Adaptive Simpson and Romberg integration methods

[Wes Loewer]

(03-24-2021 08:54 PM)robve Wrote:  In practice, Adaptive Simpson is really, really good with only two minor drawbacks:
1. Adaptive Simpson may not produce usable error estimates of the result when the function is not well approximated on points by a cubic. (note: modifications are possible that produce an error estimate, but we're talking about the straightforward method).
2. Adaptive Simpson uses closed intervals and may fail to integrate improper integrals.

I wonder if you considered an open interval version of Simpson's Rule. See Milne's Rule (Wikipedia's Open Newton–Cotes formulas). Or better yet, if you really want to stay with a 3 node method, how about the 3 node version of Gaussian Quadrature? It would be more accurate without taking any more computing effort.

I love teaching Simpson's Rule to students, but...

Quote:Where would any book on numerical analysis be without Mr. Simpson and his “rule”? The classical formulas for integrating a function whose value is known at equally spaced steps have a certain elegance about them, and they are redolent with historical association. Through them, the modern numerical analyst communes with the spirits of his or her predecessors back across the centuries, as far as the time of Newton, if not farther. Alas, times do change; with the exception of two of the most modest formulas, the classical formulas are almost entirely useless. They are museum pieces, but beautiful ones.

~Numerical Recipes in C

-wes