From 1c166fdb025725e52df23bc9e196ab9159c298ab Mon Sep 17 00:00:00 2001 From: henrydatei Date: Thu, 27 Dec 2018 14:03:54 +0100 Subject: [PATCH] NUME Kapitel 6.4 fertig --- .../revidiertes_Simplex-Verfahren.tex | 19 +++++++++++++++++- 3. Semester/NUME/Vorlesung NUME.pdf | Bin 722730 -> 724910 bytes 2 files changed, 18 insertions(+), 1 deletion(-) diff --git a/3. Semester/NUME/TeX_files/revidiertes_Simplex-Verfahren.tex b/3. Semester/NUME/TeX_files/revidiertes_Simplex-Verfahren.tex index 63137d7..f487274 100644 --- a/3. Semester/NUME/TeX_files/revidiertes_Simplex-Verfahren.tex +++ b/3. Semester/NUME/TeX_files/revidiertes_Simplex-Verfahren.tex @@ -1 +1,18 @@ -\section{Revidiertes Simplex-Verfahren} \ No newline at end of file +\section{Revidiertes Simplex-Verfahren} + +Hier wird auf die explizite Berechnung der Matrix $P$ verzichtet. Vielmehr wird beim Übergang von einer Ecke $x^r$ mit dem Indexmengen $B_r$ und $N_r$ zur nächsten Ecke $x^{r+1}$ mit den Indexmengen $B_{r+1}$ und $N_{r+1}$ ausgenutzt, dass sich die Indexmengen jeweils nur in einem Index ändern und der Unterschied der Matrix $A_{B_r}$ zur Matrix $A_{B_{r+1}}$ in der Änderung einer einzigen Spalte besteht. + +Die an der Ecke $x^{r+1}$ zur Bestimmung des neuen Pivots erforderlichen Vektoren, zum Beispiel $p^{r+1}$ und $q^{r+1}$, werden dann mit Hilfe der aus $A_{B_r}^{-1}$ mit $\mathcal{O}(m^2)$ Operationen zu gewinnenden Matrix $A_{B_{r+1}}^{-1}$ bestimmt, vergleiche die Aufdatierungsformel im Beweis von \propref{6_2_1}, zum Beispiel erhält man +\begin{align} + p^{r+1} = A_{B_{r+1}}^{-1}b\notag +\end{align} +mit +\begin{align} + A_{B_{r+1}}^{-1} &= \left(\mathbbm{1}-\frac{A_{B_r}^{-1}ve^T_{(l)}}{1+e^T_{(l)}A_{B_r}^{-1}v} \right) A_{B_r}^{-1}\notag \\ + &= A_{B_r}^{-1} - \frac{1}{1+e^T_{(l)}(A_{B_r}^{-1}v)}(A_{B_r}^{-1}v)(e^T_{(l)}A_{B_r}^{-1})\notag +\end{align} +Die Matrizen $A_{B_r}^{-1}$ müssen nicht explizit gespeichert werden. Da mit geeigneten Vektoren $u,w,\tilde{w}\in\real^m$ offenbar +\begin{align} + A_{B_{r+1}}^{-1} = (\mathbbm{1}+uw^T)A_{B_r}^{-1} = A_{B_r}^{-1} + u\tilde{w}^T\notag +\end{align} +genügt es, etwa $u,\tilde{w}$ für jeden Basiswechsel sowie $A_{B_0}$ zu speichern. Ein Restart ist hier nach einer größeren Anzahl von Basiswechseln empfehlenswert, um den Einfluss von Rundungsfehlern zu begrenzen. \ No newline at end of file diff --git a/3. Semester/NUME/Vorlesung NUME.pdf b/3. 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