a.Consider a function f : R R, f C 2 , such that x is a local minimizer

and f (x ) = 0. Suppose that we apply the algorithm

x (k+1) = x (k) k f (x (k) )

such that { k } is a positive step-size sequence that converges to 1/f (x ).

Show that if x (k) x , then the order of convergence of the algorithm

is superlinear (i.e., strictly greater than 1)

b.Given part a, what can you say about the order of convergence of the secant method?