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+\author{Leon Burgard, Christian Merten} + +\usepackage[]{gauss} + +\begin{document} + +\punkte + +\begin{aufgabe} + Sei + \[ + f\colon \begin{pmatrix} x \\ y \\ z \end{pmatrix} \mapsto \begin{pmatrix} u \\ v \\ w \end{pmatrix} + \coloneqq \begin{pmatrix} yz \\ x + 2z \\ xy \end{pmatrix} + .\] + Dann ist die Umkehrfunktion $g = f^{-1}$ gegeben als + \begin{align*} + g(u, v, w) = \begin{pmatrix} \frac{wv}{w + 2u} \\ \frac{w + 2u}{v} \\ \frac{vu}{w + 2u} \end{pmatrix} + .\end{align*} Denn $\forall x, y, z \in \R$ gilt + \begin{align*} + g(f(x,y,z)) = g(yz, x+2z, xy) = \begin{pmatrix} x \\ y \\ z \end{pmatrix} + .\end{align*} + Analog für $f(g(u,v,w)) = (u,v,w)^{T}$. + + Für die Jacobimatrizen folgt + \begin{align*} + D_f(x,y,z) = \begin{pmatrix} 0 & z & y \\ 1 & 0 & 2 \\ y & x & 0 \end{pmatrix} + \quad + \text{und} + \quad + D_g(u,v,w) = \begin{pmatrix} - 2 \frac{wv}{(w + 2u)^2} & \frac{w}{w+2u} & \frac{2uv}{(w + 2u)^2}\\ + \frac{2}{v} & -\frac{2 + 2u}{v^2} & \frac{1}{v} \\ + \frac{vw}{(w + 2u)^2} & \frac{u}{w + 2u} & -\frac{vu}{(w + 2u)^2}\end{pmatrix} + .\end{align*} + Für den Punkt $(x,y,z)^{T} = (2,1,0)$ folgt $f(2,1,0) = (0,2,2)^{T}$. Damit folgt + \begin{align*} + D_f(2,1,0) = \begin{pmatrix} + 0 & 0 & 1 \\ + 1 & 0 & 2 \\ + 1 & 2 & 0 + \end{pmatrix} + \quad + \text{und} + \quad + D_g(0,2,2) = \begin{pmatrix} + -2 & 1 & 0 \\ + 1 & -\frac{1}{2} & \frac{1}{2} \\ + 1 & 0 & 0 + \end{pmatrix} + .\end{align*} + Damit folgt + \begin{align*} + D_f(2,1,0) D_g(0,2,2) = + \begin{pmatrix} + 0 & 0 & 1 \\ + 1 & 0 & 2 \\ + 1 & 2 & 0 + \end{pmatrix} + \begin{pmatrix} + -2 & 1 & 0 \\ + 1 & -\frac{1}{2} & \frac{1}{2} \\ + 1 & 0 & 0 + \end{pmatrix} + = + \begin{pmatrix} 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1\end{pmatrix} + .\end{align*} + Also $D_f(2,1,0)^{-1} = D_g(0,2,2)$. +\end{aufgabe} + +\begin{aufgabe} + Sei + \[ + P\coloneqq \{(x,y,z)^{T} \in \R^{3} \mid x + y - z = 1\} + \quad + \text{und} + \quad + Z \coloneqq \{(x,y,z)^{T} \in \R^{3} \mid x^2 + y^2 = 1\} + .\] Setze + \begin{align*} + &f\colon \R^{3} \to \R, x \mapsto \Vert x \Vert_2^2 \\ + &g\colon \R^{3} \to \R^2, x \mapsto \begin{pmatrix} x+y-z-1 \\ x^2 + y^2 - 1 \end{pmatrix} + .\end{align*} + Dann ist die Nebenbedingung äquivalent zu $g(x) = 0$. Minimiere nun $f(x)$ unter $g(x) = 0$. + Es gilt + \begin{align*} + J_g(x) = \begin{pmatrix} 1 & 1 & -1 \\ 2x & 2y & 0 \end{pmatrix} + \quad + \text{und} + \quad + \nabla f(x) = \begin{pmatrix} 2x \\ 2y \\ 2z \end{pmatrix} + .\end{align*} + Es gilt $\text{Rg}(J_g(x)) = 2$, also ist mit Lagrangeregel notwendige Bedingung für + Minimum: + \[ + J_f(\hat{x})^{T} \lambda = \nabla f(\hat{x}) + .\] + Damit folgt das Gleichungssystem + \begin{align} + x + y - z - 1 &= 0 \\ + x^2 + y^2 - 1 &= 0 \\ + \lambda_1 + 2 x \lambda_2 &= 2x \\ + \lambda_1 + 2y \lambda_2 &= 2y \\ + - \lambda_1 &= 2z + .\end{align} + Sei zunächst $x \neq y$. Ziehe (4) von (3) ab. Damit folgt + \begin{align*} + \lambda_2 (x - y) = x - y \implies \lambda_2 = 1 + .