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

[robve]

I should add that the choice of d=1 for Exp-Sinh and c=0 for Sinh-Sinh bothered me as well, because the Exp-Sinh and Sinh-Sinh integrals are split in two parts. I have not yet looked into articles for suggestions for these values, although for Sinh-Sinh I would expect that the best c is where f has (some) symmetry e.g. f(c+x) = f(c-x).

In the meantime, I've mostly worked on testing different parameterizations e.g. h=.5 and h=1.5 initially and compared the "pure" Tanh-Sinh with the pi/2 factors etc.

(04-15-2021 05:14 PM)Albert Chan Wrote:  Nice, but we can do better ! (see previous post)

exp(0) = 1
exp(-0.01y) = 0.001       → -y/100 = ln(10^-3) = -3*ln(10) ≈ -3*2.3 ≈ -7

The derivation to select d=700 for the integrand exp(-0.01*y) is clear, except for the initial requirement that is not obvious from the integrand and interval:
exp(-0.01y) = 0.001
In this case I can empirically verify this by a rough gradient search for the value of d that produce the lowest errors, but is there a general algebraic formulation using the inverse function to determine where d might be optimal? Any literature on that?

- Rob