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].
Freitag, M. A., & Spence, A. (2011). A Newton-based method for the calculation of the distance to instability. Linear Algebra and its Applications, 435(12), 3189-3205. https://doi.org/10.1016/j.laa.2011.06.012