{
  "id": "1909.01927",
  "version": "v1",
  "published": "2019-09-04T16:23:04.000Z",
  "updated": "2019-09-04T16:23:04.000Z",
  "title": "The spectral properties of Vandermonde matrices with clustered nodes",
  "authors": [
    "Dmitry Batenkov",
    "Benedikt Diederichs",
    "Gil Goldman",
    "Yosef Yomdin"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We study rectangular Vandermonde matrices $\\mathbf{V}$ with $N+1$ rows and $s$ irregularly spaced nodes on the unit circle, in cases where some of the nodes are \"clustered\" together -- the elements inside each cluster being separated by at most $h \\lesssim {1\\over N}$, and the clusters being separated from each other by at least $\\theta \\gtrsim {1\\over N}$. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at most $$\\frac{\\pi}{2}-\\frac{c_1}{N \\theta}-c_2 N h,$$ for some constants $c_1,c_2$ depending only on the multiplicities of the clusters. As a result, spectral analysis of $\\mathbf{V}_N$ is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of $\\mathbf{V}$, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with $s$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-04T16:23:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A18",
      "65T40",
      "65F20"
    ],
    "keywords": [
      "spectral properties",
      "clustered nodes",
      "study rectangular vandermonde matrices",
      "minimal principal angle",
      "local cluster multiplicities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}