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Re: Alternating sums of large numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg91944] Re: Alternating sums of large numbers
*From*: Mikhail Lemeshko <mikhail.lemeshko at gmail.com>
*Date*: Sat, 13 Sep 2008 05:57:33 -0400 (EDT)
*References*: <gaar1q$156$1@smc.vnet.net> <gadctg$rsg$1@smc.vnet.net>
On Sep 12, 11:31 am, "David Park" <djmp... at comcast.net> wrote:
> Use the WorkingPrecision option of Plot, and set NN to a high precision
> using the backquote character, or set NN = 139/2.
>
> NN = 69.5`30;
>
> n = 7;
>
> coeff = (Gamma[2*NN - n + 1]*(2*NN - 2*n))/n!;
>
> f[z_] := Sum[((-1)^(\[Mu] + \[Nu])*Binomial[n, \[Mu]]*
> Binomial[n, \[Nu]]*
> Gamma[2*NN - 2*n + \[Mu] + \[Nu], z])/(Gamma[
> 2*NN - 2*n + \[Mu] + 1]*Gamma[2*NN - 2*n + \[Nu] + 1]), {\=
[Mu],
> 0, n}, {\[Nu], 0, n}];
>
> Plot[coeff*f[z], {z, 0, 100},
> WorkingPrecision -> 25]
>
> --
> David Park
> djmp... at comcast.nethttp://home.comcast.net/~djmpark/
>
> "Mikhail Lemeshko" <mikhail.lemes... at gmail.com> wrote in message
>
> news:gaar1q$156$1 at smc.vnet.net...
>
> > Dear friends,
>
> > I'm using a Mathematica 6, and I've faced with the following problem.
> > Here is a copy of my Mathematica notebook:
>
> > --------------------------------------------------------
>
> > NN = 69.5;
>
> > n = 7;
>
> > coeff = (Gamma[2*NN - n + 1]*(2*NN - 2*n))/n!;
>
> > f[z_] := Sum[((-1)^(\[Mu] + \[Nu])*Binomial[n, \[Mu]]*Binomial[n, \
> > [Nu]]*Gamma[2*NN - 2*n + \[Mu] + \[Nu], z])/(Gamma[2*NN - 2*n + \[Mu]
> > + 1]*Gamma[2*NN - 2*n + \[Nu] + 1]), {\[Mu],0, n}, {\[Nu], 0, n}];
>
> > Plot[coeff*f[z], {z, 0, 100}]
>
> > --------------------------------------------------------
>
> > As you can see, I want to calculate a double alternating sum,
> > consisting of large terms (products of Gamma-functions and binomial
> > coefficients). Then I want to plot the result, in dependence on
> > parameter z, which takes part in the summation as an argument of the
> > incomplete Gamma-function, Gamma[2*NN - 2*n + \[Mu] + \[Nu], z].
>
> > Apart from this, I have another parameter, n, which is an upper limit
> > for both of sums, and also takes part in Gamma functions. When this
> > parameter grows, the expression next to summation also increases. At
> > some point, Mathematica begins to show very strange results - and my
> > question is actually about this.
>
> > For instance, if the parameter n=5, everything is O.K., the plot show=
s
> > a smooth curve. When we set n=6, there appears a little "noise" at
> > 60<z<80, which is of no sense. This noise increases with n and is huge
> > for n=8.
Thank you a lot, now it seems to work!
The problem was that I was playing with WorkingPresision, without
setting an accuracy for constants, such as NN = 69.5`30.
> > A suppose that this error is caused by the huge numbers with
> > alternating signs, contributing to the summation - probably there are
> > some mistakes introduced by numerical evaluation. I tried to play with
> > Accuracy etc., but it does not help. I also investigated the
> > possibility that the error is introduced not by the summation, but by
> > the product of big numbers. According to this, I tried to compute the
> > sum of Exp[Log[Gamma]+Log[Gamma]...] (the logarithm smoothly
> > depends on z). But it does not help as well...
>
> > I would very much appreciate your advice on such problem.
>
> > Many thanks in advance,
> > Mikhail.
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