..., \sigma_k) = \Sigma$
\item $V = A U \Sigma^{-1}$
\end{enumerate}
\subsection{Pseudoinverse }
$A^+ = U \Sigma^{-1} V^T$ ; ist A regulär dann gilt $A^{-1} = A^+$
\subsection{Normalengleichung}
$|Ax-b|_2=Min!$ durch $x = A^+b$ gelöst
\end{multicols}
\section{Hessenbergform (rechte-obere Dreiecksmatrix ab der unterren Nebendiagonale)}
\subsection{Tridiagonal (Nur Haupt- und Nebendiagonale)}
\begin{align*}
\text{TeilmatrixA }&{A'}^{(j-1)} \\
w^{(j)} &= {a'}_{I}^{(j-1)} + sign({a'}_{Ii}^{(j-1)}) \cdot \|{a'}_{I}^{(j-1)}\|_2 \cdot e_I \\
{Q'}^{(j-1)} &= I - \frac {2 w^{j} w^{(j)T}} {w^{(j)T} w^{(j)}} \\
Q^{(j)} &= \begin{pmatrix} 1 & 0 \\ 0 && {Q'}^{(j-1)} \end{pmatrix} \\
H^{(j)} &= Q^{T(j)} A^{(j-1)} Q^{(j)}
\end{align*}
\subsection{Jacobi-Verfahren (Lösung von Ax =b) / Gesamtschrittverfahren}
\begin{align*}
x_m^{k+1} &= \frac 1 {A[m;m]} (b_m - \Sigma_{n \neq m} A[m,n] x_n^k) &\text{für $m=1, ..., M$} \\
x^{k+1} &= x^k + D^{-1} (b - Ax^k) & A = D + (L + U) \\
& &\text{(diagonal + (strikte linke untere / rechte obere))}
\end{align*}
\subsection{Gauß-Seidel-verfahren / Einzelschrittverfahren}
\begin{align*}
x_m^{k+1} &= \frac 1 {A[m;m]} (b_m - \Sigma_{n=1}^{m-1} A[m,n] x_n^{k+1} - \Sigma_{k=m+1}^{N} A[m,n] x_n^k) \\
x^{k+1} &= x^k + (D + L)^{-1} (b - Ax^k)
\end{align*}
\subsection{CG-Verfahren}
\begin{align*}
a
\end{align*}
\subsection{GMRES}
\begin{align*}
a
\end{align*}
Energienorm $\|x\|_A = \sqrt{x^TAx}$
SKP $<x,y> = x^TAy$
\subsection{Krylov-Raum}
\section{Spline Interpolation}
\begin{align*}
& s'(a) = v_0 \text{ und } s'(b) = v_N & \text{hermitisch} \\
& s''(a) = s''(b) = 0 & \text{natürlich} \\
& s'(a) = s'(b) \text{ und } s''(a) = s''(b) & \text{periodisch}
\end{align*}
\section{Newton-Verfahren}
\[
x^{n+1} = x^n - \frac {f(x^n)} {f'(x^n)}
\]
\section{Quadraturformel}
Gewichte $b_k \in [0,1]$, Knoten $c_k \in [0,1]$, Stützstelle $a + c_k (b-a)$
\[
\int_a^b f(x)dx \approx (b - a) \Sigma_{k=1}^s b_k f(a+c_k (b-a))
\]
\begin{tabular}{llll}
Rechteckregel & $s=1$ & $b_1=1$ & $c_1=0$ \\
Mittelpunktregel & $s=1$ & $b_1=1$ & $c_1 = \frac12$ \\
Trapezregel & $s=2$ & $b_1 = b_2 = \frac12$ & $c_1 = 0, c_2 = 1$ \\
Simpsonregel & $s=3$ & $b_1 = b_3 = \frac16, b_2 = \frac46$ & $c_1 = 0, c_2 = \frac12, c_3 = 1$
\end{tabular}
Symmetrische Quadraturformel $c_k = 1 - c_{s+1-k}$, $b_k = b_{s+1-k}$
Ordung $p$ $\frac1q = \Sigma_{k=1}^S b_k c_k^{q-1}$ für alle $q=1, .., p$ nicht für $q = p+1$!
\section{Polynom-Interpolation}
\subsection{Lagrange}
\begin{align*}
& p(x) = \Sigma_{n=0}^N f_n L_n(x) &
L_n(x) = \Pi_{j=0, j \neq n}^N \frac{x - x_j}{x_n - x_j}
\end{align*}
Lebesque-Konstante
\[
\Lambda_N := \max_{x \in [a,b]} \Sigma_{n=0}^{N} |L_n(x)|
\]
\subsection{Newton-Darstellung}
\begin{tabular}{c|c|c|c|c}
$f_n$ & 1 & 6 & -3 & 3 \\
\hline
$x_n$ & -1 & 0 & 1 & 3
\end{tabular}
\[
\begin{NiceArray}{c|cccc}
x_0 = -1 & f_0 = 1 & & & \\
x_1 = 0 & f_1 = 6 & \frac{1-6}{-1-0} = 5 & & \\
x_2 = 1 & f_2 = -3 & \frac{6+3}{0-1} = -9 & \frac{5+9}{-1-1} = -7 & \\
x_3 = 3 & f_3 = 3 & \frac{-3-3}{1-3} = 3 & \frac{-9-3}{0-3} = 4 & \frac{-7-4}{-1-3} = \frac{11}{4}
\end{NiceArray}
\]
\begin{align*}
p(x) &= 1 + 5(x-(-1)) -7(x-(-1))(x-0) + \frac{11}4 (x-(-1))(x-0)(x-1) \\
p(x) &= f_{0,0} + f_{0,1}(x-x_0) + ... + f_{0,N}(x-x_0) \cdot ... \cdot (x-x_{N-1})
\end{align*}
\end{document}