Numerical stability of the symplectic $LL^T$ factorization

TL;DR

This paper gives the detailed error analysis of two algorithms for computing the symplectic factorization of a symmetric positive definite and symplectic matrix A in the form of A=LL^T, and proves that Algorithm $W_2$ is numerically stable for a broader class of symmetricpositive definite matrices.

Abstract

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.

Figures (2)

• Figure 1: Condition numbers $\kappa_2(A)$ (denoted by ".") and $\kappa_2(A_{11})$ (denoted "x") for $A\in \mathbb R^{2n \times 2n}$ from Example \ref{['example7']}.
• Figure 3: Factorization error $\Vert A-L_1 L_1^T\Vert_2/\Vert A\Vert_2$ for Algorithm $W_1$ (denoted by ".") and $\Vert A_{11}^{-1}-S\Vert_2/\Vert A\Vert_2$ (denoted by "o") for $A\in \mathbb R^{2n \times 2n}$ from Example \ref{['example7']}.

Theorems & Definitions (52)

• Definition 1.1
• Lemma 1.2
• Theorem 1.3: Dopico
• Theorem 1.4
• Remark 1.1
• Example 1.1
• Lemma 2.1
• proof
• Remark 2.1
• Remark 2.2