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Re: problem in minimization of a matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg123023] Re: problem in minimization of a matrix
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Mon, 21 Nov 2011 04:25:29 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201111201038.FAA01248@smc.vnet.net>
  • Reply-to: drmajorbob at yahoo.com

That's too much to grasp all at once, but...

(a) For speed, I recommend = rather than := when possible, as in

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)

It's VERY inefficient to compute Integrate, Simplify, ArrayFlatten, or Det  
all over again for EVERY call of the function, which is exactly what :=  
causes to occur.

(b)

When arguments MUST be numeric, say so in the definition, as in:

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

(c)

If Integrate will work with symbolic arguments, instead of NIntegrate with  
numeric ones, use something like the above definition of d.

(d) The example in (b) might be more efficient using NDSolve, since a  
single call allows computation for ANY t in an interval. That's not as  
good as using Integrate, which computes all instances at once, but it  
avoids computing the integral every time t changes. That would go  
something like:

ClearAll[\[CapitalGamma]]
\[CapitalGamma][\[Alpha]_?NumericQ, \[Beta]_?NumericQ, \[Omega]0_?
    NumericQ, \[Lambda]_?
    NumericQ] := \[CapitalGamma][\[Alpha], \[Beta], \[Omega]0, \
\[Lambda]] =
   f /. First@
     NDSolve[{f'[s] ==
        2*\[Gamma][\[Alpha], \[Beta], \[Omega]0, \[Lambda], s],
       f[0] == 0}, f, {s, 0, tUpperLimit}]
\[CapitalGamma][\[Alpha]_?NumericQ, \[Beta]_?NumericQ, \[Omega]0_?
    NumericQ, \[Lambda]_?NumericQ,
   t_?NumericQ] := \[CapitalGamma][\[Alpha], \[Beta], \[Omega]0, \
\[Lambda]][t]

(e) Be careful with Simplify and FullSimplify. They may OVER-simplify,  
giving an expression that's not valid for every value of the arguments.  
Integrate may do this as well. The Integrate result generally works if it  
is continuous in the range of interest. If not, find an expression that  
eliminates the jump points.

Bobby

On Sun, 20 Nov 2011 04:38:28 -0600, Herman16 <btta2010 at gmail.com> wrote:

> 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.
>


-- 
DrMajorBob at yahoo.com



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