Recurrent Relations

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Atiwong Suchato

Faculty of Engineering, Chulalongkorn University

 

• Readings:

Recurrence Relations Rosen section 6.1-6.2

Atiwong Suchato

 

• A recurrence relationfor the sequence {a_n} is an equation that expresses a_n in terms of one or more of the previous terms, a₀,a₁,…,a_n-1.

• A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation.

• The initial conditions specify the terms that precede the first term where the recurrence relation takes effect.

 

• Example (Rosen p.402):

Determine whether a_n=3n, for every nonnegative integer n, is a solution of

a_n= 2a_n-1-a_n-2; n = 2, 3, …

   

How many bit strings of length n that do not have two consecutive zeros?

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Atiwong Suchato

   

Rules:

Move a disk at a time from one peg to another.

Never place a disk on a smaller disk.

The goal is to have all disk on the 2^nd peg in order of size.

Find H_n, the number of moves needed to solve the problem with n disks.

Atiwong Suchato

  

• A linear homogeneous recurrence relation can be solved in a systematic way.

Linear homogeneous recurrence relation of degree k with constant coefficients

a_n= c₁a_n-1 + c₂a_n-2+ …+c_ka_n-k where c₁, c₂, …, c_kare real numbers

and c_k ≠0

  

     

• a_n=a_n-1+a²_n-2

• H_n= 2H_n-1+1

• B_n= nB_n-1

• f_n= f_n-1 + f_n-2

  

 

• Recurrence Relation:

• Characteristic equation:

a_n- c₁a_n-1- c₂a_n-2- … - c_ka_n-k = 0

r^k- c₁r^k-1- c₂r^k-2… - c^k= 0 Example:

a_n= a_n-1+3a_n-2 b_n= 2b_n-1-b_n-2+5b_n-3 d = d +d

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Atiwong Suchato

  

 

• Let c₁, c₂, …, c_kbe real numbers. Suppose the characteristic equation:

r^k- c₁r^k-1- c₂r^k-2… - c^k = 0

has kdistinct roots r₁, r₂, …, r_k. Then a sequence {a_n} is a solution of the recurrence relation:

a_n - c₁a_n-1- c₂a_n-2- … - c_ka_n-k = 0

if and only if:

a_n= α₁r₁ⁿ+ α₂r₂ⁿ + … + α_kr_kⁿ

for n = 0, 1, 2, ….Where α₁,α₂ ,…, α_k are constants.

Atiwong Suchato

What is the solution of the recurrence relation:

a_n = a_n-1 + 2a_n-2 with a₀= 2and a₁= 7?

a_n = 6a_n-1- 11a_n-2+ 6a_n-3 with a₀= 2, a₁= 5and a₂ = 15 ?

 

• Suppose the characteristic equation has t distinct roots r₁, r₂, …, r_twith multiplicities m₁, m₂, …, m_t.

• Solution:

a_n = (α_1,0+ α_1,1n + … +α_1,m1-1n^m1-1)r₁ⁿ +(α_2,0+ α_2,1n + … +α_2,m2-1n^m2-1)r₂ⁿ + ………..

+(α_t,0+ α_t,1n + … +α_t,mt-1n^mt-1)r_tⁿ

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Atiwong Suchato

a_n= -3a_n-1- 3a_n-2 - a_n-3 with a₀= 1, a₁= -2and a₂= -1 ?

Atiwong Suchato

 

   

 

a_n= c₁a_n-1+ c₂a_n-2+ … + c_ka_n-k + F(n) Associated homogeneous

recurrence relation

{a

_n^h

} {a

_n^p

} {a

_n

} = {a

_n^h

} + {a

_n^p

}

 

   

 

• Key:

1 – Solve for a solution of the associated homogeneous part.

2 – Find a particular solution.

3 – Sum the solutions in 1 and 2

• There is no general method for finding the particular solution for every F(n)

• There are general techniques for some F(n) such as polynomials and powers of constants.

Find the solutions of a_n= 3a_n-1+2n with a₁ = 3

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Atiwong Suchato

Find the solutions of a_n= 5a_n-1 - 6a_n-2+7ⁿ

Atiwong Suchato

 

F(n) = ( b_tn^t+ b_t-1n^t-1+ … + b₁n + b₀) sⁿ where b₀, b₁, …, b_t and s are real numbers.

When s is not a root of the characteristic equation:

The particular solution is of the form:

( p_tn^t+ p_t-1n^t-1 + … + p₁n + p₀) sⁿ When s is a root of multiplicity m:

The particular solution is of the form:

n^m( p_tn^t+ p_t-1n^t-1+ … + p₁n + p₀) sⁿ

What form does a particular solution of a_n= 6a_n-1 – 9a_n-2+ F(n) have when:

F(n)= 3n, F(n)= n3ⁿ, F(n)= n²2ⁿ, F(n)= (n²+1)3ⁿ?

Find the solution of

∑

=

= ⁿ

k

n k

a

1