The tanh-sinh quadrature

[Valentin Albillo]

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Hi, Paul:

(03-08-2017 02:42 AM)Paul Dale Wrote:  I originally used a Gauss-Kronrod quadrature in the 34S. As with all quadrature methods, it didn't always work and it wasn't adaptive but it was pretty fast. Enough people pointed out specific failure cases and wanted Romberg that I changed the method over. Rapidly oscillating functions and some exponential were bad from memory.

I have my own quadrature implementation and would be very obligued if you could post (or include working links to) the specific failure cases you mention as well as any other cases you tested your quadrature against.

In particular I'd like to know:

- the actual integral: function, limits of integration, accuracy desired
- the result you got: actual result vs. exact result
- the timing: actual min.secs (as compared to the time it takes your machine to invert a 10x10 random matrix of (0..1) 12-digit reals) and/or total function evaluations:

Example (a far too easy one):

Integral(0,1,sin(x^2) dx, 1E-12) -> 0.310268246746, exact=0.310268(whatever)
Timing: 1.2 seconds. (on a machine that takes 0.57 seconds to invert said matrix), 257 function evaluations in all.

Thanks and best regards.
V.
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