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

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].