What are the limitations of using newton-raphson method?

Answer 1

There are several limitations to the Newton-Raphson method (N-R).

1. The method relies on the use of the derivative of the function whose root is being sought. If the function is not differentiable then N-R cannot be used. Even if the derivative exists, it may not be calculable analytically. In that case N-R may require huge amounts of effort or prove to be impossible.

2. If there is a stationary point in the vicinity of the root, the derivative will become 0 at that point and attempted division by zero will stop N-R. Even if the iteration does not actually hit the stationary point, the rounding errors due to division by a very small number can lead to very large errors in the N-R calculations.

3. If the first derivative is ill-behaved in the neighbourhood of the root then N-R can overshoot. For example, f(x) = |x|^a where 0 < a < 1/2.

4. A poor starting point for the N-R iteration can lead to non-convergence.

5. Where a root has a multiplicity greater than 1, then convergence will be slow (unless appropriate adjustments are made to N-R).

Answer 2

Mehtamatics gave some great reasons. You may find this Paul's Online Math Notes website useful, as well.