[guyomes]

A generalisation of this idea is known as Taylor model in 1998 [1]. It might even have been known in 1984 as neighborhood arithmetic [2]. The generalisation works by taking a Taylor expansion of the function up to order n, and then by using a bound for the remainder using bounds on the partial derivatives of order n+1 [3].

[1]: https://www.bmtdynamics.org/cgi-bin/display.pl?name=rdaic

[2]: https://books.google.fr/books?id=2zDUCQAAQBAJ

[3]: https://en.wikipedia.org/wiki/Taylor%27s_theorem#Taylor's_th...

[sfpotter]

Worth pointing out that these ideas were already well known by Moore, the founder of interval arithmetic. Chapter 3 of his monograph "Methods and Applications of Interval Analysis" has the basic ideas worked out.

The folks working on the Taylor model were coming from physics where they had some nasty (very ill-conditioned) nonlinear systems that they needed to solve and developed a package called COSY which implements it.

My understanding is that the Taylor model is effective in practice, but there might have been some confusion around what it was actually capable of. I believe the Taylor model people claimed that their interval inclusions had better than quadratic excess, but this turned out not to be the case. Other people were able to push past the quadratic limit of the well known interval inclusions using other techniques. There are some recent papers by Hormann and Yap along these lines, although I think the first interval inclusion that is better than quadratic dates back further...

[yorwba]

I assume "recent papers by Hormann and Yap" is a reference to e.g. Range functions of any convergence order and their amortized complexity analysis https://www.inf.usi.ch/hormann//papers/Hormann.2023.RFO.pdf ? Cool stuff.