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Bill Page | 3 Nov 2008 18:51
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Re: Selection of roots

On Thu, Oct 30, 2008 at 8:39 AM, Alejandro Jakubi wrote:
> I wonder how it is done in Axiom the selection of roots of a polynomial
> with some property. As in this example, select the positive roots out
> of the list of three roots generated by:
>
> radicalSolve(p^3-p+1/10=0,p)
>

Since the discriminant is positive we know all roots are real, so in
principle we should be able to solve the problem like this:

(1) -> discriminant(p^3-p+1/10)

        373
   (1)  ---
        100
                                            Type: Fraction(Integer)

(2) -> select(p+->rhs(p)::AlgebraicNumber>0,radicalSolve(p^3-p+1/10=0,p))

                             +------------------+2
                             |    +-+    +-----+
                 +---+       |- 3\|3  + \|- 373
            (- 3\|- 3  - 3)  |------------------  + 2
                            3|         +-+
                            \|      60\|3
   (2)  [p= -----------------------------------------]
                               +------------------+
                               |    +-+    +-----+
                   +---+       |- 3\|3  + \|- 373
(Continue reading)

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Martin Rubey | 3 Nov 2008 19:32
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Re: [fricas-devel] Re: Selection of roots

"Bill Page" <bill.page <at> newsynthesis.org> writes:

> As I understand it AlgebraicNumber is supposed to be able to properly
> order the roots. 

No, that's RECLOS.

Unfortunately, there is no coercion from AN to RECLOS, and this would actually
be tricky, since sqrt(-3) is not allowed in RECLOS.

Martin
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Bill Page | 3 Nov 2008 20:30
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Re: [open-axiom-devel] Re: [Axiom-math] Selection of roots

On Mon, Nov 3, 2008 at 1:32 PM, Martin Rubey wrote:
>
> Bill Page writes:
>
>> As I understand it AlgebraicNumber is supposed to be able to properly
>> order the roots.
>
> No, that's RECLOS.
>
> Unfortunately, there is no coercion from AN to RECLOS, and this would actually
> be tricky, since sqrt(-3) is not allowed in RECLOS.
>

Hmmm... So what is the meaning of < in AN? Ok, I guess that it is just
whatever is exported by Expression Integer. The fact that the positive
roots appear < 0 while the negative roots appear > 0 must be just an
accident of some strange lexical ordering rules for expressions. :-(

Regards,
Bill Page.

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Martin Rubey | 3 Nov 2008 20:53
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Re: [open-axiom-devel] Re: [Axiom-math] Selection of roots

"Bill Page" <bill.page@...> writes:

> Hmmm... So what is the meaning of < in AN? Ok, I guess that it is just
> whatever is exported by Expression Integer. The fact that the positive
> roots appear < 0 while the negative roots appear > 0 must be just an
> accident of some strange lexical ordering rules for expressions. :-(

Well, it's not so strange, it's the order used for hashing.

Yes, it's high time to separate mathematical order and internal hashing order.
Waldek proposed a programme in the thread "RFC: Orders in algebra" some time
ago, as you know.  There was a long discussion, but little result.

I believe that we wanted too much.

Here is the original programme:

> 1) create a new category, say Comparable which has functions
>    like lessThen, lessOrEqual etc. 
> 2) Change domains that use unnatural order to use comparisons from
>    Comparable
> 3) Undefine unnatural orders (remove them from OrderedSet)
> 4) If possible define natural ones
> 
> In particular for Expression w probably should _not_ define an
> order, but rather make '<' produce elements of new domain
> 'Inequality' (like Equation, but for ordered comparisons).

I guess it would be best to just go ahead and implement a simple version of
this at first.  In fact, Bill, you did already implement a version of
(Continue reading)

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Bill Page | 3 Nov 2008 21:55
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Re: [fricas-devel] Re: Selection of roots

On Mon, Nov 3, 2008 at 1:32 PM, Martin Rubey wrote:
>
> "Bill Page" <bill.page <at> newsynthesis.org> writes:
>
>> As I understand it AlgebraicNumber is supposed to be able to
>> properly order the roots.
>
> No, that's RECLOS.
>
> Unfortunately, there is no coercion from AN to RECLOS, and this would
> actually be tricky, since sqrt(-3) is not allowed in RECLOS.
>

Thanks Martin, that was a very useful reference. Actually it turns out
that RealClosure is pretty cool.

Thank you Renaud!

