Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

M A Freitag, A Spence, Paul Van Dooren

Research output: Contribution to journalArticlepeer-review

13 Citations (SciVal)
223 Downloads (Pure)


We propose a fast algorithm to calculate the $H_{\infty}$-norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a two-parameter eigenvalue problem, where, in the generic case, the critical value corresponds to a two-dimensional Jordan block. We use the implicit determinant method which replaces the need for eigensolves by the solution of linear systems, a technique recently used in [M. A. Freitag and A. Spence, Linear Algebra Appl., 435 (2011), pp. 3189--3205] for finding the distance to instability. In this paper the method takes advantage of the structured linear systems that arise within the algorithm to obtain efficient solves using the staircase reduction. We give numerical examples and compare our method to the algorithm proposed in [N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709--737].
Original languageEnglish
Pages (from-to)619-635
Number of pages17
JournalSIAM Journal On Matrix Analysis and Applications (SIMAX)
Issue number2
Early online date15 May 2014
Publication statusPublished - 31 Dec 2014


  • Newton method
  • transfer matrix
  • Hamiltonian matrix
  • H∞-norm


Dive into the research topics of 'Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method'. Together they form a unique fingerprint.

Cite this