Some Difficulties with the Newton-Raphson Method:

  1. The derivative needs to be calculated analytically. This may not be possible in situations where the function is defined by a complex formula.
  2. The initial guess may be out of the interval of convergence of the zero.
  3. If the derivative of an iterate is too close to zero, the next iterate might leave the interval of convergence.
  4. The guess may enter into a cycle, as illustrated here.

Below are two more animated examples of the Newton-Raphson Method:

cube root movie

cosx=x movie