Question on solving linear recurrence relations

  • Thread starter japplepie
  • Start date
  • #1
japplepie
93
0
Why does the characteristic equation of a linear recurrence relation always look like

an = series of constants multiplied by a number raised to n
 

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
7,741
1,527
Why does the characteristic equation of a linear recurrence relation always look like

an = series of constants multiplied by a number raised to n

Unless a specific statement is made about the relation between the characteristic equation and the recurrence, we don't have any theorem to prove. The definition of the characteristic equation gives the form of the equation. Definitions are stipulations, not things to be proven or disproven. We can explain the motivation for the definition by doing some algebra.

You didn't establish much notation, so let's use the notation of the Wikipedia article http://en.wikipedia.org/wiki/Recurrence_relation.

In that article, the recurrence relation is:
[itex] a_n = c_1a_{n-1} + c_2a_{n-2}+\cdots+c_da_{n-d} [/itex] .

Assume [itex] a_n = t^n [/itex] for some unknown [itex] t [/itex]. Substitute a [itex]t^k [/itex] for each [itex] a^k [/itex] in the recurrence relation.

This gives:
[itex] t^n = c_1t^{n-1} + c_2t^{n-2}+\cdots+c_dt^{n-d} [/itex]

Assuming [itex] t\ne 0 [/itex] multiply both sides of the above equation by [itex] t^{d-n} [/itex] :

[itex] t^d = c_1 t^{d-1} + c_2t^{d-2}+\cdots+ c_d^0 [/itex]

Which is equivalent to:
[itex] 0 = t^d - c_1 t^{d-1} - c_2t^{d-2}-\cdots- c_d^0 [/itex]

---------
The above is the same as the characteristic equation:
[itex] 0 = t^d - c_1t^{d-1} - c_2t^{d-2}-\cdots-c_{d} [/itex]

Which comes from setting zero equal to the characteristic polynomial, which is:
[itex] p(t)= t^d - c_1t^{d-1} - c_2t^{d-2}-\cdots-c_{d} [/itex]

.
 

Suggested for: Question on solving linear recurrence relations

Replies
3
Views
6K
Replies
3
Views
811
Replies
4
Views
2K
  • Last Post
Replies
12
Views
1K
  • Last Post
Replies
7
Views
6K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
13
Views
3K
  • Last Post
Replies
22
Views
8K
Top