% Aufgabe1
syms x
f=pi*cos(pi*x);% kleine Aenderung der Funktion
g=sqrt(1-x^2);

[M_Romb_f,Kon_G_f] = Romb_ver(0,1/2,7,f,1);
[M_Romb_g,Kon_G_g] = Romb_ver(0,1,7,g,pi/4);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [M_Romb,Kon_G] = Romb_ver(a,b,m,f,Int_f)

%Eingabe: das Interval [a,b], N=2^m, die Funktion f
%Ausgabe: die Matrix M_Romb
syms x
M_Romb = zeros(m+1,m+1);

% Die erste Spalte des Romberg-Schemas (iterierte Trapezsumme)
for j=1:m+1
    h = (b-a)/2^(j-1); % Schrittweite
    knot = a+h:h:b-h;  % Stuetzstelle
    M_Romb(j,1) = h*(0.5*subs(f,x,a)+sum(subs(f,x,knot)) + 0.5*subs(f,x,b));
end

% Von der zweiten bis m+1-ten Spalte des Romberg-Schemas
for k=2:m+1
    M_Romb(1:end-k+1,k)=M_Romb(2:end-k+2,k-1) + (M_Romb(2:end-k+2,k-1)-M_Romb(1:end-k+1,k-1))./(4^(k-1)-1);
end

%Konvergenzgeschwindigkeit
for j=1:5
    for s=1:3
        Kon_G{j}(:,s) = abs(M_Romb(2:end-j+1,j)-Int_f)./abs(M_Romb(1:end-j,j)-Int_f).^s;
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [X]=my_newton_2d;

% f ist symbolisch 2 Variable
syms f g
syms x y
f = x*y-1;
g = x^2 + y^2 -4;
syms D inv_D

D = [diff(f,x), diff(f,y); diff(g,x), diff(g,y)];
ID = inv(D);

X=-2:0.25:2;
Y=2:-0.25:-2;


for j=1:length(X)
    for k=1:length(Y)
        clear V;
        % Startvektor
        V(:,1)=[X(j),Y(k)]';
        temp(1)=norm([prod(V(:,1))-1, norm(V(:,1))^2-4]);
        for i=1:20; % Maximale Anzahl der Iteration
            if (rank(subs(D,{x,y},{V(1,i),V(2,i)}))~=2) % Pruefe ob D vollen Rang hat
                N(j,k)=inf % Wenn nein (der naechste Schritt ist unmoeglich)
                break;     % Abbruch der "For-Schleife (von i)"
            else           % Wenn ja
                % Ein Schritt von Newton-Verfahren
                V(:,i+1) = V(:,i) - subs(ID,{x,y},{V(1,i),V(2,i)})* [prod(V(:,i))-1, norm(V(:,i))^2-4]';
                temp(i+1)=norm([prod(V(:,i+1))-1, norm(V(:,i+1))^2-4]);% Norm von Funtionswert
                if (temp(i+1) < 10^(-6)) % Wenn die Norm klein genug ist,
                    N(j,k)=i;            % wird die Iterationszahl gespeichert.
                    break;               % Abbruch der "For-Schleife (von i)"
                end
                
            end
       end
    end
end