Recursive Matrix Row Reduction

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Christian
Christian 2011 年 10 月 31 日
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?

採用された回答

Alex
Alex 2011 年 11 月 1 日
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);
  4 件のコメント
Christian
Christian 2011 年 11 月 1 日
That makes sense, I'll keep that in mind next time.
The pivot vector contains a list of columns that have pivot values in them.
On another note, I seem to be getting a logic error on some test matrices that I've constructed, but not others. Here is an example of a matrice that is producing an incorrect RREF:
x=[0 3 -6 6 4 -5; 3 -7 8 -5 8 9; 3 -9 12 -9 6 15];
In most cases, it is usually only the last column that produces a logic error. Any ideas?

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その他の回答 (1 件)

Alex
Alex 2011 年 10 月 31 日
This is a quick look, but I think your command [~,newpivots] = echelon(M(2:m,2:n)); should be [M,newpivots] = echelon(M(2:m,2:n));
M is a matrix, and Matlab does not pass by reference matrix. So, when you are calling your recursive function, a new copy of M is made, within the scope of the new recursive function execution. That new copy must be returned to the calling function if you want the row operations to be kept.
  2 件のコメント
bym
bym 2011 年 10 月 31 日
maybe looking at this code might help:
http://www.mathworks.com/matlabcentral/answers/13667-solving-system-of-linear-equation-by-gaussian-elimination

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