Re: Re: Recursion question

*To*: mathgroup at smc.vnet.net*Subject*: [mg55529] Re: [mg55516] Re: [mg55498] Recursion question*From*: DrBob <drbob at bigfoot.com>*Date*: Mon, 28 Mar 2005 02:42:06 -0500 (EST)*References*: <200503260739.CAA21875@smc.vnet.net> <200503270742.CAA06239@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

Brilliant! This makes it easier, though, to experiment with different depths of recursion (until you find one that works): dec = # /. n -> n - 1 &; cpoly = NestList[dec, c[n] - (a[n] + b[n])/2, 2]; apoly = NestList[dec, a[n] - 4*a[n - 1]*(1 - a[n - 1]), 1]; allpolys = Join[apoly, apoly /. a -> b, cpoly]; cvars = Cases[Variables[allpolys], c[_]]; elims = Complement[Variables[allpolys], cvars]; gb = GroebnerBasis[allpolys, cvars, elims] {-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]} Bobby On Sun, 27 Mar 2005 02:42:53 -0500 (EST), Daniel Lichtblau <danl at wolfram.com> wrote: > 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 > > > > > -- DrBob at bigfoot.com

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

**Re: Recursion question***From:*Daniel Lichtblau <danl@wolfram.com>