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-\[A = -\begin{pmatrix} - 3 & 15 & 13 \\ - 6 & 6 & 6 \\ - 2 & 8 & 19 -\end{pmatrix}\] - -\textbf{Aufgabe:} LR-Zerlegung von $A$ mit Spaltenpivotwahl - -\textbf{Lösung:} - -\begin{align*} - & - & - A^{(0)} &= \begin{gmatrix}[p] - 3 & 15 & 13 \\ - 6 & 6 & 6 \\ - 2 & 8 & 19 - \rowops - \swap{0}{1} - \end{gmatrix} - &\\ - P^{(1)} &= \begin{pmatrix} - 0 & 1 & 0\\ - 1 & 0 & 0\\ - 0 & 0 & 1 - \end{pmatrix}, - & - A^{(1)} &= \begin{gmatrix}[p] - 6 & 6 & 6 \\ - 3 & 15 & 13 \\ - 2 & 8 & 19 - \rowops - \add[\cdot (-\frac{1}{2})]{0}{1} - \add[\cdot (-\frac{1}{3})]{0}{2} - \end{gmatrix} - &\\ - L^{(1)} &= \begin{pmatrix} - 1 & 0 & 0\\ - -\frac{1}{2} & 1 & 0\\ - -\frac{1}{3} & 0 & 1 - \end{pmatrix}, - & - A^{(2)} &= \begin{gmatrix}[p] - 6 & 6 & 6 \\ - 0 & 12 & 10 \\ - 0 & 6 & 17 - \rowops - \add[\cdot (-\frac{1}{2})]{1}{2} - \end{gmatrix} - &\\ - L^{(2)} &= \begin{pmatrix} - 1 & 0 & 0\\ - 0 & 1 & 0\\ - 0 & -\frac{1}{2} & 1 - \end{pmatrix}, - & - A^{(3)} &= \begin{gmatrix}[p] - 6 & 6 & 6 \\ - 0 & 12 & 10 \\ - 0 & 0 & 12 - \end{gmatrix} -\end{align*} - -Es gilt: - -\begin{align} - L^{(2)} \cdot L^{(1)} \cdot \underbrace{P^{(1)}}_{=: P} \cdot A^{0} &= \underbrace{A^{(3)}}_{=: R}\\ - \Leftrightarrow P A &= (L^{(2)} \cdot L^{(1)})^{-1} \cdot R \\ - \Rightarrow L &= (L^{(2)} \cdot L^{(1)})^{-1}\\ - &= \begin{pmatrix} - 1 & 0 & 0\\ - \frac{1}{2} & 1 & 0\\ - \frac{1}{3} & \frac{1}{2} & 1 - \end{pmatrix} -\end{align} - -Nun gilt: $P A = L R = A^{(1)}$ (Kontrolle mit \href{http://www.wolframalpha.com/input/?i=%7B%7B1%2C0%2C0%7D%2C%7B0.5%2C1%2C0%7D%2C%7B1%2F3%2C0.5%2C1%7D%7D*%7B%7B6%2C6%2C6%7D%2C%7B0%2C12%2C10%7D%2C%7B0%2C0%2C12%7D%7D}{Wolfram|Alpha}) - -\subsection*{Teilaufgabe b} - -\textbf{Gegeben:} - -\[A = -\begin{pmatrix} - 9 & 4 & 12 \\ - 4 & 1 & 4 \\ - 12 & 4 & 17 -\end{pmatrix}\] - -\textbf{Aufgabe:} $A$ auf positive Definitheit untersuchen, ohne Eigenwerte zu berechnen. - -\textbf{Vorüberlegung:} -Eine Matrix $A \in \mathbb{R}^{n \times n}$ heißt positiv definit $\dots$ -\begin{align*} - \dots & \Leftrightarrow \forall x \in \mathbb{R}^n \setminus \Set{0}: x^T A x > 0\\ - & \Leftrightarrow \text{Alle Eigenwerte sind größer als 0} -\end{align*} - -Falls $A$ symmetrisch ist, gilt: -\begin{align*} - \text{$A$ ist positiv definit} & \Leftrightarrow \text{alle führenden Hauptminore von $A$ sind positiv}\\ - & \Leftrightarrow \text{es gibt eine Cholesky-Zerlegung $A=GG^T$}\\ -\end{align*} - -\subsubsection*{Lösung 1: Hauptminor-Kriterium} - -\begin{align} - \det(A_1) &= 9 > 0\\ - \det(A_2) &= - \begin{vmatrix} - 9 & 4 \\ - 4 & 1 \\ - \end{vmatrix} = 9 - 16 < 0\\ - &\Rightarrow \text{$A$ ist nicht positiv definit} -\end{align} - -\subsubsection*{Lösung 2: Cholesky-Zerlegung} -\begin{align} - l_{11} &= \sqrt{a_{11}} = 3\\ - l_{21} &= \frac{a_{21}}{l_{11}} = \frac{4}{3}\\ - l_{31} &= \frac{a_{31}}{l_{11}} = \frac{12}{3} = 4\\ - l_{22} &= \sqrt{a_{22} - {l_{21}}^2} = \sqrt{1 - \frac{16}{9}}= \sqrt{-\frac{7}{9}} \notin \mathbb{R}\\ - & \Rightarrow \text{Es ex. keine Cholesky-Zerlegung, aber $A$ ist symmetrisch}\\ - & \Rightarrow \text{$A$ ist nicht positiv definit} -\end{align} diff --git a/documents/Numerik/Klausur1/Aufgabe2.tex b/documents/Numerik/Klausur1/Aufgabe2.tex deleted file mode 100644 index 79758ac..0000000 --- a/documents/Numerik/Klausur1/Aufgabe2.tex +++ /dev/null @@ -1,53 +0,0 @@ -\section*{Aufgabe 2} -\subsection*{Teilaufgabe a} -\textbf{Aufgabe} -Formulieren Sie einen Algorithmus in Pseudocode zum Lösen des Gleichungssystems -\[Ly = b,\] -wobei $L$ eine invertierbare, untere Dreiecksmatrix ist. - -Geben Sie die Formel zur Berechnung von $y_i$ an. - -\textbf{Lösung:} -\[y_i = \frac{b_i - \sum_{k=1}^{i-1} l_{ik} \cdot y_k}{l_{ii}}\] - -\begin{algorithm}[H] - \begin{algorithmic} - \Require Lower, invertable, triangular Matrix $L \in \mathbb{R}^{n \times n}$, Vektor $b$ - \Procedure{solve}{$L$, $b$} - \For{$i \in \Set{1, \dots n}$} - \State $y_i \gets b_i$ - \For{$k \in \Set{1, \dots, i-1}$} - \State $y_i \gets y_i - l_{ik} \cdot y_k$ - \EndFor - \State $y_i \gets \frac{y_i}{l_{ii}}$ - \EndFor - \EndProcedure - \end{algorithmic} -\caption{Calculate $y$ in $Ly = b$} -\end{algorithm} - -\subsection*{Teilaufgabe b} -\[Ax = b \Leftrightarrow PAx = Pb \Leftrightarrow LRx = Pb \] - -\begin{algorithm}[H] - \begin{algorithmic} - \Require Matrix $A$, Vektor $b$ - \Procedure{LoeseLGS}{$A$, $b$} - \State $P, L, R \gets \Call{LRZer}{A}$ - \State $b^* \gets P \cdot b$ - \State $c \gets \Call{VorSub}{L, b^*}$ - \State $x \gets \Call{RueckSub}{R, c}$ - \State \Return $x$ - \EndProcedure - \end{algorithmic} -\caption{Löse ein LGS $Ax = b$} -\end{algorithm} - -\subsection*{Teilaufgabe c} -Der Gesamtaufwand ist: -\begin{itemize} - \item LR-Zerlegung, $\frac{1}{3}n^3 - \frac{1}{3} n^2$ - \item Vektormultiplikation, $2n$ - \item Vorwärtssubstitution, $\frac{1}{2} n^2$ - \item Rückwärtssubstitution, $\frac{1}{2} n^2$ -\end{itemize} diff --git a/documents/Numerik/Klausur1/Aufgabe3.tex b/documents/Numerik/Klausur1/Aufgabe3.tex deleted file mode 100644 index d370420..0000000 --- a/documents/Numerik/Klausur1/Aufgabe3.tex +++ /dev/null @@ -1,165 +0,0 @@ -\section*{Aufgabe 3} -Die Jacobi-Matrix von $f$ lautet: -\[f' (x,y) = \begin{pmatrix} - 3 & \cos y\\ - 3 x^2 & e^y -\end{pmatrix}\] -Hierfür wurde in in der ersten Spalte nach $x$ abgeleitet und in der -zweiten Spalte nach $y$. - -Eine Iteration des Newton-Verfahren ist durch -\begin{align} -x_{k+1}&=x_{k}\underbrace{-f'(x_k)^{-1}\cdot f(x_k)}_{\Delta x} -\end{align} -gegeben (vgl. Skript, S. 35). - -Zur praktischen Durchführung lösen wir -\begin{align} - f'(x_0, y_0)\Delta x &= -f(x_0,y_0)\\ - L \cdot \underbrace{R \cdot \Delta x}_{=: c} &= -f(x_0, y_0) -\end{align} -mit Hilfe der LR Zerlegung nach $\Delta x$ auf. - -\subsection*{Lösungsvorschlag 1 (Numerische Lösung)} -\begin{align} -% - f'(x_0,y_0) &= L \cdot R \\ - \Leftrightarrow f'(-\nicefrac{1}{3}, 0) &= L \cdot R \\ - \Leftrightarrow \begin{pmatrix} - 3 & 1\\ - \frac{1}{3} & 1 - \end{pmatrix} - &= - \overbrace{\begin{pmatrix} - 1 & 0\\ - \frac{1}{9} & 1 - \end{pmatrix}}^{=: L} \cdot - \overbrace{\begin{pmatrix} - 3 & 1\\ - 0 & \frac{8}{9} - \end{pmatrix}}^{=: R}\\ -% - L \cdot c &= -f(x_0,y_0) \\ - \Leftrightarrow - \begin{pmatrix} - 1 & 0\\ - \frac{1}{9} & 1 - \end{pmatrix} - \cdot c - &= - - \begin{pmatrix} - 2\\ - \frac{26}{27} - \end{pmatrix}\\ - \Rightarrow - c &= \begin{pmatrix} - -2\\ - -\frac{20}{27} - \end{pmatrix}\footnotemark\\ -% - R\cdot \Delta x &= c\\ - \Leftrightarrow - \begin{pmatrix} - 3 & 1\\ - 0 & \frac{8}{9} - \end{pmatrix} - \cdot \Delta x &= - \begin{pmatrix} - -2\\ - -\frac{20}{27} - \end{pmatrix}\\ - \Rightarrow \Delta x &= \frac{1}{18} - \begin{pmatrix} - -7\\ - -15 - \end{pmatrix} -\end{align} -\footnotetext{Dieser Schritt wird durch Vorwärtssubsitution berechnet.