How to reduce running time of diagonal matrix multiplication with full matrix in Matlab?

I need to calculate a matrix multiplication that , where B is a full matrix with and D is a digonal matrix with .The computational complexity is

Acturally, if the matrix D is a full matrix, the computational complexity will be .

I recorded the running time for both cases in matlab, and find that the running time and time complexity are not consistent, how can I speed it up? I want to use less time to calculate the first case compared with the second case.

Thanks.

[Chunru]

The simplest case is for D=I. Then . The complexity for this matrix multiplication is rather than .

For full matrix D, the complexity is .

For diagonal D, the complexity is .

[z cy]

Thanks, I made a typo. I have modified it in the question

Answer 1

[Chunru]

n = 2000; d=500;
B = randn(d, n);
dv = randn(n, 1);
D = diag(dv);

% Normal
tic
A = B*D*B';
toc

Elapsed time is 0.052677 seconds.

% Speed up 1
tic
C = B*(dv.*B');
toc

Elapsed time is 0.012931 seconds.

% Speed up 2
tic
E = (B.*dv')*B';
toc

Elapsed time is 0.012015 seconds.

A(1:5, 1:5)

ans = 5×5

6.0809 11.2463 -58.6430 -66.3057 -24.0124 11.2463 -49.0177 52.5213 23.6997 18.0264 -58.6430 52.5213 42.9952 29.0237 23.4184 -66.3057 23.6997 29.0237 -44.6983 -0.1757 -24.0124 18.0264 23.4184 -0.1757 -61.5623

C(1:5, 1:5)

ans = 5×5

6.0809 11.2463 -58.6430 -66.3057 -24.0124 11.2463 -49.0177 52.5213 23.6997 18.0264 -58.6430 52.5213 42.9952 29.0237 23.4184 -66.3057 23.6997 29.0237 -44.6983 -0.1757 -24.0124 18.0264 23.4184 -0.1757 -61.5623

E(1:5, 1:5)

ans = 5×5

6.0809 11.2463 -58.6430 -66.3057 -24.0124 11.2463 -49.0177 52.5213 23.6997 18.0264 -58.6430 52.5213 42.9952 29.0237 23.4184 -66.3057 23.6997 29.0237 -44.6983 -0.1757 -24.0124 18.0264 23.4184 -0.1757 -61.5623

[z cy]

Thank you very much!