Understanding a Cholesky Factorisation Algorithim

Hello I am reviewing the following code for Cholesky Factorisation, trying to understand each individual step. I am new to Matlab and have tested the code (it does work). At the step which I've marked by %***%, I believe that the value produced by multiplying matrix elements L(j,k)*L(i,k) will always be zero, in particular, the element L(j,k) will always be zero since the matrix element L is 0 before the for loop and it is not redefined at the end of the for loop, and iterated k=j-1 times. I.e s=s at this step in the code and always will be.

Is there perhaps a visualisation or an effective way to understand how the elements of this code work? I have tried running the simulation by hand and it gets very large and complicated very quickly.

   function L = CholScalar(A)
  % L = CholScalar(A)
  % Cholesky factorization of a symmetric and positive definite matrix A.
  % L is lower triangular so A = L*L'.

[n,n] = size(A);
L = zeros(n,n);
for i=1:n
   % Compute L(i,1:i)
   for j=1:i
      s = A(j,i);
      for k=1:j-1  %****% This is the step I don't understand. 
         s = s - L(j,k)*L(i,k);
      end
      if j<i
         L(i,j) = s/L(j,j);
      else
         L(i,i) = sqrt(s);
      end
   end
end

Answer 1

[James Tursa]

To see if your supposition is correct, just add this to your for loop and put a break point there:

      for k=1:j-1  %****% This is the step I don't understand. 
         s = s - L(j,k)*L(i,k);
         if( L(j,k) ~= 0 )
             disp('not zero');  % <-- break point here
         end
      end

Then run your code.