On the method by Rostami for computing the real stability radius of large and sparse matrices

In a recent paper, Rostami [SIAM J. Sci. Comput, 37(2015), pp. S447-S471] has presented an interesting algorithm for the computation of the real pseudospectral abscissa and the real stability radius (aka the distance to instability) of a square matrix A ∈ ℝn, n in the spectral norm. The method is particularly well suited for large sparse matrices. The algorithm to compute the stability radius is based on the alternation of an iterative method in the rank-2 manifold of real matrices and a Lyapunov inverse iteration. The algorithm shows quadratic convergence in all experiments, but this peculiar property is only conjectured. In this paper, we provide a rigorous proof and propose a modification which replaces the Lyapunov inverse iteration and speeds up the method. Moreover, we interpret the algorithm to compute the real ϵ-pseudospectral abscissa as a nonstandard discretization of the ODE-based method given in Guglielmi and Lubich [SIAM J. Matrix Anal. Appl., 34(2013), pp. 40-66]. Several numerical illustrations comparing the modified algorithm with that proposed by Rostami conclude the paper.