Estimation of the Joint Spectral Radius

Abstract

The joint spectral radius of a set of matrices is a generalization of the concept of spectral radius of a matrix. Such notation has many applications in the computer science, and more generally in applied mathematics. It has been used, for example in graph theory, control theory, capacity of codes, continuity of wavelets, overlap-free words, trackable graphs. It is impossible to provide analytic formulae for this quantity and therefore any estimation are highly desired. The main result of this paper is to provide an estimation of the joint spectral radius in the terms of matrices norms and spectral radii.

Simple renormalization schemes for multiple scattering series expansions

“…We can then approximate A by a series expansion of order n : A n = I + K + K 2 + K 3 + K n .The absolute truncation error ε abs introduced by this approximation is then‖ A − A n ‖ ≤ ε abs ,where ‖ ‖ represents any matrix norm. Using eqn (38), the previous inequality can be written as‖( K ) n +1 ( I − K ) −1 ‖ ≤ ε abs .We can then use the property‖ XY ‖ ≤ ‖ X ‖·‖ Y ‖,which is valid for any matrix norm, as well as 18 ρ ( X n ) = ρ ( X ) n ≤ ‖ X ‖ n .Choosing the minimum norm for which 21 the inequality (39) will always be satisfied if ρ ( K n +1 )(1 − ρ ( K )) −1 ≤ ε abs .This leads to the conditionFor instance, choosing an accuracy ε abs = 0.1, we find n ∼ 43 for ρ ( K ) = 0.9, n ∼ 9 for ρ ( K ) = 0.7, while for ρ ( K ) = 0.5 n drops to ∼3.Defining the relative truncation error ε rel byand using similar arguments, we obtain an estimate for the truncation order required for convergence to within ε rel :…”