NIntegrate::zeroregion error

Question

While integrating the following function,

me = 511000/29979200(*eV*);
re = 2.81794*10^-13(*cm*);
h = 4.135667516 * 10^-15;(*eV*)
c = 29979200;(*cm/s*)

Zg = 79;
Ag = 196.96655;
\[Rho] = 19.320(*g/\[Mu]m^3*);

\[Xi][\[Nu]_?NumericQ, T_?NumericQ] := (100 me c^2 h \[Nu])/(
 T (T - h \[Nu] ) Zg^(1/3));(*Per Bremmstrahlung*)
\[Phi]1[T_?NumericQ, \[Nu]_?NumericQ] := 
 20.863 - 2 Log[1 + (0.55846 \[Xi][T, \[Nu]])^2] - 
  4 (1 - 0.6 Exp[-0.9 \[Xi][T, \[Nu]]] - 
     0.4 Exp[-105 \[Xi][T, \[Nu]]]);(*Per Bremmstrahlung*)
\[Phi]2[T_?NumericQ, \[Nu]_?NumericQ] := \[Phi]1[T, \[Nu]] - 
  2/3 1/(1 + 6.5 \[Xi][T, \[Nu]] + 
    6 \[Xi][T, \[Nu]]^2);(*Per Bremmstrahlung*)
f[Z_?NumericQ ] := (Z/137)^2 (1/(1 - (Z/137)^2) + 0.20206 - 
    0.0369 (Z/137)^2 + 0.0083 (Z/137)^4 - 
    0.002 (Z/137)^6);(*Per Bremmstrahlung*)

dEdx2[T_?NumericQ] := 
  6.022*10^23*\[Rho]/
   Ag NIntegrate[(4 Zg^2 re/
        137 1/\[Nu] (1 + (T/(T - h \[Nu]))^2) (\[Phi]1[T, \[Nu]]/4 - 
          1/3 Log[Zg] - f[Zg]) - 
       2/3 T/(T - 
         h \[Nu]) (\[Phi]2[T, \[Nu]] - 1/3 Log@Zg - f[Zg])), {\[Nu], 
      0, T/h}];

ParallelTable[{j, dEdx2[j]}, {j, 10000, 2*10^6, 10000}]

I get a very long series of errors, the first of which is

NIntegrate::zeroregion Integration region {{2.41799*10^18,2.41798934786516787199999999993411568411875634969888072969436258221*10^18}} cannot be further subdivided at the specified working precision. NIntegrate assumes zero integral there and on any further indivisible regions.

What does it mean, and how do I get rid of it?

Answer 1

Obtaining a zero length integration region means that most likely the singularity handler is not aggressive enough.

So, one way to deal with your problem is to try to change the singularity handler and/or the precision goal. Here is an answer doing that for a similar question.

Further, you can rationalize all the numerical entities and use higher precision. (Not done here.)

First let us redefine dEdx2:

ClearAll[dEdx2]
Options[dEdx2] = Options[NIntegrate];
dEdx2[T_?NumericQ, opts : OptionsPattern[]] := 
  6.022*10^23*\[Rho]/
    Ag NIntegrate[(4 Zg^2 re/
        137 1/\[Nu] (1 + (T/(T - h \[Nu]))^2) (\[Phi]1[T, \[Nu]]/4 - 
         1/3 Log[Zg] - f[Zg]) - 
      2/3 T/(T - h \[Nu]) (\[Phi]2[T, \[Nu]] - 1/3 Log@Zg - 
         f[Zg])), {\[Nu], 0, T/h}, opts];

This takes less than 40s on my laptop (without any messages):

AbsoluteTiming[
 res3 = 
   Table[
     {j, 
      dEdx2[j, 
       Method -> {"GlobalAdaptive", "SingularityHandler" -> "DoubleExponential"}, 
       MaxRecursion -> 20, PrecisionGoal -> 3]}, 
     {j, 10000, 2*10^6, 10000}];
 ]
Short[res3]

(* {37.456, Null} *)

(* {{10000, -6.33661*10^43}, {20000, -1.26732*10^44}, {30000, \
-1.92223*10^44}, {40000, -2.53464*10^44}, {50000, -3.18431*10^44}, \
{60000, -3.84447*10^44} ... } *)

The computation above with PrecisionGoal->4 takes ~15 times longer; the results are assigned to res4. Here are the relative errors:

Through[{Min, Max, Mean, Median}[
  Abs[(res4[[All, 2]] - res3[[All, 2]])/res4[[All, 2]]]]]

(* {0., 0.0134694, 0.00151681, 0.000671349} *)