On computing the symplectic $LL^T$ factorization

TL;DR

This paper compares two algorithms for computing the symplectic LL^T factorization of positive definite symplectic matrices, analyzing their properties and numerical stability through MATLAB experiments.

Contribution

It provides a detailed comparison and analysis of two algorithms for symplectic LL^T factorization, including their implementation and properties in floating-point arithmetic.

Findings

Algorithm W1 is based on the HH^T factorization method.

Algorithm W2 combines Cholesky and Reverse Cholesky decompositions.

Numerical experiments demonstrate the properties and stability of both algorithms.

Abstract

We analyze two algorithms for computing the symplectic $LL^T$ factorization $A=LL^T$ of a given symmetric positive definite symplectic matrix $A$. The first algorithm $W_1$ is an implementation of the $HH^T$ factorization from [Dopico et al., 2009], see Theorem 5.2. The second one, algorithm $W_2$ uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We presents a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on simplecticity properties of the computed matrices in floating-point arithmetic.