On Convergence Rates of Root-Seeking Methods

[Claudio L.]

I'm very interested in this subject, since I'm about to start coding the solvers for newRPL.
My question here is generic for all the people that have been doing tests:

I'm planning to have several solvers and let the user choose which one to use for their particular problem.
First thing that's needed is to separate in 2 classes of methods: single-point and bracketed.

Based on the tests reported in this thread, what methods would be a "must have" in each category?

Bracketed:
* Bisection
<add others here!>


Single starting-point:
* Newton-Raphson
<add others here!>

Keep in mind newRPL is multiprecision but limited to 2000 digits, so a very complex method that is only faster for more than a million digits wouldn't really be useful.