# A Newton-based method for the calculation of the distance to instability

### Reference:

Freitag, M. A. and Spence, A., 2011. A Newton-based method for the calculation of the distance to instability. *Linear Algebra and its Applications*, 435 (12), pp. 3189-3205.

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### Official URL:

http://dx.doi.org/10.1016/j.laa.2011.06.012

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### Abstract

In this paper, a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided. The method uses Newton’s method to find a two-dimensional Jordan block corresponding to a pure imaginary eigenvalue in a certain two-parameter Hamiltonian eigenvalue problem introduced by Byers [R. Byers, A bisection method for measuring the distance of a stable matrix to the unstable matrices, SIAM J. Sci. Statist. Comput. 9 (1988) 875–881]. This local method is augmented by a test step, previously used by other authors, to produce a global method. Numerical results are presented for several examples and comparison is made with the methods of Boyd and Balakrishnan [S. Boyd, V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L∞-norm, Systems Control Lett. 15 (1990) 1–7] and He and Watson [C. He, G.A. Watson, An algorithm for computing the distance to instability, SIAM J. Matrix Anal. Appl. 20 (1999) 101–116].

### Details

Item Type | Articles | ||||

Creators | Freitag, M. A.and Spence, A. | ||||

DOI | 10.1016/j.laa.2011.06.012 | ||||

Related URLs |
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Departments | Faculty of Science > Mathematical Sciences | ||||

Research Centres | Bath Institute for Complex Systems (BICS) | ||||

Refereed | Yes | ||||

Status | Published | ||||

ID Code | 25300 |

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