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MathGroup Archive 2004

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Problem rephrased: how to simplify summation of millions of exponentials symbolicly

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50185] Problem rephrased: how to simplify summation of millions of exponentials symbolicly
  • From: "networm" <networm8848 at yahoo.com>
  • Date: Thu, 19 Aug 2004 06:28:19 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hi all,

I have a summation of exponentials:

SUM= 1+r(a1, b1)*exp(-j*(a1*u+b1*v))+r(a2, b2)*exp(-j*(a2*u+b2*v))+r(a3,
b3)*exp(-j*(a3*u+b3*v))
           + ...
           + r(a1000000, b1000000)*exp(-j*(a1000000*u+b1000000*v))

where "j" is the imaginary sign.  a1, a2, ... a1000000, b1, b2, ... b1000000
are known constants... (a's and b's) constitute some grids on the 2D plane,
r(a, b) is the function defined on this 2D plane...
r(a, b) is known as a look-up-table, but there is no closed-form expression
for r(a, b)...
u, v are frequency variable in 2D case.

Do you think it is possible to compute the close-form of the above SUM
symbolically/analytically?

If not, is there any simple/efficient way to compute it ? 

I just need to compute this huge expression once, then if an simplified
symbolic expression is found, it will save my subsequent numerical
evaluations(that's going to tens of millions...)





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