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.

翻译：本文对计算对称正定辛矩阵$A \in \mathbb R^{2n \times 2n}$的辛分解（形式为$A=LL^T$，其中$L \in \mathbb R^{2n \times 2n}$为辛块下三角矩阵）的两种算法（记为$W_1$和$W_2$）进行了详细的误差分析。算法$W_1$是[Dopico et al., 2009]中$HH^T$分解的一种实现。算法$W_2$由[Bujok et al., 2023]提出，利用分解过程中出现的对称正定矩阵块的Cholesky分解和逆Cholesky分解。我们证明，对于更广泛的一类对称正定矩阵$A \in \mathbb R^{2n \times 2n}$，算法$W_2$是数值稳定的，其在机器精度为$u$的浮点运算中产生的计算因子$\tilde L$满足$||A-\tilde L {\tilde L}^T||_2= {\cal O}(u ||A||_2)$。然而，对于一般的对称正定辛矩阵$A$，算法$W_1$通常是不稳定的，这一结论已由[Bujok et al., 2023]中的数值实验证实。本文同时给出了算法$W_1$的相应误差界，该界较弱，因为分解误差依赖于主子矩阵$A_{11}$逆的大小。在MATLAB中进行的测试表明，我们对所考虑算法给出的误差界是相当紧的。