A relaxation scheme for computation of the joint spectral radius of matrix sets

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.Comment: 16 pages, 2 figures, corrected typos, accepted for publication in JDE

Semi-hyperbolicity and bi-shadowing in nonautonomous difference equations with Lipschitz mappings

It is shown how known results for autonomous difference equations can be adapted to definitions of semi-hyperbolicity and bi-shadowing generalized to nonautonomous difference equations with Lipschitz continuous mappings. In particular, invertibility and smoothness of the mappings are not required and, for greater applicability, the mappings are allowed to act between possibly different Banach spaces

Approximations of the Rate of Growth of Switched Linear Systems

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts, in particular it characterizes the growth rate of switched linear systems. The joint spectral radius is notoriously di#cult to compute and to approximate. We introduce in this paper the first polynomial time approximations of guaranteed precision. We provide an approximation # that is based on ellipsoid norms, that can be computed by convex optimization, and that is such that the joint spectral radius belongs to the interval [ #/ # n, #], where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of all the matrices are non-negative; in this case the approximation is proved to be within a factor at most m (m is the number of matrices) of the exact value

Approximations of the rate of growth of switched linear systems

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts, in particular it characterizes the growth rate of switched linear systems. The joint spectral radius is notoriously difficult to compute and to approximate. We introduce in this paper the first polynomial time approximations of guaranteed precision. We provide an approximation (p) over cap that is based on ellipsoid norms that can be computed by convex optimization and that is such that the joint spectral radius belongs to the interval [(p) over cap/ rootn (p) over cap] where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of all the matrices are non-negative; in this case the approximation is proved to be within a factor at most m (m is the number of matrices) of the exact value

Stability of linear problems: joint spectral radius of sets of matrices

It is wellknown that the stability analysis of step-by-step numerical methods for differential equations often reduces to the analysis of linear difference equations with variable coefficients. This class of difference equations leads to a family F of matrices depending on some parameters and the behaviour of the solutions depends on the convergence properties of the products of the matrices of F. To date, the techniques mainly used in the literature are confined to the search for a suitable norm and for conditions on the parameters such that the matrices of F are contractive in that norm. In general, the resulting conditions are more restrictive than necessary. An alternative and more effective approach is based on the concept of joint spectral radius of the family F, r(F). It is known that all the products of matrices of F asymptotically vanish if and only if r (F) < 1. The aim of this chapter is that to discuss the main theoretical and computational aspects involved in the analysis of the joint spectral radius and in applying this tool to the stability analysis of the discretizations of differential equations as well as to other stability problems. In particular, in the last section, we present some recent heuristic techniques for the search of optimal products in finite families, which constitute a fundamental step in the algorithms which we discuss. The material we present in the final section is part of an original research which is in progress and is still unpublished