CE update

ws2012/CE/Klausurzettel.rtf

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+{\rtf1\ansi\ansicpg1252\cocoartf1187\cocoasubrtf340
+{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\paperw11900\paperh16840\margl1440\margr1440\vieww10800\viewh8400\viewkind0
+\pard\tx566\tx1133\tx1700\tx2267\tx2834\tx3401\tx3968\tx4535\tx5102\tx5669\tx6236\tx6803\pardirnatural
+
+\f0\fs24 \cf0 \
+- Berechnen einer L\'f6sung einer DGL (allgemeine und spezielle L\'f6sung)\
+- Formeln f\'fcr Zeitcharakteristika (-> Steifheitsma\'df)\
+- Formeln f\'fcr alle m\'f6glichen Approximationsverfahren (Euler-, Runge-Kutta und co.)\
+	-> explizites / implizites Euler-Verfahren mit mehrdimensionalem System\
+	(vgl. \'dcbung 7 Aufgabe 2 d)\
+- PD-Regelung linearer System (\'fcberkritisch ged\'e4mpft, kritisch ged\'e4mpft, unterkritisch ged\'e4mpft)\
+	(vgl. \'dcbung 12 Aufgabe 2)\
+- R\'e4uber-Beute-Modell -> Probeklausur Aufgabe 3\
+- PQ-Formel / ABC-Formel\
+- Schritte einer Simulationsstudie\
+- Regeln f\'fcr die automatische Modellgenerierung\
+- unklare MC-Fragen\
+- Fragen aus Probeklausur zur Modellierung und Simulation\
+- Formel zu Pseudozufallszahlengenerator\
+- Formel zu Mittelwert und Varianz}

ws2012/CE/offene Fragen.rtf

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+{\rtf1\ansi\ansicpg1252\cocoartf1187\cocoasubrtf340
+{\fonttbl\f0\fswiss\fcharset0 Helvetica;}
+{\colortbl;\red255\green255\blue255;}
+\paperw11900\paperh16840\margl1440\margr1440\vieww11560\viewh11700\viewkind0
+\pard\tx566\tx1133\tx1700\tx2267\tx2834\tx3401\tx3968\tx4535\tx5102\tx5669\tx6236\tx6803\pardirnatural
+
+\f0\b\fs24 \cf0 \'dcbungen 4 Aufgabe 1b):
+\b0 \
+	- Warum ist die erste Ruhelage stabil obwohl der Realteil von Eigenwert lambda_3/4 = 0 ist ?\
+	Es hei\'dft doch hier, dass bei Re(EW) = 0 keine Aussage \'fcber Stabilit\'e4t getroffen werden kann\'85\
+	=> 3,0151i = 3 + 151i laut Wolframalpha.\
+\
+
+\b Was ist eine Referenztrajektorie? 
+\b0 \
+=> Eine Referenztrajektorie gibt einen Ruhebereich in Abh\'e4ngigkeit von der Zeit t an. (z.B. eine stabile/instabile Umlaufbahn um einen Planeten). Daher ist das Einsetzen von expliziten t_s nicht notwendig (-> durch das Einsetzen von t_s erh\'e4lt man eine Ruhelage, falls die Referenztrajektorie stabil ist.) Deshalb setzt man immer die allgemeinen Referenztrajektorie ein!! Vgl. \'dcbung 4 Aufgabe 2 a)\
+\
+\
+\
+
+\b Probleklausur WS10/11:
+\b0 \
+letzte Aufgabe: Mechanik\
+\
+
+\b Probleklausur WS11/12:
+\b0 \
+Aufgabe 4c): Eigenwert f\'fcr d=0 m\'fcsste doch 0 sein und somit keine Aussage \'fcber Stabilit\'e4t m\'f6glich?\
+Aufgabe 5d): Warum kann hier nur t_2 schalten. Laut Definition kann eine Transition schalten, wenn ihre Eing\'e4nge belegt sind.\
+Aufgabe 6 c): Komische Umformung in autonomes System.\
+-> Schaltfunktion bestimmen.\
+-> Richtungsfelder zeichnen\
+\
+\
+
+\b \'dcbung 5 Aufgabe 2a)
+\b0  Addition von Gleitkommazahlen letzte Schritt "Anpassen an die Mantisse"\
+\
+
+\b \'dcbung 6 Aufgabe 2a):
+\b0  hier wird von der "einfachen Form" des Newton-Verfahren gesprochen. Was ist die komplizierte Form? Die Form mit der Matrixschreibweise? \
+\
+
+\b \'dcbung 8 Aufgabe 1c):
+\b0  Heun-Verfahren zeichnerisch L\'f6sen\
+\
+
+\b \'dcbung 12 Aufgabe 2:
+\b0  Verschiedene Schwingungstypen. Wie kommt man von den Eigenwerten auf den Schwingungstyp? PD-Regelung ?!\
+\
+\
+\
+\
+\
+}

