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.