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 ]