ws2012/CE/sharedFolder/.dropbox

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+232391711

ws2012/CE/uebungen/9/andere gruppe/1c.txt

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+q1 = x-22
+q2 = x-18

ws2012/CE/uebungen/P3/.dropbox

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+207435091

ws2012/CE/uebungen/P3/P3Matlab/fixpoint.m

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+function [N,x] = fixpoint (x0)
+% Function fixpoint 
+% In: Start value
+% Out: number of iteration steps, vector of solution
+N=1;
+xCur = x0;
+xOld = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    yTemp = (A - x(3)*eye(2))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2-1;
+    y  = [ yTemp ; lambda];
+end
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+end
+
+function [y] = fixpoint_next(x)
+    y = x - inv(jacobi(x0)) * f(x);
+end
+
+xCur = fixpoint_next(xOld);
+x(:,N) = xCur;
+while(norm(xCur-xOld) > epsilon && N < 60)
+    N = N + 1;
+    xOld = xCur;
+    xCur = fixpoint_next(xOld);
+    x(:,N) = xCur;
+end
+end

ws2012/CE/uebungen/P3/P3Matlab/main.m

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+function main();
+
+% Main function for iteration of static values in 
+% eigenvalue problem
+%
+%
+% Grundlagen der Modellierung und Simulation
+% WiSe 2012/13
+
+
+%%%%%%%%%%%%%%%%%%%%%% Start Values %%%%%%%%%%%%%%%%%%%%
+
+x = [-3; 0; 0]
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+x_fixpunkt = x;
+x_newton = x;
+x_qnewton = x;
+
+%%%%%%%%%%%%%%%%%%% Fixpunkt %%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_f, x_f] = fixpoint(x_fixpunkt);
+
+%%%%%%%%%%%%%%%%%%% Newton %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_n, x_n] = newton(x_newton);
+
+%%%%%%%%%%%%%%%%%%% Quasi-Newton %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_qn, x_qn] = quasi_newton(x_qnewton);
+
+
+%%%%%%%%%%%%%%%%%%% Output %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+disp('Startwert x');
+x
+
+disp('Anzahl Iterationen bei Fixpunkt:');
+it_f
+disp('Loesung:');
+x_f(:,it_f)
+%x_f
+
+disp('Anzahl Iterationen bei Newton:');
+it_n
+disp('Loesung:');
+x_n(:,it_n)
+%x_n
+disp('Anzahl Iterationen bei Quasi-Newton:');
+it_qn
+disp('Loesung:');
+x_qn(:,it_qn)
+
+
+semilogy(1:it_n, abs(x_n(3,:)-2), ...
+    1:it_f, abs(x_f(3,:)-2), ...
+    1:it_qn, abs(x_qn(3,:)-2)) 
+title('Iterationsverfahren');
+legend('error (Newton) ','error (FixP)','error (Quasi-Newton)')
+xlabel('Anzahl Iterationen');
+ylabel('Fehler in lambda');
+

