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[t,x] = meshgrid(-1:0.2:1, -1:0.2:1); invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit amet. Lorem ipsum dolor sit amet.
function [X]=fkt1000(t, x)
lambda=-1000;
X = lambda*( x - exp(-t) ) - exp(-t);
end
function fRichtungsfeld(fhandle)
% create meshgrid
[t,x] = meshgrid(-1:0.2:1, -1:0.2:1);
% exec function
z = feval(fhandle, t, x);
% draw figure
figure;
quiver(t, x, ones(size(t)), z);
axis([-1,1,-1,1]);
xlabel('t');
ylabel('x');
title('Richtungsfeld, dx=dt=0.2');
legend('Richtungsfeld');
end
classdef Uebung5< handle
% Static Methods
methods(Static=true)
%% Aufgabe 15
function Aufgabe15()
A = [
0.9635, 1.4266;
1.4266, 0.0365
];
x0 = [
0;
1
];
p = PowerMethod(A);
iter = NumericIterator(p, x0);
% error < 10^-5
iter.addAbortConditionListener(ErrorCalculationListener(10^-5));
% start iteration
iter.start();
% print result
fprintf('Eigenwert: %10.10f\nEigenvektor:\n', p.getEigenwert());
Array.print(iter.getResult());
% Approximation Error < 1.000000e-005 @ 19 Iterations
% Eigenwert: 2.0000066033
% Eigenvektor:
% 0.8090129
% 0.5877909
end
%% Aufgabe 16
function Aufgabe16()
% referenz f�r tf=0.95
ref = fAufgabe16Ref();
% startwert
x0 = 1;
t0 = 0;
tf = 0.95;
% output buffer
buffer = {};
j = 1;
% schrittzahl, diskrete werte
n = [20, 100, 200, 1000, 2000, 10000, 20000, 100000];
% draw table
gui = table('Aufgabe 16');
% Verfahren anwenden
for i=1:1:length(n)
% schrittweite
h = (tf-t0)/(n(i)-1);
% set table output
buffer{j, 1} = n(i);
buffer{j, 2} = sprintf('%d', h);
duration = 0;
% euler
e = RungeKuttaMethod(@fAufgabe16, @fRKTEuler, t0, tf, h, x0);
buffer{j, 3} = sprintf('%d', abs(ref-e.getResult())/ref);
duration = duration + e.getDuration();
% heun
e = RungeKuttaMethod(@fAufgabe16, @fRKTHeun, t0, tf, h, x0);
buffer{j, 4} = sprintf('%d', abs(ref-e.getResult())/ref);
duration = duration + e.getDuration();
% mod euler
e = RungeKuttaMethod(@fAufgabe16, @fRKTMEuler, t0, tf, h, x0);
buffer{j, 5} = sprintf('%d', abs(ref-e.getResult())/ref);
duration = duration + e.getDuration();
% klassik RKV, just 4 fun ;)
e = RungeKuttaMethod(@fAufgabe16, @fRKTClassicRKM, t0, tf, h, x0);
buffer{j, 6} = sprintf('%d', abs(ref-e.getResult())/ref);
duration = duration + e.getDuration();
% set duration
buffer{j, 7} = sprintf('%d', duration);
% increment table row counter
j=j+1;
% tabelle updaten
gui.repaint(buffer);
end
% OUTPUT
% Result <h=5.000000e-002>: 6.460597267358655100
% Result <h=5.000000e-002>: 12.511638996887264000
% Result <h=5.000000e-002>: 12.048026035562726000
% Result <h=5.000000e-002>: 14.271173509689078000
% Result <h=9.595960e-003>: 17.964107899072395000
% Result <h=9.595960e-003>: 32.643393403424028000
% Result <h=9.595960e-003>: 32.155786064836953000
% Result <h=9.595960e-003>: 33.992428065470925000
% Result <h=4.773869e-003>: 25.392669207249959000
% Result <h=4.773869e-003>: 40.020849205774859000
% Result <h=4.773869e-003>: 39.753729672927300000
% Result <h=4.773869e-003>: 40.667817564725581000
% Result <h=9.509510e-004>: 41.312660057741127000
% Result <h=9.509510e-004>: 48.111744657920021000
% Result <h=9.509510e-004>: 48.088683279107258000
% Result <h=9.509510e-004>: 48.159528454354081000
% Result <h=4.752376e-004>: 45.338580595642853000
% Result <h=4.752376e-004>: 49.276209783713618000
% Result <h=4.752376e-004>: 49.269803171533532000
% Result <h=4.752376e-004>: 49.289181554821702000
% Result <h=9.500950e-005>: 49.333345578779792000
% Result <h=9.500950e-005>: 50.230354244304941000
% Result <h=9.500950e-005>: 50.230075076019553000
% Result <h=9.500950e-005>: 50.230908884512381000
% Result <h=4.750238e-005>: 49.894671915844583000
% Result <h=4.750238e-005>: 50.350966085905696000
% Result <h=4.750238e-005>: 50.350895538095877000
% Result <h=4.750238e-005>: 50.351105912323895000
% Result <h=9.500095e-006>: 50.355123772675057000
% Result <h=9.500095e-006>: 50.447664014784309000
% Result <h=9.500095e-006>: 50.447661168458090000
% Result <h=9.500095e-006>: 50.447669645483884000
end
%% Aufgabe 17
function Aufgabe17()
x0 = [
1;
0
];
% startwerte
t0 = 0;
tf = 50;
% schrittzahl
n = 1000;
% schrittweite
h = (tf-t0)/(n-1);
% Classic RKM
e = RungeKuttaMethod(@fAufgabe17, @fRKTClassicRKM, t0, tf, h, x0);
% cell array holen
result = e.getResults();
% cell array in 2 lineare double arrays konvertieren
x1 = [];
x2 = [];
for i=1:1:length(result)
x = result{i};
x1(i) = x(1);
x2(i) = x(2);
end
% phasenraum zeichnen
figure('Name', 'Aufgabe 17 - Phasenraum �ber x1, x2');
plot(x1, x2);
xlabel('x1');
ylabel('x2');
% x1(t) zeichnen
figure('Name', 'Aufgabe 17 - Abh�ngigkeiten von t');
subplot(2,1,1);
axis([0 50 -1 1]);
plot(t0:h:tf, x1);
xlabel('t');
ylabel('x1');
% x2(t) zeichnen
subplot(2,1,2);
axis([0 50 -1 1]);
plot(t0:h:tf, x2);
xlabel('t');
ylabel('x2');
end
% END STATIC METHODS
end
% END CLASS
end