ANALYSIS of EUCLIDEAN ALGORITHMS
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- dynamical systems
- v– dynamical
- euclid algorithm
- up?1 up?1
- benoıt daireaux
- iv– euclidean algorithms
- algorithms
- u1 u1
- results obtained
- euclidean algorithms
ANALYSISofEUCLIDEANALGORITHMS
AnArithmeticalInstanceofDynamicalAnalysis
DynamicalAnalysis:=
AnalysisofAlgorithms+DynamicalSystems
Brigitte
Valle´e
(CNRSandUniversite´deCaen,France)
Resultsobtainedwith:
Ali
Akhavi
,Viviane
Baladi
,Je´re´mie
Bourdon
,
Eda
Cesaratto
,Julien
Cle´ment
,Benoıˆt
Daireaux
,
Philippe
Flajolet
,Loı¨ck
Lhote
,Ve´ronique
Maume
.
PlanoftheTalk
I–TheEuclidAlgorithm,andtheunderlyingdynamicalsystem
II–TheotherEuclideanAlgorithms
III–Probabilistic–anddynamical–analysisofalgorithms
IV–Euclideanalgorithms:theunderlyingdynamicalsystems
V–DynamicalanalysisofEuclideanalgorithms
PlanoftheTalk
I–TheEuclidAlgorithm,andtheunderlyingdynamicalsystem
II–TheotherEuclideanAlgorithms
III–Probabilistic–anddynamical–analysisofalgorithms
IV–Euclideanalgorithms:theunderlyingdynamicalsystems
V–DynamicalanalysisofEuclideanalgorithms
input
(
u,v
)
,itcomputesthe
gcd
of
u
and
v
,
ehtnO
together
htiw
eht
deunitnoC
rFnoitca
xEnoisnap
fo
.v/u
ehT
(standard)EuclidAlgorithm:thegrandfatherofallthealgorithms.
,pm1+...1+2m1+1m1=vu:vufoEFC.htpedehtsip.stigidehteras’imeht,vdnaufodcgehtsipu0=1+pu0+pupm=1−pu1−pu<pu<0pu+1−pu1−pm=2−pu+...=...2u<3u<03u+2u2m=1u1u<2u<02u+1u1m=0u1u≥0u;u=:1u;v=:0u
The(standard)EuclidAlgorithm:thegrandfatherofallthealgorithms.
Ontheinput
(
u,v
)
,itcomputesthe
gcd
of
u
and
v
,
togetherwiththe
ContinuedFractionExpansion
of
u/v
.v/ufonoisnapxEnoitcarFdeunitnoC
u;u=:u;v=:u010
u
0
=
m
1
u
1
+
u
2
u
1
=
m
2
u
2
+
u
3
...
=
...
+
u
p
−
2
=
m
p
−
1
u
p
−
1
+
u
p
u
p
−
1
=
m
p
u
p
+0
u
p
isthe
≥u1
0
<u
2
<u
1
0
<u
3
<u
2
0
<u
p
<u
p
−
1
u
p
+1
=0
.htpedehtsip.stigidehteras’meht,vdnaufodcgi
FCEfouv:uv=m1+m21+.1..1+1mp,
The(standard)EuclidAlgorithm:thegrandfatherofallthealgorithms.
Ontheinput
(
u,v
)
,itcomputesthe
gcd
of
u
and
v
,
togetherwiththe
ContinuedFractionExpansion
of
u/v
C.v/ufonoisnapxEnoitcarFdeunitno
u
0
:=
v
;
u
1
:=
u
;
u
0
≥
u
1
u
0
=
m
1
u
1
+
u
2
u
1
=
m
2
u
2
+
u
3
...
=
...
