From bae5d05c67191192941ef7b36109aef4e4dd0e07 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Fri, 13 Sep 2013 18:51:12 +0200 Subject: [PATCH] =?UTF-8?q?Aufgabe=203=20sehr=20ausf=C3=BChrlich=20erkl?= =?UTF-8?q?=C3=A4rt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Numerik/Klausur1/Aufgabe3.tex | 51 ++++++++++++++++++++++-- documents/Numerik/Klausur1/Klausur1.pdf | Bin 240525 -> 242127 bytes 2 files changed, 48 insertions(+), 3 deletions(-) diff --git a/documents/Numerik/Klausur1/Aufgabe3.tex b/documents/Numerik/Klausur1/Aufgabe3.tex index b26d8bc..be143ae 100644 --- a/documents/Numerik/Klausur1/Aufgabe3.tex +++ b/documents/Numerik/Klausur1/Aufgabe3.tex @@ -8,11 +8,56 @@ Und jetzt die Berechnung \[f'(x, y) \cdot (x_0, y_0) = f(x,y)\] -LR-Zerlegung für $f'(x, y)$: +LR-Zerlegung für $f'(x, y)$ kann durch scharfes hinsehen durchgeführt +werden, da es in $L$ nur eine unbekannte (links unten) gibt. Es gilt +also ausführlich: \begin{align} - L &= \begin{pmatrix}1 &0 \\ x^2 & 1\end{pmatrix}\\ - R &= \begin{pmatrix}3 & \cos y \\ 0 & e^y - x^2 \cos y\end{pmatrix}\\ + \begin{pmatrix} + 3 & \cos y\\ + 3 x^2 & e^y + \end{pmatrix} + &= + \overbrace{\begin{pmatrix} + 1 & 0\\ + l_{12} & 1 + \end{pmatrix}}^L \cdot + \overbrace{\begin{pmatrix} + r_{11} & r_{12}\\ + 0 & r_{22} + \end{pmatrix}}^R\\ + \Rightarrow r_{11} &= 3\\ + \Rightarrow r_{12} &= \cos y\\ + \Rightarrow \begin{pmatrix} + 3 & \cos y\\ + 3 x^2 & e^y + \end{pmatrix} + &= + \begin{pmatrix} + 1 & 0\\ + l_{12} & 1 + \end{pmatrix} \cdot + \begin{pmatrix} + 3 & \cos y\\ + 0 & r_{22} + \end{pmatrix}\\ + \Rightarrow 3x^2 &\stackrel{!}{=} l_{12} \cdot 3 + 1 \cdot 0\\ + \Leftrightarrow l_{12} &= x^2\\ + \Rightarrow e^y &\stackrel{!}{=} x^2 \cdot \cos y + 1 \cdot r_{22}\\ + \Leftrightarrow r_{22} &= -x^2 \cdot \cos y + e^y\\ + \Rightarrow \begin{pmatrix} + 3 & \cos y\\ + 3 x^2 & e^y + \end{pmatrix} + &= + \begin{pmatrix} + 1 & 0\\ + x^2 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