Add adaptive integration to the plugin model docs

[pkienzle]

Here's a very minimal description of the adaptive integration scheme:

Target accuracy for the models is 5 digits of precision for SANS (R=20 nm at q=1/Å) and USANS (R=20 μm at q=0.002/Å). For USANS slit smearing, which pushes q out to 0.1/Å, the target accuracy 0.2. With regard to performance, I'm trying to limit it to 2 s to evaluate 200 q points on my mac M2 chip.

I recently revised all the models to use a simple adaptive integration scheme (#658) based on qr max for the shape along the c-axis and in the a-b plane. For the outer loop I use the max of these two, but only the a-b cross section for the inner loop. I'm limiting the (θ,φ) grid to 100 000 evaluation points, with no more than 76 points in the outer loop.

Here's an example from triaxial ellipsoid:

    const double qr_max_inner = fmax(q*radius_equat_minor, q*radius_equat_major);
    const double qr_max = fmax(qr_max_inner, q*radius_polar);
    constant double *z_outer, *w_outer;
    constant double *z_inner, *w_inner;
    int n_outer = gauss_weights(qr_max, ADAPTIVE_MAX_OUTER, &w_outer, &z_outer);
    int n_inner = gauss_weights(qr_max_inner, n_outer, &w_inner, &z_inner);

For the tetrapod, qr_max = q*sqrt(length**2 + (outer radius)**2) is probably all you need. Whether outer radius is radius or core radius + thickness depends on your parameterization.

Code in explore/check_adaptive.py compares the adaptive grid to a 5000x5000 grid on a few points and reports those that are out of tolerance.

Originally posted by @pkienzle in #705 (comment)

There are useful details in the various issues and PRs associated with the tetrahedron, octahedron and prism models:

• Adaptive integration for truncated octahedron #710 mentions that relabeling the axes so that the c-axis is the longest seems to improve accuracy. The outer loop wants the qr = max(q Rc, q Rab), but the inner loop only needs qr = q Rab to set the number of integration points.
• Adding nanoprisms model (pure python) #699 explores Fibonacci vs Gauss-Legendre grids
• Simple adaptive integration #658 discusses tradeoffs between speed and accuracy, and introduces explore/check_adaptive.py
• slit resolution calculation is too slow for usans #717 discusses issues with USANS and slit geometry
• Nanoprism model #685 shows that fringe density increases at the poles, which could explain why square grids are better than Fibonacci