why does method 1 result not equal to method 2 result even if both perform the same thing ?

a = [2 5 2 9 ; 3 2 5 0; 4 5 9 2; 1 6 1 5]
threshold = 3
%function 1
z = 0;
for n = 1: size(a,1)
  for m = 1: size(a,2)
      if a(n,m) <threshold, 
          z = z + a(n,m)^2; 
      end
  end
end

   %function 2
   z = sum((a(a<threshold)).^2)

[Stephen23]

"even if both perform the same thing ?"

Because they are not the same thing. In general floating point operations are not commutative like algebra or symbolic mathematics is.

Answer 1

[Walter Roberson]

If your actual data is floating point rather than integer then the two do not do the same thing. The double nested loop sums across rows with a definite ordering. The vectorized version extracts the matching values in linear order, down columns, and passes it to a vector addition operator. The vector addition operator is permitted to add the values in any order it wants to. For large enough vectors it would pass the values to a high performance library that would segment the values according to the number of available CPUs, create a subtotal for each, and then add the subtotals.

The difference in order of addition matters for floating point because floating point addition is not commutative. A+B+C does not have to add to exactly the same thing as A+(B+C) adds to.