The following code generate the error: The function value Omega is not a a number at (1.91862121487240651285721,1.71668875337901960520526). Its full format is:

`NMinimize::nnum`

: The function value `\[CapitalOmega][42/125,0,1.91862121487240651285721,1.71668875337901960520526]`

is not a number at `{\[CapitalDelta],\[Sigma]} = {1.91862121487240651285721,1.71668875337901960520526}`

.

So basically, what it does is, find the eigenfunction of some physical variable (a matrix). So the eig is the group of this eigenroots. There are 48 roots. One of the 48 roots has the implicit format as `Root[ #1&,8]`

. Means the 8th root of a expression. Other 47 roots are nice and explicit.

Then, an Ep function is defined as the summation of some absolute values of these eigen values. This Ep has 24 elements. Without inserting any number, it has a lot roots in the implicit format because of the Abs function. But mathematically, it should have the good behavior.

Then, another function CapitalOmega is created based on this Ep function. It is an integration of CapitalOmega together with something else. And with some of the parameters (CapitalLambda and etc.) known this CapitalOmega has only one element.

CapitalOmega has four variables and now I am search for the values of the last two variables which make the CapitalOmega smallest for given series of the first two variables. Thus, used the `NMinimize`

.

I have used ?NumericQ for the NIntegrate part in the Omega function so the Omega part will be evaluated only when Mathematica exploring different specific values for the variables.

However, the error shows up. From the physical point of view, the Omega should be well defined and should not have any “hole” at any point.

Any possible reason of having this error? Please ignore the code before eig, for people who is not very familar with Quantum physics, it looks a little bit confusing, but it is quiet simple and should cause this error.

The Code:

`tN = ( { {0, 0}, {0, 0} } ); t0 = ( { {1, 0}, {0, 1} } ); t1 = ( { {0, 1}, {1, 0} } ); t2 = ( { {0, -I}, {I, 0} } ); t3 = ( { {1, 0}, {0, -1} } ); la2 = ( { {0, -I, 0}, {I, 0, 0}, {0, 0, 0} } ); la5 = ( { {0, 0, -I}, {0, 0, 0}, {I, 0, 0} } ); la7 = ( { {0, 0, 0}, {0, 0, -I}, {0, I, 0} } ); la0 = ( { {1, 0, 0}, {0, 1, 0}, {0, 0, 1} } ); gaN = KroneckerProduct[tN, tN]; gaI = KroneckerProduct[t0, t0]; ga0 = ArrayFlatten[{{tN, t0}, {t0, tN}}]; ga1 = ArrayFlatten[{{tN, t1}, {-t1, tN}}]; ga2 = ArrayFlatten[{{tN, t2}, {-t2, tN}}]; ga3 = ArrayFlatten[{{tN, t3}, {-t3, tN}}]; ga5 = I*ga0.ga1.ga2.ga3; ga0t0 = KroneckerProduct[t0, ga0]; ga1t0 = KroneckerProduct[t0, ga1]; ga2t0 = KroneckerProduct[t0, ga2]; ga3t0 = KroneckerProduct[t0, ga3]; gaIt0 = KroneckerProduct[t0, gaI]; ga5t1 = KroneckerProduct[t1, ga5]; ga5t2 = KroneckerProduct[t2, ga5]; ga5t3 = KroneckerProduct[t3, ga5]; ga0t3 = KroneckerProduct[t3, ga0]; ga05t3 = KroneckerProduct[t3, ga0.ga5]; Matpu = p0*ga0t0 - 0*ga1t0 - 0*ga2t0 - p*ga3t0 - \[Sigma]*gaIt0 - I*0*ga5t3 - \[Mu]*ga0t0 + \[Nu]5*ga05t3; Matpula = KroneckerProduct[la0, Matpu]; Matpd = p0*ga0t0 - 0*ga1t0 - 0*ga2t0 - p*ga3t0 - \[Sigma]*gaIt0 - I*0*ga5t3 + \[Mu]*ga0t0 + \[Nu]5*ga05t3; Matpdla = KroneckerProduct[la0, Matpd]; ga1t1 = KroneckerProduct[t1, ga1]; ga2t1 = KroneckerProduct[t1, ga2]; ga3t1 = KroneckerProduct[t1, ga3]; Matde = (KroneckerProduct[la2, ga3t1] + KroneckerProduct[la5, ga2t1] + KroneckerProduct[la7, ga1t1]); MatdeC = ConjugateTranspose[Matde]; Mat = ArrayFlatten[{{Matpula, \[CapitalDelta]* Matde}, {-\[CapitalDelta]*MatdeC, Matpdla}}]; eig = p0 /. Solve[Det[Mat] == 0, p0] \[CapitalLambda] = 6134/10000; G = 225/(100*\[CapitalLambda]^2); H = G*3/8; m = 542/100000; fun[p_, \[Mu]_, \[Nu]5_, \[CapitalDelta]_, \[Sigma]_] = eig; len = Length[eig]; Ep[p_, \[Mu]_, \[Nu]5_, \[CapitalDelta]_, \[Sigma]_] = Sum[Abs[fun[p, \[Mu], \[Nu]5, \[CapitalDelta], \[Sigma]][[i]]], {i, 1, len}]; \[CapitalOmega][\[Mu]_？NumericQ, \[Nu]5_？NumericQ, \ \[CapitalDelta]_？NumericQ, \[Sigma]_？NumericQ] := (\[Sigma] - m)^2/( 2*G) + (3*\[CapitalDelta]^2)/(2*H) - 1/(2*(2*\[Pi]^2))* NIntegrate[ p^2*Ep[p, \[Mu], \[Nu]5, \[CapitalDelta], \[Sigma]]*\ \[CapitalLambda]^(2*5)/(\[CapitalLambda]^(2*5) + p^(2*5)), {p, 0, 10*\[CapitalLambda]}, PrecisionGoal -> 12, AccuracyGoal -> 12, WorkingPrecision -> 24]; FuncMin[\[Mu]_, \[Nu]5_] := NMinimize[{\[CapitalOmega][\[Mu], \[Nu]5, \[CapitalDelta], \ \[Sigma]], \[CapitalDelta] >= 0 && \[Sigma] > 0}, {\[CapitalDelta], \[Sigma]}, PrecisionGoal -> 12, AccuracyGoal -> 12, WorkingPrecision -> 24]; time = TimeUsed[] Clear[ls]; ls = {}; Do[AppendTo[ls, ParallelTable[ FuncMin[\[Mu], (i - 1)*6/1000], {\[Mu], 3/10, 51/100, 3/1000}]], {i, 1, 60}] TimeUsed[] - time `