} - -Anschließend berechnen wir -\begin{align} - \begin{pmatrix} - x_1\\ - y_1 - \end{pmatrix} &= - \begin{pmatrix} - x_0\\ - y_0 - \end{pmatrix}+\Delta x \\ - \Leftrightarrow\begin{pmatrix} - x_1\\ - y_1 - \end{pmatrix} &= - \begin{pmatrix} - -\frac{1}{3}\\ - 0 - \end{pmatrix} + - \frac{1}{18} - \begin{pmatrix} - -7\\ - -15 - \end{pmatrix} \\ - \Leftrightarrow\begin{pmatrix} - x_1\\ - y_1 - \end{pmatrix} &= - \begin{pmatrix} - -\nicefrac{13}{18}\\ - -\nicefrac{15}{18} - \end{pmatrix} -\end{align} - - -\subsection*{Lösungsvorschlag 2 (Analytische Lösung)} -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} - \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 & 1 - \end{pmatrix} \cdot - \begin{pmatrix} - 3 & \cos y\\ - 0 & -x^2 \cdot \cos y + e^y - \end{pmatrix}\\ - P &= I_2 -\end{align} diff --git a/documents/Numerik/Klausur1/Aufgabe4.tex b/documents/Numerik/Klausur1/Aufgabe4.tex deleted file mode 100644 index d0db6be..0000000 --- a/documents/Numerik/Klausur1/Aufgabe4.tex +++ /dev/null @@ -1,73 +0,0 @@ -\section*{Aufgabe 4} -\textbf{Aufgabe}: - -\[I(f) = \int_a^b f(x) \mathrm{d}x \] - -\begin{enumerate} - \item Integrand am linken und am rechten Rand interpolieren - \item Interpolationspolynom mit Quadraturformel integrieren -\end{enumerate} - -\textbf{Lösung}: - -Nutze Interpolationsformel von Lagrange: - -\begin{align} - L_i &= \frac{\prod_{j=1, j \neq i}^n (x-x_j)}{\prod_{j=1, j \neq i}^n (x_i - x_j)}\\ - p(x) &= \sum_{i=0}^{1} f_i \cdot L_i(x) -\end{align} - -Berechne Lagrangepolynome: - -\begin{align} - L_0(x) = \frac{x-b}{a-b} \\ - L_1(x) = \frac{x-a}{b-a} -\end{align} - -So erhalten wir: - -\begin{align} - p(x) &= f(a) \frac{x-b}{a-b} + f(b) \frac{x-a}{b-a}\\ - &= \frac{f(a) (b-x) + f(b) (x-a)}{b-a} \\ - &= \frac{f(a)b- f(a)x + f(b) x- f(b)a}{b-a}\\ - &=\frac{x \cdot \left (f(b)-f(a) \right ) + f(a)b- f(b)a}{b-a}\\ - &= x \cdot \underbrace{\frac{f(b)-f(a)}{b-a}}_{=:r} + \underbrace{\frac{f(a)b - f(b)a}{b-a}}_{=: s} -\end{align} - -Nun integrieren wir das Interpolationspolynom: - -\begin{align} - \int_a^b p(x) \mathrm{d} x &= \left [\frac{r}{2} x^2 + sx \right ]_a^b\\ - &= \left (\frac{a^2 r}{2} + sa \right ) - \left (\frac{b^2 r}{2} + sb \right )\\ - &= a\left (\frac{a r}{2} + s \right ) - b \left (\frac{b r}{2} + s \right )\\ - &= a\left (\frac{a \frac{f(b)-f(a)}{b-a}}{2} + \frac{f(a)b - f(b)a}{b-a} \right ) - b \left (\frac{b \frac{f(b)-f(a)}{b-a}}{2} + \frac{f(a)b - f(b)a}{b-a} \right )\\ - &= a\left (\frac{-a f(a)+2b f(a)-a f(b)}{2 \cdot(b-a)}\right ) - b \left (\frac{bf(b) + b f(a) - 2 a f(b)}{2 \cdot (b-a)} \right )\\ - & \dots \text{theoretisch sollte das zu } (b-a)(\frac{f(a)}{2} + \frac{f(b)}{2}) \text{ zu vereinfachen sein} -\end{align} - -Alternativer Rechenweg - -\[ \int_a^b p(x)dx = \int_a^b f(a) \frac{x-b}{a-b}dx + \int_a^b f(b) \frac{x-a}{b-a}dx \] -\[ = \int_a^b \frac{f(a) \cdot x}{a-b}dx - \int_a^b \frac{f(a) \cdot b}{a-b}dx + \int_a^b \frac{f(b) \cdot x}{b-a}dx - \int_a^b \frac{f(b) \cdot a}{b-a}dx \] -\[ = \frac{1}{2} \cdot \frac{f(a) \cdot b^2}{a-b} - \frac{1}{2} \cdot \frac{f(a) \cdot a^2}{a-b} - \frac{f(a) \cdot b^2}{a-b} + \frac{f(a) \cdot b \cdot a}{a-b} + \frac{1}{2} \cdot \frac{f(b) \cdot b^2}{b-a} \] -\[ - \frac{1}{2} \cdot \frac{f(b) \cdot a^2}{b-a} - \frac{f(b) \cdot a \cdot b}{b-a} + \frac{f(b) \cdot a^2}{b-a}\] -\[=(b-a)\cdot(\frac{f(a)}{2} + \frac{f(b)}{2})\] - -Betrachtet man nun die allgemeine Quadraturformel, -\[ -\int_a^b f(x)dx \approx (b-a) \sum_{i=1}^s b_i f(a+c_i(b-a)) -\] -so gilt für die hergeleitete Quadraturformel also $s=2$, $c_1=0, c_2=1$ und $b_1 = b_2 = \frac{1}{2}$. Sie entspricht damit der Trapezregel. - -\subsection*{Teilaufgabe b)} -Sei nun $f(x) = x^2$ und $a = 0$ sowie $b = 4$. Man soll die ermittelte -Formel zwei mal auf äquidistanten Intervallen anwenden. - -\textbf{Lösung:} - -\begin{align} - \int_0^4 p(x) \mathrm{d}x &= \int_0^2 p(x)\mathrm{d}x + \int_2^4 p(x)\mathrm{d}x \\ - &= (2-0)\cdot \left (\frac{0}{2} + \frac{4}{2} \right ) + (4-2) \cdot \left (\frac{4}{2} + \frac{16}{2} \right )\\ - &= 2 \cdot 2 + 2 \cdot (2+8)\\ - &= 24 -\end{align} diff --git a/documents/Numerik/Klausur1/Aufgabe5.tex b/documents/Numerik/Klausur1/Aufgabe5.tex deleted file mode 100644 index dfcfaf5..0000000 --- a/documents/Numerik/Klausur1/Aufgabe5.tex +++ /dev/null @@ -1,67 +0,0 @@ -\section*{Aufgabe 5} -\subsection*{Teilaufgabe a} -Eine Quadraturformel $(b_i, c_i)_{i=1, \dots, s}$ hat die Ordnung -$p$, falls sie exakte Lösungen für alle Polynome vom Grad $\leq p -1$ -liefert. - -\subsection*{Teilaufgabe b} -Die Ordnungsbedingungen, mit denen man zeigen kann, dass eine Quadraturformel -mindestens Ordnung $p$ hat, lautet: -\[\forall p \in \Set{1, \dots, p}: \sum_{i=1}^s b_i c_i^{q-1} = \frac{1}{q}\] - -\subsection*{Teilaufgabe c} -\paragraph{Aufgabe} Bestimmen Sie zu den Knoten $c_1 = 0$ und $c_2 = \frac{2}{3}$ Gewichte, um eine Quadraturformel -maximaler Ordnung zu erhalten. Wie hoch ist die Ordnung? - -\paragraph{Lösung} - -Nach VL kann bei Vorgabe von $s$ Knoten auch die Ordnung $s$ durch -geschickte Wahl der Gewichte erreicht werden. Nach Satz 27 ist diese -Wahl eindeutig. -Also berechnen wir die Gewichte, um die Ordnung $p=2$ zu sichern. - -Dazu stellen wir zuerst die Lagrange-Polynome auf: - -\begin{align} - L_1(x) &= \frac{x-x_2}{x_1 - x_2} = \frac{x-c_2}{c_1-c_2} = \frac{x-\nicefrac{2}{3}}{-\nicefrac{2}{3}} = -\frac{3}{2} x + 1\\ - L_2(x) &= \frac{x-x_1}{x_2 - x_1} = \frac{x-c_1}{c_2-c_1} = \frac{x}{\nicefrac{2}{3}} = \frac{3}{2} x -\end{align} - -Nun gilt für die Gewichte: -\begin{align} - b_i &= \int_0^1 L_i(x) \mathrm{d}x\\ - b_1 &= \int_0^1 -\frac{3}{2} x + 1 \mathrm{d}x = \left [ -\frac{3}{4}x^2 + x \right ]_0^1 = \frac{1}{4}\\ - b_2 &= \frac{3}{4} -\end{align} - -Nun sind die Ordnungsbedingungen zu überprüfen: -\begin{align} - \nicefrac{1}{1} &\stackrel{?}{=} b_1 c_1^0 + b_2 c_2^0 = \nicefrac{1}{4} + \nicefrac{3}{4} \text{\;\;\cmark}\\ - \nicefrac{1}{2} &\stackrel{?}{=} b_1 c_1^1 + b_2 c_2^1 = \frac{1}{4} \cdot 0 + \frac{3}{4} \cdot \frac{2}{3} \text{\;\;\cmark}\\ - \nicefrac{1}{3} &\stackrel{?