Numerical stability of the symplectic $LL^T$ factorization
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929493498658816 |
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| author | Bujok, Maksymilian Rozložník, Miroslav Smoktunowicz, Agata Smoktunowicz, Alicja |
| author_facet | Bujok, Maksymilian Rozložník, Miroslav Smoktunowicz, Agata Smoktunowicz, Alicja |
| contents | In this paper we give the detailed error analysis of two algorithms $W_1$ 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. We prove that Algorithm $W_2$ is numerically stable for a broader class of symmetric positive definite matrices $A \in \mathbb R^{2n \times 2n}$. It means that Algorithm $W_2$ is producing the computed factors $\tilde L$ in floating-point arithmetic with machine precision $\mathcal{u}$ such that $||A-\tilde L {\tilde L}^T||_{2} = {\cal O}(\mathcal{u} ||{A}||_{2})$. On the other hand, Algorithm $W_1$ is unstable, in general, for symmetric positive definite and symplectic matrix $A$. In this paper we also give corresponding bounds for Algorithm $W_1$ that are weaker. We show that the factorization error depends on the condition number $κ_2(A_{11})$ of the principal submatrix $A_{11}$.
Bounds for the loss of symplecticity of the lower block triangular matrices $L$ for both Algorithms $W_1$ and $W_2$ that hold in exact arithmetic for a broader class of symmetric positive definite matrices $A$ (but not necessarily symplectic) are also given. The tests performed in \textsl{MATLAB} illustrate that our error bounds for considered algorithms are reasonably sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07662 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical stability of the symplectic $LL^T$ factorization Bujok, Maksymilian Rozložník, Miroslav Smoktunowicz, Agata Smoktunowicz, Alicja Numerical Analysis 15B10\sep 15B57\sep 65F25 \sep65F35 In this paper we give the detailed error analysis of two algorithms $W_1$ 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. We prove that Algorithm $W_2$ is numerically stable for a broader class of symmetric positive definite matrices $A \in \mathbb R^{2n \times 2n}$. It means that Algorithm $W_2$ is producing the computed factors $\tilde L$ in floating-point arithmetic with machine precision $\mathcal{u}$ such that $||A-\tilde L {\tilde L}^T||_{2} = {\cal O}(\mathcal{u} ||{A}||_{2})$. On the other hand, Algorithm $W_1$ is unstable, in general, for symmetric positive definite and symplectic matrix $A$. In this paper we also give corresponding bounds for Algorithm $W_1$ that are weaker. We show that the factorization error depends on the condition number $κ_2(A_{11})$ of the principal submatrix $A_{11}$. Bounds for the loss of symplecticity of the lower block triangular matrices $L$ for both Algorithms $W_1$ and $W_2$ that hold in exact arithmetic for a broader class of symmetric positive definite matrices $A$ (but not necessarily symplectic) are also given. The tests performed in \textsl{MATLAB} illustrate that our error bounds for considered algorithms are reasonably sharp. |
| title | Numerical stability of the symplectic $LL^T$ factorization |
| topic | Numerical Analysis 15B10\sep 15B57\sep 65F25 \sep65F35 |
| url | https://arxiv.org/abs/2310.07662 |