# Whether NIntegrate evaluation is multithreaded or not?

I am NIntegrate the function grarm[e, 3, dimension, 1, 1, 0.005]. There is matrix operation in function grarm, the dimension set the dim of the matrix.

I give the code of gram in the end.( take notice that you will have to manually change the dimension to get different gram with different dimension variable before integration)

OK, First I set dimension equals 7, and integrate the following

In:=NIntegrate[Im[grarm[e, 3, 7, 1, 1, 0.005]^2], {e, -0.3, 0},
MaxRecursion -> 200] // AbsoluteTiming

Out:={7.454151, 0.0581345}


and If I use linux command top to monitor the cpu situation, I see it is single threaded. See now, I set dimension to 9, and integrate the following

In:= NIntegrate[Im[grarm[e, 3, 9, 1, 1, 0.005]^2], {e, -0.3, 0},
MaxRecursion -> 200] // AbsoluteTiming

Out= {83.592780, 0.0271489}


and If I use linux command top to monitor the cpu situation, I see it is multi-threaded. See And Now I force the same integration to use one thread only, integrate the following

In:=
ParallelMap[
NIntegrate[
Im[grarm[e, #[], 9, #[], 1, 0.005]^2], {e, -0.3, 0},
MaxRecursion -> 200] &,
Tuples[{Range[3, 3], Range[1, 1]}]] // AbsoluteTiming

Out= {16.575363, {0.0271489}}


Notice it is much faster, that is odd.

In summary, There are two peculiar things waited to be explained

1. why dimension=7 is single thread while dimension=9 is multi-thread? Can NIntegrate actually naturally multi-threaded or not? Or the multi-thread is actually due to the matrix operation in the integrand(well, I personally think this is not likely)?
2. why multi-thread evaluation is much slower than single thread evaluation in the case of dimension 9?

In the end, the code for grarm.(don't forget to change the dimension, before integration)

dimension = 7;

Clear[armchairibbonmat]
armchairibbonmat[kx_, n_] :=
(
h = Table[0, {i, 1, 2 (n + 2)}, {j, 1, 2 (n + 2)}];
a[m_] := 2 m - 1;
b[m_] := 2 m;
t1 = 1;
aa = 1;
Do[
h[[b[i], a[i]]] = t1 E^(I kx aa);
h[[a[i], b[i]]] = t1 E^(-I kx aa);
h[[b[i + 1], a[i]]] = t1 E^(-I kx aa/2);
h[[b[i - 1], a[i]]] = t1 E^(-I kx aa/2);
h[[a[i - 1], b[i]]] = t1 E^(I kx aa/2);
h[[a[i + 1], b[i]]] = t1 E^(I kx aa/2);
, {i, 2, n + 1}];
)

mat = Compile[{kx, n}, Evaluate@armchairibbonmat[kx, dimension]];
Clear[armchairibbonmat];
armchairibbonmat = mat;

Clear[gkarm]
gkarm[kx_?NumericQ, e_, xx_, width_, ii_, label_, \[Eta]_] :=
Inverse[(e - I \[Eta]) IdentityMatrix[2*width] -
armchairibbonmat[kx, width]][[2*(ii - 1) + label,
2*(ii - 1) + label]]

Clear[fkx]
fkx[kx_?NumericQ, xx_] := E^(I kx xx)

Clear[grarm]
grarm[e_?NumericQ, xx_, width_, ii_, label_, \[Eta]_] :=
NIntegrate[
gkarm[kx, e, xx, width, ii, label, \[Eta]]*
fkx[kx, xx], {kx, -\[Pi]/3, \[Pi]/3}, MaxRecursion -> 200]