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

Emre Mengi, Michael L. Overton

Research output: Contribution to journalArticle

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

Fingerprint

Numerical Radius
Radius
Modulus
Field of Values
Pseudospectra
Discrete-time Dynamical Systems
Hamiltonian Matrix
Hamiltonians
Computing
Robust Stability
Dynamical systems
Eigenvalue

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix. / Mengi, Emre; Overton, Michael L.

In: IMA Journal of Numerical Analysis, Vol. 25, No. 4, 10.2005, p. 648-669.

Research output: Contribution to journalArticle

@article{3f05cf4e5333484fb742a872e1e77a04,
title = "Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix",
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.",
keywords = "ε-pseudospectral radius, Backward stability, Field of values, Hamiltonian matrix, Numerical radius, Pseudospectrum, Quadratically convergent, Robust stability, Singular pencil, Symplectic pencil",
author = "Emre Mengi and Overton, {Michael L.}",
year = "2005",
month = "10",
doi = "10.1093/imanum/dri012",
language = "English (US)",
volume = "25",
pages = "648--669",
journal = "IMA Journal of Numerical Analysis",
issn = "0272-4979",
publisher = "Oxford University Press",
number = "4",

}

TY - JOUR

T1 - Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

AU - Mengi, Emre

AU - Overton, Michael L.

PY - 2005/10

Y1 - 2005/10

N2 - 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.

AB - 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.

KW - ε-pseudospectral radius

KW - Backward stability

KW - Field of values

KW - Hamiltonian matrix

KW - Numerical radius

KW - Pseudospectrum

KW - Quadratically convergent

KW - Robust stability

KW - Singular pencil

KW - Symplectic pencil

UR - http://www.scopus.com/inward/record.url?scp=26444586128&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26444586128&partnerID=8YFLogxK

U2 - 10.1093/imanum/dri012

DO - 10.1093/imanum/dri012

M3 - Article

AN - SCOPUS:26444586128

VL - 25

SP - 648

EP - 669

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

SN - 0272-4979

IS - 4

ER -