### Abstract

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].

Language | English |
---|---|

Pages | 619-635 |

Number of pages | 17 |

Journal | SIAM Journal On Matrix Analysis and Applications (SIMAX) |

Volume | 35 |

Issue number | 2 |

DOIs | |

Status | Published - 15 May 2014 |

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### Keywords

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

### Cite this

*SIAM Journal On Matrix Analysis and Applications (SIMAX)*,

*35*(2), 619-635. https://doi.org/10.1137/130933228

**Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method.** / Freitag, M A; Spence, A; Van Dooren, Paul.

Research output: Contribution to journal › Article

*SIAM Journal On Matrix Analysis and Applications (SIMAX)*, vol. 35, no. 2, pp. 619-635. https://doi.org/10.1137/130933228

}

TY - JOUR

T1 - Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

AU - Freitag, M A

AU - Spence, A

AU - Van Dooren, Paul

PY - 2014/5/15

Y1 - 2014/5/15

N2 - 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].

AB - 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].

KW - Newton method

KW - transfer matrix

KW - Hamiltonian matrix

KW - H∞-norm

U2 - 10.1137/130933228

DO - 10.1137/130933228

M3 - Article

VL - 35

SP - 619

EP - 635

JO - SIAM Journal On Matrix Analysis and Applications (SIMAX)

T2 - SIAM Journal On Matrix Analysis and Applications (SIMAX)

JF - SIAM Journal On Matrix Analysis and Applications (SIMAX)

SN - 0895-4798

IS - 2

ER -