way to avoid for loops

What is the best way to write the following code without using any kind of loop?

N=2;
[x,y]=meshgrid(1:N,1:N);
s=zeros(N);
for i=1:N
  for j=1:N
      s(i,j)=sum(sum(i*x+j*y));
  end
end

Some more: What if I make the function a little complex like below?

N=2;
[x,y]=meshgrid(1:N,1:N);
s=zeros(N);
for i=1:N
    for j=1:N
        s(i,j)=sum(sum(exp(sqrt((x-i).^2+(y-j).^2));
    end
end

Answer 1

[Roger Stafford]

If x and y are arbitrarily defined as N-by-N arrays, do this:

   [I,J] = ndgrid(1:N);
   s = sum(x(:))*I+sum(y(:))*J;

With x and y as you defined them, do this:

   s = N^2*(N+1)/2*(x+y);

[Rahul Shaw]

Thanks for your reply but my problem is little more complicated than this. I have updated the question accordingly.

Answer 2

[Roger Stafford]

This is for your revised problem involving the 'exp' function. (Note: I am assuming the x and y in your code are just as you have written them and not a general x and y.)

   N = 10 % <-- You choose N
   [I,J] = meshgrid(-N+1:N-1);
   T = cumsum(cumsum([zeros(1,2*N);zeros(2*N-1,1),exp(sqrt(I.^2+J.^2))],1),2);
   S = T(N+1:2*N,N+1:2*N)-T(N+1:2*N,1:N)-T(1:N,N+1:2*N)+T(1:N,1:N);

Array S is N-by-N with the desired values (except of course for differing rounding errors.)