function [x] = GAUSS(A,b)

%gaussova eliminace

close all; clc;


if nargin == 0
    A = [1 1 2 3; 1 3 2 2; 0 9 1 6; 1 2 3 4];
    b = [1 ; 2 ; 3 ; 6];

    %A = randi ([1,3], [6,6]);
    %b = randi ([1,3], [6,1]);


    %A=[1 3 4 2 2; 0 1 3 3 2; 0 0 2 3 1; 0 0 0 0 0; 0 0 0 0 0]
    %b=[3 ; 2 ; 1 ; 0 ; 0]
end


n = size (A,1);
p = size (A,1);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%overeni resitelnosti

if  rank (A) ~= rank ([A b])
    disp('NEMA RESENI');

    normalni=rank (A)
    rozsirena=rank ([A b])

    return;
end


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%prevod matice do HTT

for i=1:n-1


      if A(i,i)== 0
        ind = find (A(i:end,i)~=0);

            if isempty(ind)
                continue;
            else
                    p=A(i,:);
                    A(i,:) = A(ind(1)+i-1,:);
                    A(ind(1)+i-1,:) = p;

                    m=b(i,1);
                    b(i,1) = b(ind(1)+i-1,1);
                    b(ind(1)+i-1,1) = m;
            end
      end


   for  j=i+1:n

       f=A(j,i)/A(i,i)
       for k=i:n


           A(j,k)=A(j,k)-A(i,k)*f


       end
       b(j,1)=b(j,1)-f*b(i,1)

   end
end
%nyni mame matici v HORNIM tt


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%osetreni nekonecne mnoha reseni


if A(n,n)==0     %tj bude nekonecne mnoho reseni

        for i=n:-1:2

            if A(i,i)==0

                disp('nekonecne mnoho reseni, konecne x je jedno z mnoha');
                null(A)
                for j=1:n
                    a=1;        %nastaveni slozky vzsledneho vektoru
                    b(j,1)=b(j,1)-a*A(j,i)
                end

                A=A(1:n-1,1:n-1) %zmenseni rozmeru
                b=b(1:n-1,:)
                n = size (A,1);

            else
                break;

            end
        end

        disp('nasledne vypadaji matice a A a vektor b pred spustenim zpetneho chodu');
        A
        b

        %zpetny chod

        for i=n:-1:2



           for j=i-1:-1:1
               f=A(j,i)/A(i,i)


               for k=n:-1:i
                   A(j,k)=A(j,k)-A(i,k)*f
               end
               b(j,1)=b(j,1)-f*b(i,1)
           end
        end   %nyni mame matici v diagonalnim tvaru

        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %reseni

        x=ones(p,1);
        for m=1:n

            x(m)=b(m,1)/A(m,m);

        end    %vysledek

        x

else    %tj bude jedno reseni



        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %zpetny chod

        for i=n:-1:2



           for j=i-1:-1:1
               f=A(j,i)/A(i,i)


               for k=n:-1:i
                   A(j,k)=A(j,k)-A(i,k)*f
               end
               b(j,1)=b(j,1)-f*b(i,1)
           end
        end   %nyni mame matici v diagonalnim tvaru

        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        %reseni

        x=zeros(n,1);
        for m=1:n

            x(m)=b(m,1)/A(m,m);

        end    %vysledek

        A
        b
        x

end


end