Pointwise multiplication of 3d array with 2d matrix and 1d array and summation by vectorization

I have a 2 by 2 matrix

A = rand(2,2)

and a 1 by 5 vector

v = [8 9 12 10 11];

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

Call it D such that

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

How can I do the operations of

      J=zeros(2);
      K=zeros(2);
      for i = 1:5
        J = J + D(:,:,i)'*A*D(:,:,i);
        K = K + D(:,:,i)'*D(:,:,i);
      end

and

      Q = zeros(size(D));    
      for i = 1:5
        Q(:,:,i) = v(i)*D(:,:,i);
      end

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

Answer 1

[Stephen Jue]

Hi Jeff,

These operations all require multi-dimensional matrix multiplication, which is not a built-in feature of MATLAB. However, there is a File Exchange function called mtimesx which can be used instead. It makes use of MATLAB's vectorization optimization by calling the "mtimes" function internally. It can do many operations to n-dimensional arrays, for example:

 C = mtimesx(A,B) % performs the calculation C = A * B 
 C = mtimesx(A,'T',B) % performs the calculation C = A.' * B 
 C = mtimesx(A,B,'g') % performs the calculation C = A * conj(B) 
 C = mtimesx(A,'c',B,'C') % performs the calculation C = A' * B'

To calculate the sum (J = J + ...), you can use the result from "mtimesx" as an argument to cumsum.

Another alternative if you have the Parallel Computing Toolbox is to use the pagefun function.

I hope that helps.