On Tue, Nov 4, 2014 at 3:36 PM, Richard Fateman <[email protected]> wrote: > > > On Tuesday, November 4, 2014 12:57:08 PM UTC-8, Aaron Meurer wrote: >> >> On Tue, Nov 4, 2014 at 1:48 PM, Richard Fateman <[email protected]> wrote: >> > I think your citing of Karr's paper is OK; the more modern notation >> > seems >> > to be >> > sum (i in the set{M} of f(i)) which avoids the order of enumeration >> > of >> > elements of M. >> >> Except it's nice to use uniform notation for infinite and finite sums, >> and the order does matter for infinite sums. > > > I agree. The representation of sets (unordered) and indexed sets > (indexed by Z or subset) can help. sum( i=1 to N except i=j of f(i)...) > is another requirement if you want to handle the conventional notation. >> >> >> Also people expect to write it as sum from i=n to m, so we might as >> well let them enter it that way and print it that way. > > > Some people are less than rigorous, even though they think they are. > >> >> >> I think practically, if you want to be rigorous, you need to have a >> separate function/object to represent indefinite summation, which may >> work a little differently than most people would expect from Sum(). >> >> > >> > As far as the best known algorithm, what about >> > >> > Bill Gosper's? >> > e.g. >> > http://mathworld.wolfram.com/GospersAlgorithm.html >> >> My understanding is that the Karr algorithm is to summation as the >> Risch algorithm is to integration, i.e., it is complete and can prove >> that summations don't have closed-form solutions (for some definition >> of closed-form). > > > Well, the Risch "algorithm" is non constructive in detail (requires > zero-tests). And there are piles of stuff that it doesn't do. > If Karr's algorithm solves the problem, why all this subsequent > publication? Maybe it just handles rationals and log/exp extensions? > If so it might rigorously solve a problem, just not the interesting problem.
Almost any nontrivial symbolic algorithm is only an "algorithm" modulo zero equivalence testing (anything that depends on polynomial division will fail if you can't test zero equivalence of the coefficients; ditto if you want to compute the rank of a matrix). I think you hit on the real point, which is that relatively few people care about symbolic summation, especially compared to symbolic integration. Aaron Meurer >> >> >> I don't know if anyone has implemented it completely. There is >> http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.67.4482. You >> might see if the examples in that paper can be done with Gosper's >> algorithm. I do know that from Karr's paper alone it is quite >> difficult to implement, as it is quite terse and very complicated. >> AFAIK no one has implemented Risch completely either so there's that. >> >> Aaron Meurer >> >> > >> > and the umpteen related papers some of which are in the references >> > there? >> > (which don't mention Karr; >> > in fact, it seems mathematica ignores him!) >> > >> > While I have not recently read Karr's paper (the one in J. ACM) I >> > recall it >> > being quite lengthy and not contributing >> > anything new that could be programmed in Macsyma. At least not >> > programmed >> > by me. >> > Not to say that anything in it is wrong. Just not helpful. >> > >> > Other work, Zeilberger, Wilf. .. has been put into Maxima, I think. >> > >> > RJF >> > >> > >> > >> > On Monday, November 3, 2014 4:43:43 PM UTC-8, Aaron Meurer wrote: >> >> >> >> We've been calling it the Karr convention because Karr outlines it in >> >> detail in his paper, "Summation in Finite Terms" and explains in detail >> >> why >> >> it is necessary. To quote from Section 1.4. >> >> >> >>> Consider Sum_m<=i<n f(i). If m < n, this has an obvious meaning. If m >> >>> => >> >>> n, this is summation over the null set, which is customarily defined >> >>> to be >> >>> zero. >> >> >> >> >> >> (emphasis mine) >> >> >> >> I think the fact that Karr indicates that this convention is not >> >> customary, the fact that he outlines why it is necessary in detail (at >> >> least >> >> relative detail considering the rest of the paper), and the fact that >> >> he is >> >> the author of the best known algorithm for doing summation in finite >> >> terms >> >> (to my knowledge) justifies calling it that. He may not be the first >> >> person >> >> to consider this, but it would hardly be the first time something in >> >> mathematics is named after someone who wasn't the first person to >> >> discover >> >> it (nor in computer algebra; consider Groebner bases). >> >> >> >> At any rate, the way the convention is described in the documentation >> >> does >> >> not necessarily predicate that this is the standard term for the >> >> convention. >> >> It is introduced as "for finite sums (and sums with symbolic limits >> >> assumed >> >> to be finite) we follow the summation convention described by Karr [1], >> >> especially definition 3 of section 1.4." Karr's paper is the best >> >> reference >> >> for why this convention is necessary that we have found, so referencing >> >> is >> >> and calling it thereafter "the Karr convention" or "Karr's convention" >> >> seems >> >> fine to me. >> >> >> >> Aaron Meurer >> >> >> >> >> >> On Mon, Nov 3, 2014 at 4:59 PM, Sergey B Kirpichev <[email protected]> >> >> wrote: >> >>> >> >>> On Mon, Nov 03, 2014 at 02:39:41PM -0800, Richard Fateman wrote: >> >>> > I checked with Gradshteyn and Rhyzik (1960, revised various times >> >>> > later), >> >>> > and they define sum if n<m to be zero. >> >>> >> >>> Yes, that's a different convention (Mathematica uses that, as well as >> >>> Maxima). >> >>> >> >>> -- >> >>> You received this message because you are subscribed to the Google >> >>> Groups >> >>> "sympy" group. >> >>> To unsubscribe from this group and stop receiving emails from it, send >> >>> an >> >>> email to [email protected]. >> >>> To post to this group, send email to [email protected]. >> >>> Visit this group at http://groups.google.com/group/sympy. >> >>> To view this discussion on the web visit >> >>> >> >>> https://groups.google.com/d/msgid/sympy/20141103225929.GA15940%40darkstar.order.hcn-strela.ru. >> >>> For more options, visit https://groups.google.com/d/optout. >> >> >> >> >> > -- >> > You received this message because you are subscribed to the Google >> > Groups >> > "sympy" group. >> > To unsubscribe from this group and stop receiving emails from it, send >> > an >> > email to [email protected]. >> > To post to this group, send email to [email protected]. >> > Visit this group at http://groups.google.com/group/sympy. >> > To view this discussion on the web visit >> > >> > https://groups.google.com/d/msgid/sympy/429e8237-cf26-436b-9f1a-e944b18ba360%40googlegroups.com. >> > >> > For more options, visit https://groups.google.com/d/optout. > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sympy. > To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/3afb5969-977b-470e-b867-c1f7e08a5351%40googlegroups.com. > > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sympy" group. 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