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

Nicola Guglielmi, Michael L. Overton

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1166-1192
Number of pages27
JournalSIAM Journal on Matrix Analysis and Applications
Volume32
Issue number4
DOIs
StatePublished - 2011

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Abscissa
Fast Algorithm
Radius
Pseudospectra
Modulus
Costs
Rank One Perturbation
Lower bound
Eigenvalue Decomposition
Group Inverse
Spectral Norm
Singular matrix
Computing
Singular value decomposition
Time Constant
Resolvent
Convergence Results
Rate of Convergence
Fixed point
Perturbation

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

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title = "Fast algorithms for the appro ximation of the pseudospectral abscissa and pseudospectral radius of a matrix",
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.",
keywords = "Eigenvalue, Group inverse, Pseudospectrum, Reduced resolvent, Robustness of linear systems, Sparse matrix, Spectral abscissa, Spectral radius, Stability radius",
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