radix representation of numbers - komputeralgebra...

Introduction Expansivity Decision Classification Construction Research Dir’s Bibliography

Radix Representation of Numbers

Attila Kovács

Eötvös Loránd UniversityFaculty of Informatics

Department of Computer Algebra

Budapest, 12. November 2008

Attila Kovács Radix Representation of Numbers


Content

1 Introduction

2 Expansivity

3 Decision

4 Classification

5 Construction

6 Research Dir’s

7 Bibliography



NotationsLet Λ be a lattice in Rn,

M : Λ→ Λ such that det(M) 6= 0 (base), and

0 ∈ D ⊆ Λ a finite subset (digit set).

Definition (Number System)

The triple (Λ, M, D) is called a number system (GNS) ifevery element x of Λ has a unique, finite representation ofthe form

x =l∑

i=0

M idi ,

where di ∈ D and l ∈ N. The sequence (dl . . . d1 d0)denotes the expansion of x .



FactsSimilarity preserves the number system property, i.e,if M1 and M2 are similar via the matrix Q and(Λ, M1, D) is a number system then (QΛ, M2, QD) is anumber system as well.

No loss of generality in assuming that M is integralacting on the lattice Zn.

Notation

If two elements of Λ are in the same coset of thefactor group Λ/MΛ then they are said to be congruentmodulo M.



Theorem ([1])

If (Λ, M, D) is a number system then1 D must be a full residue system modulo M,2 M must be expansive,3 det(I −M) 6= ±1.

DefinitionIf a given triple (Λ, M, D) fulfills conditions (1) and (2)above then the system is called a radix system.



NotationsLet φ : Λ→ Λ, x

φ7→ M−1(x − d) for the unique d ∈ Dsatisfying x ≡ d (mod M).

The sequence x , φ(x), φ2(x), . . . is called the orbit ofx .

FactsSince M−1 is contractive and D is finite thereforethere exists a norm ‖.‖ on Λ and a constant C suchthat the orbit of every x ∈ Λ eventually enters thefinite set

S = {x ∈ Λ | ‖x‖ < C}

for the repeated application of φ.

The orbit of x is eventually periodic for all x ∈ Λ.



Theorem(Λ, M, D) is a GNS iff for every x ∈ Λ the orbit of xeventually reaches 0.

NotationsA point p is called periodic if φk(p) = p for somek > 0.The orbit of a periodic point p is called a cycle and isdenoted by C(p).All the periodic points are denoted by P.The domain of attraction of x ∈ Λ is

B(x) = {y ∈ Λ | ∃j ∈ N such that φj(y) = x}.

The decision problem for (Λ, M, D) asks if they form aGNS or not.The classification problem means finding all cycles.



NotationLet G(P) be a directed graph defined on P bydrawing an edge from p ∈ P to φ(p).

FactsG(P) is a disjoint union of directed cycles (loops areallowed).

P is finite;

if p ∈ P then Φ(p) ∈ P;

if p ∈ P then

‖p‖ ≤ L =Kr

1− r,

where r = ‖M−1‖ = sup‖x‖≤1 ‖M−1x‖ < 1, andK = maxd∈D ‖d‖.



Factsp ∈ P if and only if there is an l > 0 such that

p = d0 + Md1 + . . . + M l−1dl−1 + M lp, dj ∈ D;

If p1, p2 ∈ P then either C(p1) = C(p2) orC(p1) ∩ C(p2) = ∅;

B(P) = Λ;

If p1, p2 ∈ P then B(p1) = B(p2) if and only ifC(p1) = C(p2);



ExampleLet Λ = Z, M = −3, D = {0,−4, 7}.Then K = 7, L = 3.5, therefore P ⊆ [−3, 3].It is a radix system but not a GNS since

1 = 7 + 2 · (−3)⇒ φ(1) = 2,

2 = −4 + (−2) · (−3)⇒ φ(2) = −2

3 = 0 + (−1) · (−3)⇒ φ(3) = −1

−3 = 0 + 1 · (−3)⇒ φ(−3) = 1

−2 = 7 + 3 · (−3)⇒ φ(−2) = 3

−1 = −4 + (−1) · (−3)⇒ φ(−1) = −1

Hence G(P) = {0→ 0,−1→ −1}.



Let z ∈ Λ. If z0 := z 6∈ P then there is a unique l ∈ N andd0, d1, . . . , dl−1 ∈ D such that

zj = dj + Mzj+1 (j = 0, . . . , l − 1), zl = p ∈ P

and none of z0, z1, . . . , zl−1 do belong to P.

