1. Vladimir Protasov (Moscow State University, Russia) Invariant polyhedra for families of linear operators

2. The geometric sense : Taking the unit ball in that norm: The Joint spectral radius (JSR)

3. 1960 Rota, Strang (normed algebras) 1988-90 Barabanov, Kozyakin, Gurvits (linear switching systems) 1991 Daubechies, Lagarias, Cohen, Heil, Villemoes,…. (wavelets) 1989-92 Micchelli, Prautzsch, Dyn, Levin, Dahmen, … (approximation theory) distribution of random series (probability), asymptotics of the partition function (combinatorics, number theory), capacity of codes, counting of non-overlaping words, graph tractability problem, etc.

4. Basic properties of JSR

5. How to compute or estimate ? Blondel, Tsitsiclis (1997-2000). The problem of JSR computing for rational matrices in NP-hard The problem, whether JSR is less than 1 (for rational nonnegative matrices) is algorithmically undecidable in the dimension d = 47. There is no polynomial-time algorithm, with respect to both the dimension d and the accuracy for estimating JSR with the relative deviation The convergence to JSR is very slow

7. Extremal norms Theorem 1 (N. Barabanov, 1988) N F(N) The geometric sense:

8. Independently. The ‘’dual’’ fact: Theorem 2 (A.Dranishnikov, S.Konyagin, V.Protasov, 1996)

9. How to determine M ?

10. approximately with a given relative error The algorithm is polynomial w.r.t The key idea: to compute JSR and the extremal norm (the body) M simultaneously as a polytope. The geometric algorithm for computing JSR. Find The invariant polytope concept

12. AfterIterations we obtain the desirable approximation The total number of operations For d=2 the number of operations For one has to perform arithmetic operations. In practice it works faster Reason: in general the convergence is very slow. This is unavoidable, unless we do not know the extremal norm The algorithm iteratively approximates both and the extremal norm. In many cases this leads to the precise value of JSR The programm implementations for d =2 with pictures were done by I.Sheipak in 2000 and E.Shatokhin in 2005.

14. X Y Example 1. De Rham curves. Extremal polytope:

15. L.Euler (1728), A.Tanturri (1918), K.Mahler (1940), N.de Bruijn (1948) L.Carlitz (1965), D.Knuth (1966), R.Churchhouse (1969), B.Reznick (1990)

16. Example.

18. Necessary conditions: These conditions are still not sufficient. Example: A is a rotation of the plane by an irrational angle. Guglielmi, Wirth and Zennaro (2005) applied the concept of complex polytope norm. The CPE conjecture. Are conditions (1) and (2) sufficient for the existence of the invariant complex polytope: This is true for one operator. Guglielmi, Wirth and Zennaro (2005) proved the conjecture for some special cases. The answer is negative. Counterexamples are already for d=3 (Jungers, Protasov, 2009)

19. The cyclic tree algorithm (N.Guglielmi, V.Protasov, 2010): It appears that in practice the invariant polytope ‘’almost always’’ exists. For more than 99 % of randomly generated matrices

20. Every time we check if the new vertex is in the convex hull of the previous ones (this is a linear programming problem). The algorithm terminates, when there are no new vertices. The invariant polytope P is the convex hull of all vertices produced by the algorithm The ‘’dead’’ branches …..

21. Thank you ! This holds for the vast majority of practical cases (more than 99% of randomly generated matrices). The dimension d is up to 30-40.