\end{align*} + Aus (5) folgt direkt $\lambda_1 = - 2z$. Damit folgt mit (3): + \begin{align*} + - 2z +2x = 2x \implies -2z = 0 \implies z = 0 + .\end{align*} + Eingesetzt in (1) und in (2) ergibt das + \[ + x + y - 1 = 0 \implies x = 1 - y \stackrel{\text{(2)}}{\implies} (1-y)^2 + y^2 - 1 = 0 \implies y (y-1) = 0 \implies y_1 = 0 \land y_2 = 1 + .\] Damit ergeben sich $x_1 = 1 - y_1 = 1$ und $x_2 = 1 - y_2 = 0$, also + $P_1 = (1, 0, 0)^{T}$ und $P_2 = (0,1,0)^{T}$. Hier gilt + $f(P_1) = \Vert P_1\Vert_2^2 = 1$ und $f(P_2) =\Vert P_2 \Vert_2^2 = 1$. + + Falls nun $x = y$. Dann folgt aus (2) direkt $x = \pm \frac{1}{\sqrt{2} }$ und damit mit (1) +$z = \pm \frac{2}{\sqrt{2} } - 1$. Allerdings ist dann bereits $f(\pm \frac{1}{\sqrt{2}}, \pm \frac{1}{\sqrt{2}}, \pm \frac{2}{\sqrt{2} } - 1) > 1$, d.h. dies kann kein Minimum sein. + + Es bleiben also $P_1$ und $P_2$. Da $f(P_1) = f(P_2)$ sind beide Minima. +\end{aufgabe} + +\begin{aufgabe} + Definiere + \begin{align*} + &f(x) \colon \R^{3} \to \R, x \mapsto 2x_1^2 + 2x_1 x_3 + 2x_2^2 + x_2x_3 + 3x_3^2 + 3x_1 - 8x_2 + 2x_3 \\ + &g(x) \colon \R^{3} \to \R^2, x \mapsto \begin{pmatrix} -x_1 + 3x_2 - 2x^{3} - 7 \\ + -3x_1 + 2x_2 -x_3 -2 + \end{pmatrix} + .\end{align*} + Damit ist das gegebene Optimierungsproblem äquivalent zu der Minimierung von $f$ unter $g(x) = 0$. + + Mit + \[ + Q \coloneqq \begin{pmatrix} 4 & 0 & 2 \\ + 0 & 4 & 1 \\ + 2 & 1 & 6 + \end{pmatrix} + \quad + \text{und} + \quad + c \coloneqq \begin{pmatrix} -3 \\ 8 \\ -2 \end{pmatrix} + .\] folgt + \[ + f(x) = \frac{1}{2} x^{T} Qx - c^{T} x + .\] Mit + \[ + A \coloneqq \begin{pmatrix} -1 & 3 & -2 \\ -3 & 2 & -1 \end{pmatrix} + \quad + \text{und} + \quad + b \coloneqq \begin{pmatrix} 7 \\ 2 \end{pmatrix} + \] folgt + \[ + g(x) = Ax - b + .\] + Es ist $\forall x \in \R^{3}$ + \begin{salign*} + f(x) &= \frac{1}{2} x^{T} Q x - c^{T} x \\ + &= \frac{1}{2} \sum_{i=1}^{n} x_i (Qx)_i - \sum_{i=1}^{n} c_i x_i \\ + &= \frac{1}{2} \sum_{i=1}^{n} x_i \sum_{j=1}^{n} Q_{ij} x_j - \sum_{i=1}^{n} c_i x_i \\ + &= \frac{1}{2} \left[ \sum_{i=1}^{n} Q_{ii} x_i^2 + \sum_{i,j=1, i\neq j}^{n} Q_{ij} x_i x_j \right] + - \sum_{i=1}^{n} c_i x_i + \intertext{Da $Q$ symmetrisch, folgt $Q_{ij} = Q_{ji}$, also} + \frac{\partial f}{\partial x_i} + &= \frac{1}{2} \left[ 2 Q_{ii} x_i + 2 \sum_{j=1}^{n} Q_{ij} x_j \right] - c_i \\ + &= \sum_{j=1}^{n} Q_{ij} x_j - c_i \\ + &= (Qx)_i - c_i + \intertext{Insgesamt folgt} + \nabla f(x) &= Qx - c + .\end{salign*} + Mit der Definition von $g$ und $A$ folgt außerdem direkt $J_g(x) = A$. Wegen $\text{Rg}(A) = 2$ folgt + mit der Lagrangeregel und die Bedingung für ein Minimum für $\lambda \in \R^2$: + \begin{align*} + A^{T} \lambda &= Qx - c \\ + Ax &= b + .\end{align*} + Löse zunächst $Ax = b$: + \begin{align*} + \begin{gmatrix}[p] -1 & 3 & -2 & 7 \\ + -3 & 2 & -1 & 2 + \rowops + \add[-3]{0}{1} + \end{gmatrix} + \to + \begin{gmatrix}[p] + -1 & -3 & 2 & -7 \\ + 0 & -7 & 5 & -19 + \rowops + \mult{0}{-1} + \mult{1}{-\frac{1}{7}} + \add[3]{1}{0} + \end{gmatrix} + \to + \begin{gmatrix}[p] + 1 & 0 & -\frac{1}{7} & \frac{8}{7} \\ + 0 & 1 & -\frac{5}{7} & \frac{19}{7} + \end{gmatrix} + .