Re: [Axiom-mail] Unexpected results w.r.t. exponential operation of LODO (with july2008 release)

[Martin Rubey]

Dear Liu,

"Liu Xiaojun" <address@hidden> writes:

> Hi,
> 
>     Does exponential operation (^ or **) of a differential operator, say L,
> means repeating multiplications of L in Axiom ? If so, it seems the following
> code produced an unexpected result:
> 
> (1) -> Dx: LODO(EXPR INT, f+->D(f,x)) := D()
> (2) -> u := operator 'u
> (3) -> L := Dx + u(x)
> (4) -> L**2 = L*L
> 
>          2                2   2             ,          2
>    (4)  D  + 2u(x)D + u(x) = D  + 2u(x)D + u (x) + u(x)
> 
> L^2 produces the same result as l.h.s. of (4) shows. However, it does not
> equal to the r.h.s. generally, and r.h.s. should be correct answer. 

Quite right, it's a bug.  COuld you please file it with the example above on
MathAction/IssueTracker (click on "Bug Reports" on the Frontpage of
Mathaction)?

> I checked the .spad file, it seems the exponential operation of LODO (or its
> parent OREUP) is missed (or I guess it is directly inherited as usual
> polynomial operation). 

You are completely right:

SparseUnivariateSkewPolynomial(R:Ring, sigma:Automorphism R, delta: R -> R):
 UnivariateSkewPolynomialCategory R with
      outputForm: (%, OutputForm) -> OutputForm
        ++ outputForm(p, x) returns the output form of p using x for the
        ++ otherwise anonymous variable.
   == SparseUnivariatePolynomial R add
 
 ...

(and ** does not appear below)

The rules for inheritance spelt out for our situation are as follows:

MyCat: Category with
    foo: % -> %
    bar: % -> %
    xxx: % -> %
  add
    foo a == xxx a
    bar a == xxx a

MyDom1: MyCat ==
  add 
    xxx a == ... -- define xxx
    bar a == ... -- override bar from MyCat


MyDom2: MyCat == MyDom1
  add 
    xxx a == ... -- give a definition different from the one in MyDom1


Now, foo$MyDom2 will give the same result as xxx$MyDom2, 
 but bar$MyDom2 will use the definition given in MyDom1!

Thus, since ** is inherited from SparseUnivariatePolynomial, and it is
redefined there, we have a bug.  In my experience, reusing *all* definitions
from another domain as in

MyDom2: MyCat == MyDom1 add 

has to be done *very* carefully.  (But it is a nice feature, still!)

In our case, we may want to do

SparseUnivariateSkewPolynomial(R:Ring, sigma:Automorphism R, delta: R -> R):
 UnivariateSkewPolynomialCategory R with
      outputForm: (%, OutputForm) -> OutputForm
        ++ outputForm(p, x) returns the output form of p using x for the
        ++ otherwise anonymous variable.
   == SparseUnivariatePolynomial R add

-- We have to define ^ and **, because taking it from SUP R is wrong.  It is
-- sufficient to do it for PositiveInteger, since for n=0 the definitions
-- agree.  However, there might be a faster definition.

      import RepeatedSquaring(%)
      _^(x:%, n:PositiveInteger):% == x ** n
      x:% ** n:PositiveInteger == expt(x,n)

...

Liu, could you (in return :-) write a tiny testcase for each of the exported
functions of SparseUnivariateSkewPolynomial (or, if you prefer, for
LinearOrdinaryDifferentialOperator).  For example:

)expose UnittestCount UnittestAux Unittest
testsuite "LODO"
testcase "exponentiation"
Dx: LODO(EXPR INT, f+->D(f,x)) := D()
u := operator 'u
L := Dx + u(x)

testEquals("L*L*L", "L^3")
testEquals("L*L", "L**2")
testcase "leftQuotient"
...

Martin