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Recursive Calculation of Gaussian-Like Sequence

  • To: mathgroup at smc.vnet.net
  • Subject: [mg28018] Recursive Calculation of Gaussian-Like Sequence
  • From: "A. E. Siegman" <siegman at stanford.edu>
  • Date: Wed, 28 Mar 2001 02:40:48 -0500 (EST)
  • Organization: Stanford University
  • Sender: owner-wri-mathgroup at wolfram.com

(Apologies -- this is more a math than a Mathematica question -- but 
there seem to be a lot of helpful experts on this group.)

I'm evaluating the recursion relation:

   f[0] = 1; 
   f[1] = ((2-alpha)/2) f[0];
   f[n_] := f[n] = (2-alpha + beta (n-1)^2) f[n-1] - f[n-2]

numerically with beta = alpha^2 and with alpha having some small value 
like 2 X 10^(-5), expecting to get a gaussian-like sequence symmetric 
about n=0 that's close to f[n] approx= Exp[-(alpha/2) n^2] .

That's in fact exactly what I get, out to be about 5 or 6 standard 
deviations (n approx= 1000 for the alpha value above) -- but then the 
numerical results for larger n  suddenly turn into a growing 
Exp[+(alpha/2)n^2]-type behavior with a sign that's extraordinarily 
sensitive to small changes in the initial value of f[1]/f[0].

Is this round-off error? (probably not) -- an inherent numerical 
instability in the recursion relation?  (probably so).  Is there a way 
around it.

(The recursion form in the third line above comes from a variational 
analysis of a physical problem; the values of alpha and beta can be 
adjusted arbitrarily, but have to remain in a  k1 + k2 n^2  form.)


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