New Quadratic Integration

[ttw]

The first step of Romberg integration yields Simpson's rule. You should get the same results (if the programs both use the same number of points.)

https://en.wikipedia.org/wiki/Romberg%27s_method

The real problem with Romberg is that the precision increases faster the number of digits a computer has. A 128th order convergent method isn't really useful without lots of significant digits. (That's why I use Monte Carlo with either quadratic or cubic convergence, if possible; beats the usual square root order.)

Romberg, Gaussian quadrature (and it's extended family), etc., tend to fail in higher dimensions due to the Curse of Dimensionality. One can postpone the damage by using the hyperbolic cross-points; sacrificing a rooster during the dark of the moon doesn't help in this case.