Tri polyphasé Introduction Modèle de calcul Tri polyphasé

(Fn)n∈NF0= 0 F1= 1 Fn+2 =
Fn+1 +Fnn≥0
•M
•
•var ←lireii
•i(var)i
•avancerii
•reculerii
M1M2m1m2
M1M2
O(m1+m2)

N
•nN
Fn> M ≥N Fn+1
•t=N
Fn+1 Fn+1
Fn−1Fn
•
Fnt
Fn−1t
Fn−12t
Fn−2t·Fi+1
Fn−1t·Fi+2
i i
Fn−i+1 t·Fi
Fn−it·Fi+1
Fn−it·
Fi+t·Fi+1
Fn−i−1t·Fi+1
Fn−it·Fi+2
(n−1)
ONlog N
M

n−1
i Fn−iO(t·Fi+t·Fi+1)
O t·
n−1
X
i=1
Fn−iFi+2!
n
X
k=0
Fn−kFk=n−1
5Fn+2n 5Fn−1
Fn=1
√5(φn+¯
φn)|φ|>1|¯ φ|<1
C=t·
n−1
X
i=1
Fn−iFi+2 =t·
n−1
X
i=1
Fn+2−(i+2)Fi+2 =t·
n+1
X
k=3
Fn+2−kFk
=t· n+2
X
k=0
Fn+2−kFk−(2 ·F0Fn+2 +F1Fn+1 +F2Fn)!
C=t·n+ 1
5Fn+2 +2(n+ 2) 5Fn+1 −Fn+2=t·n−4 5Fn+2 +2(n+ 2) 5Fn+1
t=N
Fn+1
C=N·n−4
5
Fn+2
Fn+1
+2(n+ 2)
5
(Fn)n
Fn+2
Fn+1
= 1 + Fn
Fn+1
=O(1)
log(Fn) = −1
2log(5) + nlog(φ) + log(1 + o(1)) =⇒n=O(log(Fn))
C=O(Nlog(Fn))
N
Fn
> M Fn=O(N
M)
ONlog( N
M)
n≥1
P(n) :
n
X
k=0
Fn−kFk=n−1
5Fn+2n 5Fn−1
• P(1) : 2F1F0= 0F1+2
5F0= 0
• P(2) : 2F2F0+F2
1=1
5F2+4 5F1= 1

•n≥2P(n)P(n−1)
n+1
X
k=0
Fn+1−kFk=
n−1
X
k=0
Fn+1−kFk+FnF1+Fn+1F0=
n−1
X
k=0
(Fn−k+Fn−1−k)Fk+Fn
=
n−1
X
k=0
Fn−kFk+
n−1
X
k=0
Fn−1−kFk+Fn=
n
X
k=0
Fn−kFk+
n−1
X
k=0
Fn−1−kFk+Fn
=n−1
5Fn+2n 5Fn−1+n−2 5Fn−1+2(n−1) 5Fn−2+Fn
=4
5Fn+n 5(Fn+Fn−1) + 2(n−1) 5(Fn−1+Fn−2)
=n
5Fn+1 +2(n−1) 5Fn=4 5Fn+n 5Fn+1 +2(n+ 1) 5Fn
P(n+ 1)
n≥1P(n)
(un)nun+2 =un+1 +unn≥0un=
λφn+µ¯ φnφ=1+√5
2¯
φ=1−√5
2X2−X−1=0
F0=λ+µ= 0 F1=λφ +µ¯ φ=
1
2((λ+µ) + √5(λ−µ)=1 λ=−µ=1
√5Fn=1 √5(φn+¯
φn)
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