MATLAB: Multiplication of 3d array with 2d matrix and summing them by vectorization

Question

I have a 2 by 2 matrix A = rand(2,2).

I have a 3d array of dimension 2 by 2 by 5.

Call it T such that

T(:,:,1) = [1 2;3 4];
T(:,:,2) = [5 6;7 8];
T(:,:,3) = [12 11;10 9];
T(:,:,4) = [13 15;17 19];
T(:,:,5) = [21 22;23 28];

How can I do the operations of

J=zeros(2);
K=zeros(2);
for i = 1:5
  J = J + T(:,:,i)'*A*T(:,:,i);
  K = K + T(:,:,i)'*T(:,:,i);
end

by vectorization in the fastest way. I want to do it because the 3d array very huge dimension generally.

I don't think that these operations can be further vectorized. Just be sure to pre-allocate the memory for the results `J=zeros(size(T));` and run `J(:,:,i) = T(:,:,i)'*A*T(:,:,i);` If the matrix size is huge and therefore each iteration of the loop takes a significant amount of time, any further vectorisation will probably won't make the code any faster — Airidas Korolkovas, Aug 27 '16 at 22:14
Maybe you can get a bit of extra speed by using the mtimesx routine: http://fr.mathworks.com/matlabcentral/fileexchange/25977-mtimesx-fast-matrix-multiply-with-multi-dimensional-support — Airidas Korolkovas, Aug 27 '16 at 22:34
@Airidas Korolkovas can i reshape `A` to 3D and elementwise multiplication and do sum? — Ka Wa Yip, Aug 27 '16 at 22:52
Are you looking for an element-wise product `A.*T(:,:,i)` or a matrix product `A*T(:,:,i)`? — Airidas Korolkovas, Aug 27 '16 at 23:05
Possible duplicate of [Multiply a 3D matrix with a 2D matrix](http://stackoverflow.com/questions/1745299/multiply-a-3d-matrix-with-a-2d-matrix) — Airidas Korolkovas, Aug 27 '16 at 23:06
Note that generally "vectorization" = "faster at the cost of more memory". So either you want to do it in a vectorized way, which can be faster, but way more memory-intense, *or* you want to do it memory-efficiently (which yours already is). — Andras Deak -- Слава Україні, Aug 27 '16 at 23:13
@Airidas Korolkovas i think there may be another way if i want the sum but not the individuals. — Ka Wa Yip, Aug 27 '16 at 23:28

score 0 · Accepted Answer · answered Aug 30 '16 at 17:22

Memory permitting here's one vectorized approach that uses matrix-multiplication at various stages for reductions -

% Store sizes
[m1,n1] = size(A);
[m2,n2,~] = size(T);

% Perform some matrix-multiplications *magic*
AT2D = reshape(A*reshape(T,m2,[]),m1,n2,[])
T2D = reshape(permute(T,[2,1,3]),n2,[])
Jout = T2D*reshape(permute(AT2D,[1,3,2]),[],n2)
Kout = T2D*T2D.'

MATLAB: Multiplication of 3d array with 2d matrix and summing them by vectorization

1 Answers1