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\end{align*} + + Falls $|a| \geq |b|$: + \drmvspace + \begin{align*} + t = \frac{b}{a}, \smallhspace s = c \cdot t, \smallhspace c = \frac{1}{\sqrt{1 + t^2}} + \end{align*} +\end{multicols} +Es ist wichtig, dass wir das $r = \text{sign}(a) \sqrt{a^2 + b^2}$ mit Vorzeichen berechnen, um Auslöschung zu vermeiden + + +\newpage +\bg{purple}{Gram-Schmidt} + +Die Idee des Gram-Schmidt-Algorithmus ist es, orthonormale Vektoren zu konstruieren und diese dann zur Matrix $Q$ zusammenzubasteln. + +Es wurden zwei Algorithmen behandelt, beide unten in Pseudocode: +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{classicalGramSchmidt}(A)} + \begin{algorithmic}[1] + \State $n \gets \texttt{A.shape[0]}$ + \State $q \gets$ Initialisiere ein $n \times n$ array + \State $r \gets$ Initialisiere ein $n \times n$ array + \For{$k = 1, 2, \ldots, n$} + \State $v_k \gets a_k$ \Comment{Der $k$-te Spaltenvektor} + \For{$i = 1, \ldots, k - 1$} + \State $r_{ik} \gets q_i^\top a_k$ + \State $v_k \gets a_k - \sum_{i = 1}^{k - 1} r_{ik} q_i$ + \EndFor + \State $r_{kk} = ||v_k||$ + \State $q_k = v_k / r_{kk}$ \Comment{Vektor normieren} + \EndFor + \State \Return $q$, $r$ + \end{algorithmic} + \end{spacing} +\end{algorithm} + +\drmvspace +\begin{algorithm} + \begin{spacing}{1.2} + \caption{\textsc{modifiedGramSchmidt}(A)} + \begin{algorithmic}[1] + \State $n \gets \texttt{A.shape[0]}$ + \State $q \gets$ Initialisiere ein $n \times n$ array + \State $r \gets$ Initialisiere ein $n \times n$ array + \For{$k = 1, 2, \ldots, n$} + \State $v_k \gets a_k$ \Comment{Der $k$-te Spaltenvektor} + \For{$i = 1, \ldots, k - 1$} + \State $r_{ik} \gets q_i^\top v_k$ + \State $v_k \gets v_k - r_{ik} q_i$ + \EndFor + \State $q_k = v_k / r_{kk}$ \Comment{Vektor normieren} + \EndFor + \State \Return $q$, $r$ + \end{algorithmic} + \end{spacing} +\end{algorithm} +Falls $R$ nicht benötigt wird, kann viel Speicher gespart werden, indem man das $r_{ik}$ als eine scoped variable verwendet. + + +\bg{purple}{Householder-Reflektor} diff --git a/semester3/numcs/parts/04_linalg/03_svd.tex b/semester3/numcs/parts/04_linalg/03_svd.tex index e69de29..d1254b9 100644 --- a/semester3/numcs/parts/04_linalg/03_svd.tex +++ b/semester3/numcs/parts/04_linalg/03_svd.tex @@ -0,0 +1 @@ +\subsubsection{Singulärwertzerlegung}