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Comment by kragen | Hacker News Reader

when i did this i got better efficiency from ternary search

http://canonical.org/~kragen/sw/aspmisc/intervalgraph.html

you may find this interesting if you haven't seen it already: https://fredrikj.net/blog/2017/11/new-rigorous-numerical-int...

If on top of rigorous, you want them to be formally verified in Coq at the same time as they are computed: https://www.lri.fr/~melquion/doc/18-jar.pdf

is this the one mentioned in fredrik's post? he links https://www.lri.fr/~melquion/doc/16-itp-article.pdf which is presumably a different paper by the same author

I think they are the conference and journal versions of the same paper. Hadn't seen it was mentioned in the article, I should have read it more thoroughly!

i hadn't, this is fantastic!

Then you possibly will find this useful: https://en.wikipedia.org/wiki/Golden-section_search

that is an algorithm for a different problem; here we are looking for a zero, not an extremum, and we want to find all the zeroes (in a two-dimensional plane, so there are usually infinitely many zeroes), not just one of them

perhaps there is a way to apply it to this problem that is obvious to you but not to me

with autodiff we can use a zero-finding algorithm (even one that isn't derivative-free) to find extrema, but i don't know how you'd go about using an extremum-finding algorithm to find zeroes. the first step would seem to be quadrature? but that sounds impractical

The algoithm itself is about the use of golden ratio for interval search.

One can use subdivision, other can use ternary division. And another one can use golden ratio.

as i understand it, the advantage of golden-section search is that, to search for a minimum rather than a zero, you need to in some sense interpolate a parabola rather than a line, so you need three points rather than two, and you'd like them to be somewhat evenly spaced. i don't fully understand why the golden section is better for this than just dividing the interval between the two lowest points in half, but it's definitely an algorithm to solve a different problem

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