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%         Klausur Numerische Mathematik I, Sommersemester 2018            %
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% Gruppe: Nachklausur                                                     %
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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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% rueckwaerts besser als vorwaerts:
n = 30;

%format long
I = ones(60,1);
I(1) = (1/4)*(exp(2)+1);
% Vorwaertsiteration:
for i = 2:n
    I(i) = exp(2)/2-(i/2)*I(i-1);    
end
I
I_v = I(n)
% Rueckwaertsrekursion:
I(60) = 0.1173;
for i = 60:-1:n+1
    I(i-1) = (2/i)*(exp(2)/2-I(i));
end
I_r = I(n)  
% "Exakter" Wert
%I_e = quad(@(x)x.*log(x).^n,1,exp(1))
I_e = integral(@(x)x.*log(x).^n,1,exp(1))

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

f = @(x) x.^(4/3); 
x0 = 1;
n = 10;

df   = @(x) (4/3)*x.^(1/3);
xold = x0; 

for i = 1:n
    xnew = xold-f(xold)/df(xold);
    xold = xnew;
end
x_10 = xnew

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

% symmetrisch und positiv definit
M = gallery('kms',100,0.5);

% 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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