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 $X_i$?

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?

I have 2 real-valued symplectic matrices $X_{1},X_{2}$. 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 $X_i$?

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 $X_i$?