Help in NIntegration Methods - Takes too long, why?

Question

I have this code. It is a triple integral, and using the automatic method gives me a wrong answer for $T=0.1$ (the correct answer is $5.44$, while I got $3.73$ ). I've tried to change the integral method to GaussKronrodRule, but it takes too long (until now, hours) and also I received a message

Return::nofunc: Function Module not found enclosing Return[False,Module].ParallelDeveloperQueueRun::hmm: Received unexpected result .

And for $T > 1$ gives the correct results, in the automatic method mode. I think it is a problem in the method of NIntegrate, right ?

1) How can I solve that ?

2) In Fortran, I have a gausslegendre method. Is this option available in Mathematica?

σ = 0.6;

minroot[g_?NumericQ,b_?NumericQ]:=
Module[{rts,r},
   rts=r/.Solve[1-(b/r)^2-g^-2*(2/15*σ^9 (1/(r-1)^9-1/(r+1)^9-9/(8r) 
               (1/(r-1)^8-1/(r+1)^8))-σ^3 (1/(r-1)^3-1/(r+1)^3-3/(2r) 
               (1/(r-1)^2-1/(r+1)^2)))==0,r];
  rts=Select[rts,With[{nval=N[#,100]},Im[nval]==0&&nval>0]&];
  Max[rts]];

aA[g_?NumberQ,b_?NumberQ,i_]:=Pi-2 b  
  NIntegrate[
    1/(r^2*Sqrt[1-(b/r)^2-g^-2*(2/15* σ^9 (1/(r-1)^9-1/(r+1)^9-9/(8r) (1/(r-1)^8-1/(r+1)^8)) -
    σ^3 (1/(r-1)^3 - 1/(r+1)^3-3/(2r) (1/(r-1)^2-1/(r+1)^2)))]),   
    {r,minroot[g,b],Infinity},
    Exclusions-> 
       {r^2*Sqrt[1-(b/r)^2-g^-2*(2/15*σ^9 (1/(r-1)^9-1/(r+1)^9-9/(8r) (1/(r-1)^8- 1/(r+1)^8))-
        σ^3 (1/(r-1)^3-1/(r+1)^3-3/(2r) (1/(r-1)^2-  1/(r+1)^2)))]==0},
    MaxRecursion->i,Method->{Automatic,"SymbolicProcessing"->0}];

qQ[g_?NumberQ,i_]:=
  NIntegrate[2*(1-Cos[aA[g,b,i]]) b,
            {b,0,10},
             MaxRecursion->i, Method-> {Automatic,"SymbolicProcessing"->0}];

o[T_,i_,j_]:=
  (1/T^3) NIntegrate[(g^5*qQ[g,i])/E^(g^2/T),
            {g,0,Infinity},
            MaxRecursion->j,
            Method->{Automatic,"SymbolicProcessing"->0}];

Answer 1

Go to the NIntegrate documentation, then from that page go to Options -> Methods -> Integration Rules, scroll to the bottom of Integration Rules, and there are examples that suggest NIntegrate will use a Cartesian product of Gaussian points if you used Method->{“GaussKronrodRule”,”GaussKronrodRule”,”GaussKronrodRule”} It seems you did that in Fortran except with MaxRecursion->0 assuming you didn't use Kronrod's extension. However, GaussKronrodRule may not be a good method for your problem. You will find MonteCarlo methods implemented in Mma here and here. MonteCarlo methods are good for quadrature in multiple dimensions when a few digits of precision are needed. Earlier I sent you here. That describes some advanced methods that can be used for quadrature in multiple dimensions. I am not familiar those methods, I don’t have access to Mma today and I haven’t had a close look at the function you want to integrate. Hence I can’t be much more help.