MATLAB: Cos() between vectors in matrix Optimization

Question

Hello all,

I'm writing to ask for some help optimizing the following code. I'm taking a matrix, S, into my function, and I want to collect the cosine of the angle between each pair of columns in S into Dc, as you can see, I have naively done this with two for loops, what I want to know is if there is a way of doing this that takes advantage of the matrix operations of MATLAB, so that it runs faster. I am currently testing on a small(ish) matrix, Dc ends up on the order of 1300×1300, but I hope to expand the code to operate on larger matrices, between 10 and 15 times this size.

Any pointers would be much appreciated.

Dc = zeros(size(S,2));
for i = 1:length(S)
    for j = 1:length(S)
        Dc(i,j) = (S(:,i)' * S(:,j))/(norm(S(:,i))*norm(S(:,j)));
    end
end

Answer 1

This might be faster. It avoids repetition in computing norms.

 Dc = S'*S;
 N = sqrt(diag(Dc));
 Dc = Dc./(N*N');