What is the correct way to use FindMinimum with NDEigensystem?

Question

I would like to find the minimum of a function dE[me_, mh_, e0_] using FindMinimum but Mathematica shows errors from NDEigensystem. Most likely I have written a function en[m1_, m2_, e0_, B_] incorrectly. Could you point out what the problem is?

ClearAll["Global`*"]
mi[m1_, m2_] := m1*m2/(m1 + m2);
VC[r_, e0_] := -(1/(e0*r));
rmax = 10;
zmax = 10;
Hp[m1_, m2_, e0_, B_] := -1/2/mi[m1, m2]*
    Laplacian[[Psi][r, z], {r, [Theta], z}, "Cylindrical"] + 
   VC[r, e0][Psi][r, z] + r^2B^2;
en[m1_, m2_, e0_, B_] := 
  Module[{}, {vals, funs} = 
    NDEigensystem[{Hp[m1, m2, e0, B] + [Psi][r, z]}, [Psi][r, 
      z], {r, 0, rmax}, {z, -zmax, zmax}, 10, 
     Method -> {"SpatialDiscretization" -> {"FiniteElement", 

{"MeshOptions" -> {"MaxCellMeasure" -> 0.05}}}, 
       "Eigensystem" -> {"Arnoldi", "MaxIterations" -> 10000}}];
   Return[vals[[1 ;; 2]] - 1]];
(en[1,0.3,0.3,14][[1;;2]])
dE[m1_, m2_, e0_] := en[1, m1, m2, e0][[1]]
FindMinimum[{dEmin = 
   dE[m1, m2, e0], (0.1 < m1 < 10) && (0.1 < m2 < 10) && (1 < e0 < 10)}, {{m1, 0.2}, {m2, 0.2}, {e0, 5}}, AccuracyGoal -> 3, PrecisionGoal -> 3, 
 EvaluationMonitor :> {Print["m1=", m1, "   m2=", m2, "   e0=", e0, "   dE=", dE]}]

score 3 · Accepted Answer · answered Jun 17 '23 at 09:39

3

Change the definition of dE[] to

dE[m1_?NumericQ, m2_?NumericQ, e0_?NumericQ] := en[1, m1, m2, e0][[1]]

and FindMinimum[...]

FindMinimum[{dEmin =dE[m1, m2, e0], (0.1 < m1 < 10) && (0.1 < m2 <10) && (1 < e0 < 10)}, {{m1,0.2}, {m2, 0.2}, {e0, 5}}, AccuracyGoal -> 3, PrecisionGoal -> 3, 
EvaluationMonitor :> {Print["m1=", m1, "   m2=", m2, "   e0=", e0,"   dE=", dE[m1, m2, e0]]}]

answered Jun 17 '23 at 09:39

Ulrich Neumann

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Thanks a lot! It is not very clear why if it is added this part ?NumericQ the function definition, then the minimization starts to work? – Mam Mam Jun 17 '23 at 09:53
1

@MamMam Function FindMinimum is numerical, though it expects numerical values as argument. The definition dE[m1_?NumericQ, m2_?NumericQ, e0_?NumericQ]=... only evaluates if arguments are numerical. – Ulrich Neumann Jun 17 '23 at 09:58
Yes, but unless ?NumericQ is specified, doesn't Mathematica understand from the initial values that this is a numeric values? – Mam Mam Jun 17 '23 at 10:01
1

In this case Mathematics might try some symbolical transformation first... – Ulrich Neumann Jun 17 '23 at 10:07
1

@MamMam, see also User-defined functions, numerical approximation, and NumericQ. – Domen Jun 17 '23 at 10:49
@Domen, thanks! – Mam Mam Jun 17 '23 at 11:45

What is the correct way to use FindMinimum with NDEigensystem?

1 Answers1