I am trying to solve the following numerical integral

fun=E^(1/48 (-113 k[1]^2+102 Cos[x1-x3] k[1] k[3]-113 k[3]^2-2 Cos[x1-x5] k[1] k[5]+102 Cos[x3-x5] k[3] k[5]-113 k[5]^2+102 Cos[x1-x7] k[1] k[7]-2 Cos[x3-x7] k[3] k[7]+102 Cos[x5-x7] k[5] k[7]-113 k[7]^2))/(36864 \[Pi]^4);
Timing[NIntegrate[k[1]k[3]k[5]k[7]fun Cos[2(x1+x3-x5-x7)],{k[1],0,Infinity},{k[3],0,Infinity},{k[5],0,Infinity},{k[7],0,Infinity},{x1,0,2Pi},{x3,0,2Pi},{x5,0,2Pi},{x7,0,2Pi}]]

which is basically a 8-dimensional gaussian function times a cosine.

The result that I'm obtaining is

NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 5.956319478409388`*^-7 and 8.378969793277401`*^-7 for the integral and error estimates.

My problem is that the error estimates of the integral is bigger than its value which makes me doubt about the accuracy of the result and also I would like to minimize the timing of computation.

Is there a method of computing this integral in less than 40 seconds and obtaining a more or less accurate result?

I don't mind if the result is $6 \cdot 10^{-7}$ or $5 \cdot 10^{-7}$, I just want an approximate value in the less time possible.


1 Answer 1


... a more or less accurate result ? Have you tried Monte-Carlo integration? It's well suited for higher dimensional integrals and it cuts the time down a lot with about the same result:

  k[1] k[3] k[5] k[7] fun Cos[2 (x1 + x3 - x5 - x7)], {k[1], 0, 
   Infinity}, {k[3], 0, Infinity}, {k[5], 0, Infinity}, {k[7], 0, 
   Infinity}, {x1, 0, 2 Pi}, {x3, 0, 2 Pi}, {x5, 0, 2 Pi}, {x7, 0, 2 Pi},
   Method -> {"MonteCarlo", "MaxPoints" -> 10^6, "RandomSeed" -> 12345}]]

(* result: {2.09375, 5.51509*10^-7} *)

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