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NIntegrate slowly converging where Integrate+NIntegrate has no problems

Posted 3 years ago

I have an integrand

Integrand[x_, y_, 
   z_] := ((x + 1/10) Abs[
      x])/(((x + 1/10)^2 + y^2 + z^2) (y^2 + z^2 + Abs[x]^3));

which I want to integrate from -Infinity < y, z < Infinity and 1/10 < |x| < Infinity with high precision. I can approach this one of two ways. I can integrate over y and z exactly using cylindrical coordinates, and numerically integrate over the remaining coordinate

SimplifiedIntegrand[
   x_] := (\[Pi] Abs[x] (x + 1/10) Log[(x + 1/10)^2/
    Abs[x]^3])/((x + 1/10)^2 - Abs[x]^3);
NIntegrate[
 SimplifiedIntegrand[x] + SimplifiedIntegrand[-x], {x, 1/10, 
  Infinity}, WorkingPrecision -> 20]
(* Returns -0.15385769942458967071 *)

or I can just do the entire thing numerically

NIntegrate[
 Integrand[x, y, z] + Integrand[-x, y, z], {x, 1/10 , 1/10, 
  Infinity}, {y, -Infinity, 0, Infinity}, {z, -Infinity, 0, 
  Infinity}, WorkingPrecision -> 20]
(* Returns -0.15385791795768615749 and throws slwcon and eincr errors *)

The two don't agree to the precision that I want, and the latter also takes a very long time and throws several errors. I can check that these errors are unrelated to the singularity at y=z=0, x=1/10, as these problems persist if I increase the lower bound of x such that this singularity isn't even included. Increasing MaxErrorIncreases and WorkingPrecision do not help with this problem. I just want to understand what's so bad about this integral - as far as I can tell, it is not oscillatory, non-singular if I increase the lower bound of x, and ought to be convergent. Is there any way to improve the performance of the 3D integral?

In case you're wondering why I care about doing the full 3D integral numerically - this integrand is an approximation of a similar expression for which the analytic reduction to a 1D integral is not possible. I'm just trying to troubleshoot with this case here.

POSTED BY: Henry Shackleton
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