NIntegrate converging too slowly when increasing size of array

Question

I have a problem with a numerical integration. I have a 4x4x4x4 array that has for each entry an integral and I want to use NIntegrate to evaluate it. It gives me no problem, it's actually pretty fast. But if I increase the size of the array to 9x9x9x9 I get this message:

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.

I increased the working precision already but it didn't help.

EDITED:

I put here a simplified version of my code:

B9b = Array[Nspher[#2, #4] &, {9, 9, 9, 9}, {1, 0, 1, 0}]

where

Nspher[l_, k_] := 
 NIntegrate[
  SphericalHarmonicY[l, 0, θ, φ] SphericalHarmonicY[k, 0, θ, φ]*Sin[θ],
  {θ, 0, π}, {φ, 0, 2 π},
  WorkingPrecision -> 50, AccuracyGoal -> 10, MaxRecursion -> 30]

Answer 1

Your example code seems to repeatedly evaluate the same integral many times. To see this, try changing Nspher as follows:

Nspher[l_, k_] := (Print[{l, k}]; 
  NIntegrate[
   SphericalHarmonicY[l, 0, \[Theta], \[Phi]] SphericalHarmonicY[k, 
     0, \[Theta], \[Phi]]*Sin[\[Theta]], {\[Theta], 
    0, \[Pi]}, {\[Phi], 0, 2 \[Pi]}, WorkingPrecision -> 50, 
   AccuracyGoal -> 10, MaxRecursion -> 30])

To fix it, use memoization:

Nspher[l_, k_] := 
 Nspher[l, k] = (Print[{l, k}]; 
   NIntegrate[
    SphericalHarmonicY[l, 0, \[Theta], \[Phi]] SphericalHarmonicY[k, 
      0, \[Theta], \[Phi]]*Sin[\[Theta]], {\[Theta], 
     0, \[Pi]}, {\[Phi], 0, 2 \[Pi]}, WorkingPrecision -> 50, 
    AccuracyGoal -> 10, MaxRecursion -> 30])

As for NIntegrate::slwcon, this message does not necessarily indicate an error. It is simply a report that the error estimate on a sub-region did not decrease as fast as it is expected to (eventually) decrease, assuming the function is nicely behaved. This can be a hint that better methods or other special handling can help you. If no other messages ensue, the result of NIntegrate is still likely to be good.

Using something like the above code, you can find out all/some of the specific values of {l, k} for which your computation is producing slwcon. For example, {l, k} == {2, 8} is one case. Investigate these, and once you are satisfied, you could Quiet slwcon or solve those cases in some other way.