Discrete Dynamical Fibonacci

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Discrete
Dynamical
Fibonacci
Edward Early
Fibonacci Numbers
• F0 = 0, F1 = 1, Fn = Fn-1+Fn-2 for n > 1
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

1 5  
1 5 











2
2
5 





n
Fn 
1
n
Differential Equations
• inspired by undetermined coefficients
• Fn = Fn-1+Fn-2 characteristic polynomial x2-x-1
• formula C  1  5   C  1  5 
n
1


2


n
2


2


• initial conditions F0 = 0, F1 = 1 give C1 and C2
• BIG DRAWBACK: students were already
taking undetermined coefficients on faith (and
this application not in book)
Linear Algebra
• discrete dynamical systems
• A n x 0  c1 1n v 1  c 2  2n v 2
where λ1 and λ2 are the (distinct) eigenvalues
of the 2×2 matrix A with eigenvectors v1 and
v2, respectively, and x0=c1v1+c2v2
(Section 5.6 of Lay’s book)
Linear Algebra Meets Fibonacci
• Let
0

1
0
A 
1
1

1
 0   F0 
and x 0      
 1   F1 
Fn
1   Fn 1  
  Fn 
 
 


F
F

F
F
1   n   n 1
n 
 n 1 
• Thus Anx0 has top entry Fn
Linear Algebra Meets Fibonacci
• Let
0
A 
1
1

1
0 
and x 0   
1 
• det(A-λI) = λ2-λ-1
• eigenvalues
1
• eigenvectors
1
5
2
and
 1 


1  5 
 2 
and
5
2
 1 


1  5 
 2 
Linear Algebra Meets Fibonacci
• Let
0
A 
1
1

1
and
 1   1 
0 
1 


x0    
 1  5  1  5  
1
5
 






2


2



n 
1 
 1 
1 n 1  5 n 
1 1 5  
n


A x 0  c1 1 v 1  c 2 2 v 21  5 


1 5 

5  2  
5  2  
 2 
 2 
n
• top entry
n
n

1 5  
1
1 5 

Fn 
  
 

2  
5  2 



Caveat
• If
1 
x0   
1 
then…
 1 
 1 
5 5 
5 5 


x0 

1 5 
1 5 
10 
10 
 2 
 2 
• ugly enough to scare off most students!
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