Abstract
We propose a fast algorithm to calculate the $H_{\infty}$norm of a transfer matrix. The method builds on a wellknown relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a twoparameter eigenvalue problem, where, in the generic case, the critical value corresponds to a twodimensional 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. 31893205] 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. 709737].
Original language  English 

Pages (fromto)  619635 
Number of pages  17 
Journal  SIAM Journal On Matrix Analysis and Applications (SIMAX) 
Volume  35 
Issue number  2 
Early online date  15 May 2014 
DOIs  
Publication status  Published  31 Dec 2014 
Keywords
 Newton method
 transfer matrix
 Hamiltonian matrix
 H∞norm
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Profiles

Alastair Spence
 Centre for Networks and Collective Behaviour
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Institute for Mathematical Innovation (IMI)
 Department of Mathematical Sciences  Professor Emeritus
Person: Research & Teaching, Honorary / Visiting Staff