Re: Recursion question

*To*: mathgroup at smc.vnet.net*Subject*: [mg55510] Re: [mg55498] Recursion question*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Sun, 27 Mar 2005 02:42:43 -0500 (EST)*Reply-to*: hanlonr at cox.net*Sender*: owner-wri-mathgroup at wolfram.com

Direct recursion implementation used to verify results: ar[1]=1/10; ar[n_Integer?Positive]:=4*ar[n-1]*(1-ar[n-1]); br[1]=8/10; br[n_Integer?Positive]:=4*br[n-1]*(1-br[n-1]); Off[Solve::ifun]; c[n_, a1_:1/ 10, b1_:8/10] = Simplify[(a[n]+ b[n])/2 /. RSolve[{a[n]==4*a[n-1]*(1-a[n-1]), b[n]==4*b[n-1]*(1-b[n-1]), a[1]==a1, b[1]==b1 }, {a[n],b[n]}, n][[1]]] (1/4)*(-Cos[2^(n - 1)*ArcCos[1 - 2*a1]] - Cos[2^(n - 1)*ArcCos[1 - 2*b1]] + 2) Verifying that c[n] is consistent with direct recursion Plot[c[n],{n,0,5.2}, Epilog->{Red,AbsolutePointSize[4], Point/@Table[{n,(ar[n]+br[n])/2},{n,5}]}]; Bob Hanlon > > From: rbedient at hamilton.edu To: mathgroup at smc.vnet.net > Date: 2005/03/26 Sat AM 02:39:30 EST > To: mathgroup at smc.vnet.net > Subject: [mg55510] [mg55498] Recursion question > > 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 > >