ws2012/CE/uebungen/P3/P3Matlab/myeigen.m

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+function [out] = fmyeigen(x)
+% Defines a function which has eigenvector/eigenvalue as roots
+% x(1) : first component of eigenvector
+% x(2) : second component of eigenvector
+% x(3) : eigenvalue
+
+out=[(3-x(3))*x(1)+4*x(2); x(1)+(6-x(3))*x(2); x(1)*x(1)+x(2)*x(2)-1];
+
+end
+

ws2012/CE/uebungen/P3/P3Matlab/newton.m

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+function [N, x] = newton(x0)
+% Function newton
+% In: Start value
+% Out: number of iteration steps, vector of solution
+N=1;
+xOld = x0;
+xCur = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    yTemp = (A - x(3)*eye(2))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2-1;
+    y  = [ yTemp ; lambda];
+end
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+end
+
+function [y] = newton_next(x)
+    y = x - jacobi(x)\f(x);
+end
+
+xCur = newton_next(xOld);
+x(:,N) = xCur;
+while(norm(xCur-xOld) > epsilon && N < 60)
+    N = N + 1;
+    xOld = xCur;
+    xCur = newton_next(xOld);
+    x(:,N) = xCur;
+end
+end

ws2012/CE/uebungen/P3/P3Matlab/quasi_newton.m

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+function [N,x] = quasi_newton(x0)
+% Function quasi_newton
+% In: Start value
+% Out: number of iteration steps, vector of solution
+N=1;
+xCur = x0;
+xOld = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    yTemp = (A - x(3)*eye(2))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2 - 1;
+    y  = [ yTemp ; lambda];
+end
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+end
+
+function [y] = newton_next(x)
+    y = x + getDeltaX(x);
+end
+
+function [y] = getDeltaX(x)
+    y = jac\(-f(x));
+end
+
+jac = jacobi(x0);
+xCur = newton_next(xOld);
+x(:,N) = xCur;
+while(abs(norm(xCur)-norm(xOld)) > epsilon && N < 60)
+    N = N + 1;
+    xOld = xCur;
+    % anpassen der angenäherten jacobi matrix
+    % ---------------------------------------
+    % Schritt 1 - delta X ausrechnen
+    delta = getDeltaX(xOld);
+    % Schritt 2 - x^k ausrechnen
+    xCur = newton_next(xOld);
+    % Schritt 3 & 4 - f(x^k) = f(xCur) und anpassen der Jacobi Matrix
+    jac = jac + (1/norm(delta)^2)*(f(xCur) - f(xOld) - jac*delta) * delta';
+    x(:,N) = xCur;
+end
+end

ws2012/CE/uebungen/P3/P3Matlab/results.txt

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+Gruppe:
+        - Michael Scholz   (Matr.# 1576630)
+        - David Kaufmann   (Matr.# 1481864)
+        - Dennis Werner    (Matr.# 1513509)
+
+
+Ausgabe der main-Funktion in Matlab:
+
+
+1. Aufruf von main liefert:
+==============================
+
+>> main
+
+x =
+
+    -3
+     0
+     0
+
+Startwert x
+
+x =
+
+    -3
+     0
+     0
+
+Anzahl Iterationen bei Fixpunkt:
+
+it_f =
+
+    47
+
+Loesung:
+
+ans =
+
+   -0.9701
+    0.2425
+    2.0000
+
+Anzahl Iterationen bei Newton:
+
+it_n =
+
+     6
+
+Loesung:
+
+ans =
+
+   -0.9701
+    0.2425
+    2.0000
+
+Anzahl Iterationen bei Quasi-Newton:
+
+it_qn =
+
+    14
+
+Loesung:
+
+ans =
+
+   -0.9701
+    0.2425
+    2.0000
+
+>> 

