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MathGroup Archive 2007

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Re: Better recursive method?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg72831] Re: Better recursive method?
  • From: Johan Grönqvist <johan.gronqvist at gmail.com>
  • Date: Sun, 21 Jan 2007 06:14:02 -0500 (EST)
  • References: <eosh0f$9bs$1@smc.vnet.net>

nandan skrev:
> This should be pretty easy to calculate for Mathematica. I have written
> it the following way:
> pi[0,theta] := 0;
> pi[1,theta] := 1;

I believe this should be pi[0,theta_] := 0

> pi[i_, theta_] := pi[i, theta] =(2n-1)/(n-1) Cos[theta] pi[i-1, theta]
> - n/(n-1) pi[i-2,theta]

Here n is a free variable, and I get a huge expression when trying to 
evaluate this. Is it correct to guess that n should be i?

I also believe that in general it is best to add an N[...] in the 
definition of pi, to avoid a large polynomial expression in rational 
numbers and Cos[theta].

If that is the case, and after setting $RecursionLimit = Infinity, it 
takes me 0.052003 Second to calculate pi[1000,1].

My kernel quits silently when attempting pi[10000,1], and I would be 
interested in knowing why.

> in IDL, it calculates
> in fraction of a second. Even the simple calculation of 'pi[40,pi/4]'
> took more than 350 seconds.
> 
> What could be the problem in my logic. Is there any other better way to
> write recursive functions in Mathematica?
> 
I believe it is about avoiding analytical expressions when you only want 
numerical answers.

Hope this can help somewhat.

/ johan


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