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Alternating sums of large numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg91871] Alternating sums of large numbers
*From*: Mikhail Lemeshko <mikhail.lemeshko at gmail.com>
*Date*: Thu, 11 Sep 2008 06:14:13 -0400 (EDT)
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 shows
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.
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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