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Solving this type of recurrence equation

  1. Aug 3, 2011 #1

    The problem is to solve
    for [tex]P_{i}[/tex]
    with boundary condition
    [tex]P_{i}=P_{i+L}, g_{i}=g_{i+L}[/tex]
    Can anyone provide any guide of solving this type of recurence equation?
    Thank you!
    Last edited: Aug 3, 2011
  2. jcsd
  3. Aug 4, 2011 #2

    Stephen Tashi

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    Science Advisor

    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.
  4. Aug 5, 2011 #3
    Thanks for your reply.

    In my problem, [tex]g_i[/tex] is given and arbitrary, so in general [tex]g_i[/tex] is not a constant sequence.
  5. Aug 5, 2011 #4

    Stephen Tashi

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    Science Advisor

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

    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. )
  6. Aug 5, 2011 #5
    Because both [itex]\lbrace P_{i}\rbrace[/itex] and [itex]\lbrace g_{i}\rbrace[/itex] 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 [itex]g_{i}[/itex]. Then you will use the convolution theorem for the products.
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