Adaptive Simpson and Romberg integration methods

[robve]

(03-26-2021 04:30 PM)Albert Chan Wrote:  All integrals is hitting the assumption that end-points does not matter.
After u-transformation, integrand end-points mean (t1) is still not zero.

Albert, darn, you just beat me, as I was testing transformed integrals too! But I ran into issues doing this in C with the endpoints, you know.

I noticed that all of the integrals in his post have two distinguishing features: they have a U shape (or inverted) on both or either end that makes them suitable to apply a transform before numerical integration, i.e. change of variable or u-substitution. The trick to pull this off in an implementation is basically to have an initial check of a few points to find evidence of a U on either end and then transform. In that way we can use a simple quadrature e.g. Simpson's, Bode's, or more advanced Newton-Cotes formulas to obtain a good approximation with fewer points and a usable error estimate.

- Rob