How to vectorize an equation involving pair-wise distance computation, to improve the speed of computation?

Let me explain the situation, first:

I have a vector field, where each element has a:

1) 2-D co-ordinate location (x,y)

2) 2-D vector u,v (where u and v represents the displacements in x and y directions respectively)

I am trying to implement the following equation over this vector field:

where,

V' is the new vector field after applying the equation to the input vector field V. Therefore, is the updated (u,v) vector value for the element "i".

Also, "j" represents the set of all elements in the vector field, except "i" . Hence, is the (u,v) vector for element "j".

Finally, and represent the 2-D co-ordinate location (x,y) for elements i and j respectively.

I have done the following implementation (say, for 100 elements). But, it takes a lot of time to execute when the number of elements increases.

Can you help me to vectorize it, if possible?

% the variable "xyuv" contains:
% xyuv(i,1) : x-coordinate for i-th particle
% xyuv(i,2) : y-coordinate for i-th particle
% xyuv(i,3) : u-value for i-th particle
% xyuv(i,4) : v-value for i-th particle

for i=1:100 %total 100 elements
    sum_temp = 0;
    
    for j=1:100
        
        if(j~=i) %to avoid i-vs-i computation, in order to speed up
            
            if((1-pdist2(xyuv(i,3:4),xyuv(j,3:4),'cosine'))>=0.8) % if the condition is sastified, then do summation
                
                term_2 = exp(-1*(pdist2(xyuv(i,1:2),xyuv(j,1:2)))); %second term of the equation
                term_3 = exp(-1*abs(dot(xyuv(j,3:4), (xyuv(i,1:2)-xyuv(j,1:2))))); %third term of the equation
                sum_temp = sum_temp + xyuv(j,3:4)*term_2*term_3; % summation
            
            end
        
        end
    
    end
    new_V(i,:) = sum_temp; % updated vector for element i
end

Answer 1

[J. Alex Lee]

I think you should be able to loop only once over the reference point i, then take then compute difference vectorially.

Also, indexing can be "expensive", so perhaps better to split before the loop

xy = xyuv(:,1:2);
uv = xyuv(:,3:4);
for i = 1:N
    dxy = xy(i,:) - xy
    
    % not sure what the overhead is like for pdist2, or if it is vectorized, but
    d = sqrt(dxy.*dxy); % right?
    % etc
end

[Tintumon]

Thank you a lot.

This idea helped me to eliminate the innermost "for loop".

And, Sorry for the late reply.