DefinitionUsing the notations above the standard expansion ofz 6∈ P is

z = (d0, d1, . . . , dl−1 | p) ,

where l is the length of the expansion.If p ∈ P then its standard expansion is

p = (∗ | p).



Theorem (Lengh of the standard expansion)

For a given (Λ, M, D) and z ∈ Λ there is a constant c ∈ Nfor which

l(z) ≤ log ‖z‖log (1/‖M−1‖)

+ c.



DefinitionThe set of “fractions” in (Λ, M, D) is

H =

{∞∑

n=1

M−ndn : dn ∈ D

}⊆ Rk .

NotationsΓl = d0 + Md1 + · · ·+ M ldl , (di ∈ D).

Then D = Γ0 ⊆ Γ1 ⊆ · · · .Γ =

⋃Γl (all lattice points with finite expansions).



TheoremH is compact.

H has interior points. More specifically⋃

p∈P(p + H)contains a neighborhood of the origin.

H is the attractor set of the IFS {fdi}, wherefdi (z) = M−1(z + di), di ∈ D.

Rk =⋃

(H + Λ)

For every x ∈ Rk there is a z ∈ Λ and h ∈ H such thatx = z + h.

λ(H) > 0 (λ is the Lebesgue measure on Rk ).

λ(H + z1 ∩ H + z2) = 0 for all z1 6= z2 ∈ Γ.

The periodic points P lie in the set −H.



Plotting the set H

Let gd : Rk → Rk , gd(z) = Mz − d , d ∈ D andlet K (H) be a bounded subset of Rk containing H.

Function ES C A P E AL G O R I T H M( K(H) , lim)

forall z ∈ K (H) do1

S0 ← {z}, j ← 0;2

while j < lim or Sj 6= ∅ do3

Sj = {gdi (z) ∈ K (H) : z ∈ Sj−1, di ∈ D};4

end5

if j < lim then6

z /∈ H7

else8

z ∈ H9

end10



Example I. The Davis–Knuth Twin DragonM =

( −1 −11 −1

), D = {(0, 0), (1, 0)}.



Example II. Tiling with the Twin-DragonM =

( −1 −11 −1

), D = {(0, 0), (1, 0)}.



Example III.M =

(2 −11 2

), D = {(0, 0), (1, 0), (0, 1), (0,−1), (−6,−5)}.



Algorithmical problems

How to decide expansivity?

How to generate expansive operators?

How to decide the number system property?

Case study: generalized binary number systems.

How to classify the expansions?

How to construct number systems?



Expansivity

Let Λ = Zn. Given operator M examine P =charpoly(M).

Root finding methods are unnecessary and slow.A polynomial is said to be stable if

1 all its roots lie in the open left half-plane, or2 all its roots lie in the open unit disk.

The first condition defines Hurwitz stability and thesecond one Schur stability.

There is a bilinear mapping between these criterions(Möbius map) [2].



Expansivity

Schur stability: Algorithm of Lehmer-Schur.

Hurwitz stability: An n-terminating continued fractionalgorithm of Hurwitz.

ResultsFor arbitrary polinomials Lehmer-Schur is faster.

For stable polynomials Hurwitz-method is faster.

Caution: Intermediate expression swell may occur.



Comparision of the methods for stable polynomials.



Hurwitz-method works also for symbolic coefficients.Let a(x) = a0 + a1x + a2x2 + x3 ∈ Z[x ].Hurwitz-method gives that a(x) is expansive if

3a0 − a1 − a2 + 3a0 − a1 + a2 − 1

,a0 + a1 + a2 + 1

3a0 − a1 − a2 + 3

8(a20 − a0a2 + a1 − 1)

(a0 − a1 + a2 − 1)(3a0 − a1 − a2 + 3),

are all positive.



Expansivity

How to generate expansive integer polynomials with givendegree and constant term?

Using Las Vegas type randomized algorithm, whichproduces an expansive polynomial in R[x ], thenmakes round.

Using the algorithm of Dufresnoy and Pisot [3], whichworks well for small constant term.



Expansivity

Generating random expansive matrices seemsdifficult.

One can apply an integer basis transformation to thecompanion matrix of a polynomial.

This method generates all expansive matrices only ifthe class number of the order corresponding to thepolynomial is 1.



ExampleLet

M =

1 1 −1 0−1 0 1 1

1 0 −1 1−1 0 0 0

.

