{
  "id": "1904.10566",
  "version": "v1",
  "published": "2019-04-23T23:24:39.000Z",
  "updated": "2019-04-23T23:24:39.000Z",
  "title": "Time-Varying Matrix Eigenanalyses via Zhang Neural Networks and look-Ahead Finite Difference Equations",
  "authors": [
    "Frank Uhlig",
    "Yunong Zhang"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "This paper adapts look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying symmetric matrix flows $A(t)$. It is based on the Zhang Neural Network (ZNN) model for time-varying problems and uses the associated error function $E(t) = A(t)V(t) - V(t) D(t)$ or $e_i(t) = A(t)v_i(t) -\\la_i(t)v_i(t)$ with the Zhang design stipulation that $\\dot E(t) = - \\eta E(t)$ or $\\dot e_i(t) = - \\eta e_i(t)$ with $\\eta > 0$ so that $E(t)$ and $e(t)$ decrease exponentially over time. This leads to a discrete-time differential equation of the form $P(t_k) \\dot z(t_k) = q(t_k)$ for the eigendata vector $z(t_k)$ of $A(t_k)$. Convergent look-ahead finite difference formulas of varying error orders then allow us to express $z(t_{k+1})$ in terms of earlier $A$ and $z$ data. Numerical tests, comparisons and open questions complete the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-23T23:24:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65H17",
      "65L12",
      "65F15",
      "65Q10",
      "92B20"
    ],
    "keywords": [
      "look-ahead finite difference equations",
      "zhang neural network",
      "time-varying matrix eigenanalyses",
      "look-ahead finite difference formulas",
      "smooth time-varying symmetric matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}