From 1ef7e2b57640a454d0ceb464a00243aaf81ff87a Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sat, 21 Sep 2013 15:23:53 +0200 Subject: [PATCH] =?UTF-8?q?Viele=20kleine=20Nachtr=C3=A4ge?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/Numerik/Klausur2/Aufgabe3.tex | 11 +++++-- documents/Numerik/Klausur2/Aufgabe4.tex | 2 +- documents/Numerik/Klausur2/Klausur2.pdf | Bin 270652 -> 271872 bytes documents/Numerik/Klausur5/Aufgabe3.tex | 16 +++++++--- documents/Numerik/Klausur5/Aufgabe4.tex | 7 +++++ documents/Numerik/Klausur5/Aufgabe5.tex | 38 +++++++++++++++++++++++- documents/Numerik/Klausur5/Klausur5.pdf | Bin 190890 -> 198224 bytes documents/Numerik/Klausur5/Klausur5.tex | 2 +- 8 files changed, 67 insertions(+), 9 deletions(-) diff --git a/documents/Numerik/Klausur2/Aufgabe3.tex b/documents/Numerik/Klausur2/Aufgabe3.tex index 06f1e1e..0cdacfc 100644 --- a/documents/Numerik/Klausur2/Aufgabe3.tex +++ b/documents/Numerik/Klausur2/Aufgabe3.tex @@ -33,7 +33,13 @@ Anmerkung: Es ist nicht notwendig die Monomdarstellung zu berechnen. In diesem Fall hat es jedoch das Endergebnis stark vereinfacht. \subsection*{Teilaufgabe b)} -Zunächst die dividierten Differenzen berechnen: +Für die Berechnung der dividierten Differenzen gilt allgemein: + +\begin{align} + f[x_i, \dots, x_{i+k}] = \frac{f[x_i, \dots x_{(i+k)-1}] - f[x_{i+1}, \dots x_{i+k}]}{x_i - x_{i+k}} +\end{align} + +In diesem Fall bedeutet das konkret: \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\\ @@ -43,6 +49,7 @@ Zunächst die dividierten Differenzen berechnen: 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) + p(x) &= 7 + (x-\underbrace{(-1)}_{x_0}) \cdot (-6) + (x-\underbrace{(-1)}_{x_0}) \cdot (x-\underbrace{(0)}_{x_1}) \cdot 2 + (x+1) \cdot x \cdot (x-1) \end{align} +(Siehe erste Spalte mit $x_0$) diff --git a/documents/Numerik/Klausur2/Aufgabe4.tex b/documents/Numerik/Klausur2/Aufgabe4.tex index 791867b..ab37695 100644 --- a/documents/Numerik/Klausur2/Aufgabe4.tex +++ b/documents/Numerik/Klausur2/Aufgabe4.tex @@ -44,4 +44,4 @@ $\sum_{l=0}^{N-1} f(a + \frac{1}{2} \cdot h + l \cdot h)$ sind die jeweiligen mittleren Knoten der Intervalle. 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zF+72%JT+zWzuDdyOlL4X#*s<%n8X-DCdXrP+Mpen9A{LVQSnaM;-}cEIV#V8M(UP% zjFk6($UeS_)GPb;CsLp6?}teJQvFM$g0#F5X+V0>`l2YkYVABIz5f?!Ncx&0m89?a zNW)4KuOpR}R$BX2l-|9JR8^)Bv*tBr)_z19QD$Q*(x@_@8A6SG86;m)Hn-XdEOxbFn;Vy8WS7?^1RV@EIXE{r3MC~) GPeuv~Gvns~ diff --git a/documents/Numerik/Klausur5/Aufgabe3.tex b/documents/Numerik/Klausur5/Aufgabe3.tex index 60bb858..032daa2 100644 --- a/documents/Numerik/Klausur5/Aufgabe3.tex +++ b/documents/Numerik/Klausur5/Aufgabe3.tex @@ -3,14 +3,22 @@ relativer Fehler: \begin{align} - \frac{ | \frac{x}{y} - \frac{x \cdot (1 + \epsilon_x)}{y \cdot (1 + \epsilon_y}|}{|\frac{x}{y}|} - = \ldots = |\frac{\epsilon_y - \epsilon_x }{1 + \epsilon_y} | - \le \frac{|\epsilon_y | + | \epsilon_x |}{|1 + \epsilon_y|} \le \frac{2 \cdot \text{eps}}{|1 + \epsilon_y|} + \frac{ | \frac{x}{y} - \frac{x \cdot (1 + \varepsilon_x)}{y \cdot (1 + \varepsilon_y)}|}{|\frac{x}{y}|} + &= \frac{| \frac{x(1+\varepsilon_y) - x (1+\varepsilon_x)}{y(1+\varepsilon_y)} |}{|\frac{x}{y}|} \\ + &= \frac{| \frac{x(\varepsilon_y-\varepsilon_x)}{y(1+\varepsilon_y)} |}{|\frac{x}{y}|} \\ + &= \left |\frac{\varepsilon_y - \epsilon_x }{1 + \varepsilon_y} \right |\\ + &\le \frac{|\varepsilon_y | + | \varepsilon_x |}{|1 + \varepsilon_y|} \le \frac{2 \cdot \text{eps}}{|1 + \varepsilon_y|} \\ + &\approx 2 \cdot \text{eps} \end{align} Der letzte Ausdruck ist ungefähr gleich $2 \cdot \text{eps}$, da $1 + \epsilon_y$ ungefähr gleich $1$ ist. -Also: Der relative Fehler kann sich maximal verdoppeln. + +Der relative Fehler kann sich also maximal verdoppeln. \subsection*{Teilaufgabe ii} Die zweite Formel ist vorzuziehen, also $f(x) = -\ln (x + \sqrt{x^2-1})$, da es bei Subtraktion zweier annähernd gleich-großer Zahlen zur Stellenauslöschung kommt. Bei der ersten Formel, also $f(x) = \ln (x - \sqrt{x^2-1})$, tritt genau dieses Problem auf: $x$ und $\sqrt{x^2-1}$ sind für große $x$ ungefähr gleich groß. \\ Bei der zweiten Formel tritt das Problem nicht auf: $x$ ist positiv und $\sqrt{x^2 - 1}$ auch, also gibt es in dem Ausdruck keine Subtraktion zweier annähernd gleich-großer Zahlen. + +Außerdem ändert sich $\ln(x)$ stärker, je näher $x$ bei 0 ist. Es ist +also auch wegen der Ungenauigkeit der Berechnung des $\ln$ besser, +weiter von $0$ entfernt zu sein. diff --git a/documents/Numerik/Klausur5/Aufgabe4.tex b/documents/Numerik/Klausur5/Aufgabe4.tex index ef2b2e3..419eea6 100644 --- a/documents/Numerik/Klausur5/Aufgabe4.tex +++ b/documents/Numerik/Klausur5/Aufgabe4.tex @@ -1,2 +1,9 @@ \section*{Aufgabe 4} TODO + +\begin{itemize} + \item Klausur 3, Aufgabe 3 ist ähnlich + \item Klausur 4, Aufgabe 3 ist ähnlich + \item Klausur 5, Aufgabe 4 ist ähnlich + \item Klausur 6, Aufgabe 3 ist ähnlich +\end{itemize} diff --git a/documents/Numerik/Klausur5/Aufgabe5.tex b/documents/Numerik/Klausur5/Aufgabe5.tex index 75caec4..19f1734 100644 --- a/documents/Numerik/Klausur5/Aufgabe5.tex +++ b/documents/Numerik/Klausur5/Aufgabe5.tex @@ -1,2 +1,38 @@ \section*{Aufgabe 5} -TODO +\subsection*{Aufgabe} +Bestimme alle Quadraturformeln mit $s=3$ und Knoten +$0 = c_1 < c_2, c_3$ und Ordnung $p \geq 4$. + +Schreiben Sie ein Programm in Pseudocode, welches zu vorgegebenem +$c_2$ den Knoten $c_3$ und die Gewichte $b_i$ möglichst effizient +berechnet. + +Wie viele symmetrische Quadraturformeln gibt es mit diesen Eigneschaften? + +\subsection*{Lösung} +Da $c_1 = 0$ kann es keine Gauß-Quadraturformel sein. Daher kann +die Ordnung nicht $2 \cdot s = 6$ sein. Interessant sind also +\begin{itemize} + \item[(A)] Symmetrische Quadraturformeln der Ordnung 4 + \item[(B)] Unsymmetrische Quadraturformeln der Ordnung 4 + \item[(C)] Unsymmetrische Quadraturformeln der Ordnung 5 +\end{itemize} + +Die Simpson-Regel mit $c_1 = 0, c_2 = \frac{1}{2}$ und $c_3 = 1$ +mit $b_1 = b_3 = \frac{1}{6}$ und $b_2 = \frac{4}{6}$ ist die einzige +symmetrische Quadraturformel in (A). + +Für (B) müssen die Ordnungsbedingungen gelten: +\begin{align} + \nicefrac{1}{1} &\stackrel{!}{=} b_1 + b_2 + b_3\\ + \nicefrac{1}{2} &\stackrel{!}{=} b_2 \cdot c_2 + b_3 c_3\\ + \nicefrac{1}{3} &\stackrel{!}{=} b_2 \cdot c_2^2 + b_3 c_3^2\\ + \nicefrac{1}{4} &\stackrel{!}{=} b_2 \cdot c_2^3 + b_3 c_3^3 +\end{align} + +Für (C) muss zusätzlich gelten: +\begin{align} + \nicefrac{1}{5} &\stackrel{!}{=} b_2 \cdot c_2^4 + b_3 c_3^4 +\end{align} + +TODO: Und weiter? diff --git a/documents/Numerik/Klausur5/Klausur5.pdf b/documents/Numerik/Klausur5/Klausur5.pdf index 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