}{=} b_1 c_1^2 + b_2 c_2^2 = \frac{1}{4} \cdot 0 + \frac{3}{4} \cdot \frac{4}{9} \text{\;\;\cmark}\\ - \nicefrac{1}{4} &\stackrel{?}{=} b_1 c_1^3 + b_2 c_2^3 = \frac{1}{4} \cdot 0 + \frac{3}{4} \cdot \frac{8}{27} \text{\;\;\xmark}\\ -\end{align} - -Die Quadraturformel mit den Knoten $c_1 = 0$, $c_2 = \nicefrac{2}{3}$ sowie -den Gewichten $b_1 = \nicefrac{1}{4}$, $b_2 = \nicefrac{3}{4}$ erfüllt -also die 1., 2. und 3. Ordnungsbedingung, nicht jedoch die 4. -Ordnungsbedingung. Ihre maximale Ordnung ist also $p=3$. - -\textbf{Anmerkungen:} Da $c_1 = 0$ kann es sich nicht um die Gauß-QF handeln. -Somit können wir nicht Ordnung $p=4$ erreichen. - -Bei der Suche nach den Gewichten hätte man alternativ auch das folgende -LGS lösen können: - -\begin{align} - \begin{pmatrix} - c_1^0 & c_2^0\\ - c_1^1 & c_2^1 - \end{pmatrix} - \cdot x - = - \begin{pmatrix} - 1\\ - \nicefrac{1}{2} - \end{pmatrix} -\end{align} diff --git a/documents/Numerik/Klausur1/Klausur1.pdf b/documents/Numerik/Klausur1/Klausur1.pdf deleted file mode 100644 index 51fb56b6241249e18983c71f56b1c7cefc9c90d2..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 263209 zcma(2V{j(W7Pbk;w$rg~+qUiGiEZ0<(y?tj9ox3kv90Ov{ibTB>YS=M``6z8chz;R zb+1LHC@N0JM9&68Hn%jq2E#$fNN8_tMaatw!ys#BXYOJ_$i&ITM)-dY41>6(jf
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index a1ff428..0000000 --- a/documents/Numerik/Klausur1/Klausur1.tex +++ /dev/null @@ -1,49 +0,0 @@ -\documentclass[a4paper]{scrartcl} -\usepackage{amssymb, amsmath} % needed for math -\usepackage[utf8]{inputenc} % this is needed for umlauts -\usepackage[ngerman]{babel} % this is needed for umlauts -\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf -\usepackage{pdfpages} % Signatureinbingung und includepdf -\usepackage{geometry} % [margin=2.5cm]layout -\usepackage[pdftex]{hyperref} % links im text -\usepackage{color} -\usepackage{framed} -\usepackage{enumerate} % for advanced numbering of lists -\usepackage{marvosym} % checkedbox -\usepackage{wasysym} -\usepackage{braket} % for \Set{} -\usepackage{pifont}% http://ctan.org/pkg/pifont -\usepackage{gauss} -\usepackage{algorithm,algpseudocode} -\usepackage{units} -\usepackage{parskip} -\usepackage{lastpage} -\allowdisplaybreaks - -\newcommand{\cmark}{\ding{51}}% -\newcommand{\xmark}{\ding{55}}% - -\title{Numerik Klausur 1 - Musterlösung} -\makeatletter -\AtBeginDocument{ - \hypersetup{ - pdfauthor = {Martin Thoma, Peter, Felix}, - pdfkeywords = {Numerik, KIT, Klausur}, - pdftitle = {\@title} - } - \pagestyle{fancy} - \lhead{\@title} - \rhead{Seite \thepage{} von \pageref{LastPage}} -} -\makeatother - -\usepackage{fancyhdr} -\fancyfoot[C]{} - -\begin{document} - \include{Aufgabe1} - \include{Aufgabe2} - \include{Aufgabe3} - \include{Aufgabe4} - \include{Aufgabe5} -\end{document} diff --git a/documents/Numerik/Klausur1/Makefile b/documents/Numerik/Klausur1/Makefile deleted file mode 100644 index 1844f7e..0000000 --- a/documents/Numerik/Klausur1/Makefile +++ /dev/null @@ -1,8 +0,0 @@ -SOURCE = Klausur1 -make: - pdflatex -shell-escape $(SOURCE).tex -output-format=pdf - pdflatex -shell-escape $(SOURCE).tex -output-format=pdf - make clean - -clean: - rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg diff --git a/documents/Numerik/Klausur1/README.md b/documents/Numerik/Klausur1/README.md deleted file mode 100644 index 7edca00..0000000 --- a/documents/Numerik/Klausur1/README.md +++ /dev/null @@ -1,4 +0,0 @@ -Dies ist eine inoffizielle Lösung für [Klausur1.pdf von Dr. Daniel Weiß](http://www.math.kit.edu/ianm3/lehre/numainfing2013s/seite/uebnuminfing). - -Verbesserungsvorschläge gerne per [GitHub Pull Request](https://help.github.com/articles/using-pull-requests) -an mich schicken oder auch einfach per E-Mail an info@martin-thoma.de diff --git a/documents/Numerik/Klausur2/Aufgabe1.tex b/documents/Numerik/Klausur2/Aufgabe1.tex deleted file mode 100644 index cddbaec..0000000 --- a/documents/Numerik/Klausur2/Aufgabe1.tex +++ /dev/null @@ -1,87 +0,0 @@ -\section*{Aufgabe 1} -\subsection*{Teilaufgabe a)} - -\paragraph{Gegeben:} - -\[A := \begin{pmatrix} -4 & 2 & 8\\ -2 & 5 & 8\\ -8 & 8 & 29 -\end{pmatrix}\] - -\paragraph{Aufgabe:} Cholesky-Zerlegung $A = \overline{L} \cdot \overline{L}^T$ berechnen - -\paragraph{Rechenweg:} -Entweder mit dem Algorithmus: -\begin{algorithm}[H] - \begin{algorithmic} - \Function{Cholesky}{$A \in \mathbb{R}^{n \times n}$} - \State $L = \Set{0} \in \mathbb{R}^{n \times n}$ \Comment{Initialisiere $L$} - \For{($k=1$; $\;k \leq n$; $\;k$++)} - \State $L_{k,k} = \sqrt{A_{k,k} - \sum_{i=1}^{k-1} L_{k,i}^2}$ - \For{($i=k+1$; $\;i \leq n$; $\;i$++)} - \State $L_{i,k} = \frac{A_{i,k} - \sum_{j=1}^{k-1} L_{i,j} \cdot L_{k,j}}{L_{k,k}}$ - \EndFor - \EndFor - \State \Return $L$ - \EndFunction - \end{algorithmic} -\caption{Cholesky-Zerlegung} -\label{alg:seq1} -\end{algorithm} - -oder über die LR-Zerlegung: -\begin{align} - A &= L\cdot R\\ - &= L\cdot(D\cdot L^T)\\ - &= L\cdot(D^\frac{1}{2} \cdot D^\frac{1}{2})\cdot L^T\\ - &= \underbrace{(L\cdot D^\frac{1}{2})}_{=: \overline{L}} \cdot (D^\frac{1}{2} \cdot L^T) -\end{align} - -\paragraph{Lösung:} -$ -\overline{L} = -\begin{pmatrix} -2 & 0 & 0 \\ -1 & 2 & 0 \\ -4 & 2 & 3 \\ -\end{pmatrix} -$ - - -\subsection*{Teilaufgabe b)} -\textbf{Gesucht}: $\det(A)$ - -Sei $P \cdot A = L \cdot R$, die gewohnte LR-Zerlegung. - -Dann gilt: - -\[\det(A) = \frac{\det(L) \cdot \det(R)}{\det(P)}\] - -$\det(L) = 1$, da alle Diagonalelemente 1 sind und es sich um eine strikte untere Dreiecksmatrix handelt. - -$\det(R) = r_{11} \cdot \ldots \cdot r_{nn}$, da es sich um eine obere Dreiecksmatrix handelt. - - -$\det(P) \in \Set{1, -1}$ - -Das Verfahren ist also: - -\begin{algorithm}[H] - \begin{algorithmic} - \Require $A \in \mathbb{R}^{n \times n}$ - \State $P, L, R \gets \Call{LRZerl}{A}$ - \State $x \gets 1$ - \For{$i$ in $1..n$} - \State $x \gets x \cdot r_{ii}$ - \State $x \gets x \cdot p_{ii}$ - \EndFor - \end{algorithmic} -\caption{Determinante berechnen} -\label{alg:seq1} -\end{algorithm} - -Alternativ kann man auch in einer angepassten LR-Zerlegung direkt die -Anzahl an Zeilenvertauschungen zählen. Dann benötigt man $P$ nicht mehr. -Ist die Anzahl der Zeilenvertauschungen ungerade, muss das Produkt -der $r_ii$ negiert werden. diff --git a/documents/Numerik/Klausur2/Aufgabe2.tex b/documents/Numerik/Klausur2/Aufgabe2.tex deleted file mode 100644 index 003edb0..0000000 --- a/documents/Numerik/Klausur2/Aufgabe2.tex +++ /dev/null @@ -1,66 +0,0 @@ -\section*{Aufgabe 2} - -\paragraph{Voraussetzung:} -Gegeben sei eine Funktion $F$: -\begin{align*} - F: \mathbb{R} &\rightarrow [-1, 1]\\ - F(x) &:= \cos(x) -\end{align*} - -sowie eine Folge $(x)_k$ mit $x_{k+1} := F(x_k)$. - -\paragraph{Behauptung:} $\displaystyle \exists! x^*: \forall x \in \mathbb{R}: \lim_{k \rightarrow \infty} x_k = x^*$ - -\paragraph{Beweis:} über den Banachschen Fixpunktsatz. -\begin{proof} -Sei $ x \in \mathbb{R}$, so gilt:\marginpar{Teil 1: Fix\-punkte können nur in $[0,1]$ sein.