Many thanks for your responses, and for your comments.
Is there a list somewhere of integrals which can be solved in closed form
by Axiom/FriCAS, but cannot be solved by the most recent versions of
Maple/Mathematica? I would be very interested in seeing such a list - but
I don't myself have access at the moment to either Maple or Mathematica.
Many thanks,
Alasdair
On Fri, Oct 2, 2015 at 4:06 PM, Waldek Hebisch <hebi...@math.uni.wroc.pl>
wrote:
> Tim Daly wrote:
> > Alasdair McAndrew wrote:
> > >Finally, I believe that Axiom is the only open-source software which
> > >includes a complete implementation of the Risch decision algorithm for
> > >symbolic integration; done by the late Manuel Bronstein initially in the
> > >1970's and 1980's. Is this correct? I suppose that the major
> commercial
> > >systems support as complete integration routines as possible, but Axiom
> has
> > >the edge on other (free) systems as far as I know.
> >
> > As far as I know Axiom has the "most complete" implementation.
> > There are still cases which are not implemented but Manuel did more
> > that anyone else.
>
> Actually, while "most complete" when written Bronstain's implementation
> contained substantial gaps. FriCAS contains significant enhancement
> of Bronstain's code. AFAICS FriCAS is the only system which can
> resonably claim completeness in purely transcendental case.
> For the old code is is relatively to build examples that either
> signal internal errors or return unevaluated.
> Completing this case required about 1500 lines, while in Bronstain's
> version about 2500 lines handles transcendental case. So about 30%
> of code was missing.
>
> In fact in FriCAS more than 25% of integration code is new. IIUC
> only tiny part of this (a few bug fixes) is included in Axiom.
> So FriCAS can claim to have "most complete" implementation, but
> Axiom no longer can.
>
> Concerning commercial systems, Maple claim to implement
> Trager algoritm and transcendental part of Risch algorithm,
> which is about what Bronstain's claimed to implement.
> However, at least with default Maple settings old Bronstain's
> examples still return unevaluated. IIRC when testing
> FriCAS implementation of transcendental case I found
> some other cases which Maple returns unevaluated.
> Mathematica makes far reaching but imprecise claim
> ("almost any function"), but experimental results are
> similar to Maple.
>
> AFAICS commercial systems take mostly ad hoc approach to
> integration, they probably contain a lot of special case
> code (pattern, tables or routines handling some narrow
> class of functions). In simple cases this produces
> impressive result. However, it is also fairly incomplete.
> FriCAS not contains extension of Risch algorithm to
> special functions (integrals needing Ei, erf, Gamma, ...).
> When comparing FriCAS with Ma-s I noticed that once
> examples got complicated enough, then Ma-s could no
> longer do them.
>
> --
> Waldek Hebisch
>
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