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:

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