How to vectorize this code to eliminate nested For loops

Would like to know how this code be vectorized:

X=zeros(nn);
for i=1:nn
    for j=1:nn
        X(i,j)=x(i)-x(j);
    end
end

Answer 1

[Sean de Wolski]

X = bsxfun(@minus,x(1:nn).',x(1:nn));

[Cedric]

 bsxfun(@minus,x(1:nn),x(1:nn).')

Answer 2

[Jan]

This cannot compete with the BSXFUN approach, but it is 32% faster than the original loop:

nn = numel(x);
X  = zeros(nn);
for i = 1:nn
   for j = i+1:nn
      a      = x(i) - x(j);
      X(i,j) = a;
      X(j,i) = -a;
   end
end

Using a temporary variable for x(i), which is updated in the outer loop only, is slightly slower.

[Cedric]

Well then this one would be more efficient:

 X  = zeros(nn);
 for ii = 1:nn
     X(:,ii) = x - x(ii);
 end

It almost competes with bsxfun for nn=1e4.

[Jan]

@Cedric: When you post this as an answer, you get my vote.

Your version does not exploit the symmetry as the other approaches. But this modification is much slower:

X=zeros(nn);
for ii = 1:nn
   a       = x(ii:nn) - x(ii);
   X(ii:nn, ii) = a;
   X(ii, ii:nn) = -a;
end

Obviously the overhead for explicit indexing is too high.

[Cedric]

@Jan : thank you! Knowing that I would get your vote is enough, so I'll keep that as a comment ;-)

I just profiled your last version and it is very interesting indeed, because I would never have expected such an overhead. It might even lead me to review my code in a few projects, so thank you for the comment!

Answer 3

[Jan]

And a C-Mex, which has the same speed as BSXFUN for a [1 x 1e3] vector under Matlab R2009a/64, Win7, MSVC 2008:

#include "mex.h"

void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) {
   double *X, *Y;
   mwSize i, j, n;

     n = mxGetNumberOfElements(prhs[0]);
     X = mxGetPr(prhs[0]);

     plhs[0] = mxCreateDoubleMatrix(n, n, mxREAL);
     Y       = mxGetPr(plhs[0]);

     for (i = 0; i < n; i++) {
        for (j = 0; j < n; j++) {
           *Y++ = X[i] - X[j];
        }
     }
  }

Answer 4

[Azzi Abdelmalek]

x=[1 3 4 10 20 30];
X=cell2mat(arrayfun(@(y) y-x,x','un',0))

%or

X=bsxfun(@minus,repmat(x',1,numel(x)),x)

[Cedric]

Or

 [X, Xt] = meshgrid(x, x) ;  Xt-X

More efficient than cell2mat/arrayfun and less efficient than bsxfun. So the latter (bsxfun) should be the winner ;) But Sean's version without repmat beats them all.

[Sean de Wolski]

Well the for-loops will smoke arrayfun/cell2mat like a Lambo would smoke bicycle...

bsxfun will be faster than the for-loops some of the time; meshgrid will only be faster if you need to reuse the gridded matrices.

Answer 5

[Matt J]

 y=exp(x(1:nn)); 
 X=log( y * (1./y.')  );