Re: Curious behavior of Taylor series

"Igor Khavkine" <[email protected]> writes:

>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: 

In the Windows version, I get the correct answer. So this 
bug is newly introduced.
                         AXIOM Computer Algebra System
               Version of Tuesday November 30, 2004 at 
21:11:14
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(1) -> )set mess auto off
(1) -> monx := monomial(1,1)$UTS(EXPR INT,x,0)

    (1)  x
                          Type: 
UnivariateTaylorSeries(Expression Integer,x,0)
(2) -> sqrt(monx*monx)

    (2)  x
                          Type: 
UnivariateTaylorSeries(Expression Integer,x,0)
(3) -> x

    (3)  x
                                                            
  Type: Variable x

Note in particular (3), which illustrates that Axiom 
treats undefined identifiers as symbols (or variables). 
Even though x has been used in (1) and (2), it is still 
undefined! The reason is the way UTS (or any univariate 
domain) is constructed in Axiom: since it is univariate, 
Axiom designers decided that it is not necessary to 
associate the main variable to an identifier. The x that 
appears in the output (and input) is external to the UTS 
computing environment and a pure notation in I/O for the 
convenience of the user. Internally, the main variable is 
represented by a place holder (and denoted by the local 
identifier ?). One reason for this set up is to allow 
coefficient domains to include Symbol (that is, all 
identifiers) and since ? is only a local identifier, it 
will not get mixed up with anything in the coefficient 
domain. This explains why in your example, x*y gives x*x 
(which is really x*?). It is important to know that the 
line

y:=taylor x

does NOT define x, only y. You cannot use x to mean the 
main variable until you define it. So you can do this:

(1) -> y:=taylor x

    (1)  x
                          Type: 
UnivariateTaylorSeries(Expression Integer,x,0)
(2) -> x:UTS(EXPR INT,x,0):='x

    (2)  x
                          Type: 
UnivariateTaylorSeries(Expression Integer,x,0)
(3) -> x*y

          2
    (3)  x
                          Type: 
UnivariateTaylorSeries(Expression Integer,x,0)



William