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Re: [Axiom-developer] comparison operators and %pi

From: Francois Maltey
Subject: Re: [Axiom-developer] comparison operators and %pi
Date: 25 May 2007 21:00:36 +0200
User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4

Hello everybody !

> (6) -> solve(integrate(1/x,x=1..t, "noPole")::EXPR INT = 1, t)
>                +---+     +---+
>                |  2      |  2
>    (6)  [t= - \|%e  ,t= \|%e  ]
>                                        Type: List Equation Expression Integer
> Yes, we should teach Axiom that %e is positive.  
> In fact, I don't think that's too difficult in a hackish way.

When I played with expr.spad for theses questions I understood,
that Expressions aren't naive expressions over real numbers but 
a field with a local point of view.

So it's normal that asin (sin a) = a in the domain Expression of axiom.
But I explain to my students that 
asin (sin a) = a only for %pi/2 <= a <= -%pi/2 (for real a)

Someone, perhaps Waldek but I can't find the mail, explains me that
exp a * exp b isn't always exp (a+b)... it's impossible for complex number.

An other problem is that Expression and AlgebraicNumber are very close,
and it seems that sqrt 5 (almost)= - sqrt 5 in axiom because they are 
the root of the same polynom x^2-5. 
For my use I prefer the RealClosure domain, 
but in this domain there is no %i... How play around 1 + %i*sqrt 2 ?

So Expression isn't for me the good domain for calculus in analysis.
Is it a good idea to improve it for calculus ?
We risk to break it for others computations, Risch integrate I believe.

A better way is to duplicate Expression (with it's own algebraic point of 
view) in a Calculus domain (in analysis),
change simplifications (from an algebraic to an analysis way), 
and correct others problems in order to have expressions as others CAS 
maple, mupad or maxima. But this is a really big work, too long for me...

For this purpose axiom might have operations over intervals,
sin (1..real..10) = -1..real..1, and so. With this domain 
numeric tests might be sure but equal test remains indecidable.

provisos domain with the assume command (as mupad and maple) 
is also necessary. I remember that mupad takes about 3 years from 
the first buggy function assume to its last powerful abilities.

> Francois, you have already some expertise.  
> Maybe you can help here.  Although, in reality, we need a redesign.

I have kept main files of my previous tests : 
  1/ the list of false simplifications (for complex variables),
  2/ the simplifications we can add, 
  3/ tries about expand, rewrite and combine functions.

In the 2/ point it's almost easy to get right results for
sqrt (exp (2)) = sqrt (%e^2) = %e, 
but I fear it's "impossible" to simplify sqrt ((1+%e)^2)
because axiom expands sqrt((1+%e)^2) in sqrt(1+2*%e+%e^2).

The aim of 3/ is to have only theses 3 functions with parameters
and avoid removeSinSqr, sincos2tan, tan2exp and so... 

But this work is useless for me (I teach to first year students) 
until I can't do more analysis calculus.

I don't finish it because I spend much more time (about 5 times) 
to write a *.spad file than a *.input file or a command file for 
maple and mupad, and after write a pamphlet file was too much, sorry...

I hope I answer to your questions.

Francois, in France.

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