The stability radius of an n × n matrix A (or distance to instability) is a well-known measure of robustness of stability of the linear stable dynamical systemẋ = Ax. Such a distance is commonly measured either in the 2-norm or in the Frobenius norm. Even if the matrix A is real, the distance to instability is most often considered with respect to complex valued matrices (in such case the two norms turn out to be equivalent) and restricting the distance to real matrices makes the problem more complicated, and in the case of Frobenius norm-to our knowledge-unresolved. Here we present a novel approach to approximate real stability radii, particularly well-suited for large sparse matrices. The method consists of a two level iteration, the inner one aiming to compute the εpseudospectral abscissa of a low-rank (1 or 2) dynamical system, and the outer one consisting of an exact Newton iteration. Due to its local convergence property it generally provides upper bounds for the stability radii but in practice usually computes the correct values. The method requires the computation of the rightmost eigenvalue of a sequence of matrices, each of them given by the sum of the original matrix A and a low-rank one. This makes it particularly suitable for large sparse problems, for which several existing methods become inefficient, due to the fact that they require to solve full Hamiltonian eigenvalue problems and/or compute multiple SVDs.
Let A be a real matrix with all distinct eigenvalues. We propose a new method for the computation of the distance w_R(A) of the matrix A from the set of real defective matrices, i.e., the set of those real matrices with at least one multiple eigenvalue with algebraic multiplicity larger than its geometric multiplicity. For 0 < ε ≤ w_R(A), this problem is closely related to the computation of the most ill-conditioned ε-pseudoeigenvalues of A, that is, points in the ε-pseudospectrum of A characterized by the highest condition number. The method we propose couples a system of differential equations on a low-rank manifold which determines the ε-pseudoeigenvalue closest to coalescence, with a fast Newton-like iteration aiming to determine the minimal value ε such that an ε- pseudoeigenvalue becomes defective. The method has a local behavior; this means that, in general, we find upper bounds for w_R(A). However, these bounds usually provide good approximations, in those (simple) cases where we can check this. The methodology can be extended to a structured matrix, where it is required that the distance be computed within some manifold defining the structure of the matrix. In this paper we extensively examine the case of real matrices. As far as we know, there do not exist methods in the literature able to compute such distance
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.