We address the problem of the exact computation of two joint spectral characteristics of a family of linear operators, the joint spectral radius (in short JSR) and the lower spectral radius (in short LSR), which are well-known different generalizations to a set of operators of the usual spectral radius of a linear operator. In this article we develop a method which -under suitable assumptions -allows to compute the JSR and the LSR of a finite family of matrices exactly. We remark that so far no algorithm was available in the literature to compute the LSR exactly.The paper presents necessary theoretical results on extremal norms (and on extremal antinorms) of linear operators, which constitute the basic tools of our procedures, and a detailed description of the corresponding algorithms for the computation of the JSR and LSR (the last one restricted to families sharing an invariant cone). The algorithms are easily implemented and their descriptions are short.If the algorithms terminate in finite time, then they construct an extremal norm (in the JSR case) or antinorm (in the LSR case) and find their exact values; otherwise they provide upper and lower bounds that both converge to the exact values. A theoretical criterion for termination in finite time is also derived. According to numerical experiments, the algorithm for the JSR finds the exact value for the vast majority of matrix families in dimensions ≤ 20. For nonnegative matrices it works faster and finds JSR in dimensions of order 100 within a few iterations; the same is observed for the algorithm computing the LSR. To illustrate Keywords Linear operator · joint spectral radius · lower spectral radius · algorithm · polytope · extremal norm · antinorm.

For a given finite set Σ of matrices with nonnegative integer entries we study the growth ofWe show how to determine in polynomial time whether the growth with t is bounded, polynomial, or exponential, and we characterize precisely all possible behaviors. * R. Jungers and V. Blondel are with the

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We generalize the recent invariant polytope algorithm for computing the joint spectral radius and extend it to a wider class of matrix sets. This, in particular, makes the algorithm applicable to sets of matrices that have finitely many spectrum maximizing products. A criterion of convergence of the algorithm is proved.As an application we solve two challenging computational open problems. First we find the regularity of the Butterfly subdivision scheme for various parameters ω. In the "most regular" case ω = 1 16 , we prove that the limit function has Hölder exponent 2 and its derivative is "almost Lipschitz" with logarithmic factor 2. Second we compute the Hölder exponent of Daubechies wavelets of high order.If Algorithm 1 produces an invariant polytope, then our candidate s.m.p.'s are not only s.m.p.'s but also dominant products. A number of numerical experiments suggests that the situation when the algorithm terminates within finite time (and hence, there are dominant products) should be generic. 9