Solving this type of recurrence equation

[samuelandjw]

Hi,

The problem is to solve
$$(1-g_{i+1})P_{i+1}-P_{i}+g_{i-1}P_{i-1}=0$$
for $$P_{i}$$
with boundary condition
$$P_{i}=P_{i+L}, g_{i}=g_{i+L}$$
Can anyone provide any guide of solving this type of recurence equation?
Thank you!

 

[Stephen Tashi]

I don't know of any general methods for solving recurrence relations in several unknown functions.

Since setting g and P to be constant sequences gives a solution to the problem, I suggest that you state additional conditions that g and P must satistfy if having them constant isn't what you are after.

 

[samuelandjw]

I don't know of any general methods for solving recurrence relations in several unknown functions.

Since setting g and P to be constant sequences gives a solution to the problem, I suggest that you state additional conditions that g and P must satistfy if having them constant isn't what you are after.

Thanks for your reply.

In my problem, $$g_i$$ is given and arbitrary, so in general $$g_i$$ is not a constant sequence.

 

[Stephen Tashi]

Then, as I interpret the problem, it amounts to solving $L$ simultaneous linear equations with constant coefficients and unknowns $P_1,P_2,..P_L$.

Are you looking for a closed form symbolic answer instead of a numerical one?

( The wikipedia article on "recurrence relation" says that linear constant coefficient difference equations can be solved with z-transforms. I, myself, have never done that. )

 

[Dickfore]

Because both $\lbrace P_{i}\rbrace$ and $\lbrace g_{i}\rbrace$ are period with a common period of L, you should use the discrete Fourier transform:

$$P_{i} = \frac{1}{\sqrt{L}} \, \sum_{k = 0}^{L - 1}{\tilde{P}_{k} \, \exp{\left(\frac{2 \pi j k \, i}{L}\right)}}$$
and similarly for $g_{i}$. Then you will use the convolution theorem for the products.