NIntegral first "hit" is not equal to second "hit"

Question

Consider the following code:

r = 30; a = 95/100; q = (r Sqrt[1 - a^2]/a)^2/4; m = 4;
ce[m_, q_, x_] := 
  MathieuC[MathieuCharacteristicA[m, q], q, x];
1/Pi NIntegrate[Cos[x] Sin[r /a Cos[x]] ce[m, q, x], {x, 0, 2 Pi}, 
  Method -> "GaussKronrodRule"]

When I "hit" the shift+enter the first time, the result is:-0.0426046; the second "hit" gives -0.0673913. None of them is equal to the analytical result: -0.055116186075968306.

Note that to reproduce the bizarre behavior, you need change m to other integers and then change back to 4. If you make a fresh startup of Mathematica, it will give the right results no matter how many times you "hit". However, as long as you change m to other values and then change back, it will have the problem I described.

When $m\neq 4$, the numerical integration gives the right results.

MM version: 10.4.1.0

osx:10.11.6 EI Capitan

I also tried it on Linux with version 10.4.1, same problem.

Answer 1

The short answer is that you are not doing numerical calculations right. Your variables must be floats, nut integers which can be achieved by placing a dot in front of your values (e.g. a = 90/100 becomes a = 90./100.). This would fix your problem, however, there are two other issues with your coding style: function arguments have the same name as your assigned variables, and you do not clear your variables before rerunning your notebook. All three have been fixed in the following cell:

ClearAll["Global`*"]
r = 30.;
a = 95./100.;
q = (r Sqrt[1 - a^2]/a)^2/4;
m = 4;
ce[mm_, qq_, xx_] := MathieuC[MathieuCharacteristicA[mm, qq], qq, xx];

1/Pi NIntegrate[Cos[x] Sin[r/a Cos[x]] ce[m, q, x], {x, 0, 2 Pi}, 
  Method -> "GaussKronrodRu