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MathGroup Archive 2007

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Re: Minimize[] Problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76342] Re: Minimize[] Problem
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sun, 20 May 2007 02:24:43 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f2mfj4$mh1$1@smc.vnet.net>

anguzman at ing.uchile.cl wrote:
> Hello.
> 
> Why do I get?
> 
> In[4]:=
> Minimize[(x^4-6 x^2+1)^2,{x}]
> Out[4]=
> {1,{x->0}}
> 
> when..
> 
> In[3]:=
> (x^4-6 x^2+1)^2/.x->1-Sqrt[2]//Simplify
> Out[3]=
> 
> 
> x=0 is actually a local maximum.
> Any answer will be appreciated.
> 
> Atte Andres Guzman

I am afraid this may be a bug in version 5.2. Even with a constraint, 
Minimize is unable to find a minimum.

In[1]:=
$Version

Out[1]=
"5.2 for Microsoft Windows (June 20, 2005)"

In[2]:=
Minimize[(x^4 - 6*x^2 + 1)^2, {x}]

Out[2]=
{1, {x -> 0}}

In[3]:=
Simplify[Minimize[{(x^4 - 6*x^2 + 1)^2, x < 0}, {x}]]

Minimize::wksol: Warning: There is no minimum in the region described by 
the constraints; returning a result on the boundary. More...

Out[3]=
{1, {x -> 0}}

In[4]:=
Chop[NMinimize[(x^4 - 6*x^2 + 1)^2, {x}]]

Out[4]=
{0, {x -> -0.41421356237309503}}

In[5]:=
Chop[N[Solve[(x^4 - 6*x^2 + 1)^2 == 0, x]]]

Out[5]=
{{x -> -2.414213562373095},
   {x -> -2.414213562373095},
   {x -> -0.41421356237309515},
   {x -> -0.41421356237309515},
   {x -> 0.41421356237309515},
   {x -> 0.41421356237309515},
   {x -> 2.414213562373095}, {x -> 2.414213562373095}}

The behavior of Minimize is correct in version 6.0.

In[1]:= $Version

Out[1]= "6.0 for Microsoft Windows (32-bit) (April 28, 2007)"

In[2]:= Minimize[(x^4 - 6*x^2 + 1)^2, {x}]

Out[2]= {(1 - 6*(-1 - Sqrt[2])^2 + (-1 - Sqrt[2])^4)^2, {x -> -1 -
     Sqrt[2]}}

In[3]:= Simplify[Minimize[(x^4 - 6*x^2 + 1)^2, {x}]]

Out[3]= {0, {x -> -1 - Sqrt[2]}}

In[4]:= Simplify[Minimize[{(x^4 - 6*x^2 + 1)^2, x > 0}, {x}]]

Out[4]= {0, {x -> -1 + Sqrt[2]}}

Regards,
Jean-Marc


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