Initial commit
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301
.gitignore
vendored
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301
.gitignore
vendored
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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
|
||||
14
LICENSE
Normal file
14
LICENSE
Normal file
|
|
@ -0,0 +1,14 @@
|
|||
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.
|
||||
|
||||
10
Makefile
Normal file
10
Makefile
Normal file
|
|
@ -0,0 +1,10 @@
|
|||
MAIN = sheet
|
||||
FLAGS = -pdf -lualatex
|
||||
|
||||
all:
|
||||
latexmk $(FLAGS) $(MAIN)
|
||||
dev:
|
||||
latexmk $(FLAGS) -pvc $(MAIN)
|
||||
clean:
|
||||
latexmk -C
|
||||
|
||||
406
sheet.tex
Normal file
406
sheet.tex
Normal file
|
|
@ -0,0 +1,406 @@
|
|||
\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}
|
||||
Loading…
Reference in a new issue