There are several issues here. First is the basic problem of floating point arithmetic. Floating point numbers do not store most decimal fractions exactly. Thus, we see
>> .3 - .2 - .1
ans =
-2.7756e-17
does not produce zero. Even though we know it "should" be.
The answer is to never trust the least significant bits of a computation.
Next, we see that the same computation, performed in a different sequence, will often produce a different result, at the level of the least significant bits.
>> .3 - (.1 + .2)
ans =
-5.5511e-17
Identically the same, but not so. Must I repeat myself? :)
The answer is to never trust the least significant bits of a computation.
When you do perform matrix multiplies, MATLAB invokes the BLAS to do the work. And the BLAS, while making things run quickly, can also choose to do those operations in any sequence it wants internally. The net result is sometimes (even often) those adds get done in a different sequence, so you can see results that are subtly different down at the level of the least significant bits.
Again, don't presume those results are identical. The solution is to use methods that are tolerant of errors in the least significant bits. Always use tolerances when making comparisons between floating point numbers.
