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[MUG] Problems with doing a large sum
| [MUG] Problems with doing a large sum |
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Author: Robert Mann
Posted: 17/01/2001 16:29:07 GMT
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>> From: "Robert Mann"
Here's the problem: Consider the following quantities, which depend on
integers (N,t,s,n)
and on a complex variable "a":
>
Bn:=GAMMA(N-n+1-I*a)*GAMMA(n+I*a)/GAMMA(n)/GAMMA(N-n+1)/GAMMA(N-I*a)/GAMMA(N
+I*a)*(GAMMA(N))^2;
2
GAMMA(N - n + 1 - I a) GAMMA(n + I a) GAMMA(N)
Bn := -------------------------------------------------------
GAMMA(n) GAMMA(N - n + 1) GAMMA(N - I a) GAMMA(N + I a)
>
f1:=(N-t)*(t-s)*Bn/t/(N-t-I*a)/(N-s-I*a);f2:=Bn*(t-s)/(t+I*a)/(N-s-I*a);f3:=
s*(t-s)/(N-s)/(t+I*a)/(s+I*a)*Bn;
f1 := ((N - t) (t - s) GAMMA(N - n + 1 - I a) GAMMA(n + I a)
2
GAMMA(N) )/(GAMMA(n) GAMMA(N - n + 1) GAMMA(N - I a)
GAMMA(N + I a) t (N - t - I a) (N - s - I a))
2
f2 := GAMMA(N - n + 1 - I a) GAMMA(n + I a) GAMMA(N) (t - s)/(
GAMMA(n) GAMMA(N - n + 1) GAMMA(N - I a) GAMMA(N + I a)
(t + I a) (N - s - I a))
2
f3 := s (t - s) GAMMA(N - n + 1 - I a) GAMMA(n + I a) GAMMA(N) /(
(N - s) (t + I a) (s + I a) GAMMA(n) GAMMA(N - n + 1)
GAMMA(N - I a) GAMMA(N + I a))
The idea is to sum over (t,s,n) for each of f1, f2 and f3 to obtain a
complex function of
the variable "a". The poles of this function are always at imaginary
integer values of "a"
which range from -(N-1) to +(N-1). The goal is to compute the residues of
these poles.
Here's an example for N=3:
> Nn:=3;F1s:=0:F2s:=0:F3s:=0: for s from 1 to Nn-2 do for t from s+1 to Nn-1
do for n from t+1 to Nn do F1s:=subs(N=Nn,f1)+F1s od od od; for s from 1 to
Nn-2 do for t from s+1 to Nn-1 do for n from s+1 to t do
F2s:=subs(N=Nn,f2)+F2s od od od; for s from 1 to Nn-2 do for t from s+1 to
Nn-1 do for n from 1 to s do F3s:=subs(N=Nn,f3)+F3s od od od;
Fs:=I*factor(expand(F1s+F2s+F3s)/Nn);
for w from -Nn+1 to Nn-1 do
Dd[w,Nn]:=factor(simplify(residue(exp(-N*beta*lambda*I*a*z)*Fs,a=w*I))) od;
The problem I run into is that the computer runs out of memory rapidly for
N>10. By factorizing
separately the sums of f1, f2, and f3 I can get up to N=15, but I can't seem
to get much beyond that.
(For N=15 there are about 1300 terms before factorizing; afterward, about
90).
Once the sum is factorized the computer seems to have no trouble getting the
residues.
Is there some way around this problem to get to larger values of N (say
N=100)?
Robert Mann
Director, Guelph-Waterloo Physics Institute
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
tel: 519-885-1211x6285
fax: 519-746-8115
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