Unfortunately, I do not know of any library (not saying there isn't, I am just the type of person who likes to program anything himself).
Did you try posting on other forums?
Other than that, the fact that you are looking for
ρ(A) = maxi (|λi|)
suggests that your matrices will satisfy the conditions needed to run the Power iteration algorithm; the output of that algorithm is exactly what you are looking for. The algorithm is really simple to implement, and is also very simple to prove that, if it converges, the answer you get is correct.
What I would do if I were you is
1) try posting at another forum.
2) in the meantime implement the power iteration algorithm and see whether it works.
Do you need to have the answer for one particular example, for a fixed set of examples, or to create a program that runs for an arbitrary user input example?
The whole implementation would take some time between 30 minutes (if you are skillful with C++ and have good math libraries) and 3-4 days, if you have to write everything from scratch (including the linear algebra).
I would gladly try to help with point 2), if I can.
| i have a transition matrix (let it be A, for example) for a markov chain |
I believe that matrices coming from Markov chains very often satisfy the conditions needed for the power iteration algorithm to converge. I tried to decode that from the Markov chain page on Wikipedia, but I couldn't figure it out - you need to ask a specialist on the subject.
I think, without being sure, the question of whether the power iteration algorithm works is related to the notion of periodicity and/or ergodicity of a Markov chain, but unfortunately my knowledge on the subject ends here :(
[Edit:] If your matrix is right stochastic, then you don't need to write a program for it: the largest eigenvalue is 1:
, so I presume you will want to run the program for non-stochastic matrices.