How Can I replace the following for loop by vectorization?

Hi Guys,

I have a set of large sparse matrices:

W ,size=(1,10,m)

V ,size=(10,1,m)

Y ,size=(10,10,n)

and I need the output matrix P as follow:

P(i,j)=W(:,:,i)*Y(:,:,j)*V(:,:,i)

e.g if m=2 & n=3

P=[
   W(:,:,1)*Y(:,:,1)*V(:,:,1)  W(:,:,1)*Y(:,:,2)*V(:,:,1);

     W(:,:,1)*Y(:,:,2)*V(:,:,1)  W(:,:,2)*Y(:,:,2)*V(:,:,2);

     W(:,:,1)*Y(:,:,3)*V(:,:,1)  W(:,:,2)*Y(:,:,3)*V(:,:,2) 
    ]

I want to see if you have any suggestion how to write this by a single line code. I have written the code with "two for loop" but because of the large size of matrices, it is almost slow. Any faster idea is really appreciated.

Thank you in advance,

Alireza

[Jan]

Please post your code and typical input data. It matters if m=1e3 and n=1e7 or the other way around. Perhaps your code is slow due to a forgotten pre-allocation.

What does "sparse" mean? Do the arrays contain a lot of zeros or do you sue the type sparse, which does not work for 3D-arrays?

[Alireza Barzegar]

Actually, the size of matrices may be very large depend on the size of my problem, even 3000, and matrix Y is a sparse matrix, anyway we can ignore it now. As a simple example:

   W(1,:)=[5 3 1];

   W(2,:)=[0 1 4];

V(:,1)=[1;8;2];
V(:,2)=[6;4;7];

Y(:,:,1)=[3 1 5 ; 2 1 2 ; 0 3 1];

Y(:,:,2)=[5 0 2 ; 8 1 0 ; 2 3 7];

Y(:,:,3)=[7 0 4 ; 1 3 5 ; 7 2 0];

for i=1:2

           for j=1:3
                  P(i,j) = ( ( (W(i,:))*Y(:,:,j)*V(:,i) ) );    
           end
  end

Answer 1

[Roger Stafford]

Here's a single line code, but I'm not sure it's faster than, or even as fast as, your two nested for-loop solution:

     Corrected:
     P = (reshape(repmat(W,10,1,1).*repmat(V,1,10,1),[],m)).’*reshape(permute(Y,[2,1,3]),[],n);

[Alireza Barzegar]

would you update it for the example that I have added?

   W(1,:)=[5 3 1];
   W(2,:)=[0 1 4];
V(:,1)=[1;8;2];
V(:,2)=[6;4;7];
Y(:,:,1)=[3 1 5 ; 2 1 2 ; 0 3 1];
Y(:,:,2)=[5 0 2 ; 8 1 0 ; 2 3 7];
Y(:,:,3)=[7 0 4 ; 1 3 5 ; 7 2 0];
for i=1:2
           for j=1:3
                  P(i,j) = ( ( (W(i,:))*Y(:,:,j)*V(:,i) ) );    
           end
  end

[Roger Stafford]

I see I have made a mistake in my solution. It should have read:

P = (reshape(repmat(W,10,1,1).*repmat(V,1,10,1),[],m)).’*reshape(permute(Y,[2,1,3]),[],n);

[Roger Stafford]

In your example in keeping with your original statement you ought to have:

W(1,:,1) = [5 3 1];
W(1,:,2) = [0 1 4];
V(:,1,1) = [1;8;2];
V(:,1,2) = [6;4;7];

Then my code would read:

 P = (reshape(repmat(W,3,1,1).*repmat(V,1,3,1),[],2)).’*reshape(permute(Y,[2,1,3]),[],3);

[Alireza Barzegar]

Thanks, great idea and also much faster than for loop :)

[Roger Stafford]

I'm glad to hear it. I was afraid it would be too slow. Sorry about the earlier error.