{
  "id": "2211.05179",
  "version": "v1",
  "published": "2022-11-09T20:14:51.000Z",
  "updated": "2022-11-09T20:14:51.000Z",
  "title": "Variational Characterization of Monotone Nonlinear Eigenvector Problems and Geometry of Self-Consistent-Field Iteration",
  "authors": [
    "Zhaojun Bai",
    "Ding Lu"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This paper concerns a class of monotone eigenvalue problems with eigenvector nonlinearities (mNEPv). The mNEPv is encountered in applications such as the computation of joint numerical radius of matrices, best rank-one approximation of third-order partial symmetric tensors, and distance to singularity for dissipative Hamiltonian differential-algebraic equations. We first present a variational characterization of the mNEPv. Based on the variational characterization, we provide a geometric interpretation of the self-consistent-field (SCF) iterations for solving the mNEPv, prove the global convergence of the SCF, and devise an accelerated SCF. Numerical examples from a variety of applications demonstrate the theoretical properties and computational efficiency of the SCF and its acceleration.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-09T20:14:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F15",
      "65H17"
    ],
    "keywords": [
      "monotone nonlinear eigenvector problems",
      "variational characterization",
      "self-consistent-field iteration",
      "third-order partial symmetric tensors",
      "monotone eigenvalue problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}