(1) -> P:=p^3-p+1/10

         3        1
   (1)  p  - p + --
                 10
                                    Type: Polynomial Fraction Integer

(2) -> S:=select(positive?,allRootsOf(P)$RealClosure(Fraction Integer))

   (2)  [%A36,%A37]
                                    Type: List RealClosure Fraction Integer
(Continue reading)

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Alejandro Jakubi | 3 Nov 2008 23:00
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Re: [fricas-devel] Re: Selection of roots

On Mon, 3 Nov 2008, Bill Page wrote:

> Thanks Martin, that was a very useful reference. Actually it turns out
> that RealClosure is pretty cool.
>
> Thank you Renaud!
>
> (1) -> P:=p^3-p+1/10
>
>         3        1
>   (1)  p  - p + --
>                 10
>                                    Type: Polynomial Fraction Integer
>
> (2) -> S:=select(positive?,allRootsOf(P)$RealClosure(Fraction Integer))
>
>   (2)  [%A36,%A37]
>                                    Type: List RealClosure Fraction Integer
>

Thank you Bill for showing how to use 'RealClosure' in this example. In fact,
I have been looking a pair of days on how to use it, but, because of the lack
of proper documentation the best that I could get was:

L:=allRootsOf(p^3 - p + 1/10)$RECLOS(FRAC INT)

  [%A1,%A2,%A3]
                                Type: List RealClosure Fraction Integer

map(mainCharacterization,L)
(Continue reading)

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Martin Rubey | 3 Nov 2008 23:14
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Re: [fricas-devel] Re: Selection of roots

Alejandro Jakubi <jakubi <at> df.uba.ar> writes:
> Now, my next question is about selecting the two real roots out of the four
> produced by 'radicalSolve' here:
> 
> f:=(x^3+x^2-4*x-4)/(2*x^2+7*x-4)
> fp:=differentiate(f,x)
> radicalSolve(fp,x)
> 
> The problem is that
> 
> allRootsOf(fp)$RealClosure(Fraction Integer)

try

allRootsOf(numer fp)$RealClosure(Fraction Integer)

Martin
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Bill Page | 3 Nov 2008 23:38
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Re: [open-axiom-devel] Re: [Axiom-math] Selection of roots

On Mon, Nov 3, 2008 at 5:00 PM, Alejandro Jakubi wrote:
>
> Thank you Bill for showing how to use 'RealClosure' in this example.

You are welcome.

> In fact, I have been looking a pair of days on how to use it, but,
> because of the lack of proper documentation the best that I could
> get was:
>
> L:=allRootsOf(p^3 - p + 1/10)$RECLOS(FRAC INT)
>
>  [%A1,%A2,%A3]
>                               Type: List RealClosure Fraction Integer
>
> map(mainCharacterization,L)
>
>             1   1
>  [[- 4,0[,[0,-[,[-,1[]
>             2   2
> Type: List Union(RightOpenIntervalRootCharacterization(RealClosure
> Fraction Integer,SparseUnivariatePolynomial RealClosure Fraction
> Integer),"failed")
>
> which it was not so far.
>

Why, that's almost there! :-)  I agree that tutorial documentation is
a problem. If you haven't already seen it, perhaps you would be
interested in:
(Continue reading)

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Alejandro Jakubi | 4 Nov 2008 00:21
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Re: [fricas-devel] Re: Selection of roots

On Mon, 3 Nov 2008, Martin Rubey wrote:

> Alejandro Jakubi <jakubi <at> df.uba.ar> writes:
>> Now, my next question is about selecting the two real roots out of the four
>> produced by 'radicalSolve' here:
>>
>> f:=(x^3+x^2-4*x-4)/(2*x^2+7*x-4)
>> fp:=differentiate(f,x)
>> radicalSolve(fp,x)
>>
>> The problem is that
>>
>> allRootsOf(fp)$RealClosure(Fraction Integer)
>
> try
>
> allRootsOf(numer fp)$RealClosure(Fraction Integer)
>

Fine. Now, how I recover the algebraic expressions, as produced by
'radicalSolve', associated with such labels like %A1, %A2 ?

Alejandro Jakubi
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Martin Rubey | 4 Nov 2008 00:41
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Re: [fricas-devel] Re: Selection of roots

Alejandro Jakubi <jakubi <at> df.uba.ar> writes:

> Fine. Now, how I recover the algebraic expressions, as produced by
> 'radicalSolve', associated with such labels like %A1, %A2 ?

perhaps

(63) -> mainDefiningPolynomial(first allRootsOf(numer fp)$RECLOS(FRAC INT))

          4     3   3  2
   (63)  ?  + 7?  + - ?  + 4? + 22
                    2
  Type: Union(SparseUnivariatePolynomial(RealClosure(Fraction(Integer))),...)

Martin
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Gmane