I applied the Master theorem to a recurrence for a running time I encountered (this is a simplified version):$$T(n)=4T(n/2)+O(r)$$$r$ is independent of $n$. Case 1 of the Master theorem applies and tells us that $T(n)=O(n^2)$.However, this hides a constant dependent on $r$ in the big-oh notation: our recurrence has depth $O(\log_2 n)$ so at the final level we have $O(4^{\log_2 n})=O(n^2)$ subproblems, each of which takes $O(r)$ time to be handled. This means the actual running time is $O(n^2 r)$ (or worse: this analysis only talks about the lowest level).This is my actual recursion:$$T(n)=r^2T(n/r)+O(nr^2)$$Is there a method similar to the Master theorem for these kinds of recursions?
Master theorem and constants independent of $n$
algorithm analysis;asymptotics;recurrence relation;mathematical analysis;master theorem