M is expansive, its characteristic polynomial isf (x) = x4 + x2 + 2, D = {0, e1} is a complete residuesystem modulo the companion matrix CM of f (x) and(Z4, CM , D) is a number system, but it is not possible togive any digit set D′, for which (Z4, M, D′) would be anumber system, since M is not Z-similar to CM .



GNS DecisionLet (Zk , M, D) be given. For fast computation of φ weuse Smith normal form: there are lineartransformations U, V mapping Zk onto itself such thatUMV = G has diagonal form with positive integerelements g1, . . . , gk in the diagonal such that gi | gi+1

for i = 1, 2, . . . , k − 1, and

k∏i=1

gi = |det(M)| .

The Smith normal form can be obtained by doingelementary row and column operations of M. U andV have determinants ±1 and they are also invertiblehaving integer components.



Theorem (Computing φ)

Let u1, u2, . . . , uk denote the coordinates of Uz, z ∈ Zk

and let h : Zk → {0, 1, . . . , |det(M)| − 1},

h(z) =k∑

i=s+1

(ui mod gi)i−1∏

j=s+1

gj ,

where gi = 1, i = 1, . . . , s. Then z1 ≡ z2 modulo M if andonly if h(z1) = h(z2) (see [4]).



GNS Decision by a cuboid type encapsulation of P

The functions fd : Rk → Rk , fd(z) = M−1(z + d) for alld ∈ D are linear contractions.H is a self-affine attractor of {fd}-s.The cuboid method [4] uses a covering of the set offractions H in maximum norm (recall that all periodicpoints lie in the set −H. Since H is compact, lowerand upper bounds on the coordinates of periodicpoints can be calculated).It is possible to reduce the number of integers to bechecked for periodicity [1] by applying Hutchinson’stheorem (IFS method).Further, a simulated annealing type randomizedalgorithm (basis transformation method) can beapplied in order to improve the bounds [5].



Example for IFS methodM =

(2 −11 2

), D = {(0, 0), (1, 0), (0, 1), (0,−1), (−2,−3)}.



GNS Decision by basis transformationThe average improvement in the volume of the coveringset expressed in orders of magnitude.



GNS Decision by basis transformationM =

(1 −21 3

), D = {(0, 0), (1, 0), (0, 1), (4, 1), (−7, 6)}.

Changing the basis to {(1, 0), (−1, 1)} decreases thevolume from 42 to 24. |E | = 65.



GNS Decision by basis transformationM =

(0 −71 6

), D is canonical.

Replacing the basis vector (0, 1) with (−5, 1) givesvolume 4 instead of 64. |E | = 12.



GNS Decision by a successive enlarging

Brunotte’s canonical number system decisionalgorithm (i.e, M is the companion of a monic, integerpolynomial, D = {(i , 0, 0, . . . 0)T | 0 ≤ i < |detM|}, [6])can be extended [5].

Function CO N S T R U C T- S E T-E( M, D)

E ← D , E ′ ← ∅ ;1

while E 6= E ′ do2

E ′ ← E ;3

forall e ∈ E and d ∈ D do4

put φ(e + d) into E ;5

end6

end7

return E ;8



GNS Decision by a successive enlarging

The previous algorithm terminates. DenoteB = {(0, 0, . . . , 0,±1, 0, . . . , 0} the n basis vectors andtheir opposites.

Function SI M P L E- D E C I D E( M, D)

E ← CONSTRUCT-SET-E(M, D);1

forall p ∈ B ∪ E do2

if p has no finite expansion then3

return false ;4

end5

return true;6



Binary expansive polynomials



Binary number systemsDegree 2 3 4 5 6 7 8 9 10 11Expansive 5 7 29 29 105 95 309 192 623 339CNS 4 4 12 7 25 12 20 12 42 11

There are 1085 expansive polynomials of degree 12 withc0 = 2. The number of CNS-s are (probably) 66. Problem:in higher dimensions the volume of the covering set or theset E are sometimes too big.

Other results with small constant term’ polynomials arealso available [7].



GNS ClassificationTwo methods: cuboid type covering classification andsimple classify.

Function SI M P L E-C L A S S I F Y ( M, D)

D ← D;1finished ← false;2while not finished do3

E ← CONSTRUCT-SET-E(M,D) ;4finished ← true;5forall p ∈ E ∪ B do6

if p does not run eventually into D then7put newly found periodic points into D;8finished ← false;9

end10end11return D \ D (the set of non-zero periodic points);12



SIMPLE-CLASSIFY

M =(

1 −21 3

), D = {(0, 0), (1, 0), (0, 1), (4, 1), (−7, 6)}.



