I would like to write a computer script to generate a lot of symplectic matrices. How can I do this? Is there a parameterization of all symplectic matrices?
This article is about generating random matrices it includes symplectic matrices http://wwwstat.stanford.edu/~cgates/PERSI/papers/whatis.pdf 


Also check out http://www.ams.org/notices/200705/feamezzadriweb.pdf 


Choose a subgroup that is easy to generate, say $Sp(2)$, and pick a random pair of coordinates $i < j$ and a random element in $Sp(2)$ spanning the subspace spanned by those two coordinates. This gives a markov chain analogous to the Kac random walk. It is known that this procedure converges. In fact if it measures the convergence rate in the transportation distance, then the rate is $n^2 \log n$. 


Pick a random element of the Lie algebra of the symplectic group (this you can find explicitly described in Fulton and Harris), and exponentiate it. 

