Question

When it comes to applying Newton's method on a single variable function, what are the conditions that must be met in order for the method to work? Do we have to pick a good starting point and interval? How do I choose a good interval and good starting point? What are some type of functions where newton's method doesn't work for one reason or another?

Answer 1

1. Condition: Like all the fixed point iteration methods, Newton's method may or may not converge in the vicinity of a root. The convergence of the fixed point iteration methods is guaranteed only if |g'(x)| < 1 in some neighbourhood of the root. Even Newton's method can not always guarantee that condition. When the condition is satisfied, Newton's method converges, and it also converges faster than almost any other alternative iteration scheme based on other methods of converting the original f(x) to a function with a fixed point.
2. Failure: Newton's method is a lovely method that you should try to apply any time you are faced with a root-finding problem. Newton's method has one small flaw, though. To apply the method you have to be able to compute the derivative f '(x).
3. An example where Newton Method Fails:

4. Newton's method fails to find the root of

           f(x) = x^1/3

   with an initial guess of x = 1.

5. Approximate a good starting point by drawing rough graph of given function.