# What does does $O$ mean in this context?

I understand big O notation in computational complexity theory, but I don't see how it applies in the equation below.

From Pattern Recognition and Machine Learning:

If we weren't familiar with the rules of ordinary calculus, we could evaluate a conventional derivative $dy/dx$ by making a small change to the variable $x$ and then expanding in powers of $\epsilon$ , so that $y(x + \epsilon) = y(x) + \frac {dx}{dy}\epsilon + O(\epsilon^2)$

and finally taking the limit $\lim\epsilon\to 0$.

## 1 Answer

It means exactly the same thing. It means there is a constant $c$ such that the term is $\le c \cdot \epsilon^2$ (where $c$ does not depend on $\epsilon$). Big-O notation is not unique to computer science; it is also widely used in mathematics. See https://en.wikipedia.org/wiki/Big_O_notation, which explains some ways it is used in mathematics.

• There are two differences: (1) this O bounds the magnitude (absolute value) of the error term, (2) the limit is $\epsilon\to0$ rather than $n\to\infty$; so it might hold only for $\epsilon < \epsilon_0$ for some small $\epsilon_0 > 0$. – Yuval Filmus Feb 16 '15 at 3:02