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