%Given an interval matrix [n,N], this program tells whether its
%mrk is 0 1 or greater than 1

%It uses
%-the program reduce.m to eliminate the rows and the columns such that all its entries contain 0
%-the program rk1nnr.m, which says whether an interval matrix with nonnnegative entries contains a rk 1 matrix



   function a=rk01(n,N)

   % we reduce the interval matrix [n,N]; the interval matrix we obtain
   % is given by the first half of the columns of X and the second half
   % of the columns of X

   
  X=reduce(n,N)  ;
  [p,r]=size(X);


  if [p,r]==[0,0] %in this case obiously mrk is 0
    a=0;
  return
  end


  
    q=r/2;
m=X(:,1:q);
M=X(:,q+1:r);




% we construct a matrix v with 2 rows such that its columns are the
% indices of the entries of [m,M] such that either
% min negative max positive or
% min zero max positive or
% min negative max zero


v=zeros(2,0);
for i=1:p
	for j=1:q
		if m(i,j)<=0 & M(i,j)>=0 & -m(i,j)+M(i,j)>0
		v=[v [i; j]];
		end
		end
end
		

		% now we construct two tridimensional matrices A and B such that
		% for every i the slice [A(:,:,i) B(:,:,i)] is an interval matrix
		% whose entries are all either contained in the set of the
		% nonnegative real numbers or in the set of nonpositive real numbers 
		% and the union of [A(r,s,i),B(r,s,i)] with i varying is the entry
		% r,s of [m,M]
		
		A=zeros(p,q,1);
A(:,:,1) =m;
  B=zeros(p,q,1);
B(:,:,1) =M;
    for i=1:size(v,2)
	    A=cat(3,A,A);
B=cat(3,B,B);
for l=1:size(A,3)/2 
	B(v(1,i),v(2,i),l)=0;
A(v(1,i),v(2,i),l+size(A,3)/2)=0;
  end
		    end



  % Now we consider every slice of A,B and, by multyplying some of its rows and
  % some of its columns by -1, we change it in such way that all the entries
  % of the first row and of the first column of the slice are contained
  % in the set of nonnegative real numbers.

	for k=1:size(A,3)

		for i=1:size(A,1)
			if A(i,1,k)<0
			X=A(i,:,k);
Y=B(i,:,k);
			A(i,:,k)=min(-X,-Y);
                        B(i,:,k)=max(-X,-Y);
			end
			end
			
		for j=2:size(A,2)
		        if A(1,j,k)<0
			U=A(:,j,k);
V=B(:,j,k);
A(:,j,k)=min(-U,-V);
B(:,j,k)=max(-U,-V);

end
				end
		end


				

% b is the set of the indices i of the slices of A B such that at least some entry
% of [A(:,:,i) , B(:,:, i)] intersect the set of the negative real numbers
%

       

	
  b=zeros(1,0);
	for i=1:size(A,3)
		if sum(sum(A(:,:,i)<0))>0
		b=[b,i];
		end
		end

% I define again A and B by excluding the slices whose indices are in b



		x=size(A,3);		

A=A(:,:,setdiff(1:x,b));
		  B=B(:,:,setdiff(1:x,b));


	      
	for i=1:size(A,3)
		if rk1nnr(A(:,:,i),B(:,:,i))==1
		a=1;
		return
		end
		end
		a="mrk at least 2";