The tanh-sinh quadrature

[emece67]

I have compared this tanh-sinh method against the Romberg method used in many hp calculators, by means of another Python script that implements the Romberg method.

To get similar errors, the Romberg method is faster when working with less than 12-14 decimal digits. At 10 digits, Romberg can be, say, 2 times faster.

But when working with a higher number of digits, the tanh-sinh method is way faster. At 16 digits is slightly faster, at 32 is 3x faster and with even more digits it is faster by orders of magnitude. It needs much less function evaluations to get similar results.

As a side effect, the tanh-sinh method copes flawlessly with many integrals having infinite derivatives at the integration interval ends. In such cases the Romberg method converges very slowly and gives much greater errors. Even more, the tanh-sinh method also manages many cases of integrals having infinite discontinuities at the integration interval ends, the Romberg method fails in such cases.

I think it is a really promising quadrature method for machines like the wp34s, that work with 16/32 digits.

Regards.