Re: iterations, recursions and derivatives
- To: mathgroup at smc.vnet.net
- Subject: [mg22574] Re: [mg22544] iterations, recursions and derivatives
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Sat, 11 Mar 2000 17:52:45 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
On Thursday, March 9, 2000 3:53:41 PM, Andrzej Kozlowski wrote: > on 3/9/00 9:24 AM, Otto Linsuain at linsuain+ at andrew.cmu.edu wrote: > >> >> Hello. I find it hard to define a sequence of functions recursively and >> be able to differenciate them at the same time. For example >> >> f[x_,1]:=x^2 >> >> Try to differentiate Derivative[1,0][f][2,1] will not work. Changing := >> for = doesn't help. Can do: >> >> f[x_,1_]:=x^2 Notice the _ after the 1 (Kind of wierd, isn't it?) >> >> Now, however, can differentiate: Derivative[1,0][f][2,1] works fine. >> But cannot work recursions: >> >> f[x_,m_]:= SomeOperation[f[x,m-1]] confuses the recursion process. I >> have tried defining >> >> f[x_,m_]:=If[m==1,x^2,SomeOperation[f[x,m-1]], but the recursion again >> crashes. >> >> I have tried Which, Switch, Condition, Dt, D, etc, but to no avail. >> When I can take the derivative, I can't update m to m+1. >> >> Any suggestions? Thanks, Otto Linsuain. >> >> > > > You do not explain clearly enough what you want to do. For example, is the following satisfactory? > > > In[1]:= > f[1][x_] := x^2 > > In[2]:= > f[n_][x_] := Sin[f[n - 1][x^n]] > > In[3]:= > Derivative[1][f[5]][x] > > Out[3]= > 239 240 240 240 > 240 x Cos[x ] Cos[Sin[x ]] Cos[Sin[Sin[x ]]] > > 240 > Cos[Sin[Sin[Sin[x ]]]] > -- Forgot to add one thing (I am sure lots of others have already pointed this out): f[x_,1_]:=x^2 is simply f[x_,_]:=x^2, since Mathematica naturally interprets 1_ as Times[1,Blank]. So it is not surprising that you can differentiate but can't do recursion.