GNS Classification

Comparing covering classification and simple classify:

Covering classification is easily parallelizable(implementation in PVM is ready, publication is inprogress).

There is an efficient variant of the SIMPLE CLASSIFY

algorithm, which is parallelizable using a special datastructure.

Both give negative answers fast.

Either can beat the other in some cases.

Experiments show that the algorithmic complexity ofthe worst case is exponential for both.



Classification in special latticesJ.Thuswalder characterized the location and structureof periodic elements in dimension two using thepower basis and canonical digit sets.The same problem is solved for the integers of

imaginary quadratic fields Q(i√

F ) with basis {1, δ},F 6≡ 3 (mod 4), δ = i

√F , and basis {1, ω}, F ≡ 3

(mod 4), ω = (1 + i√

F )/2, andreal quadratic fields Q(

√F ) with basis {1, δ}, F 6≡ 1

(mod 4), δ =√

F , and basis {1, ω}, F ≡ 1(mod 4), ω = (1 +

√F )/2.



GNS Construction

Given lattice Λ and operator M satisfying criteria 2)and 3) in Theorem 1 is there any suitable digit set Dfor which (Λ, M, D) is a number system?

If yes, how many and how to construct them?



Theorem (Kátai)

Let Λ be the set of algebraic integers in an imaginaryquadratic field and let α ∈ Λ. Then there exists a suitabledigit set D by which (Λ, α, D) is a number system if andonly if |α| > 1, |1− α| > 1 hold.

Theorem ([8])

Let Λ be the set of algebraic integers in the real quadraticfield Q(

√2) and let 0 6= α ∈ Λ. If α, 1± α are not units and

|α| , |α| >√

2 then there exists a suitable digit set D bywhich (Λ, α, D) is a number system.



Theorem ([1])

If ‖M−1‖2 ≤ 1/(1 +√

k) then there exist a digit set D forwhich (Λ, M, D) is GNS (k = dim(Λ)).

Theorem ([9])

If ρ(M−1) < 1/2 then there exists a digit set D for which(Λ, M, D) is GNS.

Theorem ([9])

Let the polynomial c0 + c1x + · · ·+ xn ∈ Z[x ] be given andlet us denote its companion matrix by CM . If the condition|c0| > 2

∑ni=1 |ci | holds then there exists a digit set D for

which (Zn, CM , D) is GNS.



Research directions

Algebraic integers in some extension fields of therationals are special cases: M is the companion ofthe minimal polynomial.

Special digit sets: canonical, symmetrical.

Earlier results: D.E.Knuth, W.Penney, I.Kátai,B.Kovács, A.Petho, W.Gilbert, S.Körmendi.

Companion of monic, integer polynomials withcanonical digit sets: CNS-polynomials.

Results: S.Akiyama, A.Petho, H.Brunotte,K.Scheicher, J.Thuswaldner, etc.



Thank You



Kovács, A., Number expansion in lattices, Math. and Comp.Modelling, 38, (2003), 909–915.

Burcsi, P., Kovács, A., An algorithm checking a necessarycondition of number system constructions, Ann. Univ. Sci.Budapest. Sect. Comput. 25, (2005), 143–152.

Dufresnoy, J., Pisot, Ch., Etude de certaines fonctionsméromorphes bornées sur le cercle unité. Application a unensemble fermé d’entiers algébriques. Annales scientifiques del’École Normale Supérieure Sér. 3, 72 no. 1., (1955), 69–92.

Kovács, A., On computation of attractors for invertibleexpanding linear operators in Zk , Publ. Math. Debrecen56/1–2, (2000), 97–120.



Burcsi, P., Kovács, A., Papp-Varga, Zs., Decision andClassification Algorithms for Generalized Number Systems,Annales Univ. Sci. Budapest, Sect. Comp. 28, (2008), 141–156.

Brunotte, H., On trinomial bases of radix representations ofalgebraic integers, Acta Sci. Math. (Szeged), 67, (2001),407–413.

Burcsi, P., Kovács, A., Exhaustive search methods for CNSpolynomials, Monatshefte für Mathematik, to appear.

Farkas, G., Kovács, A., Digital expansion in Q(√

(2)), AnnalesUniv. Sci. Budapest, Sect. Comp. 22, (2003), 83–94.

Germán, L., Kovács, A., On number system constructions, ActaMath., Hungar., 115/1-2, (2007), 155–167.


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