  In this paper we give the detailed error analysis of two algorithms (denoted
as $W_1$ and $W_2$) for computing the symplectic factorization of a symmetric
positive definite and symplectic matrix $A \in \mathbb R^{2n \times 2n}$ in the
form $A=LL^T$, where $L \in \mathbb R^{2n \times 2n}$ is a symplectic block
lower triangular matrix. Algorithm $W_1$ is an implementation of the $HH^T$
factorization from [Dopico et al., 2009]. Algorithm $W_2$, proposed in [Bujok
et al., 2023], uses both Cholesky and Reverse Cholesky decompositions of
symmetric positive definite matrix blocks that appear during the factorization.
  We prove that Algorithm $W_2$ is numerically stable for a broader class of
symmetric positive definite matrices $A \in \mathbb R^{2n \times 2n}$,
producing the computed factors $\tilde L$ in floating-point arithmetic with
machine precision $u$, such that $||A-\tilde L {\tilde L}^T||_2= {\cal O}(u
||A||_2)$. However, Algorithm $W_1$ is unstable in general for symmetric
positive definite and symplectic matrix $A$. This was confirmed by numerical
experiments in [Bujok et al., 2023]. In this paper we give corresponding bounds
also for Algorithm $W_1$ that are weaker, since we show that the factorization
error depends on the size of the inverse of the principal submatrix $A_{11}$.
The tests performed in MATLAB illustrate that our error bounds for considered
algorithms are reasonably sharp.
