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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
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