Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2005
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2005

[Date Index] [Thread Index] [Author Index]

Search the Archive

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
> 
> 


  • Prev by Date: Re: Recursion question
  • Next by Date: Re: Recursion question
  • Previous by thread: Re: Recursion question
  • Next by thread: Re: Recursion question