Vladimir Protasov (Moscow State University , Russia )

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

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

3. Rota, Strang (normed algebras) • 1988-90 Barabanov, Kozyakin, Gurvits (linear switching systems) • 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 ? The convergence to JSR is very slow 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

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

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

9. How to determine M ?

10. The geometric algorithm for computing JSR. approximately with a given relative error Find 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 invariant polytope concept

12. After Iterations we obtain the desirable approximation The total number of operations For d=2 the number of operations one has to perform arithmetic operations. For 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. The programm implementations for d =2 with pictureswere done by I.Sheipak in 2000 and E.Shatokhin in 2005. In many cases this leads to the precise value of JSR

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

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

20. ….. The ‘’dead’’ branches 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

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