Re: [Axiom-developer] Can you depend on it?

[Martin Rubey]

Dear Tim,

I agree with many things you write, save two:

1) I think that it is not worth dying for some robot or mathematics or
   anything.  And it is certainly not the case that you can rely ("depend") on
   anything.  I'd say, that both Bill and Boing team were just lucky.

   Important note: I do believe that various efforts and tools can improve on
   the likelihood of success, literate programming, unit testing are certainly
   two of them (the latter still missing in Axiom, because nobody finds the
   time to make AldorUnit work with libaxiom. Should be quite trivial, though).

   But it won't help you if you "died", even if the probability of dying was
   very very very low.

2) The history

>   f:=taylor(1-x^2)
>   sin asin f
> 
> gave a wrong answer. A "fix" was applied that gives the answer:
> 
>   1 - x^2 + O(x^11)
> 
> This was considered wrong and a new "fix" was applied giving the
> answer:
> 
>  1 - 1/4 x^4 - 1/16 x^6 - 7/768 x^8 - 5/3072 x^10 + O(x^11)

does not seem entirely correct to me. Since I believe that you have some
versions of axiom around, could you please double check?

In any case, Ralf Hemmecke has (in the framework of our species project) a well
documented and unit-tested powerseries implementation, albeit only a partial
implementation, currently. Looking at the code of Axiom, I'm quite convinced
that *all* of it should go away. It can be done much better, if you look at
Joris van der Hoeven's and Dominik Gruntz papers.

(On Computing Limits in a Symbolic Manipulation System; Dominik Gruntz. ETH
 Diss 11432, 1996. http://www.gruntz.ch/publications2.html)

By the way, I think that most code in axiom is crap. It is a mere coincidence
that it works roughly. If you want to see a well engineered system, look at the
MuPAD sources. (I'm not saying that it couldn't be done better. But it's quite
incredible how well it works. It's a pity that MuPAD is non-free now.)


Martin

PS: does anybody happen to know how many initial values one needs to single out
a specific Taylor series solution of an algebraic equation? To get a feel for
the problem:

There need not be a Taylor solution at all:

f(x)^2-x=0

There can be several branches with equal constant term:

 map(e+->series(e, x=0), zerosOf((x^2-1)*z^2-2*(x^4-1)*z+(x^2-1),z))

                             +-+                              +-+
                +-+     2   \|2   3      4       +-+     2   \|2   3      4
   (160)  [1 - \|2 x + x  - ---- x  + O(x ),1 + \|2 x + x  + ---- x  + O(x )]
                              4                                4


I'd guess that the first d terms are enough, when the equation has degree d in
z, but I don't know and in general less should be sufficient.

Some problem for ADE's, i.e., equations of the form

p(1, z(x), z'(x), z''(x), ...) = 0, p polynomial with coefficients in K[x]?