NIntegrate inside FindRoot has evaluated to non-numerical values

I have a function Fdiff[m] that I want to put in FindRoot. However, it has given me an error saying that the integrand has evaluated to non-numerical values. I'm aware of other posts having this issue as well, but reading those did not lead to a solution in my case.

I have used ?NumericQ as well as trying to integrate first to get an expression just in terms of m before putting in FindRoot, but I think this can't be done since my integrand is a bit complicated so maybe there's no analytic result, so using NIntegrate is must, but for this to work m has to have a value. So I checked that Fdiff[m] for some specific value of m indeed works.

I believe the syntax of my code is correct but I'm not sure why this problem persist. Any help?

Clear["Global*"]
a = Rationalize[4.046];
b = Rationalize[0.01613];
c = Rationalize[0.227];
A[z_] := -a Log[b z^2 + 1]
int1[zh_] = ((Integrate[y^3 Exp[-3 A[y]], y] /. {y -> zh}) - (Integrate[y^3 Exp[-3 A[y]], y] /. {y -> 0})) // Simplify;
int2[zh_] = ((Integrate[y^3 Exp[-3 A[y] + c y^2], y] /. {y -> zh}) - (Integrate[y^3 Exp[-3 A[y] + c y^2], y] /. {y -> 0})) // Simplify;

s[zh_] := Exp[3 A[zh]]/(4 zh^3);
T[m_, zh_] := Rationalize[((zh^3 Exp[-3 A[zh]])/(4 Pi int1[zh])) (1 - ((2 c m^2)/(1 - Exp[c zh^2])^2) (Exp[c zh^2] int1[zh] - int2[zh]))] // Simplify;
Tp[m_, zh_] := D[T[m, zh], zh] // Simplify;

Fdiff[m_?NumericQ] := NIntegrate[s[y] Tp[m, y], {y, 3, 100}] - NIntegrate[s[y] Tp[m, y], {y, 7, 100}]

FindRoot[Chop[Fdiff[m]] == 0, {m, 0.048}, AccuracyGoal -> 10, WorkingPrecision -> 10]

NIntegrate::inumr: The integrand (-((y^6 (1+<<1>>)^(6069/<<3>>) <<1>> (1-(227 <<1>> (Power[<<2>>] <<1>>+<<1>>))/(500 (-1+<<1>>)^2)))/(4000000000000000000000000000000000000000000000000000000000000 <<2>> <<1>>^2))+<<1>><<1>><<1>>+<<1>>/<<1>>)/(4 y^3 (1+(1613 y^2)/100000)^(6069/500)) has evaluated to non-numerical values for all sampling points in the region with boundaries {{4,28.}}.

• If I do ListPlot[Table[{i,Chop[Fdiff[i]]},{i,-56/1000,56/1000,8/1000}]] then that does not look promising for FindRoot and I wonder whether it may have wandered far away from your starting point and possibly begun passing strange things to Fdiff and this may be related to the error message that you are seeing.
– Bill
Commented Apr 27 at 14:02
• The stack trace should show what was actually not numeric. One can't tell from the posted message. Commented Apr 27 at 14:09

\$Version

(* "14.0.0 for Mac OS X ARM (64-bit) (December 13, 2023)" *)

Clear["Global*"]

a = Rationalize[4.046];
b = Rationalize[0.01613];
c = Rationalize[0.227];
A[z_] = -a  Log[b  z^2 + 1];
int1[zh_] = Integrate[y^3  Exp[-3  A[y]], {y, 0, zh}] // Simplify;
int2[zh_] = Integrate[y^3  Exp[-3  A[y] + c  y^2], {y, 0, zh}] // Simplify;

s[zh_] = Exp[3  A[zh]]/(4  zh^3);
T[mv_, zhv_] =
((zh^3  Exp[-3  A[zh]])/(4  Pi  int1[
zh]))  (1 - ((2  c  m^2)/(1 - Exp[c  zh^2])^2)  (Exp[c  zh^2]  int1[
zh] - int2[zh])) // Simplify;


In the definition of Tp you cannot use D[T[m, zh], zh] when zh is being given a numeric value. Either use Set to evaluate the derivative before zh has a numeric value, or use the form Derivative[0, 1][T][m, zh].

Tp[m_, zh_] = D[T[m, zh], zh] // Simplify;

Fdiff[mv_?NumericQ] := Module[
{m = Rationalize[mv, 0]},
NIntegrate[s[y]  Tp[m, y], {y, 3, 7},
WorkingPrecision -> 15]]

Fdiff /@ {-0.048, 0, 0.048}

(* {-8.33896317973750*10^-6, -8.02294953359420*10^-6, -8.33896317973750*10^-6} *)


Fdiff does not have a root, it is negative

Plot[Fdiff[m], {m, -0.1, 0.1}]


FindMaximum[Fdiff[m], {m, 0}, WorkingPrecision -> 15]

(* {-8.02294953359420*10^-6, {m -> 0}} *)