Since the precision is very important in my case, I am curious how precise is the MMA result for this integral:
t = 0.2; w = 1; a = 4/w^2;
XX[lf_, mf_, l_, mg_, p_] := Abs[YY[lf, mf, l, mg, p]]^2;
YY[lf_, mf_, l_, mg_, p_] := WignerD[{lf, mf, l},
t] Exp[-((a r^2)/4)] Sum[((Abs[mg - l] + p)!/((p - j)! (Abs[mg - l] + j)! j!)) (Sqrt[2]/w)^(2 j + mg - l) (2/a)^(2 j + mg - l + 1) (Gamma[.5 (Abs[mg - mf] + mg - l + 2 + 2 j)]/(r Gamma[Abs[mg - mf] + 1])) a^(0.5 (2 j + mg - l + 1)) ((a r^2)/2)^(1/2 Abs[mg - mf] + .5) Hypergeometric1F1[(Abs[mg - mf] - 2 j - mg + l)/2, Abs[mg - mf] + 1, (a r^2)/4], {j, 0, p}];
MM[lf_, l_, mg_, p_] := Sum[XX[lf, ii, l, mg, p], {ii, -lf, lf, 1}];
NIntegrate[r M[2, 1, 3, 0], {r, 0, \[Infinity]}, MaxPoints -> 500000,PrecisionGoal -> 10, AccuracyGoal -> 5, MaxRecursion -> 50,WorkingPrecision -> 10]
MMA returns 4.159915211
But the result is working precision dependent (and not only):
If I put working precision 17 it says that the precision of the integrand is less than working precision. What does it mean? Can I still trust this result?
I wonder if there are some tools that could give me the error bar on how accurate this integration is since it seems like depending on the evaluation parameters I get different results. How to figure out the optimal set of parameters for this integral?
t = 2/10
instead of theMachinePrecision
number0.2
. Read some of the tutorials on precision for more explanation. (3) See (14500), (75426) for getting the error estimate. $\endgroup$