From c14a250043d9b063afee9127740f9f1be3ef033c Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 19 Sep 2013 18:23:05 +0200 Subject: [PATCH] =?UTF-8?q?Aufgabe=203a=20ausf=C3=BChrlicher=20beschrieben?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Numerik/Klausur2/Aufgabe3.tex | 31 +++++++++++++++++++----- documents/Numerik/Klausur2/Klausur2.pdf | Bin 259711 -> 264747 bytes 2 files changed, 25 insertions(+), 6 deletions(-) diff --git a/documents/Numerik/Klausur2/Aufgabe3.tex b/documents/Numerik/Klausur2/Aufgabe3.tex index c15b6b0..06f1e1e 100644 --- a/documents/Numerik/Klausur2/Aufgabe3.tex +++ b/documents/Numerik/Klausur2/Aufgabe3.tex @@ -1,17 +1,36 @@ \section*{Aufgabe 3} +\textbf{Gegeben:} + +\begin{table}[h!] + \begin{tabular}{l||l|l|l|l} + $f_i$ & 7 & 1 & -1 & 7 \\\hline + $x_i$ & -1 & 0 & 1 & 2 \\ + \end{tabular} +\end{table} + \subsection*{Teilaufgabe a)} +Allgemein lauten Lagrange-Polynome: + +\[L_i = \frac{\overbrace{\prod_{j=0, j \neq i}^n (x-x_j)}^\text{Produkt der Nullstellen}}{\underbrace{\prod_{j=0, j \neq i}^n (x_i - x_j)}_\text{Normalisierungsfaktor}}\] + +Im speziellen: \begin{align} - L_0(x) &= - \frac{1}{6} \cdot (x^3 - 3 x^2 + 2x)\\ - L_1(x) &= \frac{1}{2} \cdot (x^3 - 2x^2 - x + 2)\\ - L_2(x) &= - \frac{1}{2} \cdot (x^3 - x^2 - 2x)\\ - L_3(x) &= \frac{1}{6} \cdot (x^3 - x) + L_0(x) &= \frac{(x-0)(x-1)(x-2)}{(-1-0)(-1-1)(-1-2)} &&=-\frac{1}{6} \cdot (x^3 - 3 x^2 + 2x)\\ + L_1(x) &= \frac{(x+1)(x-1)(x-2)}{(0+1)(0-1)(0-2)} &&= \frac{1}{2} \cdot (x^3 - 2x^2 - x + 2)\\ + L_2(x) &= \frac{(x+1)x(x-2)}{(1+1)(1-0)(1-2)} &&=-\frac{1}{2} \cdot (x^3 - x^2 - 2x)\\ + L_3(x) &= \frac{(x+1)(x-0)(x-1)}{(2+1)(2-0)(2-1)} &&= \frac{1}{6} \cdot (x^3 - x) \end{align} -Damit ergibt sich: +Durch die Interpolationsformel von Lagrange + +\[p(x) = \sum_{i=0}^n f_i L_i(x)\] + +ergibt sich \begin{align} p(x) &= x^3 + 2x^2 - 5x + 1 \end{align} -Anmerkung: Es ist in der Klausur allerdings nicht notwendig die Monomdarstellung zu berechnen außer es wird explizit verlangt. 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