Re: [Axiom-math] Curious behavior of Taylor series

[Igor Khavkine]

On 20 Aug 2006 23:45:38 +0200, Martin Rubey <address@hidden> wrote:
> "Igor Khavkine" <address@hidden> writes:
>
> > Can someone explain the following behavior of Taylor series in Axiom?
> >
> > (113) -> y := taylor x
> >    (113)  x
> >                          Type: UnivariateTaylorSeries(Expression Integer,x,0)
> > (114) -> x*y
> >    (114)  x x
> >                          Type: UnivariateTaylorSeries(Expression Integer,x,0)
> > (115) -> coefficient(%,1)
> >    (115)  x
> >                                                      Type: Expression Integer
>
> The reason is that Axiom cannot really know whether you meant x in (114) to be
> an element of the coefficient Ring EXPR INT, or to be a univariate Taylor
> series. In case of doubt, it usually chooses the wrong possibility :-)
>
> Thus, you should help Axiom by saying
>
>  monomial(1,1)$UTS(EXPR INT,x,0) * y

OK, but the following definitely looks like a bug.

(279) -> monx := monomial(1,1)$UTS(EXPR INT,x,0)
  (279)  x
                        Type: UnivariateTaylorSeries(Expression Integer,x,0)
(281) -> sqrt(monx*monx)
(281) ->
  (281)  1
                        Type: UnivariateTaylorSeries(Expression Integer,x,0)
(282) -> serx := series x
(282) ->
  (282)  x
                       Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
(283) -> sqrt(serx*serx)
(283) ->
  (283)  x
                       Type: UnivariatePuiseuxSeries(Expression Integer,x,0)

It looks like sqrt(), when applied to UTS, shifts the coefficients so
that the leading term is always O(1). In the example above, it should
be O(x). The UPXS domain doesn't seem to suffer from the same problem.

Igor