} -\begin{align*} - -1 \leq \cos(x) \leq 1 -\end{align*} -Also genügt es $x \in [-1, 1]$ auf der Suche nach Fixpunkten zu betrachten. - -Sei nun $x \in [-1, 0)$. Dann gilt: $\cos(x) > 0$. Da $x <0$ aber $F(x) > 0$, -kann kein Fixpunkt in $[-1, 0)$ sein. Es genügt also sogar, -nur $[0, 1]$ zu betrachten. - -Sei $0 \leq x < y \leq 1$. Dann folgt:\marginnote{Teil 2: $F$ ist auf $[0,1]$ eine Kontraktion} -\begin{align} - \stackrel{\text{Mittelwertsatz}}{\Rightarrow} \exists L \in (x,y): \frac{\cos(y) - \cos(x)}{y-x} &= f'(L)\\ - \Rightarrow \exists L \in [0,1]: \| \cos y - \cos x \| &= \| - \sin(L) \cdot (y-x)\| \\ - &= \underbrace{\sin(L)}_{[0,1)} (y-x)\\ - \Rightarrow F \text{ ist Kontraktion auf [0,1]} -\end{align} - -Da $F|_{[0,1]}$ eine Selbstabbildung und eine Kontraktion ist und -offensichtlich $[0,1]$ abgeschlossen ist, greift der -Banachsche Fixpunktsatz. Es folgt direkt, dass auch für alle $x \in [0,1]$ -die Folge $(x)_k$ gegen den einzigen Fixpunkt $x^*$ konvergiert. -\end{proof} - -\subsection*{Anmerkung} -Um zu zeigen, dass es genau einen Fixpunkt $x^*$ in $(0,1)$ gibt, -braucht man den Banachschen Fixpunktsatz nicht. Nur um zu zeigen, -dass die Fixpunktiteration auf für jedes $x \in \mathbb{R}$ gegen -diesen Fixpunkt $x^*$ konvergiert, braucht man ihn. - -So kann man die existenz eines Fixpunktes zeigen: - -Offensichtlich ist $F(0) \neq 0$ und $F(1) \neq 1$, also ist der -Fixpunkt - falls vorhanden - in $(0,1)$. $F$ ist in $(0,1)$ stetig -und streng monoton fallend. Da auch $-x$ in $(0,1)$ streng monoton -fallend ist, folgt, dass $\cos(x) - x$ in $(0,1)$ streng monoton -fallend ist. - -$x=0 \Rightarrow \cos(x) - x = \cos(0) - 0 = 1$ - -$x=45^\circ = \frac{1}{4} \pi < 1 \Rightarrow \cos(45^\circ) - \frac{\pi}{4} = \frac{\sqrt{2}}{2} - \frac{\pi}{4} <0$, da -\begin{align} - 8 &< 9 < \pi^2\\ - \Rightarrow \sqrt{8} &< \pi\\ - \Leftrightarrow 2 \sqrt{2} &< \pi\\ - \Leftrightarrow \frac{\sqrt{2}}{2} &< \frac{\pi}{4} -\end{align} - -$\stackrel{\text{Zwischenwertsatz}}{\Rightarrow} \exists x^*: \cos(x^*) - x^* = 0 \Leftrightarrow \exists x^*: \cos(x^*) = x^*$. - -Dieses $x^*$ ist eindeutig, da $\cos(x)-x$ \emph{streng} monoton fallend ist. diff --git a/documents/Numerik/Klausur2/Aufgabe3.tex b/documents/Numerik/Klausur2/Aufgabe3.tex deleted file mode 100644 index 4123573..0000000 --- a/documents/Numerik/Klausur2/Aufgabe3.tex +++ /dev/null @@ -1,56 +0,0 @@ -\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{(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} - -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 nicht notwendig die Monomdarstellung zu berechnen. -In diesem Fall hat es jedoch das Endergebnis stark vereinfacht. - -\subsection*{Teilaufgabe b)} -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\\ - 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-\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)\\ - &= 7 -6 (x+1) + 2x(x+1) + x(x+1)(x-1) -\end{align} - -(Siehe erste Spalte mit $x_0$) diff --git a/documents/Numerik/Klausur2/Aufgabe4.tex b/documents/Numerik/Klausur2/Aufgabe4.tex deleted file mode 100644 index eea0cd4..0000000 --- a/documents/Numerik/Klausur2/Aufgabe4.tex +++ /dev/null @@ -1,47 +0,0 @@ -\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 $c_3 = 1$ sein (Symmetrie). 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_3 &= \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. Davon gibt es $N$ Stück. - -\subsection*{Teilaufgabe c)} -Diese Aufgabe ist nicht relevant, da Matlab nicht Klausurrelevant ist. diff --git a/documents/Numerik/Klausur2/Aufgabe5.tex b/documents/Numerik/Klausur2/Aufgabe5.tex deleted file mode 100644 index f77915a..0000000 --- a/documents/Numerik/Klausur2/Aufgabe5.tex +++ /dev/null @@ -1,99 +0,0 @@ -\section*{Aufgabe 5} - -Zunächst ist nach der Familie von Quadraturformeln gefragt, für die gilt: ($p := $ Ordnung der QF) -\begin{align} - s = 3 \\ - 0 = c_1 < c_2 < c_3 \\ - p \ge 4 -\end{align} - -Nach Satz 29 sind in der Familie genau die QFs, für die gilt: \\ -Für alle Polynome $g(x)$ mit Grad $\le m-1 = 0$ gilt: -\begin{align} - \int_0^1 M(x) \cdot g(x) \mathrm{d}x = 0 \label{a3} -\end{align} - -Da eine Quadraturformel höchstens Grad $2s=6$ (Satz 30) haben kann und es wegen -$c_1 = 0$ nicht die Gauss-Quadratur sein kann (Satz 31), kommt nur Ordnung $p=4$ -und $p=5$ in Frage. - -In dieser Aufgabe sind nur die symmetrischen QF, also die von Ordnung -$p=4$ explizit anzugeben. Für die QF von Ordnung $p=5$ hätte man nur -die Gewichte in Abhängigkeit der Knoten darstellen müssen und -eine Bedinung nur an die Knoten herleiten müssen. - -\subsection*{Ordnung 4} -Es gilt $g(x) = c$ für eine Konstante $c$, da $\text{Grad}(g(x))=0$ ist. -Also ist \ref{a3} gleichbedeutend mit: -\begin{align} - \int_0^1 M(x) \cdot c \mathrm{d}x &= 0 \\ - \Leftrightarrow c \cdot \int_0^1 M(x) \mathrm{d}x &= 0 \\ - \Leftrightarrow \int_0^1 M(x) \mathrm{d}x &= 0 \\ - \Leftrightarrow \int_0^1 (x-c_1)(x-c_2)(x-c_3) \mathrm{d}x &= 0 \\ - \Leftrightarrow \frac{1}{4} - \frac{1}{3} \cdot (c_2 + c_3) + \frac{1}{2} \cdot c_2 \cdot c_3 &= 0 \\ - \Leftrightarrow \frac{\frac{1}{4} - \frac{1}{3} \cdot c_3} - {\frac{1}{3} - \frac{1}{2} \cdot c_3} &= c_2 -\end{align} - -Natürlich müssen auch die Gewichte optimal gewählt werden. Dafür wird Satz 28 genutzt: -Sei $b^T = (b_1, b_2, b_3)$ der Gewichtsvektor. Sei zudem $C := -\begin{pmatrix} - {c_1}^0 & {c_2}^0 & {c_3}^0 \\ - {c_1}^1 & {c_2}^1 & {c_3}^1 \\ - {c_1}^2 & {c_2}^2 & {c_3}^2 -\end{pmatrix} -$. \\ -Dann gilt: $C$ ist invertierbar und $b = C^{-1} \cdot -\begin{pmatrix} - 1 \\ - \frac{1}{2} \\ - \frac{1}{3} -\end{pmatrix} -$. - -Es gibt genau eine symmetrische QF in der Familie. Begründung: \\ -Aus $c_1 = 0 $ folgt, dass $c_3 = 1$ ist. Außerdem muss $c_2 = \frac{1}{2} $ sein. Also sind die Knoten festgelegt. Da wir die Ordnung $\ge s = 3$ fordern, sind auch die Gewichte eindeutig. \\ -Es handelt sich um die aus der Vorlesung bekannte Simpsonregel. - -\subsection*{Ordnung 5} -Es gilt $g(x) = ax+c$ für Konstanten $a \neq 0, c$, da $\text{Grad}(g(x))=1$ ist. -Also ist \ref{a3} gleichbedeutend mit: -\begin{align} - \int_0^1 M(x) \cdot (ax+c) \mathrm{d}x &= 0 \\ - \Leftrightarrow a \int_0^1 x M(x) \mathrm{d}x + c \int_0^1 M(x) \mathrm{d}x &= 0 \\ - \Leftrightarrow a \int_0^1 x (x-c_1)(x-c_2)(x-c_3) \mathrm{d}x + c \int_0^1 (x-c_1)(x-c_2)(x-c_3) \mathrm{d}x &= 0 \\ - \stackrel{c_1=0}{\Leftrightarrow} a \int_0^1 x^2(x-c_2)(x-c_3) \mathrm{d}x + c \int_0^1 x(x-c_2)(x-c_3) \mathrm{d}x &= 0 \\ - \Leftrightarrow a \left (\frac{c_2 c_3}{3}-\frac{c_2}{4}-\frac{c_3}{4}+\frac{1}{5} \right ) + c \left ( \frac{c_2 c_3}{2}-\frac{c_2}{3}-\frac{c_3}{3}+\frac{1}{4} \right ) &= 0 \\ - \Leftrightarrow \left (\frac{c_2 c_3}{3}-\frac{c_2}{4}-\frac{c_3}{4}+\frac{1}{5} \right ) + \underbrace{\frac{c}{a}}_{=: d} \left ( \frac{c_2 c_3}{2}-\frac{c_2}{3}-\frac{c_3}{3}+\frac{1}{4} \right ) &= 0 -\end{align} - -Nun habe ich \href{http://www.wolframalpha.com/input/?i=(1%2F5+-+c%2F4+%2B+(b+(-3+%2B+4+c))%2F12)%2B+d*(3+-+4+c+%2B+b+(-4+%2B+6+c))%2F12%3D0}{Wolfram|Alpha} lösen lassen: -\begin{align} - c_2 &= \frac{6-\sqrt{6}}{10} \approx 0.355\\ - c_3 &= \frac{6+\sqrt{6}}{10} \approx 0.845 -\end{align} - -Wegen der Ordnungsbedingungen gilt nun: -\begin{align} - 1 &= b_1 + b_2 + b_3\\ - \frac{1}{2} &= b_2 \cdot \frac{6-\sqrt{6}}{10} + b_3 \cdot \frac{6+\sqrt{6}}{10}\\ - \frac{1}{3} &= b_2 \cdot \left (\frac{6-\sqrt{6}}{10} \right )^2 + b_3 \cdot \left (\frac{6+\sqrt{6}}{10} \right )^2\\ - \Leftrightarrow \frac{1}{3} - b_3 \cdot \left (\frac{6+\sqrt{6}}{10} \right )^2 &= b_2 \cdot \left (\frac{6-\sqrt{6}}{10} \right )^2\\ - \Leftrightarrow \frac{\frac{1}{3} - b_3 \cdot \left (\frac{6+\sqrt{6}}{10} \right )^2}{\left (\frac{6-\sqrt{6}}{10} \right )^2} &= b_2\\ - \Leftrightarrow b_2 &= \frac{100}{3 \cdot (6-\sqrt{6})^2} - b_3 \cdot \frac{(6+\sqrt{6})^2}{(6-\sqrt{6})^2}\\ - \Rightarrow \frac{1}{2} &= \left ( \frac{100}{3 \cdot (6-\sqrt{6})^2} - b_3 \cdot \frac{(6+\sqrt{6})^2}{(6-\sqrt{6})^2} \right ) \cdot \frac{6-\sqrt{6}}{10} + b_3 \cdot \frac{6+\sqrt{6}}{10}\\ - &= \left (\frac{10}{3 \cdot (6 - \sqrt{6})} - b_3 \cdot \frac{(6+\sqrt{6})^2}{10 \cdot (6 - \sqrt{6})} \right ) + b_3 \cdot \frac{6+\sqrt{6}}{10}\\ - &= b_3 \cdot \left (\frac{6+\sqrt{6}}{10} - \frac{(6+\sqrt{6})^2}{10 \cdot (6 - \sqrt{6})} \right ) + \frac{10}{3 \cdot (6 - \sqrt{6})}\\ - &= b_3 \cdot \left (\frac{30-(6+\sqrt{6})^2}{10 \cdot (6 - \sqrt{6})} \right ) + \frac{10}{3 \cdot (6 - \sqrt{6})}\\ -\Leftrightarrow \frac{1}{2} - \frac{10}{3 \cdot (6 - \sqrt{6})} &= b_3 \cdot \left (\frac{30-(6+\sqrt{6})^2}{10 \cdot (6 - \sqrt{6})} \right )\\ -\Leftrightarrow \frac{3 \cdot (6 - \sqrt{6}) - 20}{6\cdot (6 - \sqrt{6})} &= b_3 \cdot \left (\frac{30-(6+\sqrt{6})^2}{10 \cdot (6 - \sqrt{6})} \right )\\ -\Leftrightarrow b_3 &= \frac{(3 \cdot (6 - \sqrt{6}) - 20) \cdot 10 \cdot (6 - \sqrt{6})}{6\cdot (6 - \sqrt{6}) \cdot (30-(6+\sqrt{6})^2)}\\ -&= \frac{(3 \cdot (6 - \sqrt{6}) - 20) \cdot 5}{3 \cdot (30-(6+\sqrt{6})^2)}\\ -&= \frac{15 \cdot (6 - \sqrt{6}) - 100}{90-3 \cdot (6+\sqrt{6})^2}\\ - \Aboxed{b_3 &= \frac{16-\sqrt{6}}{36}} \approx 0.3764\\ - \Aboxed{b_2 &= \frac{16+\sqrt{6}}{36}} \approx 0.5125\\ - \stackrel{\text{Ordnungsbedinung 1}}{\Rightarrow} \Aboxed{b_1 &= \frac{1}{9}}\\ - \frac{1}{4} &\stackrel{?}{=} \frac{16+\sqrt{6}}{36} \cdot \left (\frac{6-\sqrt{6}}{10} \right )^3 + \frac{16-\sqrt{6}}{36} \cdot \left (\frac{6+\sqrt{6}}{10} \right)^3 \text{ \cmark}\\ - \frac{1}{5} &\stackrel{?}{=} \frac{16+\sqrt{6}}{36} \cdot \left (\frac{6-\sqrt{6}}{10} \right )^4 + \frac{16-\sqrt{6}}{36} \cdot \left (\frac{6+\sqrt{6}}{10} \right)^4 \text{ \cmark}\\ - \frac{1}{6} &\stackrel{?}{=} \frac{16+\sqrt{6}}{36} \cdot \left (\frac{6-\sqrt{6}}{10} \right )^5 + \frac{16-\sqrt{6}}{36} \cdot \left (\frac{6+\sqrt{6}}{10} 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@@ -1,48 +0,0 @@ -\documentclass[a4paper]{scrartcl} -\usepackage{amssymb, amsmath} % needed for math -\usepackage[utf8]{inputenc} % this is needed for umlauts -\usepackage[ngerman]{babel} % this is needed for umlauts -\usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf -\usepackage{pdfpages} % Signatureinbingung und includepdf -\usepackage{geometry} % [margin=2.5cm]layout -\usepackage[pdftex]{hyperref} % links im text -\usepackage{color} -\usepackage{framed} -\usepackage{enumerate} % for advanced numbering of lists -\usepackage{marvosym} % checkedbox -\usepackage{wasysym} -\usepackage{braket} % for \Set{} -\usepackage{pifont}% http://ctan.org/pkg/pifont -\usepackage{gauss} -\usepackage{algorithm,algpseudocode} -\usepackage{parskip} -\usepackage{lastpage} -\allowdisplaybreaks - -\newcommand{\cmark}{\ding{51}}% -\newcommand{\xmark}{\ding{55}}% - -\title{Numerik Klausur 4 - Musterlösung} -\makeatletter -\AtBeginDocument{ - \hypersetup{ - pdfauthor = {Martin Thoma, Peter, Felix}, - pdfkeywords = {Numerik, KIT, Klausur}, - pdftitle = {\@title} - } - \pagestyle{fancy} - \lhead{\@title} - \rhead{Seite \thepage{} von \pageref{LastPage}} -} -\makeatother - -\usepackage{fancyhdr} -\fancyfoot[C]{} - -\begin{document} - \input{Aufgabe1} - \input{Aufgabe2} - \input{Aufgabe3} - \input{Aufgabe4}\clearpage - \input{Aufgabe5} -\end{document} diff --git a/documents/Numerik/Klausur4/Makefile b/documents/Numerik/Klausur4/Makefile deleted file mode 100644 index 02cfc7f..0000000 --- a/documents/Numerik/Klausur4/Makefile +++ /dev/null @@ -1,8 +0,0 @@ -SOURCE = Klausur4 -make: - pdflatex -shell-escape $(SOURCE).tex -output-format=pdf - pdflatex -shell-escape $(SOURCE).tex -output-format=pdf - make clean - -clean: - rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg diff --git a/documents/Numerik/Klausur5/Aufgabe1.tex b/documents/Numerik/Klausur5/Aufgabe1.tex deleted file mode 100644 index a969bbb..0000000 --- a/documents/Numerik/Klausur5/Aufgabe1.tex +++ /dev/null @@ -1,106 +0,0 @@ -\section*{Aufgabe 1} -\paragraph{Gegeben:} - -\[A = \begin{pmatrix} - 2 & 3 & -1\\ - -6 & -5 & 0\\ - 2 & -5 & 6 -\end{pmatrix},\;\;\; b = \begin{pmatrix}20\\-41\\-15\end{pmatrix}\] - -\paragraph{LR-Zerlegung:} - -\begin{align} - &&A^{(0)} &= \begin{gmatrix}[p] - 2 & 3 & -1\\ - -6 & -5 & 0\\ - 2 & -5 & 6 - \rowops - \swap{0}{1} - \end{gmatrix}\\ - P^{(1)} &= \begin{pmatrix} - 0 & 1 & 0\\ - 1 & 0 & 0\\ - 0 & 0 & 1 - \end{pmatrix} - &A^{(1)} &= - \begin{gmatrix}[p] - -6 & -5 & 0\\ - 2 & 3 & -1\\ - 2 & -5 & 6 - \rowops - \add[\cdot \frac{1}{3}]{0}{1} - \add[\cdot \frac{1}{3}]{0}{2} - \end{gmatrix}\\ - L^{(2)} &=\begin{pmatrix} - 1 & 0 & 0\\ - \nicefrac{1}{3} & 