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Re: Recursive Calculation of Gaussian-Like Sequence

• To: mathgroup at smc.vnet.net
• Subject: [mg28054] Re: [mg28018] Recursive Calculation of Gaussian-Like Sequence
• From: Tomas Garza <tgarza01 at prodigy.net.mx>
• Date: Thu, 29 Mar 2001 03:24:19 -0500 (EST)
• References: <200103280740.CAA25791@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

I was unable to reproduce the Exp[+(alpha/2)n^2]-type behavior with n as
large as 1400. In fact, the sequence f[n] becomes negative at n = 888
(f[888] = -7.66101 x 10^-6) and grows negatively (in an apparently
monotonous way) from there on. For example, f[1400] = -29.1429. The
approximation Exp[-(alpha/2) n^2] works reasonably well (of course, only
when f[n] is positive), say with error of magnitude 10^-4. By the way, I
tried to solve your recurrence equation using RSolve, but it flatly refused
to.

Tomas Garza
Mexico City
----- Original Message -----
From: "A. E. Siegman" <siegman at stanford.edu>
To: mathgroup at smc.vnet.net
Subject: [mg28054] [mg28018] Recursive Calculation of Gaussian-Like Sequence


> (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.)
>

• References:
  • Recursive Calculation of Gaussian-Like Sequence
    • From: "A. E. Siegman" <siegman@stanford.edu>