%                                                                              %
%                D-MATH    Numerische Mathematik I     SS 2002                 %
%                                                                              %
%                               Serie 9 - Tips                                 %
%                                                                              %
% Diese Datei fasst die Matlab-Funktionen fuer die Aufgabe 3 zusammen.         %
%                                                                              %

%%%%%%%%%%%%%%%%%%%%% Aufgabe 3 ->skript aufgabe3.m <-%%%%%%%%%%%%%%%%%%%%%%%%%%

maxit = 12;                        % Anzahl der Iterationen

R1 = zeros(maxit,1);
D1 = zeros(maxit,1);
R2 = zeros(maxit,1);
D2 = zeros(maxit,1);

x = ???                            % Initialisierung

for k=1:maxit                      % Newton-Raphson-Verfahren


% Hier Newton-Raphson programmieren



  D1(k) = norm(f(x));              % Residuum der Normalenglgn
  R1(k) = norm(g(x));              % Residuum in den Messwerten
end;


x = ???                            % Initialisierung


for k=1:maxit                      % mit sym. Matrix-Update


% Hier Newton mit sym. Matrix-Update programmieren



  D2(k) = norm(f(x));              % Residuum der Normalenglgn
  R2(k) = norm(g(x));              % Residuum in den Messwerten
end;

x                                  % Ausgabe Ergebnis


figure(1);                         % Plot norm(g(x))
plot(1:maxit,R1,1:maxit,R2);

figure(2);                         % Plot norm(f(x))
semilogy(1:maxit,D1,1:maxit,D2);


%%%%%%%%%%%%%%%%%%%%%%%% Aufgabe 3 ->function g.m <-%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function erg = g(x);

% Berechnet das Residuum des ueberbestimmten  Systems g(x) = 0:
% g_k(x) = z(t_k; x) - z_k,  k=1, ..., 9
% mit  x := [a_1, a_2, a_3 ]'

t = [   0;   .5;    1;  1.5;    2;    3;    5;    8;   10];
z = [3.85; 2.95; 2.63; 2.33; 2.24; 2.05; 1.82; 1.80; 1.75];

erg = x(1) + x(2)*exp(x(3)*t) - z;

%%%%%%%%%%%%%%%%%%%%%%%% Aufgabe 3 ->function Dg.m <-%%%%%%%%%%%%%%%%%%%%%%%%%%%

function erg = Dg(x);

% Berechnet die Jacobimatrix des ueberbestimmten  Systems g(x) = 0
% Dg_(k,i) = dg_k / dx_i
% wobei g_k(x) = z(t_k; x) - z_k,  k=1, ..., 9
% mit   x := [a_1, a_2, a_3 ]'

t = [   0;   .5;    1;  1.5;    2;    3;    5;    8;   10];
z = [3.85; 2.95; 2.63; 2.33; 2.24; 2.05; 1.82; 1.80; 1.75];

erg = [ones(9,1), exp(x(3)*t), x(2)*t.*exp(x(3)*t) ];

%%%%%%%%%%%%%%%%%%%%%%%% Aufgabe 3 ->function f.m <-%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function erg = f(x)

% Berechnet den Wert der Normalengleichungen
% (Noch zu vervollstaendigen)

erg = ???

%%%%%%%%%%%%%%%%%%%%%%%% Aufgabe 3 ->function Df.m <-%%%%%%%%%%%%%%%%%%%%%%%%%%%

function erg = Df(x);

% Berechnet die Jacobimatrix der Normalengleichungen
% (Noch zu vervollstaendigen)

 ...

erg = ???

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%