ws2012/CE/uebungen/P3/P3Scilab/Dave/fixpoint.sc

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+function [N,x] = fixpoint (x0)
+// Function fixpoint 
+// In: Start value
+// Out: number of iteration steps, vector of solution
+N=1;
+xCur = x0;
+xOld = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    // eye(A) muss in matlab durch eye(2) ersetzt werden
+    yTemp = (A - x(3)*eye(A))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2-1;
+    y  = [ yTemp ; lambda];
+endfunction
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+endfunction
+
+function [y] = fixpoint_next(x)
+    y = x - inv(jacobi(x0)) * f(x);
+endfunction
+
+xCur = fixpoint_next(xOld);
+x(:,N) = xCur;
+while(abs(norm(xCur)-norm(xOld)) > epsilon & N < 60)
+    N = N + 1;
+    xOld = xCur;
+    xCur = fixpoint_next(xOld);
+    x(:,N) = xCur;
+end
+endfunction

ws2012/CE/uebungen/P3/P3Scilab/Dave/main.sc

@@ -0,0 +1,65 @@
+function main();
+
+// Main function for iteration of static values in 
+// eigenvalue problem
+//
+//
+// Grundlagen der Modellierung und Simulation
+// WiSe 2012/13
+
+
+//%%%%%%%%%%%%%%%%%%%%%% Start Values %%%%%%%%%%%%%%%%%%%%
+
+x = [-3; 0; 0]
+
+//%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+x_fixpunkt = x;
+x_newton = x;
+x_qnewton = x;
+
+//%%%%%%%%%%%%%%%%%%% Fixpunkt %%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_f, x_f] = fixpoint(x_fixpunkt);
+
+//%%%%%%%%%%%%%%%%%%% Newton %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_n, x_n] = newton(x_newton);
+
+//%%%%%%%%%%%%%%%%%%% Quasi-Newton %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_qn, x_qn] = quasi_newton(x_qnewton);
+
+
+//%%%%%%%%%%%%%%%%%%% Output %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+disp('Startwert x');
+x
+
+disp('Anzahl Iterationen bei Fixpunkt:');
+it_f
+disp('Loesung:');
+x_f(:,it_f)
+%x_f
+
+disp('Anzahl Iterationen bei Newton:');
+it_n
+disp('Loesung:');
+x_n(:,it_n)
+%x_n
+disp('Anzahl Iterationen bei Quasi-Newton:');
+it_qn
+disp('Loesung:');
+x_qn(:,it_qn)
+
+
+semilogy(1:it_n, abs(x_n(3,:)-2), ...
+    1:it_f, abs(x_f(3,:)-2), ...
+    1:it_qn, abs(x_qn(3,:)-2)) 
+title('Iterationsverfahren');
+legend('error (Newton) ','error (FixP)','error (Quasi-Newton)')
+xlabel('Anzahl Iterationen');
+ylabel('Fehler in lambda');
+
+endfunction
+

ws2012/CE/uebungen/P3/P3Scilab/Dave/myeigen.sc

@@ -0,0 +1,10 @@
+function [out] = fmyeigen(x)
+// Defines a function which has eigenvector/eigenvalue as roots
+// x(1) : first component of eigenvector
+// x(2) : second component of eigenvector
+// x(3) : eigenvalue
+
+out=[(3-x(3))*x(1)+4*x(2); x(1)+(6-x(3))*x(2); x(1)*x(1)+x(2)*x(2)-1];
+
+endfunction
+

ws2012/CE/uebungen/P3/P3Scilab/Dave/newton.sc

@@ -0,0 +1,37 @@
+function [N, x] = newton(x0)
+// Function newton
+// In: Start value
+// Out: number of iteration steps, vector of solution
+N=1;
+xOld = x0;
+xCur = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    // eye(A) muss in matlab durch eye(2) ersetzt werden
+    yTemp = (A - x(3)*eye(A))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2-1;
+    y  = [ yTemp ; lambda];
+endfunction
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+endfunction
+
+function [y] = newton_next(x)
+    y = x - jacobi(x)\f(x);
+endfunction
+
+xCur = newton_next(xOld);
+x(:,N) = xCur;
+while(abs(norm(xCur)-norm(xOld)) > epsilon & N < 60)
+    N = N + 1;
+    xOld = xCur;
+    xCur = newton_next(xOld);
+    x(:,N) = xCur;
+end
+endfunction

