about nested for loop

i have three matrices A,B,C of size N*N. and I have a nested for loop code as given here,

D=0;
for i=1:N
    for j=1:N
        for k=1:N
            for l=1:N
                D = D + A(k,l) * B(i,j) * C(i,k) * C(j,l);
            end
        end
    end
end

Now for a large value of N (say N=500), this code is taking a lot of time to execute. I want to know what is the procedure to improve this code so that it executes faster.

thanks in advance for any help.

Answer 1

[Teja Muppirala]

D=sum(sum(B.*(C*A*C')))

[Andrei Bobrov]

+1

Answer 2

[Walter Roberson]

B(i,j) * C(i,k) can be calculated at the "for k" level. Call that BC. Then you have simplified to

for l=1:N
  D = D + BC * A(k,l) * C(j,l);
end

and that can be vectorized along l without a loop.

D = D + BC * sum(A(k,:) .* C(j,:));

After that you may be able to make further optimizations.

Can you rewrite what you are doing in terms of matrix multiplication?