\end{align*} + Damit folgt als Lösungsmenge + \[ + L = \begin{pmatrix} \frac{8}{7} \\ \frac{19}{7} \\ 0 \end{pmatrix} + + \text{Lin}\left( \begin{pmatrix} -\frac{1}{7} \\ -\frac{5}{7} \\ -1 \end{pmatrix} \right) + = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \text{Lin}\left( \begin{pmatrix} 1 \\ 5 \\ 7 \end{pmatrix} \right) + .\] Also folgt für $a \in \R$: + \[ + x = \begin{pmatrix} 1 + a \\ 2 + 5a \\ -1 + 7a \end{pmatrix} + .\] Setze jetzt in $A^{T} \lambda = Qx - c$ ein: + \begin{align*} + A^{T} \lambda &= Qx -c + = \begin{pmatrix} 5 + 18a \\ -1 + 27 a \\ 49 a \end{pmatrix} + .\end{align*} + und löse + \[ + \begin{pmatrix} -1 & -3 & -18 \\ + 3 & 2 & -27 \\ + -2 & -1 & -49 + \end{pmatrix} + \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ a \end{pmatrix} + = + \begin{pmatrix} 5 \\ -1 \\ 0 \end{pmatrix} + .\] Kurze Rechnung ergibt $a = 0$. Damit folgt + \[ + x = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + .\] Da $f(1, 2, -1) = -12$ und für $y = (2,7,6)^{T} \in L$ ist $f(y) = 242 > -12$ folgt + $x = (1,2,-1)^{T}$ löst das Minimierungsproblem. +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Definiere: + \begin{align*} + x_1 &\coloneqq v_1(t) \\ + x_2 &\coloneqq v_1'(t) \\ + x_3 &\coloneqq v_2(t) \\ + x_4 &\coloneqq v_2'(t) \\ + x_5 &\coloneqq v_2''(t) \\ + x_6 &\coloneqq v_2'''(t) + .\end{align*} + Das gegebene Gleichungssystem lässt sich dann so formulieren + \begin{align*} + x_6'(t) - a ( g(t) - b x_3(t)) &= f(t) \\ + x_2'(t) + b x_3(t) &= g(t) + .\end{align*} + Dann folgt für das gegebene Gleichungssystem das äquivalente Gleichungssystem 1. Ordnung: + \begin{align*} + x' = \begin{pmatrix} x_2 \\ + g(t) - b x_3(t) \\ + x_4 \\ + x_5 \\ + x_6 \\ + a (g(t) - b x_3(t)) + f(t) + \end{pmatrix} + .\end{align*} + \item Seien $u_1$, $u_2$ Lösungen von $(*)$. Dann gilt + \[ + W(t) = \text{det} \begin{pmatrix} u_1(t) & u_2(t) \\ + \frac{\d}{\d t} u_1(t) & \frac{\d}{\d t} u_2(t) + \end{pmatrix} + = u_1 \frac{\d}{\d t} u_2(t) - u_2(t) \frac{\d}{\d t} u_1(t) + .\] Damit folgt + \begin{salign*} + \frac{\d}{\d t} W(t) &= \frac{\d}{\d t}\left( u_1 \frac{\d}{\d t}u_2 \right) + - \frac{\d}{\d t} \left( u_2 \frac{\d}{\d t} u_1 \right) \\ + &= \left( \frac{\d}{\d t} u_1 \right)\left( \frac{\d}{\d t} u_2 \right) + + u_1 \frac{\mathrm{d}^2}{\d t^2} u_2 - \left( \frac{\d}{\d t} u_2 \right) + \left( \frac{\d}{\d t} u_1 \right) + - u_2 \frac{\mathrm{d}^2}{\d t^2} u_1 \\ + &= + u_1 \frac{\mathrm{d}^2}{\d t^2} u_2 + - u_2 \frac{\mathrm{d}^2}{\d t^2} u_1 \\ + &\stackrel{(*)}{=} + u_1 \left[ - p \frac{\d}{\d t} u_2 - q u_2 \right] + - u_2 \left[ - p \frac{\d}{\d t} u_1 - q u_1 \right] \\ + &= - u_1 p \frac{\d }{\d t}u_2 - u_1 q u_2 + u_2 p \frac{\d }{\d t} u_1 + u_2 q u_1 \\ + &= -p \left( u_1 \frac{\d}{\d t} u_2 - u_2 \frac{\d }{\d t} u_1\right) \\ + &= -p W(t) + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\end{document}