Recently, I've been surveying the methods of solving nonlinear equations (both: methods that solve one equation and methods that solve a set of equations).. According to some references, I found that there are two main categories; 1- Bracketing methods which are 1-The bisection method, 2-the false position method, and 3- the incremental search. 2- Open methods which are 1-Newton Raphson method, 2-Fixed Point iteration, 3-Secant Method, 4-Brent Method, 5-Muller Method, 6-Bairstow Method, 7-Broyden Method.
The problem is that I found there are more other methods that I can't put in a category like 1-BFGS, 2-Shamanskii method, 3-Chord method etc.. I also found hybrid methods that mixes other methods..
My question is: 1-I just want to know all the methods that can solve nonlinear algebraic equations so is there a source that helps me and that covers all the methods? (An understandable source as I'm not a mathematician) 2-I also want to know how to put these methods in the suitable category and if it is enough to make only these two categories (Bracketing and open methods)? Thank you..