Vladimir Protasov (Moscow State University, Russia)

Invariant polyhedra for families of linear operators

d1, , are linear operators in mA A R

11

1/

1,..., {1,..., }

ˆ ( , , ) lim maxk

k

k

m d dk d d m

A A A A

The geometric sense:

ˆ 1 there exists a norm in

such that 1 for all 1, ... ,

d

iA i m

g fR

Taking the unit ball in that norm:

ˆ 1 there exists a symmetric convex body such that int , 1, ... , diM A M M i m g fR

M2A M

1A M

1 11

ˆ inf { 0 | , , are all contractions in some norm}mA A

The Joint spectral radius (JSR)

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.

1/

1,...,ˆ1. For one operator (A) = (A) = lim max | | (Gelfand's formula)

kkjk j d

m A

1If ,... , are commutative, or all symmetric, or all upper (lower) triangular,

or all orthogonal, or all stochastic, then

mA A

1 1ˆ ( ,..., ) max ( ) ,..., ( )m mA A A A

1 1ˆIn general, however ( ,..., ) max ( ) ,..., ( )m mA A A A

Basic properties of JSR

How to compute or estimate ?

11

1/

,..., {1,..., }ˆmax ,

Daubechies, Lagarias, Heil, Collela, Gripenberg.

Exponential complexit

By exhaust

y. Collela

ion ( by definiti

and Heil (1994) to compute JSR for spec

o )

ial

nk

k

k

d dd d m

A A k

2 2-matrices

with relative accuracy = 0.05 looked over all matrix products up to the length k = 19.

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 algorithmicallyundecidable in the dimension d = 47.

There is no polynomial-time algorithm, with respect to both the dimension d and the accuracyfor estimating JSR with the relative deviation

1

1However, there are algorithms polynomial separately in d or in (we shall s .ee)

The convergence to JSR is very slow

11

1/

,..., {1,..., }

d

The convergence to JSR is slow:

ˆmax ,

because our initial norm in may not be suitable for our operators.

The rate of convergence is ,

where C is a consta

kk

k

d dd d m

C

k

A A k

dR

nt, that may be very large.

Extremal norms

Theorem 1 (N. Barabanov, 1988)

1

1

1

a) For any irreducible family of operators ,..., there exists 0

and a norm such that

A , ... , A =

for any

ˆb) For any such a norm one has

max

( ,..., ).

m

m

d

m

A A

x

A A

x x x

RN

F(N)

f

f

1,...,

Let ( ) be the set of all norms in . is an infinite-dimensional pointed convex cone.

: , ( || . || )[ ] max || || ,

For any || . || N the functional ( | . | ) is also

d d

dj

j m

N N N

F N N F u A u u

F

R R

R

a norm.

The geometric sense:

1

1a) For an irreducible pair of operators ,..., there exists 0

and a symmetric convex body (invariant body) s.t.

ˆb) For any invariant body one has

( ,...,

(

)

def

m

m

AM Conv A M

A A

A M M

M

1 ,..., ).mA A

1A M

mA M

M

1 ( ,... , )def

mAM Conv A M A M

Independently. The ‘’dual’’ fact:

M

Theorem 2 (A.Dranishnikov, S.Konyagin, V.Protasov, 1996)

*

* *1

M = B (the polar to B), where B is the unit ball of Barabanov's norm for

the adjoint opeartors ,.

Du

..,

ali

ty:

mA A

How to determine M ?

1

is an arbitrary point , 0,

( ) { , 1,2, 1, , }

( ) , { } , ( ) , lim

is the set of accomulation points of the orbit .

Let ( ) ( ( ), ( )); then ( )

{ }

k

j j j

d

k d d j

dj j k m kj

x x

O x A A x d j k

x y k x O x x x

M x Conv x x x

d

dN

R

R

is nonempty,

dim ( ) and M x d

1 2ˆLet m =2. After possible noramalization we can assume ( , ) 1A A

1 2( ) ( ) ( )A x A x x

1 2fractal or self-similarity property ( )A K A K K

1 2( , )Conv A M A M M

Thus, ( ) ( )x x M x

x

1A x 2A x

1( ) ,O x 2 ( ) ,O x 3( ) ,O x , ( )O x

( )x

approximately with a given relative error 0.

1/ .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

1ˆ We assume ( , ... , ) 1.

We need a polytope such that , 1, ...., .

m

dj

A A

P A P P j m

R

0

1 1 2

1 1

is an arbitrary polytope, centrally-symmetric w.r.t. the origin,

the sequence { } is produced iteratively: ( , ).

The polytope approximates with the relative devi

d

k k k k k

k k

P

P Q AP Conv A P A P

P Q

R

(1 ) / 21 1 1

ation

(1 ) contains at most vertices. dk k k dQ P Q N C

1If we put , then the polytope would may have 2 verices,

and the complexity would be exponential. Actually we do not need to keep

all the vertices of , we always can make a selection so

nk k n

n

P A P P

P

that . n nP Q

kP

1 kA P

m kA P

k kQ AP

(1 ) kQ

1kP

2kP

After 10С Iterations we obtain the desirable approximation 1/ ˆ( ) n

ndiam P

The total number of operations ( 1) / 21 2( , , ) dC d A A

For d=2 the number of operations3/ 2C

For 0.0001 one has to perform 610 arithmetic operations.

In practice it works faster

Reason: in general the convergence1

1

1/

, ,ˆmax

nm

n

d dd d

A A

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.

1

0

ˆAssume we conjecture that = ( ).

