Author 
Comment/Response 
Bill Simpson

10/18/12 00:48am
Modifying your code slightly
In[1]:= Δ=4;
Ux=1;
Uy=1+Δ/3;
Uz=2 Δ;
e1=1/2;
e2=12;
f1=1/2;
f2=1f1;
Solve[f1 ((e1eff1)/(1+Da*(e1eff1)))+f2 ((e2eff1)/(1+Da*(e2eff1)))==0,eff1]
Out[9]= {
{eff1 > (2 + 25*Da  Sqrt[4 + 529*Da^2])/(4*Da)},
{eff1 > (2 + 25*Da + Sqrt[4 + 529*Da^2])/(4*Da)}
}
Likewise for eff2,3.
Using these sidesteps the numerical process of trying to find a solution on every iteration.
Next I rearrange your code to try to sidestep the nonnumeric problem.
In[20]:= fa[eff1_/; NumericQ[eff1], eff2_/; NumericQ[eff2], eff3_/; NumericQ[eff3]] := 1/(4*π)*NIntegrate[(Sin[θ]^3*Cos[ϕ]^2)/(Ux^2*((Sin[θ]^2*Cos[ϕ]^2*eff1)/Ux^2 + (Sin[θ]^2*Sin[ϕ]^2*eff2)/Uy^2 + (Cos[ϕ]^2*eff3)/Uz^2)), {ϕ, 0, 2 π}, {θ, 0, π}];
fb[eff1_/;NumericQ[eff1], eff2_/; NumericQ[eff2], eff3_/; NumericQ[eff3]] := 1/(4*π)*NIntegrate[(Sin[θ]^3*Sin[ϕ]^2)/(Uy^2*((Sin[θ]^2*Cos[ϕ]^2*eff1)/Ux^2 + (Sin[θ]^2*Sin[ϕ]^2*eff2)/Uy^2 + (Cos[ϕ]^2*eff3)/Uz^2)), {ϕ, 0, 2 π}, {θ, 0, π}];
fc[eff1_/; NumericQ[eff1], eff2_/; NumericQ[eff2], eff3_/; NumericQ[eff3]] := 1/(4*π)*NIntegrate[(Sin[θ]*Cos[ϕ]^2)/(Uz^2*((Sin[θ]^2*Cos[ϕ]^2*eff1)/Ux^2 + (Sin[θ]^2*Sin[ϕ]^2*eff2)/Uy^2 + (Cos[ϕ]^2*eff3)/Uz^2)), {ϕ, 0, 2 π}, {θ, 0, π}]; NMinimize[
Da = fa[eff1, eff2, eff3];
Db = fb[eff1, eff2, eff3];
Dc = fc[eff1, eff2, eff3];
neweff1 = (2+25*DaSqrt[4+529*Da^2])/(4*Da);
neweff2 = (2+25*DbSqrt[4+529*Db^2])/(4*Db);
neweff3 = (2+25*DcSqrt[4+529*Dc^2])/(4*Dc);
Norm[neweff1eff1]+Norm[neweff2eff2] + Norm[neweff3eff3], {eff1, eff2, eff3}
]
That has substituted the Rho expression inside each NIntegrate, included conditions so that NMinimize can't attempt a symbolic evaluation, which would make NIntegrate fail, and tries to minimize the change in eff1,2,3 as you go round your circular dependencies.
That is slow and it complains that your integrals may be oscillatory, which is not surprising, but if I haven't made a mistake then it may have dealt with your nonnumeric issue.
That still leaves the issue of potential negative real parts, but perhaps you can see if you can get any solution at all out of this and then look into which of the two conjugate solutions to use for solving each of eff1,2,3.
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