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

[robve]

Hi Albert & emece67,

I tested the qthsh and wp34s C implementations against 818 integrals from the spreadsheet downloaded from https://newtonexcelbach.com/2020/10/29/n...ture-v-5-0

I rewrote the 818 functions in C (using a bit of sed wizardry) then ran all 818 test cases in C to use IEEE double fp with n=6 levels and eps=1e-9 for qthsh and eps=1e-15 for wp34s (because eps=1e-15 sets thr=3e-7 in wp34s which is about the same threshold as qthsh's 1e-8).

The zip file includes a spreadsheet with the results for the the 818 integrands, produced with some bash scripts that generate C code that is compiled and run on each integrand. The relative errors are reported by the methods and the exact result is compared as an absolute error. An ALERT is shown when the relative error AND the absolute error are both larger than 1e-9. Many ALERT are close to 1e-9 so can in principle be ignored, typically indicating difficulties with Tanh-Sinh convergence. However, there are several ALERT that indicate more serious issues with convergence, including some NaN. I hope this is useful.

The qthsh C (and BASIC) source code is updated in this thread, because "fixing" the underflow "problem" in the inner loop caused problems with non-convergence for a few functions, notably with 25*exp(-25*x) over [0,10] (row 63 in the spreadsheet). I also updated the draft report to correct this.

The 818 integrals are mostly functions composed of the usual trig and other transcendental functions. There are a couple with ABS and one with SIGN. But nothing as difficult as a step function, for example.

Note that the inf/NaN singularities are ignored in these implementations (i.e. "interpolated" by using previous points to produce "upward pressure"), but I agree that this is something to look into, as Albert pointed out.

Compared to other integration methods, there is nothing "standard" about Tanh-Sinh rules and implementations, they are all different! The Michalski & Mosig rule is a bit cheaper to execute. But as a consequence, this rule produces point clouds that are a little bit more dense in the middle of the interval (see the chart in the draft report). Other Tanh-Sinh points and weights could be defined to produce even denser "point clouds" between the endpoints, perhaps to improve convergence for integrands with more "mass" in the middle. However, other transformations might be more appropriate as Albert suggests.

- Rob