Re: [Axiom-math] newbie type problem

[Martin Rubey]

Frank Thieme writes:

 > I'm brand new to axiom, trying to solve my homework.

Welcome!

> My idea is to have a function abl(f,x) which simply does n*x^m ->
> n*m*x^(m-1). So I defined:
> 
> apl(f,x) == coefficient(f,degree(f,x))*degree(f,x)*x^(degree(f,x)-1)
> 
> the same thing works great with maxima (but I can't get any rules defined
> there). But axiom does some error messages when I try abl(x^5,x) but then the
> correct result is displayed 5x^4
> 
> When I try abl(x,x) it does not work, abl(5,x) either...

The reason, that abl(x,x) does not work is displayed by axiom:

   There are 7 exposed and 5 unexposed library operations named 
      coefficient having 2 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                           )display op coefficient
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
 
   Cannot find a definition or applicable library operation named 
      coefficient with argument type(s) 
                                 Variable x
                               PositiveInteger
      
      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

So, what does this mean? if you look at the output of 

)di op coefficient

axiom will respond with a (long) list of operations called
"coefficient". Usually, you are only interested in the "exposed" operations,
that is, those that are available "directly". (I could expand on this, but I
think for the moment it should be enough)

There are 11 exposed functions called coefficient :
   [1] (D,D2) -> D1 from D
            if D has AMR(D1,D2) and D2 has OAMON and D1 has RING
   [2] (CliffordAlgebra(D3,D1,D4),List PositiveInteger) -> D1
            from CliffordAlgebra(D3,D1,D4)
            if D1 has FIELD and D3: PI and D4: QFORM(D3,D1)
   [3] (D2,D) -> D1 from D
            if D has FAMONC(D2,D1) and D2 has SETCAT and D1 has CABMON
            
   [4] (D,D2) -> D1 from D
            if D has FMCAT(D1,D2) and D2 has SETCAT and D1 has RING
   [5] (D,NonNegativeInteger) -> D1 from D if D has MLO D1 and D1 has 
            RING
   [6] (D,List D4,List NonNegativeInteger) -> D from D
            if D has MTSCAT(D3,D4) and D3 has RING and D4 has ORDSET
         
   [7] (D,D1,NonNegativeInteger) -> D from D
            if D has MTSCAT(D3,D1) and D3 has RING and D1 has ORDSET
         
   [8] (D,NonNegativeInteger) -> D1 from D
            if D has OREPCAT D1 and D1 has RING
   [9] (D,List D5,List NonNegativeInteger) -> D from D
            if D has POLYCAT(D3,D4,D5) and D3 has RING and D4 has 
            OAMONS and D5 has ORDSET
   [10] (D,D1,NonNegativeInteger) -> D from D
            if D has POLYCAT(D3,D4,D1) and D3 has RING and D4 has 
            OAMONS and D1 has ORDSET
   [11] (TaylorSeries D3,NonNegativeInteger) -> Polynomial D3
            from TaylorSeries D3 if D3 has RING

Note that they are all different, they take different numbers of arguments, and
the types of those arguments are also different. For example, [11] takes a
(univariate) TaylorSeries over a Ring as first, and a NonNegativeInteger as
second argument.

Your rule gave axiom a "Variable x" as first and a PositiveInteger as second
argument. In the list above, there are seven operations with two arguments:
[1],[2],[3],[4],[5],[8],and [11], but none of them applies.

So, this is the low level reason. The "real", high level reason is: what would
you expect as answer from

coefficient(x*y,1) ?

Surely, you should make your question more precise, by specifying the variable:

coefficient(x*y,x,1).

So, I'd write:

abl(f,x) == coefficient(f,x,degree(f,x))*degree(f,x)*x^(degree(f,x)-1)

or, perhaps better

abl(f:POLY FRAC INT,x:Symbol):POLY FRAC INT ==
  coefficient(f,x,degree(f,x))*degree(f,x)*x^(degree(f,x)-1)

This makes sure that the first argument must be a polynomial, guarding against
input like abl(sin(x),x). On the other hand, the second variant won't work with
abl(2.0*x^2,x), whereas the first version does. The "correct" answer to this
problem is to define a package (which has to be put into a file and compiled):

)abbrev package ABL ableitung
ableitung(R:Ring): Exports == Implementation where
    Exports == with
      abl: (Polynomial R, Symbol) -> Polynomial R

    Implementation == add
      abl(f,x) == 
        d := degree(f,x)
        if zero? d 
        then 0
        else coefficient(f,x,d)
            *monomial(d::Polynomial R,x,(d-1)::NonNegativeInteger)

There are several things to notice:

* this definition works for any polynomial ring

* we had to check whether the degree of f is zero, because otherwise d-1 might
  be -1 and x^(-1) is not a polynomial anymore

* we have to tell the compiler explicitely that d-1 is nonnegative by
  (d-1)::NonNegativeInteger

* we had to use the operation "monomial" from the domain "Polynomial R" because
  the compiler is not smart enough to understand x^((d-1)::NonNegativeInteger)

* we have to write d::Polynomial R, because d is an integer, and the ring R
  could be anything. However, an integer can always be "coerced" into an
  element of a ring. Alternatively, we could write d*1. (In this case, the
  compiler would be smart enough to understand that 1 comes from the ring)

Martin