1 & 0\\ - \nicefrac{1}{3} & 0 & 1 - \end{pmatrix}, - & A^{(2)} &= \begin{gmatrix}[p] - -6 & -5 & 0\\ - 0 & \frac{4}{3} & -1\\ - 0 & -\frac{20}{3} & 6 - \rowops - \swap{1}{2} - \end{gmatrix}\\ - P^{(3)} &= \begin{pmatrix} - 1 & 0 & 0\\ - 0 & 0 & 1\\ - 0 & 1 & 0 - \end{pmatrix}, - & A^{(3)} &= \begin{gmatrix}[p] - -6 & -5 & 0\\ - 0 & -\frac{20}{3} & 6\\ - 0 & \frac{4}{3} & -1 - \rowops - \add[\cdot \frac{1}{5}]{1}{2} - \end{gmatrix}\\ - L^{(4)} &= \begin{pmatrix} - 1 & 0 & 0\\ - 0 & 1 & 0\\ - 0 & \nicefrac{1}{5} & 1 - \end{pmatrix}, - & A^{(4)} &= \begin{gmatrix}[p] - -6 & -5 & 0\\ - 0 & -\frac{20}{3} & 6\\ - 0 & 0 & \nicefrac{1}{5} - \end{gmatrix} =:R -\end{align} - -Es gilt nun: - -\begin{align} - P :&= P^{(3)} \cdot P^{(1)}\\ - &= \begin{pmatrix} - 1 & 0 & 0\\ - 0 & 0 & 1\\ - 0 & 1 & 0 - \end{pmatrix} \cdot \begin{pmatrix} - 0 & 1 & 0\\ - 1 & 0 & 0\\ - 0 & 0 & 1 - \end{pmatrix} \\ - &= - \begin{pmatrix} - 0 & 1 & 0\\ - 0 & 0 & 1\\ - 1 & 0 & 0 - \end{pmatrix}\\ - L^{(4)} \cdot P^{(3)} \cdot L^{(2)} \cdot P^{(1)} \cdot A &= R\\ -L^{-1} &= L^{(4)} \cdot \hat{L_1}\\ - \hat{L_1} &= P^{(3)} \cdot L^{(2)} \cdot (P^{(3)})^{-1}\\ -&= P^{(3)} \cdot L^{(2)} \cdot P^{(3)}\\ -&= \begin{pmatrix} - 1 & 0 & 0\\ - \nicefrac{1}{3} & 1 & 0\\ - \nicefrac{1}{3} & 0 & 1 - \end{pmatrix}\\ - L &= (L^{(4)} \cdot \hat{L_1})^{-1}\\ - &= \begin{pmatrix} - 1 & 0 & 0\\ - -\frac{1}{3} & 1 & 0\\ - -\frac{1}{3} & -\frac{1}{5} & 1 -\end{pmatrix} -\end{align} - -Überprüfung mit \href{http://www.wolframalpha.com/input/?i=%7B%7B1%2C+0%2C+0%7D%2C+%7B-1%2F3%2C+1%2C+0%7D%2C+%7B-1%2F3%2C+-1%2F5%2C+1%7D%7D*%7B%7B-6%2C-5%2C0%7D%2C%7B0%2C-20%2F3%2C6%7D%2C%7B0%2C0%2C1%2F5%7D%7D}{Wolfram|Alpha}. diff --git a/documents/Numerik/Klausur5/Aufgabe2.tex b/documents/Numerik/Klausur5/Aufgabe2.tex deleted file mode 100644 index 980e807..0000000 --- a/documents/Numerik/Klausur5/Aufgabe2.tex +++ /dev/null @@ -1,78 +0,0 @@ -\section*{Aufgabe 2} -Zeige die Aussage für $2\times2$ Matrizen durch Gauß-en mit -Spaltenpivotwahl. - -\subsection*{Lösung} -\subsubsection*{Behauptung:} -Für alle tridiagonalen Matrizen gilt: -\begin{enumerate} - \item[(i)] Die Gauß-Elimination erhält die tridiagonale Struktur - \item[(ii)] $\rho_n(A) := \frac{\alpha_\text{max}}{\max_{i,j} |a_{ij}|} \leq 2$ -\end{enumerate} - -\subsubsection*{Beweis:} -\paragraph{Teil 1: (i)} -\begin{align} - A &= \begin{gmatrix}[p] - * & * & & \\ - * & \ddots & \ddots & \\ - & \ddots & \ddots & * \\ - & & * & * - \rowops - \add[\cdot \frac{-a_{21}}{a_{11}}]{0}{1} - \end{gmatrix} -\end{align} - -Offensichtlich ändert diese Operation nur Zeile 2. $a_{21}$ wird zu 0, -$a_{22}$ ändert sich irgendwie, alles andere bleibt unverändert. -Die gesammte Matrix ist keine tridiagonale Matrix mehr, aber die -um Submatrix in $R^{(n-1) \times (n-1)}$ ist noch eine. - -Muss man zuvor Zeile 1 und 2 tauschen (andere Zeilen kommen nicht in -Frage), so ist später die Stelle $a_{21} = 0$, $a_{22}$ ändert sich -wieder irgendwie und $a_{23}$ ändert sich auch. Dies ändert aber nichts -an der tridiagonalen Struktur der Submatrix. - -\paragraph{Teil 2: (ii) für $A \in \mathbb{R}^{2 \times 2}$} -Sei $\begin{pmatrix}a_{11} & a_{12}\\a_{21} & a_{22} \end{pmatrix} \in \mathbb{R}^{2 \times 2}$ -beliebig. - -O.B.d.A sei die Spaltenpivotwahl bereits durchgeführt, also $|a_{11}| \geq |a_{21}|$. - -Nun folgt: - -\begin{align} - \begin{gmatrix}[p] - a_{11} & a_{12}\\ - a_{21} & a_{22} - \rowops - \add[\cdot \frac{-a_{21}}{a_{11}}]{0}{1} - \end{gmatrix}\\ - \leadsto - \begin{gmatrix}[p] - a_{11} & a_{12}\\ - 0 & a_{22} - \frac{a_{12} \cdot a_{21}}{a_{11}} - \end{gmatrix} -\end{align} - -Wegen $|a_{11}| \geq |a_{21}|$ gilt: -\begin{align} - \|\frac{a_{21}}{a_{11}}\| \leq 1 -\end{align} - -Also insbesondere - -\begin{align} - \underbrace{a_{22} - a_{12} \cdot \frac{a_{21}}{a_{11}}}_{\leq \alpha_\text{max}} \leq 2 \cdot \max_{i,j}|a_{ij}| -\end{align} - -Damit ist Aussage (ii) für $A \in \mathbb{R}^{2 \times 2}$ gezeigt. - -\paragraph{Teil 3: (ii) für allgemeinen Fall} - -Aus Teil 2 folgt die Aussage auch direkt für größere Matrizen. -Der worst case ist, wenn man beim Addieren einer Zeile auf eine -andere mit $\max_{i,j}|a_{ij}|$ multiplizieren muss um das erste nicht-0-Element -der Zeile zu entfernen und das zweite auch $\max_{i,j}|a_{ij}|$ ist. -Dann muss man aber im nächsten schritt mit einem Faktor $\leq \frac{1}{2}$ -multiplizieren, erhält also nicht einmal mehr $2 \cdot \max_{i,j}|a_{ij}|$. diff --git a/documents/Numerik/Klausur5/Aufgabe3.tex b/documents/Numerik/Klausur5/Aufgabe3.tex deleted file mode 100644 index 032daa2..0000000 --- a/documents/Numerik/Klausur5/Aufgabe3.tex +++ /dev/null @@ -1,24 +0,0 @@ -\section*{Aufgabe 3} -\subsection*{Teilaufgabe i} -relativer Fehler: - -\begin{align} - \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. - -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 deleted file mode 100644 index 419eea6..0000000 --- a/documents/Numerik/Klausur5/Aufgabe4.tex +++ /dev/null @@ -1,9 +0,0 @@ -\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 deleted file mode 100644 index d917356..0000000 --- a/documents/Numerik/Klausur5/Aufgabe5.tex +++ /dev/null @@ -1,51 +0,0 @@ -\section*{Aufgabe 5} -\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\label{eq:i}\\ - \nicefrac{1}{2} &\stackrel{!}{=} b_2 \cdot c_2 + b_3 c_3\label{eq:ii}\\ - \nicefrac{1}{3} &\stackrel{!}{=} b_2 \cdot c_2^2 + b_3 c_3^2\label{eq:iii}\\ - \nicefrac{1}{4} &\stackrel{!}{=} b_2 \cdot c_2^3 + b_3 c_3^3\label{eq:iv}\\ - \stackrel{\ref{eq:ii}}{\Rightarrow} \frac{\nicefrac{1}{2} - b_3 c_3}{c_2}&\stackrel{!}{=} b_2 \label{eq:ii2}\\ - \stackrel{\ref{eq:ii2} \text{ in } \ref{eq:iii}}{\Rightarrow} \nicefrac{1}{3} &\stackrel{!}{=} (\nicefrac{1}{2} - b_3 c_3) \cdot c_2 + b_3 c_3^2\label{eq:iii2}\\ - \Leftrightarrow \nicefrac{1}{3} - \nicefrac{1}{2} \cdot c_2 &\stackrel{!}{=} b_3 (- c_3 c_2 + c_3^2)\\ - \Leftrightarrow \frac{\nicefrac{1}{3} - \nicefrac{1}{2} \cdot c_2}{c_3^2 - c_3 c_2} &\stackrel{!}{=} b_3 \label{eq:iii3}\\ - \stackrel{\ref{eq:ii2} \text{ in } \ref{eq:iv}}{\Rightarrow} \nicefrac{1}{4} &\stackrel{!}{=} (\nicefrac{1}{2} - b_3 c_3) \cdot c_2^2 + b_3 c_3^3\label{eq:iv2}\\ - \Leftrightarrow \nicefrac{1}{4} - \nicefrac{1}{2} c_2^2&\stackrel{!}{=} b_3 (c_3^3 - c_3 \cdot c_2^2) \label{eq:iv2}\\ - \stackrel{\ref{eq:iii3} \text{ in } \ref{eq:iv2}}{\Rightarrow} \nicefrac{1}{4} - \nicefrac{1}{2} c_2^2&\stackrel{!