+
u
p
−
2
=
m
p
−
1
u
p
−
1
+
u
p
u
p
−
1
=
m
p
u
p
+0
0
<u
2
<u
1
0
<u
3
<u
2
0
<u
p
<u
p
−
1
u
p
+1
=0
u
p
isthe
gcd
of
u
and
v
,the
m
i
’sarethe
digits
.
p
isthe
depth
.
uCFEof:
v
1u=,1v+m11+m2.1..+mp
0[(metsySlacimanyDehtfoyrotcejartlanoitarA=)0,...,)x(2T,)x(T,x(mhtiroglAnaedilcuEehtfonoitucexenA0=)0(T,0=6xrofx1−x1=:)x(T,]1,0[→−]1,0[:Terehw,)ix(T=1+ixroix1−ix1=1+ixsanettirwnehtsi1+iu+iuim=1−iunoisividTheunderlyingEuclideandynamicalsystem(I).
e(
u
1
,u
0
)
is:
hThetraceoftheexecutionoftheEuclidAlgorithmon
T(
u
1
,u
0
)
→
(
u
2
,u
1
)
→
(
u
3
,u
2
)
→
...
→
(
u
p
−
1
,u
p
)
→
.(
u
p
+1
,u
p
)=
1(0
,u
p
)
−iuiu=:ixlanoitarehtyb)1−iu,iu(riapregetniehtecalpeR.mhtiroglaehtfonoisnetxesuounitnocasimetsyslacimanydehT.0sehcaertahtyrotcejarta=)T,]1,
.mhtiroglaehtfonoisneTheunderlyingEuclideandynamicalsystem(I).
tfor
x
6
=0
,T
(0)=0
x11T
:[0
,
1]
−→
[0
,
1]
,T
(
x
):=
x
−
x
e11x
i
+1
=
x
−
x
or
x
i
+1
=
T
(
x
i
)
,
where
ii
sThedivision
u
i
−
1
=
m
i
u
i
+
u
i
+1
isthenwrittenas
uuReplacetheintegerpair
(
u
i
,u
i
−
1
)
bytherational
x
i
:=
i
.
u1−i
oThetraceoftheexecutionoftheEuclidAlgorithmon
(
u
1
,u
0
)
is:
u(
u
1
,u
0
)
→
(
u
2
,u
1
)
→
(
u
3
,u
2
)
→
...
→
(
u
p
−
1
,u
p
)
→
(
u
p
+1
,u
p
)=(0
,u
p
)
nitnocasimetsyslacimanydehT.0sehcaertahtyrotcejarta=)T,]1,0[(metsySlacimanyDehtfoyrotcejartlanoitarA=)0,...,)x(2T,)x(T,x(mhtiroglAnaedilcuEehtfonoitucexenA
TheunderlyingEuclideandynamicalsystem(I).
ThetraceoftheexecutionoftheEuclidAlgorithmon
(
u
1
,u
0
)
is:
(
u
1
,u
0
)
→
(
u
2
,u
1
)
→
(
u
3
,u
2
)
→
...
→
(
u
p
−
1
,u
p
)
→
(
u
p
+1
,u
p
)=(0
,u
p
)
uReplacetheintegerpair
(
u
i
,u
i
−
1
)
bytherational
x
i
:=
i
.
u1−i
Thedivision
u
i
−
1
=
m
i
u
i
+
u
i
+1
isthenwrittenas
11x
i
+1
=
x
−
x
or
x
i
+1
=
T
(
x
i
)
,
where
ii
11T
:[0
,
1]
−→
[0
,
1]
,T
(
x
):=
x
−
x
for
x
6
=0
,T
(0)=0
An
execution
oftheEuclideanAlgorithm
(
x,T
(
x
)
,T
2
(
x
)
,...,
0)
=A
rationaltrajectory
oftheDynamicalSystem
([0
,
1]
,T
)
=a
trajectory
thatreaches
0
.
Thedynamicalsystemisacontinuousextensionofthealgorithm.
11T
(
x
):=
x
−
x
T
[
m
]
:]
m
1+1
,m
1[
−→
]0
,
1[
,
h1T
[
m
]
(
x
):=
x
−
m
[]m
]:0,[111−→
]
m
+1
,m
[
1h
[
m
]
(
x
):=
m
+
x
