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

[robve]

(04-03-2021 01:40 AM)Albert Chan Wrote:  

(04-02-2021 02:46 PM)robve Wrote:  Functions that are typically integrated over non-[0,1] finite domains are normalized to the [0,1] domain in experiments.
It is simpler that way to conduct the tests and for comparisons.

Here is a link to convert integral limits to [0, 1]: http://fmnt.info/blog/20180818_infinite-integrals.html

Very nice to know. Just for clarification: we're only considering definite integrals at this time with Tanh-Sinh. A simple linear transformation suffices to rescale the bounds. The same area is computed with the same function points, but via a transformation. No tricks, just easier to analyze the point distributions in a normalized [0,1] domain. For example, NR2ed p.134 is transformed:
$$ \int_0^2 x^4 \log(x+\sqrt{x^2+1})\,dx = \int_0^1 32u^4 \log(2u+\sqrt{4u^2+1})\,du $$

I've tested other integrals including the one above that are not included in the article, but several of these are not very interesting for Tanh-Sinh, e.g. trig and other transcendental functions. We already know that Tanh-Sinh converges with double exponential rate for holomorphic complex-differentiable functions bounded on the interior of the unit disk within the Hardy space. The point distribution makes this method more resilient to singularities at the endpoints and is especially good when most of the area is located to the endpoint(s), e.g. L- and U-shaped functions.

As some have said, this method might never reach the level of a true general-purpose integrator, but who knows...

It's interesting to analyze the implementation differences, apply code optimizations and fixes when applicable to public code. Perhaps we can find a way to make the method stronger when exceptions occur due to singularities that kill the method (thanks Albert for pointing this out!). The approach that might be best to do this is to interpolate/extrapolate, see the last part of the (draft) article: "The integral of function 15 is now the same as the integral of function 14 as it should be, up to the last digit 1.77245382409889. Both are the same for the same specified eps." Though anecdotal at this early stage, this corroborates the interpolation approach. Inverting the integration bounds should give the same integration result, which for Tanh-Sinh is NOT necessarily the case for domains such as [0,1].

I did not have time today to look at emece67 Python code and results yet. As always, suggestions and comments are much appreciated.

- Rob