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%         Klausur Numerische Mathematik I, Wintersemester 2017/18         %
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% Gruppe: 1                                                               %
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% Bitte fuegen Sie Ihre Loesungen in die fuer die Aufgabe vorgesehenen    %
% Abschnitte ein. Vermeiden Sie die Nutzung von 'function' und achten Sie %
% darauf, dass das Skript ausfuehrbar ist.                                %
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%% Aufgabe 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% vorwaerts besser als rueckwarts:
 n = 20; c = 1/2;

%format long
I = ones(40,1);
I(1) = 1-10*log((c+1)/c);
% Vorwaertsiteration:
for i = 2:n
    I(i) = 1/i-c*I(i-1);   
end
I_v = I(n)
% Rueckwaertsiteration:
I(40) = 0.0165;
for i = 40:-1:n+1
    I(i-1) = (1/c)*(1/i-I(i));   
end
I_r = I(n)
% "Exakter" Wert:
%I_e = quad(@(x)(x.^n)./(x+c),0,1)
I_e = integral(@(x)(x.^n)./(x+c),0,1)

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%% Aufgabe 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

f = @(x)x.^(1/4);
n = 5;
x0 = 0.5;

x = zeros(1,n+1);
x(1) = x0;

for i = 1:n
    x(i+1) = f(x(i));
end
x_5 = x(n+1)         

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%% Aufgabe 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% symmetrisch, aber nicht positiv definit
M = gallery('fiedler',100); 

% Pruefen auf Symmetrie und pos. Def
if issymmetric(M) && min(eig(M)) > 0
    L = chol(M,'lower');
else 
    [L,U,P] = lu(M);
end

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