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..

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    $\begingroup$ Newton's method is a fixed-point method. Brent's and Muller's methods are bracketed methods. Bairstow is a variant of Newton's method to find quadratic factors of polynomials. $\endgroup$ – LutzL Aug 23 '17 at 14:41
  • $\begingroup$ Most methods are going to be "open" in the sense that they are not bracketed. This is a bad thing in terms of reliability of solution but it is a good thing because a bracket is frequently difficult to furnish in the first place. $\endgroup$ – Ian Aug 23 '17 at 14:43
  • $\begingroup$ (Cont.) The biggest thing distinguishing such methods are: do they use analytical derivatives? Do they use approximate derivatives? Or do they not use derivatives at all? Or do they not use derivatives at all? Newton uses analytical derivatives; various Newton-like methods including BFGS use approximate derivatives (these are basically multidimensional versions of the secant method); some methods like bisection don't use derivatives at all. $\endgroup$ – Ian Aug 23 '17 at 14:44
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    $\begingroup$ see here i hope this will help you math.niu.edu/~dattab/MATH435.2013/ROOT_FINDING.pdf $\endgroup$ – Dr. Sonnhard Graubner Aug 23 '17 at 14:51

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