A Rational Iteration Function for Solving Equations

is fitted to f(x) at three points, two of which are coincident. Thus a step in the iteration consists of matching / and y at the points xn_, and xn and / ' and y' at xn only, the next approximation being given by the zero of (1.2), i.e. by xn+l — a. Ostrowski showed that the order of the process so obtained is 2-414 and as each step requires the evaluation o f / and / ' , it therefore compares favourably with the second-order NewtonRaphson method. However, since the derivative f'n_ i is always available once an iteration has been started, it seems natural to try to obtain more rapid convergence = x — xn, we construct the rational function