In this paper, we introduce a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary precision. Our approximation procedure is based on semidefinite liftings and can be implemented in a recursive way. For two matrices even the first step of the procedure gives an approximation, whose relative quality is at least 1/ √ 2, that is, more than 70%. The subse...

The joint spectral radius ρ of 2 matrices is related to the boundedness of all their products. Calculating ρ is known to be NP-hard. In this work we estimate the joint spectral radius associated to a bidimensional separable multiwavelet, in order to analyze its Hölder continuity. To the author’s knowledge this has not been done. The analysis aims at testing the aplicability of the multiwavelet ...

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is indepe...

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 but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of m...

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-comple...

The existing stability analysis of particle swarm optimization (PSO) algorithm is chiefly concluded by the assumption of constant transfer matrix or time-varying random transfer matrix. Firstly, one counterexample is provided to show that the existing convergence analysis is not possibly valid for PSO system involving random variables. Secondly, the joint spectral radius, mainly calculated by t...

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real d × d matrices {S1, S2}, one of which has rank 1, where 2 ≤ d < +∞. Let ρ(A) denote the spectral radius of a square matrix A. Then we prove that there always exists a finite-length word (i1, . . . , i ∗ l) ∈ {1, 2}, for some finite l ≥ 1, such that l √ ρ(Si1 · · ·Si∗l ) = sup n≥1 { max (i1,...,in)∈...

we give further results for perron-frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. we indicate two techniques for establishing the main theorem ofperron and frobenius on the numerical range. in the rst method, we use acorresponding version of wielandt&apos;s lemma. the second technique involves graphtheory.

In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.