Efficient multiplication of very large matrices in MATLAB

Question

I don't have enough memory to simply create a diagonal D-by-D matrix, since D is large. I keep getting an 'out of memory' error.

Instead of performing M x D x D operations in the first multiplication, I do M x D operations, but still my code takes ages to run.

Can anybody find a more effective way to perform the multiplication A'*B*A? Here's what I've attempted so far:

D=20000
M=25

A = floor(rand(D,M)*10);
B = floor(rand(1,D)*10);

for i=1:D
    for j=1:M
        result(i,j) = A(i,j) * B(1,j);
    end
end    

manual = result * A';
auto = A*diag(B)*A';
isequal(manual,auto)

Answer 1

One option that should solve your problem is using sparse matrices. Here's an example:

D = 20000;
M = 25;
A = floor(rand(D,M).*10);    %# A D-by-M matrix
diagB = rand(1,D).*10;       %# Main diagonal of B
B = sparse(1:D,1:D,diagB);   %# A sparse D-by-D diagonal matrix
result = (A.'*B)*A;         %'# An M-by-M result

Another option would be to replicate the D elements along the main diagonal of B to create an M-by-D matrix using the function REPMAT, then use element-wise multiplication with A.':

B = repmat(diagB,M,1);   %# Replicate diagB to create an M-by-D matrix
result = (A.'.*B)*A;    %'# An M-by-M result

And yet another option would be to use the function BSXFUN:

result = bsxfun(@times,A.',diagB)*A;  %'# An M-by-M result