### 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 language | English (US) |
---|---|

Pages (from-to) | 648-669 |

Number of pages | 22 |

Journal | IMA Journal of Numerical Analysis |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2005 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*IMA Journal of Numerical Analysis*, vol. 25, no. 4, pp. 648-669. https://doi.org/10.1093/imanum/dri012

}

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 -