Re: an equation
- From: Jean-Marc Gulliet <jeanmarc.gulliet@xxxxxxxxx>
- Date: Sat, 10 Nov 2007 15:54:46 +0100
dimitris wrote:
Hello to all of you!
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?
Last but not least, how we could obtain the solution of the equation
in Maple?
Dimitris Anagnostou
Hi Dimitris,
According to _The Mathematica Book_,
"For polynomial systems Reduce uses cylindrical algebraic decomposition for real domains and Gröbner 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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