[Axiom-developer] Re: integration

[Martin Rubey]

C Y writes:

 > Re: fixing bug - Well, I can give it a shot if it's simple to find... 

Well, what you need to find out is how the Risch algorithm would handle
sin(x^2). Maybe it is simply handled by pattern matching, as exp(-x^2) is... In
this case, fixing it would be easy. Mathematica also has a representation for
sin(x^n), but I don't know whether it is "useful". (It's in terms of incomplete
Gamma functions)

 > I've got some learning to do though.  Me thinks the Risch algorithm
 > would be a rather ambitious way to try getting the hang of the Axiom
 > source code...

Note that Axioms source is very nicely split into two parts: the "Algebra",
implementing its mathematical capabilities, and the rest (compiler,
interpreter, display, graphics, ...)

 > Ah, yes - the Risch algorithm.  Do you happen to know if the full
 > algorithm is described online anywhere?  Ideally both Risch's algorithm
 > and the extensions by Cherry et. al. for special functions would be
 > formally implimented (or documented since a fair part of it must
 > already be present.)  Apparently nobody impliments the full algorithm
 > due to complexity?:  http://mathworld.wolfram.com/RischAlgorithm.html  
 > I wonder if it's too complex for the Axiom system, given the structure
 > it has in place.  I've never studied it, but given how critical it is
 > to both systems it's high time I learned about it.

If you are interested, it's certainly worth the effort.

 > > Unfortunately Manuel Bronstein died. I don't know anybody who would
 > > be able to document these routines.
 > 
 > Well, when the great fall the lesser must pick up the torch.  

With some luck (and brains), the lesser become great that way.

 > I need to get ahold of some basic materials (Bronstein's books would be
 > useful, I'd wager :-) and get as much up to speed as my brain allows.  I
 > need to learn Axiom's guts anyway and this sounds like a possibly useful
 > way.  Maybe I could warm up by implimenting a physical units domain in
 > Axiom, although lord knows how I'd convince it to display things in the form
 > (numbers and variables)*units, or recognize Newtons out of a*kg*b*s^-2*c*m.
 > Does Axiom have provisions for non-standard display conventions and
 > substitution like that?

Output is easy. 

)sh OUTFORM

tells you everything you want to know about it.

Pattern matching is there too. 

)sh PATMATCH

or rather, use HyperDoc.

Martin