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

Melina A Freitag, Alastair Spence

Research output: Contribution to journalArticle

22 Citations (Scopus)

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].
Original languageEnglish
Pages (from-to)3189-3205
Number of pages17
JournalLinear Algebra and its Applications
Volume435
Issue number12
Early online date19 Jul 2011
DOIs
Publication statusPublished - 15 Dec 2011

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Unstable
Bisection Method
Jordan Block
Computing
Hamiltonians
Singular Values
Transfer Matrix
Fast Algorithm
Eigenvalue Problem
Newton-Raphson method
Two Parameters
Regularity
Eigenvalue
Norm
Numerical Results
Control systems
Gas

Cite this

A Newton-based method for the calculation of the distance to instability. / Freitag, Melina A; Spence, Alastair.

In: Linear Algebra and its Applications, Vol. 435, No. 12, 15.12.2011, p. 3189-3205.

Research output: Contribution to journalArticle

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