Summation by PartsDate: 01/07/2004 at 13:36:04 From: Doruk Subject: Series Problem Using 'E' to represent sigma, is there an approximate solution to E(Ai*Bi) = ? where i = 0,1,...,n if Ai is known explicitly and E(Bi) is known? Date: 01/07/2004 at 15:12:33 From: Doctor Vogler Subject: Re: Series Problem Hi Durok, I think you are looking for a technique called "summation by parts" (analogous to the calculus technique "integration by parts"). Essentially, in integration by parts, you change int(u*dv) = u*v - int(v*du). In summation by parts, you change sum(u*dv) = u*v - sum(v*du), where u is the Ai, and dv is the Bi (so that v = sum(Bi)). More explicitly, it works like this: Suppose that S(k) = sum(B(i),i=1 to k), so that S(0) = 0. sum(A(i)*B(i), i=1 to n) = sum(A(i)*(S(i)-S(i-1)),i=1 to n) = sum(A(i)*S(i),i=1 to n) - sum(A(i)*S(i-1),i=1 to n) = sum(A(i)*S(i),i=1 to n) - sum(A(i+1)*S(i),i=0 to n-1) = sum(A(i)*S(i),i=1 to n) - sum(A(i+1)*S(i),i=1 to n-1) = A(n+1)*S(n) + sum((A(i)-A(i+1))*S(i),i=1 to n) = A(n+1)*S(n) - sum((A(i+1)-A(i))*S(i),i=1 to n) Does that make sense? See also references on the Internet (found by searching Google for "summation by parts", with the quotes): Summation By Parts http://mathworld.wolfram.com/SummationbyParts.html Lemma: Summation By Parts http://www.shu.edu/projects/reals/numser/proofs/sumparts.html See if you can take it from there and write back if you need more help. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 01/07/2004 at 17:39:04 From: Doruk Subject: Thank you (Series Problem) Thanks for the quick reply, it was very useful information. I'm working on it and now I have another question. If I want to use only S(n), not S(i) i=1 to n, what would be the best approximation to the given exact solution? Date: 01/07/2004 at 20:09:53 From: Doctor Vogler Subject: Re: Series Problem Hi Doruk, Knowing only the A's and the sum of all of the B's is not enough information. The sum of the products could differ wildly according to the individual values of the B's. So you need more information to get any kind of approximation. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 01/08/2004 at 00:14:55 From: Doruk Subject: Thank you (Series Problem) Thanks again for all the useful information and help. |
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