(PC-12xx~14xx) qthsh Tanh-Sinh quadrature

[emece67]

(04-05-2021 05:34 PM)robve Wrote:  I noticed a few things in the Python code, including:

      try:
        fpl = f(bpa2 + bma2*r)
      except Exception:
        fpl = 0;
      p = fpl*w
      try:
        fmi = f(bpa2 + bma2/r) if expsinh else f(bpa2 - bma2*r)
      except Exception:
        fmi = 0

I may be mistaken, but this sets fpl and fmi to zero when an exception occurs. This may put some downward pressure on the quadrature towards zero and could lead to delayed convergence if the weight isn't yet insignificant, in turn requiring more points that in turn could cause more exceptions in the same area. I did some comparisons and it looks like interpolating/extrapolating the missing function values is best (rather than terminating the iterations). A simple way to interpolate is to reuse the previous fpl and fmi, although more elaborate schemes could be used. Think about functions that have a large nonzero limit at an endpoint.

On the other hand, the Python code does not get as close to the endpoints (due to tmax) as the other implementations. By comparison, the VB implementation uses a fixed tmax=6.56, which I am looking into.

- Rob

I've just tested re-using the previous fpl or fmi when the exception raises, but the end results are the same.

The idea behind tmax is described succinctly in another thread here (the expression for N, although I later reworked it to directly get tmax). With such tmax one ensures that the integrand is never evaluated at the extremes. Maybe this is the reason the modification above (reusing fpl/fmi) does not show any benefits.

Regards.