Hybrid expansion-contraction: A robust scaleable method for approximating the H norm

Tim Mitchell, Michael L. Overton

Research output: Contribution to journalArticlepeer-review


We present a new scaleable algorithm for approximating the H∞ norm, an important robust stability measure for linear dynamical systems with input and output. Our spectral-value-set-based method uses a novel hybrid expansion-contraction scheme that, under reasonable assumptions, is guaranteed to converge to a stationary point of the optimization problem defining the H∞ norm, and, in practice, typically returns local or global maximizers. We prove that the hybrid expansion-contraction method has a quadratic rate of convergence that is also confirmed in practice. In comprehensive numerical experiments, we show that our new method is not only robust but exceptionally fast, successfully completing a large-scale test set 25 times faster than an earlier method by Guglielmi, Gürbüzbalaban & Overton (2013, SIAM J. Matrix Anal. Appl., 34, 709-737), which occasionally breaks down far from a stationary point of the underlying optimization problem.

Original languageEnglish (US)
Pages (from-to)985-1014
Number of pages30
JournalIMA Journal of Numerical Analysis
Issue number3
StatePublished - Jul 1 2016


  • complex stability radius
  • pseudospectra
  • robust stability

ASJC Scopus subject areas

  • General Mathematics
  • Computational Mathematics
  • Applied Mathematics


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