  We analyze two algorithms for computing the symplectic $LL^T$ factorization
$A=LL^T$ of a given symmetric positive definite symplectic matrix $A$. The
first algorithm $W_1$ is an implementation of the $HH^T$ factorization from
[Dopico et al., 2009], see Theorem 5.2. The second one, algorithm $W_2$ uses
both Cholesky and Reverse Cholesky decompositions of symmetric positive
definite matrices. We presents a comparison of these algorithms and illustrate
their properties by numerical experiments in MATLAB. A particular emphasis is
given on simplecticity properties of the computed matrices in floating-point
arithmetic.
