I have two related questions about numerical methods for root solving:


1) $f: R \to R$ is continuous and piece-wise smooth, with $f(a)f(b) < 0$.  

 $f$ has very high number of knot-points and computing analytic expression for $f'$ is not possible. 
 To find a root of $f$ in $[a,b]$, I can try the following: 


* Use bracketing methods like bisection, secant, Brent
* Use Newton's method with numerical approximation of derivative ( being careful about choice of $\delta$, so that $(x-\delta)$ and $(x + \delta)$ falls between two consecutive knot-points )


My question is: Does the second option have any advantage over first ?


My intuition is that having to compute numerical derivative compensates any advantage Newton's method has over (say) secant method, in terms of stability and convergence rate.


2) $f: R^2 \to R^2$, continuous, piece-wise smooth, known to have a root in some given rectangle. Writing $f=(f\_1, f\_2)$ and $g = f\_1^2 + f\_2^2$ , I can try


* Nelder-Mead on $g$
* Gradient descent with numerical gradient estimation


Is it correct to think that second method will perform worse than first because of numerical approximation ?  

Also is there any other method *without derivative* that I can try ? ( possibly some analogue of secant or Brent method )  

[ I am a newbie to numerical analysis, so any reference would be helpful. Most sources on higher order root solving seems to talk about gradients ]



root solving without analytic derivative
Obviously, within the realm of piecewise-smooth functions one can find examples where any derivative-based approach fails. I believe you're looking for the term "**derivative free optimization**". 


* [Here is the relevant page on Wikipedia](https://en.wikipedia.org/wiki/Derivative-free_optimization), which lists about a dozen algorithms.
* [Here is a nice review article](http://www2.peq.coppe.ufrj.br/Pessoal/Professores/Arge/COQ897/dfo-Sahinidis.pdf).
* [Here is a whole book on the topic](https://books.google.com.sa/books/about/Introduction_to_Derivative_free_Optimiza.html?id=7c7X6tlcaHEC&redir_esc=y&hl=en).


