Summary of Newton-Raphson

You must learn the following facts about the Newton-Raphson method.

1. The algorithm is
   
   $$x_{n+1} = x_{n} - {F(x_{n})\over F^{\prime}(x_{n})}.$$
   
   
   You must know this, how it is derived, how to use it and when to stop.

2. It is a second order scheme with
   
   $$\epsilon_{n+1} \approx -{F^{\prime\prime}(r)\over 2 F^{\prime}(r)}\epsilon_{n}^{2}.$$
   
   
   You must know how to calculate this error estimate

3. Drawbacks. There are a couple of drawbacks in that the method may not converge to the root you want if the initial guess is not good enough (you may be starting on the wrong side of a turning point) and you need to calculate $F^{\prime}(x)$ and this may be hard.

4. The algorithm is best done on a computer.