}{=} \frac{\nicefrac{1}{3} - \nicefrac{1}{2} \cdot c_2}{c_3^2 - c_3 c_2} (c_3^3 - c_3 \cdot c_2^2)\\ - \Leftrightarrow \frac{1}{2} \left (\frac{1}{2} - c_2^2 \right ) &= \frac{(\nicefrac{1}{3} - \nicefrac{1}{2} \cdot c_2)(c_3^2 - c_2^2)}{c_3 - c_2}\\ - \Leftrightarrow \frac{1}{2} - c_2^2 &= (\nicefrac{2}{3} - c_2)(c_3 + c_2)\\ - \Leftrightarrow \frac{1}{2} - c_2^2 &= \nicefrac{2}{3} c_3 + \nicefrac{2}{3} c_2 - c_2 c_3 - c_2^2\\ - \Leftrightarrow \frac{1}{2} - \frac{2}{3} c_2 &= \nicefrac{2}{3} c_3- c_2 c_3\\ - \Leftrightarrow \frac{1}{2} - \frac{2}{3} c_2 &= c_3(\nicefrac{2}{3}- c_2)\\ - \Leftrightarrow c_3 &= \frac{\frac{1}{2} - \frac{2}{3} c_2}{\nicefrac{2}{3}- c_2} = \frac{2c_2-\nicefrac{3}{2}}{3c_2 - 2}\\ -\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 diff --git a/documents/Numerik/Klausur5/Klausur5.pdf b/documents/Numerik/Klausur5/Klausur5.pdf deleted file mode 100644 index f7ee35eb5c5229223bb6b6a5db26e3acf102cc77..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 216085 zcma%?Q>-XK+oiW{+qUg}wr$(CZQJJAwr$(CZF9~{W|B$1|KjVU(%m<8QI)Fo)LQkD z%Ks6iWu#+;BAr_rUV~yIU?8wFv?So+fuff-u{CoxCt&1cXCnAN4-~zqg|)MZBLTgr 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z?6UWSXxlx8A$W`&X8i;32BK?(vT20FuZ7Tiz|0!p<_*KfEr7V~qW6Sx+@@6@uP;&N z;kK|?m{%k(0aPT|6hoC)Ei~pD3JmAZ**|q5TRLrTXj;2{KZ;V^H#e1OIzDrh&mA^r zsnNHj(M#o?*Xw_gKj#YkGJSpy{-t}JAD+^+<=%Q-(eYhJt;(0mQc56C*hl^7^#&+f zI;$J>OyuQ#SyGGf>^U}m@_CG6b=rBfQ84vp<~sspz~mVt$*c8 mKYt>O>^2hlUw>+6mmjBx$B#n-h8e)d@ h(1)$ -gilt, müssen sich $g$ und $h$ im Intervall mindestens ein mal schneiden. -Da beide Funktionen streng monoton sind, schneiden sie sich genau -ein mal. - -Ein Schnittpunkt der Funktion $g,h$ ist äquivalent zu einer -Nullstelle der Funktion $f$. Also hat $f$ genau eine Nullstelle -und diese liegt in $[0,1]$. - -\subsection*{Teilaufgabe ii} - \begin{align} - 2x - e^{-x} &= 0\\ - \Leftrightarrow 2x &= e^{-x}\\ - \Leftrightarrow x &= \frac{1}{2} \cdot e^{-x} = F_1(x) \label{a2iif1}\\ - \stackrel{x \in \mathbb{R}^+}{\Rightarrow} \ln(2x) &= -x\\ - \Leftrightarrow x &= - \ln(2x) = F_2(x)\label{a2iif2} - \end{align} - -Gleichung \ref{a2iif1} zeigt, dass der Fixpunkt von $F_1$ mit der -Nullstelle von $f$ übereinstimmt. - -Gleichung \ref{a2iif2} zeigt, dass der Fixpunkt von $F_1$ mit der -Nullstelle von $f$ übereinstimmt. Da es nur in $[0,1]$ eine Nullstelle -gibt (vgl. Teilaufgabe i), ist die Einschränkung von $x$ auf $\mathbb{R}^+$ -irrelevant. - -Man sollte $F_1$ zur Fixpunktiteration verwenden, da $\ln(x)$ nur für -$x>0$ definiert ist. Bei der Iteration kommt man aber schnell in -einen Bereich, der nicht erlaubt ist (das erlaubte Intervall ist klein; -Rechenungenauigkeit) - -$F_1$ ist auf $[0,1]$ eine Kontraktion mit Kontraktionszahl $\theta = \frac{1}{2}$: - -Nach dem Mittelwertsatz der Differentialrechnung ex. ein $\xi \in (a,b)$ mit $ 0 \leq a < b \leq 1$, sodass -gilt: - - -\begin{align} - \frac{F(b) - F(a)}{b-a} &= f'(\xi) \\ - \Leftrightarrow \frac{F(b) - F(a)}{b-a} &= - \frac{1}{2} e^{- \xi} \\ - \Leftrightarrow \frac{\|F(b) - F(a)\|}{\|b-a\|} &= \frac{1}{2} \frac{1}{e^{\xi}} < \frac{1}{2 e^a} \\ - \Leftrightarrow \|F(b) - F(a)\| &< \frac{1}{2 e^a} |b-a|\\ - \Rightarrow \forall x, y \in [0,1]: |F(x) - F(y)| &< \frac{1}{2} |x-y| -\end{align} - -Die Ableitung $F_2' = -\frac{1}{x}$. Da $F_2(1) \neq 1$ ist $x^* \neq 1$. -Also ist $|F_2'(x^*)| > 1$. Deshalb konvergiert das Iterationsverfahren -definiert durch $F_2$ nicht gegen $x^*$ für Startwerte ungleich $x^*$. - -Gegen $F_2$ spricht auch, dass $\log$ nur auf $\mathbb{R}^+$ definiert -ist. Das kann bei Rundungsfehlern eventuell zu einem Fehler führen. -(vgl. Python-Skript) - -\subsection*{Teilaufgabe iii} -\[x_{k+1} = x_k - \frac{2x_k - e^{-x_k}}{2 + e^{-x_k}}\] - -Laut \href{http://www.wolframalpha.com/input/?i=2x-e%5E(-x)%3D0}{Wolfram|Alpha} ist die Lösung etwa 0.35173371124919582602 diff --git a/documents/Numerik/Klausur6/Aufgabe3.tex b/documents/Numerik/Klausur6/Aufgabe3.tex deleted file mode 100644 index ff336f9..0000000 --- a/documents/Numerik/Klausur6/Aufgabe3.tex +++ /dev/null @@ -1,37 +0,0 @@ -\section*{Aufgabe 3} - -\begin{table}[H] - \begin{tabular}{l|l|l|l|l} - $f_i$ & 8 & 3 & 4 & 8 \\ \hline - $x_i$ & -1 & 0 & 1 & 3 \\ - \end{tabular} -\end{table} - -\subsection*{Teilaufgabe i} -\begin{align} - p(x) = \sum_{i=0}^3 f_i \cdot L_i(x) -\end{align} -mit -\begin{align} - L_0(x) &= \frac{(x-x_1)(x-x_2)(x-x_3)}{(x_0 - x_1)(x_0-x_2)(x_0-x_3)} - = \ldots = \frac{x^3 - 4x^2 + 3x}{-8} \\ - L_1(x) &= \frac{x^3 - 3x^2 - x + 3}{3} \\ - L_2(x) &= \frac{x^3 - 2x^2 - 3x}{-4} \\ - L_3(x) &= \frac{x^3 - x}{24} -\end{align} - -\subsection*{Teilaufgabe ii} -Anordnung der dividierten Differenzen im so genannten Differenzenschema: -\begin{table}[H] - \begin{tabular}{llll} - $f[x_0]=f_0=8$ & ~ & ~ & ~ \\ - $f[x_1]= 3$ & $f[x_0,x_1] = \frac{f[x_0] - f[x_1]}{x_0-x_1} = -5$ & ~ & ~ \\ - $f[x_2] = 4$ & $1$ & $3$ & ~ \\ - $f[x_3] = 8$ & $2$ & $\frac{1}{3}$ & $- \frac{2}{3} $ \\ - \end{tabular} -\end{table} -Also: -\begin{align} - p(x) &= f[x_0] + f[x_0,x_1] \cdot (x-x_0) + f[x_0, x_1, x_2] \cdot (x-x_0) \cdot (x-x_1) \\ & + f[x_0, x_1, x_2, x_3] \cdot (x-x_0) \cdot (x-x_1) \cdot (x-x_2) \\ - &= 8 - 5 \cdot (x-x_0) + 3 \cdot (x-x_0) \cdot (x-x_1) \\ & - \frac{2}{3} \cdot (x-x_0) \cdot (x-x_1) \cdot (x-x_2) -\end{align} diff --git a/documents/Numerik/Klausur6/Aufgabe4.tex b/documents/Numerik/Klausur6/Aufgabe4.tex deleted file mode 100644 index 5e4698c..0000000 --- a/documents/Numerik/Klausur6/Aufgabe4.tex +++ /dev/null @@ -1,14 +0,0 @@ -\section*{Aufgabe 4} -Nach der Substitutionsregel gilt: -\[\int_{x_2}^{x_3} f(x) \mathrm{d}x = (x_3 - x_2) \cdot \int_0^1 f(x_2 + \tau (b-a)) \mathrm{d} \tau\] - -Wenn $f$ ein Polynom vom Grad $q$ war, so ist auch das neue Integral ein Polynom -vom Grad $q$. - -Ein Polynom, das vier Punkte interpoliert, hat maximal Grad 3. -Da wir das Integral über dieses Polynom im Bereich $[x_2, x_3]$ -exakt berechnen sollen, muss die Quadraturformel vom Grad $p=4$ sein. - - - -TODO diff --git a/documents/Numerik/Klausur6/Aufgabe5.tex b/documents/Numerik/Klausur6/Aufgabe5.tex deleted file mode 100644 index 1647fbd..0000000 --- a/documents/Numerik/Klausur6/Aufgabe5.tex +++ /dev/null @@ -1,7 +0,0 @@ -\section*{Aufgabe 5} - -Das \href{https://de.wikipedia.org/wiki/Explizites_Euler-Verfahren}{explizite Euler-Verfahren} -dient der numerischen Lösung eines Anfangswertproblems (Differentialgleichungen). - -Wir haben das nicht in der Vorlesung gemacht, also wird das wohl nicht -relevant sein. diff --git a/documents/Numerik/Klausur6/Klausur6.pdf b/documents/Numerik/Klausur6/Klausur6.pdf 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z6|DYe&f667(^YitJKkL`N*Vsrf8wHlu5bCrhMmq3y+L403V|33xM%_pZA~{2*q(He z`1=q9#sMx600uxfnP;d}IsidvbDaFl?