I have been working with computing and numerical integration, and I have to juxtapose the operations required for Trapezoidal and Romberg for $n$ points to a certain error tolerance. For trapezoidal it is obviously $O(n)$. However, I am unsure of the operation count for Romberg, which i assume is either $O\left(n^2\right)$ or $O\left(2^n\right)$.
Can someone teach me what the operation count for Romberg integration is for $n$ points?