### Abstract

The ε-pseudospectral abscissa and radius of an n × n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for ε sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rank-one perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with ε, and may be less than 10 in many practical cases of interest.

Original language | English (US) |
---|---|

Pages (from-to) | 1166-1192 |

Number of pages | 27 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 32 |

Issue number | 4 |

DOIs | |

State | Published - 2011 |

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### Keywords

- Eigenvalue
- Group inverse
- Pseudospectrum
- Reduced resolvent
- Robustness of linear systems
- Sparse matrix
- Spectral abscissa
- Spectral radius
- Stability radius

### ASJC Scopus subject areas

- Analysis

### Cite this

**Fast algorithms for the appro ximation of the pseudospectral abscissa and pseudospectral radius of a matrix.** / Guglielmi, Nicola; Overton, Michael L.

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 32, no. 4, pp. 1166-1192. https://doi.org/10.1137/100817048

}

TY - JOUR

T1 - Fast algorithms for the appro ximation of the pseudospectral abscissa and pseudospectral radius of a matrix

AU - Guglielmi, Nicola

AU - Overton, Michael L.

PY - 2011

Y1 - 2011

N2 - The ε-pseudospectral abscissa and radius of an n × n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for ε sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rank-one perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with ε, and may be less than 10 in many practical cases of interest.

AB - The ε-pseudospectral abscissa and radius of an n × n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. Existing techniques compute these quantities accurately, but the cost is multiple singular value decompositions and eigenvalue decompositions of order n, making them impractical when n is large. We present new algorithms based on computing only the spectral abscissa or radius of a sequence of matrices, generating a sequence of lower bounds for the pseudospectral abscissa or radius. We characterize fixed points of the iterations, and we discuss conditions under which the sequence of lower bounds converges to local maximizers of the real part or modulus over the pseudospectrum, proving a locally linear rate of convergence for ε sufficiently small. The convergence results depend on a perturbation theorem for the normalized eigenprojection of a matrix as well as a characterization of the group inverse (reduced resolvent) of a singular matrix defined by a rank-one perturbation. The total cost of the algorithms is typically only a constant times the cost of computing the spectral abscissa or radius, where the value of this constant usually increases with ε, and may be less than 10 in many practical cases of interest.

KW - Eigenvalue

KW - Group inverse

KW - Pseudospectrum

KW - Reduced resolvent

KW - Robustness of linear systems

KW - Sparse matrix

KW - Spectral abscissa

KW - Spectral radius

KW - Stability radius

UR - http://www.scopus.com/inward/record.url?scp=84856242290&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856242290&partnerID=8YFLogxK

U2 - 10.1137/100817048

DO - 10.1137/100817048

M3 - Article

AN - SCOPUS:84856242290

VL - 32

SP - 1166

EP - 1192

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 4

ER -