## How can i speed up this large matrix multiplications?

### Sinwoo Jeong (view profile)

on 15 Oct 2018
Latest activity Commented on by Sinwoo Jeong

on 15 Oct 2018

### Bruno Luong (view profile)

Hello.
The purpose of my code is solving O.D.E. and in the function I need to do large matrix multiplication every single time step.
Therefore, almost all CPU time taken is consumed in this matrix multiplications.
I think mentioning every detail of my code is not necessary. So, only simplified code is uploaded as follows:
C = rand(10000, 5000);
B = rand(10000, 1000);
for i = 1:1000
A = rand(10000, 1);
D = (A.*B)'*C;
end
Here, B and C are constant matrix but A is changed in every single for loop. That's why I put A into a for loop. (I used function rand here just for a simple example) I've tried GPU, mex file, etc. But I have not been able to find the way which is faster than normal MATLAB .* or * operation. I think I can pre-calculate the value of the multiplication between B and C to avoid multiplying B to A and C to (A.*B) again. But I have no idea. Could you guys please help me?
(The matrix are defined as a sparse matrix in my actual code)

#### 1 Comment

Stephen Cobeldick

on 15 Oct 2018

### Bruno Luong (view profile)

on 15 Oct 2018
Edited by Bruno Luong

### Bruno Luong (view profile)

on 15 Oct 2018

There is also another way to precompute B*C as explained my answer on this similar question
It's actually very memory consuming for full matrix and noticeable slower due to inefficient memory access, even the FPU requirement is less.
However in some cases of SPARSE matrices, it can win, as showed in the test case bellow:
m = 5000;
n = 1000;
p = 10000;
C = sprand(p,m,1e-2);
B = sprand(p,n,1e-2);
A = rand(p,1);
tic
(A.*B).'*C;
toc % Elapsed time is 0.186504 seconds.
% Preparation step, must be carried out only once, since B and C are known
p = size(B,1);
[ib,jb,b] = find(B.');
[ic,jc,c] = find(C.');
commonj = intersect(jb,jc);
keepb=ismember(jb,commonj);
keepc=ismember(jc,commonj);
ib = ib(keepb);
jb = jb(keepb);
b = b(keepb);
ic = ic(keepc);
jc = jc(keepc);
c = c(keepc);
[~,locb]=ismember(jb,commonj);
[~,locc]=ismember(jc,commonj);
cb = accumarray(jb,ib,[p 1],@(x) {x});
cc = accumarray(jc,ic,[p 1],@(x) {x});
ifun = @(cb,cc) reshape(cb(:)+(cc(:).'-1)*size(B,2),[],1);
ibc = cellfun(ifun, cb, cc, 'unif', 0);
jfun = @(j) j+zeros(size(ibc{j}));
jbc = arrayfun(jfun, (1:p).', 'unif', 0);
b = accumarray(jb,b,[p 1],@(x) {x});
c = accumarray(jc,c,[p 1],@(x) {x});
vfun = @(b,c) reshape(b(:).*(c(:).'),[],1);
vbc = cellfun(vfun, b, c, 'unif', 0);
ibc = cat(1,ibc{:});
jbc = cat(1,jbc{:});
vbc = cat(1,vbc{:});
BC = sparse(ibc,jbc,vbc,size(B,2)*size(C,2),p);
%%this should be in the loop
tic
reshape(BC*A,size(B,2),[]);
toc % Elapsed time is 0.072390 seconds.

Stephan

### Stephan (view profile)

on 15 Oct 2018
Hi,
i think it should be:
Sinwoo Jeong

### Sinwoo Jeong (view profile)

on 15 Oct 2018
OK. Thank you so much. I'm gonna look over it and tell you what I think. In my problem, matrix B and C are quite sparse. So, I think I will get a nice result thanks to your comment.
Sinwoo Jeong

### Sinwoo Jeong (view profile)

on 15 Oct 2018
I think it works very well. You r a great genius.
Well, how did you come up with this brilliant idea?