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Re: no message from Minimize[] on a weird function(x^x) !?!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96771] Re: no message from Minimize[] on a weird function(x^x) !?!
  • From: dh <dh at metrohm.com>
  • Date: Tue, 24 Feb 2009 05:47:23 -0500 (EST)
  • References: <gnr1d2$ep0$1@smc.vnet.net>


Hi Roger,

your function is more "weird" than you think, but it is not 

discontinuous and non-differentiable, except at zero where it is not 

defined. What you have is a multi valued complex function. For numerical 

calculations mathematica has to pick a branch, what creates problems 

with branch cuts. Therefore, it does not make sense to minimize x^x, but 

you may e.g. minimize Abs[f[x]].

Therefore, you may minimize e.g. Abs[f[x]].

hope this helps, Daniel



congruentialuminaire at yahoo.com wrote:

> Hello MathGroup:

> 

> I have:

> 

> f[x_]=x^x

> Plot[f[x],{x,-3,3.}]

> 

> What makes this a weird function is that when x<0, the function is

> discontinuous and non-differentiable and has a global minimum at -1.

> 

> To answer the question: "what is the minimum of this function", I

> tried

> 

> FindMinimum[f[x],{x,2}] (* this appears correct *)

>> {0.692201, {x -> 0.367879}}

> FindMinimum[f[x],{x,2}] (* this complains about the gradient, but

> appears correct *)

>> FindMinimum::nrgnum: The gradient is not a vector of real numbers at {x} = {-1.}. >>

>> {-1., {x -> -1.}}

> NMinimize[f[x], x] (* this gives the minimum in the positive domain *)

>> {0.692201, {x -> 0.367879}}

> Minimize[f[x], x] (* this gives no answer and no error message *)

>> Minimize[x^x, x]

> 

> Is this expected behavior?

> 

> TIA.

> 

> Roger Williams

> Franklin Laboratory

> 




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