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MathGroup Archive 2001

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Re: How Do I Solve This System of 6 Inequalities?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg30249] Re: [mg30222] How Do I Solve This System of 6 Inequalities?
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Sat, 4 Aug 2001 01:14:20 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Mathematica has very powerful functions for dealing with algebraic 
inequalities of many kinds, but to use them in reasonable time  one has 
to transform your expression into something managable.
The first thing is to replace floating values by exact ones. If we do 
that (and if there is no mistake in your posting) your  expression 
becomes:

expr = w - x > 0 && (z/v)(u - y) < 1/4 && (z*x/v) >
       1 && (z*w/v)y/((z*w/v) - 1) - u < 0 && x^2 + y^2 - u^2 < 0 &&
     w - Sqrt[15]u/4 < 0 && x > 0 && y > 0 && z > 0 && w > 0 && u > 0 && 
v > 0

One could already try to use one of the built in functions but to speed 
up the solution further transformations are desirable. First of all, 
since we are assuming all the variables are >0, we can replace the 
condition  (z/v)(u - y) < 1/4 by the z(u-y)<v/4, the condition 
(z*x/v) >   1 by z*x>v. There is a problem with the condition  
(z*w/v)y/((z*w/v) - 1) - u < 0. This reduces to ((z*w)*y)/(w*z - v) - 
u < 0. There are two cases here, w*z<v and w*z>v. In the first case we 
get u*v-u*w*z+w*y*z<0 and in the second  u*v-u*w*z+w*y*z>0. So we shall 
use two pairs of conditions instead of your original one. We could now 
try to solve the problem using CylindricalAlgebraicDecomposition or 
InequalitySolve but it will take a very long time. To do it faster we 
need to replace Sqrt[15] by a rational approaximation. I shall take
In[50]:=
Rationalize[Sqrt[15],0.01]
Out[50]=
31
--
8

Finally we load the Experimantal context:

<< Experimental`

Now we can use

GenericCylindricalAlgebraicDecomposition[
   w - x > 0 && z*(u - y) < v/4 && z*x > v &&
    u*v - u*w*z + w*y*z > 0 && x^2 + y^2 - u^2 < 0 &&
    w - Rationalize[Sqrt[15], 0.01]*(u/4) < 0 && x > 0 &&  y > 0 && z > 
0 && w > 0 && u > 0 && v > 0 && z*w < v, {x, y, z, w, u, v}]

Out[2]=
{False, False}

and

GenericCylindricalAlgebraicDecomposition[
   w - x > 0 && z*(u - y) < v/4 && z*x > v &&
    u*v - u*w*z + w*y*z < 0 && x^2 + y^2 - u^2 < 0 &&
    w - Rationalize[Sqrt[15], 0.01]*(u/4) < 0 && x > 0 &&  y > 0 && z > 
0 && w > 0 && u > 0 && v > 0 && z*w > v, {x, y, z, w, u, v}]

Out[3]=
{False, False}



Andrzej Kozlowski

On Thursday, August 2, 2001, at 04:16  PM, Mentor wrote:

> How do I solve these 6 nonlinear inequalities
> w.r.t. x,y,z,w,u,v?
> My guess is that there is no positive solution.
> How do I rely on Mathematica to prove this?
>
>    [{w - x > 0, (z/v)(u - y) < .25, (zx/v) > 1., 	
>    ((zw/v)y/((zw/v) - 1.)) - u < 0, x^2 + y^2 - u^2 < 0.,
>     w - Sqrt[15.]u/4. < 0., x > 0., y > 0., z > 0., w > 0., u > 0.,
>     v > 0.}, {x, y, z, w, u, v}]
>
> Thanks a million!
>
> Laurie
>
>
>

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/


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