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.

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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