Speed up double sum and product

Dear all,

I would like to speed up the code below. It is executed thousands of times with different values for the matrix v, so it slows down my program. Basically, I have to compute the variable etaa as the sum over i of the sum over j of the product for r from 1 to N (with r different than j) of the elements of the matrix A indexed by the matrix v (below is the formula in latex):

\eta = \frac{1}{M}\frac{1}{N} \sum_{i=1}^M \sum_{j=1}^N \prod_{r=1, r\neq j}^N A(1+v(i,j),1+(v(i,r))

I have written the code in many different forms, but I am unable to speed up the overall execution. Do you have suggestion for improving execution speed? Your help is really appreciated!

Thank you in advance.

tmpsum = 0;
for i=1:M
    for j=1:N
        tmpprod = 1;
        for r=1:N
            if(r~=j)
                tmpprod = tmpprod*A(1+v(i,j),1+v(i,r));
            end
        end
        tmpsum = tmpsum + tmpprod;
    end
end
etaa = tmpsum/M/N;

As an example, the code can be executed with the following initialization:

M = 500;
N = 5;
% This 7x7 matrix is fixed
A = [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0; ...
     1.0, 0.9, 0.8, 1.0, 1.0, 1.0, 1.0; ...
     1.0, 0.8, 0.7, 1.0, 1.0, 1.0, 1.0; ...
     1.0, 1.0, 1.0, 0.7, 0.6, 1.0, 1.0; ...
     1.0, 1.0, 1.0, 0.6, 0.5, 1.0, 1.0; ...
     1.0, 1.0, 1.0, 1.0, 1.0, 0.7, 0.6; ...
     1.0, 1.0, 1.0, 1.0, 1.0, 0.6, 0.5];
% v is a matrix of indexes used to access the elements of A. Here I wrote
% ony the first 10 rows as an example.
v = zeros(M, N);
v(1:10,:) = [1, 1, 1, 1, 3; ...
             1, 1, 2, 3, 0; ...
             2, 2, 3, 0, 0; ...
             2, 2, 3, 0, 0; ...
             3, 3, 3, 3, 3; ...
             3, 3, 3, 3, 3; ...
             3, 4, 4, 0, 0; ...
             4, 4, 0, 0, 0; ...
             4, 4, 0, 0, 0; ...
             4, 4, 0, 0, 0 ...
            ];

Answer 1

[Tony Mohan Varghese]

Modify the algorithm to use parfor loop. It can execute the loop iterations on parallel workers.

https://in.mathworks.com/help/distcomp/convert-for-loops-into-parfor-loops.html

https://in.mathworks.com/help/distcomp/parfor.html