In mathematics, the big $O$ notation is used to describe the limiting behavior of a function. It is abuse of notation to say $ f(x)=O(g(x)). $ But this is understandable. However, in the class of numerical analysis, I found that the teacher used the big $O$ notation as the following:
If $\kappa = O(10^{-6})$, $\epsilon_{machine} = O(10^{-16})$, then we can only expect $O(10^{-10})$ accuracy.
I am surprised that they regard $O(10^{-6})$, $O(10^{-16})$, $O(10^{-10})$ as different things. Since according to the definition, they are nothing but $O(1)$.
I guess this is another kind of abuse of notions: when one says $O(10^{-6})$, s/he actually means $c\times 10^{-6}$ where $1
So here are my questions:
Is there anyone who has seen this kind of usage before? Can anyone come up with the references (books, paper, etc.) that have the similar usage I mentioned above of the big $O$ notation?