{
  "id": "2203.03930",
  "version": "v1",
  "published": "2022-03-08T08:51:16.000Z",
  "updated": "2022-03-08T08:51:16.000Z",
  "title": "Integral representations for higher-order Fréchet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number",
  "authors": [
    "Marcel Schweitzer"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We propose an integral representation for the higher-order Fr\\'echet derivative of analytic matrix functions $f(A)$ which unifies known results for the first-order Fr\\'echet derivative of general analytic matrix functions and for higher-order Fr\\'echet derivatives of $A^{-1}$. We highlight two applications of this integral representation: On the one hand, it allows to find the exact value of the level-2 condition number (i.e., the condition number of the condition number) of $f(A)$ for a large class of functions $f$ when $A$ is Hermitian. On the other hand, it also allows to use numerical quadrature methods to approximate higher-order Fr\\'echet derivatives. We demonstrate that in certain situations -- in particular when the derivative order $k$ is moderate and the direction terms in the derivative have low-rank structure -- the resulting algorithm can outperform established methods from the literature by a large margin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-08T08:51:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F35",
      "65F60",
      "15A16",
      "65D30"
    ],
    "keywords": [
      "condition number",
      "integral representation",
      "higher-order fréchet derivatives",
      "quadrature algorithms",
      "higher-order frechet derivative"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}