I have 2 symplectic matrices $X\_{1},X\_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X\_{1} \cdot X\_{2}$ is also symplectic. 


**Question:** Are there any theorems which allow me to express eigenvalues of $X$ if I know eigenvalues of each $X\_i$? 


**P.S.** Actually I am interested in some numerical algorithms. I cannot mupliply these matrices direcly because time-to-time this leads to overflow (or accuracy loss) problems. Maybe there are some iterative procedures? Some helpful decomposition of matrix?



Eigenvalues of product of symplectic matrices
Answering to your edit: there are numerical methods to compute the eigenvalues of a matrix product without forming it first. This has stability advantages with respect to forming the product explicitly. There is an excellent review by D.S. Watkins, *Product eigenvalue problems*, 2005, Siam review. As far as I know there is nothing relative to symplectic matrices specifically.


