non linear recurrence equations
in [Math]
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From: OwlHoot on 13 Jul 2010 12:51 On Jul 13, 5:44 pm, OwlHoot <ravensd...(a)googlemail.com> wrote: > On Jul 10, 5:17 pm, Panita Gomez <gatitablan...(a)gmail.com> wrote: > > > > > I'm serching to solve this non linear recurrence equations of this > > form with no hope :(. > > Anyone has an idea how to solve this kind of equations? > > > a+ b X_{i-1} +c X_{i} + d X_{i+1} + e X_{i-1} X_{i+1}=0 > > > without the last term, I can use the guassian elimination method to > > solve it. But when the last term appear I don't know how to deal with > > the multiplication between two reccurence anymore. help appreciated. > > If e != 0 you can simplify it a bit by dividing the > other coefficients by e and (symbolically) taking e = 1. > > (This might not be a good idea in practice, if you are > working on it numerically, for example if e is very > small compared with the other coefficients.) > > Assuming e = 1, you can factor it as follows: > > (X_{i-1} + d) (X_{i+1} + b) + c X_{i} + a - b d = 0 > > In this if c = 0 then you effectively have two separate > recurrences of opposite parity, each a continued fraction. > > Otherwise you can divide throughout by c^2 and then taking > X_i = c Y_i gives: > > (Y_{i-1} + A) (Y_{i+1} + B} + (Y_i + C) = 0 > > where A, B, C are rational functions of a, b, c, d. > > Next defining Z_i = Y_i + B gives the following with > C', A' = C - B, A - B : > > Z_{i+1} = (Z_i + C') / (Z_i + A') Woops, I was doing OK up to here, and then the phone rang! The last should be: Z_{i+1} = - (Z_i + C') / (Z_{i-1} + A') which isn't a continued fraction. So please ignore the last part of my post. Cheers John Ramsden |