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InequalitySolve with algebraic numbers and Simplify
*To*: mathgroup at smc.vnet.net
*Subject*: [mg22987] InequalitySolve with algebraic numbers and Simplify
*From*: Gianluca Gorni <gorni at dimi.uniud.it>
*Date*: Sun, 9 Apr 2000 01:45:52 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Hello!
I had a system of *linear* inequalities in two variables x,y,
with simple algebraic numbers as coefficients,
to solve with InequalitySolve[], and I naively assumed
that the solution would be a set of likewise *linear*
inequalities. So I was surprised at results like this:
Needs["Algebra`InequalitySolve`"]
InequalitySolve[{y <= x*Sqrt[2], y <= x}, {x, y}]
x <= 0 && y <= -(Sqrt[2]*Sqrt[x^2]) || x > 0 && y <= x
Strictly speaking the result is correct, but it does not look
good, because of that Sqrt[x^2]. I tried with
Experimental`CylindricalAlgebraicDecomposition,
but it gives exactly the same answer.
Trying to reduce the results to a more manageable form, I
met some somewhat disappointing behaviour of Simplify[]
with assumptions (of Mathematica version 4):
In: Simplify[Sqrt[x^2], x == 1 + Sqrt[2]]
Out: Sqrt[x^2]
It seems that Mathematica doesn't notice that 1+Sqrt[2] is real and
positive! So let's teach Mathematica that it is real, at least:
In: Simplify[Sqrt[x^2], {Element[x, Reals], x == 1 + Sqrt[2]}]
Out: x
Somehow I would have expected the answer to be 1+Sqrt[2], what
about you? Even if x has a smaller LeafCount than 1+Sqrt[2].
Best regards,
Gianluca Gorni
--
+---------------------------------+
| Gianluca Gorni |
| Universita` di Udine |
| Dipartimento di Matematica |
| e Informatica |
| via delle Scienze 208 |
| I-33100 Udine UD |
| Italy |
+---------------------------------+
| Ph.: (39) 0432-558422 |
| Fax: (39) 0432-558499 |
| mailto:gorni at dimi.uniud.it |
| http://www.dimi.uniud.it/~gorni |
+---------------------------------+
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