Exploiting symmetry in multidimensional arrays

Dear all, In a symmetric matrix A of size [n,n], for two indices i and j such that 1<=i<=n and 1<=j<=n we have that A(i,j)=A(j,i). Suppose I know A(i,j) for i>=j how can I efficiently set A(j,i)?

More generally, if I have a hypercube A of size n^m, how can I, on the basis of A(i1,i2,...,im) with i1>=i2>=i3....>=im, set all A for all permutations of i1,i2,...,im instead of recalculating the entry for each permutation.

Thanks,

Answer 1

[Roger Stafford]

For a general m-dimensional n^m array, A, do this:

 [I1,I2,...,Im] = ndgrid(1:n);
 P = [I1(:),I2(:),...,Im(:)];
 Q = sort(P,2,'descend');
 A(sub2ind(size(P),P(:,1),P(:,2),...,P(:,m))) = ...
   A(sub2ind(size(Q),Q(:,1),Q(:,2),...,Q(:,m)))

Note: The four three-dot ellipses (...) shown above (but not the one following the '=' sign) are to be filled in by the user.

[Alfonso Nieto-Castanon]

without "..." it might look something like:

 clear I;
 [I{1:m}] = ndgrid(1:n);
 P = reshape(cat(m+1,I{:}),[],m);
 Q = sort(P,2,'descend');
 A(:) = A(1+(Q-1)*(n.^(0:m-1)'));

EDIT: replaced sub2ind which can be too slow at times...

[Patrick Mboma]

@ Roger, What you suggest is very helpful but there is, I think, a mistake in what you wrote. I believe the last command should be

 A(sub2ind(size(A),P(:,1),P(:,2),...,P(:,m))) = ...
   A(sub2ind(size(A),Q(:,1),Q(:,2),...,Q(:,m)))

That is, I replaced size(P) and size(Q) with size(A);

other than that it seems to work. Basically, instead of replacing just the missing indexes, you replace the whole array.

Many thanks,

[Patrick Mboma]

@Alfonso What you suggest would really be neat and seems to work well in the two-dimensional case that I tested. I still need to investigate higher dimensions and the speed.

[Roger Stafford]

@Patrick. Yes, you are quite right. I should have used size(A). I must have had a mental black out at that point! :-)

Answer 2

[Roger Stafford]

For a 2D array it's easy:

 B = tril(A,-1)
 A = B+B.'+diag(A);