MATLAB: Recursive Matrix Row Reduction

gaussianmatrixrref

Hi,
I'm having trouble debugging a program that I'm writing to reduce a matrix into Reduced Row-Echelon form (RREF). I'm using recursion and Gaussian Elimination to accomplish this task.
The problem is that the last values of the Matrix 'M' in the recursive loop don't seem to carry outside of the recursion; that is, the operations that I perform on them do not seem to stick. This is easier to show with part of the debug output:
    Getting out of the recursion, M is...
        1.0000    0.8333
    Check on m
        1.0000    0.8333
    At the end of the function, M is...
        1.0000    0.8333
    Getting out of the recursion, M is...
         1     3     2
         0     6     5
Full code of the Gaussian Elimination recursive program as below:
    function [M,pivots] = echelon(M)
        %Reduce matrix to reduced row-echelon form
        %ECHELON(M) Returns reduced row-echelon matrix M and pivots vector 'pivots'
        %Check if input is numerical matrix.  If it isn't, give error.
            narginchk(1,1);
            if ~(ismatrix(M) && isnumeric(M))
                error('Not a numeric array');
            end
        %Check if the matrix is empty.  If it is, return
            pivots=[];
            if isempty(M)
                disp('found empty');
                pivots = [];
                return;
            end
        %Find size of Matrix, calculate tolerance level  
            [m,n] = size(M);
            tolerance = length(M)*norm(M)*eps;
        %Find the pivot variable - it is value x, at position p
            [x,p] = max(abs(M(:,1)));
        if x>tolerance
            M = Exchange(M,1,p);
            M = Divide(M,1,M(1,1));
            for i=2:m
                if M(i,1)==0
                    %do nothing
                else
                  Subtract(M,1,i,M(i,1)/M(1,1))
                end
            end
            disp('Recurring now.');
            [~,newpivots] = echelon(M(2:m,2:n));
            disp('Getting out of the recursion, M is...');
            disp(M);
            pivots(1) = 1;
            for z=1:length(newpivots)
                pivots(end+1) = newpivots(z)+1; 
            end
                for z=2:length(pivots)
                    currentpivot = pivots(z);
                    M = Subtract(M,z,1,M(1,currentpivot)/M(z,currentpivot)) 
                end
                  fprintf('Check on m\n');
                    disp(M);
        else
            disp('the else has been called.');
            [~,newpivots] = echelon(M(:,2:n));
            for z=1:length(newpivots)
                pivots(end+1) = newpivots(z) + 1;
            end
        end
        disp('At the end of the function, M is...');
        disp(M);
    end
If you're familiar with Gaussian row reduction, the subtract, exchange, and divide functions used in the program are all elementary row operations (subtracting row i2 by row i1 * a scalar).
Functions as follows:
function returnMatrix = Subtract(M,i1,i2,r)
if ~(nargin == 4)
    error('# of arguments incorrect.  Exiting.');
end
if ~((i1 >= 1) && (i1 <= size(M,1)) && (i2 >= 1) && (i2 <= size(M,1)))
    error('Row i not in bounds of matrix.  Exiting');
end
if r==0
    error('r is zero.  Exiting.');
end
returnMatrix = M;
returnMatrix(i2,:) = returnMatrix(i2,:) - (r*returnMatrix(i1,:));
end
function returnMatrix = Exchange(M,i1,i2)
if ~(nargin == 3)
    error('Incorrect arguments.  Exiting.');
end
returnMatrix = M;
if (i1==round(i1)) && (i2==round(i2)) && (i1 >= 1) && (i2 >= 1) && (i1 <= size(M,1)) && (i2 <= size(M,1))
    returnMatrix([i1 i2],:) = returnMatrix([i2 i1],:);
else
    error('Incorrect arguments entered for row values.  Exiting.');
end
end
function returnMatrix = Divide(M,i,r)
if ~(nargin == 3)
    error('# of arguments incorrect.  Exiting.');
end
if ~((i >= 1) && (i <= size(M,1)))
    error('Row i not in bounds of matrix.  Exiting');
end
if r==0
    error('r value is zero.  Exiting');
end
returnMatrix = M;
returnMatrix(i,:) = returnMatrix(i,:)/r;
end
Sorry, this is a pretty big post, and I've been working on this small error for hours. I was wondering if someone can help me solve the problem I'm having?

Best Answer

Ok, I have your solution.
1. Your subtract function in the beginning did not return the subtraced matrix.
2. You needed to return the previously calculated matrix (as I commented in the first response). However, since the returned matrix is smaller than the current matrix, you have to account for that.
3. The whole pivot point vector is not needed. The current pivot point is calcuated in the beginning of each cycle, meaning that when later on you need to subtract out the other values of the current row, you just use the current row.
Now, in the solution, I assumed that you wanted the generalized format for solving systems of non-zero equations.
I.e. The format of the input M is assumed to be
[matrix A] = [Column solution B] - such that the last column of the input matrix is the vector B.
I'm using an older verion of matlab, so I commented out some of your debugging to make it easier on myself.
if isempty(M)
    disp('found empty');
    %     pivots = [];
    return;
end        %Find size of Matrix, calculate tolerance level
[rows,cols] = size(M);
tolerance = length(M)*norm(M)*eps;
%Find the pivot variable - it is value x, at position p
[x,p] = max(abs(M(:,1)));
if x>tolerance
    M = Exchange(M,1,p);
    M = Divide(M,1,M(1,1));
    for i=2:rows
        if M(i,1)==0
            %do nothing
        else
            M = Subtract(M,1,i,M(i,1)/M(1,1));
        end
    end
    disp('Recurring now.');
    [new_sub_matrix] = echelon(M(2:rows,2:cols));
    disp('Getting out of the recursion, M is...');
      if(~isempty(new_sub_matrix))
          %check to make sure the sub matrix exists - cannot add an empty
          %matrix
          M(2:rows,2:cols) = new_sub_matrix;
      end
      disp(M);
          %we index over colums, but we count based on the # rows
          for z=2:rows
              %         currentpivot = pivots(z);
              subtract_factor = M(1,z);
              M = Subtract(M,z,1,subtract_factor);
          end
  %     end

      fprintf('Check on m\n');
      disp(M);
      % else
      %     disp('the else has been called.');
      %     [new_sub_matrix,newpivots] = echelon(M(:,2:cols));
      %     for z=1:length(newpivots)
      %         pivots(end+1) = newpivots(z) + 1;
      %     end
  end
  disp('At the end of the function, M is...');
  disp(M);
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