Optimal order and efficiency for iterations with two evaluations

The problem is to calculate a simple zero of a nonlinear function f. We consider rational iterations without memory which use two evaluations of f or its derivatives. It is shown that the optimal order is 2. This settles a conjecture of Kung and Traub that an iteration using n evaluations without memory is of order at most $2^{n - 1} $, for the case $n = 2 $.Furthermore we show that any rational two-evaluation iteration of optimal order must use either two evaluations of f or one evaluation of f and one of ${f'}$. From this result we completely settle the question of the optimal efficiency, in our efficiency measure, for any two-evaluation iteration without memory. Depending on the relative cost of evaluating f and ${f'}$, the optimal efficiency is achieved by either Newton iteration or the iteration $\psi $ defined by \[ \psi (f)(x) = x - \frac{{f^2 (x)}}{{f(x + f(x)) - f(x)}}. \]