mirror of
https://github.com/MartinThoma/LaTeX-examples.git
synced 2025-04-26 06:48:04 +02:00
73 lines
1.3 KiB
TeX
73 lines
1.3 KiB
TeX
\section*{Aufgabe 1}
|
|
\textbf{Gegeben:}
|
|
|
|
\[
|
|
A = \begin{pmatrix}
|
|
1 & 2 & 3\\
|
|
2 & 8 & 14\\
|
|
3 & 14 & 34
|
|
\end{pmatrix}\]
|
|
|
|
\textbf{Aufgabe:} Durch Gauß-Elimination die Cholesky-Zerlegung $A = \overline{L} \overline{L}^T$
|
|
berechnen
|
|
|
|
\textbf{Lösung mit Gauß-Elimination:}
|
|
\begin{align*}
|
|
A &=
|
|
\begin{gmatrix}[p]
|
|
1 & 2 & 3\\
|
|
2 & 8 & 14\\
|
|
3 & 14 & 34
|
|
\rowops
|
|
\add[\cdot (-2)]{0}{1}
|
|
\add[\cdot (-3)]{0}{2}
|
|
\end{gmatrix}\\
|
|
\leadsto
|
|
L^{(1)} &=
|
|
\begin{pmatrix}
|
|
1 & 0 & 0\\
|
|
-2 & 1 & 0\\
|
|
-3 & 0 & 1
|
|
\end{pmatrix},&
|
|
A^{(1)} &=
|
|
\begin{gmatrix}[p]
|
|
1 & 2 & 3\\
|
|
0 & 4 & 8\\
|
|
0 & 8 & 25
|
|
\rowops
|
|
\add[\cdot (-2)]{1}{2}
|
|
\end{gmatrix}\\
|
|
\leadsto
|
|
L^{(2)} &=
|
|
\begin{pmatrix}
|
|
1 & 0 & 0\\
|
|
0 & 1 & 0\\
|
|
0 & -2 & 1
|
|
\end{pmatrix},&
|
|
A^{(2)} &=
|
|
\begin{gmatrix}[p]
|
|
1 & 2 & 3\\
|
|
0 & 4 & 8\\
|
|
0 & 0 & 9
|
|
\rowops
|
|
\add[\cdot (-2)]{1}{2}
|
|
\end{gmatrix}\\
|
|
\end{align*}
|
|
|
|
TODO: Und wie gehts weiter?
|
|
|
|
|
|
\textbf{Lösung ohne Gauß-Elimination:}
|
|
\[
|
|
A =
|
|
\underbrace{
|
|
\begin{pmatrix}
|
|
1 & 0 & 0\\
|
|
2 & 2 & 0\\
|
|
3 & 4 & 3
|
|
\end{pmatrix}}_{=: L} \cdot \underbrace{\begin{pmatrix}
|
|
1 & 2 & 3\\
|
|
0 & 2 & 4\\
|
|
0 & 0 & 3
|
|
\end{pmatrix}}_{=: L^T}
|
|
\]
|