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
- References:
- Re: problem in minimization of a matrix
- From: Herman16 <btta2010@gmail.com>
- Re: problem in minimization of a matrix