From f4422549d96c4826905bce10e7379dca20b9b4b7 Mon Sep 17 00:00:00 2001 From: FelixBenzBaldas Date: Sun, 15 Sep 2013 13:34:57 +0200 Subject: [PATCH] =?UTF-8?q?Ein=20paar=20Kleinigkeiten=20ge=C3=A4ndert.=20A?= =?UTF-8?q?u=C3=9Ferdem=20ist=20das=20ein=20"Test-PullRequest"?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Numerik/Klausur2/Aufgabe3.tex | 1 + documents/Numerik/Klausur2/Aufgabe3.tex~ | 28 +++++++++++++ documents/Numerik/Klausur2/Aufgabe4.tex | 17 +++----- documents/Numerik/Klausur2/Aufgabe4.tex~ | 47 ++++++++++++++++++++++ documents/Numerik/Klausur2/Klausur2.pdf | Bin 210841 -> 220087 bytes documents/Numerik/Klausur3/Aufgabe1.tex | 46 ++++++++++++++++++++++ documents/Numerik/Klausur3/Aufgabe1.tex~ | 46 ++++++++++++++++++++++ documents/Numerik/Klausur3/Aufgabe2.tex | 17 ++++++++ documents/Numerik/Klausur3/Aufgabe2.tex~ | 17 ++++++++ documents/Numerik/Klausur3/Klausur3.tex | 38 ++++++++++++++++++ documents/Numerik/Klausur3/Klausur3.tex~ | 48 +++++++++++++++++++++++ documents/Numerik/Klausur3/Makefile | 8 ++++ documents/Numerik/Klausur3/Makefile~ | 8 ++++ 13 files changed, 310 insertions(+), 11 deletions(-) create mode 100644 documents/Numerik/Klausur2/Aufgabe3.tex~ create mode 100644 documents/Numerik/Klausur2/Aufgabe4.tex~ create mode 100644 documents/Numerik/Klausur3/Aufgabe1.tex create mode 100644 documents/Numerik/Klausur3/Aufgabe1.tex~ create mode 100644 documents/Numerik/Klausur3/Aufgabe2.tex create mode 100644 documents/Numerik/Klausur3/Aufgabe2.tex~ create mode 100644 documents/Numerik/Klausur3/Klausur3.tex create mode 100644 documents/Numerik/Klausur3/Klausur3.tex~ create mode 100644 documents/Numerik/Klausur3/Makefile create mode 100644 documents/Numerik/Klausur3/Makefile~ diff --git a/documents/Numerik/Klausur2/Aufgabe3.tex b/documents/Numerik/Klausur2/Aufgabe3.tex index 665acbe..c15b6b0 100644 --- a/documents/Numerik/Klausur2/Aufgabe3.tex +++ b/documents/Numerik/Klausur2/Aufgabe3.tex @@ -11,6 +11,7 @@ Damit 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. (Das spart viel Zeit) % Anmerkung hinzugefügt von Felix Benz-Baldas \subsection*{Teilaufgabe b)} Zunächst die dividierten Differenzen berechnen: diff --git a/documents/Numerik/Klausur2/Aufgabe3.tex~ b/documents/Numerik/Klausur2/Aufgabe3.tex~ new file mode 100644 index 0000000..665acbe --- /dev/null +++ b/documents/Numerik/Klausur2/Aufgabe3.tex~ @@ -0,0 +1,28 @@ +\section*{Aufgabe 3} +\subsection*{Teilaufgabe a)} +\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) +\end{align} + +Damit ergibt sich: +\begin{align} + p(x) &= x^3 + 2x^2 - 5x + 1 +\end{align} + +\subsection*{Teilaufgabe b)} +Zunächst die dividierten Differenzen berechnen: +\begin{align} + f[x_0] &= 7, &f[x_1] &= 1, & f[x_2] &= -1, & f[x_3] = 7\\ + f[x_0, x_1] &= -6, &f[x_1, x_2] &= -2, &f[x_2, x_3] &= 8\\ + f[x_0, x_1, x_2] &= 2, &f[x_1, x_2, x_3] &= 5\\ + f[x_0, x_1, x_2, x_3] &= 1 +\end{align} + +Insgesamt ergibt sich also +\begin{align} + p(x) &= 7 - (x+1) \cdot 6 + (x+1) \cdot x \cdot 2 + (x+1) \cdot x \cdot (x-1) +\end{align} + diff --git a/documents/Numerik/Klausur2/Aufgabe4.tex b/documents/Numerik/Klausur2/Aufgabe4.tex index af8766f..6ef97a1 100644 --- a/documents/Numerik/Klausur2/Aufgabe4.tex +++ b/documents/Numerik/Klausur2/Aufgabe4.tex @@ -30,23 +30,18 @@ Als erstes ist festzustellen, dass es sich hier um die Simpsonregel handelt und \end{align} ist. Wenn diese nun auf $N$ Intervalle aufgepflittet wird gilt folgendes: - \begin{align} - \int_a^b f(x) \mathrm{d}x &= (b-a) \cdot \frac{1}{6} \cdot \left [ f(a) + f(b) + 2 \cdot \sum_{i=1}^{N-1} f(i \cdot \frac{1}{N}) + 4 \cdot \sum_{i=1}^N f(i \cdot \frac{1}{2N})\right ] + h &= \frac{(b-a)}{N} \\ + \int_a^b f(x) \mathrm{d}x &= h \cdot \frac{1}{6} \cdot \left [ f(a) + f(b) + 2 \cdot \sum_{i=1}^{N-1} f(a + i \cdot h) + 4 \cdot \sum_{l=0}^{N-1} f(a + \frac{1}{2} \cdot h + l \cdot h)\right ] \end{align} -$\sum_{i=1}^{N-1} f(i \cdot \frac{1}{N})$ sind die Grenzknoten der Intervalle +$\sum_{i=1}^{N-1} f(a + i \cdot h)$ steht für die Grenzknoten (deshalb werden sie doppelt gezählt). Von den Grenzknoten gibt es -insgesamt $s-2$ Stück, da die tatsächlichen Integralgrenzen $a$ und $b$ +insgesamt $N-2$ Stück, da die tatsächlichen Integralgrenzen $a$ und $b$ nur einmal in die Berechnung mit einfließen. -$\sum_{i=1}^N f(i \cdot \frac{1}{2N})$ sind die jeweiligen -mittleren Knoten der Intervalle. Davon gibt es $s-1$ Stück. - -\begin{figure}[h] - \centering - \includegraphics*[width=\linewidth, keepaspectratio]{aufgabe4-b.png} -\end{figure} +$\sum_{l=0}^{N-1} f(a + \frac{1}{2} \cdot h + l \cdot h)$ sind die jeweiligen +mittleren Knoten der Intervalle. Davon gibt es $N$ Stück. \subsection*{Teilaufgabe c)} TODO diff --git a/documents/Numerik/Klausur2/Aufgabe4.tex~ b/documents/Numerik/Klausur2/Aufgabe4.tex~ new file mode 100644 index 0000000..e67786c --- /dev/null +++ b/documents/Numerik/Klausur2/Aufgabe4.tex~ @@ -0,0 +1,47 @@ +\section*{Aufgabe 4} +\subsection*{Teilaufgabe a)} +\begin{enumerate} + \item Ordnung 3 kann durch geschickte Gewichtswahl erzwungen werden. + \item Ordnung 4 ist automatisch gegeben, da die QF symmetrisch sein soll. + \item Aufgrund der Symmetrie gilt Äquivalenz zwischen Ordnung 5 und 6. + Denn eine hätte die QF Ordnung 5, so wäre wegen der + Symmetrie Ordnung 6 direkt gegeben. Ordnung 6 wäre aber + bei der Quadraturformel mit 3 Knoten das Maximum, was nur + mit der Gauß-QF erreicht werden kann. Da aber $c_1 = 0$ gilt, + kann es sich hier nicht um die Gauß-QF handeln. Wegen + erwähnter Äquivalenz kann die QF auch nicht Ordnung 5 haben. +\end{enumerate} + +Da $c_1 = 0$ gilt, muss $c_3 = 0$ sein. Und dann muss $c_2 = \frac{1}{2}$ +sein. Es müssen nun die Gewichte bestimmt werden um Ordnung 3 zu +garantieren mit: + +\begin{align} + b_i &= \int_0^1 L_i(x) \mathrm{d}x\\ + b_1 &= \frac{1}{6},\\ + b_2 &= \frac{4}{6},\\ + b_4 &= \frac{1}{6} +\end{align} + +\subsection*{Teilaufgabe b)} +Als erstes ist festzustellen, dass es sich hier um die Simpsonregel handelt und die QF +\begin{align} + \int_a^b f(x) \mathrm{d}x &= (b-a) \cdot \frac{1}{6} \cdot \left ( f(a) + 4 \cdot f(\frac{a+b}{2}) + f(b) \right ) +\end{align} + +ist. Wenn diese nun auf $N$ Intervalle aufgepflittet wird gilt folgendes: +\begin{align} + h &= \frac{(b-a)}{N} \\ + \int_a^b f(x) \mathrm{d}x &= h \cdot \frac{1}{6} \cdot \left [ f(a) + f(b) + 2 \cdot \sum_{i=1}^{N-1} f(a + i \cdot h) + 4 \cdot \sum_{l=0}^{N-1} f(a + \frac{1}{2} \cdot h + l \cdot h)\right ] +\end{align} + +$\sum_{i=1}^{N-1} f(a + i \cdot h)$ steht für die Grenzknoten + (deshalb werden sie doppelt gezählt). Von den Grenzknoten gibt es +insgesamt $N-2$ Stück, da die tatsächlichen Integralgrenzen $a$ und $b$ +nur einmal in die Berechnung mit einfließen. + +$\sum_{l=0}^{N-1} f(a + \frac{1}{2} \cdot h + l \cdot h)$ sind die jeweiligen +mittleren Knoten der Intervalle. 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