Some Difficulties with the Newton-Raphson Method:
- The derivative needs to be calculated analytically. This may not be possible in situations where the function is defined by a complex formula.
- The initial guess may be out of the interval of convergence of the zero.
- If the derivative of an iterate is too close to zero, the next iterate might leave the interval of convergence.
- The guess may enter into a cycle, as illustrated here.
Below are two more animated examples of the Newton-Raphson Method: