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Re: problem with LessEqual::nord:

  • To: mathgroup at smc.vnet.net
  • Subject: [mg115878] Re: problem with LessEqual::nord:
  • From: Peter Pein <petsie at dordos.net>
  • Date: Sun, 23 Jan 2011 17:32:47 -0500 (EST)
  • References: <ihbjvn$dl3$1@smc.vnet.net> <ihe451$q9r$1@smc.vnet.net>

On 22.01.2011 09:22, tarun dutta wrote:
> On Jan 21, 2:34 pm, tarun dutta<tarundut... at gmail.com>  wrote:
>> I am using "NMINIMIZE" to get the result of a function which contains
>> complex term.after executing the program it is showing the following
>> error--
>> LessEqual::nord: Invalid comparison with 155.698+0. I attempted.>>
>>
>> can you explain the reason and how will I be free from this kind of
>> error?
>> thanks in advance.
>> regards,
>> tarun
>
> n = 5;
> p = 86/1000;
> q = 1;
> Print["P= ", p, ";Q=", q];
> d[n + 1] = 0;
> d[i_] = re[i] + I*im[i];
> conj[a_] := ComplexExpand[Conjugate[a]]; f =
>   Together[Sum[
>     4*p*(Sqrt[i + 1]*d[i]*
>          conj[d[i + 1]]*(Sum[
>            Sqrt[i + 1]*conj[d[i]]*d[i + 1], {i, 0, n}]) +
>         Sqrt[i + 1]*conj[d[i]]*
>          d[i + 1]*(Sum[
>            Sqrt[i + 1]*d[i]*conj[d[i + 1]], {i, 0, n}])) + (-q*i +
>         i (i - 1))* d[i]*conj[d[i]], {i, 0, n}]];
> c = Expand@ComplexExpand[Sum[Abs[d[i]]^2, {i, 0, n}]] == 1;
> v = Join[Array[re, n + 1, 0], Array[im, n + 1, 0]];
...

Hi,

the value of f depends only on the radii of d[i]:

In[23]:= fpolar =
  Collect[f /. {re -> (r[#] Cos[#] &), im -> (r[#] Sin[#] &)},
   r /@ Range[0, n],
   FullSimplify[#,
     cpolar =
      And @@ Flatten[{Sum[r[k]^2, {k, 0, n}] == 1,
         Table[0 <= r[k] <= 1 && -Pi <= t[k] <= Pi, {k, 0, n}]}]] &]

Out[23]= 86/125 r[0]^2 r[1]^2 + r[1]^2 (-1 + (172 r[2]^2)/125) +
  258/125 r[2]^2 r[3]^2 + r[3]^2 (3 + (344 r[4]^2)/125) + (
  344 r[3] r[4]^2 r[5])/(25 Sqrt[5]) + 15 r[5]^2 +
  r[2] (344/125 Sqrt[3] r[3]^2 r[4] +
     172/25 Sqrt[3/5] r[3] r[4] r[5]) + r[4]^2 (8 + (86 r[5]^2)/25) +
  r[1] (172/125 Sqrt[6] r[2]^2 r[3] +
     r[2] (344/125 Sqrt[2] r[3] r[4] + 172/25 Sqrt[2/5] r[4] r[5])) +
  r[0] (172/125 Sqrt[2] r[1]^2 r[2] +
     r[1] (172/125 Sqrt[3] r[2] r[3] + 344/125 r[3] r[4] + (
        172 r[4] r[5])/(25 Sqrt[5])))

In[24]:= Clear@target;
target[rr : {__?NumericQ}] :=
  fpolar /. Thread[(r /@ Range[0, n]) -> rr]

In[26]:= sol =
  NMinimize[{target[r /@ Range[0, n]],
    DeleteCases[cpolar, _t, \[Infinity]]}, r /@ Range[0, n]]

Out[26]= {-1., {r[0] -> 0., r[1] -> 1., r[2] -> 0., r[3] -> 0.,
   r[4] -> 0., r[5] -> 0.}}

so you are free to select any d[1] on the unit-circle.

Cheers,
   Peter


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