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Re: problem in minimization of a matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg123016] Re: problem in minimization of a matrix
  • From: Herman16 <btta2010 at gmail.com>
  • Date: Sun, 20 Nov 2011 05:38:28 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

I want to minimize det \tau over all Covariance matrix of \[Sigma]t[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] but At and Ct are depend on real numbers that is, alpha, beta, lambda, t & r are constants. 
please look into the notebook below. 

Many thanks for any comments

s[\[Omega]_, t_] := 
 FullSimplify[
  Integrate[
   Sin[\[Omega] \[Tau]] Sin[\[Omega]0 \[Tau]], {\[Tau], 0, t}]]


c[\[Omega]_, t_] := 
 FullSimplify[
  Integrate[
   Cos[\[Omega] \[Tau]] Cos[\[Omega]0 \[Tau]], {\[Tau], 0, t}]]


d[\[Omega]_, t_] := 
 FullSimplify[
  Integrate[
   Cos[\[Omega] \[Tau]] Sin[\[Omega]0 \[Tau]], {\[Tau], 0, t}]]

J[\[Omega]_, \[Lambda]_] := \[Omega]/(\[Omega]^2 + \[Lambda]^2);

J1[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, 
   t_] := (\[Alpha]^2*\[Beta]*\[Pi])/(
   2 (\[Omega]0^2 + \[Lambda]^2)) (Exp [-\[Lambda]*
        t] ( \[Omega]0/\[Lambda]*Sin[t \[Omega]0] - 
        Cos[t \[Omega]0]) + 1);

\[CapitalDelta][\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  J1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t];



J2[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_] := (\[Alpha]^2*\[Pi]*\
\[Omega]0*\[Beta])/(2 \[Lambda] (\[Omega]0^2 + \[Lambda]^2));

J3[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, 
   t_] := -(\[Alpha]^2*\[Beta]*\[Pi]*Cos[t \[Omega]0])/(
    2 (\[Omega]0^2 + \[Lambda]^2)) (Sin[
      t \[Omega]0] + \[Omega]0/\[Lambda] Exp [-\[Lambda]*t]);


J4[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, 
   t_] := (\[Alpha]^2*\[Beta]*\[Pi]*Sin[t \[Omega]0])/(
   2 (\[Omega]0^2 + \[Lambda]^2)) (Cos[t \[Omega]0] - 
     Exp [-\[Lambda]*t]);

J5[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, 
   t_] := (\[Alpha]^2*\[Beta])/(\[Omega]0^2 + \[Lambda]^2) \
((SinIntegral [\[Omega]0 t] + \[Pi]/2) + 
     1/2 Exp [-\[Lambda] t] 
      ExpIntegralEi[\[Lambda] t] (\[Omega]0/\[Lambda] 
         Cos[t \[Omega]0] - Sin[t \[Omega]0]) - 
     1/2 Exp [\[Lambda] t] 
      ExpIntegralEi[-\[Lambda] t] (\[Omega]0/\[Lambda] 
         Cos[t \[Omega]0] + Sin[t \[Omega]0]));

\[CapitalPi][\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  J2[\[Alpha], \[Beta], \[Omega]0, \[Lambda]] + 
   J3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t] + 
   J4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t];



\[Gamma][\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  J5[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t];


\[CapitalGamma][\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  NIntegrate[
   2*\[Gamma][\[Alpha], \[Beta], \[Omega]0, \[Lambda], s], {s, 0, 
    t}];


\[CapitalDelta]\[CapitalGamma][\[Alpha]_, \[Beta]_, \[Omega]0_, \
\[Lambda]_, t_] := 
  NIntegrate[
   J1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s], {s, 0, t}];



\[CapitalDelta]co[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  NIntegrate[
   J1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s]*
    Cos[2 \[Omega]0 (t - s)], {s, 0, t}];



\[CapitalDelta]si[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  NIntegrate[
   J1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s]*
    Sin[2 \[Omega]0 (t - s)], {s, 0, t}];



\[CapitalPi]co[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  NIntegrate[(J2[\[Alpha], \[Beta], \[Omega]0, \[Lambda]] + 
      J3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s] + 
      J4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s])*
    Cos[2 \[Omega]0 (t - s)], {s, 0, t}];


\[CapitalPi]si[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_] := 
  NIntegrate[(J2[\[Alpha], \[Beta], \[Omega]0, \[Lambda]] + 
      J3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s] + 
      J4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], s])*
    Sin[2 \[Omega]0 (t - s)], {s, 0, t}];




Clear[A0]
A0[r_] = {{1/2 Cosh[2 r], 0}, {0, 1/2 Cosh[2 r]}};

Clear[At, Ct]




At[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[ArrayFlatten[
    A0[r]*(1 - \[CapitalGamma][\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
         t]) + {{\[CapitalDelta]\[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], 
         t] + (\[CapitalDelta]co[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t] - \[CapitalPi]si[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t]), -(\[CapitalDelta]si[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t] - \[CapitalPi]co[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t])}, {-(\[CapitalDelta]si[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t] - \[CapitalPi]co[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t]), \[CapitalDelta]\[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], 
         t] - (\[CapitalDelta]co[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
           t] - \[CapitalPi]si[\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], t])}}]];


