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MathGroup Archive 2007

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Re: simple sequence problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74423] Re: simple sequence problem
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Wed, 21 Mar 2007 02:44:10 -0500 (EST)
  • References: <etnkfe$gqo$1@smc.vnet.net>

=CF/=C7 traz =DD=E3=F1=E1=F8=E5:
> Hi, this might be a silly question, but how do you do this in mathematica.
>
> x1 == Cos[1];
> x2 == Cos[x1];
> x3 == Cos[x2];
>
> and so on ... ... so basically x(n)= Cos[x(n - 1)]
>
> I want to plot n against x(n). How do you do the code for this type of se=
quence in mathematica?


>I want to plot n against x(n). How do you do the code for this type of seq=
uence in mathematica?


Are you interested in cosine iterations?

In[1]:=
(*change some default settrings*)
s = Options[Plot];
$TextStyle = {FontFamily -> "Times"};
SetOptions[Plot, Frame -> {True, True, False, False}, Axes -> {True,
False}, AxesStyle -> GrayLevel[0.6], ImageSize -> 400, PlotRange ->
All];

(*your function*)
In[2]:=
f[x_][i_] := Nest[Cos, x, i]

In[5]:=
Print["example of application of f"]
f[4*(Pi/7)][10]

In[7]:=
Print["another example of application of f"]
(f[2.][#1] & ) /@ Range[40]

In[20]:=
Show[Block[{$DisplayFunction = Identity}, (Plot[f[x][#1], {x, -Pi,
Pi}, PlotStyle -> Hue[#1/7]] & ) /@ Range[1, 6]],
  PlotLabel -> StyleForm["Cosine Iteration", FontSize -> 14],
FrameTicks -> {Range[-Pi, Pi, Pi/2], Range[-1, 1, 1/2]}]

Note that

In[30]:=
(f[#1][100] & ) /@ Range[1., 100, 10]
Out[10]=
{0.739085,0.739085,0.739085,0.739085,0.739085,0.739085,0.739085,0.739085,0.\
739085,0.739085}

because cos(x)=x has a fixed point at x ~ 0.73908513.

In[31]:=
({#1, Cos[#1]} & )[FixedPoint[Cos, 10.]]
Out[31]=
{0.739085,0.739085}

In[12]:=
(*restore original settings of Plot*)
$TextStyle ={};
(SetOptions[Plot,#1]&)/@s;


Best Regards
Dimitris



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