-JfrPU<&k{Z5$zF8bOq6C_MWOGgupgqWaV zS~^HH3~C00pM=A8bWBW*4T1kI;l!~fQhYcp1&1R4*#G}9VxriK#9;B_Ei3GXkio;| z*~Y=Rmc7{hrdY{?u6tgjIv5A4So3k0=RI%!WTRdZcRfZaeE)Z8$2BJg1 zcllT0C3`4B{nz|7vgI!L@6AMp?}78Vr=>bXu&?;a`pRYip$pEQWs@mP6Jtl#lElsD z`>%BMP$^d&eWPdr$26XsEwU@E0uG-F;HDVarR`2F=;e#mO{%&^&2`(TjYHR(mDgH_ znGH-~%1PAd)x9->A(I*hnne!0jP^?`EcE)s6c1mN^Ds$X0R3h*0#7+*jafale z`g3*7J@ZDcp+%k;M)v+qRSf5|TZc1nMF#6Dm+Z0ayM+Y|j4^)D1EmCN+SDhC|Jqr{ z78QPMO`$^6M4Yd*f=nHOO0HWHagAKzKU+T}J;bVWsfk#f3PSZkq(kOqBi~sh|Npy^ ZM(2=uD2JLsP)#TjhTzPHn_ 0: - x = iterate(x, times-1) - - return x - -print(iterate(0.5, 6)) diff --git a/documents/Numerik/README.md b/documents/Numerik/README.md index 9f3f6e6..35985a1 100644 --- a/documents/Numerik/README.md +++ b/documents/Numerik/README.md @@ -1,23 +1 @@ -Diese Lösungen sind noch im Aufbau. - -Wenn du einen Fehler findest (auch Textsetzungs- und Rechtschreibfehler -oder missverständliche Stellen) -oder selbst eine Lösung geschrieben hast, kannst du mir eine E-Mail -schreiben (info@martin-thoma.de). Oder du machst direkt einen Pull-Request. - -Credits -======= -Detailliert in den Commit-Nachrichten, aber die beitragenden waren: - -* Felix Benz-Baladas -* Martin Thoma -* Peter Merkert - -Korrektur gelesen -================= -Da diese Lösungen von Studenten geschrieben wurden, wäre es toll wenn viele -Leute sie korrektur-lesen würden! Wenn ihr eine **komplette** Klausur -korrektur-gelesen habt, dann schreibt mir doch bitte eine E-Mail, in der steht -welche Version (z.B. bae5d05c67191192941ef7b36109aef4e4dd0e07 von Klausur1.pdf) -ihr korrektur gelesen habt und wo ihr denkt Fehler zu sehen. Diese können wir -dann gemeinsam besprechen korrigieren. +See https://github.com/MartinThoma/KIT-Musterloesungen \ No newline at end of file diff --git a/documents/Numerik/UB11/Aufgabe31.tex b/documents/Numerik/UB11/Aufgabe31.tex deleted file mode 100644 index 1f28fba..0000000 --- a/documents/Numerik/UB11/Aufgabe31.tex +++ /dev/null @@ -1,100 +0,0 @@ -\section*{Aufgabe 31} -\subsection*{Gesucht:} -Eine Quadraturformel maximaler Ordnung mit: -\begin{align} - s &= 3\\ - c_1 &= 0\\ - c_3 &= 1\\ -\end{align} - -\subsection*{Lösung:} - -Nach Satz 28 können Ordnungen $\geq s = 3$ erreicht werden. - -Die Ordnung kann nach Satz 31 höchstens $2s = 6$ sein. Da $c_1 = 0$ -ist, kann es jedoch keine Gauß-Quadraturformel sein. Also kann -die Ordnung höchstens $5$ sein. - -\subsubsection*{Ordnung 5} - -Es gibt mindestens zwei Möglichkeiten, zu zeigen, dass es keine -QF der Ordnung 5 mit den Knoten $c_1 = 0$ und $c_3 = 1$ gibt: -Mit hilfe von Satz 29 oder über die Ordnungsbedingungen. - -\paragraph*{Mit Satz 29} - -\begin{align} - M(x) &= (x-c_1) (x-c_2) (x-c_3)\\ - &= x (x-c_2) (x-1)\\ - &= (x^2- x) (x-c_2)\\ - &= x^3 - (1+c_2)x^2 + c_2 x\\ - \int_0^1 M(x) \cdot g(x) \mathrm{d} x &\stackrel{!}{=} 0 -\end{align} - -Da wir Ordnung $5 = s + 2$ erreichen wollen, muss $g$ ein beliebiges -Polynom vom Grad $\leq 2-1 = 1$ sein. Also: -\begin{align} - g(x) &= ax + b\\ - M(x) \cdot g(x) &= ax^4 + (b-a-ac_2)x^3 + (ac_2-bc_2-b)x^2 + b c_2 x\\ - \int_0^1 M(x) g(x) \mathrm{d} x &= \frac{a}{5} + \frac{b-a-ac_2}{4} + \frac{ac_2 - bc_2-b}{3} + \frac{b c_2}{2}\\ - &= \frac{a c_2}{12}-\frac{a}{20}+\frac{b c_2}{6}-\frac{b}{12}\\ - 0 &\stackrel{!}{=}\frac{a c_2}{12}-\frac{a}{20}+\frac{b c_2}{6}-\frac{b}{12}\\ - \Leftrightarrow 0 &\stackrel{!}{=} 5 a c_2 - 3a + 10 b c_2 - 5 b\\ - \Leftrightarrow -5 a c_2 - 10 b c_2&\stackrel{!}{=} - 3a - 5 b\\ - \Leftrightarrow 5 a c_2 + 10 b c_2&\stackrel{!}{=} 3a + 5 b\\ - \Leftrightarrow c_2(5 a + 10 b)&\stackrel{!}{=} 3a + 5 b\\ - \Leftrightarrow c_2 &\stackrel{!}{=} \frac{3a + 5 b}{5 a + 10 b} -\end{align} - -Da diese Bedingung für alle $a, b \in \mathbb{R}$ gelten soll, muss -sie auf jeden Fall für $a=1, b=0$ sowie für $a=1, b=1$ gelten. Aber: - -\begin{align} - \frac{2\cdot1+5\cdot0}{5\cdot1+10\cdot0} = \frac{3}{5} &\neq \frac{8}{15} = \frac{3\cdot1+5\cdot1}{5\cdot1+10\cdot1} -\end{align} - -Offensichtlich gibt also es kein $c_2$, dass diese Bedingung für jedes $a,b \in \mathbb{R}$ -erfüllt. Daher kann es keine Quadraturformel der Ordnung $5$ mit den Knoten -$0$ und $1$ geben. - -\paragraph*{Mit Ordnungsbedingungen} -Wir kennen $c_1 = 0$ und $c_3=1$, was die Ordnungsbedingungen -sehr vereinfacht: -\begin{align} - 1 &\stackrel{!}{=} b_1 + b_2 + b_3\\ - \nicefrac{1}{2} &\stackrel{!}{=} b_2 \cdot c_2 + b_3 \label{eq:bed2}\\ - \nicefrac{1}{3} &\stackrel{!}{=} b_2 \cdot c_2^2 + b_3 \label{eq:bed3}\\ - \nicefrac{1}{4} &\stackrel{!}{=} b_2 \cdot c_2^3 + b_3\\ - \nicefrac{1}{5} &\stackrel{!}{=} b_2 \cdot c_2^4 + b_3 -\end{align} - -Aus \ref{eq:bed2} folgt: -\begin{align} - c_2 &= \frac{\nicefrac{1}{2} - b_3}{b_2} -\end{align} - -Und damit: -\begin{align} - \nicefrac{1}{3} &\stackrel{!}{=} b_2 \cdot \left (\frac{\nicefrac{1}{2} - b_3}{b_2} \right )^2 + b_3\\ - &= \frac{(\nicefrac{1}{2} - b_3)^2}{b_2} + b_3\\ -\Leftrightarrow \frac{1}{3} b_2 - b_2 b_3&= (\nicefrac{1}{2} - b_3)^2\\ -\Leftrightarrow b_2 (\frac{1}{3} - b_3) &= (\nicefrac{1}{2} - b_3)^2\\ -\Leftrightarrow b_2 &= \frac{(\nicefrac{1}{2} - b_3)^2}{\frac{1}{3} - b_3} -\end{align} - -Nun könnte man das ganze in die 4. Ordnungsbedinung einsetzen \dots aber ich -glaube nicht, dass das schön wird. Mache das, wer will. - -\subsubsection*{Ordnung 4} -Die Simpson-Regel erfüllt offensichtlich alle Bedinungen und hat -Ordnung 4: - -\begin{align} - c_2 &= \nicefrac{1}{2}\\ - b_1 &= \nicefrac{1}{6}\\ - b_2 &= \nicefrac{4}{6}\\ - b_3 &= \nicefrac{1}{6} -\end{align} - -Dass die Simpson-Regel Ordnung 4 hat, lässt sich schnell über -die Ordnungsbedingungen zeigen. diff --git a/documents/Numerik/UB11/Makefile b/documents/Numerik/UB11/Makefile deleted file mode 100644 index 82e184d..0000000 --- a/documents/Numerik/UB11/Makefile +++ /dev/null @@ -1,8 +0,0 @@ -SOURCE = UB11 -make: - pdflatex -shell-escape $(SOURCE).tex -output-format=pdf - pdflatex -shell-escape $(SOURCE).tex -output-format=pdf - make clean - -clean: - rm -rf $(TARGET) *.class *.html *.log *.aux *.out *.pyg diff --git a/documents/Numerik/UB11/UB11.pdf b/documents/Numerik/UB11/UB11.pdf deleted file mode 100644 index 68766587aaf0151f75c3fa517c2b13e41ce832d4..0000000000000000000000000000000000000000 GIT binary patch literal 0 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