Ats[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[A0[r]*(1 - \[CapitalGamma][\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], 
        t]) + {{\[CapitalDelta]\[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], t], 
      0}, {0, \[CapitalDelta]\[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], t]}}];



Ct[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  ArrayFlatten[{{1/
      2 Sinh[2 r]  Cos [
       2 \[Omega]0 t]*(1 - \[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], t]), 
     1/2 Sinh[2 r] Sin [
       2 \[Omega]0 t]*(1 - \[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], t])}, {1/
      2 Sinh[2 r]  Sin [
       2 \[Omega]0 t]*(1 - \[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], t]), -1/2 Sinh[2 r]  Cos [
       2 \[Omega]0 t]*(1 - \[CapitalGamma][\[Alpha], \[Beta], \
\[Omega]0, \[Lambda], t])}}];


\[Sigma]t[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  ArrayFlatten[{{At[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r], 
     Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
      r]}, {Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
       r]\[Transpose], 
     At[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]}}];



\[Sigma]ts[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  ArrayFlatten[{{Ats[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r], 
     Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
      r]}, {Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
       r]\[Transpose], 
     Ats[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]}}];


I1[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
 Det[At[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]]

I1s[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
 Det[Ats[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]]

I3[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
 Det[Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]]


I4[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Det[\[Sigma]t[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]];

I4s[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Det[\[Sigma]ts[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]];

C1[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[\[Sqrt](1/(
      2 I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r]) (I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
        I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
        I4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] + 
        Sqrt[(I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
           I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
           I4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2 - (2 I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r]*
           I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2        ]))];

C1s[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[\[Sqrt](1/(
      2 I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r]) (I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
        I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
        I4s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] + 
        Sqrt[(I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
           I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
           I4s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2 - (2 I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t,
             r]*I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2        ]))];

C2 [\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[\[Sqrt](1/(
      2 I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r]) (I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
        I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
        I4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
        Sqrt[(I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
           I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
           I4[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2 - (2 I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r]*I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2        ]))];

C2s [\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[\[Sqrt](1/(
      2 I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r]) (I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
        I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
        I4s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
        Sqrt[(I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 
           I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 - 
           I4s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2 - (2 I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t,
             r]*I3[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
            r])^2        ]))];

an[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[Sqrt[I1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]]];

ans[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[Sqrt[I1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]]];

\[Kappa]1[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
 Re[Sqrt[(an[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
     C1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
      r])*(an[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
     C2[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r])]]

\[Kappa]1s[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
 Re[Sqrt[(ans[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
     C1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
      r])*(ans[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
     C2s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r])]]

xm[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[(\[Kappa]1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 1/
    4)/(2 \[Kappa]1[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r])];

xms[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[(\[Kappa]1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r]^2 + 1/
    4)/(2 \[Kappa]1s[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r])];
 

g1[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, r_] := 
  Re[(an[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] + 1/2) Log [
      an[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] + 1/
       2] - (an[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 1/
       2)  Log [
      an[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 1/2]];



\[Sigma]M[\[Rho]_, \[Phi]_] := 
 Cosh[2 \[Rho]]/
  2 ({{1 + Tanh[2 \[Rho]] Cos[\[Phi]], -Tanh [
        2 \[Rho]] Sin[\[Phi]] }, {-Tanh [2 \[Rho]] Sin[\[Phi]], 
     1 - Tanh[2 \[Rho]] Cos[\[Phi]]}})

\[Tau][\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, 
  r_, \[Rho]_, \[Phi]_] := 
 FindMinimum[{Det At[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] - 
    Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, r] 
     Inverse[(At[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
         r] + \[Sigma]M[\[Rho], \[Phi]])] 
     Ct[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
       r]\[Transpose], \[Rho] >= 0, 
   0 <= \[Phi] <= 2 \[Pi]}, {\[Rho], \[Phi]}]

k[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, 
  r_, \[Rho]_, \[Phi]_] := 
 Re[Sqrt[\[Tau][\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
   r, \[Rho], \[Phi]]]]

k1[\[Alpha]_, \[Beta]_, \[Omega]0_, \[Lambda]_, t_, 
   r_, \[Rho]_, \[Phi]_] := 
  Re[(k[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r, \[Rho], \[Phi]] + 1/2) Log [
      k[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r, \[Rho], \[Phi]] + 1/
       2] - (k[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r, \[Rho], \[Phi]] - 1/2)  Log [
      k[\[Alpha], \[Beta], \[Omega]0, \[Lambda], t, 
        r, \[Rho], \[Phi]] - 1/2]];


I would like to minimize \[Tau][\[Alpha], \[Beta], \[Omega]0, \
\[Lambda], t, 
  r, \[Rho], \[Phi]] the variables \[Alpha], \[Beta], \[Omega]0, \
\[Lambda], t & r  are constants.



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