On Fri, Oct 2, 2015 at 4:06 PM, Waldek Hebisch <address@hidden> 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
--