Re: Recursion question

*To*: mathgroup at smc.vnet.net*Subject*: [mg55516] Re: [mg55498] Recursion question*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sun, 27 Mar 2005 02:42:53 -0500 (EST)*References*: <200503260739.CAA21875@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

rbedient at hamilton.edu wrote: > I have a set of single step recursion equations that I want to simplify > into a single multi-step equation. Here's what it looks like: > > a[n]=4*a[n-1]*(1-a[n-1]) > b[n]=4*b[n-1]*(1-b[n-1]) > c[n]=(a[n]+b[n])/2 > a[1]=.1 <-arbitrary starting value > b[1]=.8 <-arbitrary starting value > > What I'm hoping for is something like: > > c[n]=some function of c[n-1], c[n-2]... > > I've tried various combinations of Solve, RSolve, Simplify etc. to no > avail. Any help would be appreciated. > > Fairly Newbie > > Dick You need to acquire sufficiently many polynomials to eliminate all a[...] and b[...] variables. One can observe that this is accomplished if we go to {a,b,c}[n-2]. polys = {a[n]-4*a[n-1]*(1-a[n-1]), b[n]-4*b[n-1]*(1-b[n-1]), c[n]-(a[n]+b[n])/2, c[n-1]-(a[n-1]+b[n-1])/2}; polysm1 = polys /. n->n-1; allpolys = Union[polys,polysm1] cvars = Cases[Variables[allpolys],c[_]]; elims = Complement[Variables[allpolys], cvars]; Here is the desired relation. In[24]:= InputForm[gb = GroebnerBasis[allpolys, cvars, elims]] Out[24]//InputForm= {-64*c[-2 + n] + 320*c[-2 + n]^2 - 512*c[-2 + n]^3 + 256*c[-2 + n]^4 + 20*c[-1 + n] - 64*c[-2 + n]*c[-1 + n] + 64*c[-2 + n]^2*c[-1 + n] - 4*c[-1 + n]^2 - c[n]} Daniel Lichtblau Wolfram Research

**Follow-Ups**:**Re: Re: Recursion question***From:*DrBob <drbob@bigfoot.com>

**References**:**Recursion question***From:*rbedient@hamilton.edu