ws2012/CE/uebungen/P3/P3Scilab/Dave/quasi_newton.sc

@@ -0,0 +1,46 @@
+function [N, x] = quasi_newton(x0);
+// Function quasi_newton
+// In: Start value
+// Out: number of iteration steps, vector of solution
+N=1;
+xCur = x0;
+xOld = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    // eye(A) muss in matlab durch eye(2) ersetzt werden
+    yTemp = (A - x(3)*eye(A))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2 - 1;
+    y  = [ yTemp ; lambda];
+endfunction
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+endfunction
+
+function [y] = newton_next(x)
+    y = x + getDeltaX(x);
+endfunction
+
+function [y] = getDeltaX(x)
+    y = jac\(-f(x));
+endfunction
+
+jac = jacobi(x0);
+xCur = newton_next(xOld);
+x(:,N) = xCur;
+while(abs(norm(xCur)-norm(xOld)) > epsilon & N < 60)
+    N = N + 1;
+    xOld = xCur;
+    // anpassen der angenäherten jacobi matrix
+    delta = getDeltaX(xOld); //schritt 1
+    xCur = newton_next(xOld); //schritt 2
+    //delta'*delta in matlab evtll durch norm(delta,2) austauschen
+    jac = jac + (1/(delta'*delta))*(f(xCur) - f(xOld) - jac*delta) * delta'; //schritt 4
+    x(:,N) = xCur;
+end
+endfunction

ws2012/CE/uebungen/P3/P3Scilab/Dave/results.txt

@@ -0,0 +1,59 @@
+-->[N, X] = newton([-3;0;0])
+ X  =
+ 
+  - 1.6666667  - 1.1121653  - 0.9790265  - 0.9701824  - 0.9701425  - 0.9701425  
+    0.2777778    0.265897     0.2443321    0.2425437    0.2425356    0.2425356  
+    1.2962963    1.7950236    1.976989     1.9998       2.           2.         
+ N  =
+ 
+    6.  
+    
+    
+-->[N, X] = fixpoint([-3;0;0])
+ X  =
+ 
+ 
+         column 1 to 7
+ 
+  - 1.6666667  - 1.3575103  - 1.2033799  - 1.1161244  - 1.0636206  - 1.0308881  - 1.0100171  
+    0.2777778    0.2862654    0.2746085    0.2632281    0.2551597    0.2499909    0.2468379  
+    1.2962963    1.5519547    1.6869252    1.7754082    1.8382905    1.8841094    1.917574   
+ 
+         column  8 to 14
+ 
+  - 0.9965065  - 0.9876671  - 0.9818386  - 0.9779729  - 0.9753975  - 0.9736757  - 0.9725214  
+    0.2449727    0.2438948    0.2432839    0.2429434    0.2427563    0.2426548    0.2426001  
+    1.941856     1.9593059    1.9717187    1.980464     1.986573     1.9908095    1.9937297  
+ 
+         column 15 to 21
+ 
+  - 0.9717459  - 0.9712241  - 0.9708726  - 0.9706355  - 0.9704755  - 0.9703675  - 0.9702945  
+    0.2425708    0.2425551    0.2425466    0.2425420    0.2425394    0.2425379    0.2425371  
+    1.995733     1.9971018    1.9980344    1.9986683    1.9990984    1.99939      1.9995874  
+ 
+         column 22 to 28
+ 
+  - 0.9702453  - 0.9702119  - 0.9701894  - 0.9701742  - 0.9701639  - 0.970157   - 0.9701523  
+    0.2425366    0.2425362    0.2425360    0.2425359    0.2425358    0.2425357    0.2425357  
+    1.999721     1.9998114    1.9998725    1.9999138    1.9999417    1.9999606    1.9999734  
+ 
+         column 29 to 35
+ 
+  - 0.9701491  - 0.9701470  - 0.9701455  - 0.9701445  - 0.9701439  - 0.9701434  - 0.9701431  
+    0.2425357    0.2425357    0.2425357    0.2425356    0.2425356    0.2425356    0.2425356  
+    1.999982     1.9999878    1.9999918    1.9999944    1.9999962    1.9999975    1.9999983  
+ 
+         column 36 to 42
+ 
+  - 0.9701429  - 0.9701428  - 0.9701427  - 0.9701426  - 0.9701426  - 0.9701426  - 0.9701425  
+    0.2425356    0.2425356    0.2425356    0.2425356    0.2425356    0.2425356    0.2425356  
+    1.9999988    1.9999992    1.9999995    1.9999996    1.9999998    1.9999998    1.9999999  
+ 
+         column 43 to 46
+ 
+  - 0.9701425  - 0.9701425  - 0.9701425  - 0.9701425  
+    0.2425356    0.2425356    0.2425356    0.2425356  
+    1.9999999    1.9999999    2.           2.         
+ N  =
+ 
+    46.

