Return values differ depending on vector size

Question

I noticed that for example the log(x) function returns slightly different values when called with vectors of different sizes in MATLAB.

Here is a minimal working example:

x1 = 0.1:0.1:1;
x2 = 0.1:0.1:1.1;

y1 = log(x1);
y2 = log(x2);

d = y1 - y2(1:length(x1));

d(7)

Executing returns:

>> ans = 
  -1.6653e-16

The behaviour seems to start when the vector becomes greater than 10 entries.

Although the difference is very small, when being scaled by a lot of operations using large vectors, the errors became big enough to notice.

Does anyone have an idea what is happening here?

Answer 1

The differences exist in x1 and x2 and those errors are propagated and potentially accentuated by log.

max(abs(x1 - x2(1:numel(x1))))
% 1.1102e-16

This is due to the inability of floating point number to represent your data exactly. See here for more information.