Math 6610 • Analysis of Numerical Methods I

{Newton Type Methods}
Suppose \(f(x)\) is a real-valued function defined for all values of the real variable
\(x\), and that \(f'\) is Lipschitz continuous. Consider the discrete Newton method:
Guess \(x_0\) and for \(k=0,1,2,\ldots\), do:
\[
  a_k=\frac{f(x_k+h_k)-f(x_k)}{h_k}
\]
\[
  x_{k+1}=x_k-\frac{f(x_k)}{a_k}
\]
Analyze the local convergence properties of this method. How does the speed of
convergence depend on the choice of \(h_k\) (ignore any effect of round-off errors in
this analysis)? Now taking roundoff into account what would be a reasonable size
for \(h_k\) in practice?