Re: [Axiom-mail] Type of Expression Problem

[Martin Rubey]

Hans Peter Würmli <address@hidden> writes:

> (1) -> e2:=binomial(n-1,k)/binomial(n,k)
> 
>          n - 1
>         (     )
>            k
>    (1)  -------
>            n
>           ( )
>            k
>                                                      Type: Expression Integer
> (2) -> retractIfCan(e2)@Union(Fraction Polynomial Integer, "failed")
> 
>    (2)  "failed"
>                                                     Type: Union("failed",...)
> (3) -> retractIfCan(normalize(e2))@Union(Fraction Polynomial Integer, 
> "failed")
> 
>         n - k
>    (3)  -----
>           n
>                                  Type: Union(Fraction Polynomial Integer,...)

this follows the documentation: binomial(n-1,k)/binomial(n,k) is not a rational
function, but normalize(e2) is:

        n - k
   (3)  -----
          n
                                                     Type: Expression Integer

Note the type information: it is still EXPR INT. After retraction it becomes
Fraction Integer!

> I manage now, but then my naive approach was to take a polynomial, say, p[n],
> and substitute n for N: subst(p,n=N) or eval(p,n=N), something that fails
> miserably. Nevertheless, the idea of substitution is so general that one
> would expect that Axiom would allow it.

I hope you use HyperDoc:

browse POLY
select operations
notice that there is no subst, but there is eval
select eval
select descriptions...

So what you want is eval(p,n,N) where N has to be of Type Poly. For example, it
could be N::POLY INT

(In the interpreter you don't have to specify anything, it does all the type
coercions for you)

> my primary goal would not be to have some representation, but to be able to
> represent the recurrences as naturally as possibly, natural meaning as close
> to standard math language as possible.

This is a different matter. I think that you should decide what you want to
implement: Zeilbergers algorithm or coercion of holonomic functions to 
expressions. The latter might be more tricky, but I don't know. 

> But as an aside: maybe 3 years ago when I encountered by chance Aldor I 
> became 
> quite enthusiastic about it, but stayed off really using it, because the 
> project was not free. Nevertheless, as some of my colleagues knew Manuel 
> Bronstein from his time in Zurich, I approached him and asked whether the 
> Algebra library would remain closed source. He answered very positively that 
> he only needed to brush up (to clean some sins, as I understood), before he 
> was going to publish the source. It seemed to me at the time that Stephen 
> Watt "owned" the compiler and Manuel the Algebra libraries, but with much 
> more openness on. Are they really lost with INRIA?

I don't know about Algebra, but SumIt is said to be open source in the near
future.

Martin