I am afraid, I do not understand the question. The Routh-Hurwitz criterion (or the Hurwitz criterion) gives necessary and sufficient conditions for a real polynomial of arbitrary degree to have all roots in the left half-plane. Do you look for a different criterion or for alternative forms of this criterion?
In the latter case, I find the form according to Hurwitz
http://www.ece.rutgers.edu/~npuri/research/stable_poly.pdf
quite useful for the exposition, although not for calculation as it would imply the
calculation of large determinants. So the recursive form
http://en.wikipedia.org/wiki/Routh%E2%80%93Hurwitz_stability_criterion
is more useful for polynomials of degree>3.
There is theory of Lyapunov, see
http://en.wikipedia.org/wiki/Lyapunov_stability
Nyquist plot method see
http://en.wikipedia.org/wiki/Nyquist_stability_criterion
and many others.
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