Math 6610 • Analysis of Numerical Methods I

{Newton's Method}
Suppose \(f(x)\) is a real-valued function defined for all values of the real variable
\(x\), and that \(f'\) is Lipschitz continuous. State and prove a local convergence
theorem for Newton's method for finding a simple zero of \(f\). How does this result
change if the root is a double root? Is Newton's method globally convergent? If you
answered yes, prove your claim. If you answered no, sketch an example which
illustrates your claim, and outline a method that has the Newton method's local
convergence properties but which is globally convergent.