I would like to compare different methods of finding roots of functions in python (like Newton's methods or other simple calc based methods). I don't think I will have too much trouble writing the algorithms
What would be a good way to make the actual comparison? I read up a little bit about Big-O. Would this be the way to go?
You can get wildly different answers for the same problem just by changing starting points.  Pick an initial guess that's close to the root and Newton's method will give you a result that converges quadratically.  Choose another in a different part of the problem space and the root finder will diverge wildly.
What does this say about the algorithm?  Good or bad?
Comparing Root-finding (of a function) algorithms in Python