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] > <javascript:>> 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. > > 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] <javascript:>. > > To post to this group, send email to [email protected] > <javascript:>. > > 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.