ws2012/CE/uebungen/P3/P3Scilab/fixpoint.sc

@@ -0,0 +1,37 @@
+function [N,x] = fixpoint (x0)
+// Function fixpoint 
+// In: Start value
+// Out: number of iteration steps, vector of solution
+N=1;
+xCur = x0;
+xOld = x0;
+epsilon = 10^(-8);
+A = [3,4;1,6];
+
+function [y] = f(x)
+    // eye(A) muss in matlab durch eye(2) ersetzt werden
+    yTemp = (A - x(3)*eye(A))*[x(1);x(2)];
+    lambda = x(1)^2+x(2)^2-1;
+    y  = [ yTemp ; lambda];
+endfunction
+
+function [y] = jacobi(x)
+    y1 = [A(1,1)-x(3) , A(1,2)      , -x(1)];
+    y2 = [A(2,1)      , A(2,2)-x(3) , -x(2)];
+    y3 = [2*x(1)      , 2*x(2)      , 0];
+    y  = [y1;y2;y3];
+endfunction
+
+function [y] = fixpoint_next(x)
+    y = x - inv(jacobi(x0)) * f(x);
+endfunction
+
+xCur = fixpoint_next(xOld);
+x(:,N) = xCur;
+while(norm(xCur-xOld) > epsilon & N < 60)
+    N = N + 1;
+    xOld = xCur;
+    xCur = fixpoint_next(xOld);
+    x(:,N) = xCur;
+end
+endfunction

ws2012/CE/uebungen/P3/P3Scilab/main.sc

@@ -0,0 +1,65 @@
+function main();
+
+// Main function for iteration of static values in 
+// eigenvalue problem
+//
+//
+// Grundlagen der Modellierung und Simulation
+// WiSe 2012/13
+
+
+//%%%%%%%%%%%%%%%%%%%%%% Start Values %%%%%%%%%%%%%%%%%%%%
+
+x = [-3; 0; 0]
+
+//%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+x_fixpunkt = x;
+x_newton = x;
+x_qnewton = x;
+
+//%%%%%%%%%%%%%%%%%%% Fixpunkt %%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_f, x_f] = fixpoint(x_fixpunkt);
+
+//%%%%%%%%%%%%%%%%%%% Newton %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_n, x_n] = newton(x_newton);
+
+//%%%%%%%%%%%%%%%%%%% Quasi-Newton %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+[it_qn, x_qn] = quasi_newton(x_qnewton);
+
+
+//%%%%%%%%%%%%%%%%%%% Output %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+disp('Startwert x');
+x
+
+disp('Anzahl Iterationen bei Fixpunkt:');
+it_f
+disp('Loesung:');
+x_f(:,it_f)
+%x_f
+
+disp('Anzahl Iterationen bei Newton:');
+it_n
+disp('Loesung:');
+x_n(:,it_n)
+%x_n
+disp('Anzahl Iterationen bei Quasi-Newton:');
+it_qn
+disp('Loesung:');
+x_qn(:,it_qn)
+
+
+semilogy(1:it_n, abs(x_n(3,:)-2), ...
+    1:it_f, abs(x_f(3,:)-2), ...
+    1:it_qn, abs(x_qn(3,:)-2)) 
+title('Iterationsverfahren');
+legend('error (Newton) ','error (FixP)','error (Quasi-Newton)')
+xlabel('Anzahl Iterationen');
+ylabel('Fehler in lambda');
+
+endfunction
+

