{
  "id": "1512.05292",
  "version": "v1",
  "published": "2015-12-16T19:32:00.000Z",
  "updated": "2015-12-16T19:32:00.000Z",
  "title": "Accurate Computations of Eigenvalues of Extremely Ill-conditioned Matrices and Differential Operators",
  "authors": [
    "Qiang Ye"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "This paper is concerned with computations of a few smaller eigenvalues (in absolute value) of a large extremely ill-conditioned matrix. It is shown that smaller eigenvalues can be accurately computed for a diagonally dominant matrix or a product of diagonally dominant matrices by combining a standard iterative method with the accurate inversion algorithms that have been developed for such matrices. Applications to the finite difference discretization of differential operators are discussed. In particular, a new discretization is derived for the 1-dimensional biharmonic operator that can be written as a product of diagonally dominant matrices. Numerical examples are presented to demonstrate the accuracy achieved by the new algorithms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-16T19:32:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F15",
      "65F35",
      "65N06",
      "65N25"
    ],
    "keywords": [
      "differential operators",
      "accurate computations",
      "diagonally dominant matrix",
      "smaller eigenvalues",
      "finite difference discretization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151205292Y"
    }
  }
}