Leap-frogging Newton's method

Using Newton’s method as an intermediate step, we introduce an iterative method that approximates numerically the solution of f…x † ˆ 0. The method is essentially a leap-frog Newton’s method. The order of convergence of the proposed method at a simple root is cubic and the computational e ciency in general is less, but close to that of Newton’s method. Like Newton’s method, the new method requires only function and ®rst derivative evaluations. The method can easily be implemented on computer algebra systems where high machine precision is available. 1. A higher order Newton-type method Solutions to single non-linear equations that have no solutions in closed form are of much interest in physics. In these problems we seek methods that lead to approximate solutions. Newton’s method is often the method of choice for approximating such solutions. Newton’s method of iteration also appears in physics in the analysis of the chaotic behaviour of dynamical systems determined by rational maps [1]. The convergence of the iterates and the rate of convergence