{
  "id": "2310.07662",
  "version": "v1",
  "published": "2023-10-11T17:02:39.000Z",
  "updated": "2023-10-11T17:02:39.000Z",
  "title": "Numerical stability of the symplectic $LL^T$ factorization",
  "authors": [
    "Maksymilian Bujok",
    "Miroslav Rozložník",
    "Agata Smoktunowicz",
    "Alicja Smoktunowicz"
  ],
  "comment": "24 pages, 3 figures, 2 tables",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-11T17:02:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B10",
      "15B57",
      "65F25"
    ],
    "keywords": [
      "factorization",
      "numerical stability",
      "symmetric positive definite matrix",
      "positive definite matrix blocks",
      "symplectic block lower triangular matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}