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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
>
>
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