Comment #24 on issue 3128 by [email protected]: Sum and Product manipulations
http://code.google.com/p/sympy/issues/detail?id=3128
Agree, reversed limits is definitely a corner case for students. I think
it's a corner case for almost everybody. I think that's a good reason not
to break with intuition.
2. Let S(k) be the set of integers between a(k) and b(k), including the
end points. Let M be the multiset composed of the join of the sets S(k),
for k = 0 .. n. How big is M? This is a very reasonable general math
question. In conventional notation, the answer should be Sum(
Sum(1,(i,a(k),b(k))), (k,0,n) ). However, the Karr convention imposes a
directionality on the sum, such that sometimes this will give the wrong
answer now. An explicit case is when a(k)=k-2, b(k)=2-k, n=4.
[ Sum((Sum(1,(i,-2+k,2-k))),(k,0,j)).doit() for j in [0,1,2,3,4]]
[5, 8, 9, 8, 5]
The answer should be [5, 8, 9, 12, 17]. Note that the problem is not
``easily fixed'' either, by say, inserting an absolute value. In that case,
[ Sum(abs(Sum(1,(i,-2+k,2-k))),(k,0,j)).doit() for j in range(5)]
[5, 8, 9, 10, 13]
which is still wrong. Of course, for this particular example, we can do
the math out by hand, but that would be missing the point. And there is
another version of this problem for which the Karr convention will give the
correct solution.
The Karr convention appeats to me to be a very nice self-consistent
resolution to a potential ambiguity for some problems where oriented sums
are needed. But it is not the unique solution, nor is it the ``intuitive
solution'' (IMO, data needed).
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