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Re: greetings and a question!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg83117] Re: greetings and a question!
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Mon, 12 Nov 2007 05:08:25 -0500 (EST)
*Organization*: The Open University, Milton Keynes, UK
*References*: <fh6df9$4ou$1@smc.vnet.net>
dimitris wrote:
> Hello to all of you!
>
> Unfortunately, family, working and research issues
> prevent me from participating to the forum as frequent
> as I used to.
>
> Anyway...
>
> It is my first post since a long time so everybody be patient!
>
> A student of mine came across the following
> equation in a mathematical contest:
>
> In[1]:=
> eq = x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 - a - b + 1==0=
;
>
> (all variables are assumed real)
>
> Of course for Mathematica the solution is rather trivial.
>
> In[1]:=
> $Version
>
> Out[1]=
> "5.2 for Microsoft Windows (June 20, 2005)"
>
> In[2]:=
> eq = x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 - a - b + 1==0=
;
>
> In[3]:=
> Reduce[eq, {a, b, x, y}, Reals]
> ToRules[%]
> eq /. %
>
> Out[3]=
> a == 1 && b == 1 && x == -1 && y == 0
>
> Out[4]=
> {a -> 1, b -> 1, x -> -1, y -> 0}
>
> Out[5]=
> True
>
> Can somebody explain concisely the mathematica concept
> behind this solution? In fact I would be much obliged if somebody
> pointed me out how to obtain the result by hand. Also, by curiosity,
> how Mathematica reaches the result?
>
> Dimitris
Hi Dimitris,
Nice to hear from you.
According to _The Mathematica Book_,
"For polynomial systems Reduce uses cylindrical algebraic decomposition
for real domains and Gr=C3=B6bner basis methods for complex domains."
See http://documents.wolfram.com/mathematica/book/section-A.9.5
For more insight about the Cylindrical Algebraic Decomposition (CAD)
algorithm, you could check the following web sites:
http://mathworld.wolfram.com/CylindricalAlgebraicDecomposition.html
http://planning.cs.uiuc.edu/node292.html
Best regards,
--
Jean-Marc
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