In this paper we give the detailed error analysis of two algorithms \(\varvec{W}_{\varvec{1}}\) and \(\varvec{W}_{\varvec{2}}\) for computing the symplectic factorization of a symmetric positive definite and symplectic matrix \(\varvec{A} \varvec{\in } \mathbb {R}^{\varvec{2n \times 2n}}\) in the form \(\varvec{A}\varvec{=}\varvec{LL}^{\varvec{T}}\) , where \(\varvec{L} \varvec{\in } \mathbb {R}^{\varvec{2n \times 2n}}\) is a symplectic block lower triangular matrix. We prove that Algorithm \(\varvec{W}_{\varvec{2}}\) is numerically stable for a broader class of symmetric positive definite matrices \(\varvec{A} \varvec{\in } \mathbb {R}^{\varvec{2n \times 2n}}\) . It means that Algorithm \(\varvec{W}_{\varvec{2}}\) is producing the computed factors \(\varvec{\tilde{L}}\) in floating-point arithmetic with machine precision \(\varvec{u}\) such that \(\varvec{\Vert }{\varvec{A-\tilde{L} {\tilde{L}}}^{\varvec{T}}}\varvec{\Vert }_{\varvec{2}}\varvec{=} \varvec{\mathcal {O}}\varvec{(n}^{\varvec{2}} \varvec{u} \varvec{\Vert A\Vert _2}\varvec{)}\) . On the other hand, Algorithm \(\varvec{W}_{\varvec{1}}\) is unstable, in general, for symmetric positive definite and symplectic matrix \(\varvec{A}\) . In this paper we also give corresponding bounds for Algorithm \(\varvec{W}_{\varvec{1}}\) that are weaker. We show that the factorization error depends on the conditioning of the principal submatrix \(\varvec{A}_{\varvec{11}} \varvec{\in } \mathbb {R}^{{\varvec{n}} \varvec{\times } \varvec{n}}\) , located in the upper-left block of \(\varvec{A}\) . Bounds for the loss of symplecticity of the lower block triangular matrices \(\varvec{L}\) for both Algorithms \(\varvec{W}_{\varvec{1}}\) and \(\varvec{W}_{\varvec{2}}\) that hold in exact arithmetic for a broader class of symmetric positive definite matrices \(\varvec{A}\) (but not necessarily symplectic) are also given. The tests performed in MATLAB illustrate that our error bounds for considered algorithms are reasonably sharp.