To prove this conjecture we try to construct an invariant politope , where

= {

The ide

, }

a.

k

A

M

M Conv v v

11

1 1

1

, = { , 1, , },

where = ( ) , is the corresponding eigenvector .

ˆIf for some we have , then is an invarint polytope, and = .

k j k

k k k

M Conv A M j m

A v A v v

k M M M

0

0

A paremeter (0 ,1/2) and a polygon are given.

Iterative ``cutting-angle'' algorithm with parameter .

Converges to a continuous curve ( , ).

De Rham (1949-53). Implemented in curve design,

P

P

numerical methods of extrapolations, approximation theory.

Generalized to ``subdivision schemes'', studied since 1985

by N.Dyn, A.Levin, S.Dubuc, C.Micchelli, W.Dahmen, H.Prautzsch, K.DeBoor, etc.

1 2

1 2

De Rh

0

am ma

; , (0, 0.5

tri

)

ce

0 2

s

1

A A

X

Y

( )M

v

Example 1. De Rham curves.

0What is the smoothness (Holder exponent) of Proble (m ) ?: , P

is reduced to computing JSR of two special 2 x The answer 2 -matric es.

1

21 2

For 0.25 we have

For 0.25 we have

Computation of JSR:

ˆ = ( ) max {1 2 , }.

1ˆ = ( ) 4

For all the polygon

7 .

M has 6 vertices depending2

on .

A

A A

Extremal polytope: , 1, , .iA P P i m

0P

0( , )P

0( , )P

( ) is the tEuler binary otal number opartition f binary f eunct xpanion sionsdb k

1 2 10 1 2 1 , where 2 2 2 {0,1, , 1 }m

m jk d d d d d d

2Clearly, ( ) 1. For 3 one needs to estimate the growth of ( ) as . db k d b k k

2 L.( E) u1 l er , 1( 8)2 7b k

3 (S te( ) rn, 1858( 1 ))b k s k

4 Klosinsky, Alexanderson, Hill( ) / man ( ), 19842 1b k k

What is the asymptotic growth of ( ) as ?db k k

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)

The asymptotics of the binary Euler partitioExample 2. n func tion.

1 2

1 2

The asympotic growth is expressed by the JSR of two matrices , .

, are ( 1

Answer:

1, 2 2 1( )

) ( 1) matrices of zeros and ones:

iff

otherwise. 0,i j k

T T

T T

k j i dT

d d

1

1 1 1 0

0 1 1 0

0 1 1 1

0 0 1 1

T

2

1 1 0 0

1 1 1 0

0 1 1 0

0 1 1 1

T

For 5d Example.

1 2 1It appears that either or .

For every the polytope has 4 ( 2) dimensional

ˆ ˆ

face

( )

.

)

s

(

d M d d

T T T

1

Adjasency matrices in the problem of capacity of codes.

The dimension is 2 .

For {(0, , )} we have two 4 4-matrices of zeros and ones. For t

Example 3.

ˆ (1hem

the extremal polyt

5) / 2,

ope

md

D

M

4

1

5 ( ), ( 2, 5 1 ,2 , 5 1 ) , has 32 vertices.

For {(0, , )} we have the extremal polytope ( ),

(2, 5 1, 5 1, 2), has 40 vertices.

For {( , , , )} we have 8 8-matrices,

ˆ (1 5) / 2,

ˆ (

M x v M

D M M x

v M

AD A

1

2

1the extremal polytope ( ), it has 528 vertices (E.S

) 1.8668

hatokhi

...

n, 200

,

5).M M x

1

Does the extremal polytope norm always exist ?

ˆWe assume ( , ... , ) 1. We need a polytope such that , 1, ...., .dm jA A P A P P j m R

Necessary conditions:

1(1) There exists a finite product ... such that ( ) = 1 (the finiteness property).

kd dA A

1(2) The family is product bounded, i.e., || ... || C for all .

kd dA A k

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.

11,...,

Let , ... , , then , , | | 1 , 1,....,

is a complex polytope.

dk j j j j

j k

a a P z a z z j k

R C

The CPE conjecture. Are conditions (1) and (2) sufficient for the existence of the invariant complex polytope: , 1, ...., ?jA P P j m

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)

1ˆWe assume ( , ... , ) 1. We need a polytope such that , 1, ...., .dm jA A P A P P j m R

The cyclic tree algorithm (N.Guglielmi, V.Protasov, 2010):

1

1

1/

We look over all products ... of length k N.

Take the product ... , for which ( ) is maximal.

Normalze the opertors so tha

Step 1.

t ( ) = 1.

k

k

d d

kd d

A A

A A

1ˆWe try to prove that ( ,..., ) = . 1mA A

It appears that in practice the invariant polytope ‘’almost always’’ exists.

For more than 99 % of randomly generated matrices

1 We construct an invariant polytope for St ,.ep .., 2. .mP A A

1

2

1

j 1

We take the leading eigenvector of .

Set v , 2, , .k

k j k

d d

d d

v A A

A A v j k

1v 2v

kv

s j rv A v

p i qv A v

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

3v…..

Thank you !

1 be the closed semigroup generated by the operators ,..., mLet A A A

The algorithm converges within finite time, i.e., the invariant polytope

exists, iff all eigenvalues of operators from , except for the leading eigenvaues of

the product and of

Theorem.

its c li

yc

A

c permutations (that equal to 1) are strictly less than 1.

This holds for the vast majority of practical cases (more than 99% of randomly generated matrices).

The dimension d is up to 30-40.