Solution of the reccurrence relation

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In summary, the problem given is to find the solution of the recurrence relation an=3an-1 - 3an-2 + an-3 if a0=2, a1=2, and a2=4. The approach to solving this is through "Linear homogeneous recurrence relations" using the characteristic equation, and the solution is an=n2-n+2. The Z-transform method can also be used to solve this type of equation.
  • #1
romo84
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Homework Statement


Find the solution of the recurrence relation an=3an-1 - 3an-2 + an-3 if a0=2, a1=2, and a2= 8.


Homework Equations





The Attempt at a Solution


I know that I need to express an in terms of an-1 but don't know how to conclude.
 
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  • #2
romo84 said:
I know that I need to express an in terms of an-1 but don't know how to conclude.


Um an is already in terms of an-1 so what do you mean? What are you covering in class now that might make the solution clearer? Usually solving for a recurrence relation means finding the general (non-recursive) form. Is that what you're supposed to do here?
 
  • #3
Yes, I am to find the non-recurrsive form (closed form).

Thank You!
 
  • #4
These types of equations are solvable using the Z-transform which is the dicrete math version of the Laplace transform. Are you familiar with this approach?

The basic idea is to take the Z-transform of your difference equation, then solve, then take the inverse Z-transform.
 
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  • #5
It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.
 
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  • #6
romo84 said:
It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.

Oh, too bad they make you do it the hard way. A couple of turns of the Z-crank yield an=n2-n+2.

Anyway, having the answer will help you check that you have your method right.
 
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  • #7
romo84 said:
It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.
What did you get for the characteristic equation and its roots?
 

1. What is a recurrence relation?

A recurrence relation is a mathematical equation that defines a sequence of numbers, in which each term is defined in terms of previous terms in the sequence. This allows for the calculation of subsequent terms without explicitly listing all the terms in the sequence.

2. How is a recurrence relation solved?

The solution of a recurrence relation depends on the specific equation and the order of the relation. For first-order recurrence relations, a closed-form solution can typically be found through substitution or by using a generating function. For higher-order recurrence relations, techniques such as iteration or characteristic roots can be used to find the solution.

3. What is the difference between a homogeneous and non-homogeneous recurrence relation?

A homogeneous recurrence relation is one in which the right-hand side of the equation is equal to zero, while a non-homogeneous recurrence relation has a non-zero right-hand side. This distinction is important in finding the solution, as the techniques used for homogeneous relations may not work for non-homogeneous ones.

4. Can all recurrence relations be solved?

Not all recurrence relations have a closed-form solution. In some cases, the solution may only be expressible in terms of an infinite series or may require numerical methods to approximate the solution. However, there are certain types of recurrence relations that have known solutions, such as geometric or arithmetic sequences.

5. How are recurrence relations used in real-world applications?

Recurrence relations are commonly used in various fields of science and engineering to model and solve problems involving repeated or recursive processes. They can be applied to analyze the growth of populations, predict future values in financial markets, and optimize algorithms in computer science, among many other applications.

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