Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

Emre Mengi, Michael L. Overton

Research output: Contribution to journalArticlepeer-review

Abstract

Two useful measures of the robust stability of the discrete-time dynamical system x(k)(+1) = Ax(k) are the ε-pseudospectral radius and the numerical radius of A. The ε-pseudospectral radius of A is the largest of the moduli of the points in the ε-pseudospectrum of A, while the numerical radius is the largest of the moduli of the points in the field of values. We present globally convergent algorithms for computing the ε-pseudospectral radius and the numerical radius. For the former algorithm, we discuss conditions under which it is quadratically convergent and provide a detailed accuracy analysis giving conditions under which the algorithm is backward stable. The algorithms are inspired by methods of Byers, Boyd-Balakrishnan, He-Watson and Burke-Lewis-Overton for related problems and depend on computing eigenvalues of symplectic pencils and Hamiltonian matrices.

Original languageEnglish (US)
Pages (from-to)648-669
Number of pages22
JournalIMA Journal of Numerical Analysis
Volume25
Issue number4
DOIs
StatePublished - Oct 2005

Keywords

  • Backward stability
  • Field of values
  • Hamiltonian matrix
  • Numerical radius
  • Pseudospectrum
  • Quadratically convergent
  • Robust stability
  • Singular pencil
  • Symplectic pencil
  • ε-pseudospectral radius

ASJC Scopus subject areas

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

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