## 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) |
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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 |

## 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

- General Mathematics
- Computational Mathematics
- Applied Mathematics