Full modal integral does not give correct results

[chrisbrahms]

I have two separate combination of modes, both TM01+HE21, but with HE21 rotated by π/4 in one case and -π/4 in the other. This creates LP11 modes with different orientations:

If I now calculate the plasma nonlinear polarisation for both of these using TransModal with full=true, I don't get the same result. I can fix this, however, by changing from pcubature_v to hcubature_v:

The reason appears to be that pcubature starts by evaluating at three angles, θ = [0, π, 2π], and then never goes any further (I got this by printing xs inside pointcalc!). In the vertical case, it only actually runs pointcalc! once, so I get

julia> transform_vert(nl_vert, Eω_vert, 0.0);
pointcalc eval at 
        r=[6.25e-5, 6.25e-5, 6.25e-5, 0.000125, 0.000125, 0.000125, 0.0, 0.0, 0.0]
        θ/π=[1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0]

[chrisbrahms]

It appears this is the exact same problem as in JuliaMath/Cubature.jl#24

[chrisbrahms]

I think the problem is actually more general than this, to the extent that the full modal integration doesn't actually work at all. I cannot seem to get pcubature to evaluate the integrand anywhere other than 0, π and 2π for any combination of modes. For the horizontal mode example above, I get several iterations of the integral, but still only at those angles. Same for a diagonal version

pointcalc eval at 
        r=[0.00012498117616851275, 0.00012498117616851275, 0.00012498117616851275, 1.8823831487239906e-8, 1.8823831487239906e-8, 1.8823831487239906e-8, 0.00012483065354241814, 0.00012483065354241814, 0.00012483065354241814, 1.693464575818664e-7, 1.693464575818664e-7, 1.693464575818664e-7, 0.00012452997091241938, 0.00012452997091241938, 0.00012452997091241938, 4.700290875806249e-7, 4.700290875806249e-7, 4.700290875806249e-7, 0.00012407985264930883, 0.00012407985264930883, 0.00012407985264930883, 9.201473506911763e-7, 9.201473506911763e-7, 9.201473506911763e-7, ...]
        θ/π=[1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0, ...]

This can also be seeen in the difference between pcubature and hcubature in the plot above: not only do the curves agree for hcubature, but the peak value of the nonlinear polarisation is different as well!