ws2012/CE/uebungen/P3/P3Scilab/myeigen.sc

@@ -0,0 +1,10 @@
+function [out] = fmyeigen(x)
+// Defines a function which has eigenvector/eigenvalue as roots
+// x(1) : first component of eigenvector
+// x(2) : second component of eigenvector
+// x(3) : eigenvalue
+
+out=[(3-x(3))*x(1)+4*x(2); x(1)+(6-x(3))*x(2); x(1)*x(1)+x(2)*x(2)-1];
+
+endfunction
+

ws2012/CE/uebungen/P3/P3Scilab/newton.sc

@@ -0,0 +1,46 @@
+function [N, matrixX] = newton(x0)
+
+// Function newton
+// In: Start value
+// Out: number of iteration steps, vector of solution
+
+// Todo: Replace below code with Newton method
+
+
+bool=1;
+N=1;
+x=x0;
+xTmp = x0;
+e = 10^(-8);
+matrixX = []; //matrix we use for iterations
+matrixA = [3 4;1 6];
+
+//define function f(x) as vectorfunction:
+function [y] = f(x)
+      y = [(matrixA(1,1)-x(3))*x(1) + matrixA(1,2)*x(2) ;
+            matrixA(2,1)*x(1) + (matrixA(2,2)-x(3))*x(2) ;
+            (x(1)^2-1) + (x(2)^2 -1)];
+endfunction
+
+
+while(bool == 1)
+    //calculate x (of current iteration)
+    jac = numdiff(f, x); //jacobi-matrix
+    deltaX = jac\(-f(x));    
+    x = x + deltaX;
+    
+    //add x to output matrix
+    matrixX(:,N) = x; 
+    
+    //check criteria
+    if(N == 60 | (abs(norm(xTmp)-norm(x)) < e & N >1)) 
+        bool = 0;      //-> will cancel the while-loop
+    end
+
+    N = N+1; //increase N
+    xTmp = x;
+
+end
+
+N = N-1;
+endfunction

ws2012/CE/uebungen/P3/P3Scilab/quasi_newton.sc

@@ -0,0 +1,56 @@
+function [N, matrixX] = quasi_newton(x0);
+
+// Function quasi_newton
+// In: Start value
+// Out: number of iteration steps, vector of solution
+
+// Todo: Replace below code with quasi-Newton method
+
+
+bool=1;
+N=1;
+x=x0;
+xTmp = x0;
+e = 10^(-8);
+matrixX = []; //matrix we use for iterations
+matrixA = [3 4;1 6];
+a = 1;
+
+
+//define function f(x) as vectorfunction:
+function [y] = f(x)
+      y = [(matrixA(1,1)-x(3))*x(1) + matrixA(1,2)*x(2) ;
+            matrixA(2,1)*x(1) + (matrixA(2,2)-x(3))*x(2) ;
+            (x(1)^2) + (x(2)^2) - 1];
+endfunction
+
+
+jac = numdiff(f, x); //jacobi-matrix
+
+
+while(bool == 1)
+    //calculate x (of current iteration)
+
+    deltaX = jac\(-f(x));    
+    x = x + deltaX;
+    
+    //add x to output matrix
+    matrixX(:,N) = x;
+    
+    //calculate jacobi-matrix for next iteration
+    deltaF = f(x) - f(x-deltaX);
+    incr = 1 / (norm(deltaX)^2)*(deltaF-jac*a*deltaX)*(a.*deltaX)';
+    jac = jac + incr;
+    
+    //check criteria
+    if(N == 30 | (abs(norm(xTmp)-norm(x)) < e & N >1)) 
+        bool = 0;      //-> will cancel the while-loop
+    end
+
+    N = N+1; //increase N
+    xTmp = x;
+
+end
+
+N = N-1;
+endfunction

ws2012/CE/uebungen/P4/.dropbox

@@ -0,0 +1 @@
+219911329