I am running very computationally intensive tasks and wish to adjust the parameters respective of how long it takes.
The function I am running is PLINK - for those who don't know, it is used for genotype data.
The function is said to follow a $O(nm^2)$ w.r.t. big O.
I have the run time for two time points with different parameters for $m$ and a constant $n$, they are: 3 hours and 648 hours.
From this I wish to estimate the run-time for different parameters of $m$, that would respect the $O(nm^2)$ relationship.
Can anybody provide some insight as to methods for estimating run-time with the constant $n$ parameters? In other words, we know for run-time function $R$: $R(n_0, m_1)=3$ and $R(n_0, m_2)=648$, and from this I wish to estimate $R(n_0, m)$ for a general $m$; knowing that it follows a growth function $O(nm^2)$.
Okay, if $m^2$ were the only relevant changing value, then I could guess your ratio of $m$ values was roughly, 14.7 (sqrt of 216 roughly) . The notation, is mostly used as a way to show a term or set of terms, that approaches the value of the overall function as it's variable(s) get larger. For example: $$4z^2+ 20876549321768543219865740$$ has a relatively huge constant term. But, by $$z=2284543133854.586647726934526$$ that only accounts for roughly half the value of the polynomial. Above this, the term with a variable, starts to take over half. By :$$z=22845431338545.86647726934526$$ the constant term is now at under 0.1 percent of the value of that function. Above this, a good approximation is simply $$4z^2$$ and above $$z=2$$ this estimate is estimated by $$z^2$$. With $n$ constant, your function if large enough to overwhelm other terms will roughly scale with the square of the ratio of $m$ values.