[Axiom-math] Re: partfrac, expand, combine, rewrite and simplify.

[Tim Daly]

Dylan Thurston wrote:

>On Tue, Oct 07, 2003 at 10:19:14AM -0400, Tim Daly wrote:
>> > // 2 //
>> > I call partfrac the transform as 2x^3 / (1-x^2) =3D 1/(1-x) - 1/(1+x) -=
> 2x
>> >=20
>> > What is its name in axiom ?
>>=20
>> The partialFraction function will create a partial fraction thus:
>>=20
>> -> partialFraction(7,12)
>>=20
>>        3   1
>>   1 - -- + -
>>        2   3
>>       2
>>                   Type: PartialFraction Integer
>
>Right, but this doesn't help with the actual problem!  Apparently to
>convince partialFraction to work on polynomials you need to provide some
>type information.
>
>Furthermore, I think there's a bug here:
>
>------
>(8) -> partialFraction((2*x^3), (1-x^2))
>   There are 1 exposed and 1 unexposed library operations named=20
>      partialFraction having 2 argument(s) but none was determined to=20
>      be applicable. [snippage]
>(8) -> partialFraction((2*x^3)::UP(x, FRAC INT), (1-x^2))
>
>                 2x
>   (8)  - 2x - ------
>                2
>               x  - 1
>               Type: PartialFraction UnivariatePolynomial(x,Fraction Intege=
>r)
>(9) -> partialFraction((2*x^3), (1-x^2)::UP(x, FRAC INT))
>
>                 1       1
>   (9)  - 2x - ----- - -----
>               x - 1   x + 1
>               Type: PartialFraction UnivariatePolynomial(x,Fraction Intege=
>r)
>------
>
>Why didn't input (8) above actually do the partial fraction expansion?

Dylan,

(1) -> )d op partialFraction

There is one exposed function called partialFraction :
   [1] (D1,Factored D1) -> PartialFraction D1 from PartialFraction D1
            if D1 has EUCDOM

There are 2 unexposed functions called partialFraction :
   [1] (Fraction Polynomial D4,Symbol) -> Any from 
            PartialFractionPackage D4
            if D4 has Join(EuclideanDomain,CharacteristicZero)
   [2] (Polynomial D5,Factored Polynomial D5,Symbol) -> Any
            from PartialFractionPackage D5
            if D5 has Join(EuclideanDomain,CharacteristicZero)

First, I make a couple variables to hold the expressions.
This lets me figure out what the interpreter type is by default:

(1) -> a:=2*x^3

          3
   (1)  2x
                                                     Type: Polynomial Integer
(2) -> b:=1-x^2

           2
   (2)  - x  + 1
                                                     Type: Polynomial Integer

The signature listed above requires that the first argument type 
must be a EuclideanDomain (EUCDOM):

(3) -> Polynomial(Integer) has EUCDOM

   (3)  false
                                                                Type: Boolean
POLY(INT) is not a EUCDOM.
However, the type you provided is a EUCDOM.

(4) -> UP(x,FRAC INT) has EUCDOM

   (4)  true

                                                                Type: Boolean
So we can make variables that are the correct type:

(5) -> aa:UP(x,FRAC INT):=2*x^3

          3
   (5)  2x
                               Type: UnivariatePolynomial(x,Fraction Integer)
(6) -> bb:UP(x,FRAC INT):=1-x^2

           2
   (6)  - x  + 1
                               Type: UnivariatePolynomial(x,Fraction Integer)

And the call succeeds:

(7) -> partialFraction(aa,bb)

                 1       1
   (7)  - 2x - ----- - -----
               x - 1   x + 1
               Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)

So partialFraction can create the correct call signature. I suspect the
interpreter created some intermediate type that it couldn't coerce to
the required target type. I agree that it should have been able to 
give the factored output but did not.

If you really want to see the struggles the interpreter is going thru try:

)lisp (setq |$monitorNewWorld| t)

and you can see the interpreter trying to find the correct signature
using various coercions. 

I'll put this problem on the list of things to look at but it won't be
in the near future as I'm trying to correct a major distribution problem
and get some documentation available. If you get any insight I'd like
to know.

Tim
address@hidden
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