You need to use actual mathematics to explain what you need. Clear wording, instead of mathematical jargon that you make up on the fly. Sorry, but you do. For example, that you "know SOME of the limit numbers of the eigenvalues" makes no sense at all in mathematics, at least to anyone but you.
Anyway, even if I try to make some wild guesses at what you need to do, computing the eigenvalues of a symbolic matrix (i.e., a function of an unknown parameter s) that is as large as the one you describe will be essentially impossible. And since you apparently know only SOME of those eigenvalues (in the form of a limit?) your question really does not make any mathematical sense.
Finally, computation of the eigenvalues of such a matrix is highly nonlinear, since those eigenvalues are essentially the roots of a polynomial of seriously high order. Since that polynomial is of higher order than 4, there is provably NO analytical solution to such a high order polynomial, so the only approach will be a numerical one. Worse, do not forget that the eigenvalues can effectively change order as you change such a parameter. What that does is make your problem non-differentiable, so it plays hell with any numerical optimizer.
I would recommend that you spend some serious time describing your problem far more clearly. As it is, your question is far too vague to have any answer at all. Even if you do manage to word it in a valid mathematical context, I'll still claim that your problem will be virtually insolvable. That suggests you need to do some serious re-thinking about your problem.